Fermi-Dirac statistics

The Fermi-Dirac statistics (after the Italian physicist Enrico Fermi (1901–1954) and the British physicist Paul Dirac (1902–1984)) is a term used in physical quantum statistics . It describes the macroscopic behavior of a system that consists of many identical particles of the Fermion type . B. for the electrons that ensure electrical conductivity in metals and semiconductors.

The starting points of the Fermi-Dirac statistics are:

None of the states of the individual particles can be occupied by more than one particle ( Pauli principle ).
If you swap two particles with each other, you don't get a new state (which would have to be counted separately in the statistical analysis), but the same as before (principle of indistinguishability of the same particles ).

The Fermi distribution indicates the probability with which a state of energy is occupied by one of the particles in an ideal Fermigas at a given absolute temperature . In statistical physics, the Fermi distribution is derived from the Fermi-Dirac statistics for fermions of the same type for the important special case of freedom from interaction . ${\ displaystyle W}$ ${\ displaystyle T}$ ${\ displaystyle E}$

For a full description of Fermi-Dirac statistics, see quantum statistics . For a simplified derivation see ideal Fermigas .

description

Fermi distribution for different temperatures,
increasing rounding with increasing temperature
(red line: T = 0 K)

General formula

In a system of temperature , the Fermi distribution , which measures the occupation probability, is: ${\ displaystyle T \! \,}$ ${\ displaystyle W (E)}$

{\ displaystyle W (E) = {\ frac {1} {\ exp {\ left ({\ frac {E- \ mu} {k _ {\ mathrm {B}} T}} \ right)} + 1}} }

With

the energy for the state of a particle,

{\ displaystyle E}

the chemical potential ( where applies , where is referred to as the Fermi level ), ${\ displaystyle \ mu}$ ${\ displaystyle T = 0 \, \ mathrm {K}}$ ${\ displaystyle \ mu = E_ {F}}$ ${\ displaystyle E _ {\ rm {F}}}$

the thermal energy , where is the Boltzmann constant . ${\ displaystyle k _ {\ mathrm {B}} T}$ ${\ displaystyle k _ {\ mathrm {B}} = 8 {,} 617 \; 3303 \; (50) \ cdot 10 ^ {- 5} \, \ mathrm {eV} / \ mathrm {K}}$

If the energy is calculated from the lowest possible single-particle state, it is also called Fermi energy . The occupation probability for a state with the energy of the Fermi level is at all temperatures: ${\ displaystyle E \,}$ ${\ displaystyle E _ {\ rm {F}} \,}$ ${\ displaystyle W}$ ${\ displaystyle E = E _ {\ rm {F}} \! \,}$

{\ displaystyle W (E = E _ {\ rm {F}}) = {\ frac {1} {e ^ {0} +1}} = {\ frac {1} {2}} \.}

To calculate the particle density prevailing in the energy , e.g. B. for electrons in a metal , the Fermi distribution has to be multiplied by the density of states : ${\ displaystyle E \,}$ ${\ displaystyle \ langle n (E) \ rangle}$ ${\ displaystyle D (E) \! \,}$

{\ displaystyle \ langle n (E) \ rangle = W (E) \ cdot D (E) \.}

At absolute zero temperature

At the absolute zero point of temperature , the Fermi gas as a whole is in its energetically lowest possible state, i.e. in the basic state of the many-particle system . Since (with a sufficiently large number of particles), according to the Pauli principle, not all particles can occupy the single-particle ground state, particles must also be in excited single-particle states at absolute temperature zero . This can be clearly described with the idea of a Fermi lake : each added fermion occupies the lowest possible energy state , which is not yet occupied by another fermion. The “filling level” is determined from the density of the occupable states and the number of particles to be accommodated. ${\ displaystyle T = 0 \, \ mathrm {K}}$ ${\ displaystyle T = 0 \, \ mathrm {K}}$

Accordingly, the Fermi distribution for the temperature has a sharp jump in the Fermi energy , which is therefore also called the Fermi edge or Fermi limit (see figure). ${\ displaystyle T = 0 \, \ mathrm {K}}$ ${\ displaystyle E _ {\ mathrm {F}} = \ mu \! \,}$

All states with are occupied, since the following applies here:, d. H. the probability of encountering one of the fermions in such a state is one. ${\ displaystyle E <E _ {\ rm {F}}}$ ${\ displaystyle W (E) = 1}$
None of the states with is occupied, since the following applies here:, d. H. the probability of encountering one of the fermions in such a state is zero. ${\ displaystyle E> E _ {\ rm {F}}}$ ${\ displaystyle W (E) = 0}$

