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 ${\textstyle {\text{ℰ}}=\varepsilon -\varepsilon _{\rm {C}}.}$ We can define a parameter ζ^[8] that references the Fermi level with respect to the band edge:

$${\displaystyle \zeta =\mu -\epsilon _{\rm {C}}.}$$

It follows that the Fermi–Dirac distribution function can be written as

$${\displaystyle f({\mathcal {E}})={\frac {1}{e^{({\mathcal {E}}-\zeta )/k_{\mathrm {B} }T}+1}}.}$$

The band theory 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). 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 (even more bands in other materials);^[9] 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 (again), the name Fermi energy sometimes is used to refer to ζ at non-zero temperature.

Terminology problems

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. 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 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). A similar ambiguity exists between the terms, chemical potential and electrochemical potential.

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 usually refers to the maximum kinetic energy of a fermion in an idealized non-interacting, disorder free, zero temperature Fermi gas. 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. 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.^[10]

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 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.

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 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 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. The value of chemical potential µ can be said to be fixed by the electrode, and the number of electrons N 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.):
  $${\displaystyle \mu (\left\langle N\right\rangle ,T)=\left({\frac {\partial F}{\partial \left\langle N\right\rangle }}\right)_{T},}$$

  
  where F(N, T) 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. 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 N electrons,^[12]
  $${\displaystyle \mu '(N,T)=F(N+1,T)-F(N,T),}$$

  
  where F(N, T) is the free energy function of the canonical ensemble, alternatively,
  $${\displaystyle \mu ''(N,T)=F(N,T)-F(N-1,T)=\mu '(N-1,T).}$$

  

These chemical potentials are not equivalent, µ ≠ µ′ ≠ µ″, except in the thermodynamic limit. The distinction is important in small systems such as those showing Coulomb blockade.^[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. 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.