**2.3 Density of states calculation**

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

- Introduction
- 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, *E*_{min} 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*^{*}/*m*_{0} = 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
E*_{0} 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
D**E* =
20 *E*_{0} as a function of the
normalized energy *E*/*E*_{0}. (*E*_{0} 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 *E*_{0} (*E*_{0} is the
lowest energy in an 1-dimensional quantum well).

2.2
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© Bart J. Van Zeghbroeck, 1996, 1997