(1) Institute of Applied Physics of the Academy of Sciences of Moldova

Academy str5 Chisinau MD 2028 Moldova

(2) Department of Chemistry Texas AampM University P O Box 30012College StationTX 77843-3012 USA

(3) Chemistry Department Ben-Gurion University of the Negev Beer-Sheva 84105 Israel

Origin of Magnetic Anisotropyin Single Molecule and Single Chain Magnets

Containing Ions with Unquenched Orbital Angular Momenta

S Klokishner (1) A Palii (1) SOstrovsky (1) OReu (1) PTregenna-Piggott K Dunbar (2) BTsukerblat(3)

OUTLINE

1 General introduction

2 Magnetic anisotropy in the ReII4MnII

4single molecule magnet quantum-spin and classical-spin approach

3 Experimental study and theoretical modeling of the CuII

2TbII2 single molecule

magnet

4 Basic ingredients of single-chain magnet behavior

5 Highly anisotropic Co(II)-based single-chain magnet

Antiferromagnetic coupling between

Mn3+ and Mn4+

Ground state S=10

SINGLE MOLECULAR MAGNETS- DISCOVERY OF THE PHENOMENON

R Sessoli D Gatteschi A Caneschi M A Novak Nature 1993365 141

4S(Mn4+)-

ferromagnetic

8S(Mn3+)-

ferromagnetic

Mn12O12 (CH3COO)16 (H2O)4] ndashmolecule Mn12-acetateeight Mn3+ ions (si =2)

and four Mn4+ ions (si =32)

Quenched orbital angular momenta for the manganese ions

Mn(III) ions large spin S=2 and single ion easy axis anisotropy (single-ion ZFS parameter D~-35 cm-1) due to the Jahn-Teller elongation of the coordination octahedron

Zero-field splitting is a second-order effect Weak anisotropy

Mn(IV) ions very small zero-field splitting parameter

Zero-field splitting DSMS2 with DSlt0 uniaxial magnetic anisotropy

Mechanism of the formation of spin reversal barrier in the Mn12 cluster

Barrier heightU =S2DS

E(MS)=DS(MS2 S(S+1)3)

DS is the effective (molecular) zero-field

splitting parameter for the ground S-multiplet

DS=-064 K S=10

Barrier height E =S2DSThe widely used design rule rdquoto increase Srdquo is not as efficient as promised

bull The zero-field splitting parameter DS for the ground state decreases with increasing S

bull The barrier height does not increase with S as S2 but as S0

(OWaldmann InorgChem 2007 46 10035)

bull The strategy tordquo increase the total spin S of the ground staterdquo by synthesis of big spin-clusters has not produced better SMMs yet

SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta

KR Dunbar et al AngewChemIntEd

2003 421523

)1(3 lmE

)0(23 lmA

)( 421

3 tTMn(III)

Mn(II)

CN

Cyano-bridged cluster [MnIII(CN)6]2[MnII(tmphen2]3

Incorporation of orbitally degenerate 3d ndashmetal ions

Strongly magnetically anisotropic ground E-state

The interplay between strong single ion anisotropy arising from the trigonal crystal field combined with SO interaction and antiferromagnetic Heisenberg-type exchangerarr

appreciable barrier for reversal of magnetization

SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta

Trigonally distorted molecular cube[MnCl]4[Re(triphos)(CN)3]4

JAmChemSoc 2004 126 15004-15005

Rendashlight blue Mnndashred C-grayNndashdark blue Clndashyellow P-purple

Replacing 3d transition metal ions with 4d or 5d ones

bullSpin-orbit interaction in 4d and 5d ions is at least one order of magnitude larger than in 3d ions

bull Much stronger exchange interaction than in the case of 3d-ions

bullThe barrier for reversal of magnetization can be significantly increased compared to clusters of 3d-ions

bullDespite the prospect of stronger magnetic interactions only few SMMs incorporating heavier transition metal ions are known

Mononuclear Lanthanide Single Molecule Magnets

Introduction of lanthanide metal ions

SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy

ECoronado et al JACS20081308874 InorgChem2009483467

Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions

Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization

Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694

3d-4f Single Molecule Magnets

Linear trinuclear heterobimetallic Co2Gd complex

VChandrasekhar et alInorgChem2007465140

Ground term of the Gd3+-ion ndash8S72

Spin -ion

Co ndashions unquenched orbital angular momenta

Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]

VChandrasekhar et alInorgChem2008474918

bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought

by the Dy3+ ion

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2

Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are

omitted

Temperature dependenceof the out-of-phase ac

susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and

[NiIILTbIII(hfac)2]2 compounds

[CuLTb(hfac)2]2

SOsa et alJACS2004126420

MOTIVATION OF THE WORK

Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters

Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems

Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster

2

1

5

73

4

6

8

Z

C3

J

J J

J

J J

J

J

J

J

J

J

Re(II) Mn(II)

Molecular cube compressedalong one of the trigonal axes

No zero field splitting for the Re(II) ion with spin frac12

Mn(II) ions do not carry anymagnetic anisotropy

The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways

J J - exchange integrals J ne J

Trigonally distorted mixed-ligand surrounding of the Re(II) ion

Re(II)

P

PP

C

CC

Local trigonal Z-axis

Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5

Htrig=(lz 23)2^

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

gt0

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

||

lt0

Relevant case

Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||

2E (ml=1)||

5+6mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )

Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0

bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12

bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z

Exchange Interaction for the Re(II)-Mn(II) Pair

Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators

J(Re-Mn)rarr exchange interaction parameter2E (ml=1)

||5+6

mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|

Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian

bull One can restrict the consideration of the exchange interaction by the ground Kramers space

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

OUTLINE

1 General introduction

2 Magnetic anisotropy in the ReII4MnII

4single molecule magnet quantum-spin and classical-spin approach

3 Experimental study and theoretical modeling of the CuII

2TbII2 single molecule

magnet

4 Basic ingredients of single-chain magnet behavior

5 Highly anisotropic Co(II)-based single-chain magnet

Antiferromagnetic coupling between

Mn3+ and Mn4+

Ground state S=10

SINGLE MOLECULAR MAGNETS- DISCOVERY OF THE PHENOMENON

R Sessoli D Gatteschi A Caneschi M A Novak Nature 1993365 141

4S(Mn4+)-

ferromagnetic

8S(Mn3+)-

ferromagnetic

Mn12O12 (CH3COO)16 (H2O)4] ndashmolecule Mn12-acetateeight Mn3+ ions (si =2)

and four Mn4+ ions (si =32)

Quenched orbital angular momenta for the manganese ions

Mn(III) ions large spin S=2 and single ion easy axis anisotropy (single-ion ZFS parameter D~-35 cm-1) due to the Jahn-Teller elongation of the coordination octahedron

Zero-field splitting is a second-order effect Weak anisotropy

Mn(IV) ions very small zero-field splitting parameter

Zero-field splitting DSMS2 with DSlt0 uniaxial magnetic anisotropy

Mechanism of the formation of spin reversal barrier in the Mn12 cluster

Barrier heightU =S2DS

E(MS)=DS(MS2 S(S+1)3)

DS is the effective (molecular) zero-field

splitting parameter for the ground S-multiplet

DS=-064 K S=10

Barrier height E =S2DSThe widely used design rule rdquoto increase Srdquo is not as efficient as promised

bull The zero-field splitting parameter DS for the ground state decreases with increasing S

bull The barrier height does not increase with S as S2 but as S0

(OWaldmann InorgChem 2007 46 10035)

bull The strategy tordquo increase the total spin S of the ground staterdquo by synthesis of big spin-clusters has not produced better SMMs yet

SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta

KR Dunbar et al AngewChemIntEd

2003 421523

)1(3 lmE

)0(23 lmA

)( 421

3 tTMn(III)

Mn(II)

CN

Cyano-bridged cluster [MnIII(CN)6]2[MnII(tmphen2]3

Incorporation of orbitally degenerate 3d ndashmetal ions

Strongly magnetically anisotropic ground E-state

The interplay between strong single ion anisotropy arising from the trigonal crystal field combined with SO interaction and antiferromagnetic Heisenberg-type exchangerarr

appreciable barrier for reversal of magnetization

SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta

Trigonally distorted molecular cube[MnCl]4[Re(triphos)(CN)3]4

JAmChemSoc 2004 126 15004-15005

Rendashlight blue Mnndashred C-grayNndashdark blue Clndashyellow P-purple

Replacing 3d transition metal ions with 4d or 5d ones

bullSpin-orbit interaction in 4d and 5d ions is at least one order of magnitude larger than in 3d ions

bull Much stronger exchange interaction than in the case of 3d-ions

bullThe barrier for reversal of magnetization can be significantly increased compared to clusters of 3d-ions

bullDespite the prospect of stronger magnetic interactions only few SMMs incorporating heavier transition metal ions are known

Mononuclear Lanthanide Single Molecule Magnets

Introduction of lanthanide metal ions

SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy

ECoronado et al JACS20081308874 InorgChem2009483467

Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions

Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization

Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694

3d-4f Single Molecule Magnets

Linear trinuclear heterobimetallic Co2Gd complex

VChandrasekhar et alInorgChem2007465140

Ground term of the Gd3+-ion ndash8S72

Spin -ion

Co ndashions unquenched orbital angular momenta

Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]

VChandrasekhar et alInorgChem2008474918

bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought

by the Dy3+ ion

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2

Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are

omitted

Temperature dependenceof the out-of-phase ac

susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and

[NiIILTbIII(hfac)2]2 compounds

[CuLTb(hfac)2]2

SOsa et alJACS2004126420

MOTIVATION OF THE WORK

Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters

Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems

Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster

2

1

5

73

4

6

8

Z

C3

J

J J

J

J J

J

J

J

J

J

J

Re(II) Mn(II)

Molecular cube compressedalong one of the trigonal axes

No zero field splitting for the Re(II) ion with spin frac12

Mn(II) ions do not carry anymagnetic anisotropy

The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways

J J - exchange integrals J ne J

Trigonally distorted mixed-ligand surrounding of the Re(II) ion

Re(II)

P

PP

C

CC

Local trigonal Z-axis

Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5

Htrig=(lz 23)2^

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

gt0

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

||

lt0

Relevant case

Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||

2E (ml=1)||

5+6mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )

Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0

bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12

bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z

Exchange Interaction for the Re(II)-Mn(II) Pair

Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators

J(Re-Mn)rarr exchange interaction parameter2E (ml=1)

||5+6

mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|

Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian

bull One can restrict the consideration of the exchange interaction by the ground Kramers space

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

Antiferromagnetic coupling between

Mn3+ and Mn4+

Ground state S=10

SINGLE MOLECULAR MAGNETS- DISCOVERY OF THE PHENOMENON

R Sessoli D Gatteschi A Caneschi M A Novak Nature 1993365 141

4S(Mn4+)-

ferromagnetic

8S(Mn3+)-

ferromagnetic

Mn12O12 (CH3COO)16 (H2O)4] ndashmolecule Mn12-acetateeight Mn3+ ions (si =2)

and four Mn4+ ions (si =32)

Quenched orbital angular momenta for the manganese ions

Mn(III) ions large spin S=2 and single ion easy axis anisotropy (single-ion ZFS parameter D~-35 cm-1) due to the Jahn-Teller elongation of the coordination octahedron

Zero-field splitting is a second-order effect Weak anisotropy

Mn(IV) ions very small zero-field splitting parameter

Zero-field splitting DSMS2 with DSlt0 uniaxial magnetic anisotropy

Mechanism of the formation of spin reversal barrier in the Mn12 cluster

Barrier heightU =S2DS

E(MS)=DS(MS2 S(S+1)3)

DS is the effective (molecular) zero-field

splitting parameter for the ground S-multiplet

DS=-064 K S=10

Barrier height E =S2DSThe widely used design rule rdquoto increase Srdquo is not as efficient as promised

bull The zero-field splitting parameter DS for the ground state decreases with increasing S

bull The barrier height does not increase with S as S2 but as S0

(OWaldmann InorgChem 2007 46 10035)

bull The strategy tordquo increase the total spin S of the ground staterdquo by synthesis of big spin-clusters has not produced better SMMs yet

SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta

KR Dunbar et al AngewChemIntEd

2003 421523

)1(3 lmE

)0(23 lmA

)( 421

3 tTMn(III)

Mn(II)

CN

Cyano-bridged cluster [MnIII(CN)6]2[MnII(tmphen2]3

Incorporation of orbitally degenerate 3d ndashmetal ions

Strongly magnetically anisotropic ground E-state

The interplay between strong single ion anisotropy arising from the trigonal crystal field combined with SO interaction and antiferromagnetic Heisenberg-type exchangerarr

appreciable barrier for reversal of magnetization

SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta

Trigonally distorted molecular cube[MnCl]4[Re(triphos)(CN)3]4

JAmChemSoc 2004 126 15004-15005

Rendashlight blue Mnndashred C-grayNndashdark blue Clndashyellow P-purple

Replacing 3d transition metal ions with 4d or 5d ones

bullSpin-orbit interaction in 4d and 5d ions is at least one order of magnitude larger than in 3d ions

bull Much stronger exchange interaction than in the case of 3d-ions

bullThe barrier for reversal of magnetization can be significantly increased compared to clusters of 3d-ions

bullDespite the prospect of stronger magnetic interactions only few SMMs incorporating heavier transition metal ions are known

Mononuclear Lanthanide Single Molecule Magnets

Introduction of lanthanide metal ions

SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy

ECoronado et al JACS20081308874 InorgChem2009483467

Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions

Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization

Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694

3d-4f Single Molecule Magnets

Linear trinuclear heterobimetallic Co2Gd complex

VChandrasekhar et alInorgChem2007465140

Ground term of the Gd3+-ion ndash8S72

Spin -ion

Co ndashions unquenched orbital angular momenta

Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]

VChandrasekhar et alInorgChem2008474918

bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought

by the Dy3+ ion

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2

Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are

omitted

Temperature dependenceof the out-of-phase ac

susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and

[NiIILTbIII(hfac)2]2 compounds

[CuLTb(hfac)2]2

SOsa et alJACS2004126420

MOTIVATION OF THE WORK

Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters

Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems

Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster

2

1

5

73

4

6

8

Z

C3

J

J J

J

J J

J

J

J

J

J

J

Re(II) Mn(II)

Molecular cube compressedalong one of the trigonal axes

No zero field splitting for the Re(II) ion with spin frac12

Mn(II) ions do not carry anymagnetic anisotropy

The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways

J J - exchange integrals J ne J

Trigonally distorted mixed-ligand surrounding of the Re(II) ion

Re(II)

