Fermi level

Not to be confused with Fermi energy.

In the context of solid state physics, the total chemical potential for electrons (or electrochemical potential) is known as the Fermi level, usually denoted by µ or E_F. Identical in meaning to the electrochemical potential, the Fermi level of a body is the thermodynamic work required to add one electron to it (not counting the work required to remove the electron from wherever it came from). A precise understanding of the Fermi level---how it relates to electronic band structure in determining electronic properties, how it relates to the voltage and flow of charge in an electronic circuit---is essential to an understanding of solid state physics.

In a band structure picture, the Fermi level can be considered to be a hypothetical energy level of an electron, such that at thermodynamic equilibrium this energy level would have a 50% probability of being occupied at any given time. The Fermi level does not necessarily correspond to an actual energy level (in an insulator the Fermi level lies in the band gap), nor does it even require the existence of a band structure. Nonetheless, the Fermi level is a precisely defined thermodynamic quantity, and differences in Fermi level can be measured simply with a voltmeter.

The Fermi level and voltage

In oversimplified descriptions of electric circuits it is said that electric currents are driven by differences in electrostatic potential (Galvani potential), but this is not exactly true. As a counterexample, multi-material devices such as p-n junctions contain internal electrostatic potential differences at equilibrium, without any accompanying current. Also, if a voltmeter is attached to the junction, one simply measures zero volts. Clearly, the electrostatic potential is not the only factor influencing the flow of charge in a material.

In fact, the quantity called "voltage" as measured in an electric circuit is more closely related to the chemical potential for electrons (Fermi level). When the leads of a voltmeter are attached to two points in a circuit, the displayed voltage is a measure of the work that can be obtained by allowing a tiny unit of charge to flow from one point to the other. If a simple wire is connected between two points of differing voltage (forming a short circuit), current will flow from positive to negative voltage, converting the available work into heat.

The electrochemical potential (Fermi level) of a body expresses precisely the work required to add an electron to it, or equally the work obtained by removing an electron. Therefore, the observed difference (V_A-V_B) in voltage between two points "A" and "B" in an electronic circuit is exactly related to the corresponding difference (µ_A-µ_B) in electrochemical potential by the formula

$(V_{\mathrm{A}}-V_{\mathrm{B}}) = -(\mu_{\mathrm{A}}-\mu_{\mathrm{B}})/e$

where -e is the electron charge.

From the above discussion it can be seen that electrons will move from a body of high µ (low voltage) to low µ (high voltage) if a simple path is provided. This flow of electrons will cause the lower µ to increase (due to charging or other repulsion effects) and likewise cause the higher µ to decrease. Eventually, µ will settle down to the same value in both bodies. This leads to an important fact regarding the equilibrium (off) state of an electronic circuit:

An electronic circuit in thermodynamic equilibrium will have a constant Fermi level throughout its connected parts. No current flows in this circuit.

This also means that the voltage (measured with a voltmeter) between any two points will be zero, at equilibrium. Note that thermodynamic equilibrium here requires that the circuit should be internally connected and not contain any batteries or other power sources, nor any variations in temperature.

Fermi level referencing and the location of zero Fermi level

Much like the choice of origin in a coordinate system, the zero point of energy can be defined arbitrarily, since observable phenomena only depend on energy differences. When comparing distinct bodies, however, it is important that they are all consistent in their choice of the location of zero energy, or else nonsensical results will be obtained. It can therefore be helpful to explicitly name a common point to ensure that different components are in agreement. On the other hand, if a reference point is chosen ambiguously (such as "the vacuum", see below) it will instead cause more problems.

A practical and well-justified choice of common point is a bulky, physical conductor, such as the electrical ground or earth. Such a conductor can be considered to be in a good thermodynamic equilibrium and so its µ is well defined. It provides a reservoir of charge, so that large numbers of electrons may be added or removed without incurring charging effects. It also has the advantage of being accessible, so that the Fermi level of any other object can be measured simply with a voltmeter.

Why it is not advisable to use "the energy in vacuum" as a reference zero

In principle, one might consider using the state of a stationary electron in the vacuum as a reference point for electrochemical potential. This approach is not advisable unless one is careful to define exactly where "the vacuum" is. The problem is that not all points in the vacuum are equivalent.

At thermodynamic equilibrium, it is typical for electrical potential differences of order 1 V to exist in the vacuum (Volta potentials). The source of this vacuum potential variation is the variation in work function between the different conducting materials exposed to vacuum. 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).

The parameter that gives the best approximation to universality is the "Earth-referenced electrochemical potential" used earlier. This also has the advantage that it can be measured with a voltmeter.

