Spin transfer torque in Mn$_3$Ga-based ferrimagnetic tunnel junctions from first principles

We report on first-principles calculations of spin-transfer torque (STT) in epitaxial magnetic tunnel junctions (MTJs) based on ferrimagnetic tetragonal Mn$_3$Ga electrodes, both as analyzer in an Fe/MgO stack, and also in an analogous stack with a second Mn$_3$Ga electrode (instead of Fe) as polarizer. Solving the ballistic transport problem (NEGF + DFT) for the nonequilibrium spin density in a scattering region extended to over 7.6 nm into the Mn$_3$Ga electrode, we find long-range spatial oscillations of the STT decaying on a length scale of a few tens of angstroms, both in the linear response regime and for finite bias. The oscillatory behavior of the STT in Mn$_3$Ga is robust against variations in the stack geometry and the applied bias voltage, which may affect the phase and the amplitude of the spatial oscillation, but the wave number is only responsive to variations in the longitudinal lattice constant of Mn$_3$Ga (for fixed in-plane geometry) without being commensurate with the lattice. Our interpretation of the long-range STT oscillations is based on the bulk electronic structure of Mn$_3$Ga, taking also into account the spin-filtering properties of the MgO barrier. Comparison to a fully Mn$_3$Ga-based stack shows similar STT oscillations, but a significant enhancement of both the TMR effect at the Fermi level and the STT at the interface, due to resonant tunneling for the mirror-symmetric junction with thinner barrier (three monoatomic layers). From the calculated energy dependence of the spin-polarized transmissions at 0 V, we anticipate asymmetric or symmetric TMR as a function of the applied bias voltage for the Fe-based and the all-Mn$_3$Ga stacks, respectively, which also both exhibit a sign change below 1 V. In the latter (symmetric) case we expect a TMR peak at zero, which is larger for the thinner barriers because of a spin-polarized resonant tunneling contribution.