symmetry and mechanism of multiferroicity in frustrated magnets 黃迪靖 and 牟中瑜 resonant...

Symmetry and Symmetry and Mechanism of Mechanism of

Multiferroicity in Multiferroicity in Frustrated MagnetsFrustrated Magnets

黃迪靖黃迪靖 and and 牟中牟中瑜瑜

• Resonant soft x-ray scattering• Ginzburg-Landau approach

2

Collaborators

Soft x-ray scattering: J. Okamoto, H.-J. Lin, and C. T. Chen (NSRRC, Taiwan)

K. S. Chao (National Chiao-Tung University, Taiwan)

TbMn2O5: S. Park, S. W. Cheong (Rutgers University, USA)

Acknowledgement

L. L. Lee, H. W. Fu, and S. C. Chung (NSRRC, Taiwan)

S. Ishihara (Tohoku University, Japan)

Y. Tokura (Univ. of Tokyo, Japan)

C. H. Chen (University of Taiwan)

T. K. Lee (Academia Sinica)

3

• Introduction

• Resonant soft x-ray scattering

• Ginzburg-Landau approach

4

Magnetism: ordering of spins

Ferroelectricity: polar arrangement of charges

strain E T EPZT“Fire”(-Pyro)electricity“Pressure”(-piezo) electricity

5

(Ferro)magnetism vs. (Ferro)electricityPerovskite structure

(La,Sr)MnO3: spins from : 3d3 or 3d4

BaTiO3: polarization from cation/anion paired diploes

O-2

Ti+4

Magnetic moment:

-Ba+2

0.10 Å

0.05 Å0.04 Å

+

+

Ti 3d0 O 2p2

unfilled d bandsimpurities

inversion symmetry broken

6

• BiMnO3, BiFeO3, Pb(Fe2/3W1/3)O3:

6s2 lone pairs off-center distortion polar behavior

uncorrelated with magnetism ?

Mechanism of 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)

7

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)

Three Scenarios

1. Magnetic and Ferroelectric Transition

occurrs at the same T

3 7 13 2 4

:

boracite (Ni B O I) & Cr BeOExample

Ni I

2 2( - ) ( - ) - - qF a T Tc P b T Tc S EP MH

qorder parameter = (P, S )

0, /( )

F

P E T Tc EP

m both dive a rgnd t Tce a e

2. Ferroelectric Transition occurs first

2 2( - ) - - qF a T Tc P bS EP MH

m

Only at Tc,

but may have an

omaly at Tm

diverges

.

e

3

:

RMnO , R=Sc, Y, In, Ho-Lu

Example

Lawes et al., PRL 91, 257208, 2003

Hexagonal RMnO3 :TC: 570-900 K, TN=70-130 K

BiFeO3: ferroelectric, ferroelastic, and weakly ferromagnetic; rhombohedrally distorted; TC=1100 K, TN= 650 K

2 2( - ) - - qF aP b T Tc S EP MH

3. Magnetic Transition occurs first

23 2 8 3 5

Example:

Ni V O , RMnO , and R=Tb, DRM yn O , , Mo

e

Only at Tc,

but may have an

omaly at Tm

diverges

.

m

c

N

N

11

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

“multiferroic” systems

Recent Discoveries

• Frustrated magnetic systems.• The magnetic phases are complicated; incommensurate AF orders seem to be common.• Strong coupling between ferroelectricty and magnetism.

R=Tb, Dy, Mo

TbMnO3, Nature 426, 55, (2003); PRL 95, 087206 (2005).

RMn2O5, Nature 429, 392 (2004) (Tb);

PRL 96, 067601 (2006) (Y).

3 2 8Ni V O , Lawes et al., PRL 95 087205 (2005)

13

*Can the ME be enhanced by the internal fields?

Require

of mag

coexistenc

netism and

e and strongly coupl

ferroelectricity (

ing

r )are

Multiferroicity:coexistence of magnetism and ferroelectricity with cross coupling

How to enhance the coupling?

14

Geometric ferroelectrics: hexagonal RMnO3

BaNi(Mn,Co,Fe)F4For example:

YMnO3: A-type AF, lacking lone pairs,

bulking of MnO5 pyramids & displacement of Y polarization

Origin of ferroelectricity

van Aken et al., Nature Materials 3, 164, (2004)

Y

MnO5

15

Site-centered charge order

• Electronic ferroelectrics:Combination of bond-centered and site-centered charge order

Efremov et al., Nature Materials 3, 853, (2004)

Bond-centered charge order

Ferroelectric intermediate state

O

TM

16

• Magnetic ferroelectrics: (frustrated spin systems)

Dzyaloshinskii-Moriya interaction (=D . S1X S2)

spin current

“electromagnon” (spin waves excited by ac E fields)