P

PP

C

CC

Local trigonal Z-axis

Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5

Htrig=(lz 23)2^

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

gt0

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

||

lt0

Relevant case

Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||

2E (ml=1)||

5+6mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )

Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0

bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12

bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z

Exchange Interaction for the Re(II)-Mn(II) Pair

Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators

J(Re-Mn)rarr exchange interaction parameter2E (ml=1)

||5+6

mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|

Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian

bull One can restrict the consideration of the exchange interaction by the ground Kramers space

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

Quenched orbital angular momenta for the manganese ions

Mn(III) ions large spin S=2 and single ion easy axis anisotropy (single-ion ZFS parameter D~-35 cm-1) due to the Jahn-Teller elongation of the coordination octahedron

Zero-field splitting is a second-order effect Weak anisotropy

Mn(IV) ions very small zero-field splitting parameter

Zero-field splitting DSMS2 with DSlt0 uniaxial magnetic anisotropy

Mechanism of the formation of spin reversal barrier in the Mn12 cluster

Barrier heightU =S2DS

E(MS)=DS(MS2 S(S+1)3)

DS is the effective (molecular) zero-field

splitting parameter for the ground S-multiplet

DS=-064 K S=10

Barrier height E =S2DSThe widely used design rule rdquoto increase Srdquo is not as efficient as promised

bull The zero-field splitting parameter DS for the ground state decreases with increasing S

bull The barrier height does not increase with S as S2 but as S0

(OWaldmann InorgChem 2007 46 10035)

bull The strategy tordquo increase the total spin S of the ground staterdquo by synthesis of big spin-clusters has not produced better SMMs yet

SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta

KR Dunbar et al AngewChemIntEd

2003 421523

)1(3 lmE

)0(23 lmA

)( 421

3 tTMn(III)

Mn(II)

CN

Cyano-bridged cluster [MnIII(CN)6]2[MnII(tmphen2]3

Incorporation of orbitally degenerate 3d ndashmetal ions

Strongly magnetically anisotropic ground E-state

The interplay between strong single ion anisotropy arising from the trigonal crystal field combined with SO interaction and antiferromagnetic Heisenberg-type exchangerarr

appreciable barrier for reversal of magnetization

SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta

Trigonally distorted molecular cube[MnCl]4[Re(triphos)(CN)3]4

JAmChemSoc 2004 126 15004-15005

Rendashlight blue Mnndashred C-grayNndashdark blue Clndashyellow P-purple

Replacing 3d transition metal ions with 4d or 5d ones

bullSpin-orbit interaction in 4d and 5d ions is at least one order of magnitude larger than in 3d ions

bull Much stronger exchange interaction than in the case of 3d-ions

bullThe barrier for reversal of magnetization can be significantly increased compared to clusters of 3d-ions

bullDespite the prospect of stronger magnetic interactions only few SMMs incorporating heavier transition metal ions are known

Mononuclear Lanthanide Single Molecule Magnets

Introduction of lanthanide metal ions

SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy

ECoronado et al JACS20081308874 InorgChem2009483467

Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions

Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization

Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694

3d-4f Single Molecule Magnets

Linear trinuclear heterobimetallic Co2Gd complex

VChandrasekhar et alInorgChem2007465140

Ground term of the Gd3+-ion ndash8S72

Spin -ion

Co ndashions unquenched orbital angular momenta

Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]

VChandrasekhar et alInorgChem2008474918

bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought

by the Dy3+ ion

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2

Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are

omitted

Temperature dependenceof the out-of-phase ac

susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and

[NiIILTbIII(hfac)2]2 compounds

[CuLTb(hfac)2]2

SOsa et alJACS2004126420

MOTIVATION OF THE WORK

Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters

Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems

Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster

2

1

5

73

4

6

8

Z

C3

J

J J

J

J J

J

J

J

J

J

J

Re(II) Mn(II)

Molecular cube compressedalong one of the trigonal axes

No zero field splitting for the Re(II) ion with spin frac12

Mn(II) ions do not carry anymagnetic anisotropy

The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways

J J - exchange integrals J ne J

Trigonally distorted mixed-ligand surrounding of the Re(II) ion

Re(II)

P

PP

C

CC

Local trigonal Z-axis

Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5

Htrig=(lz 23)2^

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

gt0

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

||

lt0

Relevant case

Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||

2E (ml=1)||

5+6mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )

Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0

bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12

bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z

Exchange Interaction for the Re(II)-Mn(II) Pair

Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators

J(Re-Mn)rarr exchange interaction parameter2E (ml=1)

||5+6

mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|

Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian

bull One can restrict the consideration of the exchange interaction by the ground Kramers space

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

Barrier height E =S2DSThe widely used design rule rdquoto increase Srdquo is not as efficient as promised

bull The zero-field splitting parameter DS for the ground state decreases with increasing S

bull The barrier height does not increase with S as S2 but as S0

(OWaldmann InorgChem 2007 46 10035)

bull The strategy tordquo increase the total spin S of the ground staterdquo by synthesis of big spin-clusters has not produced better SMMs yet

SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta

KR Dunbar et al AngewChemIntEd

2003 421523

)1(3 lmE

)0(23 lmA

)( 421

3 tTMn(III)

Mn(II)

CN

Cyano-bridged cluster [MnIII(CN)6]2[MnII(tmphen2]3

Incorporation of orbitally degenerate 3d ndashmetal ions

Strongly magnetically anisotropic ground E-state

The interplay between strong single ion anisotropy arising from the trigonal crystal field combined with SO interaction and antiferromagnetic Heisenberg-type exchangerarr

appreciable barrier for reversal of magnetization

SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta

Trigonally distorted molecular cube[MnCl]4[Re(triphos)(CN)3]4

JAmChemSoc 2004 126 15004-15005

Rendashlight blue Mnndashred C-grayNndashdark blue Clndashyellow P-purple

Replacing 3d transition metal ions with 4d or 5d ones

bullSpin-orbit interaction in 4d and 5d ions is at least one order of magnitude larger than in 3d ions

bull Much stronger exchange interaction than in the case of 3d-ions

bullThe barrier for reversal of magnetization can be significantly increased compared to clusters of 3d-ions

bullDespite the prospect of stronger magnetic interactions only few SMMs incorporating heavier transition metal ions are known

Mononuclear Lanthanide Single Molecule Magnets

Introduction of lanthanide metal ions

SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy

ECoronado et al JACS20081308874 InorgChem2009483467

Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions

Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization

Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694

3d-4f Single Molecule Magnets

Linear trinuclear heterobimetallic Co2Gd complex

VChandrasekhar et alInorgChem2007465140

Ground term of the Gd3+-ion ndash8S72

Spin -ion

Co ndashions unquenched orbital angular momenta

Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]

VChandrasekhar et alInorgChem2008474918

bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought

by the Dy3+ ion

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2

Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are

omitted

Temperature dependenceof the out-of-phase ac

susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and

[NiIILTbIII(hfac)2]2 compounds

[CuLTb(hfac)2]2

SOsa et alJACS2004126420

MOTIVATION OF THE WORK

Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters

Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems

Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster

2

1

5

73

4

6

8

Z

C3

J

J J

J

J J

J

J

J

J

J

J

Re(II) Mn(II)

Molecular cube compressedalong one of the trigonal axes

No zero field splitting for the Re(II) ion with spin frac12

Mn(II) ions do not carry anymagnetic anisotropy

The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways

J J - exchange integrals J ne J

Trigonally distorted mixed-ligand surrounding of the Re(II) ion

Re(II)

