Nonadiabatic spin-transfer torque in real materials

The motion of simple domain walls and of more complex magnetic textures in the presence of a transport current is described by the Landau-Lifshitz-Slonczewski (LLS) equations. Predictions of the LLS equations depend sensitively on the ratio between the dimensionless material parameter $\beta$ which characterizes nonadiabatic spin-transfer torques and the Gilbert damping parameter $\alpha$. This ratio has been variously estimated to be close to zero, close to one, and large compared to one. By identifying $\beta$ as the influence of a transport current on $\alpha$, we derive a concise, explicit, and relatively simple expression which relates $\beta$ to the band structure and Bloch state lifetimes of a magnetic metal. Using this expression we demonstrate that intrinsic spin-orbit interactions lead to intraband contributions to $\beta$ which are often dominant, and can be (i) estimated with some confidence and (ii) interpreted using the “breathing Fermi-surface” model.

Figure 1

Feynman diagrams for (a) $\alpha$ and (b) $\beta \left(\mathbf{q}\cdot {\mathbf{v}}_s\right)$, the latter with a heuristic consideration of the electric field (for a more rigorous treatment, see Appendix ). Solid lines correspond to the Green’s functions of the band quasiparticles in the Born approximation, dashed lines stand for the magnon of frequency $\omega$ and wave vector $\mathbf{q}$, ${\omega }_n$ is the Matsubara frequency, and $e{V}_{a,b}$ is the difference in the transport deviation energies.

Figure 2

(Color online) Magnetized two-dimensional electron gas (M2DEG): interband contribution, intraband contribution, and the total nonadiabatic STT for a M2DEG. In this figure the exchange field dominates over the spin-orbit splitting. At higher disorder the interband part (proportional to resistivity) dominates while at low disorder the interband part (proportional to conductivity) overtakes. For simplicity, the scattering time $\tau$ is taken to be the same for all subbands.

Figure 3

(Color online) GaMnAs: ${\beta }^{\left(0\right)}$ for $\mathbf{E}$ perpendicular to the easy axis of magnetization $\left(\widehat{z}\right)$. $x$ and $p$ are the Mn fraction and the hole density, respectively. The intraband contribution is considerably larger than the interband contribution due to the strong intrinsic spin-orbit interaction. Since the four-band model typically overestimates the influence of intrinsic spin-orbit interaction, it is likely that the dominion of intraband contributions be reduced in the more accurate six-band model. By evaluating $\beta$ for $\mathbf{E}\parallel \widehat{z}$ (not shown) we infer that it does not depend significantly on the relative direction between the magnetic easy axis and the electric field.

Figure 4

(Color online) Comparison of $\alpha$ and $\beta$ in (Ga,Mn)As for $x=0.08$ and $p=0.4{\text{nm}}^{-3}$. It follows that $\beta /\alpha \simeq 8$, with a weak dependence on the scattering rate off impurities. If we use the torque-correlation formula (Sec. 7), we obtain $\beta /\alpha \simeq 10$.

Figure 5

(Color online) M2DEG: comparing $S$ and $K$ matrix element expressions for the nonadiabatic STT formula in the weakly spin-orbit coupled regime. Both formulations agree in the clean limit, where the intraband contribution is dominant. In more disordered samples interband contributions become more visible, and $S$ and $K$ begin to differ; the latter is known to be more accurate in the weakly spin-orbit coupled regime.

Figure 6

(Color online) M2DEG: In the strongly spin-orbit coupled limit the intraband contribution reigns over the interband contribution, and accordingly $S$ and $K$ matrix element expressions display a good (excellent in this figure) agreement. Nevertheless, this agreement does not guarantee quantitative reliability because for strong spin-orbit interactions impurity vertex corrections may play an important role.

Figure 7

(Color online) GaMnAs: comparison between $S$ and $K$ matrix element expressions for $\mathbf{E}\perp \widehat{z}$. The disagreement between both formulations stems from interband transitions, which are less important as $\tau$ increases. Little changes when $\mathbf{E}\parallel \widehat{z}$.

Figure 8

Feynman diagram for ${\chi }_{S,S,j}$. The dashed lines correspond to magnons, whereas the wavy line represents a photon.

Figure 9

Feynman diagram for the first-order vertex correction. The dotted line with a cross represents the particle-hole correlation mediated by impurity scattering.