EFFECTS OF SPIN-ORBIT INTERACTION ON THE ENVELOPE-FUNCTION EQUATIONS FOR SEMICONDUCTOR HETEROSTRUCTURES

The effects of the spin-orbit interaction on the envelope-function equ ations for semiconductor heterostructures are investigated. The wave f unctions are expanded over a complete orthonormal set of spinors. The entire derivation is based solely on the Schrodinger equation as well as the completeness and orthonormality relations of the expansion base s. The envelope-function equations are found to be a set of integral e quations, rather than a set of integrodifferential equations. It is ve rified that two aspects of the model including the spin-orbit interact ion are the same as the spinless model. First, the exact envelope-func tion equations can be localized to be a set of differential equations if the envelope functions are slowly varying. Second, the localization effectively smooths the abruptness of the material transitions at sem iconductor heterojunctions. However, in the vicinity of the heterojunc tion, the expressions for the coefficients of the localized envelope-f unction equations differ from the spinless model by a sine [sinc(t)=si n(t)/t] term due to the spatial gradient in the spin-orbit-interaction operator. Finally, the exact envelope-function equations for the syst em, which is non-lattice-matched in the growth direction, are derived for the three-dimensional model with spin-orbit interaction.