  To study the effect of spin-orbit coupling (SOC) on spin-transfer torque in
magnetic materials, we have implemented two theoretical formalisms that can
accommodate SOC. Using the "charge-pumping" formalism, we find two
contributions to the out-of-plane spin-transfer torque parameter $\beta$ in
ballistic Ni domain walls (DWs). For short DWs, the nonadiabatic reflection of
conduction electrons caused by the rapid spatial variation of the exchange
potential results in an out-of-plane torque that increases rapidly with
decreasing DW length. For long DWs, the Fermi level conduction channel
anisotropy that gives rise to an intrinsic DW resistance in the presence of SOC
leads to a linear dependence of $\beta$ on the DW length. To understand this
counterintuitive divergence of $\beta$ in the long DW limit, we use the
"nonequilibrium magnetization" formalism to examine the spatially resolved
spin-transfer torque. The SOC-induced out-of-plane torque in ballistic DWs is
found to be quantitatively consistent with the values obtained using the
charge-pumping calculations indicating the equivalence of the two theoretical
methods.
