Spin-Orbit Coupling in Solids

In solids, the spin-orbit coupling effect has shot to prominence recently in the context of spin-electronics and the attempts to build a spin-transistor. The spin-orbit coupling Eq. (II.3.6),

$${\color{col1}\hat{H}_{\rm SO} = \frac{e\hbar}{4m^2c^2}{\bf\sigma} [ {\bf E}({\bf r})\times {\bf p}]},$$ (3.13)

leads to a spin-splitting for electrons moving in solids (e.g., semiconductors) even in absence of any magnetic field. Symmetries of the crystal lattice then play a role (Dresselhaus effect), and in artificial heterostructures or quantum wells, an internal electric field ${\bf E}({\bf r})$ can give rise to a coupling to the electron spin. This latter case is called Rashba effect.

For a two-dimensional sheet of electrons in the $x$- $y$-plane (two-dimensional electron gas, DEG), the simplest case is a Hamiltonian

$$\hat{H}_{\rm SO} = -\frac{\alpha}{\hbar} \left[ {\bf p} \times {\bf\sigma}\right]_z,$$ (3.14)

where the index $_z$ denotes the $z$ component of the operator in the vector product ${\bf p} \times {\bf\sigma}$ and $\alpha$ is the Rashba parameter. In the case of the hydrogen atom, this factor was determined by the Coulomb potential. In semiconductor structures, it is determined by many factors such as the geometry.

The Rashba parameter $\alpha$ can be changed externally by, e.g., applying additional `back-gate' voltages to the structure. This change in $\alpha$ then induces a change of the spin-orbit coupling which eventually can be used to manipulate electron spins.