Fermi level

[4] Clearly, the electrostatic potential is not the only factor influencing the flow of charge in a material—Pauli repulsion, carrier concentration gradients, electromagnetic induction, and thermal effects also play an important role.

Note that thermodynamic equilibrium here requires that the circuit be internally connected and not contain any batteries or other power sources, nor any variations in temperature.

In semiconductors and semimetals the position of μ relative to the band structure can usually be controlled to a significant degree by doping or gating.

We can define a parameter ζ[9] that references the Fermi level with respect to the band edge:

The band theory of metals was initially developed by Sommerfeld, from 1927 onwards, who paid great attention to the underlying thermodynamics and statistical mechanics.

In this article, the terms conduction-band referenced Fermi level or internal chemical potential are used to refer to ζ. ζ is directly related to the number of active charge carriers as well as their typical kinetic energy, and hence it is directly involved in determining the local properties of the material (such as electrical conductivity).

For this reason it is common to focus on the value of ζ when concentrating on the properties of electrons in a single, homogeneous conductive material.

The Fermi level, μ, and temperature, T, are well defined constants for a solid-state device in thermodynamic equilibrium situation, such as when it is sitting on the shelf doing nothing.

When the device is brought out of equilibrium and put into use, then strictly speaking the Fermi level and temperature are no longer well defined.

Fortunately, it is often possible to define a quasi-Fermi level and quasi-temperature for a given location, that accurately describe the occupation of states in terms of a thermal distribution.

The quasi-equilibrium approach allows one to build a simple picture of some non-equilibrium effects as the electrical conductivity of a piece of metal (as resulting from a gradient of μ) or its thermal conductivity (as resulting from a gradient in T).

One cannot define the quasi-Fermi level or quasi-temperature in this case; the electrons are simply said to be non-thermalized.

The term Fermi level is mainly used in discussing the solid state physics of electrons in semiconductors, and a precise usage of this term is necessary to describe band diagrams in devices comprising different materials with different levels of doping.

It is common to see scientists and engineers refer to "controlling", "pinning", or "tuning" the Fermi level inside a conductor, when they are in fact describing changes in ϵC due to doping or the field effect.

This concept is very theoretical (there is no such thing as a non-interacting Fermi gas, and zero temperature is impossible to achieve).

However, it finds some use in approximately describing white dwarfs, neutron stars, atomic nuclei, and electrons in a metal.

[11] Much like the choice of origin in a coordinate system, the zero point of energy can be defined arbitrarily.

When comparing distinct bodies, however, it is important that they all be consistent in their choice of the location of zero energy, or else nonsensical results will be obtained.

A practical and well-justified choice of common point is a bulky, physical conductor, such as the electrical ground or earth.

It also has the advantage of being accessible, so that the Fermi level of any other object can be measured simply with a voltmeter.

In principle, one might consider using the state of a stationary electron in the vacuum as a reference point for energies.

Just outside a conductor, the electrostatic potential depends sensitively on the material, as well as which surface is selected (its crystal orientation, contamination, and other details).

In cases where the "charging effects" due to a single electron are non-negligible, the above definitions should be clarified.

If the capacitor is uncharged, the Fermi level is the same on both sides, so one might think that it should take no energy to move an electron from one plate to the other.

In this case one must be precise about the thermodynamic definition of the chemical potential as well as the state of the device: is it electrically isolated, or is it connected to an electrode?

[13] The parameter, μ, (i.e., in the case where the number of electrons is allowed to fluctuate) remains exactly related to the voltmeter voltage, even in small systems.

Filling of the electronic states in various types of materials at equilibrium . Here, height is energy while width is the density of available states for a certain energy in the material listed. The shade follows the Fermi–Dirac distribution ( black : all states filled, white : no state filled). In metals and semimetals the Fermi level E F lies inside at least one band.
In insulators and semiconductors the Fermi level is inside a band gap ; however, in semiconductors the bands are near enough to the Fermi level to be thermally populated with electrons or holes . "intrin." indicates intrinsic semiconductors .
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A voltmeter measures differences in Fermi level divided by electron charge .
Fermi-Dirac distribution vs. energy , with μ = 0.55 eV and for various temperatures in the range 50 K ≤ T ≤ 375 K .
Example of variations in conduction band edge E C in a band diagram of GaAs/AlGaAs heterojunction -based high-electron-mobility transistor .
When the two metals depicted here are in thermodynamic equilibrium as shown (equal Fermi levels E F ), the vacuum electrostatic potential ϕ is not flat due to a difference in work function .