Fermi–Dirac statistics

The Fermi -Dirac statistics ( after the Italian physicist Enrico Fermi and the British physicist Paul Dirac ) is a concept of physical quantum statistics. It describes the macroscopic behavior of a system consisting of many identical particles of type fermion, and applies, for example, the electrons in metals and semiconductors provide for the electrical conductivity.

The starting points of the Fermi -Dirac statistics are:

• None of the states of the individual particles can have more than one particle to be occupied ( Pauli principle).
• Swaps are two particles, one obtains no new condition (the count in the statistical consideration would be an extra ), but the same as before (the principle of indistinguishability of identical particles).

The Fermi distribution indicates the probability for a given absolute temperature T, a state of the energy E of the particles is occupied by a W in an ideal Fermi gas. In statistical physics, the Fermi distribution of the Fermi -Dirac statistics for identical fermions is derived for the important special case of the interaction freedom.

For a full description of the Fermi -Dirac statistics see quantum statistics. For a simplified derivation see ideal Fermi gas.

Description

General formula

In a system of temperature is the Fermi distribution, which measures the probability of occupation:

With

• The energy for the state of a particle
• The chemical potential ( at, applies is being referred to as the Fermi level )
• The thermal energy the Boltzmann constant contains.

If the energy is calculated from the lowest single-particle state, however, does also Fermi energy. The occupation probability W is a state with the energy of the Fermi level is at all temperatures:

To calculate the dominant energy in the particle, e.g., for electrons at a metal Fermi distribution needs to be multiplied by the density of states:

At absolute zero temperature

At absolute zero temperature is the Fermi gas as a whole in its lowest possible energy state, ie the ground state of the many-body system. Since (for a sufficiently large number of particles ) not all particles can occupy the Einteilchengrundzustand according to the Pauli principle, particles must be in the excited single-particle states is at absolute zero temperature. Can be graphically describe it with the notion of a Fermi sea: each added fermion occupies the lowest energy state, which is not yet occupied by another fermion. The " filling level " is determined by the density of the positions to be occupied states and the number of particles to be housed.

Accordingly, the Fermi distribution has a sharp jump at the Fermi energy, which is therefore also called the Fermi level or Fermi level has for the temperature T = 0 K ( see figure).

• All states with E < EF are occupied, since the following applies: W ( E) = 1, that is, the probability of finding one of the fermions in such a state, is one.
• None of the states with E> EF is busy, because here applies: W ( E) = 0, that is, the probability of finding one of the fermions in such a state is zero.

The Fermi level at, therefore, is determined by the number and energy distribution of the states and the number of fermions, which are to be accommodated in these states. In the formula, only an energy difference appears. Is it the size of the Fermi energy alone, it is the energy difference between the highest occupied single-particle state at the lowest possible. For illustration, or for fast estimation of temperature-dependent effects, this size is often as a temperature value - the Fermi temperature - expressed:

At the Fermi temperature, the thermal energy would be equal to the Fermi energy. This term has nothing to do with the real temperature of the fermions, it only serves to characterize energy ratios.

At finite temperatures

In the temperature range is referred to the system as a degenerate Fermi gas, because it is largely determined by the exclusion principle, which states that can not be greater than one, the occupation number of a state. Starting from T = 0 (T = 0 K) are occupied with fermions when heated states above the Fermi energy EF. For many states remain the same empty below the Fermi energy and are referred to as holes. The Fermi distribution gives the probability of occupation in the equilibrium state the temperature T> 0 K.

In the field of the degenerate Fermi gas, ie at temperatures well below the Fermi temperature TF, the sharp Fermi edge in a region of width ~ 2kBT rounded ( "softened ", see figure). States with lower energies are still almost fully occupied (), which states at higher energies only very weak ().

Since there is still the same number of particles to be spread over the possible states, the Fermi energy can shift with temperature. Is the density in the region of the excited particles are smaller than the holes, the Fermi energy must rise, fall in the opposite case.

At very high temperatures

At very high energies and / or at very high temperatures () is the Fermi distribution gets closer by the classical Boltzmann distribution:

This always applies. The Fermi gas behaves like a classical gas, it is not degenerate. The Fermi energy is then well below the lowest level positions to be occupied.

Fermi distribution of metals

For the conduction electrons in a metal, the Fermi energy is a few electron volts, corresponding to a temperature of several Fermi 10,000 K. As a result, the thermal energy is much smaller than the typical width of the conduction band. Is a degenerate electron gas. The contribution of the electrons to the thermal capacity is therefore negligible, even at room temperature and can be taken into account interference theoretically. The temperature dependence of the Fermi energy, is very low ( MeV range ), and is often neglected.

Fermi distribution for semiconductors and insulators

For semiconductors and insulators, the Fermi level lies in the forbidden zone. In the area of the Fermi level, therefore, there are no states whose occupation can significantly depend on the temperature. This means that at a temperature T = 0 K, the valence band completely filled with electrons and the conduction band is empty and that it > 0 K are very few holes or excited electrons T. By introducing impurities with additional charge carriers ( donor or acceptor ) can be shifted the Fermi level downwards or upwards, which greatly increases the conductivity. In this case moves with the temperature, the Fermi level significantly. Therefore, for example, electronic circuits operate on the basis of semi-conductors (such as computer ) correctly only within a narrow temperature range.

Derivation of the Fermi -Dirac statistics of a minimum free energy

From the condition, that is in thermal equilibrium ( with a fixed and volume), the free energy assumes a minimum, the Fermi -Dirac statistics can be derived in nice way. For this we consider fermions - such as electrons - that are spread over several levels. The levels have energies and are each - fold degenerate (see figure), thus can absorb maximum electron ( Pauli principle). The number of electrons in the -th level is denoted by. For the macroscopic state of the system is irrelevant which of the electrons in the -th level and occupy the states in which they. The macro state is therefore determined entirely by the sequence of numbers.

For an arbitrary distribution of electrons in the levels of the following applies:

Equation (1) indicates the total number of the particles again, which is to be kept constant, while the individual can be varied to find the minimum of. Equation (2) shows the distribution for the present associated energy of the system as it is inserted into the formula. Equation ( 3) ( according to Boltzmann ) the entropy of the status of the system ( macro-state ), the thermodynamic probability of the relevant sequence of the occupation numbers, indicating, that the number of possible distributions ( micro-states ) of each electron of squares for all levels together.

To find the distribution in which, by varying the under Nebendingung the free energy is minimal, we use the method of Lagrange multipliers. The result is

This is the ( independent of ) Lagrange multiplier. For the calculation of the derivative of formula is used for the explicit need:

It is

The binomial coefficient, that is, the number of ways to select different among objects.

Using the simplified Stirling formula gives further

And thus

Overall, the equation ( 2)

With the definition of the occupation probability and the identification of the Lagrange factor with the chemical potential arises

, And from the Fermi -Dirac statistics

.

Observations

In solid state, the Fermi distribution can be observed very well, if the electronic population density of the conduction band as a function of energy measured. A particularly good example of the ideal Fermi gas is available in aluminum. Such studies can also be the resolution of a measuring apparatus determined by measuring the profile of the distribution at a given temperature and compares it with the formula for the Fermi distribution.

Further examples of the meaning, refer to the Fermi energy.