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We develop a numerically exact scheme for resumming certain classes ofFeynman diagrams in the self-consistent perturbation expansion for the electronand magnon self-energies in the nonequilibrium Green function formalism appliedto a coupled electron-magnon (\mbox{e-m}) system which is driven out ofequilibrium by the applied finite bias voltage. Our scheme operates with theelectronic and magnonic GFs and the corresponding self-energies viewed asmatrices in the Keldysh space, rather than conventionally extracting theirretarded and lesser components. This is employed to understand the effect ofinelastic \mbox{e-m} scattering on charge and spin current vs. bias voltage$V_b$ in F/I/F magnetic tunnel junctions (MTJs), which are modeled on aone-dimensional (1D) tight-binding lattice for the electronic subsystem and 1DHeisenberg model for the magnonic subsystem. For this purpose, we evaluate Fockdiagram for the electronic self-energy and the electron-hole polarizationbubble diagram for the magnonic self-energy. The respective electronic andmagnonic GF lines within these diagrams are the fully interacting ones, therebyrequiring to solve the ensuing coupled system of nonlinear integral equationsself-consistently. Despite using the 1D model and treating \mbox{e-m}interaction in many-body fashion only within a small active region consistingof few lattice sites around the F/I interface, our analysis captures essentialfeatures of the so-called zero-bias anomaly observed in both MgO- andAlO$_x$-based realistic 3D MTJs where the second derivative $d^2 I/dV_b^2$(i.e., inelastic electron tunneling spectrum) of charge current exhibits sharppeaks of opposite sign on either side of the zero bias voltage.

physical review. b, condensed matter and materials physics

Signatures of electron-magnon interaction in charge and spin currents in  magnetic tunnel junctions: A nonequilibrium many-body perturbation theory  approach