Fermi level

The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic 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. The concept of the Fermi level is an important component of the electronic band structure model for determining electronic properties, especially as it relates to the voltage and flow of charge in electronic circuits.
In band structure theory, used in solid state physics 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 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, 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 with a voltmeter.

Voltage measurement

Sometimes it is said that electric currents are driven by differences in electrostatic 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, yet without any accompanying net current; if a voltmeter is attached to the junction, one simply measures zero volts.
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.
In fact, the quantity called voltage as measured in an electronic circuit has a simple relationship to the chemical potential for electrons.
When the leads of a voltmeter 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, 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
where −e is the electron charge.
From the above discussion it can be seen that electrons will move from a body of high μ to low μ if a simple path is provided.
This flow of electrons will cause the lower μ to increase 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 state of an electronic circuit:
This also means that the voltage between any two points will be zero, at equilibrium.
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.

Band structure of solids

In the band theory of solids, electrons 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,, gives the probability that a state having energy ϵ is occupied by an electron:
Here, T is the absolute temperature and k_B is the Boltzmann constant. If there is a state at the Fermi level, 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, 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 entire band structure to shift up and down. For further information about the Fermi levels of semiconductors, see 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 We can define a parameter ζ that references the Fermi level with respect to the band edge:It follows that the Fermi–Dirac distribution function can be written as: The band theory of metals was initially developed by Sommerfeld, from 1927 onwards, who focused on 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.
File:HEMT-band structure scheme-en.svg|thumb|270px|Example of variations in conduction band edge E_C in a band diagram of GaAs/AlGaAs heterojunction-based high-electron-mobility transistor.
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.
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, 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. In a multi-band material, ζ may even take on multiple values in a single location.
For example, in a piece of aluminum there are two conduction bands crossing the Fermi level ; 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, sometimes written ζ₀. Confusingly, the name Fermi energy sometimes is used to refer to ζ at non-zero 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 of a piece of metal or its thermal conductivity. The quasi-μ and quasi-T can vary in any non-equilibrium situation, such as:

If the system contains a chemical imbalance.
If the system is exposed to changing electromagnetic fields.
Under illumination from a light-source with a different temperature, such as the sun,
When the temperature is not constant within the device,
When the device has been altered, but has not had enough time to re-equilibrate.

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. Even then, the values of μ and T may jump discontinuously across a material interface when a current is being driven, and be ill-defined at the interface itself.