Magnetic octupole tensor decomposition and second-order magnetoelectric effect

TL;DR

This paper explores the role of magnetic octupoles in the second-order magnetoelectric effect, using tensor decomposition and first-principles calculations to reveal hidden magnetic order in Cr₂O₃.

Contribution

It introduces a tensor decomposition approach for magnetic octupoles and demonstrates their non-zero presence in Cr₂O₃ through first-principles calculations.

Findings

Magnetic octupoles are non-zero in Cr₂O₃.

Magnetic octupoles have an anti-ferroic arrangement.

Hidden magnetic order can be revealed by second-order effects.

Abstract

We discuss the second-order magnetoelectric effect, in which a quadratic or bilinear electric field induces a linear magnetization, in terms of the ferroic ordering of magnetic octupoles. We present the decomposition of a general rank-3 tensor into its irreducible spherical tensors, then reduce the decomposition to the specific case of the magnetic octupole tensor, $\mathcal{M}_{ijk} = \int \mu_i (\mathbf{r}) r_j r_k d^3 \mathbf{r}$. We use first-principles density functional theory to compute the size of the local magnetic multipoles on the chromium ions in the prototypical magnetoelectric Cr$_2$O$_3$, and show that, in addition to the well established local magnetic dipoles and magnetoelectric multipoles, the magnetic octupoles are non-zero. The magnetic octupoles in Cr$_2$O$_3$ have an anti-ferroic arrangement, so their net second-order magnetoelectric response is zero. Therefore they form a kind of hidden order, which could be revealed as a linear magnetic (antiferromagnetic) response to a non-zone-center (uniform) quadratic electric field.