Spin injection from a half-metal at finite temperatures

Spin injection from a half-metallic electrode in the presence of thermal spin disorder is analyzed using a combination of random matrix theory, spin-diffusion theory, and explicit simulations for the tight-binding $s$-$d$ model. It is shown that efficient spin injection from a half-metal is possible as long as the effective resistance of the normal metal does not exceed a characteristic value, which does not depend on the resistance of the half-metallic electrode but, rather, is controlled by spin-flip scattering at the interface. This condition can be formulated as $\alpha \lesssim l/l_{sf}^N{T}_c^{-1}$, where $\alpha$ is the relative deviation of the magnetization from saturation, $l$ and $l_{sf}^N$ are the mean-free path and the spin-diffusion length in the nonmagnetic channel, and $T_c$ is the transparency of the tunnel barrier at the interface (if present). The general conclusions are confirmed by tight-binding $s$-$d$ model calculations. A rough estimate suggests that efficient spin injection from true half-metallic ferromagnets into silicon or copper may be possible at room temperature across a transparent interface.

Figure 1

Equivalent resistor circuit for spin injection from a half-metal.

Figure 2

Schematic of the band alignment for the spin-injection device without spin disorder. Darker and lighter bands correspond to majority and minority spins. The minority-spin band in the half-metallic region is shifted up beyond the energy range shown here. The dashed horizontal line shows the Fermi level, and the vertical black bars show the amplitudes of random disorder.

Figure 3

(a) Conductance polarization as a function of the inverse resistance-area product for the reduced magnetization $m=0.8$. Inset: The same quantity as a function of the thickness of the half-metal at $m=0.9$. (b) Effective resistance ${\tilde{r}}_F$ as a function of ${\alpha }^{-1}$, where $\alpha =(1-m)/2$.

Figure 4

Effective resistance ${\tilde{r}}_F$ as a function of $T_c^{-1}$ for fixed magnetization $m=0.6$.