A quasi Fermi level (also called imref, which is "fermi" spelled backwards) is a term used in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
and especially 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 l ...
for the
Fermi level (
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 species ...
of electrons) that describes the population of
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
s separately in the
conduction band
In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in w ...
and
valence band
In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in w ...
, when their
populations
Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...
are displaced from
equilibrium. This displacement could be caused by the application of an external voltage, or by exposure to light of
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
, which alter the populations of electrons in the conduction band and valence band. Since
recombination rate (the rate of equilibration between bands) tends to be much slower than the energy relaxation rate within each band, the conduction band and valence band can each have an individual population that is internally in equilibrium, even though the bands are not in equilibrium with respect to exchange of electrons. The displacement from equilibrium is such that the carrier populations can no longer be described by a single Fermi level, however it is possible to describe using concept of separate quasi-Fermi levels for each band.
Definition
When a semiconductor is in
thermal equilibrium
Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
, the distribution function of the electrons at the energy level of ''E'' is presented by a
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 ...
function. In this case the Fermi level is defined as the level in which the probability of occupation of electron at that energy is . In thermal equilibrium, there is no need to distinguish between conduction band quasi-Fermi level and valence band quasi-Fermi level as they are simply equal to the Fermi level.
When a disturbance from a thermal equilibrium situation occurs, the populations of the electrons in the conduction band and valence band change. If the disturbance is not too great or not changing too quickly, the bands each relax to a state of quasi thermal equilibrium. Because the
relaxation time
In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium.
Each relaxation process can be categorized by a relaxation time τ. The simplest theoretical description of relaxation as function of time ' ...
for electrons within the
conduction band
In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in w ...
is much lower than across 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 (in ...
, we can consider that the electrons are in thermal equilibrium in the conduction band. This is also applicable for electrons in the
valence band
In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in w ...
(often understood in terms of
hole
A hole is an opening in or through a particular medium, usually a solid body. Holes occur through natural and artificial processes, and may be useful for various purposes, or may represent a problem needing to be addressed in many fields of en ...
s). We can define a quasi Fermi level and quasi temperature due to thermal equilibrium of electrons in conduction band, and quasi Fermi level and quasi temperature for the valence band similarly.
We can state the general
Fermi function
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" an ...
for electrons in conduction band as
and for electrons in valence band as
where:
*
is 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 ...
function,
*
is the conduction band quasi-Fermi level at location ''r'',
*
is the valence band quasi-Fermi level at location ''r'',
*
is the conduction band temperature,
*
is the valence band temperature,
*
is the probability that a particular conduction-band state, with
wavevector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
''k'' and position ''r'', is occupied by an electron,
*
is the probability that a particular valence-band state, with wavevector ''k'' and position ''r'', is occupied by an electron (i.e. ''not'' occupied by a hole).
*
is the energy of the conduction- or valence-band state in question,
*
is
Boltzmann's 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 constant, ...
.
p-n junction
As shown in the figure below, the
conduction band
In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in w ...
and
valence band
In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in w ...
in a
p-n junction is indicated by blue solid line in the left, and quasi Fermi level is indicated by the red dashed line.
When there is no external voltage(bias) applied to a p-n junction, the quasi Fermi levels for electron and holes overlap with one another. As bias increase, the valence band of the p-side gets pulled down, and so did the hole quasi Fermi level. As a result separation of hole and electron quasi Fermi level increased.
![PN band](https://upload.wikimedia.org/wikipedia/commons/2/22/PN_band.gif)
Application
This simplification will help us in many areas. For example, we can use the same equation for electron and hole densities used in thermal equilibrium, but substituting the quasi-Fermi levels and temperature. That is, if we let
be the spatial density of conduction band electrons and
be the spatial density of holes in a material, and if the
Boltzmann approximation holds, i.e. assuming the electron and hole densities are not too high, then
where
is the spatial density of conduction band electrons that would be present in thermal equilibrium if the Fermi level were at
, and
is the spatial density of holes that would be present in thermal equilibrium if the Fermi level were at
.
A current (due to the combined effects of
drift
Drift or Drifts may refer to:
Geography
* Drift or ford (crossing) of a river
* Drift, Kentucky, unincorporated community in the United States
* In Cornwall, England:
** Drift, Cornwall, village
** Drift Reservoir, associated with the village
...
and
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
) will only appear if there is a variation in the Fermi or quasi Fermi level. The current density for electron flow can be shown to be proportional to the gradient in the electron quasi Fermi level. For if we let
be the
electron mobility, and
be the quasi fermi energy at the spatial point
, then we have
Similarly, for holes, we have
Further reading
*{{Cite book, url=https://books.google.com/books?id=s5NN34HLWO8C, title=The Physics of Solar Cells, last=Nelson, first=Jenny, date=2003-01-01, publisher=Imperial College Press, isbn=9781860943492, language=en
Electronic band structures
Fermi–Dirac statistics