Cornell University
“I would found an institution where any person can find instruction in any study.” − Ezra Cornell, 1868
Competing Ferroic Orders The magnetoelectric effect
Basic Training in Condensed Matter Theory 2009
Craig J. Fennie School of Applied and Engineering Physics [email protected]
Basic Training 2009– Lecture 04
2 Basic Training 2009– Lecture 04
Module Outline 1. Overview and Background
• Ferro ordering, the magnetoelectric effect
2. ME revisited, and basic oxide physics • ME effect revisited: Toroidal moments • Complex oxides basics: Types of insulators (i.e., ZSA classifications), Coordination chemistry
3. Structure and Ferroelectricity • Basics of space groups • Soft mode theory, lattice dynamics, group theoretic methods • Competing lattice instabilities • microscopic mechanisms, improper FE • Modern theory of polarization (Berry Phase)
4. Magnetism • Basics, exchange interactions, superexchange, Dzyaloshinskii-Moria • How spins couple to the lattice! Phenomenology and microscopics (spin-phonon, spin-lattice, etc) • Competing magnetic orders • Systems: ZnCr2O4, EuTiO3, SeCuO4, TeCuO4
2
3 Basic Training 2009– Lecture 04
Magnetism and how it couples to the lattice
Tuning AFM ⇔FM – The Goodenough-Kanamori rules
Spin-phonon coupling – ZnCr2O4 → spin-induced phonon anisotropy – EuTiO3 → magneto-capacitance
Spin-lattice coupling – Weak ferromagnetism Fe2O3 – Spin spiral Lifshitz invariant
3
Isotropic Exchange
Dzyaloshinskii-Moria Exchange
4 Basic Training 2009– Lecture 04
Magnetic properties of localized systems
4
Eu2O3
Eu: 4f7 5d0 6s2 → Eu3+: 4f6 5d0 6s0
Magnetism arises from an incomplete shell
Question: is Eu2O3 magnetic? What do I mean by magnetic? For now I mean, Does a magnetic field couple to the fairly large S=3µB, NO! why? Well magnetic field couples to J=|S-L|
Hund’s Rules 1) Max S =3 2) Max L =3 3) J=|S-L| =0 less than half filled (J=|S+L| if more …)
But is it magnetic? Yes!
〈0|µ|0〉 = gJµB〈0|J |0〉 = 0
χ =N
V
(2µ2
B
∑
n
|〈0|Lz + gSz|0〉|2
En − E0− e2mu0
6me
Z∑
i=1
〈r2i 〉
)
Van Vleck Paramagnetism diamagnetism
Both terms, small and temperature independent (χVV ~ 1/Δ)
5 Basic Training 2009– Lecture 04
Magnetic properties of localized systems
Currie Weiss Susceptibility
5
Temperature ΘCW = Tc
0
χ-1
TN -ΘCW
AFM PM FM
χ ∝ 1T − θCW
E = −∑
ij
JijSi · Sj + gµB
∑
i
Si · H
kBθCW =23
∑
n
JnznS(S + 1)
Magnetic exchange interactions Mean-field
6 Basic Training 2009– Lecture 04
Properties of common Antiferromagnets
6
7 Basic Training 2009– Lecture 04
Effect of strong magnetic field (AFM)
Susceptibility below TN and Spin flop – Hard vs easy axis
7
8 Basic Training 2009– Lecture 04
Origin of easy/hard axis
Spin-orbit interaction
1. Single-ion anisotropy
2. Anisotropic exchange coupling
8
λ"L · "S
1. Crystal field couple to charge density 2. Charge density couple to spin E-field
overlap
Different energy
9 Basic Training 2009– Lecture 04
General form of spin-spin coupling
9
+kxS1xS2x + kyS1yS2y + kzS1zS2z
Isotropic symmetric exchange (l=0)
Dzyaloshinskii-Moriya Antisymmetric exchange (l=1)
Anisotropic symmetric exchange (l=2) D and K both spin-orbit effects
D ∼ λJ ≈ (∆g/g)J
k ∼ λ2J ≈ (∆g/g)2J
H = J!S1 · !S2 + !D · (!S1 × !S2)
H = !S1 · J · !S2
10 Basic Training 2009– Lecture 04
10
Dzyaloshinskii-Moriya
E = |J| 〈Si⋅Sj〉 + ∑ Dij⋅〈SixSj〉
Collinear AFM Spin canting
D = 0 M
D ≠ 0
Relativistic correction to Anderson’s superexchange
• Direction of canting determined by the sign of D
11 Basic Training 2009– Lecture 04
Weak ferromagnetism vs ME effect
11
Cr2O3
!"!
#$"
#$"
%
#
!
&
4321 MMMML −+−=
'"
0≠zL
+!!Ez
++!Lz
+!+Hz
3z2xI
zzzzz HEHEL ||αλ =
()**+,-.+(/01/23-3*345+,.6/273(+
( )yyxxz HEHEL +'583-.35,(9
zL∝⊥αα ,||
:;</0-=+-/23-3*+,+-/ >? @/!ABC
m3
z
z
m 3),(3,1
3),2(3,1
±
±
⊥
⊥
point group
Cr2O3
L = M1-M2+M3-M4
Cr2O3
!"!
#$"
#$"
%
#
!
