Numerical calculation of the Wannier Functions GaAs/Al(0.25)Ga(0.75)As superlattice structure

Abstract

This paper presents the numerical calculation of the Wannier functions (a sub n (Z)) of the GaAs/Al(0.25)Ga(0.75)As superlattice structure. The Wannier functions are linear combinations of the Bloch functions (Psi sub nk (Z)) that can be viewed as a convenient mathematical instrument to get around the lack of orthogonality of the tight binding formulation and are useful tool when the position of an electron has physical importance. However, except for finding a variational principal for the energy levels of one dimensional crystals in terms of the Wannier functions and simple cubic lattice calculations for instructional reasons, models have not been calculated or plotted showing the theory can be expanded to more complex superlattice structures. We develop an algorithm to numerically calculate and plot the Wannier functions of the GaAs/Al(0.25)Ga(0.75)As superlattice structure and hence prove that these functions can be calculated even for complex structures. By using the plots of our numerical modelling, we displayed the peculiar properties of the Wannier functions: a. Wannier functions are real. b. They fall off exponentially. c. The are either symmetric or anti-symmetric about z=0,