The Fermi level at is therefore determined by the number and energetic distribution of the states and the number of fermions that are to be accommodated in these states. Only one energy difference appears in the formula. If one gives the size of the Fermi energy alone, it is the energy difference between the highest occupied and the lowest possible single-particle state. To illustrate or to quickly estimate temperature-dependent effects, this quantity is often expressed as a temperature value - the Fermi temperature : ${\ displaystyle T = 0 \, \ mathrm {K}}$

{\ displaystyle T _ {\ mathrm {F}} = {\ frac {E _ {\ mathrm {F}}} {k _ {\ mathrm {B}}}} \! \,}

.

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 relationships. ${\ displaystyle k _ {\ mathrm {B}} T \! \,}$

At finite temperatures

The Fermi distribution indicates the probability of occupation in equilibrium with temperature . Starting from , states above the Fermi energy are filled with fermions when heated . Instead, the same number of states below the Fermi energy remain empty and are called holes . ${\ displaystyle T> 0 \, \ mathrm {K}}$ ${\ displaystyle T = 0 \, \ mathrm {K}}$ ${\ displaystyle E _ {\ mathrm {F}} (T = 0 \, \ mathrm {K})}$

The sharp Fermi edge is rounded off in a symmetrical interval of the total width ("softened", see fig.). States with lower energies are still almost fully occupied ( ), the states with higher energies are only very weak ( ). ${\ displaystyle E _ {\ mathrm {F}}}$ ${\ displaystyle \ sim 4k _ {\ mathrm {B}} T}$ ${\ displaystyle W \ lessapprox 1}$ ${\ displaystyle 0 <W \ ll 1}$

As still the same number of particles on the possible states with the density of states is to be distributed, the Fermi energy can shift with temperature: If the state density in the region of the excited particles are smaller than the holes, the Fermi energy increases in the opposite case it sinks. ${\ displaystyle D (E) \! \,}$

In the temperature range , the system is called a degenerate Fermi gas , because the occupation of the states is largely determined by the Pauli principle (exclusion principle ). This means that all states have the same probability (of almost one) to be occupied; this concerns a large energy range compared to the softening interval. ${\ displaystyle T \ ll T _ {\ mathrm {F}} \,}$ ${\ displaystyle E <E _ {\ mathrm {F}}}$

For energies of at least a few above , i.e. for , the Fermi distribution can be approximated by the classic Boltzmann distribution : ${\ displaystyle E}$ ${\ displaystyle k _ {\ mathrm {B}} T}$ ${\ displaystyle E _ {\ mathrm {F}}}$ ${\ displaystyle E-E _ {\ mathrm {F}} \ gg k _ {\ mathrm {B}} T}$

{\ displaystyle W (E) \ propto \ exp {\ left (- {\ frac {E-E _ {\ mathrm {F}}} {k _ {\ mathrm {B}} T}} \ right)}}

.

At very high temperatures

"Very high temperatures" are those well above the Fermi temperature, ie . Because this makes the softening interval very large, so that the population probability is noticeably different from zero even for energies far above the Fermi energy, the conservation of the number of particles leads to the Fermi energy being below the lowest possible level. The Fermi gas then behaves like a classic gas; it is not degenerate . ${\ displaystyle T \ gg T _ {\ mathrm {F}} \ Leftrightarrow k _ {\ mathrm {B}} T \ gg E _ {\ mathrm {F}}}$

Fermi distribution in metals

For the conduction electrons in a metal, the Fermi energy is a few electron volts , corresponding to a Fermi temperature of several 10,000 K. As a result, the thermal energy is much smaller than the typical width of the conduction band. It is a degenerate electron gas . The contribution of the electrons to the heat capacity is therefore negligible even at room temperature and can be taken into account in terms of perturbation theory . The temperature dependence of the Fermi energy is very low ( meV range) and is often neglected. ${\ displaystyle E _ {\ rm {F}} \! \,}$ ${\ displaystyle T _ {\ mathrm {F}} \! \,}$ ${\ displaystyle k _ {\ mathrm {B}} T}$

