Spin-Transfer Torque in Double Barrier Magnetic Tunnel Junction.

The transport properties in double-barrier magnetic tunnel junctions (DBMTJ) are determined by the spin polarized quantum well states(QWS) formed in the middle ferromagnetic(FM) region. Using tight-binding approach to the Keldysh formalism, we have studied the effect of the QWS on spin-transfer torque(STT) exerting on the middle FM region in non-collinear DBMTJ, with components parallel,$T_{\vert \vert } $, and perpendicular,$T_\bot $, to the interface. Our results reveal that both \textit{local} STT $T_{\vert \vert (\bot ),i} $exerting on atomic layer i in the middle FM region can be dramatically enhanced for values of the thickness of the middle FM region, b, for which spin-up and spin-down QWS are in close proximity to each other and lie within the bias window. This enhancement though, is cancelled out for the \textit{total} STT, $T_{\vert \vert (\bot )} =\sum\limits_{i=1}^b {T_{\vert \vert (\bot ),i} } $ due to the oscillations of $T_{\vert \vert ,i} $and $T_{\bot ,i} $, as a function of i. In addition we show that the bias dependence of $T_{\vert \vert } $ for different b and different orientations of the magnetizations of the leads varies due to the QWS and symmetry respectively. We also show that the angular dependence of both $T_{\vert \vert } $ and $T_\bot $deviates from the sinusoidal behavior. Interestingly$T_\bot $, which measures the non-equilibrium exchange coupling, exhibits an enhanced biquadratic term in its angular dependence.