Quantum mechanical theory of current-induced domain wall torques in ferromagnetic materials

Calculations are made of the spin transfer torque acting on the magnetization in a ferromagnetic domain wall when an electric current traverses the wall. The magnetization is assumed to lie in a plane, as in current experiments on flat nanowires. A nonadiabatic out-of-plane torque is found, which can be regarded as an addition to the Zhang-Li out-of-plane torque that does not appear in our calculation since we neglect spin-orbit coupling. There is also an in-plane nonadiabatic component of torque together with a correction to the standard adiabatic torque. It is shown that the calculated nonadiabatic torques are fitted rather well by two additional terms in the phenomenological Landau-Lifshitz-Gilbert equation. The electronic structure of the ferromagnetic metal is treated within the tight-binding approximation both for a one-band model and for realistic multiband models of Co and Permalloy, but with spin-orbit coupling neglected. The torque is deduced from calculations of spin current in the ballistic limit by using the Keldysh formalism, as in previous work on magnetic trilayers. The wall dynamics due to the current-induced torque is investigated and the out-of-plane torque contributes an additional term to the current-driven wall velocity. The sign of this term differs between Co and Permalloy, and it can be significant in the case of a narrow wall and/or weak damping.