P

PP

C

CC

Local trigonal Z-axis

Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5

Htrig=(lz 23)2^

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

gt0

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

||

lt0

Relevant case

Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||

2E (ml=1)||

5+6mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )

Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0

bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12

bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z

Exchange Interaction for the Re(II)-Mn(II) Pair

Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators

J(Re-Mn)rarr exchange interaction parameter2E (ml=1)

||5+6

mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|

Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian

bull One can restrict the consideration of the exchange interaction by the ground Kramers space

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta

KR Dunbar et al AngewChemIntEd

2003 421523

)1(3 lmE

)0(23 lmA

)( 421

3 tTMn(III)

Mn(II)

CN

Cyano-bridged cluster [MnIII(CN)6]2[MnII(tmphen2]3

Incorporation of orbitally degenerate 3d ndashmetal ions

Strongly magnetically anisotropic ground E-state

The interplay between strong single ion anisotropy arising from the trigonal crystal field combined with SO interaction and antiferromagnetic Heisenberg-type exchangerarr

appreciable barrier for reversal of magnetization

SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta

Trigonally distorted molecular cube[MnCl]4[Re(triphos)(CN)3]4

JAmChemSoc 2004 126 15004-15005

Rendashlight blue Mnndashred C-grayNndashdark blue Clndashyellow P-purple

Replacing 3d transition metal ions with 4d or 5d ones

bullSpin-orbit interaction in 4d and 5d ions is at least one order of magnitude larger than in 3d ions

bull Much stronger exchange interaction than in the case of 3d-ions

bullThe barrier for reversal of magnetization can be significantly increased compared to clusters of 3d-ions

bullDespite the prospect of stronger magnetic interactions only few SMMs incorporating heavier transition metal ions are known

Mononuclear Lanthanide Single Molecule Magnets

Introduction of lanthanide metal ions

SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy

ECoronado et al JACS20081308874 InorgChem2009483467

Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions

Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization

Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694

3d-4f Single Molecule Magnets

Linear trinuclear heterobimetallic Co2Gd complex

VChandrasekhar et alInorgChem2007465140

Ground term of the Gd3+-ion ndash8S72

Spin -ion

Co ndashions unquenched orbital angular momenta

Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]

VChandrasekhar et alInorgChem2008474918

bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought

by the Dy3+ ion

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2

Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are

omitted

Temperature dependenceof the out-of-phase ac

susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and

[NiIILTbIII(hfac)2]2 compounds

[CuLTb(hfac)2]2

SOsa et alJACS2004126420

MOTIVATION OF THE WORK

Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters

Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems

Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster

2

1

5

73

4

6

8

Z

C3

J

J J

J

J J

J

J

J

J

J

J

Re(II) Mn(II)

Molecular cube compressedalong one of the trigonal axes

No zero field splitting for the Re(II) ion with spin frac12

Mn(II) ions do not carry anymagnetic anisotropy

The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways

J J - exchange integrals J ne J

Trigonally distorted mixed-ligand surrounding of the Re(II) ion

Re(II)

P

PP

C

CC

Local trigonal Z-axis

Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5

Htrig=(lz 23)2^

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

gt0

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

||

lt0

Relevant case

Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||

2E (ml=1)||

5+6mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )

Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0

bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12

bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z

Exchange Interaction for the Re(II)-Mn(II) Pair

Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators

J(Re-Mn)rarr exchange interaction parameter2E (ml=1)

||5+6

mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|

Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian

bull One can restrict the consideration of the exchange interaction by the ground Kramers space

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

SMMs Containing Metal Ions with Unquenched Orbital Angular Momenta

Trigonally distorted molecular cube[MnCl]4[Re(triphos)(CN)3]4

JAmChemSoc 2004 126 15004-15005

Rendashlight blue Mnndashred C-grayNndashdark blue Clndashyellow P-purple

Replacing 3d transition metal ions with 4d or 5d ones

bullSpin-orbit interaction in 4d and 5d ions is at least one order of magnitude larger than in 3d ions

bull Much stronger exchange interaction than in the case of 3d-ions

bullThe barrier for reversal of magnetization can be significantly increased compared to clusters of 3d-ions

bullDespite the prospect of stronger magnetic interactions only few SMMs incorporating heavier transition metal ions are known

Mononuclear Lanthanide Single Molecule Magnets

Introduction of lanthanide metal ions

SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy

ECoronado et al JACS20081308874 InorgChem2009483467

Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions

Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization

Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694

3d-4f Single Molecule Magnets

Linear trinuclear heterobimetallic Co2Gd complex

VChandrasekhar et alInorgChem2007465140

Ground term of the Gd3+-ion ndash8S72

Spin -ion

Co ndashions unquenched orbital angular momenta

Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]

VChandrasekhar et alInorgChem2008474918

bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought

by the Dy3+ ion

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2

Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are

omitted

Temperature dependenceof the out-of-phase ac

susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and

[NiIILTbIII(hfac)2]2 compounds

[CuLTb(hfac)2]2

SOsa et alJACS2004126420

MOTIVATION OF THE WORK

Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters

Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems

Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster

2

1

5

73

4

6

8

Z

C3

J

J J

J

J J

J

J

J

J

J

J

Re(II) Mn(II)

Molecular cube compressedalong one of the trigonal axes

No zero field splitting for the Re(II) ion with spin frac12

Mn(II) ions do not carry anymagnetic anisotropy

The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways

J J - exchange integrals J ne J

Trigonally distorted mixed-ligand surrounding of the Re(II) ion

Re(II)

P

PP

C

CC

Local trigonal Z-axis

Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5

Htrig=(lz 23)2^

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

gt0

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

||

lt0

Relevant case

Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||

2E (ml=1)||

5+6mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )

Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0

bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12

bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z

Exchange Interaction for the Re(II)-Mn(II) Pair

Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators

J(Re-Mn)rarr exchange interaction parameter2E (ml=1)

||5+6

mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|

Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian

bull One can restrict the consideration of the exchange interaction by the ground Kramers space

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

Mononuclear Lanthanide Single Molecule Magnets

Introduction of lanthanide metal ions

SMMs Based on Polyoxometalates encapsulating lanthanides Er HoDy

ECoronado et al JACS20081308874 InorgChem2009483467

Much stronger single-ion anisotropy and much slower relaxation as compared with those exhibiting by 3d 4d and 5d ions

Presence of an axial crystal field acting on the 4f-ion and stabilizing a Stark sublevel with a large absolute value of the total angular momentum projection |MJ| thus achieving an easy axis of magnetization

Lanthanide Double-Decker Complexes Functioning as SMMs DyHoTb NIshikawa et al JACS20031258694

3d-4f Single Molecule Magnets

Linear trinuclear heterobimetallic Co2Gd complex

VChandrasekhar et alInorgChem2007465140

Ground term of the Gd3+-ion ndash8S72

Spin -ion

Co ndashions unquenched orbital angular momenta

Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]

VChandrasekhar et alInorgChem2008474918

bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought

by the Dy3+ ion

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2

Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are

omitted

Temperature dependenceof the out-of-phase ac

susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and

[NiIILTbIII(hfac)2]2 compounds

[CuLTb(hfac)2]2

SOsa et alJACS2004126420

MOTIVATION OF THE WORK

Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters

Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems

Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster

2

1

5

73

4

6

8

Z

C3

J

J J

J

J J

J

J

J

J

J

J

Re(II) Mn(II)

