Numerically stable Hermitian secular equation for the envelope-function approximation for superlattices

A method is developed for implementing the coupled-band envelope-function-approximation (EFA) formalism for the calculation of the electronic structure of superlattices. The approach overcomes the difficulties in handling exponentially growing and decaying wave-function components, in particular, the so-called wing solutions, as is the case with existing secular equations. As importantly, the secular equation, which, in general, is general complex, is recast into a Hermitian form, which makes it easy to separate degenerate eigensolutions of the superlattice problem. Therefore, it is not necessary to first find a unitary transformation to eliminate the Kramers degeneracy of the starting k{·}p Hamiltonian. In the case of the simple Kronig-Penney model, the present formalism recasts the characteristic equation into a form that directly exhibits its parentage to the underlying quantum-well problem. The 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 technologically important semiconducting InAs/Inₓ{Ga}_{1{{-}}x}$Sb type-II superlattice. {} 1996 The American Physical Society.