Effect of capping layer on spin-orbit torques

In order to enhance the magnitude of spin-orbit torque (SOT), considerable experimental works have been devoted to studying the thickness dependence of the different layers in multilayers consisting of heavy metal (HM), ferromagnet (FM), and capping layers. Here, we present a theoretical model based on the spin-drift-diffusion formalism to investigate the effect of the capping layer properties such as its thickness on the SOT observed in experiments. It is found that the spin Hall-induced SOT can be significantly enhanced by incorporating a capping layer with an opposite spin Hall angle to that of the HM layer. The spin Hall torque can be maximized by tuning the capping layer thickness. However, in the absence of the spin Hall effect (SHE) in the capping layer, the torque decreases monotonically with the capping layer thickness. Conversely, the spin Hall torque is found to decrease monotonically with the FM layer thickness, irrespective of the presence or absence of the SHE in the capping layer. All these trends are in correspondence with experimental observations. Finally, our model suggests that capping layers with a long spin diffusion length and high resistivity would also enhance the spin Hall torque.