Molecular cube compressedalong one of the trigonal axes

No zero field splitting for the Re(II) ion with spin frac12

Mn(II) ions do not carry anymagnetic anisotropy

The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways

J J - exchange integrals J ne J

Trigonally distorted mixed-ligand surrounding of the Re(II) ion

Re(II)

P

PP

C

CC

Local trigonal Z-axis

Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5

Htrig=(lz 23)2^

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

gt0

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

||

lt0

Relevant case

Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||

2E (ml=1)||

5+6mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )

Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0

bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12

bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z

Exchange Interaction for the Re(II)-Mn(II) Pair

Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators

J(Re-Mn)rarr exchange interaction parameter2E (ml=1)

||5+6

mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|

Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian

bull One can restrict the consideration of the exchange interaction by the ground Kramers space

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

3d-4f Single Molecule Magnets

Linear trinuclear heterobimetallic Co2Gd complex

VChandrasekhar et alInorgChem2007465140

Ground term of the Gd3+-ion ndash8S72

Spin -ion

Co ndashions unquenched orbital angular momenta

Trinuclear heterobimetalic complex [L2Ni2Dy][ClO4]

VChandrasekhar et alInorgChem2008474918

bullThe Ni2La complex is not a SMMbullMagnetic anisotropy is brought

by the Dy3+ ion

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2

Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are

omitted

Temperature dependenceof the out-of-phase ac

susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and

[NiIILTbIII(hfac)2]2 compounds

[CuLTb(hfac)2]2

SOsa et alJACS2004126420

MOTIVATION OF THE WORK

Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters

Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems

Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster

2

1

5

73

4

6

8

Z

C3

J

J J

J

J J

J

J

J

J

J

J

Re(II) Mn(II)

Molecular cube compressedalong one of the trigonal axes

No zero field splitting for the Re(II) ion with spin frac12

Mn(II) ions do not carry anymagnetic anisotropy

The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways

J J - exchange integrals J ne J

Trigonally distorted mixed-ligand surrounding of the Re(II) ion

Re(II)

P

PP

C

CC

Local trigonal Z-axis

Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5

Htrig=(lz 23)2^

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

gt0

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

||

lt0

Relevant case

Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||

2E (ml=1)||

5+6mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )

Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0

bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12

bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z

Exchange Interaction for the Re(II)-Mn(II) Pair

Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators

J(Re-Mn)rarr exchange interaction parameter2E (ml=1)

||5+6

mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|

Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian

bull One can restrict the consideration of the exchange interaction by the ground Kramers space

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

Molecular structure of the cyclic compound [CuIILTbIII(hfac)2]2

Yellow TbIII reddish pink CuII red O blue N white C pink F The H atoms are

omitted

Temperature dependenceof the out-of-phase ac

susceptibility (Mrsquorsquo) for the [CuIILTbIII(hfac)2]2 and

[NiIILTbIII(hfac)2]2 compounds

[CuLTb(hfac)2]2

SOsa et alJACS2004126420

MOTIVATION OF THE WORK

Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters

Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems

Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster

2

1

5

73

4

6

8

Z

C3

J

J J

J

J J

J

J

J

J

J

J

Re(II) Mn(II)

Molecular cube compressedalong one of the trigonal axes

No zero field splitting for the Re(II) ion with spin frac12

Mn(II) ions do not carry anymagnetic anisotropy

The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways

J J - exchange integrals J ne J

Trigonally distorted mixed-ligand surrounding of the Re(II) ion

Re(II)

P

PP

C

CC

Local trigonal Z-axis

Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5

Htrig=(lz 23)2^

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

gt0

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

||

lt0

Relevant case

Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||

2E (ml=1)||

5+6mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )

Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0

bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12

bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z

Exchange Interaction for the Re(II)-Mn(II) Pair

Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators

J(Re-Mn)rarr exchange interaction parameter2E (ml=1)

||5+6

mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|

Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian

bull One can restrict the consideration of the exchange interaction by the ground Kramers space

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

MOTIVATION OF THE WORK

Experimental ndash explanation of the origin of single molecule magnetism in the [MnCl]4[Re(triphos)(CN)3]4and [CuLTb (hfac)2]2 clusters

Theoretical ndashdevelopment of models of the magnetic anisotropy in 3d-5d and 3d-4f systems

Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster

2

1

5

73

4

6

8

Z

C3

J

J J

J

J J

J

J

J

J

J

J

Re(II) Mn(II)

Molecular cube compressedalong one of the trigonal axes

No zero field splitting for the Re(II) ion with spin frac12

Mn(II) ions do not carry anymagnetic anisotropy

The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways

J J - exchange integrals J ne J

Trigonally distorted mixed-ligand surrounding of the Re(II) ion

Re(II)

P

PP

C

CC

Local trigonal Z-axis

Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5

Htrig=(lz 23)2^

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

gt0

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

||

lt0

Relevant case

Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||

2E (ml=1)||

5+6mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )

Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0

bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12

bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z

Exchange Interaction for the Re(II)-Mn(II) Pair

Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators

J(Re-Mn)rarr exchange interaction parameter2E (ml=1)

||5+6

mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|

Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian

bull One can restrict the consideration of the exchange interaction by the ground Kramers space

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

Illustration for the trigonal overall symmetry of the Re4Mn4 -cluster

2

1

5

73

4

6

8

Z

C3

J

J J

J

J J

J

J

J

J

J

J

Re(II) Mn(II)

Molecular cube compressedalong one of the trigonal axes

No zero field splitting for the Re(II) ion with spin frac12

Mn(II) ions do not carry anymagnetic anisotropy

The unquenched orbital angular momenta of the low-spin Re(II) ions are supposed to be responsible for the observed SMM propertiesTwo types of exchange pathways

J J - exchange integrals J ne J

Trigonally distorted mixed-ligand surrounding of the Re(II) ion

Re(II)

P

PP

C

CC

Local trigonal Z-axis

Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5

Htrig=(lz 23)2^

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

gt0

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

||

lt0

Relevant case

Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||

2E (ml=1)||

5+6mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )

Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0

bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12

bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z

Exchange Interaction for the Re(II)-Mn(II) Pair

Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators

J(Re-Mn)rarr exchange interaction parameter2E (ml=1)

||5+6

mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|

Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian

bull One can restrict the consideration of the exchange interaction by the ground Kramers space

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

Trigonally distorted mixed-ligand surrounding of the Re(II) ion

Re(II)

P

PP

C

CC

Local trigonal Z-axis

Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5

Htrig=(lz 23)2^

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

gt0

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

||

lt0

Relevant case

Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||

2E (ml=1)||

5+6mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )

Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0

bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12

bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z

Exchange Interaction for the Re(II)-Mn(II) Pair

Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators

J(Re-Mn)rarr exchange interaction parameter2E (ml=1)

||5+6

mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|

Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian

bull One can restrict the consideration of the exchange interaction by the ground Kramers space

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

Trigonal crystal field splitting of the ground 2T2(t2 ) term of the low-spin Re(II) ion5

Htrig=(lz 23)2^

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

gt0

2T2(t2 )5

2A1 (ml=0)

2E (ml=1)

||

lt0

Relevant case

Spin-orbital splitting of the ground trigonal 2E-termin the limit of strong negative trigonal crystal field ||gtgt||

2E (ml=1)||

5+6mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )

Principal values of the effective g-tensor for the ground Kramers doublet 4 in the limit of the strong negative crystal field rarr g||=ge+2 g=0

bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12

bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z

Exchange Interaction for the Re(II)-Mn(II) Pair

Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators

J(Re-Mn)rarr exchange interaction parameter2E (ml=1)

||5+6

mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|

Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian

bull One can restrict the consideration of the exchange interaction by the ground Kramers space

