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

We develop a numerically exact scheme for resumming certain classes of Feynman diagrams in the self-consistent perturbative expansion for the electron and magnon self-energies in the nonequilibrium Green function formalism applied to a coupled electron-magnon (e-m) system driven out of equilibrium by the applied finite bias voltage. Our scheme operates with the electronic and magnonic GFs and the corresponding self-energies viewed as matrices in the Keldysh space, rather than conventionally extracting their retarded and lesser components, which greatly simplifies translation of diagrams into compact mathematical expressions and their computational implementation. This is employed to understand the effect of inelastic e-m scattering on charge and spin current vs bias voltage $V_b$ in F/I/F (F-ferromagnet; I-insulating barrier) magnetic tunnel junctions (MTJs), which are modeled on a quasi-one-dimensional (quasi-1D) tight-binding lattice for the electronic subsystem and quasi-1D Heisenberg model for the magnonic subsystem. For this purpose, we evaluate the Fock diagram for the electronic self-energy and the electron-hole polarization bubble diagram for the magnonic self-energy. The respective electronic and magnonic GF lines within these diagrams are the fully interacting ones, thereby requiring to solve the ensuing coupled system of nonlinear integral equations self-consistently. Despite using the quasi-1D model and treating e-m interaction in many-body fashion only within a small active region consisting of few lattice sites around the F/I interface, our analysis captures essential features of the so-called zero-bias anomaly observed [V. Drewello, J. Schmalhorst, A. Thomas, and G. Reiss, Phys. Rev. B 77, 014440 (2008)] in both MgO- and ${\mathrm{AlO}}_x-$based realistic 3D MTJs where the second derivative $d^{2}I/d{V}_b^2$ (i.e., inelastic electron tunneling spectrum) of charge current exhibits sharp peaks of opposite sign on either side $V_b=0$. We show that this is closely related to a substantially modified magnonic density of states (DOS) after the e-m interaction is turned on—the magnonic bandwidth over which DOS is nonzero becomes broadened, thereby making e-m scattering at arbitrary small bias voltage possible, while DOS also acquires peaks (on the top of a continuous background) signifying the formation of quasibound states of magnons dressed by the cloud of electron-hole pair excitations. We also demonstrate that the sum of electronic spin currents in all of the semi-infinite leads attached to the active region quantifies the loss of spin angular momentum carried away from the active region by the magnonic spin current.

Figure 1

Schematic view of a quasi-1D model of F/I/F MTJ where the left semi-infinite ideal F lead, modeled as a spin-split tight-binding lattice of size $\infty \times N_y$, is attached via hopping $\gamma$ to an active region consisting of $N_x\times N_y$ lattice sites (we use $N_x=3$ and $N_y=1$ or $N_y=3$ in the calculations below). The right semi-infinite lead is attached to the active region via smaller hopping ${\gamma }_I=0.1\gamma$ that simulates the tunnel barrier $I$. The same sites also host localized spins, which are coupled to each other via the ferromagnetic coupling $J>0$ in the Heisenberg model. The local coupling between the spin of conduction electrons and localized spin on each site is of strength $g$. The left and right semi-infinite leads are assumed to terminate into macroscopic Fermi liquid reservoirs held at electrochemical potentials ${\mu }_L$ and ${\mu }_R$, respectively, whose difference sets the bias voltage $e{V}_b={\mu }_L-{\mu }_R$. The voltage profile across MTJ is shown on the top. The left F layer is assumed to be attached at infinity to a macroscopic reservoir of magnons held at temperature $T$.

Figure 2

Diagrammatic representation of the Dyson equation in the Keldysh space: (a) the electron case, $\check{\mathbf{G}}={\check{\mathbf{G}}}_0+{\check{\mathbf{G}}}_0{\check{\mathit{\Sigma }}}_{\mathrm{e}}-\mathrm{m}\check{\mathbf{G}}$, in Eq. (8); and (b) the magnon case, $\check{\mathbf{B}}={\check{\mathbf{B}}}_0+{\check{\mathbf{B}}}_0{\check{\mathit{\Omega }}}_{\mathrm{m}}-\mathrm{e}\check{\mathbf{B}}$, in Eq. (9). The perturbation expansion for the electronic self-energy ${\check{\mathit{\Sigma }}}_{\mathrm{e}}-\mathrm{m}$ in (a) retains the Hartree and Fock diagrams, while the expansion in (b) for the magnonic self-energy ${\check{\mathit{\Omega }}}_{\mathrm{m}}-\mathrm{e}$ retains the e-h polarization bubble diagrams. The single straight line denotes the noninteracting electronic GF, ${\check{\mathbf{G}}}_0$ [which includes the self-energies due to the leads in Eq. (10)]; the single wavy line denotes the noninteracting magnonic GF, ${\check{\mathbf{B}}}_0$ [which includes the self-energy due to the left lead in Eq. (11)]; the double straight line denotes the interacting electronic GF, $\check{\mathbf{G}}$; and the double wavy line denotes the interacting magnonic GF, $\check{\mathbf{B}}$. The solid circles denote vertices that are integrated out. The electron spin is flipped at each vertex, which is illustrated by spin $\uparrow$ (before the vertex) being flipped into spin $\downarrow$ (after the vertex), while a magnon is being created. The same process applies to flipping of spin $\downarrow$ to spin $\uparrow$, where the direction of magnon propagation (indicated by arrow on the wavy lines) is reversed. Note that the Hartree diagram in (a) contains a single (rather than double) wavy line in order to avoid double counting.