&
4321 MMMML −+−=
'"
0≠zL
+!!Ez
++!Lz
+!+Hz
3z2xI
zzzzz HEHEL ||αλ =
()**+,-.+(/01/23-3*345+,.6/273(+
( )yyxxz HEHEL +'583-.35,(9
zL∝⊥αα ,||
:;</0-=+-/23-3*+,+-/ >? @/!ABC
m3
z
z
m 3),(3,1
3),2(3,1
±
±
⊥
⊥
point group
L = M1+M2-M3-M4
Fe2O3
Neel temperature ~500K Morin temp ~260K
12 Basic Training 2009– Lecture 04
Exchange: Background
12
Hund’s rule ➟ like FM
Covalent bond ➟ like AFM
bonding
Anti-bonding
13 Basic Training 2009– Lecture 04
Exchange: Background
13
If they are the same orbital ➟ AFM
If they are the different orbital ➟ FM
Direct exchange
TM-d O TM-d
Superexchange 90˚ SE
90˚ Hund’s rule J_H coupling on the anion
14 Basic Training 2009– Lecture 04
Exchange: Background
14
NiO
Competing interactions 1. Strong
AFM Ni-O-Ni 180
2. Weaker FM 90˚ SE
15 Basic Training 2009– Lecture 04
Exchange: Background
15
Arbitrary angle
J =J90 sin2θ + J180 cos2θ
90˚ 180˚
Empirically 135˚
SeCuO3 FM TeCuO3 AFM
16 Basic Training 2009– Lecture 04
ACr2X4: Cubic Spinel Structure
16
eg
t2g ⇒ S=3/2
Cr3+: 3d3 4s0
What is Moment? µ ~ S =3/2 Why -> orbital dof quenched
Superexchange pathways
17 Basic Training 2009– Lecture 04
ACr2X4: Cubic Spinel Structure
Network of edge sharing octahedra
17
18 Basic Training 2009– Lecture 04
Background: Spinels
18
ACr2X4
• A= Zn, Cd, Hg
• X= O, S, Se
eg
t2g ⇒ S=3/2
a (Å) Theory Exp.
CdCr2S4 10.12 10.24 CdCr2Se4 10.63 10.74
HgCr2Se4 10.65 10.74
ZnCr2O4 8.26 8.31
CdCr2O4 8.54 8.59
Ferromagnetic Insulators
Tc ~100K,
TΘ ~200K
Anti-ferromagnetic Insulators
TN ~10K,
TΘ ~ 100K
Cr3+: 3d3 4s0
19 Basic Training 2009– Lecture 04
Exchange interactions Direct Cr-Cr exchange → AFM 90° Cr-O-Cr SE → FM
19 oxygen
chromium
AFM → FM a) as lattice constant increases b) as electronegativity of anion decreases
20 Basic Training 2009– Lecture 04
Huge spin-phonon coupling
20
3-fold T1u phonon mode
Singlet A2u
Doublet Eu
Temperature (K)
Tn= 12.5K
Symmetry lowering at Tn
Oh → D4h Cubic to tetragonal
Phonon splitting at Tn
T1u → A2u ⊕ Eu
AFM-ZnCr2O4
Exp: λ =6-10 cm-1
21 Basic Training 2009– Lecture 04
21
Spin-phonon coupling
J(u)
J(u+δ)
Phonon modulated exchange interaction Baltensperger and Helman, Helvetica physica acta 1968.
e.g. can understand large spin-phonon coupling in ZnCr2O4 Fennie and Rabe, Phys. Rev Lett. May 2006
Eph = 1/2 ω02u2
E = E0 +Ephonon+ Espin
Esp = -∑Jij 〈Si⋅Sj〉
⇒ ω2∝ ω02 - ∂2J/∂u2 〈Si⋅Sj〉
renormalized bare magnetic contribution phonon phonon
J(u) ≈ J(0) + 1/2 ∂2J/∂u2 〈Si⋅Sj〉 u2
22 Basic Training 2009– Lecture 04
22
Spin-phonon coupling
J(fu)
J(fu+δ)
Phonon modulated exchange interaction Baltensperger and Helman, Helvetica physica acta 1968.
e.g. can understand large spin-phonon coupling in ZnCr2O4 Fennie and Rabe, Phys. Rev Lett. May 2006 to lowest order in Si Sj
23 Basic Training 2009– Lecture 04
Origin of large anisotropy
23
spin up
spin down
Cr ions
Direct-exchange length scale, α-1
Exp: α=8.9 Å-1
Theory: α=9.05 Å-1
f3 has significant anisotropy in force constant matrix
x y z
f3: One set of T1u Partner functions
z-mode x-mode (or equivalently y-mode)
24 Basic Training 2009– Lecture 04
24
TN
Bulk Eu2+Ti4+O3: Ground state antiferromagnetic paraelectric
• r(Eu2+) ~ r(Sr2+); Cubic perovskite • Eu2+ → J=S=7/2; Tn ~ 5.5K, G-type AFM
25 Basic Training 2009– Lecture 04
25
Perovskites and the Period Table
Substitutions on A, B or both (A1-xA’x)(B1-yB’y)O3 Random distribution or ordered
26 Basic Training 2009– Lecture 04
Phonon anomaly at Tc
FM- CdCr2S4
26
Symmetry lowering at Tc
Oh → Oh
Cubic to cubic (ignoring LS)
No phonon splitting at Tc
T1u → T1u
