The Fermi energy is a concept 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, ...
usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s at
absolute zero
Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibration ...
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
.
In a
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 spin. ...
, the lowest occupied state is taken to have zero kinetic energy, whereas in a metal, the lowest occupied state is typically taken to mean the bottom of 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 ...
.
The term "Fermi energy" is often used to refer to a different yet closely related concept, the
Fermi ''level'' (also called
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 ...
).
[The use of the term "Fermi energy" as synonymous with ]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 ...
(a.k.a. 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 ...
) is widespread in semiconductor physics. For example:
''Electronics (fundamentals And Applications)''
by D. Chattopadhyay
''Semiconductor Physics and Applications''
by Balkanski and Wallis.
There are a few key differences between the Fermi level and Fermi energy, at least as they are used in this article:
* The Fermi energy is only defined at absolute zero, while the Fermi level is defined for any temperature.
* The Fermi energy is an energy ''difference'' (usually corresponding to a
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 accele ...
), whereas the Fermi level is a total energy level including kinetic energy and potential energy.
* The Fermi energy can only be defined for
non-interacting fermions (where the potential energy or band edge is a static, well defined quantity), whereas the Fermi level remains well defined even in complex interacting systems, at thermodynamic equilibrium.
Since the Fermi level in a metal at absolute zero is the energy of the highest occupied single particle state,
then the Fermi energy in a metal is the energy difference between the Fermi level and lowest occupied single-particle state, at zero-temperature.
Context
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, ...
, a group of particles known as
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s (for example,
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,
proton
A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
s and
neutron
The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
s) obey the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
. This states that two fermions cannot occupy the same
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
. Since an idealized non-interacting Fermi gas can be analyzed in terms of single-particle
stationary state
A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (instead of a quantum superposition of different energies). It is also called energy eigenvector, energy eigenstate, ener ...
s, we can thus say that two fermions cannot occupy the same stationary state. These stationary states will typically be distinct in energy. To find the ground state of the whole system, we start with an empty system, and add particles one at a time, consecutively filling up the unoccupied stationary states with the lowest energy. When all the particles have been put in, the Fermi energy is the kinetic energy of the highest occupied state.
As a consequence, even if we have extracted all possible energy from a Fermi gas by cooling it to near
absolute zero
Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibration ...
temperature, the fermions are still moving around at a high speed. The fastest ones are moving at a velocity corresponding to a kinetic energy equal to the Fermi energy. This speed is known as the Fermi velocity. Only when the temperature exceeds the related Fermi temperature, do the particles begin to move significantly faster than at absolute zero.
The Fermi energy is an important concept in the
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 ...
of metals and
superconductors. It is also a very important quantity in the physics of
quantum liquids like low temperature
helium
Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic table. ...
(both normal and superfluid
3He), and it is quite important to
nuclear physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter.
Nuclear physics should not be confused with atomic physics, which studies the ...
and to understanding the stability of
white dwarf stars
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 fro ...
against
gravitational collapse
Gravitational collapse is the contraction of an astronomical object due to the influence of its own gravity, which tends to draw matter inward toward the center of gravity. Gravitational collapse is a fundamental mechanism for structure formatio ...
.
Formula and typical values
The Fermi energy for a ''non-interacting'' ensemble of
identical
Two things are identical if they are the same, see Identity (philosophy).
Identical may also refer to:
* ''Identical'' (Hopkins novel), a 2008 young adult novel by Ellen Hopkins
* ''Identical'' (Turow novel), a 2013 legal drama novel by Scott T ...
spin- fermions in a three-dimensional (non-
relativistic) system is given by
where ''N'' is the number of particles, ''m''
0 the
rest mass
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, i ...
of each fermion, ''V'' the volume of the system, and
the reduced
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
.
Metals
Under the
free electron model
In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quant ...
, the electrons in a metal can be considered to form a Fermi gas. The number density
of conduction electrons in metals ranges between approximately 10
28 and 10
29 electrons/m
3, which is also the typical density of atoms in ordinary solid matter. This number density produces a Fermi energy of the order of 2 to 10
electronvolts
In physics, an electronvolt (symbol eV, also written electron-volt and electron volt) is the measure of an amount of kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt in vacuum. ...
.
White dwarfs
Stars known as
white dwarfs
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 fro ...
have mass comparable to our
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
, but have about a hundredth of its radius. The high densities mean that the electrons are no longer bound to single nuclei and instead form a degenerate electron gas. Their Fermi energy is about 0.3 MeV.
Nucleus
Another typical example is that of the
nucleon
In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number).
Until the 1960s, nucleons were ...
s in the nucleus of an atom. The
radius of the nucleus admits deviations, so a typical value for the Fermi energy is usually given as 38
MeV
In physics, an electronvolt (symbol eV, also written electron-volt and electron volt) is the measure of an amount of kinetic energy gained by a single electron accelerating from rest through an Voltage, electric potential difference of one volt i ...
.
Related quantities
Using this definition of above for the Fermi energy, various related quantities can be useful.
The Fermi temperature is defined as
where
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 constant, ...
, and
the Fermi energy. The Fermi temperature can be thought of as the temperature at which thermal effects are comparable to quantum effects associated with
Fermi statistics
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" and ...
. The Fermi temperature for a metal is a couple of orders of magnitude above room temperature.
Other quantities defined in this context are Fermi momentum
and Fermi velocity
These quantities are respectively the
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
and
group velocity
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space.
For example, if a stone is thrown into the middl ...
of a
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
at the
Fermi surface In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature. The shape of the Fermi surface is derived from the periodicity and symmetry of the cryst ...
.
The Fermi momentum can also be described as
where
, called the Fermi wavevector, is the radius of the Fermi sphere.
These quantities may ''not'' be well-defined in cases where the
Fermi surface In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature. The shape of the Fermi surface is derived from the periodicity and symmetry of the cryst ...
is non-spherical.
See also
*
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 Pauli exclusion principle. A result is the Fermi–Dirac di ...
: the distribution of electrons over stationary states for non-interacting fermions at ''non-zero'' temperature.
*
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 ...
*
Quasi Fermi level
A quasi Fermi level (also called imref, which is "fermi" spelled backwards) is a term used in quantum mechanics and especially in solid state physics for the Fermi level (chemical potential of electrons) that describes the population of electrons ...
Notes
References
Further reading
*
{{Authority control
Condensed matter physics
Fermi–Dirac statistics