First Principles Study of Angular Dependence of Spin-Orbit Torque in Pt/Co and Pd/Co Bilayers

TL;DR

This study uses first principles calculations to analyze the angular dependence of spin-orbit torque in Pt/Co and Pd/Co bilayers, revealing the dominant Fermi surface contributions and strain effects, with results aligning with experimental data.

Contribution

It provides a first-principles, angular-dependent analysis of SOT in HM/FM bilayers without assuming prior angular forms, highlighting the Fermi surface dominance and strain effects.

Findings

Fermi surface contributions dominate SOT

Strain influences FL- and DL-SOT components

Results agree with experimental measurements

Abstract

Spin-orbit torque (SOT) induced by spin Hall and interfacial effects in heavy metal(HM)/ferromagnetic(FM) bilayers has recently been employed to switch the magnetization direction using in-plane current injection. In this paper, using the Keldysh Green's function approach and first principles electronic structure calculations we determine the Field-Like (FL) and Damping-Like (DL) components of the SOT for the HM/Co (HM = Pt, Pd) bilayers. Our approach yields the angular dependence of both the FL- and DL-SOT on the magnetization direction without assuming a priori their angular form. Decomposition of the SOT into the Fermi sea and Fermi surface contributions reveals that the SOT is dominated by the latter. Due to the large lattice mismatch between the Co and the HM we have also determined the effect of tensile biaxial strain on both the FL- and DL-SOT components. The calculated dependence of FL- and DL-SOT on the HM thickness is overall in good agreement with experiment. The dependence of the SOT with the position of the Fermi level suggests that the DL-SOT dominated by the Spin Hall effect of the bulk HM.