Multiferroics – Combining Magnetism with Ferroelectricity€¦ · Multiferroics • Type I...
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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!