Self-consistent nonequilibrium many-body theory of electron-magnon interaction in realistic three-dimensional magnetic tunnel junctions: reproducing the zero-bias anomaly, finite-bias TMR, and inelastic spin-transfer torque

In magnetic tunnel junctions (MTJs), inelastic electron-magnon (e-m) scattering shapes the measured inelastic electron tunneling spectra (IETS): experiments show a zero-bias anomaly (ZBA) with two peaks of the second derivative $d^2I/dV_b^2$ symmetric about zero bias and a dip in the differential conductance $dI/dV_b$ at $V_b=0$ , while elastic theories of the spin-transfer torque (STT) deviate from experiment at high bias. Mahfouzi and Nikolić [Phys. Rev. B 90, 045115 (2014)] developed a conserving, self-consistent nonequilibrium Green function many-body perturbation treatment of coupled e-m transport, but could only solve it in quasi-one-dimensional models with a few transverse sites: the Hartree diagram was dropped, the electron-hole polarization bubble was excluded from the widest calculation, the IETS peaks shifted away from zero bias, the zero-bias dip in $dI/dV_b$ was not reproduced at all, and the bias dependence of the STT parameters was not computed. The open problem is whether the framework itself — extended to realistic multi-channel (three-dimensional, transverse- $k$ -sampled) junctions with the conserving diagram set (self-consistent Fock electronic self-energy with fully dressed lines, bubble-dressed magnon Green function, and the previously omitted Hartree term either stably included or justifiably neglected) — can reproduce the essential features of these measured spectra.

Background

A magnetic tunnel junction (MTJ) is a ferromagnet/insulator/ferromagnet (F/I/F) stack whose resistance depends on the relative orientation of the two magnetizations; the effect is called tunneling magnetoresistance (TMR). Inelastic electron tunneling spectroscopy (IETS) measures the second derivative $d^2I/dV_b^2$ of the current with respect to bias voltage $V_b$ : inelastic channels open at thresholds set by excitation energies, producing characteristic peaks. In CoFeB/MgO and AlO $_x$ -based MTJs, IETS shows a zero-bias anomaly (ZBA): peaks of opposite sign at $V_b\simeq\pm 10$ mV attributed to magnons (quantized spin-wave excitations of the ferromagnetic electrodes), together with a dip in the differential conductance $dI/dV_b$ at $V_b=0$ , and TMR decreases with increasing temperature (V. Drewello et al., Phys. Rev. B 77, 014440 (2008); A. A. Khan et al., Phys. Rev. B 82, 064416 (2010)). Separately, the spin-transfer torque (STT) — the torque exerted by the spin-polarized tunneling current on the magnetization, parameterized by field-like and damping-like coefficients $b_J$ and $a_J$ — deviates from purely elastic phase-coherent theories at high bias, especially for $b_J$ (H. Kubota et al., Nat. Phys. 4, 37 (2008); C. Wang et al., Nat. Phys. 7, 496 (2011)), and inelastic scattering has been invoked to explain the discrepancy. An early alternative theory attributed the ZBA to inelastic tunneling via two-level impurities in the barrier (L. Sheng, D. Y. Xing, and D. N. Sheng, Phys. Rev. B 70, 094416 (2004)), so the microscopic origin of the ZBA features has competing candidate mechanisms.

Mahfouzi and Nikolić (Phys. Rev. B 90, 045115 (2014)) formulated a nonequilibrium many-body perturbation theory of the coupled e-m system, based on the nonequilibrium Green function (NEGF) formalism on the Keldysh contour, with the electron and magnon Green functions and self-energies handled as matrices in Keldysh space. The localized spins were mapped to magnons via the Holstein-Primakoff transformation, the e-m coupling arises from the local $s$ - $d$ exchange interaction (the contact exchange coupling between itinerant electron spins and the localized magnetic moments), and the electronic subsystem was a tight-binding lattice modeling F/I/F MTJs. To conserve charge current at finite bias they used a self-consistent Born approximation (SCBA) — resumming the Fock diagram for the electronic self-energy with fully dressed Green function lines, mutually self-consistent with the electron-hole polarization bubble diagram for the magnon self-energy (analogous to self-consistent GW for electron-electron interaction). Their key physical finding was that the bubble diagram substantially broadens the magnon density of states to low energies, enabling inelastic e-m scattering even at zero bias and producing a two-peak structure in the partial (elastic or inelastic) IETS around $V_b=0$ .

