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 fennie@cornell.edu

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