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

bull All matrix elements of the operators sX(Re) and sY(Re)are vanishing in the basis |ml= 1 ms= 12

bull The matrix of the operator sz(Re) coincides with the matrix of the pseudo-spin-12 operator z=(12)z

Exchange Interaction for the Re(II)-Mn(II) Pair

Hex= 2J(Re-Mn)s(Re)s(Mn)s(Re) s(Mn) rarr single-ion spin operators

J(Re-Mn)rarr exchange interaction parameter2E (ml=1)

||5+6

mj=32 (ml=1 ms=12 )

4 mj=12 (ml=1 ms= 12 )|| =2100 cm-1|| gtgt | J(Re-Mn)|

Hex= 2J(Re-Mn)Z(Re)sZ(Mn)Anisotropic Ising form of the exchange Hamiltonian

bull One can restrict the consideration of the exchange interaction by the ground Kramers space

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

Hamiltonian of exchange interaction

Local (red) and molecular (blue) Z-axes and the network of exchange pathways

for the Re4Mn4 ndashcluster

8ˆ6ˆ4ˆ

8ˆ5ˆ3ˆ

6ˆ5ˆ2ˆ2

8ˆ7ˆ6ˆ4ˆ

8ˆ6ˆ5ˆ1ˆ2ˆ

444

333

222

4444

1111

ZZZ

ZZZ

ZZZ

ZZZZ

ZZZZex

ss

ss

ssJ

sss

sssJH

Re(II) Mn(II)

2

1

5

73

4

6

8

Z1

C3(Z)

J

J J

J

J J

J

J

J

J

J

J

Z2

Z3

Z4

All operators are defined in the local frames

iiZ - is the z -component

5ˆ1Zs - is the z -component

of the spin operator of the 5-th Mn(II) ion defined in the local frame associated with the 1-st Re(II) ion etc

of the pseudo ndash spin -12 operator of i ndashth Re(II) iondefined in its local frame

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

0 50 100 150 200 250 30012

14

16

18

20

22

24

g|| Re = 4 gMn = 195

J = -35 cm-1

J = -105 cm-1

T c

m3 K

mol

-1

Temperature K

Comparison between theory and experiment

Temperature K

J = 35 cm 1 J = 105 cm 1 gMn = 195Best fit parameters

0 1 2 3 4 5 6 7 8

0

2

4

6

8

10

12

3K4K

2KT

M

B

H THT

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

0 50 100 150 200 250 3000

10

20

30

40

50

T

||T

T c

m3 K

mol

-1

Temperature K

Evidences for the existence of the barrier for the reversal of magnetization1) Parallel and perpendicular components

of the magnetic susceptibility calculated with the set of the best-fit

parameters

Temperature K

|| T

T

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80

90

Ene

rgy

cm

-1

MS

2) Low-lying energy levels as functions of MS (MS is the projection of the

total spin of four Mn(II) ions)calculated with the set of the best-

fit parameters

Pseudo-spin-12 projections for the Re(II) ions are neglected as compared to the true spin projections for the Mn(II) ions (only for

illustration but not in calculations )

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

Quantum-classical approach for the Re4Mn4 cluster The spins 52 of the Mn(II) ions are regarded as classical ones

0 50 100 150 200 250 3000

10

20

30

40

50

60

T (quantum)

T (quantum-classical)

||T (quantum)

||T (quantum-classical)

T c

m3 K

mol

-1

Temperature K0 50 100 150 200 250 300

14

16

18

20

22

24Quantum-classical approach

Quantum approach

T c

m3 K

mol

-1

Temperature K

quantum-classical approach

quantum approach

quantum-classical approach

quantum approach

quantum-classical approachquantum approach

cos1 kkkZ ssks cossin1 kkkkX ssks

kkkkY sinsinssks 1

25ks

The pseudo-spins of the Re(II) ions are treated quantum-mechanically

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

The proposed model not only provides a satisfactorydescription of the dc susceptibilty and magnetization databut it is also compatible with the observed SMMbehavior of the Re4Mn4 cluster

The presence of nd-ions with non-cubic stronglymagnetically anisotropic ground states 2s+1E in combinationwith superexchange rarrbarrier for the reversal ofmagnetization

The quantum-classical approach can be successfullyused for the evaluation of the magnetic properties oflarge 3d-nd clusters

SUMMARY

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2Tb3+(1)

Tb3+(2)Cu2+(2)

Cu2+(1)

J1 J1

J2

J2

Exchange interactions in the [CuIILTbIII(hfac)2]2 complex

x

y

z

Z

N1

N2 O2

O3

O1

O4

O5

O6

O7

O8

XY

Cu

Tb

Cluster fragment

bullVery low local symmetry of Tb ions

bullIsolated tetranuclear molecules in the crystal

bullDifferent strength of exchange interaction for two types of Cu-Tb pairs

bullNegligible through-space interactions in the Cu-Cu and Tb-Tb pairs

The [CuIILTbIII(hfac)2]2 Single Molecule Magnet

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

INTERACTIONS INCLUDED IN THE MODEL )2()1(

2(2)Tb1

(1)Tb21

(2)Tb2

(1)Tb1

(2)Tb

(1)Tb

CuCuZeZe

HHJJHHH sjsjsjsj

Crystal field and Zeeman interaction for

Tb-ions

Exchange interaction

between Tb and Cu ions

Zeeman interaction for

Cu-ions

Crystal Field Potential Acting on the Electronic Shell of the Tb3+ Ion

iim

lmli

ml

i

opi

iicc YBWVV

pRrr )()( ecm

lpcm

lml

BBB

Exchange charges (Malkin BZ) pp

ml

p p

plecml C

RRSeB

)(5

2 2

)( pπlπpσσpsspl RSγGRSGRSGRS 222

00404 mpnmfRSsnmfRS pps

114 mpnmfRS p

S ndashoverlap integrals G-phenomenological parameters of the model

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

MAGNETIC SUSCEPTIBILITY OF

THE [ CuIILTbIII(hfac)2]2 AND [ NiIILTbIII(hfac)2]2 CLUSTERS

0 50 100 150 200 250 300

0

20

40

60

80

100

120

140

xxT

yyT

zzT

T e

mu

K m

ol-1

T K

Temperature dependences of the XXT YYT and ZZT components for the

[CuIITbIII(hfac)2]2 cluster calculated with the set of the best fit parameters

Gs=01 Gσ=02 Gπ=53 J1=283 cm-1J2=064 cm-1

0 50 100 150 200 250 300

20

25

30

35

40

45

50

55T

em

u K

mol

-1

T KT as a function of temperature

for the [CuIITbIII(hfac)2]2 and [NiIITbIII(hfac)2]2

clusters calculated with the set of the best fit parameters Gs=01Gσ=02 Gπ=53

solid line-theory circles-experimental data for [CuIITbIII(hfac)2]2exchange parameters J1=283 cm-1 J2=064 cm-1 squares ndashexperimental data for [NiIITbIII(hfac)2]2

