Intraband and interband spin-orbit torques in noncentrosymmetric ferromagnets

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 ${\mathbf{T}}_{\mathrm{FL}}={\tau }_{\mathrm{FL}}\mathbf{m}\times {\mathbf{u}}_{\mathrm{so}}$ (${\mathbf{u}}_{\mathrm{so}}$ 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 ${\mathbf{T}}_{\mathrm{DL}}={\tau }_{\mathrm{DL}}\mathbf{m}\times ({\mathbf{u}}_{\mathrm{so}}\times \mathbf{m})$. 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.

Figure 1

(a) Fermi surface of a nonmagnetic Rashba 2DEG: under the application of an external electric field (thick gray arrow), a nonequilibrium field $\delta \mathbf{B}$ is produced (thick blue arrow) that distorts the spin direction out of the plane (red arrows). After averaging, the spin density vanishes. (b) Fermi surface of a magnetic Rashba 2DEG in the strong ferromagnetic regime: in this case, since the spin directions (pink arrows) are initially mostly aligned along the magnetization m (black arrow), the resulting nonequilibrium spin density (red arrows) does not vanish and is aligned along $\mathbf{z}$.

Figure 2

(a) Intraband and (b),(c) interband contributions to the SOT field as a function of spectral broadening $\Gamma$ for an otherwise typical (Ga,Mn)As sample (doping concentration $x=5\%$, lattice-mismatch strain ${\epsilon }_{zz}=-0.3\%$). The inset of panel (a) shows that $h_{\parallel }^{\mathrm{intra}}\propto 1/\Gamma$ holds over a broad range of $\Gamma$. Only lattice-mismatch strain is considered, so that ${\epsilon }_{xy}=0$ in Eq. (20).

Figure 3

(a) Intraband and (b),(c) interband SOT field as a function of exchange interaction $J_{\mathrm{ex}}=J_{pd}{N}_{\text{Mn}}$. Varied values of $J_{\mathrm{ex}}$ can be understood as a proxy to different Mn doping concentrations, e.g., $x=5\%$ corresponds to $J_{\mathrm{ex}}=0.06$ eV, the spectral broadening is set to 50 meV, and other parameters are the same as in Fig. 2.

Figure 4

(a) Intraband and (b),(c) interband SOT field as a function of hole concentration for different lattice-mismatch strain ${\epsilon }_{zz}$. Inset in (c): interband SOT field in the parabolic model. The dashed lines in panel (a) are calculated using Eq. (17) in Ref. [6] and follow a $p^{1/3}$ law. Parameters are the same as in Fig. 3 except for $J_{pd}{N}_{\text{Mn}}$ fixed to a value corresponding to Mn doping $x=5\%$.

Figure 5

Intraband and interband SOT field as a function of the magnetization direction for different models labeled (i), (ii), and (iii) in the text. The red ($◯$), blue ($▵$), and black ($\square$) data stand for the full four-band Luttinger model, its spherical approximation, and the parabolic model, respectively. The parameters are the same as in Fig. 4 except for fixed $p=1.0{\text{nm}}^{-3}$ and ${\epsilon }_{zz}=-0.3\%$.