Electronic scattering off a magnetic hopfion

TL;DR

This paper investigates how electrons scatter off a magnetic hopfion in a 3D metal, revealing skew-scattering effects and Hall phenomena even when the average emergent magnetic field is zero, challenging previous assumptions.

Contribution

It provides a detailed analysis of electron scattering off a hopfion, showing skew-scattering and Hall effects without a net magnetic field, using eikonal and Born approximations for different hopfion sizes.

Findings

Skew-scattering leads to Hall effect within hopfion plane.

Hall effect occurs even with zero average emergent magnetic field.

Different scattering contributions relate to hopfion anisotropy and moments.

Abstract

We study scattering of itinerant electrons off a magnetic hopfion in a three-dimensional metallic magnet described by a magnetization vector $\mathbf S(\mathbf r)$. A hopfion is a confined topological soliton of $\mathbf S(\mathbf r)$ characterized by an {\it emergent} magnetic field $B_\gamma(\mathbf r) \equiv \epsilon_{\alpha\beta\gamma} \,\mathbf S\cdot(\nabla_\alpha \mathbf S\times \nabla_\beta \mathbf S)/4 \neq 0$ with vanishing average value $\langle \mathbf B(\mathbf r)\rangle = 0$. We evaluate the scattering amplitude in the opposite limits of large and small hopfion radius $R$ using the eikonal and Born approximations, respectively. In both limits, we find that the scattering cross-section contains a skew-scattering component giving rise to the Hall effect within a hopfion plane. That conclusion contests the popular notion that the topological Hall effect in non-collinear magnetic structures necessarily implies $\langle \mathbf B(\mathbf r)\rangle \neq 0$. In the limit of small hopfion radius $pR \ll 1$, we expand the Born series in powers of momentum $p$ and identify different expansion terms corresponding to the hopfion anisotropy, toroidal moment, and skew-scattering.