Cu2Tb2Ni2Tb2

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

Barrier for Magnetization Reversal in the [CuIILTbIII(hfac)2]2 Cluster

Wave function of the th state of the cluster

ZJ JcZ

)2(2

Tb)1(1

TbZCu

ZZCu

ZZ sjsjJ

Expectation value of the cluster total angular momentum projection

ZZ JJ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Ground cluster state

13876102

ZJ JCZ

First excited state

0864202

ZJ JCZ

212166 )2()1()2()1( ZCu

ZCu

ZTb

ZTb ssjj

Probability of the one-phonon transitionground statelt-gtfirst excited state

02)2()1( excitedfirstHHground Tb

eLTbeL

Energy level diagram for the [CuIILTbIII(hfac)2]2 cluster

calculated with the parameters GS=01 Gσ=02 Gπ=53 J1=283 cm-

1J2=064 cm-1 determined from the best-fit procedure for the magnetic susceptibility and INS spectra The states are labeled according to the

expectation value of the operator Jz

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

jTb=6 sCu =12 Tb rarr7F6 J=13

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

Role of the Exchange Interaction in the Formation of the Barrier for Magnetization Reversal

Single Tb-ionNon-degenerate Stark levels vanishing mean value of the z- projection of the total angular momentum for each Stark level no barrier for magnetization reversal in a Tb-ion

Tb-Cu pair

Doubly degenerate energy levels approximately described by a definite eigenvalue jz of the total angular momentum operator jz= jz

Tb +szCu of the pair

Cluster energy pattern indicative of a system with almost axial anisotropy

Suppression of the rhombic component of the crystal field by isotropic exchange interaction barrier for magnetization reversal in a pair of exchange-coupled Tb and Cu ions

Stark structure of the ground 7F6multiplet of the TbIII ion in

[CuIILTbIII(hfac)2]2 (a) and low-lying energy levels of an exchange coupled

Tb-Cu pair (b) calculated with the best fit parameters Gs=01 Gσ=02 Gπ=53

J1=283 cm-1

0

100

200

300

400zj

plusmn 18plusmn 29plusmn 38

plusmn 01plusmn 03

plusmn 05plusmn 013plusmn 26plusmn 22plusmn 21plusmn 17

plusmn 54plusmn 64

b)a)

Ene

rgy

cm

-1

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

Low-energy INS Spectrum of the [CuII LTbIII(hfac)2]2 Cluster as a Function of Temperature in the Region ~ -30 to 30 cm-1

0

5

10

15

20

25

30

35

40

45

I

0

plusmn 107

plusmn 117plusmn 09plusmn 117

0

plusmn 127

Ene

rgy

cm

-1

Calculated energy level diagram for the Cu2Tb2 cluster

Cold peak I rarrtransition between the groundlevel (Jz=127) and the energy level with Jz=117

Hot peak rarrtransition between the first excited level (Jz=0) and the doublet with Jz= 09The simulated spectrum reproduces the main features of the observed one decrease of the intensity of the central peak and the increase of the intensity of the hot peak with temperature The observed INS spectra do not show any peaks for the Ni2Tb2 cluster in all available range of measurements

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

SUMMARY

The proposed model describes simultaneously the dc susceptibility and INS spectra and explains on this basis the observed SMM behavior of the [CuIITbIII(hfac)2]2 cluster

The interplay between the crystal field acting on the Tb ion and the ferromagnetic Heisenberg-type exchange between Tb and Cu ions produces a barrier for the reversal of magnetization

Intermetallic 4f-4d and 4f-5d complexes arepromising candidates for SMMs rarr the exchange interaction is stronger than that in 3d-4f complexesand the rhombic part of weak crystal field acting on a lanthanide ion can be easier suppressed by the exchange interaction

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

SINGLE ndashCHAIN MAGNETSDGatteschi et al AngewChemIntEd 2001401760

Single-chain magnets (SCMs) are made up of magnetically isolated chains possessing a finite magnetization that can be frozen in the

absence of an applied magnetic field At low temperatures the relaxation of the magnetization becomes so slow that these systems

can be considered as a magnet

BASIC INGREDIENTS OF SINGLE-CHAIN MAGNET BEHAVIOUR

hellip STRONG UNIAXIAL ANISOTROPY OF EACH MAGNETIC UNIT WHICH CREATES A BARRIER ABLE TO BLOCK OR ldquoFREEZErdquo MAGNETIZATION IN ONE DIRECTION THE BARRIER ORIGIN IS SIMILAR TO THAT IN SINGLE MOLECULE MAGNETS

hellip INDIVIDUAL MAGNETIC MOMENTS IN THE CHAIN MUST NOT CANCEL OUT

THE INTERCHAIN INTERACTIONS MUST BE AS SMALL AS POSSIBLE (IN COMPARISON TO THE INTRACHAIN ONES)

hellip THE CHAIN NEEDS TO EXHIBIT SPONTANEOUS MAGNETIZATION TO BE CALLED A MAGNET

RJ Glauber JMathPhys 19634294

ji

ZZex jiJH 2

Z is the operator for the Z component of the spin or pseudospin J is the coupling constant

TkT

Bbexp)( 0

Barrier to reverse the magnetization direction energy loss in one spin flip-flop process

Jb 2

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

SINGLE CHAIN MAGNETS CONTAINING IONS WITH QUENCHED ORBITAL ANGULAR MOMENTA

Barrier height |DS|S2

CONVENTIONAL MODEL

J parameter of ferromagnetic exchange between the spin units DS single-unit zero-

field splitting parameter

i

ziSii

iF SDSSJH 212

MAGNETIC UNIT

Ener

gyEn

ergy

DSlt0

S ndash spin of the ground state

BARRIER HEIGHT FOR A SINGLE CHAIN MAGNET

4JF S2 -4JF S2

|DS|S2 +8JFS2

Schematic view of the chain structure showing themagnetic units (Mn-Ni-Mn trimers) in

[Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2

Miysaka H et al Chem-Eur J 2005

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

COBALT(II) DIPHOSPHONATE [Co(H2L)(H2O)] COMPOUNDWITH a 1D STRUCTURE

KRDunbar et al JApplPhys2005

0 50 100 150 200 250 300

05

10

15

20

25

30

35

0 10 20 30

05

10

15

20

25

30

35

Temperature K

MT

cm

3 Km

ol-1

Temperature K

MT

cm

3 Km

ol-1

Temperature dependence of T for the Co(H2L)(H2O) compound

evidencesa) unquenched orbital angular momentum

T=300 K T =32 emumiddotmol-1 spin-only value T=300 K T =18 emumiddotmol-1

S=32b) antiferromagnetic exchange interaction

between Co ionsc) T maximum at 25 K rarr spin canting

(noncolinear spin structure)

ORTEP representation of the

Co(H2L)(H2O) unit

View of a 1D zigzag chain of Co(H2L)(H2O)

The cobalt octahedra and CPO3tetrahedra are shaded in green and

pink respectively

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

MOTIVATION OF THE WORK

Experimental ndashexplanation of the magnetic ( SCM ) behavior of the Co(H2L)(H2O) compound with a 1D structureTheoretical ndashelaboration of a new approach for analysis of the SCM properties of 1D chains containing ions with unquenched orbital angular momenta

PRELIMINARY REMARKSTetragonal crystal field splitting of the ground 4T1 (ml=0plusmn1) term of the

Co(II) ion ∆( lZ2 ndash 23)l=1 ml=0rarr4A2 ml=plusmn1rarr4E

4A2

4E

4T1 0

Zero-field splitting of the isolated trigonal 4A2 term

Providing gt 0 the barrier

for the reversal of magnetization

in a single magnetic unit cannot exist

MS=32

MS=12

2D4A2 Dgt04E

4A2

4T1

0

Orbital magneticcontribution isfully suppressed

Orbital angular momentum of theCo-ion is partially reduced butthe first order orbital magneticcontribution still remains

