Why the effective-mass approximation works so well for nano-structures

Abstract

The reason why the effective-mass approximation, derived using wave functions of infinite periodic systems, works so well with nanoscopic structures, has been an enigma and a challenge for theorists. To explain this issue, we first show that the essential “only-one-band” and “band-edge” assumptions that are behind the standard derivation of the effective mass approximation are better justified for nano-structures. We show then that the effective-mass approximation can also be derived using, instead of Bloch-type wave functions, the eigenfunctions and eigenvalues obtained in the theory of finite periodic systems, where the finiteness of the number of primitive cells in nanoscopic layers is a prerequisite and a crucial condition. We also show, with specific calculations of the optical response, that the rapidly varying eigenfunctions of the one-band wave functions , can be safely dropped out for the calculation of inter-band transition matrix elements.