Table of Contents - Glossary - Study Aids - ¬ ®

In this Section

- Introduction
- 3-D Density of states
- Fermi function
- General expression
- Approximation for non-degenerate semiconductors
- Approximation for degenerate semiconductors

- (f46)

Since holes correspond to empty states in the valence band, the probability of having a hole equals the probability that a particular state is not filled, so that the hole density per unit energy equals:

- (f47)

The density of carriers is then obtained by integrating the density of carriers per unit energy over all possible energies within a band. The general expression is derived as well as an approximate analytic solution which is valid for non-degenerate semiconductors. In addition we also present the Joyce-Dixon approximation.

For an electron which behaves as a free particle with effective mass,
*m ^{*}*, the
density of states in three dimensions is given by:

- (f24a)

where *E _{c}* is the bottom of the conduction band below which the density of states is zero. The
density of states for holes in the valence band is given by:

- (f24b)

The Fermi-Dirac probability density function provides the probability
that an energy level is occupied by a fermion which is in thermal
equilibrium with a large reservoir. Fermions are by definition
particles with half-integer spin (1/2, 3/2, 5/2 ...). A unique characteristic
of fermions is that they obey
the Pauli exclusion principle which states that only one
fermion can occupy a state which is defined by its set of
quantum numbers *n,k,l* and *s*. The definition of
fermions could therefore
also be particles which obey the Pauli exclusion principle. All
such particles also happen to have a half-integer spin.

Electrons as well as holes have a spin 1/2 and obey the Pauli exclusion
principle. As these
particles are added to an energy band, they will fill the available
states in an energy band
just like water fills a bucket. The states with the lowest energy
are filled first, followed by the
next higher ones. At absolute zero temperature (*T* = 0 K),
the energy levels are
all filled up to a maximum energy which we call the Fermi level. No states
above the Fermi level are filled. At higher temperature one finds that the
transition between completely filled states and completely empty states is
gradual rather than abrupt. The Fermi function which describes this
behavior, is given by:

- (f18)

This function is plotted in the figure below for an ambient temperature of 150 K (red curve), 300 K (blue curve) and 600 K (black curve).

fermi.tcl - fermi.gif

The Fermi function has a value of one for energies which are more than
a few times *kT* below the Fermi energy, equals 1/2 if the energy equals
the Fermi energy and decreases exponentially for energies which are a few
times *kT* larger than the Fermi energy. While at
*T* = 0 K the Fermi function
equals a step function, the transition is more gradual at
finite temperatures and more
so at higher temperatures.

A more detailed description of the Fermi function and other distribution functions can be found in the section on distribution functions

Where *N _{c}(E)* is the density of states in
the conduction band and

This general expression is illustrated with the figure below for a parabolic
density of states function with *E _{c}* = 0. The figure shows the density
of states function (green curve), the Fermi function (blue curve) as well as
the product of both which is the density of electrons per unit volume and
per unit energy,

density.tcl - density.gif

- (f53)

For a three-dimensional semiconductor this integral becomes:

- (f54)

While this integral can not be solved analytically we can either obtain a numeric solution or an approximate analytical solution. Similarly for holes one obtains:

- (f56)

and

- (f57)

These equations were used to obtain the figure 2.5.2.

- (f55)

with

- (f5)

where *N _{c}* is the effective density of states
in the

- (f58)

with

- (f6)

where *N _{v}* is the

distrib1.tcl - carrier.gif

- (f51)

for electrons and by:

- (f52)

for holes.

© Bart J. Van Zeghbroeck, 1996, 1997