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

Wavefunction of the ground Kramers doublet of the

Co2+ion

Pseudo-spin-12 Hamiltonian for a pair of octahedrally coordinated Co(II) ions exhibiting spin canting

Single-ion Hamiltonian (local frames)

2

2

XA ZA

XB

ZB

Z

X

A B

Local (XCZC C=AB) and molecular (XZ) coordinates

21|)21(252321231

14

jj

jgr

T

mjCjSl

-1500 -1000 -500 0 500 1000 1500

-1000

-500

0

500

1000

3212

12123252

|mj |

Ene

rgy

cm

-1

cm-1

Energy levels of the Co-ion in the axial surrounding

Zero-fieldspliitingparameterpositive

Spin-orbit interactionAxial crystal field

BAplpHpZCo sl23322

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

BABA

BABABA

XZZX

ZZXXYYex

ssss

ssssssJH

sin

cos2

mmsmm

mmsmm

slXsl

slXsl

A

A

0231231

0231231

Isotropic Heisenberg-Dirac-Van Vleck Hamiltonian

BsAsBsAsBsAsJJH

ZZYYXX

BAex

22 ss

molecular frame

local frames of ions A and B

Ising pseudo-spin-12 Hamiltonian for the exchange

problem

BA ZZeffex JH 2

||

cos3cos92

JJJeff

components |12gt of the pseudo-spin correspond to the states |ml= 1 ms= 32gt

23

23

23

ml=1 ms= 32

ml=1 ms= 12

ml=1 ms= 12

ml=1 ms= 324E (ml=1)

STRONG NEGATIVE TETRAGONALFIELD ON Co(II)

Spin-orbital splitting of the tetragonal 4E term

pp ZZ

pSO slEH 234

Axial form of the spin-orbit interaction

J |||| λκ

0 3|| ggg e

Principal values of the g-tensor for the ground Kramers doublet

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

TOTAL HAMILTONIAN OF A CHAIN (local frames ||gtgt |J| )

A

B

A

B

ZB

Z

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

A

B

A

B

ZB

Z

ZA

XA

ZA

XA

XB

X

Noncolinear spin structure of the chain and illustration

for a single spin flip-flop process

BZBZAZAZi

AZBZBZAZeff iigiiiiJH HH ||12

)1(2 cos2 21

i

AZ

BZ

BZ

AZeff iiiiJH

Antiferromagnetic interaction (Jeff =9Jcos Jlt0π2 )

Ferromagnetic interaction

12sin2 22 iiiiJH A

XBX

BX

AX

ieff

Height of the Barrier b=E[B(1)= 12]-E[ B(1)= 12 ]=2|Jeff|

Molecular frames

H=H1+H2+H3

H3 - antisymmetric Dzyaloshinsky-Moria exchange

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

0

5

10

15

20

25

1 10 100 1000Frequency (Hz)

(

emu

mol

)

22 K21 K20 K19 K18 K

0

1

2

3

4

5

6

7

1 10 100 1000Frequency (Hz)

(em

um

ol)

22 K21 K20 K19 K18 K

Frequency dependenceof the prime and primeprime

components of the ac magnetic susceptibility

measured at various temperatures

BARRIER FOR MAGNETIZATION REVERSAL

TEMPERATURE DEPENDENCE OF THE

RELAXATION TIME

044 046 048 050 052 054 056 0584

5

6

7

8

9

Ln(1

)

1T (K -1) 01 ln ln

TBkb

The triangles and diamonds represent the relaxation times obtained from the frequency dependence of rsquo and rsquorsquo

respectively The solid line corresponds to the best fit of the data to the Arrhenius

expression

b=2|Jeff |=194 cm-1

Jeff =-97 cm-1

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

MAGNETIC BEHAVIOR OF THE CO CHAIN

011

YYXX

XXZZ

ZZ XFZF ||H||HHHHH

Principal values of the magnetic susceptibility tensor

0 10 20 30 40 5000

05

10

15

20

25

30

e

mu

K m

ol-1

T K0 10 20 30 40 50

0

1

2

3

4

5

6

7

8

9

ZZT

XXT

e

mu

K m

ol-1

T K

Temperature dependence of T

circles ndashexperimental data solid line ndashtheoretical curve calculated

with =-180 cm-1=08Jeff=-97 cm-1=15 o

Nonzero diagonal components of the T tensor calculated using the

parameter values =-180 cm-1=08Jeff=-97cm-1=15o

3XXZZ

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

AXIAL LIMIT||gtgt ||gtgt|J| 2BA ZZeffex JH

MAIN QUESTION What is the possible range of the values of the axial field parameter for which the exchange Hamiltonian is close to the Ising form

BABABABABA XZZXZZXXYYex ssssssssssJH sincos2

Isotropic exchange (local frames)

PERTURBATION THEORYsplittings due to the spin-orbit coupling and axial crystal field exceed those caused

by exchange and Zeeman interactions

UNPERTURBED HAMILTONIAN PERTURBATION

BAi

iZEex HHV

EFFECTIVE PSEUDO-SPIN-12 HAMILTONIAN

BApsllH ppZBAp p

))23()32(( 2

0

21effeffeff HHH

Heff rarrrestricted space of ground Kramers doublets Heff(1) Heff

(2)first and second order terms with respect to V

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

EFFECTIVE HAMILTONIAN

BAppXpXpXpXpZpZBXAZBXAZ

BZAXBZAXBZAZBZAZBYAYBYAYBXAXBXAXeff

HHgHgJ

JJJJH

|| )(2

2222

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

BAZXJ

BA XZJ

BAZZJ

BAYYJBAXXJ

-3000 -2000 -1000 0 1000-10

-8

-6

-4

-2

0

2

Exch

ange

para

met

ers

cm-1

cm-1

Exchange parameters as functions of J=-11cm-1=-180 cm-1=08=150

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

-3000 -2000 -1000 0 1000

1

2

3

4

5

6

7

8

g fa

ctor

s

cm-1

||g

g

Principal values of the g-tensor as functions of

bullOnly the exchange parameter JZAZB is nonvanishing for lt0||1500 cm-1

bullg||gtgtg in this range of valuesbullValues of in the range 1500 cm-1 || 2000 cm-1 are realistic for transition metal ionsbull The adopted approximation is well-justified for reasonable values of the axial crystal field parameters

11 111579cos9 cmJcmJJ oeff

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

SUMMARY

The developed quantum-mechanical approach represents the first attempt to explain by theory the SCM behavior and spin-canting phenomenon in the zigzag-chain compound [Co(H2L)(H2O)]

Ferro- and antiferromagnetic contributions to the deduced pseudo-spin-12 Hamiltonian give rise to a canted spin structure and subsequently to an uncompensated magnetic moment

The model is applicable not only to the spin-canted Co(II) chains but also to chains composed of other Kramers ions it is not restricted to the case of antiferromagnetic exchange

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian

The main ingredient of SMM and SCM behavior of the systems containing metal ions with unquenched orbital

angular momenta is the strong negative first order single-ion anisotropy which manifests itself in the

anisotropy of the parameters of the effective Hamiltonian acting within the ground manifold of the system From

this point of view these systems are quite different from the conventional spin systems exhibiting weak magnetic

anisotropy described by the zero-field splitting spin Hamiltonian