  An applied electric current through a space-inversion asymmetric magnet
induces spin-orbit torques (SOTs) on the magnetic moments, which holds much
promise for future memory devices. We discuss general Green's function
expressions suitable to compute the linear-response SOT in disordered
ferromagnets. The SOT can be decomposed into an even and an odd component with
respect to magnetization reversal, where in the limit of vanishing disorder the
even SOT is given by the constant Berry curvature of the occupied states, while
the odd part exhibits a divergence with respect to disorder strength. Within
this formalism, we perform first principles density-functional theory
calculations of the SOT in Co/Pt(111) and Mn/W(001) magnetic bilayers. We find
the even and odd torque components to be of comparable magnitude. Moreover, the
odd torque depends strongly on an additional capping layer, while the even
torque is less sensitive. We show that the even torque is nearly entirely
mediated by spin currents in contrast to the odd torque, which can contain an
important contribution not due to spin transfer. Our results are in agreement
with experiments, showing that our linear response theory is well-suited for
the description of SOTs in complex ferromagnets.
