Carrier distribution functions

The distribution or probability density functions describe the probability that particles occupy the available energy levels in a given system. Of particular interest is the probability density function of electrons, called the Fermi function. The derivation of such probability density functions can be found in one of the many statistical thermodynamics references . However, given the importance of the Fermi distribution function, we will carefully examine an example as well as the characteristics of this function. It is also derived in section 2.5.5. Other distribution functions such as the impurity distribution functions, the Bose-Einstein distribution function and the Maxwell Boltzmann distribution are also provided.

Fermi-Dirac distribution function
The Fermi-Dirac distribution function, also called Fermi function, provides the probability of occupancy of energy levels by Fermions. Fermions are half-integer spin particles, which obey the Pauli exclusion principle. The Pauli exclusion principle postulates that only one Fermion can occupy a single quantum state. Therefore, as Fermions 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.

Electrons are Fermions. Therefore, the Fermi function provides the probability that an energy level at energy, E, in thermal equilibrium with a large system, is occupied by an electron. The system is characterized by its temperature, T, and its Fermi energy, EF. The Fermi function is given by:

This function is plotted in Figure 2.5.1 at 150, 300 and 600 K.

Figure 2.5.1 : The Fermi function at three different temperatures

The Fermi function has a value of one for energies, which are more than a few times kT below the Fermi energy. It 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.

Example
To better understand the origin of distribution functions, we now consider a specific system with equidistant energy levels at 0.5, 1.5, 2.5, 3.5, 4.5, 5.5, .... eV. Each energy level can contain two electrons. Since electrons are indistinguishable from each other, no more than two electrons (with opposite spin) can occupy a given energy level. This system contains 20 electrons.

The minimum energy of this system corresponds to the situation where all 20 electrons occupy the ten lowest energy levels without placing more than 2 in any given level. This situation occurs at T = 0 K and the total energy equals 100 eV.

Since we are interested in a situation where the temperature is not zero, we arbitrarily set the total energy at 106 eV, which is 6 eV more than the minimum possible energy of this system. This ensures that the thermal energy is not zero so that the system must be at a non-zero temperature.

There are 24 possible and different configurations, which satisfy these particular constraints. Eight of those configurations are shown in Figure 2.5.2, where the filled circles represent the electrons:

Figure 2.5.2 : Eight of the 24 possible configurations in which 20 electrons can be placed having a total energy of 106 eV.

We now apply the basic postulate of statistical thermodynamics, namely that all possible configurations are equally likely to occur. The expected configuration therefore equals the average occupancy of all possible configurations.

The average occupancy of each energy level taken over all (and equally probable) 24 configurations is compared in Figure 2.5.3 to the Fermi-Dirac distribution function. A best fit was obtained using a Fermi energy of 9.998 eV and kT = 1.447 eV or T = 16,800 K. The agreement is surprisingly good considering the small size of this system.

Figure 2.5.3 : Probability versus energy averaged over the 24 possible configurations (circles) fitted with a Fermi-Dirac function (solid line) using kT = 1.447 eV and EF = 9.998 eV.

Based on the construction of the distribution function in this example, one would expect the distribution function to be dependent on the density of states. This is the case for small systems. However, for large systems and for a single energy level in thermal equilibrium with a larger system, the distribution function no longer depends on the density of states. This is very fortunate, since it dramatically simplifies the carrier density calculations.

Impurity distribution functions
The distribution function of impurities, as introduced in section 2.6.4, differs from the Fermi-Dirac distribution function even though the particles involved are still Fermions. The difference is due to the fact that an ionized donor energy level still contains one electron with either spin. The donor energy level cannot be empty since this would leave a doubly positively charged atom, which would have an energy different from that of the singly ionized donor level. The distribution function for donors therefore differs from the Fermi function and is given by:

The distribution function for acceptors differs also because of the different possible ways to occupy the acceptor level. The neutral acceptor contains no electrons. The ionized acceptor contains one electron, which can have either spin, while the doubly negatively charged state is not allowed since this would require a different energy. This restriction would yield a factor of 2 in front of the exponential term. In addition, one finds that most commonly used semiconductors have a two-fold degenerate valence band, which causes this factor to increase to four, yielding: