Fermi–Dirac statistics
Fermi–Dirac statistics is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle.
It is named after Enrico Fermi and Paul Dirac, each of whom derived the distribution independently in 1926.
Fermi–Dirac statistics applies to identical and indistinguishable particles with half-integer spin (1/2, 3/2, etc.
For the case of negligible interaction between particles, the system can be described in terms of single-particle energy states.
Fermi–Dirac statistics is most commonly applied to electrons, a type of fermion with spin 1/2.
In classical physics, Maxwell–Boltzmann statistics is used to describe particles that are identical and treated as distinguishable.
Before the introduction of Fermi–Dirac statistics in 1926, understanding some aspects of electron behavior was difficult due to seemingly contradictory phenomena.
In other words, it was believed that each electron contributed to the specific heat an amount on the order of the Boltzmann constant kB.
Fermi–Dirac statistics was first published in 1926 by Enrico Fermi[1] and Paul Dirac.
[7] Fermi–Dirac statistics was applied in 1926 by Ralph Fowler to describe the collapse of a star to a white dwarf.
where kB is the Boltzmann constant, T is the absolute temperature, εi is the energy of the single-particle state i, and μ is the total chemical potential.
In the case of a spectral gap, such as for electrons in a semiconductor, the point of symmetry μ is typically called the Fermi level or—for electrons—the electrochemical potential, and will be located in the middle of the gap.
[15] Since the Fermi–Dirac distribution was derived using the Pauli exclusion principle, which allows at most one fermion to occupy each possible state, a result is that
[nb 2] The variance of the number of particles in state i can be calculated from the above expression for
:[21] so that The Fermi–Dirac distribution approaches the Maxwell–Boltzmann distribution in the limit of high temperature and low particle density, without the need for any ad hoc assumptions: The classical regime, where Maxwell–Boltzmann statistics can be used as an approximation to Fermi–Dirac statistics, is found by considering the situation that is far from the limit imposed by the Heisenberg uncertainty principle for a particle's position and momentum.
For example, in physics of semiconductor, when the density of states of conduction band is much higher than the doping concentration, the energy gap between conduction band and fermi level could be calculated using Maxwell-Boltzmann statistics.
Otherwise, if the doping concentration is not negligible compared to density of states of conduction band, the Fermi–Dirac distribution should be used instead for accurate calculation.
It can then be shown that the classical situation prevails when the concentration of particles corresponds to an average interparticle separation
For the case of conduction electrons in a typical metal at T = 300 K (i.e. approximately room temperature), the system is far from the classical regime because
Although the temperature of white dwarf is high (typically T = 10000 K on its surface[23]), its high electron concentration and the small mass of each electron precludes using a classical approximation, and again Fermi–Dirac statistics is required.
[8] The Fermi–Dirac distribution, which applies only to a quantum system of non-interacting fermions, is easily derived from the grand canonical ensemble.
In other words, each single-particle level is a separate, tiny grand canonical ensemble.
[24] The variance in particle number (due to thermal fluctuations) may also be derived (the particle number has a simple Bernoulli distribution): This quantity is important in transport phenomena such as the Mott relations for electrical conductivity and thermoelectric coefficient for an electron gas,[25] where the ability of an energy level to contribute to transport phenomena is proportional to
Consider a many-particle system composed of N identical fermions that have negligible mutual interaction and are in thermal equilibrium.
[29] A result can be achieved by directly analyzing the multiplicities of the system and using Lagrange multipliers.
Suppose each level contains gi distinct sublevels, all of which have the same energy, and which are distinguishable.
The Pauli exclusion principle states that only one fermion can occupy any such sublevel.
The number of ways of distributing ni indistinguishable particles among the gi sublevels of an energy level, with a maximum of one particle per sublevel, is given by the binomial coefficient, using its combinatorial interpretation: For example, distributing two particles in three sublevels will give population numbers of 110, 101, or 011 for a total of three ways which equals 3!/(2!1!).
The number of ways that a set of occupation numbers ni can be realized is the product of the ways that each individual energy level can be populated: Following the same procedure used in deriving the Maxwell–Boltzmann statistics, we wish to find the set of ni for which W is maximized, subject to the constraint that there be a fixed number of particles and a fixed energy.
We constrain our solution using Lagrange multipliers forming the function: Using Stirling's approximation for the factorials, taking the derivative with respect to ni, setting the result to zero, and solving for ni yields the Fermi–Dirac population numbers: By a process similar to that outlined in the Maxwell–Boltzmann statistics article, it can be shown thermodynamically that
