# 2.3 Density of states calculation

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:
(f41)
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:
(f42)
Elimination of k using the E(k) relation above then yields the desired density of states functions, namely:
(f43)
for a three-dimensional semiconductor,
(f44)
for a two-dimensional semiconductor such as a quantum well in which particles are confined to a plane, and
(f45)
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

Fig. 2.3.1 Density of states per unit volume and energy for a 3-D semiconductor (blue curve) , a 10 nm quantum well with infinite barriers (red curve) and a 10 nm by 10 nm quantum wire with infinite barriers (green curve). m*/m0 = 0.8.
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

Table 2.3.1 Degeneracy (not including spin) of the lowest 10 energy levels in a quantum well, a quantum wire with square crosssection and a quantum cube with infinite barriers. The energy E0 equals the lowest energy in a quantum well which has the same size

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

dens3d.gif
Fig. 2.3.2 Number of states within a range DE = 20 E0 as a function of the normalized energy E/E0. (E0 is the lowest energy in an 1-dimensional quantum well).
The first point corresponds to the number of states

dens3d1.gif
Fig. 2.3.2 Number of states with energy less than or equal to E as a function of E0 (E0 is the lowest energy in an 1-dimensional quantum well).

2.2 ¬ ­ ® 2.4

© Bart J. Van Zeghbroeck, 1996, 1997