Intraband and interband spin-orbit torques in non-centrosymmetric ferromagnets

TL;DR

This paper theoretically investigates the different contributions to spin-orbit torque in non-centrosymmetric ferromagnets, highlighting both extrinsic and intrinsic mechanisms through analytical and numerical methods.

Contribution

It provides a comprehensive theoretical analysis of intraband and interband spin-orbit torques, including analytical expressions and numerical results for specific material models.

Findings

Identification of field-like and anti-damping-like torque components.

Analytical expressions for Rashba 2D electron gas.

Numerical results for (Ga,Mn)As semiconductor.

Abstract

Intraband and interband contributions to the current-driven spin-orbit torque in magnetic materials lacking inversion symmetry are theoretically studied using Kubo formula. In addition to the current-driven field-like torque ${\bf T}_{\rm FL}= \tau_{\rm FL}{\bf m}\times{\bf u}_{\rm so}$ (${\bf u}_{\rm so}$ being a unit vector determined by the symmetry of the spin-orbit coupling), we explore the intrinsic contribution arising from impurity-independent interband transitions and producing an anti-damping-like torque of the form ${\bf T}_{\rm DL}= \tau_{\rm DL}{\bf m}\times({\bf u}_{\rm so}\times{\bf m})$. Analytical expressions are obtained in the model case of a magnetic Rashba two-dimensional electron gas, while numerical calculations have been performed on a dilute magnetic semiconductor (Ga,Mn)As modeled by the Kohn-Luttinger Hamiltonian exchanged coupled to the Mn moments. Parametric dependences of the different torque components and similarities to the analytical results of the Rashba two-dimensional electron gas in the weak disorder limit are described.