Spin-Polarized Current Induced Torque in Magnetic Tunnel Junctions
  We present tight-binding calculations of the spin torque in non-collinear magnetic tunnel junctions based on the non-equilibrium Green functions approach. We have calculated the spin torque via the effective local magnetic moment approach and the divergence of the spin current. We show that both methods are equivalent, i.e. the absorption of the spin current at the interface is equivalent to the exchange interaction between the electron spins and the local magnetization. The transverse components of the spin torque parallel and perpendicular to the interface oscillate with different phase and decay in the ferromagnetic layer (FM) as a function of the distance from the interface. The period of oscillations is inversely proportional to the difference between the Fermi-momentum of the majority and minority electrons. The phase difference between the two transverse components of the spin torque is due to the precession of the electron spins around the exchange field in the FM layer. In absence of applied bias and for a relatively thin barrier the perpendicular component of the spin torque to the interface is non-zero due to the exchange coupling between the FM layers across the barrier. 
A. Kalitsov, I. Theodonis, N. Kioussis, M. Chshiev, W. H. Butler, A.   Vedyayev