Magnetic octupole tensor decomposition and second-order magnetoelectric effect

Andrea Urru; Nicola A. Spaldin

doi:10.1016/j.aop.2022.168964

paper

Magnetic octupole tensor decomposition and second-order magnetoelectric effect

arXiv:2206.00522 · doi:10.1016/j.aop.2022.168964

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, $M_{ij k} = \int μ_{i} (r) r_{j} r_{k} d^{3} r$ . 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.

Magnetic octupole tensor decomposition and second-order magnetoelectric effect

Abstract

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