Fermi distribution in semiconductors and insulators

For semiconductors and insulators , the Fermi level is in the forbidden zone . In the area of the Fermi edge, therefore, there are no states whose occupation can clearly depend on the temperature. This means that at one temperature the valence band is completely occupied by electrons and the conduction band is unoccupied, and that there are only very few holes or excited electrons. By introducing foreign atoms with additional charge carriers ( donor or acceptor doping ), the Fermi level can be shifted downwards or upwards, which greatly increases the conductivity. In this case, the Fermi level also shifts significantly with temperature. Therefore, z. B. electronic circuits based on semiconductors (as in computers) are only correct within a narrow temperature range. ${\ displaystyle T = 0 \, \ mathrm {K}}$ ${\ displaystyle T> 0 \, \ mathrm {K}}$

Derivation from a minimum of free energy

Schematic state, energy and occupation diagram for a system of 7 energy levels , each -fold degenerate and -fermionically occupied.

{\ displaystyle E_ {1} \ dots E_ {7}}

{\ displaystyle D_ {i}}

{\ displaystyle N_ {i}}

The Fermi-Dirac statistics can be derived in a nice way from the condition that the free energy assumes a minimum in thermal equilibrium (with solid and volume ) . To do this, we consider fermions - for example electrons - that are distributed over levels . The levels have energies and are each - times degenerate (see Fig.), So they can take up a maximum of electrons ( Pauli principle ). The number of electrons in the -th level is denoted by. For the macrostate of the system it is irrelevant which of the electrons are in the -th level and which of the states they occupy. The macrostate is therefore entirely determined by the sequence of numbers . ${\ displaystyle T, N}$ ${\ displaystyle V}$ ${\ displaystyle F = E-TS}$ ${\ displaystyle N}$ ${\ displaystyle i = 1,2,3 \ dots I}$ ${\ displaystyle E_ {1}, E_ {2}, E_ {3} \ dots E_ {I}}$ ${\ displaystyle D_ {1}, D_ {2}, D_ {3} \ dots D_ {i}}$ ${\ displaystyle D_ {i}}$ ${\ displaystyle i}$ ${\ displaystyle N_ {i}}$ ${\ displaystyle N}$ ${\ displaystyle i}$ ${\ displaystyle D_ {i}}$ ${\ displaystyle N_ {1}, N_ {2}, \ dots}$

For any distribution of electrons on the levels, the following applies:

{\ displaystyle {\ begin {aligned} N & = \ sum _ {i = 1} ^ {I} N_ {i} & \ qquad (1) \\ E & = \ sum _ {i = 1} ^ {I} N_ {i} E_ {i} & \ qquad (2) \\ S & = k _ {\ rm {B}} \ ln W & \ qquad (3) &. \ end {aligned}}}

Equation (1) gives the total number of particles that should be kept constant while varying each to find the minimum of . Equation (2) gives the energy of the system belonging to the present distribution , as it is to be inserted into the formula for . Equation (3) is (according to Ludwig Boltzmann ) the entropy of the state of the system (macrostate) , indicating the thermodynamic probability for the relevant sequence of occupation numbers, i.e. the number of possible distributions (microstates) of electrons in places for all Levels together. ${\ displaystyle N_ {i}}$ ${\ displaystyle F}$ ${\ displaystyle E}$ ${\ displaystyle F}$ ${\ displaystyle W = \ prod _ {i = 1} ^ {I} W_ {i}}$ ${\ displaystyle N_ {1}, N_ {2}, \ dots}$ ${\ displaystyle N_ {i}}$ ${\ displaystyle D_ {i}}$ ${\ displaystyle i = 1,2,3 \ dots I}$

In order to find the distribution in which the free energy becomes minimal by varying the under the constraint , we use the method of the Lagrange multipliers . It turns out ${\ displaystyle N_ {i}}$ ${\ displaystyle N = const}$ ${\ displaystyle F}$

{\ displaystyle {\ frac {\ partial F} {\ partial N_ {i}}} = \ lambda {\ frac {\ partial N} {\ partial N_ {i}}} \ equiv \ lambda}

for everyone .