However, the solved models were quasi-one-dimensional: a strictly 1D active region ( $3\times1$ sites) produced an overly complex total IETS (interference between few spin-resolved conducting channels), and the widest calculation ( $3\times3$ sites) recovered the simple two-peak structure but only after dropping the bubble diagram for computational expense, so the peaks shifted away from $V_b=0$ . Three limitations were explicitly flagged: (1) the Hartree diagram was omitted because its direct evaluation is numerically unstable when the junction is invariant under spin rotations about the magnetization axis (the spin-flip rate in the Hartree term can acquire an arbitrary phase); (2) the experimentally observed dip of $dI/dV_b$ at $V_b=0$ was not reproduced in any of their calculations, plausibly due to the small number of transverse channels; (3) extending the approach to fully 3D MTJs requires transverse $k$ -point sampling, turning the self-consistent coupled nonlinear integral equations into prohibitively expensive five-dimensional integrals, so carefully crafted approximations are needed. They also did not compute the bias dependence of the STT parameters $a_J$ and $b_J$ (the announced follow-up never appeared in peer-reviewed form), and the TMR temperature dependence was shown only in the linear-response limit $V_b\to 0$ .

Problem Statement

Extend the conserving, self-consistent nonequilibrium many-body perturbation theory of the coupled electron-magnon system in ferromagnet/insulator/ferromagnet magnetic tunnel junctions — electronic Fock self-energy with fully dressed Green function lines, mutually self-consistent with the electron-hole polarization bubble self-energy for the magnons — to a realistic multi-channel junction that stands in for measured 3D MTJs (active region whose number of transverse sites, or transverse momentum $k_{y},k_{z}$ mesh density, is converged in the sense that further enlargement leaves the computed spectra qualitatively unchanged, with suitable controlled approximations to keep the five-dimensional self-consistency integrals tractable), and settle the status of the previously omitted Hartree diagram — either by including it in a numerically stable, charge-conserving way, or by establishing within the same framework that its effect on the observables below is negligible, as is customarily assumed for the analogous Hartree term in electron-phonon SCBA. Within that framework, determine whether inelastic electron-magnon scattering alone reproduces the essential features of the measured inelastic magnetotransport signatures of MgO- and AlO $_x$ -based MTJs, namely: (a) the zero-bias anomaly in the total IETS — the experimentally measured $d^2I/dV_b^2$ — a two-peak structure centered at $V_b=0$ on the meV scale of the observed ZBA, together with the dip in $dI/dV_b$ at $V_b=0$ , i.e., $dI/dV_b$ rising in absolute value as $|V_b|$ increases away from the zero-bias minimum; (b) the TMR ratio as a function of both temperature and finite bias — within the temperature range in which the low-magnon-occupation Holstein-Primakoff approximation underlying the framework remains valid ( $\langle \hat{b}_i^\dagger \hat{b}_i\rangle \ll S$ ) — including its interplay with the ZBA; and (c) the bias dependence of the spin-transfer torque parameters $a_J$ and $b_J$ , where purely elastic theories are known to deviate from experiment at high bias. All three sub-targets (a), (b) and (c) are required, as are the multi-channel extension and the settlement of the Hartree diagram (by either of the two options stated above), and each sub-target is resolved by whether the extended framework, with its converged active region, reproduces the qualitative feature it names — for (a) the two zero-bias IETS peaks together with the zero-bias dip in $dI/dV_b$ , for (b) the temperature- and finite-bias dependence of the TMR ratio together with its interplay with the ZBA, and for (c) a bias-dependent $a_J$ and $b_J$ — the agreement being qualitative reproduction of these enumerated features rather than a quantitative fit to a single measured curve. A definite demonstration, obtained within the same converged framework, that inelastic electron-magnon scattering alone cannot reproduce one or more of these features resolves the corresponding sub-target equally; partial reproduction of a sub-target, or a conclusion resting on a non-converged or uncontrolled calculation, does not.

References

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