Microscopic theory of spin-polarization coupling in multiferroic transition metal oxides

A systematic microscopic theory of magnetically induced ferroelectricity and lattice modulation is presented for various electron configurations of Mott-insulating transition metal oxides. The origin of polarization is classified as the spin-orbit interaction effective (i) within the magnetic $t_{2g}$ orbitals, (ii) between the $t_{2g}$ and $e_g$ orbitals, and (iii) within the ligand ion’s $p$ orbitals. Predictions for x-ray and neutron scattering experiments are proposed to clarify the microscopic mechanism of the spin-polarization coupling in different materials. Semiquantitative agreements with the multiferroic $\mathrm{Tb}\mathrm{Mn}{\mathrm{O}}_3$ are obtained.

Figure 1

(Color online) (a) The $\mathrm{TM}\text{−}L\text{−}\mathrm{TM}$ cluster model with the rod-type $d_{3x^2-r^2}∕d_{3y^2-r^2}$ staggered orbital order under a noncollinear spin configuration (${\mathit{m}}_{\mathit{l}}$ and ${\mathit{m}}_{\mathit{r}}$) with the associated electric polarization $P_z$. $V_l$ and $V_r$ denote the $pd$ hybridization. (b) The lattice structure and the staggered orbital order of $\mathrm{Tb}\mathrm{Mn}{\mathrm{O}}_3$ within the $xy$ (or $ab$) plane. (c) The level scheme for the $t_{2g}^3{e}_g^1$ high-spin $(S=2)$ configuration.

Figure 2

(Color online) (a) Elliptic spiral spins (arrows at magnetic $M$ ions) with the ellipticity $e=0.6$ and the associated local dipole moments (at oxygen O sites) arising from oxygen spin-orbit interaction are shown as $P_z^{\mathrm{sp}}$ (along the $z$ direction, enlarged by ${10}^3$ times) and $P_x^{\mathrm{ms}}$ (along the $x$ direction) for the observed spin modulation period $(Q\approx 0.28\pi )$ as in $\mathrm{Tb}\mathrm{Mn}{\mathrm{O}}_3$ at low temperature. Two types of $M$ sites exist (even and odd $i$) due to the orbital ordering. Statistical weights are introduced to account for elliptic spins. $P_x^{\mathrm{ms}}$ alternates its direction due to the staggered orbital order. (b) Induced dipole moments $P_{z,i}^{\mathrm{sp}}$ and $\mid P_{x,i}^{\mathrm{ms}}\mid$ at the oxygen site ${\mathrm{O}}_i$ in ${\mathrm{TM}}_i\text{−}{\mathrm{O}}_i\text{−}{\mathrm{TM}}_{i+1}$ cluster in units of ${10}^{-3}{L}_z$ and ${10}^{-3}{L}_x$, respectively, with $L_z=⟨d_{zx}\mid z\mid p_x⟩$ and $L_x=⟨d_{3x^2-r^2}\mid x\mid p_x⟩$. [(c) and (d)] Main Fourier components of the dipole moments, at $q=2Q$ for $\mid P_{x,i}^{\mathrm{ms}}\mid$ and at $q=0$ for $P_{z,i}^{\mathrm{sp}}$, as functions of $e$ for $T<T_L$ in (c) and $S$ with $e=1$ for $T_L<T<T_N$ in (d). Note that $e$ in (c) and $1∕2-S$ in (d) decrease with decreasing temperature. This figure is worked out for the orbital ordered model of $\mathrm{Tb}\mathrm{Mn}{\mathrm{O}}_3$ (Fig. 1) with $(U,{\Delta }_{cf},E_{\mathrm{JT}},V_{pd\sigma },{\lambda }_p,{\lambda }_d)=(3.0,2.0,1,1.2,0.025,0.048)\mathrm{eV}$.