Figure 3

Elastic and inelastic contributions to charge current in the left and right leads of MTJ in Fig. 1 with an $N_x\times N_y\equiv 3\times 1$ active region at different iteration numbers within the self-consistent loop for solving the coupled Eqs. (8), (9), (12), and (14). The orientation of the magnetizations of two F layers in Fig. 1 is parallel in (a) and antiparallel in (b). The bias voltage is set as $V_b=-60$ mV and the temperature is $T=12$ K.

Figure 4

Total spin current dissipated (at $T=12$ K) inside the $N_x\times N_y\equiv 3\times 1$ active region of MTJ in Fig. 1 as a function of the bias voltage for the parallel and antiparallel orientations of the magnetizations of two F layers. The spin current is obtained from Eq. (28) using the electronic GF computed by solving the coupled Eqs. (8), (9), and (12).

Figure 5

The electronic density of states within the active region of MTJ model in Fig. 1 of size: (a) $N_x\times N_y\equiv 3\times 1$ and (b) $N_x\times N_y\equiv 3\times 3$. The magnonic density of states within the same active regions is shown in (c) and (d), respectively. These quantities are computed at finite bias voltage $V_b=-60$ mV and at temperature $T=12$ K, in the absence $(g=0$ for dashed line) or the presence $(g\ne 0$ for solid line) of e-m interaction in the Hamiltonian in Eq. (3c). The respective DOS is obtained from the retarded electronic GF, using $-\mathrm{Tr}\left[\Im \left({\mathbf{G}}^r\right)\right]/\pi$, or the retarded magnonic GF, using $-\mathrm{Tr}\left[\Im \left({\mathbf{B}}^r\right)\right]/\pi$, after solving coupled Eqs. (8), (9), (12), and (14), which take into account influence of electrons on magnons. The arrows in (a) and (b) point at the kinks (located at the Fermi energies of MTJ model with two different sizes of the active region, respectively) in the interacting electronic DOS (solid line) due to e-m coupling.

Figure 6

Elastic and inelastic charge currents in Eq. (27) and their sum, as well as the corresponding first and second derivatives, vs the bias voltage in the model of MTJ in Fig. 1 with active region $N_x\times N_y\equiv 3\times 1$ for parallel and antiparallel orientations of its magnetizations. These charge currents are obtained from the electronic GF computed in F-SCBA, which includes (in self-consistent fashion) the noninteracting magnonic GF. The temperature is set as $T=12$ K.

Figure 7

Elastic and inelastic charge currents in Eq. (27) and their sum, as well as the corresponding first and second derivatives, vs the bias voltage in the model of MTJ in Fig. 1 with active region $N_x\times N_y\equiv 3\times 1$ for the parallel and antiparallel orientations of its magnetizations. These charge currents are obtained from the electronic GF computed in F-SCBA, which includes (in self-consistent fashion) the interacting magnonic GF with the e-h polarization bubble diagram in Fig. 2. The temperature is set as $T=12$ K.

Figure 8

Elastic and inelastic charge currents in Eq. (27) and their sum, as well as the corresponding first and second derivatives, vs the bias voltage in the model of MTJ in Fig. 1 with active region $N_x\times N_y\equiv 3\times 3$ for parallel and antiparallel orientations of its magnetizations. These charge currents are obtained from the electronic GF computed in F-SCBA, which includes (in self-consistent fashion) the noninteracting magnonic GF. The temperature is set as $T=12$ K.

Figure 9

The TMR vs bias voltage $V_b$ in the model of MTJ in Fig. 1 with an active region $N_x\times N_y\equiv 3\times 3$. The inset shows TMR as a function of temperature in the linear-response limit $V_b\to 0$. These results are obtained from an electronic GF computed by solving the coupled Eqs. (8), (9), and (12).