{\ displaystyle i}

This is the ( independent) Lagrange multiplier. For the calculation of the derivative the explicit formula is required for: ${\ displaystyle \ lambda}$ ${\ displaystyle i}$ ${\ displaystyle {\ tfrac {\ partial F} {\ partial N_ {i}}}}$ ${\ displaystyle S}$

{\ displaystyle S = k _ {\ rm {B}} \ ln W = k _ {\ rm {B}} \ ln \ prod _ {i = 1} ^ {I} W_ {i} = k _ {\ rm {B }} \ sum _ {i = 1} ^ {I} \ ln W_ {i}}

It is

{\ displaystyle {\ begin {aligned} W_ {i} & = {\ binom {D_ {i}} {N_ {i}}} \\ & = {\ frac {D_ {i}!} {N_ {i} ! [D_ {i} -N_ {i}]!}}. \ End {aligned}}}

the binomial coefficient, d. H. the number of possibilities to choose different objects among objects . ${\ displaystyle D_ {i}}$ ${\ displaystyle N_ {i}}$

With the help of the simplified Stirling formula, the result is further ${\ displaystyle \ ln k! \ approx k \ ln kk}$

{\ displaystyle {\ begin {aligned} \ ln W_ {i} & \ approx D_ {i} \ ln D_ {i} -D_ {i} -N_ {i} \ ln N_ {i} + N_ {i} - [D_ {i} -N_ {i}] \ ln [D_ {i} -N_ {i}] + [D_ {i} -N_ {i}] \\ & = D_ {i} \ ln D_ {i} -N_ {i} \ ln N_ {i} - [D_ {i} -N_ {i}] \ ln [D_ {i} -N_ {i}] \ end {aligned}}}

and thus

{\ displaystyle {\ begin {aligned} {\ frac {\ partial \ ln W_ {i}} {\ partial N_ {i}}} & \ approx - \ ln N_ {i} -1+ \ ln [D_ {i } -N_ {i}] + 1 \\ & = - \ ln N_ {i} + \ ln [D_ {i} -N_ {i}] \\ & = \ ln ([D_ {i} / N_ {i } -1]). \ End {aligned}}}

Overall equation (2) becomes

{\ displaystyle \ lambda = {\ frac {\ partial F} {\ partial N_ {i}}} = {\ frac {\ partial E} {\ partial N_ {i}}} - T {\ frac {\ partial S } {\ partial N_ {i}}} = E_ {i} -k _ {\ rm {B}} T {\ frac {\ partial \ ln W_ {i}} {\ partial N_ {i}}} = E_ { i} -k _ {\ rm {B}} T \ ln ([D_ {i} / N_ {i} -1])}

.

Insertion of the given occupation probability and conversion results in: ${\ displaystyle f_ {i}: = {\ frac {N_ {i}} {D_ {i}}}}$ ${\ displaystyle f_ {i}}$

{\ displaystyle f_ {i} = {\ frac {1} {\ exp {\ frac {E_ {i} - \ lambda} {k _ {\ rm {B}} T}} + 1}}}

.

This is the Fermi-Dirac statistic. The Lagrange multiplier turns out to be their chemical potential . ${\ displaystyle \ mu = \ lambda}$

Observations

The Fermi distribution can be observed very well in solids if the electronic population density of the conduction band is measured as a function of the energy. A particularly good example of the ideal Fermigas is aluminum. Such studies can also be used to determine the resolution of a measuring apparatus by measuring the course of the distribution at a certain temperature and comparing it with the formula for the Fermi distribution.

For more examples of the meaning see under Fermi energy .

literature

Ellen Ivers-Tiffée, Waldemar von Münch: Materials in electrical engineering . 10th edition. Vieweg + Teubner, 2007, ISBN 978-3-8351-0052-7 .
Michael Reisch: Semiconductor components . 2nd Edition. Springer-Verlag, Berlin 2004, ISBN 3-540-21384-8 .
U. Krey, A. Owen: Basic Theoretical Physics - a Concise Overview . Springer-Verlag, Berlin, Heidelberg, New York 2007, ISBN 978-3-540-36804-5 (English).

Individual evidence

↑ ^a ^b Enrico Fermi: On the quantization of the monatomic ideal gas. In: Journal of Physics. Volume 36, 1926, pp. 902-912, doi: 10.1007 / BF01400221 .
^ PAM Dirac: On the Theory of Quantum Mechanics. In: Proceedings of the Royal Society of London. Series A Volume 112, 1926, pp. 661-677, doi: 10.1098 / rspa.1926.0133 .

[Fermi_ZeitPhys-1] Enrico Fermi: On the quantization of the monatomic ideal gas. In: Journal of Physics. Volume 36, 1926, pp. 902-912, doi: 10.1007 / BF01400221 .

[2] PAM Dirac: On the Theory of Quantum Mechanics. In: Proceedings of the Royal Society of London. Series A Volume 112, 1926, pp. 661-677, doi: 10.1098 / rspa.1926.0133 .