Electronic scattering off a magnetic hopfion
Sergey S. Pershoguba, Domenico Andreoli, Jiadong Zang
Abstract
We study scattering of itinerant electrons off a magnetic hopfion in a three-dimensional metallic magnet described by a magnetization vector $\mathbit{S}(\mathbit{r})$. A hopfion is a confined topological soliton of $\mathbit{S}(\mathbit{r})$ characterized by an emergent magnetic field ${B}_{\ensuremath{\gamma}}(\mathbit{r})\ensuremath{\equiv}{\ensuremath{\epsilon}}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}}\phantom{\rule{0.16em}{0ex}}\mathbit{S}\ifmmode\cdot\else\textperiodcentered\fi{}({\ensuremath{\nabla}}_{\ensuremath{\alpha}}\mathbit{S}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\nabla}}_{\ensuremath{\beta}}\mathbit{S})/4\ensuremath{\ne}0$ with vanishing average value $\ensuremath{\langle}\mathbit{B}(\mathbit{r})\ensuremath{\rangle}=0$. 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 Hall effect in noncollinear magnetic structures necessarily implies $\ensuremath{\langle}\mathbit{B}(\mathbit{r})\ensuremath{\rangle}\ensuremath{\ne}0$. In the limit of small hopfion radius $pR\ensuremath{\ll}1$, 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.