Carrier densities

Now that we have discussed the density of states and the distribution functions, we have all the necessary tools to calculate the carrier density in a semiconductor.

General discussion
The density of electrons in a semiconductor is related to the density of available states and the probability that each of these states is occupied. The density of occupied states per unit volume and energy, n(E), ), is simply the product of the density of states in the conduction band, gc(E) and the Fermi-Dirac probability function, f(E), (also called the Fermi function):

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, p(E), equals:

Where gv(E) is the density of states in the valence band. The density of carriers is then obtained by integrating the density of carriers per unit energy over all possible energies within a band. A 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, an approximate solution useful when describing degenerate semiconductors.

The density of states in a semiconductor was obtained by solving the Schrödinger equation for the particles in the semiconductor. Rather than using the actual and very complex potential in the semiconductor, we use the simple particle-in-a box model, where one assumes that the particle is free to move within the material.

For an electron which behaves as a free particle with effective mass, m*, yielding:

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:

Calculation of the Fermi integral
The carrier density in a semiconductor, is obtained by integrating the product of the density of states and the probability density function over all possible states. For electrons in the conduction band the integral is taken from the bottom of the conduction band, labeled, Ec, to the top of the conduction band:

Where gc(E) is the density of states in the conduction band and f(E) is the Fermi function.

This general expression is illustrated with Figure 2.6.1 for a parabolic density of states function with Ec = 0. The figure shows the density of states function, gc(E), the Fermi function, f(E), as well as the product of both, which is the density of electrons per unit volume and per unit energy, n(E). The integral corresponds to the crosshatched area.

Figure 2.6.1 : The carrier density integral. Shown are the density of states, gc(E), the density per unit energy, n(E), and the probability of occupancy, f(E). The
carrier density, no, equals the crosshatched area.

The actual location of the top of the conduction band does not need to be known as the Fermi function goes to zero at higher energies. The upper limit can therefore be replaced by infinity. We also relabeled the carrier density as no to indicate that the carrier density is the carrier density in thermal equilibrium.