Magnetic octupole tensor decomposition and second-order magnetoelectric effect

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, Mijk=∫μi(r)rjrkd3r. We use first-principles density functional theory to compute the size of the local magnetic multipoles on the chromium ions in the prototypical magnetoelectric Cr2O3, 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 Cr2O3 have an anti-ferroic arrangement, so the 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.