Reflection and refraction of an electron spin at the junction between two two-dimensional regions with and without spin-orbit interaction

Abstract

We derive the reflection and refraction laws for an electron spin incident from a quasi-two-dimensional medium with no spin-orbit interaction on another with both Rashba and Dresselhaus spin-orbit interaction using only energy conservation. We obtain the well-known result that for an incident angle, there can be generally two different refraction angles for refraction into the two spin eigenstates in the refraction medium, resulting in two different 'spin refractive indices' and two critical angles for total internal reflection. We derive expressions for the spin refractive indices, which are not constant for a given medium but depend on the incident electron's energy. If the effective mass of an electron in the refraction medium is larger than that in the incidence medium, then we show that for some incident electron energies and potential barrier at the interface, the spin refractive index of the incidence medium can lie between the two spin refractive indices of the refraction medium, resulting in only one critical angle. In that case, if the incident angle exceeds that critical angle, then refraction can occur into only one spin eigenstate in the refraction medium. If the system is engineered to make this happen, then it will be possible to obtain a very high degree of spin-polarized injection into the refraction medium. The amplitudes of reflection of the incident spin into its own spin eigenstate and the orthogonal spin eigenstate (due to spin flip at the interface), as well as the refraction amplitudes into the two spin eigenstates in the refraction medium are derived for an incident electron (with arbitrary spin polarization and incident energy) as a function of the angle of incidence.