Multiferroics – Combining Magnetism with Ferroelectricity

Amnon Aharony, BGU and TAU

M. Sc. Thesis of Shlomi Matityahu

Work with Ora Entin-Wholman (earlier also A Brooks Harris)

Multiferroics

• Type I multiferroics – ferroelectricity and magnetism have different sources and are independent

Weak coupling, different transition temperatures

• Type II multiferroics – ferroelectricity and magnetism are

strongly coupled Magnetic order causesferroelectricity and therefore giant magnetoelectric

effect ; can change magnetic (electric) moment by electric (magnetic) field

SPINTRONICS

Symmetry considerations in multiferroics

• Multiferroic state requires both broken symmetries!

IT

+-Magnetic moment

-+Electric moment

++Strain

Landau theory:

(1) Use symmetry group theory to identify

order parameters, then incorporate inversion

symmetry and analyze general Landau expansion.

(2) Start from microscopic Hamiltonian, “derive”

Landau expansion in spins.

Landau Theory for magnetic part

• Energy –

• Entropy –

• Magnetic free energy –

Inverse susceptibility matrix -

3

, ' 1 , 1, '

1, ' ' ' '

2

n

R R

H J R R S R S R

2 4

1 1

1

2

n n

R R

S a S R b S R

3

1 4

, ' 1 , 1, '

1, ' ' ' '

2

n

mag

R R

F R R S R S R O S

1

, ' ,, ', ' ' , ' '

R RR R aT J R R

• Magnetic free energy per unit cell –

• Fourier transform of the inverse susceptibility matrix –

• For each , The inverse susceptibility matrix has N= eigenvalues which depend on temperature and eigenvectors

3

1 4

, ' 1 , 1

1; , ' , , '

2

n

mag

q

f q S q S q O S

q

'1 1; , ' , ' 3 3iq r

r

q r e n n HermitianMatrix

3n

3

1,

n

i iq T

3

1

n

ii

S q

3n

Landau: Coming down from paramagnetic phase – only one irrep orders

Dimension of irrep determines the degeneracy of the eigenvalues and the # of

coefficients of degenerate eigenvectors in N-component spin structure vector

Components of normalized eigenvectors related by symmetry: (N-1) ratios

Higher order terms in F determine actual order parameters

Interesting critical phenomena with a large number of order parameter

components, N

Diagonalize ; Eigenvalues correspond to irreducible

representations (irreps) of symmetry

group of paramagnetic phase,

incorporate inversion

+…

• Some of the coefficients in the magnetoelectric term vanishdue to symmetry considerations

• Minimization with respect to gives the induced polarization in a magnetically ordered state

23

01 ,2

ele cell

E

Pf V

3 3

, ' 1 , 1 1

1; , ' , , '

2

n

int

q

f A q S q S q P

P

Coupling to ferroelectric mode

1st approach: use symmetry, group theory and

Landau theory, emphasizing the role of inversion.

This approach was first used to

explain multiferroicity in

multiferroic NVO, G. Lawes et al.,

PRL 95, 087205 (2005)

P HTI LTI C

LTI is ferroelectric

Approach reviewed with many examples:

A. B. Harris, PRB 76, 054447 (2007)

For group theory, see also

P. G. Radaelli and L. C. Chapon, PRB 76, 054428 (2007)

823 OVNi

),0,( zx qq)4

1,0,( xq)4

1,0,2

1(q =

),0,( zx qq

magnetism

ferroelectricity

R=Y, Ho, Tb, Er, Tm

Our results: Generic

phase diagram,

T versus control

parameters

52ORMn Generic phase diagram

),0,( zx qq , 1 1D irrep

),0,( zx qq

, 2 1D irreps

, 1 2D irrep

, 1 2D irrep

)4

1,0,( xq , 1 1D irrep

)4

1,0,( xq

, 2 1D irreps

Shlomi Matityahu: Multiferroic MnWO4

• Crystal structure – monoclinic with alternate zigzag chains of

• Two magnetic ions per unit cell N=4

2 6,Mn W

2 5, 0

2Mn S L

Taniguchi et al., PRB 77, 064408 (2008)

Multiferroic MnWO4

• Three magnetic phase transitions –

StructureWave vectorTemperature

Sinusoidal13.5KAF3

Spiralelliptical

12.3-12.7KAF2

7-8KAF1

0.214,0.5,0.457ICq

0.214,0.5,0.457ICq

1,2 0.25,0.5,0.5Cq

AF1 AF2

Taniguchi et al., PRB 77, 064408 (2008)

The model – use 2nd approach

• Heisenberg model with nine exchange couplings

• Single ion anisotropy which favor the easy axis

• We will omit the hard axis

Ehrenberg et al., J. Phys. Condens. Matter 11, 2649 (1999)

Frustration

• Fourier transform of the inverse susceptibility matrix –

• are determined by the superexchange interaction couplings

1

;1,1 ;1,2 0 0

;1,2 ;1,1 0 0

0 0 ;1,1 ;1,2

0 0 ;1,2 ;1,1

aT J q J q

J q aT J qq

aT D J q J q

J q aT D J q

;1,1 , ;1,2J q J q

• Eigenvalues and eigenvectors of the matrix: 1 q

, ,

, ,, ;

;,

,

,

b

xx

bq T

q T a

q T aaT D T D

q

q

T qT aT q

q

,

,,

,

1

1

2 0

0

01

12

0

01

1

0

2

1

1;

2 0

0

;

i q

b

i q

x

i

x

b

q

i qe

Se

S q

q

e

q

q

S

e

S

;1,2

;1,1 ;1,2

;1,2

i q J qe

q J q J q

J q

Magnetic field

• The critical wave vector of the first transition is determined by maximizing

• Order parameters for the first two transitions:

• Order parameters for the third transition lock-in at commensurate wave vector:

qICq

1

2

, ,

1

2

,

,

,

,

x IC

x IC

x IC x IC b IC b IC

b IC

b IC

S q

S qq S q q S q

S q

S q

1

2

,

1

2

,

,

,

,

x C

x C

x C x C

b C

b C

S q

S qq S q

S q

S q

Results - Phase diagram of MnWO4

Theory Experiment

Arkenbout et al., PRB 74, 184431 (2006)

Results - Phase diagram of MnWO4

Theory Experiment

Arkenbout et al., PRB 74, 184431 (2006)

Dilution

Results - Phase diagram of Mn0.965Fe0.035WO4

Theory Experiment

Ye et al., PRB 78, 193101 (2008)

Results– Electric polarization of MnWO4

ExperimentTheory

Arkenbout et al., PRB 74, 184431 (2006)

Results - Electric polarization of MnWO4

Theory Experiment

Arkenbout et al., PRB 74, 184431 (2006)

Conclusions

• Input: Experimental exchange couplings plus very few

additional parameters: spin anisotropy D, quartic coefficient

b and dilution coefficients ci.

• Output: several H-T phase diagrams of pure and dilute MnWO4.

• Output: the induced ferroelectric polarization.

• Landau theory works excellently well for phase diagrams!

• Can exclude other sets of proposed exchange couplings!