Spin torque from tunneling through impurities in a magnetic tunnel junction

We calculate the contribution to the spin-transfer torque from sequential tunneling through impurities in a magnetic tunnel junction. For a junction with weakly polarized ferromagnetic contacts, the torque is found to be in the plane spanned by the magnetizations of the ferromagnetic contacts and proportional to $\text{sin}\theta$, where $\theta$ is the angle between the magnetic moments. If the polarization is larger, the torque acquires a significant out-of-plane component and a different dependence on $\theta$.

Figure 1

(Color online) Schematic picture showing the impurity energy level and the chemical potentials of the ferromagnetic reservoirs. The left panel is for the case ${\mu }_R<\epsilon <{\mu }_L$, in which the occupation of the impurity site fluctuates between 0 and 1. The right panel is for the case ${\mu }_R<\epsilon +U<{\mu }_L$, in which the impurity-site occupation varies between 1 and 2. These two cases are labeled I and II in the main text.

Figure 2

(Color online) Spectral current $i^{\prime }\left(\epsilon \right)$ and torque components $t_{R\parallel }^{\prime }\left(\epsilon \right)$, $t_{L\parallel }^{\prime }\left(\epsilon \right)$, and $t_{\perp }^{\prime }\left(\epsilon \right)$ versus the angle $\theta$ between the magnetizations of the left and right ferromagnetic contacts. The curves shown are for polarization $p=0.2$ (solid), 0.5 (short dash), 0.9 (dash dot), and 0.99 (long dash). For definiteness, the exchange field parameters have been set to ${\eta }_L={\eta }_R=1$.