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Multiferroics – Combining Magnetism with Ferroelectricity

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Multiferroics – Combining Magnetism with Ferroelectricity 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 causes ferroelectricity and therefore giant magnetoelectric effect ; can change magnetic (electric) moment by electric (magnetic) field SPINTRONICS Symmetry considerations in multiferroics T I Magnetic moment - + Electric moment + - Strain + + • Multiferroic state requires both broken symmetries! 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 – 1 n 3 HJRRSRSR ,'''' 2 RR,', ' 1 , 1 • Entropy – 1 nn S a S24 R b S R 2 RR11 • Magnetic free energy – n 3 1 14 FRRSRSROSmag ,'''' 2 RR,', ' 1 , 1 Inverse susceptibility matrix - 1 R ,'','' R aT RR,' ,', J R R • Magnetic free energy per unit cell – n 3 1 14 fmag q;,',,' S q S q O S 2 , ' 1 , 1 q • Fourier transform of the inverse susceptibility matrix – 11iq r ' q; , ' , r ' e 3 n 3 n Hermitian Matrix • For each , Ther inverse susceptibility matrix has N= eigenvaluesq which depend on temperature3n 3n and eigenvectors qT, i i1 3n Sqi i1 +… Diagonalize ; Eigenvalues correspond to irreducible representations (irreps) of symmetry group of paramagnetic phase, incorporate inversion 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 Coupling to ferroelectric mode 3 P 2 fV ele cell 0 1 2E, n 33 1 fint A q;,',,' S q S q P 2 , ' 1 , 1 1 q • Some of the coefficients in the magnetoelectric term vanish due to symmetry considerations • Minimization with respect to P gives the induced polarization in a magnetically ordered state 1st approach: use symmetry, group theory and Landau theory, emphasizing the role of inversion. Ni3V2O8 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) R=Y, Ho, Tb, Er, Tm ferroelectricity magnetism (1 ,0, 1 ) (q ,0, 1 ) (q ,0,q ) q = 2 4 x 4 x z Our results: Generic phase diagram, T versus control parameters Generic phase diagram RMn2O5 , 1 1D irrep , 2 1D irreps , 1 1D irrep , 2 1D irreps , 1 2D irrep , 1 2D irrep (q ,0, 1 ) (q ,0,q ) x 4 x z Shlomi Matityahu: Multiferroic MnWO4 • Crystal structure – monoclinic with alternate zigzag chains of Mn26, W 2 5 • Two magnetic ions Mn S ,0 L per unit cell N=4 2 Taniguchi et al., PRB 77, 064408 (2008) Multiferroic MnWO4 • Three magnetic phase transitions – Temperature Wave vector Structure AF3 13.5K qIC 0.214,0.5,0.457 Sinusoidal AF2 12.3-12.7K Spiral elliptical q 0.25,0.5,0.5 AF1 7-8K C1,2 Taniguchi et al., PRB AF1 AF2 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 – aT J q;1,1 J q ;1,2 0 0 J q;1,2 aT J q ;1,1 0 0 1 q 0 0aT D J q ;1,1 J q ;1,2 0 0J q ;1,2 aT D J q ;1,1 • are determined by the superexchange Jinteraction q;1,1 , J q couplings ;1,2 • Eigenvalues and eigenvectors of the 1 q matrix: ,x q,;T aT q ,x q, T aTq ,b qT, aT D q ; ,b q, T aTD q 1 1 iq iq 1 e 1 e S,x q ; S,x q 2 0 2 0 q J q;1,1 J q ;1,2 0 0 iq Jq ;1,2 0 0 e Jq ;1,2 1 0 1 0 S q ; S q ,b 2 1 ,b 2 1 iq iq e e Magnetic field • The critical wave vector q IC of the first transition is determined by maximizing q • Order parameters for the first two transitions: Sqx IC ,1 Sqx IC , 2 xq IC S,, x q IC b q IC S b q IC Sq, b IC 1 Sqb IC , 2 • Order parameters for the third transition lock-in at commensurate wave vector: SqxC ,1 SqxC , 2 xq C S, x q C Sq, bC 1 SqbC , 2 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 Theory Experiment 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!.
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