Self-consistent theory of current-induced switching of magnetization

A self-consistent theory of the current-induced switching of magnetization using nonequilibrium Keldysh formalism is developed for a junction of two ferromagnets separated by a nonmagnetic spacer in the ballistic limit. It is shown that the spin-transfer torques responsible for current-induced switching of magnetization can be calculated from first principles in a steady state when the magnetization of the switching magnet is stationary. A steady state is achieved when the spin-transfer torque, proportional to bias voltage in the linear response regime, is balanced by the torque due to anisotropy fields. The spin-transfer torque is expressed in terms of one-electron surface Green functions for the junction cut into two independent parts by a cleavage plane immediately to the left and right of the switching magnet. The surface Green functions are calculated using a tight-binding Hamiltonian with parameters determined from a fit to anab initio band structure. This treatment yields the spin transfer torques taking into account rigorously contributions from all the parts of the junction. The spin-transfer torque has two components, one with the torque vector Ti in the plane containing the magnetizations of the two magnetic layers and another with the torque vector T’ perpendicular to this plane. It is shown that, in general, Ti and T’ may be comparable in magnitude and they both tend to finite values independent of the spacer thickness in the limit of a thick spacer. T’ is shown to be small when the exchange splitting of the majority- and minority-spin bands in both ferromagnets tends to infinity or in the case when the junction has a plane of reflection symmetry at the center of the spacer. The torques T’ and Ti are comparable for a Co/ Cu/ Cos111d junction when the switching Co layer is one or two atomic planes thick. T’ is <27% of Ti even for a switching Co magnet of ten atomic planes. Depending on material parameters of the junction, the relative sign of T’ and Ti can be negative as well as positive. In particular, T’ / Ti , 0 for Co/ Cu/ Cos111d with switching Co magnet of one atomic plane and T’ / Ti . 0 for two atomic planes of Co. A negative sign of the ratio T’ / Ti has a profound effect on the nature of switching, particularly in the realistic case of easy-plane sshaped anisotropy much larger than in-plane uniaxial anisotropy. To calculate the hysteresis loops of resistance versus current, and hence to determine the critical current for switching, the microscopically calculated spintransfer torques are used as an input into the phenomenological Landau-Lifshitz equation with Gilbert damping. In the absence of an applied magnetic field, an ordinary hysteresis loop is the only possible switching scenario when T’ / Ti . 0. However, for T’ / Ti , 0, a normal hysteretic switching occurs only at relatively low current densities. When the current exceeds a critical value, there are no stable steady states and the system thus remains permanently in a time dependent state. This is analogous to the observed precession of the switching magnet magnetization caused by a dc current in the presence of an applied magnetic field. The present calculations for Co/ Cu/ Cos111d show that the critical current for switching in the hysteretic regime is <10 7 A/c m 2 , which is in good agreement with experiment.