Skyrmion-size dependence of the topological Hall effect: A real-space calculation

Abstract

Motivated by the recent discoveries of magnets harboring short-pitch skyrmion lattices, we investigate the skyrmion-size dependence of the topological Hall effect. By means of large-scale real-space calculations, we find that the Hall conductivity takes its extreme value in the crossover region where both the real-space and momentum-space Berry curvature play a crucial role. We also investigate how the optimum skyrmion size ($λ_{\rm sk}^*$) depends on the lifetime of itinerant electrons ($τ$) and coupling constant between electrons and localized spins ($J$). For the former, we show that $λ_{\rm sk}^{*}$ is proportional to $\sqrtτ$, which indicates that $λ_{\rm sk}^{*}$ is much less sensitive to $τ$ than the conventional expectation that $λ_{\rm sk}^{*}$ is proportional to the mean-free path $\propto τ$. For the latter, we show that the non-adiabaticity considerably suppresses the topological Hall effect when the time scale determined by the skyrmion size and Fermi velocity is shorter than $1/J$. However, its effect on $λ_{\rm sk}^{*}$ is not so siginificant and $λ_{\rm sk}^{*}$ is about ten times the lattice constant in a wide range of $J$ and $τ$.