Magnetic transitions of multiferroics revealed by photons 黃迪靖
同步輻射研究中心清華大學物理系
May 9, 2007
• Multiferroicity• Soft x-ray magnetic scattering• Magnetic transitions and switch of spin chirality
CollaboratorsNational Synchrotron Research Center: J. Okamoto, K. S. Chao, H. H. Wu, H.-J. Lin, and C. T. Chen
National Tsing Hua Univ. : C. Y. Mou
Rutgers Univ. : S. Park, J. Y. Choi, and S-W. Cheong
Acknowledgement
C. D. Hu, National Taiwan University
趙國勝 吳雪鴻 林宏基 陳建德
牟中瑜
胡崇德
Magnetism: ordering of spins
Ferroelectricity: polar arrangement of charges
Magnetization can be induced by H field
Electric polarization can be induced by E field
Induction of magnetization by an electric field; induction of polarization by a magnetic field.
- first presumed to exist by Pierre Curie in 1894 on the basis of symmetry considerations
However, the effects are typically too small to be useful in applications!
Magnetoelectric effect
Materials exhibiting ME effect:Cr2O3
BiMnO3
BiFeO3
…..
M. Fiebig, J. Phys. D: Appl. Phys 38, R123 (2005)
(Ferro)magnetism vs. (Ferro)electricity
(La,Sr)MnO3: spins from : 3d3 or 3d4
Magnetic moment:
-
unfilled d bandsimpurities
Pauli vs. Coulomb
Exchange interactions: i jJS S
superexchange
double exchangeMn
TM O TM
Large displacement often involved in ferroelectricity•PbTiO3: Pb-O covalent bond cubic
800 K
tetragonal300 K
Pb-O plane Ti-O plane
Pb
O
Kuroiwa et al, PRL87 217601 (2001)
TC=763 K
Ti4+
(Ferro)magnetism vs. (Ferro)electricity
Classic examples: BaTiO3 or PbTiO3
polarization from cation/anion paired diploes
-
O-2
Ba+2
0.10 Å
0.05 Å0.04 Å
+
+Ti+4
Ti+4 3d0 O 2p2
Filled or empty d band, no room for magnetism!
Two contrasting order parameters
:
t t M M
P P
Magnetization: time-reversal symmetry broken
Polarization: inversion symmetry broken
: ,
r P MP Mr P qr
Recently discovery in the coexistence and gigantic coupling of antiferromagnetism and ferroelectricity in frustrated spin systems such RMnO3 and RMn2O5 (R=Tb, Ho , …)
revived interest in “multiferroicity”
•TC < TN
• Frustrated magnetic systems.
TbMnO3: Kimura et al., Nature 426, 55, (2003)
TbMn2O5: Hur et al., Nature 429, 392 (2004)
Magnetism and ferroelectricity coexist in materials called “multiferroics.”
Normal antiferromagnetgeometrically frustrated magnetization
?
FMAFM frustrated spin chains
spin frustration
J >0
T=35 K
T=15 K
TN=42 K
TC=27 K
TbMnO3
antiferromagneticTN=42 K
Kenzelmann et al., PRL 95, 087206 (2005)
collinear magnetic order, inversion symmetric
non-collinear magnetic order, inversion symmetry broken
Kimura et al. Nature, 426, 55 (2003)
H // b
TbMn2O5 Nature, 429, 392 (2004)
antiferromagneticTN=42 K
x
y
y
Issues: -What is the underlying mechanism of the gigantic ME effect?
-What kind of spin configurations supports electric polarization?
MostovoyPRL (2006)
Frustrated magnets: RMnO3, RMn2O5
TC < TN polarization governed by magnetism ?
spiral ? collinear
?
2q
Phenomenological Ginzburg-Landau approach
212
e inEF P P
Lowest order in the expansion of the free energy:
ji q rq j
j
S S e
magnetization at modulation vector
0 e inPF
PE
q
How to construct in terms of ?in qE S
internal field from spins
Symmetry properties
1 1; j jiq r iq r
q j j qj q
S S e S S eN
1[ ] jiq r
q j qj
Inv S S e SN
Inversion invariant: q qS S
Phenomenological Ginzburg-Landau approach
ˆ ( ˆ: ( ) , , or )
in q q q q q qu S S SS q SE S
, is invariant. ����������������������������
P P F
eventime num reversa ber ofl:
qS
homogeneous: and -
P q q
212
e inEF P P
interal field: , if . ����������������������������
in inE E r r
How to construct ?
inE
Mostovoy PRL(2006)
noncollinear spins, e.g. spiral
Okamoto et al., PRL 98, 157202 (2007)
Microscopic theory
•Atomic displacement + antisymmetric exchange interaction Sergienko and Dagotto PRB 73, 094434 (2006)
Sergienko,Sen and Dagotto PRL 97, 227204(2007)
•Spin current Katsura et. al., PRL 95, 057205 (2005) Jia et. al. PRB 74, 224444
How to induce polarization without involving atomic displacement?
Essential Physics: Motion of magnetic moments induces electric dipoles! – the intrinsic Aharonov-Casher Effect
Einstein and Laub (1908):A magnetic dipole moment m that moves with constant velocity should develop an electric dipole moment
Hirsch, PRL 83, 1834 (1999)
What is spin current?
