Gilbert damping and current-induced torques on a domain wall: A simple theory based on itinerant3delectrons only

Abstract

Many electronic theories of Gilbert damping in ferromagnetic metals are based on the $s\text{\ensuremath{-}}d$ exchange model, where localized $3d$ magnetic spins are exchanged-coupled to itinerant $4s$ electrons, which provide the needed spin relaxation. Recently, Tserkovnyak et al. have obtained Gilbert damping from itinerant $3d$ electrons alone, which have their own spin relaxation. We show that simple semiclassical equations of motion for precessing itinerant $3d$ spins predict exactly the same formula $\ensuremath{\alpha}=1/({\ensuremath{\omega}}_{d}{\ensuremath{\tau}}_{sr}^{d})$ for the Gilbert damping constant as the full Green's function quantum treatment by Tserkovnyak et al. Here, ${\ensuremath{\omega}}_{d}$ is the precession frequency of $3d$ spins in the $d\text{\ensuremath{-}}d$ mutual exchange field, and ${\ensuremath{\tau}}_{sr}^{d}$ the $3d$ spin-relaxation time. A correct form for the spin-relaxation torque is crucial for success: The spins relax toward an instantaneous direction which is that of the vector sum of external field and $d\text{\ensuremath{-}}d$ exchange field. Remarkably, $d\text{\ensuremath{-}}d$ exchange torques disappear completely from the equations of motion for the total $3d$ magnetization, and exchange plays only an indirect role through the spin relaxation. This purely $3d$ model is simpler than the traditional $s\text{\ensuremath{-}}d$ model. We also present a theory of current-induced torques on a domain wall, based on the $3d$ model. We find equivalents to the so-called adiabatic and nonadiabatic torques. They are given by formulas similar to those holding for the $s\text{\ensuremath{-}}d$ exchange model.