March Meeting 2007 Spin-polarization coupling in multiferroic transition-metal oxides Shigeki Onoda (U. Tokyo) Chenglong Jia (KIAS) Jung Hoon Han (SKKU) Naoto Nagaosa (U. Tokyo)

March Meeting 2007 Multiferroics with noncollinear magnetic and ferroelectric phase Materiald-electron Polarization (  C/m 2 ); Q Specifics TbMnO 3 (Kimura et al Nature 2003; Kenzelman et al PRL 2005) d 4 (t 2g ) 3 (e g ) 1 800; q~0.27Orbital order Ni 3 V 2 O 8 (Lawes et al PRL 2005) d 8 (t 2g ) 6 (e g ) 2 100; q~0.27Kagome Ba 0.5 Sr 1.5 Zn 2 Fe 12 O 22 (Kimura et al PRL 2005) d 5 [(t 2g ) 3 (e g ) 2 ]?150 (B=1T); N/AN/A CoCr 2 O 4 (Yamasaki et al. PRL 2006) Co 2+ : d 7 (e) 4 (t 2 ) 3 Cr 3+ : d 3 (t 2g ) 3 2; [qq0] q~0.63ferrimagnetic MnWO 4 (Taniguchi et al. PRL 2006) d 5 (t 2g ) 3 (e g ) 2 50; q=(-.214,.5,.457) N/A CuFeO 2 (Kimura et al PRB 2006) d 5 (t 2g ) 3 (e g ) (B>10T); 1/5<q<1/4 2D triangular; Field-driven LiCuVO 4 (Naito et al cond-mat/ ) d 9 (t 2g ) 6 (e g ) 3 N/A; q~0.5321D chain LiCu 2 O 2 (Park et al PRL 2007) d 9 (t 2g ) 6 (e g ) 3 <10; q~0.1741D chain RED = magnetic ions

March Meeting 2007 Common & uncommon features CommonNot Common Involves 3d transition metal magnetic ions (Fe, Mn, Ni, Co, Cr, Cu) Onset of ferroelectricity concomitant with that of noncollinear magnetic order Spin-polarization coupling seems universal, irrespective of d- electron number Different d-electron configurations (d 1 -d 9, t 2g, e g, mixed t 2g -e g )

March Meeting 2007 A microscopic theory of spin-polarization coupling for arbitrary d-electron configuration is presented In this talk

March Meeting 2007 Existing theories Phenomenological theories of Mostovoy & Harris provide general ground for writing down the spin-polarization coupling Microscopic derivation first given by Katsura, Nagaosa, Balatsky (2005) Physical origin: spin-orbit interaction

March Meeting 2007 Developing a Microscopic Theory A linear chain consisting of alternating M(agnetic) and O(xygen) atoms is a reasonable model for magneto-electric insulators M O M O M O M O M O M O M O M The building block is a single M-O-M cluster. We solve this model as exactly as possible for realistic d-electron configurations: t 2g, e g, mixed t 2g -e g Previous theory of KNB based on t 2g orbitals

March Meeting 2007 Our Result Jia, Onoda, Nagaosa, Han, cond-mat/

March Meeting 2007 d-electronMechanism t 2g Spin-orbit within 3-fold degenerate t 2g subspace (SO M ) egeg No SO allowed within eg subspace, invoke SO at ligand ion (SO L ) t 2g -e g Partially filled t 2g & e g ; SO M can mix t 2g and e g levels and the new e g levels become “active” Classification of spin-orbit interactions In all instances we find non-zero P sp associated with noncollinear magnetic order P sp is reallly UNIVERSAL

March Meeting 2007 t 2g p Both P orb and P sp are found

March Meeting 2007 egeg eg p Only P sp exists due to oxygen p-orbital spin-orbit interaction Relevant to d 8 NVO; d 9 LiCu 2 O 2,LiCuVO 4

March Meeting 2007 Mixed t 2g -e g : Model for TbMnO 3 Ingredients: (1) Orbital ordering takes place at high temperature -> Inversion symmetry is broken; two-sublattice structure to begin with -> Need to generalize theory for two-sublattice orbitals (2) d 4 (t 2g ) 3 (e g ) 1 configuration gives rise to (t 2g )-(e g ) mixing and polarization (3) Spin-orbit coupling at oxygen gives rise to polarization

March Meeting 2007 A new term along the cluster axis due to lack of inversion symmetry; No spin-orbit interaction is required Loss of Inversion Symmetry

March Meeting 2007 Mixing of occupied spin-up e g state and unoccupied spin-down t 2g state Gives rise to P sp t 2g -e g mixing (C.D.Hu, cond-mat/060470; Our work) Numerical estimate using realistic parameters of TbMnO 3 consistent with experimentally measured polarization Same mechanism may be relevant for CuFeO 2 (t 2g ) 3 (e g ) 2 MnWO 4 (t 2g ) 3 (e g ) 2

March Meeting 2007 For a helical spin pattern at wavevector Q, there arises lattice modulations due to induced polarization at various wavevectors; Relevance for X-ray scattering Two-sublattice structure gives further peaks at Spin-current type of polarization is the only UNIFORM POLARIZATION Fourier harmonicsPolarization

March Meeting 2007 Summary A general theory of magnetism-induced dipole moment is presented The mechanisms can be classified according to t 2g, e g, and mixed t 2g - e g configurations One can identify the origin of improper ferroelectricity in diverse d-electron configurations as follows: d-electron configurationMechanism (t 2g ) 3 (e g ) n (n=1,2,3)t 2g -e g mixing; Ligand SO (t 2g ) 6 (e g ) n (n=1,2,3)Ligand SO (t 2g ) n (n=1,2,4,5)Magnetic SO