Hiroshima Nov 2006 Electric Polarization induced by Magnetic order Jung Hoon Han Sung Kyun Kwan U. (SKKU) Korea Collaboration Chenglong Jia (KIAS) Shigeki Onoda (U. Tokyo) Naoto Nagaosa (U. Tokyo) Cond-mat/ , PRB

Hiroshima Nov 2006 Spin-orbit coupling is usually a minor, relativistic phenomenon in solids. However, some things can only happen due to SO. Examples are Anomalous Hall Effect (->Spintronics) Spin Hall Effect (->Spintronics) Etc. Most recently we might have a new item on the list: Magnetism-induced Electric Polarization S*L in solid state physics

Hiroshima Nov 2006 Physics of S*L in Magnetic States If spin orientations are fixed due to magnetic ordering, S*L ~ *L acts like Zeeman field and polarize the orbital states. Some orbitals are more occupied than others. The results may be polarization of the electronic wave function.

Hiroshima Nov 2006 Ni 3 V 2 O 8 Lawes et al PRL 05 TbMnO 3 Kenzelman et al PRL 05 Collinear to non-collinear spin transition accompanied by onset of polarization

Hiroshima Nov 2006 CoCr 2 O 4 Tokura group PRL 06 Co spins have ferromagnetic + spiral (conical) components Emergence of spiral component accompanied by P

Hiroshima Nov 2006 Phenomenon seems rather general. Can we understand this on simple theoretical ground? What is the origin/mechanism of non-collinear spin inducing electric polarization? What is polarization due to? Ionic shift, or wave function polarization? Key Issues of Multiferroic Materials Magnetism-induced Ferroelectricity

Hiroshima Nov 2006 Step 1: Ginzburg-Landau theory

Hiroshima Nov 2006 Spin and polarization break different symmetries: breaks time-inversion symmetry breaks space-inversion symmetry Naively, lowest-order coupling occurs at 2 2. Lower-order terms involving spatial gradient, 2 or, are possible. Spin-polarization Coupling via GL Theory

Hiroshima Nov 2006 Spin-polarization coupling via GL theory Generally one can write down GL terms like that result in the induced polarization Mostovoy PRL 06 in the presence of magnetic ordering For instance…

Hiroshima Nov 2006 Spin-polarization Coupling via GL Theory For spiral spins induced polarization has a uniform component given by Mostovoy PRL 06

Hiroshima Nov 2006 Uniform induced polarization depends on the product M 1 M 2 - Collinear (M 1 M 2 =0) spin cannot induce polarization - Only non-collinear, spiral spins have a chance Quite consistent with experimental facts No microscopic derivation of Mostovoy’s free energy exists yet. Spin-polarization Coupling via GL Theory

Hiroshima Nov 2006 Step 2: Microscopic theory & Mechanism

Hiroshima Nov 2006 At the microscopic level, one can write down spin-polarization interaction terms such as Fully consistent with symmetries Consistent with GL theory P ij SiSi SjSj

Hiroshima Nov 2006 Where does it come from?

Hiroshima Nov 2006 Microscopic Theory of Spin-induced Polarization 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. One tries to solve this model as exactly as possible to see if noncollinear-spin-induced polarization can be understood.

Hiroshima Nov 2006 Microscopic Theory – Further Details In magnetic atoms, d-orbital electrons are responsible for magnetism. Keep the outermost d-orbitals and truncate out the rest. Five-fold d-orbitals are further split into 3 t 2g and 2 e g orbitals with a large energy gap of a few eV due to crystal field effects. Keep the t 2g or e g levels only. M O M Crystal Field d egeg t 2g Magnetic Oxygen

Hiroshima Nov 2006 Microscopic Theory – Further Details Electrons can “hop” between M and O sites as represented by a hopping integral V. KEY ELEMENT: SPIN-ORBIT INTERACTION Each magnetic site is subject to spin-orbit interaction. If the spin state is polarized, so is the orbital state. M O M

Hiroshima Nov 2006 Theory of Katsura, Nagaosa, Balatsky (KNB) The cluster Hamiltonian KNB Hamiltonian is solved assuming  (spin-orbit) > U (Hund) KNB PRL 05

Hiroshima Nov 2006 Results of KNB Polarization orthogonal to the spin rotation axis and modulation wave vector develops

Hiroshima Nov 2006 Results of KNB The results may be generalized to the lattice case; consistent with phenomenological theories RED: spin orientation BLACK: polarization

Hiroshima Nov 2006 Our Recent Results We revisited the KNB Hamiltonian in the limit of large Hund coupling U and small spin-orbit interaction, which is presumably more realistic. A new (longitudinal) component of the polarization is found which was absent in past theories which only predicted transverse polarization. BEFORE AFTER

Hiroshima Nov 2006 Our Results Spontaneous polarization exists ALONG the bond direction - LONGITUDINAL. Only possible for non-collinear spins

Hiroshima Nov 2006 Numerical Approach KNB and our results probe different regions of parameter space. We decided to compute polarization numerically without ANY APPROXIMATION Exact diagonalization of the cluster Hamiltonian (only 16 dimensional) for arbitrary parameters ( /V,U/V) Both longitudinal and transverse polarizations were found !

Hiroshima Nov 2006 Numerical Results for Polarization Transverse and longitudinal components exist which we were able to fit using very simple empirical formulas: KNB Px (longitudinal) Py (transverse) JONH

Hiroshima Nov 2006 Uniform vs. non-uniform KNB term is responsible for macroscopic polarization JONH term averages out Locally, JONH >> KNB Detecting such local ordering of polarization will be interesting.

Hiroshima Nov 2006 Consideration of e g orbitals Real Manganese oxides have inert t 2g levels e.g. TbMnO 3 (t 3 2g e 1 g ) Inert t 2g orbitals cannot produce electronic polarization e g orbital alone is insensitive to spin-orbit coupling due to angular momentum mismatch A way out: (1) t 2g -e g level mixing due to octahedral tilting (2) Assuming oxygens carry the spin-orbit interaction (According to some LDA calculations, SO(oxygen) is nontrivial fraction of SO(Mn))

Hiroshima Nov 2006 Model for TbMnO 3 Ingredients: (1)Orbital ordering takes place at high temperature (2)Only one e g orbital level is assumed, which hybridize with oxygen p orbital (2) Spin-orbit coupling at oxygen site mixes all three p orbitals

Hiroshima Nov 2006 Results for e g orbitals and oxygen SO Transverse polarization still possible with oxygen SO We propose this as a possible mechanism of polarization for RMnO 3 Jia, Onoda, Han, To be published

Hiroshima Nov 2006 Summary Microscopic mechanism for magnetism-induced electronic polarization is investigated. We show that spin-orbit interaction alone gives rise to electronic polarizations of both transverse and longitudinal types Both t 2g and e g orbitals can produce such polarization Comparison with real multiferroic material is in progress