Spin-transfer torque for continuously variable magnetization

We report quantum and semiclassical calculations of spin current and spin-transfer torque in a free-electron Stoner model for systems where the magnetization varies continuously in one dimension. Analytic results are obtained for an infinite spin spiral and numerical results are obtained for realistic domain wall profiles. The adiabatic limit describes conduction electron spins that follow the sum of the exchange field and an effective, velocity-dependent field produced by the gradient of the magnetization in the wall. Nonadiabatic effects arise for short domain walls but their magnitude decreases exponentially as the wall width increases. Our results cast doubt on the existence of a recently proposed nonadiabatic contribution to the spin-transfer torque due to spin-flip scattering.

Figure 1

(Color online) Equilibrium (zero-current) results: (a) Cartesian components of an arbitrarily chosen magnetization $\mathbf{M}\left(x\right)$ (lines); Cartesian components of the calculated spin density ${\mathbf{s}}_{\mathrm{eq}}\left(x\right)$ (solid dots); (b) exchange torque (solid line) and calculated spin-transfer torque (solid dots).

Figure 2

(Color online) (a) Cartesian components of an imposed magnetization $\mathbf{M}\left(x\right)$ used in the other panels; (b) comparison of quantum (lines) to semiclassical (solid dots) calculations for the Cartesian components of the spin density $\mathbf{s}\left(x\right)$; (c) same comparison for the Cartesian components of the spin-current density $\mathbf{Q}\left(x\right)$.

Figure 3

(Color online) Cartoon of a spin spiral where the magnetization (arrows) rotates uniformly in the $x\text{−}z$ plane of a fixed coordinate system where $\widehat{\mathbf{M}}\left(x\right)\bullet \widehat{\mathbf{z}}=\cos \theta$. The inset shows a local coordinate system where $\mathbf{M}\left(x\right)$ always points along the $z^{\prime }$ axis.

Figure 4

(Color online) Electrons (wave vector $k_x$) and holes (wave vector $-k_x$) move in an effective field that is the sum of the exchange field ${\mathbf{B}}_{\mathrm{ex}}\left(x\right)\Vert {\widehat{\mathbf{z}}}^{\prime }$ and a fictitious, velocity-dependent “gradient” field induced by the spatial dependence of the exchange field. The spins align to the total effective field in an infinite spin spiral. The $x$ axis lies in $x^{\prime }\text{−}{z}^{\prime }$ plane.

Figure 5

Distributed spin-transfer torque for a long Néel domain wall with $w=50$ and $L=6.25$ ($k_F=1$ and $k_B=0.4$): semiclassical calculation of ${\mathbf{N}}_{\mathrm{st}}$ (solid dots) compared to Eq. (33) for ${\mathbf{N}}_{\mathrm{ad}}\left(x\right)$ (solid curve).

Figure 6

(Color online) Distributed spin-transfer torque for a short Néel domain wall with $w=4$ and $L=6.25$ ($k_F=1$ and $k_B=0.4$): in-plane piece $a\left(x\right)$ (heavy solid curve); out-of-plane piece $b\left(x\right)$ (dashed curve); adiabatic prediction (light solid curve); second term in Eq. (1) scaled to match the maximum of $b\left(x\right)$ (light solid curve).

Figure 7

Nonadiabaticity in Eq. (35) versus wall width scaled by the characteristic length in Eq. (32). Note the logarithmic scale.

Figure 8

Dependence of $\gamma$ in Eq. (36) on domain wall sharpness: Squares are calculated points. Straight line is a guide to the eye that passes through the origin.

Figure 9

The semiclassical weighting factor for the spin density. Solid (dotted) curve is the expression on the left (right) side of the ≈ symbol in Eq. (A10).