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

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 (λₛₖ*), which separates the above two regions in the adiabatic region, depends on the lifetime of itinerant electrons (τ) and coupling constant between electrons and localized spins (J). For the former we show that λₛₖ* is proportional to √τ, which indicates that λₛₖ* is much less sensitive to τ than the conventional expectation that λₛₖ* is proportional to the mean-free path ∝τ. For the latter we show that the nonadiabaticity considerably suppresses the topological Hall effect when the timescale determined by the skyrmion size and Fermi velocity is shorter than $1/J$. However, its effect on λₛₖ* is not so significant and λₛₖ* is about ten times the lattice constant in a wide range of J and τ.