• Geometric ferroelectrics: hexagonal RMnO3

BaNi(Mn, Co, Fe)F4

Origin of ferroelectricity

Sergienko and Dagotto, Phys. Rev. B 73, 094434 (2006)

Katsura, Nagaosa and Baltasky, Phys. Rev. Lett. 95, 057205 (2006)

• Electronic ferroelectrics:Combination of bond-centered and site-centered charge order

Efremov et al., Nature Materials 3, 853, (2004)

Pimenov et al. Nature Phys (2006)

17

Nature, 426, 55 (2003)

TbMnO3

incommensurate AF order ME effect

18

T=35 KT=15 K

TN=42 K

TC=27 K

TbMnO3

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

IC sinusoidally modulated collinear magnetic order, inversion symmetric

IC noncollinear magnetic order, inversion symmetry broken

19

• 3 transitions on cooling. • Magnetic field induces a sign reversal of the electric polarization.

TbMn2O5 Nature, 429, 392 (2004)

20

Difficulty in microscopic measurement2

max 40 /P nC cm

4/ 10 is extremely sma

0.56 ,

ll

0.85 ,

!

0.73

a nm b nm c nm

L. C. Chapon et al. Phys. Rev. Lett. 93, 177402, 2004

Scattering amplitude accumulates microscopic effects and is macroscopic:

ii q rq i

i

f S S e

1210 r m

21

Nature of the transition

N. Hur et al. Phys. Rev. Lett. 93, 107207, 2004

22

Tb

O

Mn4+O6 octahedronq

P

• orthorhombic structure (a b c, = = = 90˚)

•AFM insulator (TN=42 K )magnetization in the ab plane

•AFM square lattice with asymmetrical next-nearest-neighbor interactions, i.e., geometrically frustrated

•Spontaneous polarization P // b

q P

Mn3+O5 pyramid

TbMn2O5

Tb3+ Mn4+ , Mn3+ O2-

Chapon et al, PRL (2004)

Blake et al, PRB (2005)

23Blake et al., PRB (2005)

20°

ab planeab plane

6° 31°17°

TbMnTbMn22OO55

Mn4+O6Mn3+O5

Tb3+ O2-

The spins lie in the ab plane. Within the ab plane, two zigzag chains of AFM-coupled nearest-neighbor Mn4+ and Mn3+ run in a direction parallel to the a axis.

orthorhombic structure a=7.3233 Å, b=8.5205 Å c=5.6601 Å

24

qx

qz

0 20 40 60Temperature (K)

0.25

0.30

0.50

0.55


(½ 0 ¼) commensurateincommensurate

33

incommensurate

24 42

Neutron diffraction: complex spin structure AFM, TN= 42 K, q = (qx 0 qz)

Mechanism of ME effect ?

25

qx

qz

0 20 40 60Temperature (K)

0.25

0.30

0.50

0.55


(½ 0 ¼) commensurateincommensurate

33

incommensurate

24 42

Neutron diffraction: complex spin structure

Kobayashi et al, JPSJ(2004)

3724 42

AFM, TN= 42 Kmodulation vector q = (qx 0 qz)

26

Issues: -What is the underlying mechanism of

thegigantic ME effect?

-Is a spiral-spin configuration necessary?

-Can collinear spins lead to a polarization?

27

• Introduction



28

Elastic x-ray scattering

2

qfd

d

scattering form factor

k

'k

k'kq

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

reqi

r)er( r df qi

q 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. r)er( r

dSS qiq

detectable?

29

X-ray scatteringkkq '

drrrq

rqrnf i

2)sin(

)(4

k

'k

)(rni : electron density

charge scattering

EFe emagnetic-moment scattering

B)mFm (

32

e

m 10~~F

F

mc

6

e

mag 10~

Non-resonant X-ray magnetic scattering is very weak.

(for ~ 600 eV)

30

Resonant X-ray magnetic scattering

1 lm electric dipole transitions

0ε ε • F1,1 F1,-1

• scattering amplitudes enhanced

)(ˆ)εε(8

31,11,10

* FFzif res

mag

Hannon et al., PRL(1988)

2p3/2

1 lm 1 lm

3d

2p1/2

(F1,±1 => Sq )

31beamline4-m long elliptically polarized undulator

UHV, two-circlediffractometer

detector

Soft X-ray Scattering Set-up at NSRRC, Taiwan

H. J. Lin & C. T. Chen et al.

32

Soft x-ray scattering of TbMn2O5

3d

2p3/2

Mn

cm

TbMn2O5 single crystal

h = 637.7 eVE bT = 30 K

) 0 ( 41

21q

'k

k

q )(0 σε

)(ε

[100]

[001]

[010]

33(½ 0 ¼) commensurateincommensurate incommensurate

Coexistence of ICM and CM AF order

3d

2p3/2

Mn

640 eV

) 0 ( 41

21

zxq

34

qx

qz

Kobayashi et al, JPSJ(2004)

CM

CM

3d

2p3/2

Mn

) 0 ( 41

21

zxq

qz = 1/4

IC

) 0 ( 41

21

35

AF transitions closely resembles ferroelectric transitions

ICM AF order

ICM

CM

36

'k

kq 001

100

010

2aqS

2bqS

Antiferromagnetism and ferroelectrics are strongly coupled.

