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 longrange 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.
1
1. Introduction
Multiferroics:
The kind of material which have several long-range orders
such as antiferromagnetism and ferroelectricity.
http://physics.aps.org/
articles/v2/20
Physics 2, 20 (2009)
Daniel Khomskii
Illustration:
Alan Stonebraker
2
Ferroelectricity: Material which has net electric polarization.
BiFeO3
BaTiO3
movement
of Ti(4+)
lone-pair of
6s-electrons of
Bi(3+)
YMnO3
TbMn2O5
tilting of MnO5
dimerization
3
Antiferromagnetism:
Material which has no net magnetization but has long-range
magnetic order.
Sinusoidal
Cycloidal spiral
Proper screw
4
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 multiferroics
Low Tc and small electric polarization, but strong
coupling between electricity and magnetism.
5
K.F. Wan, J.-M. Liu and Z.F. Ren, Advances in Physics 58, 321–448 (2009)
6
7
8
2. Experiments
RMnO3 orthorhombic
c
b
a
O: Oxygen
R: Rare earth elements
Mn: Manganese
9
Kimura et. al. PRB 71 224425 (2005), TN=41K, Tlock=28K.
10
Kenzelmann et. al. PRL 95, 087206 (2005)
neutron scattering
11
12
3. Models
Why are electric polarization and magnetic order
coupled?
 D  
 B 0
Maxwell's equations
E  
B
t
H  j
D
t
13
A. Atomic displacement (lattice distortion)
Ca3CoMnO6
Choi et. al., PRL 100, 047601 (2008)
face-sharing octahedra
along c-axis
14
H  J 1  i S i S i 1  J 2  i S i S i  2 ,
4+ 2+
Mn Co
4+
2+
4+
2+
J 1  0,
4+
J2  0
2+
4+
15
B. Spin current model
z
TM1
H. Katsura, N. Nagaosa and A. V. Balatsky, PRL 95, 057205
O
TM2
S1
y
S2
x

eg
p
V
eg
16
d-orbitals under crystal field of cubic symmetry
17
H 
  p p x , p x ,     d d x 2  y 2 , d x 2  y 2 ,   U  s j  S j


 V  ( p x , d x 2  y 2 ,  H .c .)

hybridization
Hund’s coupling
Sj: local spin
Direction: S j  (sin  j , 0, cos  j ),  j  q R j
sj: mobile spin
U  s j  S j
S
j
an d s
j
are p arallel.
18
   a L  L ,d , x

2
y
2
 a p  p,x
cos( L / 2) 


  sin( L / 2) 
 cos( R / 2) 


 a R  R ,d , x 2  y 2  a p  p , x 

  sin( R / 2) 
spin-orbit interaction
 l s    zx
l s
 sin( / 2) 



cos(

/
2)


19
3
V  1
P   e r   eI  
eˆ  S 1  S 2
2 12
 S

I    d , zx ( r ) z p , x ( r ) d r
3



eˆ12 || bond direction
*
electronic origin




P
20
z
Result
y
x
spiral spin
P
eˆ z
21
spin current
H  J  S i S j

j s  S j  i[ S j , H ]  J S j  S j  1  S j 1  S j

spin current  electric field  electric polarization
AC (Aharonov-Casher) effect
E
E
charge accumulation
22
Spiral Spin and Orbital Ordering
The spin current model gives a very small electric polarization.
It is due to cancellation of the contribution of entire energy band.

j
 e  i / 2 cos( q  R j / 2)   [ A  B cos( Q  R j )] ( r  R j ) 
d1




 e  i / 2 sin( q  R / 2)  
B
sin(
Q

R
)

(
r

R
)
j
j
j
d
2


 
q is the w ave vector of spiral spins. Q is the w ave vector of O O .
23
P 
2 eIV  a 0
 ( d   p ) 

Q  s j  s j 1

3
V  1
P   eI  
eˆ  S 1  S 2
2 12
 S

24

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.
25
Polarization and Dzyaloshinskii-Moriya interaction
I. Dzyaloshinskii, J. Phys. Chem. Solids 4, 241 (1958).
Tôru Moriya, Phys. Rev. 120, 91 (1960)
exchange interaction with spin-orbit coupling
E
SO
SO
V
V
D(/E) (t2/U), D/JSEg/g
26
 (  l  S i ) 0 m ( JS i  S j ) m 0 ( JS i  S j ) 0 m (  l  S i ) m 0
E   m 

E0  Em
E0  Em

 m
H DM 
( l ) 0 m  ( J [ S i , S i  S j ]) m 0

E0  Em
D ij  S i  S j
 m
D 



( l ) 0 m  ( iJS i  S i ) m 0
iJ 
E
E0  Em
l
,J
0m
V
4
 dp
3
27
Relation between multiferroics and DM
H V  V  (  1) [cos  j d j c pj , l ,   e sin  d j c pj , l ,  ]  H .c .
P   r 


l ,m ,n
i

l
pm H V d n 
(  d   p )  E cf

dn l dl
n s l
d l r pm
Im  n s  l  ( s j  s j 1 )
i  p m H V ,i d n
Pk   2 e 
l ,m ,n
H DM 

D ij  S i  S j 
Pk   2 e 
l ,m ,n
 2e
(  d   p )  E cf
(  d   p )  E cf
M H DM 0
E0  EM
iJ 

i  p m H V ,i d n
d n l j d l ( s N  s N  1 ) j d l rk p m
 E cf
li  Si  S j
d n l j d l ( s N  s N  1 ) j d l rk p m
0 l rk M
28
H DM 
DM interaction
j
Spin current J i  aV
4i

aV
2
Gauge field
i
Aj


AC effect
H DM 

D ij  S i  S j 



iJ 
 E cf
li  Si  S j
( d N 1 d N  d N  d N 1 ) e B |i i: bond direction
j
j
 ( s N  1  s N ) | j e B |i
2 iJ 
eaV  E cf

i
A j ji
8 Jm c 
l j e B |i  
* 2
4m c
 ijk E k
j
* 2
E 
eV a  E cf
eB  l
29
z
y
x
Eeff,z
s
Mn
jxy
O
Mn
30