crystal eld and magnetism with wannier...

Crystal field and magnetism with Wannierfunctions

Pavel NovakInstitute of Physics, Prague

Seminar talk, Vienna, March 2014

Pavel Novak Institute of Physics, Prague Crystal field and magnetism with Wannier functions

Outline

Introduction: electronic structure of rare-earth oxides

Atomic-like hamiltonian

Crystal field hamiltonian and its parameters

Calculation of crystal field parameters.

Determination of multiplet splitting and magnetism

Results

Energy levelsMagnetism

Limitations of method

Conclusions and outlook


Electron structure of oxide matrix: YAlO3

Density of states calculated with WIEN2k package

-6 -4 -2 0 2 4 6 8 10E [eV]

0

4

8

12

16

20

24

28

Den

sity

of

stat

es [s

tate

s/eV

]

total DOS

EF

valence bandgap

conduction band


Tb:YAlO3 (TbY23Al24O72)

Tb(4f ) states treated as core levels (WIEN2k)

Core states feel only spherical potential,they are not split by crystal field and do not contribute to it.

-6 -4 -2 0 2 4 6 8 10E [eV]

0

4

8

12

16

20

24

28

Den

sity

of

stat

es [s

tate

s/eV

]

total DOS

EF

Tb3+

4f states



Expected position of Tb(4f ) levels, 4f treated as valence states

-8 -4 0 4 8 12E [eV]

0

5

10

15

20

25

30

Den

sity

of

stat

es

[sta

tes/

eV]

total DOS

EF



Splitting of Tb(4f ) levels by crystal field

7 orbital singlets - complete lifting of degeneracy

0 20 40 60 80 100E [meV]

0

200

400

600

800

Tb(4

f) d

ensi

ty o

f st

ates

[s

tate

s/eV

]

Ecf


Crystal field parameters from density of states

Cubic symmetry

4f level split on three states (singlet, 2 triplets).Crystal field characterized by two parameters=⇒ energies are sufficient information to determine crystal field.PrO2, PrBaO3 Novak, Divis, phys.stat.sol (b) 244, 3168 (2007)

Cs symmetry of Tb3+ site in YAlO3

7 singlets, 15 parameters of crystal field=⇒ energies are not sufficient to determine crystal field.Additional information needed: orbital composition of singlets.Problem: too many information (48) =⇒ use least squares.Ambiguous, many local minima, which one is correct?

Way out: Wannier functions


Atomic-like hamiltonian for well localized f electrons

Sum of atomic (free ion), crystal field and Zeeman parts

H = HA + HCF + HZ ; HZ = µB ~BN∑i=1

(li + 2si )

Free ion hamiltonian

HA = Eavg +∑

k=2,4,6

F k fk + ξ4f

N∑i=1

si li +

αL2 + βG (G2) + γG (R2) +∑j=0,2,4

M jmj +∑

k=2,4,6

Pk pk +∑

r=2,3,4,6,7,8

T r t(r),

HA has spherical symmetry, depends on 20 parameters.For rare-earth ions approximate values of parameters are known, orthey may be calculated ab initio.


Crystal field hamiltonian

HCF =2L∑k=2

k∑q=−k

B(k)q C

(k)q

C(k)q : spherical tensor operator of rank k.

C(k)q form full orthogonal system.

B(k)q : crystal field parameters.

HCF acts on one-electron states, it does not depend on spin.

Values of q and k for which B(k)q 6= 0 depend on site symmetry.

f electronscubic symmetry: 2 parameters.Cs symmetry of orthorhombic perovskites: 15 parameters.LaF3 no symmetry: 27 parameters.


Determination of crystal field parameters

Steps

1 Selfconsistent band calculation with f electrons in core.Standard WIEN2k calculation.

2 f states and oxygen ligand treated as valence statesin a nonselfconsistent calculation, all other states moved away.Relative position of 4f and oxygen states is adjusted using’hybridization’ parameter ∆ (the only parameter of method).WIEN2k with orbital energy shift potential.

3 Transformation of 4f band states to Wannier basis.wien2wannier, wannier90 packages.

4 Extraction of local 4f hamiltonian and its expansion in seriesof spherical tensor operators. program Bkqcoefficients of expansion are crystal field parameters.


Hybridization parameter ∆

∆ is the charge transfer energy between 4f and ligand statesAnalog in 3d compounds (Co:ZnO): Kuzian et al. Phys. Rev. B 74,155201 (2006).

R(4fn) ox(2p

6)

R(4fn+1

) ox(2p5)

∆

In oxides semiempirical analysis of optical absorption data gives3 eV < ∆ < 10 eV. ∆ can also be estimated using WIEN2k.


Multiplet splitting and magnetic properties

1 REcfp. Solves eigenproblem for hamiltonian H,originated from lanthanide package (Edwardsson and Aberg,2001).Result: energies and eigenvectors of all states of (4f )n

electron configuration, and their dependence on externalmagnetic field.

2 g chi. Calculation of eigenstates magnetic moments, g and χtensors including their canonical form.

3 temp. Temperature dependence of magnetic moments andsusceptibility, optionally averaging for polycrystals.


Multiplet splitting in Er:YAlO3

Splitting of the lowest four multiplets by crystal field.Experiment: Duan et al., Phys. Rev. B 75, 195130 (2007).

4 8 12 16 20 24n (index of eigenstate)

0

25

50

75

100

En

[m

eV]

exp.