37

• Introduction



38

Magnetoelectric Effect (ME)

0 Si ij j iji iM E EM

*Induction of M by E, induction of P by H

0 Si ij j iji jP E HP

*Can the ME be enhanced by the internal fields?

39

Symmetry consideration

time reversal, -t t

q qM M S PS P

Two important symmetries:

inversion, -r r

q qS S PM M P

40

Symmetry properties

1;

i iiq r iq r

q i i qi q

S S e S S eN

* *

iiq rq

q

Si Si real S e

*

q qS S

*1[ ]

iiq rq i q q

i

I S S e S SN

*Inversion invar ant: i

q qS S

41

Inversion symmetry broken in magnetic phases

3 2 8Ni V O

4BaMnF3TbMnO

42Blake et al., PRB (2005)

20°

ab planeab plane

6° 31°17°

TbMnTbMn22OO55

Mn4+O6Mn3+O5

Tb3+ O2-

The spins lie in the ab plane. Within the ab plane, two zigzag chains of AFM-coupled nearest-neighbor Mn4+ and Mn3+ run in a direction parallel to the a axis.

a

b

c

a

d

a

b

c

d d

c

b

43

( ) ( )

i ii q r i q ri q qS S e S e

Inversion symmetric magnetic phase

Common phase can be removed

by choosing the correct center

(2 ,2 ,2 ) cos( ) x y zq q q iS S S q r

0 / 2 q

44

Inversion Symmetry versus collinearity

*

i iiq r iq ri q qS S e S e

*collinear =0

q qS S

( ) ( )* ] [

i i i iiq r r iq r ri j q qS S S S e e

*Inversion symmetric [: ]

q q qS I S S

*collinear (inversion symmetri c)

q qS S

45

Inversion symmetric broken - spiral

Example of non-common phases

ˆ ˆcos( ) sin( )

x yi q qS S qx x S qx y

xy

*

q qS S

ˆ ˆ ˆ ˆ=

2 2

x y x yq q q qiqx iqxS x iS y S x iS y

e e

46

Other spirals

ˆ ˆcos( ) sin( )

x yi q qS S qz x S qz y

z

47

2 (3 ) ˆ( 2 )

iqa iqa iq naq

n

S e e e s

Inversion symmetric broken - collinear

s

*

q qS S

a

48

Coupling to polarization

Inversion Symmetry in Magnetic Phase

Odd orders of P involved:

Inversion symmetry is broken

Even orders of P involved:Inversion symmetry is not broken

2[ ,..other parameter]

qS P aP bP0( )O P

Uniform M( , )Neel order on sqaure l e: attic

Examples S

49

Magnetic induced polarization with inversion symmetry broken

Lowest order -- the existence of internal fields

2 20/ 2 ( ) ( )

in q q q qF P J q S S u S SE P

0 e inPF

PE

( ) 2 ( )0

q

inq q q q

q

EJ q S u S

SSS

FS P

50

Symmetry Constraints

magnetic:

q qq S S

ˆ ( , , or ˆ( ) )

in q q q q q qu S S SS q SE S

under inversion

in inE E

ˆlattice: a b c

eventime num reversa ber ofl:

qShomogeneous: an -d

q q

51

2 / 2

e inEF P P

0 e inPF

PE

ˆ , ( )

q q q q q qq q

P i S S i u S S

q ˆor ( )

q qS S q

sinq q q q

not consistent with expts.

52

Electric Polarization Change and Reversal

ˆ ( )

q qP i u S S

ˆ (1,0,0)Take u a

( , ,0)

a bq q qS S S

ˆ| || |- ( * sin )* ˆ) (

a bq q a

a b a bq q q q bbP i S S SS S S b

Strong magnetic field along a or b axis:

/ / - !

a b a bq q PS PS

53

Nature 429, 392, 2004

Consistent with our result but why it does not happen for b-axis?Possibility: a-axis is an easy axis, and it is very hard to reverse Sb

P must be odd in Sa!

