1.4.1 Rashba Spin-orbit Coupling

The study on the spin-orbit effects arising from structural inversion asym-metry was pioneered by Emmanuel Rashba. You know, in the nearly-free electron model for semiconductors, the spin-orbit coupling is inversely pro-portional to the band gap energy Egwhose typical value is Eg～1eV. So spin-orbit coupling for electrons in solids is considerably stronger than that in free space.

Along with the modern development in material science, the two-dimensional electron gas (2DEG) can be engineered by various means such as metal-oxide-semiconductor field-effect transistor(MOSFET), high-electron-mobility-transistor(HEMTs)and rectangular quantum wells etc. In those systems, the 2DEG is confined within the interface of semiconduc-tor materials, which is illustrated in Fig.1.7. If a large potential gradient is applied perpendicular to the semiconductor hetero interface(saying-∇V along direction), a spin-orbit coupling in proportion to potential gradient is produced. . Thus the two-dimensional electron gas in x-y plane with Rashba spin-orbit coupling is described by the following Hamiltonian

Here the strength of Rashba spin-orbit coupling α is related to the mag-nitude of the gate voltage. The eigenvalues and eigenfunctions of the Hamiltonian (1.60) can be solved (left for readers as an exercise). The result in one dimension is simply

where kR= mα/ħ2. Clearly, Rashba term brings about a momentum-dependent spin splitting that is different from the conventional Zeeman effect; their differences are illustrated in Fig. 1.8. Note that the up ar-row and down arrow in (1.61) do not mean the conventional spin up and spin down that refer to the eigenstates of σz. Whereas, they refer to the eigenstates of σyif the one-dimensional system is along x-axis.