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Title: Multiferroics

台灣大學物理系 胡崇德 (C. D. Hu)

Abstract

Multiferroics is the type of material which possesses several long-range orders. These long-range orders, such as ferroelectric order, (anti)ferromagnetic order, ferroelastic order are coupled with each other and give rise to interesting physical phenomena. Experiments such as magnetic susceptibility measurement, spin-polarized neutron scattering and synchrotron radiation can give us important information of the cause of multiferrroics. Several theoretical models of the mechanism will be introduced. The emphasis will be on the "spin-current" model in which spin-orbit interaction will play the essential role.

1. Introduction. 2. Experiments. 3. Models. 4. Conclusion.

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1. Introduction

Multiferroics:

The kind of material which have several long-range orders

such as antiferromagnetism and ferroelectricity.

Illustration: Alan Stonebraker

http://physics.aps.org/articles/v2/20Physics 2, 20 (2009) Daniel Khomskii

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Ferroelectricity: Material which has net electric polarization.

BiFeO3

lone-pair of 6s-electrons of Bi(3+)

YMnO3

tilting of MnO5

BaTiO3

movement of Ti(4+)

TbMn2O5

dimerization

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Antiferromagnetism:Material which has no net magnetization but has long-rangemagnetic order.

Sinusoidal

Cycloidal spiral

Proper screw

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There are more than 100 multiferroic compounds.

Type I multiferroics

High Tc and large electric polarization, but weak coupling between electricity and magnetism.

Type II multiferroicsLow Tc and small electric polarization, but strong coupling between electricity and magnetism.

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K.F. Wan, J.-M. Liu and Z.F. Ren, Advances in Physics 58, 321–448 (2009)

7

8

9

RMnO3 orthorhombic

O: OxygenR: Rare earth elementsMn: Manganese

c

b

a

2. Experiments

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Kimura et. al. PRB 71 224425 (2005), TN=41K, Tlock=28K.

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Kenzelmann et. al. PRL 95, 087206 (2005) neutron scattering

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13

3. Models

Why are electric polarization and magnetic order coupled?

0

BE

t

DH j

t

D

B

����������������������������

������������������������������������������

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Maxwell's equations

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A. Atomic displacement (lattice distortion)

Ca3CoMnO6 Choi et. al., PRL 100, 047601 (2008)

face-sharing octahedra along c-axis

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4+ 2+ 4+ 2+ 4+ 2+ 4+ 2+ 4+

Mn Co

21 21 12 0,, 0i i i i i iJ S SH J JS S J

16

x

y

z

H. Katsura, N. Nagaosa and A. V. Balatsky, PRL 95, 057205

TM1 O TM2

eg eg

p

V

S1 S2

B. Spin current model

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d-orbitals under crystal field of cubic symmetry

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(sin ,0,cos ), jj j j jS q R ������������������������������������������

Sj: local spin Direction:

sj: mobile spin

2 2 2 2

2 2

, , , ,

, ,( . .)

p x x d j jx y x y

x x y

H p p d d U s S

V p d H c

j jU s S

hybridization Hund’s coupling

and are parallel.j jS s����������������������������

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2 2

2 2

,, ,

,, ,

cos( / 2)

sin( / 2)

cos( / 2)

sin( / 2)

LL p p xL d x y

L

RR p p xR d x y

R

a a

a a

l s

sin( / 2)

cos( / 2)zxl s

spin-orbit interaction

20

3

1 2122

1ˆ

VP e r eI e S S

S

���������������������������� 12ˆ ||e bond direction

* 3

, ,( ) ( )d zx p xI r z r d r

P

electronic origin

21ˆzP e

��������������

spiral spin

z

y

x

Result

22

1 1[ , ]

i j

s j j j j j j

H J S S

j S i S H J S S S S

spin current

spin current electric field electric polarization

E E����������������������������

AC (Aharonov-Casher) effect

charge accumulation

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Spiral Spin and Orbital Ordering

/ 21

/ 22

[ cos( )] ( )cos( / 2)

sin( ) ( )sin( / 2)

ij jj d

j ij jj d

A B Q R r Re q R

B Q R r Re q R

��������������������������������������������������������������������� �

��������������������������������������������������������������������� �

q is the wave vector of spiral spins. Q is the wave vector of OO.������������� �

The spin current model gives a very small electric polarization.

It is due to cancellation of the contribution of entire energy band.

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01

2

( )j j

d p

eIV aP Q s s

���������������������������� 3

1 2122

1ˆ

VP eI e S S

S

����������������������������

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Conclusion

1. Multiferroics is a field of rich physics.

2. Multiferroics is a field driven by experiments.

3. The theoretical models are not complete.

4. Multiferroics is related to a lot other physical phenomena such as, lattice distortion, charge order, orbital order, spintronics, and magnetic properties.

26D(/E) (t2/U), D/JSEg/g

VSOE

exchange interaction with spin-orbit coupling

Polarization and Dzyaloshinskii-Moriya interaction

I. Dzyaloshinskii, J. Phys. Chem. Solids 4, 241 (1958).Tôru Moriya, Phys. Rev. 120, 91 (1960)

SO V

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DM ij i jH D S S 4

30,

mdp

iJ VD l J

E

����������������������������

0 0 0 0

0 0

0 0 0 0

0 0

( ) ( ) ( ) ( )

( ) ( [ , ]) ( ) ( )

m i j m i j m m

m

m m

m i i j m m i

m

i i

i mm m

m

l S l S

l l

JS S JS SE

E E E E

J S S S iJS S

E E E E

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Relation between multiferroics and DM

, , ( )m V n

n l l ml m n d p cf

n l

p H dP r d l d ds r p

E

������������������������������������������

1Im ( )j jn ls s s

, , , ,( 1) [cos sin ] . .l i

V j j jpj l pj lH V d c e d c H c

,1

, ,

2 ( )( )

m V i nN Nk n j l j l k m

l m n d p cf

i p H dP e d l d s s d r p

E

iijcf

i i jDM jS S lE

S SDHiJ

1

,

0

,

,

2 ( )

0

( )

2 0

m V i nn j l

d p

N Nk j l k ml m n

DMl k

M

cf

P e s s d r p

M He

i p H dd

r

d

ME

E

E

l

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DM interaction iDM ijcf

i j i jS S lE

S SiJ

H D

i jDM j iH A j

Spin current 1 1

1

( ) |4

( ) | |2

j j ji BN N N N i

N N Bj i

aVJ d d d d e

iaV

s s e

i: bond direction

Gauge field

AC effect

* 2

2|

4i

Bj j i ijk kcf

iJA l e E

eaV E m c

* 28B

cf

JE e l

eV

m

a

c

E

����������������������������

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z

y

x

Mn O Mn

jxys

Eeff,z