The Fermi level and band structure

Simplified diagram of the filling of electronic band structure in various types of material, relative to the Fermi level E_F (materials are shown in equilibrium with each other). In metals and semimetals the Fermi level lies inside at least one band, with semimetals containing far fewer charge carriers. In insulators the Fermi level is deep inside a forbidden gap, while in semiconductors the bands near the Fermi level are populated by thermally activated electrons and holes.

In the band theory of solids, electrons are considered to occupy a series of bands composed of single-particle energy eigenstates each labelled by ϵ. Although this single particle picture is an approximation, it greatly simplifies the understanding of electronic behaviour and it generally provides correct results when applied correctly.

The Fermi-Dirac distribution $f(\epsilon)$ gives the probability that (at thermodynamic equilibrium) an electron will occupy a state having energy ϵ. Alternatively, it gives the average number of electrons that will occupy that state given the restriction imposed by the Pauli exclusion principle:

$f(\epsilon) = \frac{1}{e^{(\epsilon-\mu) / (k T)} + 1}$

Here, T is the absolute temperature and k is Boltzmann's constant. If there is a state at the Fermi level (ϵ = µ), then this this level will have a 50% chance of being occupied at any given time.

The location of µ within a material's band structure is important in determining the electrical behaviour of the material.

• In an insulator µ lies within a large band gap, far away from any states that are able to carry current.
• In a metal, semimetal or degenerate semiconductor, µ lies within a delocalized band. A large number of states nearby µ are thermally active and readily carry current.
• In an intrinsic or lightly doped semiconductor, µ is close enough to a band edge that there are a dilute number of thermally excited carriers residing near that band edge.

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. These controls do not change µ which is fixed by the electrodes, but rather they cause the the entire band structure to shift up and down (sometimes also changing the band structure's shape). For further information about the Fermi levels of semiconductors, see (for example) Sze.

Local conduction band referencing, internal chemical potential, and the parameter ζ

Simple band diagram with denoted vacuum energy E_VAC, conduction band edge E_C, Fermi level E_F, valence band edge E_V, electron affinity E_ea, work function Φ and band gap E_g

If the symbol ℰ is used to denote an electron energy level measured relative to the energy of the bottom of its enclosing band, ϵ_C, then in general we have ℰ = ϵ – ϵ_C, and in particular we can define the parameter ζ by referencing the Fermi level to the band edge:

$\zeta = \mu - \epsilon_{\rm C}.$

It follows that the Fermi-Dirac distribution function can also be written

$f(\mathcal{E}) = \frac{1}{1 + \mathrm{exp}[(\mathcal{E}-\zeta)/k_{\mathrm{B}} T]}.$

The band theory of metals was initially developed by Sommerfeld, from 1927 onwards, who paid great attention to the underlying thermodynamics and statistical mechanics. He describes ζ as the "free enthalpy of an electron", but this name is not now in common use. Confusingly, in some contexts ζ may be called the "Fermi level", "chemical potential" or "electrochemical potential", leading to ambiguity with the globally-referenced quantity µ. 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. By analogy to the energy states of a free electron, the ℰ of a state is the kinetic energy of that state and ϵ_C is its potential energy. With this in mind, the parameter ζ could also be labelled the "Fermi kinetic energy".

Unlike µ, the parameter ζ is not a constant at equilibrium, taking on multiple values due to variations in ϵ_C. ζ usually varies from location to location in a material, depending on factors such as material quality and impurities/dopants. Near the surface of a semiconductor or semimetal, ζ can be strongly controlled by externally applied electric fields, as is done in a field effect transistor. ζ in a multi-band material may even take on multiple values in a single location. For example, in a piece of aluminum metal there are two conduction bands crossing the Fermi level (even more bands in other materials); each band has a different edge energy ϵ_C and a different value of ζ.

The value of ζ at zero temperature is widely known as the Fermi energy, sometimes written ζ₀. Confusingly (again), the name "Fermi energy" sometimes is used to refer to ζ at nonzero temperature.

The Fermi level and temperature out of equilibrium

See also: Quasi-Fermi level

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 device is said to be in 'quasi-equilibrium' when such a description is possible.

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 in μ) or its thermal conductivity (as resulting from a gradient in T). The quasi-μ and quasi-T can vary (or not exist at all) in any non-equilibrium situation, such as:

• If the system contains a chemical imbalance (as in a battery).
• If the system is exposed to changing electromagnetic fields. (as in capacitors, inductors, and transformers).
• Under illumination from a light-source with a different temperature, such as the sun (as in solar cells),
• When the temperature is not constant within the device (as in thermocouples),
• When the device has been altered, but has not had enough time to re-equilibrate (as in piezoelectric or pyroelectric substances).

