2.3 Density of states calculation


Table of Contents - Glossary - Study Aids - ¬ ­ ®
In this Section

  1. Introduction
  2. Calculation of the density of states in 1, 2, and 3 dimensions

2.3.1 Introduction

The goal of this section is to find the density of electron states in a semiconductor as a function of the electon energy. The result is a density per unit volume and energy in the case of a three-dimensional semiconductor. We will find this density by first calculating the density of states as a function of the wave vector k. Keep in mind that these states are simply solutions to the Schödinger equation applied to our material. The solutions are waves with a specific wave vector k and corresponding energy E(k). The E(k) relation requires to solve the equation, while the possible values of k are determined by the boundary conditions.


2.3.1 Calculation of the density of states in 1, 2 and 3 dimensions

We will here postulate that the density of electrons in k is constant and equals the physical length of the sample divided by 2p and that for each dimension. The number of states between k and k + dk in 3, 2 and 1 dimension(s) then equals: we now assume that the electrons in a semiconductor are close to a band minimum, Emin and can be described as free particles with a constant effective mass, or: Elimination of k using the E(k) relation above then yields the desired density of states functions, namely: for a three-dimensional semiconductor, for a two-dimensional semiconductor such as a quantum well in which particles are confined to a plane, and for a one-dimensional semiconductor such as a quantum wire in which particles are confined along a line.

An example of the density of states in 3, 2 and 1 dimension is shown in the figure below:


states.xls - states.gif

The above figure illustrates the added complexity of the quantum well and quantum wire: Even though the density in two dimensions is constant, the density of states for a quantum well is a step function with steps occuring at the energy of each quantized level. The case for the quantum wire is further complicated by the degeneracy of the energy levels: for instance a two-fold degeneracy increases the density of states associated with that energy level by a factor of two. A list of the degeneracy (not including spin) for the 10 lowest energies in a quantum well, a quantum wire and a quantum box, all with infinite barriers, is provided in the table below:


qwdegen.gif


Comparison of the number of discreet states in a 3-dimensional cube with the analytic expressions derived above.

dens3d.gif
The first point corresponds to the number of states

dens3d1.gif

2.2 ¬ ­ ® 2.4

© Bart J. Van Zeghbroeck, 1996, 1997