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The Fermi level of a
solid-state Solid state, or solid matter, is one of the four fundamental states of matter. Solid state may also refer to: Electronics * Solid-state electronics, circuits built of solid materials * Solid state ionics, study of ionic conductors and their ...
body is the
thermodynamic work In thermodynamics, work is one of the principal processes by which a thermodynamic system can interact with its surroundings and exchange energy. An exchange of energy is facilitated by a mechanism through which the system can spontaneously exer ...
required to add one electron to the body. It is a
thermodynamic Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of ...
quantity usually denoted by ''µ'' or ''E''F for brevity. The Fermi level does not include 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 solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called '' band gaps'' or ...
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
band structure In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or ...
theory, used in
solid state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the ...
to analyze the energy levels in a solid, the Fermi level can be considered to be a hypothetical energy level of an electron, such that at
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...
this energy level would have a ''50% probability of being occupied at any given time''. The position of the Fermi level in relation to the band energy levels is a crucial factor in determining electrical properties. The Fermi level does not necessarily correspond to an actual energy level (in an insulator the Fermi level lies in the
band gap In solid-state physics, a band gap, also called an energy gap, is an energy range in a solid where no electronic states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference ( ...
), nor does it 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 A voltmeter is an instrument used for measuring electric potential difference between two points in an electric circuit. It is connected in parallel. It usually has a high resistance so that it takes negligible current from the circuit. ...
.


Voltage measurement

Sometimes it is said that electric currents are driven by differences in
electrostatic potential Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...
(
Galvani potential In electrochemistry, the Galvani potential (also called Galvani potential difference, or inner potential difference, Δφ, delta phi) is the electric potential difference between two points in the bulk of two phases. These phases can be two diffe ...
), but this is not exactly true. As a counterexample, multi-material devices such as
p–n junction A p–n junction is a boundary or interface between two types of semiconductor materials, p-type and n-type, inside a single crystal of semiconductor. The "p" (positive) side contains an excess of holes, while the "n" (negative) side contai ...
s contain internal electrostatic potential differences at equilibrium, yet without any accompanying net current; 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—
Pauli repulsion In chemistry and physics, the exchange interaction (with an exchange energy and exchange term) is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classical ...
, carrier concentration gradients, electromagnetic induction, and thermal effects also play an important role. In fact, the quantity called ''voltage'' as measured in an electronic circuit has a simple relationship to the
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a speci ...
for electrons (Fermi level). When the leads of a
voltmeter A voltmeter is an instrument used for measuring electric potential difference between two points in an electric circuit. It is connected in parallel. It usually has a high resistance so that it takes negligible current from the circuit. ...
are attached to two points in a circuit, the displayed voltage is a measure of the ''total'' work transferred when a unit charge is allowed to move from one point to the other. If a simple wire is connected between two points of differing voltage (forming a
short circuit A short circuit (sometimes abbreviated to short or s/c) is an electrical circuit that allows a current to travel along an unintended path with no or very low electrical impedance. This results in an excessive current flowing through the circu ...
), current will flow from positive to negative voltage, converting the available work into heat. The Fermi level of a body expresses the work required to add an electron to it, or equally the work obtained by removing an electron. Therefore, ''V''A − ''V''B, the observed difference in voltage between two points, ''A'' and ''B'', in an electronic circuit is exactly related to the corresponding chemical potential difference, ''µ''A − ''µ''B, in Fermi level by the formula V_\mathrm - V_\mathrm = \frac where −''e'' is the
electron charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
. 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: This also means that the voltage (measured with a voltmeter) between any two points will be zero, at equilibrium. Note that
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...
here requires that the circuit be internally connected and not contain any batteries or other power sources, nor any variations in temperature.


Band structure of solids

In the
band theory In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
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 Fermi–Dirac may refer to: * Fermi–Dirac statistics Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pa ...
, f(\epsilon), gives the probability that (at
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...
) a state having energy ''ϵ'' is occupied by an electron: f(\epsilon) = \frac Here, ''T'' is the
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic ...
and ''k''B is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
. If there is a state at the Fermi level (''ϵ'' = ''µ''), then this state will have a 50% chance of being occupied. The distribution is plotted in the left figure. The closer ''f'' is to 1, the higher chance this state is occupied. The closer ''f'' is to 0, the higher chance this state is empty. 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 In solid-state physics, a band gap, also called an energy gap, is an energy range in a solid where no electronic states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference ( ...
, far away from any states that are able to carry current. * In a metal,
semimetal A semimetal is a material with a very small overlap between the bottom of the conduction band and the top of the valence band. According to electronic band theory, solids can be classified as insulators, semiconductors, semimetals, or metals ...
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 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''

