Spin-transfer torque and spin-polarization in topological-insulator-based magnetic tunnel junctions

We derive a nonequilibrium Green function-based formula for spin-transfer torque (STT) exerted by the conduction electrons on the magnetization of a free ferromagnetic (F) layer where {\em strong} spin-orbit coupling (SOC) is present either in the bulk or at the interface of the F layer. This nonequilibrium Born-Oppenheimer approximation-type formula is employed to predict unconventional STT in N$|$TI$|$F semi- magnetic tunnel junction (MTJ) containing a three-dimensional topological insulator (TI). The STT is driven by the SOC on the surface of TI, as well as by the charge current becoming spin-polarized in the direction of transport as it flows from the normal metal (N) through the bulk of the TI layer. The in- plane and perpendicular STT components in N$|$TI$|$F semi-MTJ are an order of magnitude larger than in conventional F$^\prime|$I$|$F MTJ, or N$|$I$|$F semi-MTJ with the strong Rashba SOC at the I$|$F interface, assuming comparable resistance of all three junctions.