In physics and Fermi-Dirac statistics, the Fermi energy (E_F) of a system of non-interacting fermions is the smallest possible increase in the ground state energy when exactly one particle is added to the system. It is equivalent to the chemical potential of the system in its ground state at absolute zero. It can also be interpreted as the maximum energy of an individual fermion in this ground state. The Fermi energy is one of the central concepts of condensed matter physics.

Derivation for three dimensions

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If a system of fermions exists in a cube that has a side length L, the total volume of this system is V = L³. The wavefunction for a particle in such a system is the three-dimensional version of a particle in an infinite square well. This wavefunction is:

$\backslash psi\; =\; A\; sin\backslash left(\backslash frac\{n\_x\; \backslash pi\; x\}\{L\}\backslash right)\; sin\backslash left(\backslash frac\{n\_y\; \backslash pi\; x\}\{L\}\backslash right)\; sin\backslash left(\backslash frac\{n\_z\; \backslash pi\; x\}\{L\}\backslash right)\; \backslash ,$

where

A is a constant (found by normalizing the wavefunction) and

n_x, n_y, n_z are are positive integers.

The energy of a particle in a certain energy level is given by:

$E\_n\; =\; \backslash frac\{\backslash hbar^2\; \backslash pi^2\}\{2m\; L^2\}\; \backslash left(\; n\_x^2\; +\; n\_y^2\; +\; n\_z^2\backslash right)\; \backslash ,$

Now for a system of these fermions in a box at absolute zero, there is a fermion the has the highest energy, the fermi energy, and we say it is in a specific state n_f. For this system to hold N fermions, this highest level, n_f, must be given by:

$N\; =\; \backslash frac\{1\}\{8\}\; \backslash times\; 2\; \backslash times\; \backslash frac\{4\}\{3\}\; \backslash pi\; n\_f^3\; \backslash ,$

Or, more simply:

$n\_f\; =\; \backslash left(\; \backslash frac\{3\; N\}\{\backslash pi\}\; \backslash right)^\{1/3\}$

Finally, one can obtain the fermi energy with

{| $E\_f\; \backslash ,$ $=\; \backslash frac\{\backslash hbar^2\; \backslash pi^2\}\{2m\; L^2\}\; n\_f^2$ $=\; \backslash frac\{\backslash hbar^2\; \backslash pi^2\}\{2m\; L^2\}\; \backslash left(\; \backslash frac\{3\; N\}\{\backslash pi\}\; \backslash right)^\{2/3\}$

Which results in a relationship between the fermi energy and the number of particles per volume (when you replace L² with V^2/3):

{|cellpadding="2" style="border:2px solid #ccccff" $E\_f\; =\; \backslash frac\{\backslash hbar^2\}\{2m\}\; \backslash left(\; \backslash frac\{3\; \backslash pi^2\; N\}\{V\}\; \backslash right)^\{2/3\}\; \backslash ,$

Fermi level

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The Fermi level is the top of the collection of electron energy levels at absolute zero temperature. Since fermions cannot exist in identical energy states (see the exclusion principle), at absolute zero, electrons pack into the lowest available energy states and build up a "Fermi sea" of electron energy states. ^* In this state (at 0 K), the average energy of an electron is given by:

$E\_\{av\}\; =\; \backslash frac\{3\}\{5\}\; E\_f$

where $E\_f$ is the fermi energy.

The Fermi momentum is the momentum of fermions at the Fermi surface. The Fermi momentum is given by:

$p\_F\; =\; \backslash sqrt\{2\; m\_e\; E\_f\}$

where $m\_e$ is the mass of the electron.

This concept is usually applied in the case of dispersion relations between the energy and momentum that do not depend on the direction. In more general cases, one must consider the Fermi energy.

The Fermi velocity is the average velocity of an electron in an atom at absolute zero. This average velocity corresponds to the average energy given above. The Fermi velocity is defined by:

$V\_f\; =\; \backslash sqrt\{\backslash frac\{2\; E\_f\}\{m\_e\}\}$

where $m\_e$ is the mass of the electron.

Below the Fermi temperature, a substance gradually expresses more and more quantum effects of cooling. The Fermi temperature is defined by:

$T.f\; =\; \backslash frac\{E\_f\}\{k\}$

where k is the Boltzmann constant.

Quantum mechanics

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According to quantum mechanics, fermions -- particles with a half-integer spin, usually 1/2, such as electrons -- follow the Pauli exclusion principle, which states that multiple particles may not occupy the same quantum state. Consequently, fermions obey Fermi-Dirac statistics. The ground state of a non-interacting fermion system is constructed by starting with an empty system and adding particles one at a time, consecutively filling up the lowest-energy unoccupied quantum states. When the desired number of particles has been reached, the Fermi energy is the energy of the highest occupied molecular orbital (HOMO). Within conductive materials, this is equivalent to the lowest unoccupied molecular orbital (LUMO), however within other materials there will be a significant gap between the HOMO and LUMO on the order of 2-3 eV. This gap does exist in conductors, however it is infinitesimally small.

Free electron gas

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In the free electron gas, the quantum mechanical version of an ideal gas of fermions, the quantum states can be labelled according to their momentum. Something similar can be done for periodic systems, such as electrons moving in the atomic lattice of a metal, using something called the "quasi-momentum" (see Bloch wave). In either case, the Fermi energy states reside on a surface in momentum space known as the Fermi surface. For the free electron gas, the Fermi surface is the surface of a sphere; for periodic systems, it generally has a contorted shape (see Brillouin zones). The volume enclosed by the Fermi surface defines the number of electrons in the system, and the topology is directly related to the transport properties of metals, such as electrical conductivity. The study of the Fermi surface is sometimes called Fermiology. The Fermi surfaces of most metals are well studied both theoretically and experimentally.

The Fermi energy of the free electron gas is related to the chemical potential by the equation

$\backslash mu\; =\; E\_F\; \backslash left1-\; \backslash frac\{\backslash pi\; ^2\}\{12\}\; \backslash left(\backslash frac\{kT\}\{E\_F\}\backslash right)\; ^2\; +\; \backslash frac\{\backslash pi^4\}\{80\}\; \backslash left(\backslash frac\{kT\}\{E\_F\}\backslash right)^4\; +\; \backslash cdots\; \backslash right$

where E_F is the Fermi energy, k is the Boltzmann constant and T is temperature. Hence, the chemical potential is approximately equal to the Fermi energy at temperatures of much less than the characteristic Fermi temperature E_F/k. The characteristic temperature is on the order of 10⁵ K for a metal, hence at room temperature (300 K), the Fermi energy and chemical potential are essentially equivalent. This is significant since it is the chemical potential, not the Fermi energy, which appears in Fermi-Dirac statistics.

See also

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• fermi gas
• semiconductors
• electrical engineering
• electronics
• thermodynamics

References

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• Table of fermi energies, velocities, and temperatures for various elements.
• a discussion of fermi gases and fermi temperatures.

Condensed matter physics | Statistical mechanics

Energia de Fermi | Fermiverteilung | Energía de Fermi | Énergie de Fermi | Energia di Fermi | Fermi-niveau | フェルミエネルギー | Poziom Fermiego | Энергия Ферми