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, 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
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© Bart J. Van Zeghbroeck, 1996, 1997