Coupled spinors orthonormalization criterion in multiband systems

Abstract

Some fundamental physical quantities are determined by solving the eigenvalue problem that comes from a system of N coupled second order linear differential equations. An uncommon scenario evolves from the second order derivatives that appear in most multiband Hamiltonians, which leads to wave function spaces with non orthogonal axes. This notorious property has often been ignored by many authors. In this paper we discuss a possible criterion for the orthonormalization of eigenspinors (N$\times$1) derived from the eigenvalue quadratic problem associated to the differential equation system. Such eigenspinors are taken as the basis on which the propagating wave modes system is built. When the norm of the new space is reformulated, the non-standard character of the weighted internal product comes to the forefront. This scheme has been successfully applied to the study of hole tunneling as it is described by the (4$\times$4) Kohn Lüttinger model.