In some situations, such as immediately after a material experiences a high-energy laser pulse, the electron distribution cannot be described by any thermal distribution. One cannot define the quasi-Fermi level or quasi-temperature in this case; the electrons are simply said to be "non-thermalized". In less dramatic situations, such as in a solar cell under constant illumination, a quasi-equilibrium description may be possible but requiring the assignment of distinct values of μ and T to different bands (conduction band vs. valence band). Even then, the values of μ and T may jump discontinuously across a material interface (e.g., p-n junction) when a current is being driven, and be ill-defined at the interface itself.

Terminology problems

Unfortunately, the definitions of the terms "Fermi level", "Fermi energy", "chemical potential", and "electrochemical potential" are by no means universal. This can lead to some confusion when comparing scientific or engineering literature between different authors.

• Chemical potential and Electrochemical potential: In some parts of the literature the term "chemical potential" is used instead of "electrochemical potential". In the past there has been no consensus as to whether these two terms should mean the same thing. Some textbooks continue to make a distinction (and, worse, there are alternative conventions as to what each term means). The more modern view^{[citation needed]} is that "chemical potential" should mean the same thing as "electrochemical potential", – but that in some contexts there is a separate concept – called here the "internal chemical potential" – that is the energy left when the "purely electrostatic component of electrochemical potential" is subtracted out. (In other contexts it may not be possible make a division into components in any sensible way.) In any case, it is usually only the total combined thermodynamic potential that can be measured. As already noted, it is thought less confusing here to use the name "electrochemical potential" for the total thermodynamic potential.

• Alternative uses of the name "Fermi energy". It is normal in solid-state physics to use the term "Fermi energy" as a name for ζ₀, as done here. However, particularly in semiconductor physics and engineering, the term "Fermi energy" is sometimes used as a synonym for "Fermi level".

Discrete charging effects

In cases where the "charging effects" due to a single electron are non-negligible, the above definitions should be clarified. For example, consider an capacitor made of two identical parallel-plates. 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. But when the electron has been moved, the capacitor has become (slightly) charged, so this does take a slight amount of energy. In a normal capacitor, this is negligible, but in a nano-scale capacitor it can be more important.

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

• If the charge on a body is fixed and known, but the body is thermally connected to a reservoir, then it is in the canonical ensemble. We can define a "chemical potential" in this case as the work required to add one electron to a body that already has exactly $N$ electrons,

  $\mu(N,T) = F(N+1,T) - F(N,T),$

  where $F(N+1,T)$ is the free energy with $N+1$ electrons, and $F(N,T)$ is the free energy with $N$ electrons. The "chemical potential" here has a slightly different meaning than the Fermi level; the occupation of electron energy levels in the canonical ensemble is not described by the Fermi-Dirac distribution, as that distribution implies that $N$ can fluctuate.
• When the body is also able to exchange charge with the reservoir (electrode), it enters the grand canonical ensemble. The value of chemical potential $\mu$ is fixed by the electrode, and the charge $N$ on the body may fluctuate. In this case $\mu$ corresponds to the notion of Fermi level in this article, as it is constant in the device at equilibrium, and the electron statistics are described by the Fermi-Dirac distribution. This $\mu$ is not determined by a discrete charging event; rather, it gives the infinitesimal amount of work needed to increase the average number of electrons ($\langle N\rangle$) by an infinitesimal amount:

  $\mu(\langle N\rangle,T) = \left(\frac{\partial F}{\partial \langle N\rangle}\right)_{T}$

In the example of the nano-scale capacitor we can therefore consider two distinct situations of charging. Let us label the two plates A and B, and note that the chemical potential of each plate will have some interdependence on the status of the other plate:

• Electrically isolated plates (canonical ensemble): The work to move one electron from A to B will be determined by the process of removing then adding the electron (or adding then removing). This work is the difference

  $\begin{align} W & = \mu_{\rm B}(N_{\rm A}-1,N_{\rm B},T) - \mu_{\rm A}(N_{\rm A}-1,N_{\rm B},T) \\ & = F(N_{\rm A}-1,N_{\rm B}+1,T) - F(N_{\rm A},N_{\rm B},T). \end{align}$

• Reservoir-connected plates (grand canonical ensemble): We do not directly move the charge, but we may instead apply a voltage to each plate and change the average number of electrons $\langle N\rangle$ by one. For each plate, $\mu$ is a continuous function of $\langle N\rangle$ and the work performed is determined by integrals of $\mu$, or

  $W = F(\langle N_{\rm A}\rangle-1,\langle N_{\rm B}\rangle+1,T) - F(\langle N_{\rm A}\rangle,\langle N_{\rm B}\rangle,T).$