Spin Polarization of Electrons by Nonmagnetic Heterostructures: The Basics of Spin Optics

We propose to use the lateral interface between two regions with different strengths of the spin-orbit interaction(s) to spin polarize the electrons in gated two-dimensional semiconductor heterostructures. For a beam with a nonzero angle of incidence, the transmitted electrons will split into two spin polarization components propagating at different angles. We analyze the refraction at such an interface and outline the basic schemes for filtration and control of the electron spin.

Figure 1

(a) The refraction of electrons at the interface between the regions with ($SO$) and without ($N$) spin-orbit interaction. The refracted beam split into two beams. (b) An angle of refraction for the two polarization modes as a function of an angle of incidence. ${\phi }^c$ is an angle for total internal reflection for $+$ mode. ${\theta }^c$ determines the aperture for $-$ mode. Other angles are related to the spin filtration and are explained in the text. We use $\alpha =0.1$ and $B(x)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$.

Figure 2

A sharp $N\mathrm{\text{−}}SO$ interface; $\alpha =0.1$. The intensities per unit outgoing angle of the electrons transmitted without change of their chirality $\sim (d{\theta }^{+}/d\phi {)}^{-1}|t_{++}{|}^2$ and $\sim (d{\theta }^{-}/d\phi {)}^{-1}|t_{--}{|}^2$ as a function of an angle of incidence. The dashed line indicates the level of spin polarization, $|r_{--}/r_{++}{|}^2$, of the reflected electrons for an unpolarized incident beam.

Figure 3

A spin polarizing junction. Two $SO$ regions are separated by a stripe with a reduced strength of Bychkov-Rashba term. The stripe leads to the total reflection of one of the spin components. The kinematical construction is explained in the text.

Figure 4

A spin transistor. A spin current at terminals A and B is controlled by modulation of the magnitude of $\alpha$.