Fermi energy

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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.

1 Derivation for three dimensions
2 Typical fermi energies
- 2.1 White dwarfs
- 2.2 Nucleus
3 Fermi level
4 Quantum mechanics
5 Free electron gas
6 See also
7 References

[edit] Derivation for three dimensions

If a system of non-interacting 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:

$\psi = A \sin\left(\frac{n_x \pi x}{L}\right) \sin\left(\frac{n_y \pi y}{L}\right) \sin\left(\frac{n_z \pi z}{L}\right) \,$

where

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

n_x, n_y, n_z are positive integers.

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

$E_n = \frac{\hbar^2 \pi^2}{2m L^2} \left( n_x^2 + n_y^2 + n_z^2\right) \,$

Now for a system of these fermions in a box at absolute zero, there is a fermion that 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:

$N = \frac{1}{8} \times 2 \times \frac{4}{3} \pi n_f^3 \,$

Or, more simply:

$n_f = \left( \frac{3 N}{\pi} \right)^{1/3}$

Finally, one can obtain the fermi energy with

$E_f \,$	$= \frac{\hbar^2 \pi^2}{2m L^2} n_f^2$
	$= \frac{\hbar^2 \pi^2}{2m L^2} \left( \frac{3 N}{\pi} \right)^{2/3}$

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

$E_f = \frac{\hbar^2}{2m} \left( \frac{3 \pi^2 N}{V} \right)^{2/3} \,$

[edit] Typical fermi energies

[edit] White dwarfs

Stars known as White dwarfs, have mass comparible to our Sun, but have a radius about 100 times smaller. The high densities means that the electrons are no longer bound to single nuclei and instead form a degenerate electron gas. The number density of electrons in a White dwarf are on the order of 10³⁶ electrons/m³. This means their fermi energy is:

$E_f = \frac{\hbar^2}{2m_e} \left( \frac{3 \pi^2 (10^{36})}{1 \ \mathrm{m}^3} \right)^{2/3} \approx 3 \times 10^5 \ \mathrm{eV} \,$

[edit] Nucleus

Another typical example is that of the particles in a nucleus of an atom. The radius of the nucleus is roughly:

$R = \left(1.5 \times 10^{-15} \mathrm{m} \right) \times A^{1/3}$

where A is the number of nucleons.

The number density of nucleons in a nucleus is therefore:

$n = \frac{A}{\begin{matrix} \frac{4}{3} \end{matrix} \pi R^3 } \approx 7 \times 10^{43} \ \mathrm{m}^{-3}$

Now since the fermi energy only applies to fermions of the same type, one must divide this density in two. This is because the presence of neutrons does not affect the fermi energy of the protons in the nucleus.

So the fermi energy of a nucleus is about:

$E_f = \frac{\hbar^2}{2m_p} \left( \frac{3 \pi^2 (7 \times 10^{43})}{1 \ \mathrm{m}^3} \right)^{2/3} \approx 33 \times 10^6 \ \mathrm{eV} = 33 \ \mathrm{MeV}$

[edit] Fermi level

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. [1] In this state (at 0 K), the average energy of an electron is given by:

$E_{av} = \frac{3}{5} E_f$

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:

$p_F = \sqrt{2 m_e E_f}$

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:

$V_f = \sqrt{\frac{2 E_f}{m_e}}$

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:

$T_f = \frac{E_f}{k}$

where k is the Boltzmann constant.

[edit] Quantum mechanics

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.

[edit] Free electron gas

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

$\mu = E_F \left[ 1- \frac{\pi ^2}{12} \left(\frac{kT}{E_F}\right) ^2 + \frac{\pi^4}{80} \left(\frac{kT}{E_F}\right)^4 + \cdots \right]$

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.