2 i jH J S S
1[ , ]
s nn
d Sj S H JS S
dt i
Heisenberg Model:
S
Electric polarization induced by “spin current”
12ˆ sP e j
The spin-current model
1 2sj S S
Katsura et. al., PRL 95, 057205 (2005)
Magnetic transitions and switch of spin chirality in CoCr2O4
Yamasaki et al. PRL 96, 207204(2006)
CoCr2O4
ferrimagnetic
TC= 93 K
conical spinstructure
TS= 26 K
spinel
field cooling0.01 T
P//[-110]
[001]
q// [110]
CoCr2O4
Yamasaki et al. PRL (2006)
The spin-current model
ˆ ˆcos( ) sin( ˆ)
zA r x B r yq SS q z
a* (2/a)
b* (
2/a
)
110 [1,1,0]2
a
a
interlayer spacing of (110) lattice planes
q=(2/3,2/3,0)
real space
reciprocal space
(2,2,0)
(4,4,0)
Bragg peaks of CoCr2O4
3
2 2
a
(110)
110
2 2[1,1,0] 2
q
a a
3 3110 2 2
2 2 2 2[ , ,0]
3
q
a a
Tomiyasu et al. PRB (2006)
(2-, 2-, 0)=0.63
Yamasaki et al. PRL (2006)
Cor
rela
tion
leng
th (
nm)
P
Soft x-ray magnetic scattering
Elastic x-ray scattering
2
qfd
d
scattering form factor
k
'k
q k' k
momentum transfer
q
sin
4sin2 kq
2
A volume element at will contribute an amount to the scattering field with a phase factor .
r3d r
reiq
jrj(r )eiq
qj
f Fourier transform of
charge distribution.
Bragg condition:q = modulation vector of charge, spin , or orbital order
r
rk r'k
' kk
elastic scattering
Fourier transform of spin distribution.
jrj(r )eiq
qj
S S
Scattering accumulates microscopic effects and reveals macroscopic properties.
ii q rq i
i
S S e
X-ray scattering
kkq 'k
'k
magnetization at modulation vector
q
Elastic x-ray scattering
2
qfd
d
scattering form factor
k
'k
q k' k
momentum transfer
q
sin
4sin2 kq
2
A volume element at will contribute an amount to the scattering field with a phase factor .
r3d r
reiq
jrj(r )eiq
qj
f Fourier transform of
charge distribution.
Bragg condition:q = modulation vector of charge, spin , or orbital order
r
rk r'k
' kk
elastic scattering
Fourier transform of spin distribution.
jrj(r )eiq
qj
S S
Detectable with x-ray?
X-ray magnetic scattering
( ) jiq rq j
j
S s r e
kkq 'k
'k
( )js r : spin density
2 22( )
2
( )
e ej j j jmc c
j j
ece
mc
A
p AE v s E
A
m
s P st
c
2
21
int 2
2( )
2
e ej
e ej j cm
jm c mcj j
cj
s E
H A
P
P s A
A
Relevant interactions:
1c
AE
tB A
Spin-orbit interactions:
2 2
2 2 2
2
2 22 )int (( ) ( )e e
j jmc me e
j jmc mcj
cj j j
AA r A P AH A
ts s
2
222
int
2
2 2
2
2 2 ' 'j jiq
j
rq r
j
ijf e
c e
V mi f
f H i
mcce s ii
kinetic energy m.B SO
Non-resonant
mag 32
charge
~ ~ 10f
f mc
6
e
mag 10~
for ~ 600 eV
X-ray magnetic scattering
Resonant
int i
2
nt2( )
n i ni f
f H n n HE
iE
E E
Blume, J. Appl. Phys. (1985)
Resonant X-ray magnetic scattering
1 lm
electric dipole transitions
1 lm
0ε ε
F1,1 F1,-1
'k
kq 0ε
ε
001
100
010
scattering amplitudes )(ˆ)εε(8
31,11,10
* FFzif res
mag
Hannon et al., PRL(1988)
As a result of spin-orbit and exchange interactions,magnetic ordering manifests itself in resonant scattering.
1 lm
X-ray absorption
X-ray scatteringq=(0.63, 0.63, 0)
3d
2p3/2Co
778 eV
Resonant soft x-ray scattering of CoCr2O4
0
2
0ˆˆS x cos( ) S si
( )
n( )
q q qS S
q y qx
S
S x2
qS
[001]
q // [110]
P // [-110]
ˆ ( )
q q qq
P i q S S
2
qS2
qS
( ) cos( ) sin( )S r A q r B q r
=2 2
iq r iq rA iB A iBe e
2q q
iS S A B
0A B
( ) cos( ) sin( )S r A q r B q r
chirality change:
changes signq qS S
q q
( ) cos( ) sin( )S r A q r B q r
110 110ˆ ˆq q x y
110 110 q r q x y
reciprocal space
ˆ [110]y
For a given x, switch of chirality:
110
110 q
110 110 110 110= cos( ) sin( ) cos( )+ sin( ) cos( ) sin( )A q x B q x y A q x B q x y
' cos( ) ' sin( ) A y B y
' 0 'A B
ˆ [110]x
We can “see” spin order of TMO’s by using photons.
Multiferroicity
• ME from an internal field determined by .
q qS S
Summary
CoCr2O4• Magnetic lock-in transition revealed with resonant soft
x-ray scattering• Switch of spin chirality.