If the symbol ''ℰ'' is used to denote an electron energy level measured relative to the energy of the edge of its enclosing band, ''ϵ''C, then in general we have ''ℰ'' = ''ϵ'' – ''ϵ''C. We can define a parameter ''ζ'' that references the Fermi level with respect to the band edge: \zeta = \mu - \epsilon_. It follows that the Fermi–Dirac distribution function can be written as f(\mathcal) = \frac. The
band theory In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
of metals was initially developed by Sommerfeld, from 1927 onwards, who paid great attention to the underlying thermodynamics and statistical mechanics. Confusingly, in some contexts the band-referenced quantity ''ζ'' may be called the ''Fermi level'', ''chemical potential'', or ''electrochemical potential'', leading to ambiguity with the globally-referenced Fermi level. 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 Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
). 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 In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
of that state and ''ϵ''C is its
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potenti ...
. 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, but rather varies from location to location in a material due to variations in ''ϵ''C, which is determined by 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 The field-effect transistor (FET) is a type of transistor that uses an electric field to control the flow of current in a semiconductor. FETs (JFETs or MOSFETs) are devices with three terminals: ''source'', ''gate'', and ''drain''. FETs contro ...
. In a multi-band material, ''ζ'' 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 ''ζ''. The value of ''ζ'' at zero temperature is widely known as the
Fermi energy The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature. In a Fermi ga ...
, sometimes written ''ζ''0. Confusingly (again), the name ''Fermi energy'' sometimes is used to refer to ''ζ'' at non-zero temperature.


Temperature out of equilibrium

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 and where 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 Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
of a piece of metal (as resulting from a gradient of ''μ'') or its
thermal conductivity The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
(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 Battery most often refers to: * Electric battery, a device that provides electrical power * Battery (crime), a crime involving unlawful physical contact Battery may also refer to: Energy source *Automotive battery, a device to provide power t ...
). *If the system is exposed to changing electromagnetic fields (as in
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...
s,
inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s, and
transformer A transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple circuits. A varying current in any coil of the transformer produces a varying magnetic flux in the transformer' ...
s). *Under illumination from a light-source with a different temperature, such as the sun (as in
solar cell A solar cell, or photovoltaic cell, is an electronic device that converts the energy of light directly into electricity by the photovoltaic effect, which is a physical and chemical phenomenon.
s), *When the temperature is not constant within the device (as in
thermocouple A thermocouple, also known as a "thermoelectrical thermometer", is an electrical device consisting of two dissimilar electrical conductors forming an electrical junction. A thermocouple produces a temperature-dependent voltage as a result of th ...
s), *When the device has been altered, but has not had enough time to re-equilibrate (as in
piezoelectric Piezoelectricity (, ) is the electric charge that accumulates in certain solid materials—such as crystals, certain ceramics, and biological matter such as bone, DNA, and various proteins—in response to applied mechanical stress. The word '' ...
or
pyroelectric Pyroelectricity (from the two Greek words ''pyr'' meaning fire, and electricity) is a property of certain crystals which are naturally electrically polarized and as a result contain large electric fields. Pyroelectricity can be described as the a ...
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 A p–n junction is a boundary or interface between two types of semiconductor materials, p-type and n-type, inside a single crystal of semiconductor. The "p" (positive) side contains an excess of holes, while the "n" (negative) side contai ...
) when a current is being driven, and be ill-defined at the interface itself.


