Non-equilibrium thermodynamic study of magnetization dynamics in the presence of spin-transfer torque

The dynamics of magnetization in the presence of spin-transfer torque was studied. We derived the equation for the motion of magnetization in the presence of a spin current by using the local equilibrium assumption in non-equilibrium thermodynamics. We show that, in the resultant equation, the ratio of the Gilbert damping constant, $\alpha$, and the coefficient, $\beta$, of the current-induced torque, called non-adiabatic torque, depends on the relaxation time of the fluctuating field $\tau_{c}$. The equality $\alpha=\beta$ holds when $\tau_c$ is very short compared to the time scale of magnetization dynamics. We apply our theory to current-induced magnetization reversal in magnetic multilayers and show that the switching time is a decreasing function of $\tau_{c}$.