Some Properties of Exponentially Damped Wave Functions

Abstract

We know from Bloch’s theorem that the wave function ψ of an electron in a semiconducting or insulating crystal can be written

$$\psi ={{e}^{ikx}}u\left( x \right),$$

(1)

where k is the wave vector of the state and u(x) is a periodic function with the lattice period of the crystal. Over reasonable distances the u(x) part of the function averages out, and we can view the propagating part e ^ikx as a quasi-particle behaving in much the same way as a particle in free space. The dynamics of this particle are given by the dispersion relation between the wave vector k and the energy E. For a semiconductor or insulator with a forbidden gap the dispersion relation is something like the one shown in Fig. 1. Note that we have plotted E versus k ² rather than the more conventional E versus k. The reason is as follows: It can be shown rather generally that near one of the band edges (E _c or E _υ ) the energy is quadratic in k,and is usually written E=ћ ² k ²/2m* by analogy with the free electron. Here m* is the effective mass of the quasi-particle. This form is shown by the straight line in Fig. 1. For E>E _c, and E<E _υ , k ² is positive; therefore k is real, and propagating solutions result. For energies in the forbidden gap E _c <E<E _υ , k ² is negative, k is imaginary, and exponentially damped solutions result. It is these damped solutions with which we are concerned in tunneling problems.