∆ = 5.4 eV∆ = 8.2 eV∆ = 10.9 eVno hybridization

4I15/2

4I13/2

- 832 meV

4I11/2

- 1293

4I9/2

-1553

In orthoperovskites ∆ was fixed at 8.2 eV (0.6 Ry)


Comparison with experiment: energy levels

NdGaO3: splitting of ground 4H9/2 multiplet by crystal field.Experiment: Podlesnyak et al., J. Phys.:Condens. Matter 5, 8973 (1993).

1 2 3 4 5i (index of eigenstate)

0

10

20

30

40

50

60

70

Ei

[m

eV]

experiment

theory



PrGaO3: splitting of ground 3H4 multiplet by crystal field.Only six of eight excited singlets were detected experimentally.Experiment: Podlesnyak et al., J. Phys.: Condens. Matter 6, 4099 (1994).

1 2 3 4 5 6 7 8 9i (index of eigenstate)

0

20

40

60

80

Ei

[m

eV]

theory

exp. A (5 and 7 singlet undetected, χ = 4.2 meV)

exp. B (Podlesnyak, 6 and 9 singlet undetected, χ=10.3 meV)



Nd:LaF3: splitting of low lying multiplets by crystal field.No local symmetry, 27 crystal field parameters calculated.Experiment: Carnall et al., J. Chem. Phys 90, 3443 (1989).

10 20 30 40 50 60i (index of eigenstate)

0

200

400

600

800

Ei

[cm

-1]

Experiment

Theory with ∆=0.4 RyI9/2

I11/2

-1978

I13/2

-3918

I15/2

-5816

F3/2

H9/2

, F5/2

F7/2

,S3/2

F9/2

H11/2

G5/2

,G7/2

∆calculated

=0.42 Ry


Nd:LaF3. Experiment vs. calculation

Mean quadratic deviation χ as function of ∆

χ =1

Nlevels

√√√√Nlevels∑i=1

(E calci − E exp

i )2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1∆ [Ry]

0

20

40

60

80

χ

[c

m-1

]


RE:LaF3. Calculated ∆ parameter

∆ as function of number of 4f electrons

Calculation with RE(4f ) treated as core electrons∆ = Etot(4f n, N val. electrons) - Etot(4f n+1, N − 1 val. electrons)

1 2 3 4 5 6 7 8 9 10 11 12 13n

4f

0

0.1

0.2

0.3

0.4

0.5

∆ [

Ry]

n4f

(Nd) = 3, ∆(Nd) = 0.42 Ry


Manganites - magnetism in excited states

NdMnO3: splitting of the lowest three multiplets by crystal fieldand ga factor of nine Kramers doublets.Experiment: Jandl et al., Phys. Rev. B 71, 024417 (2005).

0 4 8 12 16i (index of eigenstate)

0

100

200

300

400

Ei

[m

eV]

experiment

theory

5.2 4.6

5.8 4.9

8.4 8.6

2.0 2.5

5.2 5.9

2.4 2.0

6.4 6.4

10.2 11.7

2.6 3.4

4I9/2

4I11/2

- 486 meV4I13/2

- 1436 meV


NdGaO3: temperature dependence of inverse susceptibility.Experiment: Novak et al., J. Phys.: Condens. Matter 25, 446001 (2013).

0 40 80 120 160 200 240 280 320T [K]

0

20

40

60

80

100

120

1/χ

[T/µ

B]

B || aB || bB || cpolycrystal average

experiment

NdGaO3


Limitations, outlook

Limitations

Hybridization of f electrons should be weak,as it is not taken into account selfconsistently.If the f -oxygen states hybridization is strong:multiplet ligand-field theory using Wannier orbitals.M.W. Haverkort et al. Phys. Rev. B 85, 165113 (2012)

Atomic-like, model hamiltonian is limited to f space.

Outlook

Intensity of f − f transitions. Instead of Judd-Ofelt methodab-initio approach.

Application to other rare-earth compounds.

Is the method applicable to the d and 5f states?


Conclusions

New method to calculate crystal field parameters proposed.Method contains single adjustable parameter, which can beindependently estimated.

Until now method applied to more than sixty compounds:

orthorhombic aluminates, gallates, cobaltites and manganites.Rare-earth impurities in aluminium and gallium garnets.Rare-earth layered hexagonal cobaltates.R:LaF3; R=Ce, Pr ...Yb.R2Fe14B (cooperation with Tohoku Univ.).CeCu2Si2 (cooperation with MPI Dresden).

Agreement with experimental multiplet splitting within meV.

Good agreement with magnetic properties.

Present version applicable to arbitrary local symmetry.

With few modifications method may be used bynon-specialists.


Acknowledgements

Cooperation, co-authors

Eva Mihokova: Application of method to rare-earth impurities inaluminium and galium garnets.Jan Kunes: Wannier functions, problem of hybridization.Zdenek Jirak: correction of numerous bugs and omissions.Karel Knızek: help with orthorhombic perovskite structure

and WIEN2k calculations.Mirek Marysko: measurements of magnetic susceptibility.

Constructive criticism (all co-authors).

We acknowledge useful discussion with prof. M. F. Reid (Univ.Christchurch) during f electron conference in Udine.

Financial support

Grant Agency of the Czech RepublicProjects No. 204/11/0713, P204/10/0284, and 13-25251S.


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