Effects of spin-orbit interaction on the envelope-function equations for semiconductor heterostructures.

Abstract

The effects of the spin-orbit interaction on the envelope-function equations for semiconductor heterostructures are investigated. The wave functions are expanded over a complete orthonormal set of spinors. The entire derivation is based solely on the Schr\"odinger equation as well as the completeness and orthonormality relations of the expansion bases. The envelope-function equations are found to be a set of integral equations, rather than a set of integrodifferential equations. It is verified that two aspects of the model including the spin-orbit interaction are the same as the spinless model. First, the exact envelope-function 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 semiconductor heterojunctions. However, in the vicinity of the heterojunction, the expressions for the coefficients of the localized envelope-function equations differ from the spinless model by a sinc [sinc(t)=sin(t)/t] term due to the spatial gradient in the spin-orbit-interaction operator. Finally, the exact envelope-function equations for the system, which is non-lattice-matched in the growth direction, are derived for the three-dimensional model with spin-orbit interaction.