Spin-transfer torque in magnetic tunnel junctions

Abstract

We present a theoretical study of the spin-transfer torque vector and the tunneling magnetoresistance (TMR) for symmetric magnetic tunnel junctions (MTJ) using the single-band tight-binding model and the nonequilibrium Keldysh formalism. We provide a comprehensive analysis of the effect of band filling and exchange splitting of the FM leads on the bias behavior of the spin-transfer component, ${T}_{\ensuremath{\parallel}}$, in the plane containing the magnetizations of the two magnetic layers, and the fieldlike component, ${T}_{\ensuremath{\perp}}$, perpendicular to this plane. We demonstrate that both components of the spin torque and the TMR can exhibit a wide range of interesting and unusual bias behavior. We show that ${T}_{\ensuremath{\parallel}}(V)$ satisfies an expression involving the difference in spin currents between the ferromagnetic (FM) and antiferromagnetic (AF) configurations, which is general and independent of the details of the electronic structure. The spin current for the FM (AF) alignment is shown to have a linear (quadratic) bias dependence, whose origin lies in the symmetric (asymmetric) nature of the barrier. On the other hand, the bias dependence of ${T}_{\ensuremath{\perp}}$ is quadratic with ${d}^{2}{T}_{\ensuremath{\perp}}/d{V}^{2}<0$, and it can change sign at finite bias. Finally, we show that the exchange splitting and band filling have a large effect on the bias dependence of the TMR.