Adiabatic and nonadiabatic spin-transfer torques in the current-driven magnetic domain wall motion

A consistent theory to describe the correlated dynamics of quantum-mechanical itinerant spins and semi-classical local magnetization is given. We consider the itinerant spins as quantum-mechanical operators, whereas local moments are considered within classical Lagrangian formalism. By appropriately treating fluctuation space spanned by basis functions, including a zero-mode wave function, we construct coupled equations of motion for the collective coordinate of the center-of-mass motion and the localized zero-mode coordinate perpendicular to the domain wall plane. By solving them, we demonstrate that the correlated dynamics is understood through a hierarchy of two time scales: Boltzmann relaxation time tau(el), when a nonadiabatic part of the spin-transfer torque appears, and Gilbert damping time tau(DW), when adiabatic part comes up.