Intraband and interband spin-orbit torques in noncentrosymmetric ferromagnets

Abstract

Intraband and interband contributions to the current-driven spin-orbit torque in magnetic materials lacking inversion symmetry are theoretically studied using the Kubo formula. In addition to the current-driven fieldlike torque ${\mathbf{T}}_{\mathrm{FL}}={\ensuremath{\tau}}_{\mathrm{FL}}\mathbf{m}\ifmmode\times\else\texttimes\fi{}{\mathbf{u}}_{\mathrm{so}}$ (${\mathbf{u}}_{\mathrm{so}}$ 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 ${\mathbf{T}}_{\mathrm{DL}}={\ensuremath{\tau}}_{\mathrm{DL}}\mathbf{m}\ifmmode\times\else\texttimes\fi{}({\mathbf{u}}_{\mathrm{so}}\ifmmode\times\else\texttimes\fi{}\mathbf{m})$. 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 exchange coupled to the Mn moments. Parametric dependencies 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.