Emergent odd-parity multipoles and magnetoelectric effects on a diamond structure: Implication for the $5d$ transition metal oxides $A{\mathrm{OsO}}_4(A=\mathrm{K},\text{Rb},\text{and Cs})$

We report our theoretical predictions on the linear magnetoelectric (ME) effects originating from odd-parity multipoles associated with spontaneous spin and orbital ordering on a diamond structure. We derive a two-orbital model for $d$ electrons in $e_g$ orbitals by including the effective spin-orbit coupling which arises from the mixing between $e_g$ and $t_{2g}$ orbitals. We show that the model acquires a net antisymmetric spin-orbit coupling once staggered spin and orbital orders occur spontaneously. The staggered orders are accompanied by odd-parity multipoles: magnetic monopole, quadrupoles, and toroidal dipoles. We classify the types of the odd-parity multipoles according to the symmetry of the spin and orbital orders. Furthermore, by computing the ME tensor using the linear response theory, we show that the staggered orders induce a variety of the linear ME responses. We elaborate all possible ME responses for each staggered order, which are useful to identify the order parameter and to detect the odd-parity multipoles by measuring the ME effects. We also elucidate the effect of lowering symmetry by a tetragonal distortion, which leads to richer ME responses. The implications of our results are discussed for the $5d$ transition metal oxides, $A{\mathrm{OsO}}_4(A=\text{K},\text{Rb},\text{and Cs})$, in which the order parameters are not fully identified.

Figure 1

(a) Schematic picture of the atomic energy levels for $5d$ orbitals. The cubic CEF and SOC represent the cubic crystalline electric field ${\mathrm{\Delta }}_c$ and the atomic spin-orbit coupling $\lambda$, respectively. We take into account the lowest-energy levels ${\tilde{d}}_{x^2-y^2}$ and ${\tilde{d}}_{z^2}$ [Eqs. (1) and (2)], which are derived from the $e_g$ orbitals $d_{x^2-y^2}$ and $d_{z^2}$ with a modulation through the mixing between $e_g$ and $t_{2g}$ orbitals by the SOC. (b) and (c) Schematic pictures of the diamond structure in the (b) absence and (c) presence of a tetragonal distortion. The lowest panel shows the schematic energy levels for the ${\tilde{d}}_{x^2-y^2}$ and ${\tilde{d}}_{z^2}$ orbitals in the absence and presence of the tetragonal CEF ${\mathrm{\Delta }}_t$. In (b) and (c), schematic pictures of ${\mathrm{OsO}}_4$ tetrahedra are partly shown.

Figure 2

The ME coefficients $K_{\mu \nu }$ in the staggered ordered phases with the order parameters (a) $\langle {\sigma }_x\rangle$, (b) $\langle {\sigma }_{xy}\rangle =\sqrt{{\langle {\sigma }_x\rangle }^2+{\langle {\sigma }_y\rangle }^2}$, (c) $\langle {\sigma }_d\rangle =\sqrt{{\langle {\sigma }_x\rangle }^2+{\langle {\sigma }_y\rangle }^2+{\langle {\sigma }_z\rangle }^2}$, (d) $\langle {\tau }_y\rangle$, (e) $\langle {\sigma }_x{\tau }_x\rangle$, (f) $\langle {\sigma }_z{\tau }_x\rangle$, (g) $\langle {\sigma }_x{\tau }_z\rangle$, and (h) $\langle {\sigma }_z{\tau }_z\rangle$. The horizontal axis is the corresponding order parameter in each phase. The closed (open) symbols represent the results in the absence (presence) of the tetragonal crystalline electric field. The data are calculated at $t_0=1.0,t_1=0.5,\delta t_0=0\left(\delta t_0=0.1\right)$, $\delta t_1=0\left(\delta t_1=0.05\right)$, ${\mathrm{\Delta }}_t=0\left({\mathrm{\Delta }}_t=0.5\right)$, temperature $T=0.01$, and the damping factor $\delta =0.01$ by using Eq. (54) for the cubic (tetragonal) system.