Free-electron model of current-induced spin-transfer torque in magnetic tunnel junctions

Abstract

Spin and charge transport in a biased planar tunnel junction with two ferromagnetic electrodes are analyzed theoretically in the spin-polarized free-electron-like model. Tunneling current and both in-plane and out-of-plane components of the current-induced spin torque exerted on ferromagnetic components of the junction are determined as a function of the angle between magnetic moments of the electrodes, barrier height and thickness, and spin polarization of the electrodes. The out-of-plane component of the torque is found to be comparable in magnitude to the in-plane component. Numerical results for junctions with sufficiently high barriers show that the in-plane torque exerted on the sink electrode increases monotonically with increasing spin polarization (spin splitting) of the electron band in the source electrode, while the out-of-plane torque increases with increasing spin polarization of the electron bands in both electrodes. In junctions with thick and low barriers, the torque components can change their sign. The bias voltage dependence of the in-plane torque is not symmetric with respect to the bias reversal, even in symmetric junctions. When one of the ferromagnetic electrodes is replaced by a ferromagnetic layer of finite thickness (followed by a nonmagnetic electrode), both components of the spin torque depend significantly on the layer thickness due to the size effect and quantum well states formed in the thin magnetic layer.