Semiclassical theory of spin-orbit torques in disordered multiband electron systems

We study spin-orbit torques (SOT) in nondegenerate multiband electron systems in the weak-disorder limit. In order to have better physical transparency, a semiclassical Boltzmann approach equivalent to the Kubo diagrammatic approach in the noncrossing approximation is formulated. This semiclassical framework accounts for the interband-coherence effects induced by both the electric field and static impurity scattering. Using the two-dimensional Rashba ferromagnet as a model system, we show that the antidamping-like SOT arising from disorder-induced interband-coherence effects is very sensitive to the structure of disorder potential in the internal space and may have the same sign as the intrinsic SOT in the presence of spin-dependent disorder. While the cancellation of this SOT and the intrinsic one occurs only in the case of spin-independent short-range disorder.

Figure 1

${\chi }_{yy}$ vs $\zeta ={\tau }_A/{\tau }_B$ for fixed values of ${\epsilon }_F/J_{\mathrm{ex}}$. ${\chi }_{yy}$ is measured in units of $-e{\alpha }_R{D}_0\hslash /J_{\mathrm{ex}}$. The plot shows the crossover from the class A dominated regime to the class B dominated regime as $\zeta$ increases. Here, we set ${\epsilon }_R/J_{\mathrm{ex}}=0.1$.

Figure 2

${\chi }_{yy}$ vs ${\epsilon }_F$ for fixed values of $\zeta$. ${\chi }_{yy}$ is measured in units of $-e{\alpha }_R{D}_0\hslash /J_{\mathrm{ex}}$ and ${\epsilon }_F$ is measured in units of $J_{\mathrm{ex}}$. We have chosen ${\epsilon }_R/J_{\mathrm{ex}}=0.1$ in plotting the curves.