Giant resonances in topological spin Hall effect due to electron-skyrmion scattering in two-dimensional Rashba spin-orbit ferromagnets

Abstract

We study a topological spin Hall effect where conduction electrons are scattered by N\'eel-type skyrmions in two-dimensional Rashba spin-orbit ferromagnets. We find the resonance structures in direct and Hall electric conductivities. The resonances strongly depend on the skyrmion size, the Rashba spin-orbit coupling vector value, and the relative direction between magnetization and Rashba vectors. The antiparallel arrangement of the Rashba spin-orbit coupling vector with respect to the magnetization determines the resonance structure and increases the direct conductivity by about two orders of magnitude. For the parallel arrangement the electric conductivities decrease by one to three orders of magnitude and the Hall conductivity changes the sign. The Rashba spin-orbit coupling also modifies electron bands depending on the ratio ${\ensuremath{\varepsilon}}_{R}/J$, where ${\ensuremath{\varepsilon}}_{R}$ is Rashba energy and $J$ is an exchange integral between conduction and localized electrons. If ${\ensuremath{\varepsilon}}_{R}/J\ensuremath{\le}1$, each energy band has the single minimum while for ${\ensuremath{\varepsilon}}_{R}/J>1$ the lower band has the ``Mexican-hat'' shape with the maximum at $k=0$. We focus on the dependencies of direct and Hall electric conductivities on Fermi energy, ${\ensuremath{\varepsilon}}_{F}$, skyrmion sizes, and Rashba spin-orbit coupling constant values and its relative direction with the magnetization. Analyzing both types of the energy bands, in general, we find two types of resonances: at the minimum of the upper band and behaviors, we employ the scattering pattern analysis. The resonance dependencies on skyrmion sizes and Rashba, a spin-orbit coupling contestant, can be qualitatively explained in terms of the Ramsauer-Townsend scattering of the upper band electrons by the skyrmion quantum well. The resonance properties can be used in spin transistors. To discover the resonances it is necessary to know the specific range of the parameters. At some values of the parameters the electric conductivity changes by about two orders of magnitude in the narrow range of ${\ensuremath{\varepsilon}}_{F}$ ($<0.01$ eV). To detect the resonances it is also important to identify the relative direction of the Rashba spin-orbit coupling vector with respect to the magnetization.