NUMERICALLY STABLE HERMITIAN SECULAR EQUATION FOR THE ENVELOPE-FUNCTION APPROXIMATION FOR SUPERLATTICES

A method is developed for implementing the coupled-hand envelope-funct ion-approximation (EFA) formalism for the calculation of the electroni c structure of superlattices. The approach overcomes the difficulties in handling exponentially growing and decaying wave-function component s, in particular, the so-called wing solutions, as is the case with ex isting secular equations. As importantly, the secular equation, which, in general, is general complex, is recast into a Hermitian form, whic h makes it easy to separate degenerate eigensolutions of the superlatt ice problem. Therefore, it is not necessary to first find a unitary tr ansformation to eliminate the Kramers degeneracy of the starting k . p Hamiltonian. In the case of the simple Kronig-Penney model, the prese nt formalism recasts the characteristic equation into a form that dire ctly exhibits its parentage to the underlying quantum-well-problem. Th e present method can be used in conjunction with Burt's EFA formalism in the form of a coupled differential equation with piecewise-constant coefficients. The method is demonstrated on the example of the techno logically important semiconducting InAs/InxGal-xSb type-II superlattic e.