Technicalities


Terminology problems

The term ''Fermi level'' is mainly used in discussing the solid state physics of electrons in
semiconductor A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way ...
s, and a precise usage of this term is necessary to describe
band diagram In solid-state physics of semiconductors, a band diagram is a diagram plotting various key electron energy levels (Fermi level and nearby energy band edges) as a function of some spatial dimension, which is often denoted ''x''. These diagrams ...
s in devices comprising different materials with different levels of doping. In these contexts, however, one may also see Fermi level used imprecisely to refer to the ''band-referenced Fermi level'', ''µ'' − ''ϵ''C, called ''ζ'' above. 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 ''ϵ''C due to doping or the field effect. In fact,
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...
guarantees that the Fermi level in a conductor is ''always'' fixed to be exactly equal to the Fermi level of the electrodes; only the band structure (not the Fermi level) can be changed by doping or the field effect (see also
band diagram In solid-state physics of semiconductors, a band diagram is a diagram plotting various key electron energy levels (Fermi level and nearby energy band edges) as a function of some spatial dimension, which is often denoted ''x''. These diagrams ...
). A similar ambiguity exists between the terms, ''
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a speci ...
'' and ''
electrochemical potential In electrochemistry, the electrochemical potential (ECP), ', is a thermodynamic measure of chemical potential that does not omit the energy contribution of electrostatics. Electrochemical potential is expressed in the unit of J/ mol. Introductio ...
''. It is also important to note that Fermi ''level'' is not necessarily the same thing as Fermi ''energy''. In the wider context of quantum mechanics, the term
Fermi energy The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature. In a Fermi ga ...
usually refers to ''the maximum kinetic energy of a fermion in an idealized non-interacting, disorder free, zero temperature
Fermi gas An ideal Fermi gas is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer ...
''. 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 dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes ...
s,
neutron star A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. w ...
s,
atomic nuclei The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron ...
, and electrons in a
metal A metal (from ancient Greek, Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electrical resistivity and conductivity, e ...
. On the other hand, in the fields of semiconductor physics and engineering, ''Fermi energy'' often is used to refer to the Fermi level described in this article.


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. Observable phenomena only depend on energy differences. 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. 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 inherently ambiguous (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 In electrical engineering, ground or earth is a reference point in an electrical circuit from which voltages are measured, a common return path for electric current, or a direct physical connection to the Earth. Electrical circuits may be co ...
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 energies. 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 potential The Volta potential (also called Volta potential difference, contact potential difference, outer potential difference, Δψ, or "delta psi") in electrochemistry, is the electrostatic potential difference between two metals (or one metal and one ele ...
s). The source of this vacuum potential variation is the variation in
work function In solid-state physics, the work function (sometimes spelt workfunction) is the minimum thermodynamic work (i.e., energy) needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. Here "immediately ...
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 Fermi level suggested above. This also has the advantage that it can be measured with a voltmeter.


Discrete charging effects in small systems

In cases where the "charging effects" due to a single electron are non-negligible, the above definitions should be clarified. For example, consider a
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...
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 The nanoscopic scale (or nanoscale) usually refers to structures with a length scale applicable to nanotechnology, usually cited as 1–100 nanometers (nm). A nanometer is a billionth of a meter. The nanoscopic scale is (roughly speaking) a ...
capacitor it can be more important. 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? * When the body is able to exchange electrons and energy with an electrode (reservoir), it is described by the
grand canonical ensemble In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibriu ...
. The value of chemical potential can be said to be fixed by the electrode, and the number of electrons on the body may fluctuate. In this case, the chemical potential of a body is the infinitesimal amount of work needed to increase the ''average'' number of electrons by an infinitesimal amount (even though the number of electrons at any time is an integer, the average number varies continuously.): \mu(\left\langle N \right\rangle,T) = \left(\frac\right)_T, where is the free energy function of the grand canonical ensemble. * If the number of electrons in the body is fixed (but the body is still thermally connected to a heat bath), then it is in the
canonical ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the hea ...
. We can define a "chemical potential" in this case literally as the work required to add one electron to a body that already has exactly electrons, \mu'(N, T) = F(N + 1, T) - F(N, T), where is the free energy function of the canonical ensemble, alternatively, \mu''(N, T) = F(N, T) - F(N - 1, T) = \mu'(N - 1, T). These chemical potentials are not equivalent, , except in the
thermodynamic limit In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundel ...
. The distinction is important in small systems such as those showing
Coulomb blockade In mesoscopic physics, a Coulomb blockade (CB), named after Charles-Augustin de Coulomb's electrical force, is the decrease in electrical conductance at small bias voltages of a small electronic device comprising at least one low-capacitance t ...
. The parameter, , (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. To be precise, then, the Fermi level is defined not by a deterministic charging event by one electron charge, but rather a statistical charging event by an infinitesimal fraction of an electron.


Footnotes and references

{{Authority control Electronic band structures Fermi–Dirac statistics de:Fermienergie th:ระดับพลังงานแฟร์มี vi:Mức Fermi