54

qThe dependence of is weak.q

Direct comparison with expt

commensurate+incommensurateincommensu

rate

incommensurate

55


0

q qS S

0

q qS S

collinear orNoncollinear and Inversion-symmetry broken Inversion-symmetric

56


N. Hur et al. Phys. Rev. Lett. 93, 107207, 2004

57

Effect of External Fields - the dielectric constant

Nature 429, 392, 2004

ˆ|| , expect only b-axis is soft

P b

1 4 e eP E D E

58

2( ) ( )ine

PF P E O P

P P P

22 *

2( )2 q

eq q

PO P F E S SP J

magneto-elastic coupling

position of the intermediate atomir

Possible origin: ( , )ij i i jJ R r S S

Expansion of Free Energy

59

2

1 2( ) ( )2q

PJ g q E P g q

Change of exchange energy

0 Re

FP E

P

*1

*2

1 ( )

1 ( )

e q qRe

e q q

g q S S

g q S S

60

Dielectric Anomaly

1 2( ) 0, ( ) 0c cg q g q

(Lawes et al., RPL 25, 257208, 2003)

*2

1

1 ( )e

q

R

e qg q S S

Small step

2*

1

1 ( )R

qe q

eg Sq S

61

Large Step at IC transition

1 2( ) 0, ( ) 0IC ICg q g q

*

2

*11

1

)

(

(

)

q qeRe

e q qg q S S

g q S S

62

0 0( , ) 0 determines ( )J q T q T

Higher orders may also

*0 ( , )

q qF J q T S S

Wavenumber change of IC moments

* Lock-in transition ( with 2)

nm q qS S nqF G

0 ( )

nc c q qT T S S

* Change of q ( , )( )( )

m c Q Q q qF J q Q S S S S

0 1 2( . . cos cos 2 ) e g J dkT J q J q

0help in selecting q

0 ( , )( )

c Q QJ J J q Q S S

63

Connection with Microscopic Theories

Sergienko and Dagotto, Phys. Rev. B 73, 094434 (2006)

Dzyaloshinskii-Moriya interaction with atomic displacement

ˆ ˆ and take q u x

xexpt: y z

*ˆ ( ) | || | sin ˆ( )

x zq q q q q q x zi u S S S SP z

64

Katsura, Nagaosa and Baltasky, Phys. Rev. Lett. 95, 057205 (2006)

Theory of Katsura, Nagaosa and Baltasky

1. Polarization without atomic displacement

Efremov et al., Nature Materials 3, 853, 2004

2.

65

How to induce polarization without involving atomic displacement?

Essential Physics: Motion of magnetic moments induces electric dipoles! – the intrinsic Aharonov-Casher Effect

02

Lorentz contraction (1 / )

x

dld

c

-

+

2

P

c

66

( )ij s ij i jij

P e j e S S

*ˆ(sin ) ( )x q qq

P i q a S S

(1,0,0f )I e a

sinq xq

Katsura, Nagaosa and Baltasky : Motion of magnetic moments= spin current

67

Aharonov-Casher Effect in condensed matter (Meier and Loss, PRL 90, 167204,2003)

2

ij iji i

i j i jij

JH S S e S S e

= phase that the magnetic dipole ˆ experience s- bij g z

2ˆ( ) /

j

i

rb

n ir i n

gdr E z B

cdt EB c

is generated!

HPE

68

2

i j i j i jij

JH S S S S D S S

if 2

ˆij ij

iji

i

j iij

i

jS S SJ

e S D Dze

ˆˆ ˆ( ) j

i

r

ij ijrdr E z e z E

zziijj

Sj

t

( )ij ij

i j

i izi i

ijj jj i S eS Se

HS

ˆˆ( )ij zi ijj ij

ijs

H HP j P ez e

E Ej

69

Conclusion

70

Conclusion

For magnetic phases with inversion symmetry broken

Consistent well with our expts on TbMn2O5TbMn2O5

ˆ ( )

in q q qE i u S S*Response to the internal field:

ˆ ( )

q q qP i u S S

sumarized in the dieletric constant *Response to the external electric field:

Consistent with magneto-elastic effect on exchange energy

71

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

Magnetoelectric effect

0 Si ij j iji iM E EM

0 Si ij j iji jP E HP

0 0 0

1 1

2 2

S Si i i i ij i j ij i j

ij i jE H

F F P E M H E E H H

&

i i

i i

F FP M

E H

M. Fiebig, J. Phys. D: Appl. Phys 38, R123 (2005)

72

Old Examples

Ferromagnetic, ferroelectric, and ferroelastic at Tc=61.5K

3 7 13 boracite (Ni B O I)Ni I

[Ascher et al., J. Appl. Phys. 37, 1404 (1966)]

73

Antiferromagnetic (spiral) & ferroelectric at Tc=28K

2 4Cr BeO

[Newnham et al., J. Appl. Phys. 49, 6088 (1979)]

4BaMF , M=Mg, Mn, Fe, Co, Ni, Zn

[Fox et al., Phys. Rev. B 21, 2926 (1980)]

Ferroelastic at high temperature but Antiferromagnetic at Tc=25-70K

symmetry and mechanism of multiferroicity in frustrated magnets 黃迪靖 and 牟中瑜 resonant...

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