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:
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:
where Ec 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:
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:
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).
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 Nc(E) is the density of states in the conduction band and f(E) is the Fermi function.
This general expression is illustrated with the figure below for a parabolic density of states function with Ec = 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, n(E) (red curve). The integral corresponds to crosshatched area under the red curve.
For a three-dimensional semiconductor this integral becomes:
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:
These equations were used to obtain the figure 2.5.2.
where Nc is the effective density of states in the conduction band. Similarly for holes one can approximate the hole density integral as:
where Nv is the effective density of states in the valence band.
for electrons and by: