The topological Hall effect is the result of spin-asymmetric deflection of charge carriers flowing through a noncollinear spin system. Effective manipulation of the topological Hall conductivity (THC) in skyrmions is currently a vigorous area of research with an eye towards potential spintronics application. Here, we show that the band topology and the THC in a skyrmion crystal can be tuned by changing the strength of the Rashba spin-orbit coupling (SOC), which can be accomplished via a perpendicular electric field. This results in the change of the subband Chern numbers and a transition between ordinary insulator and Chern insulator as the Rashba SOC is varied. For partially filled subbands, the Rashba SOC can tune the THC and reverse its sign, so that the direction of the Hall current is flipped. The critical Rashba strength for this depends on the skyrmion type and the carrier density. We extend our analysis to the cases of Dresselhaus and Weyl SOC as well and show that they can be directly mapped to the Rashba SOC case and therefore lead to similar results. Our work opens up the scope to go beyond the existing avenues for the control of charge transport in skyrmion crystals.

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• Berry curvature
• Skyrmions
• Spin-orbit coupling
• Topological Hall effect
• Band structure methods
• Tight-binding model