In this Section
The distribution or probability density functions describe the probability with which one can expect particles to occupy the available energy levels in a given system. While the actual derivation belongs in a course on statistical thermodynamics it is of interest to understand the initial assumptions of such derivations and therefore also the applicability of the results.
The derivation starts from the basic notion that any possible distribution of particles over the available energy levels has the same probability as any other possible distribution, which can be distinguished from the first one.
In addition, one takes into account the fact that the total number of particles as well as the total energy of the system has a specific value.
Third, one must acknowledge the different behavior of different particles. Only one Fermion can occupy a given energy level (as described by a unique set of quantum numbers including spin). The number of bosons occupying the same energy levels is unlimited. Fermions and Bosons all "look alike" i.e. they are indistinguishable. Maxwellian particles can be distinguished from each other.
The derivation then yields the most probable distribution of particles by using the Lagrange method of indeterminate constants. One of the Lagrange constants, namely the one associated with the average energy per particle in the distribution, turns out to be a more meaningful physical variable than the total energy. This variable is called the Fermi energy, EF.
An essential assumption in the derivation is that one is dealing with a very large number of particles. This assumption enables to approximate the factorial terms using the Stirling approximation.
The resulting distributions do have some peculiar characteristics, which are hard to explain. First of all the fact that a probability of occupancy can be obtained independent of whether a particular energy level exists or not. It would seem more acceptable that the distribution function does depend on the density of available states, since it determines where particles can be in the first place.
The fact that the distribution function does not depend on the density of states is due to the assumption that a particular energy level is in thermal equilibrium with a large number of other particles. The nature of these particles does not need to be described further as long as their number is indeed very large. The independence of the density of states is very fortunate since it provides a single distribution function for a wide range of systems.
A plot of the three distribution functions, the Fermi-Dirac distribution, the Maxwell-Boltzmann distribution and the Bose-Einstein distribution is shown in the figure below, where the Fermi energy was set equal to zero.
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Fig. 2.4.1 Occupancy probability versus energy of the Fermi-Dirac (red curve), the Bose-Einstein (green curve) and the Maxwell-Boltzman (blue curve) distribution.
All three distribution functions are almost equal for large energies (more than a few kT beyond the Fermi energy). The Fermi-Dirac distribution reaches a maximum of 1 for energies which are a few kT below the Fermi energy, while the Bose-Einstein distribution diverges at the Fermi energy and has no validity for energies below the Fermi energy.
To better understand the general derivation without going through it, we now consider a system with equidistant energy levels at 0.5, 1.5, 2.5, 3.5, 4.5, 5.5, .... eV, which each can contain two electrons. The electrons are Fermions so that they are indistinguishable from each other and no more than two electrons (with opposite spin) can occupy a given energy level. This system contains 20 electrons and we arbitrarily set the total energy at 106 eV, which is 6 eV more than the minimum possible energy of this system. There are 24 possible and different configurations, which satisfy these particular constraints. Six of those configurations are shown in the figure below, where the red dots represent the electrons:
Fig. 2.4.2 Six of the 24 possible configurations in which 20 electrons can be placed having an energy of 106 eV.
A complete list of the 24 configurations is shown in the table below:
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Table 2.4.1 All 24 possible configurations in which 20 electrons can be placed having an energy of 106 eV.
The average occupancy of each energy level as taken over all (and equally probable) 24 configurations is compared in the figure below to the expected 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.
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Fig. 2.4.3 Probability versus energy averaged over the 24 possible configurations of the example (red squares) fitted with a Fermi-Dirac function (green curve) using kT = 1.447 eV and EF= 9.998 eV.
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.
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Fig. 2.4.4 Fermi function at 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. 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.
The distribution function of impurities differs from the Fermi-Dirac distribution function although the particles involved are Fermions. The difference is due to the fact that a filled donor energy level contains only one electron which can have either spin (spin up or spin down) , while having two electrons with opposite spin occupy this one level is not allowed since this would leave a negatively charge atom which would have a different energy as the donor energy. This yields a modified distribution function for donors as given by:
The main difference is the factor 1/2 in front of the exponential term.
The distribution function for acceptors differs also because of the different possible ways to occupy the acceptor level. The neutral acceptor contains two electrons with opposite spin, the ionized acceptor still contains one electron which can have either spin, while the doubly positive 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 4 yielding:
In order to understand the carrier distribution functions one must be familiar with a variety of thermodynamic concepts. These include thermal equilibrium, the difference between the total energy and heat, work and particle energy and the meaning of the Fermi energy. These and other related topics are discussed in the section on semiconductor thermodynamics. An ideal electron gas is discussed in more detail as an example.