Editing Fermi level (section)

===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''<sub>C</sub>, then in general we have <math display="inline">\text{ℰ} = \varepsilon - \varepsilon_{\rm C}.
</math> We can define a parameter ''ζ''<ref>{{cite book | author=Sommerfeld, Arnold | title= Thermodynamics and Statistical Mechanics | publisher=Academic Press | year=1964}}</ref> that references the Fermi level with respect to the band edge:<math display="block">\zeta = \mu - \epsilon_{\rm C}.</math>It follows that the Fermi–Dirac distribution function can be written as<math display="block">f(\mathcal{E}) = \frac{1}{e^{(\mathcal{E} - \zeta)/k_\mathrm{B} T} + 1}. </math>The [[electronic band structure|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''.

[[File:HEMT-band structure scheme-en.svg|thumb|270px|Example of variations in conduction band edge ''E''<sub>C</sub> 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 (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''<sub>C</sub> 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''<sub>C</sub>, 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);<ref>{{cite web|url=http://www.phys.ufl.edu/~tschoy/r2d2/Fermi/Fermi.html |title=3D Fermi Surface Site |publisher=Phys.ufl.edu |date=1998-05-27 |access-date=2013-04-22}}</ref> each band has a different edge energy, ''ϵ''<sub>C</sub>, and a different ''ζ''.

The value of ''ζ'' at [[absolute zero|zero temperature]] is widely known as the [[Fermi energy]], sometimes written ''ζ''<sub>0</sub>. Confusingly (again), the name ''Fermi energy'' sometimes is used to refer to ''ζ'' at non-zero temperature.