Chapter 4 - Electron states

Abstract

This chapter focuses on the nearly free electron model. The nearly free electron method deals with electrons that spend most of their time outside the atomic cores where they propagate as plane waves. This is on the opposite side of the spectrum of electrons closely bound with their atoms that spend most of their time inside atoms. Quantitatively, the dispersion relation for nearly free electrons can be investigated on the basis of the simplified two-plane waves model. First, the effective mass near k =0 is the free electron mass m. Secondly, at the zone boundary one has the energy gap, which is equal to 2UG, where UG is the Fourier transform of the lattice potential. Thirdly, the slope of the dispersion relation ɛk, proportional to the wave velocity is zero at the zone boundary where the incoming wave function interfering with the reflected wave function forms the standing wave. One more difference between the electron wave in a free space and a nearly free electron wave in a crystal is the quantization of the possible values of vector k in the crystal. The wave function in a crystal has to be periodic.