Emergent odd-parity multipoles and magnetoelectric effects on a diamond structure: implication to 5$d$ transition metal oxides $A$OsO$_4$ ($A=$ K, Rb, and Cs)

TL;DR

This paper predicts how odd-parity multipoles and spontaneous spin and orbital orders in a diamond-structured model lead to magnetoelectric effects, with implications for 5d transition metal oxides like AOsO4.

Contribution

It introduces a two-orbital model including spin-orbit coupling to describe emergent odd-parity multipoles and their associated magnetoelectric responses in diamond-structured materials.

Findings

Staggered spin and orbital orders induce antisymmetric spin-orbit coupling.

Various linear magnetoelectric responses are classified and computed.

Lowering symmetry enhances the magnetoelectric effects.

Abstract

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 $5d$ transition metal oxides, $A$OsO$_4$ ($A=$ K, Rb, and Cs), in which the order parameters are not fully identified.