Single Band Effective Mass Equation and Envolvent Functions

Abstract

The single-band effective mass Schrödinger equation to calculate the envelope functions is described and its grounds are shown. These envelope functions are used to multiply periodic part of the Bloch functions to obtain approximate eigenfunctions of the Hamiltonian of a nanostructured semiconductor. The Bloch functions, which are the product of a periodic function and a plane wave, constitute the exact solution of a homogeneous semiconductor; they are taken as a basis to represent the nanostructured Hamiltonian. The conditions that make possible the use of this single band effective mass Schrödinger equation are explained. The method is applied to the calculation of the energy spectrum of quantum dots for wavefunctions belonging to the conduction band. A box-shaped model of the quantum dots is adopted for this task. The results show the existence of energy levels detached from this band as well as eigenfunctions bound totally or partially around the quantum dot. The absorption coefficients of photons in the nanostructured semiconductor, which is our ultimate goal, are calculated. The case of spherical quantum dots is also considered.Keywords Solar cells Á Quantum calculations Á Quantum dots Á Energy spectrum Á Absorption coefficientsThe easiest way to approach to the behavior of electrons in semiconductors is the use of effective mass equations. This is presented in this chapter for the specific cases of nanostructured semiconductors. The approximate treatment presented here applies to the case that the nanostructure has a mesoscopic size, much larger than the size of the semiconductor microscopic structure, that is, where the individual atoms are located.We shall see that the effective mass equation is only of application when we are dealing with electrons in a single semiconductor band, namely the conduction band. Therefore, the bipolar behavior, crucial for the understanding of most semiconductor devices, in which both electrons and holes enter into the game-or in other words, when electrons in the CB and in the VB are to be considered-cannot be explained with the resources of the simple effective mass treatment studied in this chapter. They will be properly studied in the following chapter. Nevertheless, the