Spin-orbit interaction in 2D Dirac-like and Kane semiconductors

The single particle equations describing motion of carriers in external potential in 2D Dirac-like and Kane intrinsic semiconductors are obtained within second quantization method. The terms renormalizing external potential in these equations, referred to as spin-orbit (SO) terms, are compared with their classical counterpart. The well-known expression for SO obtained in relativistic Dirac theory arises in considered approach in the second order in $\gamma k\mathrm{/}{E}_g$ ($\gamma$ - characteristic velocity, $E_g$ - energy gap) parameter if electron-hole pair production terms are neglected. It is shown that in Kane problem the modifying terms are of standard SO functional Dirac-like form only for electrons in the case of positive'' energy gap and for light holes in semiconductors with negative’’ energy gap. The general expression for renormalizing terms has in all cases non-local character. The arising of correction terms to single particle potentials which do not depend on band parameters is demonstrated for 2D gapless Dirac problem (graphene) and for Kane model. The origin of such ``topological’’ terms is attributed to the presence of degenerate bands in considered problems.