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.