Microscopic determination of interface-magnon tunneling parameters and the magnon spectrum at realistic ferromagnet/insulator interfaces: extending the theory of inelastic (magnon-assisted) tunneling beyond the low-bias regime in magnetic tunnel junctions

The 1997 experiment–theory work of Zhang, Levy, Marley and Parkin explained the rapid drop of tunneling magnetoresistance (TMR) with applied bias voltage in ferromagnetic tunnel junctions — including a zero-bias anomaly at low temperature that is stronger for antiparallel than parallel electrode magnetization — by inelastic tunneling assisted by magnons (spin-wave quanta) localized at the ferromagnet/insulator interfaces. The theory fits R(V,T) data up to about 200 mV, but only by leaving its key microscopic ingredients undetermined: the transfer matrix elements (spin-independent $T^d$ and spin-flip exchange $T^J$), the interfacial densities of states, the magnon spectrum at rough, disordered interfaces, and the low-energy cutoff $E_c$ that regularizes the two-dimensional magnon infrared divergence were all treated as adjustable or phenomenological. Later work substantially confirmed and spectroscopically identified interface-magnon excitations (inelastic electron tunneling spectroscopy in AlOx, MgO, CoFeB/MgO and FePt/MgO junctions) and produced model-level and some first-principles elastic-channel treatments, but a genuinely microscopic — ideally first-principles — calculation of the magnon-assisted tunneling matrix elements and their energy/bias dependence, of the magnon spectrum at realistic disordered tunnel interfaces, and of the origin, magnitude and temperature dependence of $E_c$, together with a parameter-free prediction of R(V,T) and inelastic tunneling spectra beyond the low-bias regime reconciled with competing inelastic mechanisms, remains open.

Background

A magnetic tunnel junction (MTJ) is a two-terminal device in which two ferromagnetic electrodes are separated by a thin insulating barrier (typically Al2O3 or MgO), through which electrons tunnel. Its tunneling magnetoresistance — the change of junction resistance between parallel (P) and antiparallel (AP) electrode magnetization — is the basis of magnetic sensors, hard-disk read heads and magnetoresistive random-access memory. A long-standing limitation of MTJs is that the TMR falls off rapidly as the bias voltage V across the barrier is increased, limiting device operating margins. Already in early junctions, the current–voltage characteristics at low temperature showed a zero-bias anomaly (ZBA): a sharp dip in dynamic resistance at V = 0, larger for AP than for P alignment.

Zhang, Levy, Marley and Parkin (1997) combined Parkin's IBM junction data (Co/Al2O3/CoFe at 4.2 K and 210 K) with a transfer-Hamiltonian theory of tunneling. In the transfer-Hamiltonian (Bardeen) approach, the tunneling current is expressed through matrix elements connecting electronic states on the two sides of the barrier. The authors extended it to inelastic tunneling in which the tunneling electron emits or absorbs a magnon — a quantized spin wave, i.e. a collective precession excitation of the spin lattice — that is localized at the ferromagnet/barrier interface. Two types of transfer matrix element appear: $T^d$, the spin-conserving (direct) transmission, and $T^J$, the exchange (spin-flip) transmission that accompanies magnon emission or absorption; their ratio $|T^d|^2/|T^J|^2 = 17$ was the single adjustable parameter of the fit. Interface magnons in two dimensions have a nearly flat, gapless density of states, which produces an infrared (low-energy) divergence in the inelastic current; the authors introduced a phenomenological low-energy cutoff $E_c \approx 4$ meV — made temperature-dependent to reproduce the disappearance of cusplike features in R(V) between 4.2 K and 210 K — to regularize it, and a characteristic interface magnon energy $E_m^\alpha = 3k_B T_c^\alpha/(S^\alpha+1) \approx 110$ meV (with interface spin $S^\alpha = 3/2$), where $T_c^\alpha$ is an effective interface Curie-like temperature. The spin-dependent density of states (DOS) ratios of majority to minority itinerant states were fixed from separate symmetric-junction measurements rather than computed. With these ingredients the model reproduced the P and AP resistance–voltage curves and the stronger bias quenching of the magnetoresistance in the AP state up to about 200 mV.

The open question left by that paper is that none of the microscopic ingredients is actually derived: the matrix elements $T^d$ and $T^J$ and their energy dependence, the detailed interfacial DOS, the magnon spectrum at the rough, disordered (amorphous-oxide) interfaces of real junctions, and the origin, magnitude and temperature dependence of the cutoff $E_c$ are all input parameters rather than predictions, and the theory is explicitly restricted to incoherent tunneling at biases below roughly 200 mV with energy-independent matrix elements. Inelastic electron tunneling spectroscopy (IETS), the second derivative $d^2I/dV^2$ of the current, which shows peaks at the bias $eV$ matching excitation energies at the interface, is the standard experimental probe of such inelastic channels.

Problem Statement

Determine microscopically — ideally from first-principles electronic-structure and spin-dynamics calculations for realistic ferromagnet/insulator interfaces — the quantities that the Zhang–Levy–Marley–Parkin interface-magnon theory of inelastic tunneling in magnetic tunnel junctions introduced as adjustable or phenomenological, and use them to extend that theory beyond the low-bias regime. Concretely: (i) compute the transfer matrix elements $T^d$ (spin-conserving) and $T^J$ (spin-flip exchange) and their energy dependence, together with the spin-resolved interfacial itinerant-electron densities of states, rather than fixing their ratio ($|T^d|^2/|T^J|^2$) and DOS ratios by fitting; (ii) derive the magnon spectrum at realistic disordered or rough tunnel interfaces, including the microscopic origin, magnitude and temperature dependence of the low-energy cutoff $E_c$ that regularizes the two-dimensional magnon infrared divergence (candidates include interface anisotropy gaps and finite magnetic coherence length); (iii) extend the theory to bias voltages beyond roughly 200 mV by including the energy and bias dependence of the matrix elements and densities of states, and assess the role of other inelastic-tunneling channels (magnetic impurities with phonon emission, electron–electron interaction) alongside interface-magnon emission and absorption; and (iv) confront the resulting parameter-free predictions with measured R(V,T) and inelastic electron tunneling spectra across material systems, extracting interface spin parameters ($S^\alpha$, $T_c^\alpha$) from independent measurements rather than fits. Parts (i)–(iv) are interdependent: a parameter-free prediction confronted with measured R(V,T) and inelastic tunneling spectra requires the derived matrix elements, interfacial densities of states, magnon spectrum and the cutoff $E_c$. The overarching objective is a predictive, microscopically grounded account of the quenching of tunneling magnetoresistance by hot-electron excitation of interface magnons.

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