Topological Hall effect in three-dimensional centrosymmetric magnetic skyrmion crystals

Centrosymmetric skyrmions attract much attention from the research community because of their small sizes and high concentrations. These features can be useful for applications. Such skyrmions are originated due to the Ruderman-Kittel-Kasuya-Yosida and Kondo interactions rather than the Dzyaloshinski-Moria interaction. We study a topological Hall effect in such systems using the Boltzmann equation for a nonequilibrium distribution function. For the relaxation, we choose the electron-acoustic phonon and electron-skyrmion interactions. We find that the topological Hall resistivity exhibits nonlinear behavior depending on chemical potential. Because of noncubic lattice symmetry, we investigate the dependence of the resistivity tensor components along the $x$ direction (parallel to an applied electric field), ${\ensuremath{\rho}}_{xx}$; ${\ensuremath{\rho}}_{zz}$ (the component along the $z$ axis); and ${\ensuremath{\rho}}_{xy}$ (a topological Hall component) with respect to the effective mass ratio, ${m}_{z}/{m}_{x}$. We assume that the skyrmions are spaced on the $\mathit{xy}$ plane and stretched out along the $z$ direction. The temperature dependence of the resistivity tensor reveals the monotonic growth for all components. There is some concern in the interpretation of experiments. Sometimes it can be very difficult to measure the topological Hall resistivity. Indeed, we find that ${\ensuremath{\rho}}_{xy}$ is one to two orders of magnitude less than ${\ensuremath{\rho}}_{xx}$ and ${\ensuremath{\rho}}_{zz}$. Additionally, there is another important factor, which complicates the problem. The $z$ axis and an applied electric field are not exactly perpendicular because of experimental conditions. Thus, the perpendicular to the electric field resistivity contains a linear combination of ${\ensuremath{\rho}}_{xy}$ and $|{\ensuremath{\rho}}_{xx}\ensuremath{-}{\ensuremath{\rho}}_{zz}|$. To determine ${\ensuremath{\rho}}_{xy}$ from an experiment, we propose an experimental setup to measure the topological Hall effect and provide the equations that allow us to determine ${\ensuremath{\rho}}_{xy}$.