Unified Description of Bulk and Interface-Enhanced Spin Pumping

We describe a mechanism for generating nonequilibrium electron-spin accumulation in semiconductors or metals by rf magnetic field pumping. With a semiclassical model we show that a rotating applied magnetic field (or the precessing magnetization inside a weak ferromagnet) generates a dc spin accumulation. For bulk systems this spin accumulation is in general given by a small fraction of $\hslash \omega$, where $\omega$ is the rotation or precession frequency. With the addition of a neighboring, field-free region, and allowing for the diffusion of spins across the interface, the spin accumulation is dramatically enhanced towards $\hslash \omega$ near the interface. The interface-enhanced spin accumulation obtained within our bulk-oriented model is surprisingly similar to predictions based on interface-scattering theory [A. Brataas et al., Phys. Rev. B 66, 060404(R) (2002)].

Figure 1

(a) Magnetic field vector diagram. An applied rf field $B_{xy}$ rotates with angular frequency $\omega$ in the $x\mathrm{\text{−}}y$ plane. A static magnetic field $B_z$ is applied perpendicular to $B_{xy}$ generating a field vector $\overrightarrow{B}$ that rotates around $B_z$ with cone angle $\theta$. (b) The hybrid system with an interface at $x=0$. The fields just described are applied in the pumped region I, $x<0$. $\overrightarrow{B}=0$ in region II, $x>0$.

Figure 2

The spin accumulation $f_z/\hslash \omega$ as a function of $x/\lambda$ away from the interface, for $\tau$ ranging in decades from 100 ps to 10 s. Both regions are unbounded. For (a), $B_{xy}=10\mathrm{mT}$ and $B_z=0$, for (b) $B_{xy}=1\mathrm{T}$ and $B_z=100\mathrm{T}$.

Figure 3

Spin accumulation at the interface for the symmetrically bounded system plotted as a function of the normalized bounding length $L/{\lambda }_{\omega }$, where ${\lambda }_{\omega }=\sqrt{2\pi D/\omega }\simeq 1.1\mu \mathrm{m}$, for $\tau$ ranging in decades from ${10}^{-10}$ (bottom curve) to 10 s (top curve).

Figure 4

The spin accumulation at the interface vs $\tau$ for specific bounding lengths indicated as $L_1,L_2$ (curves A through E). The open circles are based on Ref. 2 with $\eta ={10}^{-5}\frac{1}{\sqrt{\tau }}\tanh \sqrt{\frac{2\pi }{\omega \tau }}$ (see text for details).