Edwards, D.M.; Federici, F.; Mathon, J. and Umerski, A.
Self-consistent theory of current-induced switching of magnetization.
Physical Review B, 71(5) (054407)1-(054407)16.
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 an ab 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 T 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, T 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 T are comparable for a Co/Cu/Co(111) junction when the switching Co layer is one or two atomic planes thick. T is 27% of T even for a switching Co magnet of ten atomic planes. Depending on material parameters of the junction, the relative sign of T and T can be negative as well as positive. In particular, T/T<0 for Co/Cu/Co(111) with switching Co magnet of one atomic plane and T/T > 0 for two atomic planes of Co. A negative sign of the ratio T/T has a profound effect on the nature of switching, particularly in the realistic case of easy-plane (shape) 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 spin-transfer 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/T > 0. However, for T/T<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/Co(111) show that the critical current for switching in the hysteretic regime is 107 A/cm2, which is in good agreement with experiment.
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