In
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
, the density of states (DOS) of a system describes the number of allowed modes or
states
State most commonly refers to:
* State (polity), a centralized political organization that regulates law and society within a territory
**Sovereign state, a sovereign polity in international law, commonly referred to as a country
**Nation state, a ...
per unit energy range. The density of states is defined as where
is the number of states in the system of volume
whose energies lie in the range from
to
. It is mathematically represented as a distribution by a
probability density function
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
, and it is generally an average over the space and time domains of the various states occupied by the system. The density of states is directly related to the
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the ...
s of the properties of the system. High DOS at a specific energy level means that many states are available for occupation.
Generally, the density of states of matter is continuous. In
isolated system
In physical science, an isolated system is either of the following:
# a physical system so far removed from other systems that it does not interact with them.
# a thermodynamic system enclosed by rigid immovable walls through which neither ...
s however, such as atoms or molecules in the gas phase, the density distribution is
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
* Discrete group, ...
, like a
spectral density
In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be decomposed into ...
. Local variations, most often due to distortions of the original system, are often referred to as local densities of states (LDOSs).
Introduction
In quantum mechanical systems, waves, or wave-like particles, can occupy modes or states with wavelengths and propagation directions dictated by the system. For example, in some systems, the interatomic spacing and the atomic charge of a material might allow only electrons of certain wavelengths to exist. In other systems, the crystalline structure of a material might allow waves to propagate in one direction, while suppressing wave propagation in another direction. Often, only specific states are permitted. Thus, it can happen that many states are available for occupation at a specific energy level, while no states are available at other energy levels.
Looking at the density of states of electrons at the band edge between the
valence and conduction bands in a semiconductor, for an electron in the conduction band, an increase of the electron energy makes more states available for occupation. Alternatively, the density of states is discontinuous for an interval of energy, which means that no states are available for electrons to occupy within the band gap of the material. This condition also means that an electron at the conduction band edge must lose at least the band gap energy of the material in order to transition to another state in the valence band.
This determines if the material is an
insulator or a
metal
A metal () is a material that, when polished or fractured, shows a lustrous appearance, and conducts electrical resistivity and conductivity, electricity and thermal conductivity, heat relatively well. These properties are all associated wit ...
in the dimension of the propagation. The result of the number of states in a
band is also useful for predicting the conduction properties. For example, in a one dimensional crystalline structure an odd number of
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s per atom results in a half-filled top band; there are free electrons at the
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 re ...
resulting in a metal. On the other hand, an even number of electrons exactly fills a whole number of bands, leaving the rest empty. If then the Fermi level lies in an occupied band gap between the highest occupied state and the lowest empty state, the material will be an insulator or
semiconductor
A semiconductor is a material with electrical conductivity between that of a conductor and an insulator. Its conductivity can be modified by adding impurities (" doping") to its crystal structure. When two regions with different doping level ...
.
Depending on the quantum mechanical system, the density of states can be calculated for
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s,
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
s, or
phonon
A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. In the context of optically trapped objects, the quantized vibration mode can be defined a ...
s, and can be given as a function of either energy or the
wave vector
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) ...
. To convert between the DOS as a function of the energy and the DOS as a function of the wave vector, the system-specific energy dispersion relation between and must be known.
In general, the topological properties of the system such as the band structure, have a major impact on the properties of the density of states. The most well-known systems, like
neutron matter in
neutron star
A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
s and
free electron gases in metals (examples of
degenerate matter and a
Fermi gas
A Fermi gas is an idealized model, 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. These statis ...
), have a 3-dimensional
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdot ...
. Less familiar systems, like
two-dimensional electron gases (2DEG) in
graphite
Graphite () is a Crystallinity, crystalline allotrope (form) of the element carbon. It consists of many stacked Layered materials, layers of graphene, typically in excess of hundreds of layers. Graphite occurs naturally and is the most stable ...
layers and the
quantum Hall effect
The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance exhi ...
system in
MOSFET
upright=1.3, Two power MOSFETs in amperes">A in the ''on'' state, dissipating up to about 100 watt">W and controlling a load of over 2000 W. A matchstick is pictured for scale.
In electronics, the metal–oxide–semiconductor field- ...
type devices, have a 2-dimensional Euclidean topology. Even less familiar are
carbon nanotubes
A carbon nanotube (CNT) is a tube made of carbon with a diameter in the nanometre range (nanoscale). They are one of the allotropes of carbon. Two broad classes of carbon nanotubes are recognized:
* ''Single-walled carbon nanotubes'' (''SWC ...
, the
quantum wire and
Luttinger liquid with their 1-dimensional topologies. Systems with 1D and 2D topologies are likely to become more common, assuming developments in
nanotechnology
Nanotechnology is the manipulation of matter with at least one dimension sized from 1 to 100 nanometers (nm). At this scale, commonly known as the nanoscale, surface area and quantum mechanical effects become important in describing propertie ...
and
materials science
Materials science is an interdisciplinary field of researching and discovering materials. Materials engineering is an engineering field of finding uses for materials in other fields and industries.
The intellectual origins of materials sci ...
proceed.
Definition
The density of states related to volume and countable energy levels is defined as:
Because the smallest allowed change of momentum
for a particle in a box of dimension
and length
is
, the volume-related density of states for continuous energy levels is obtained in the limit
as
Here,
is the spatial dimension of the considered system and
the wave vector.
For isotropic one-dimensional systems with parabolic energy dispersion, the density of states is In two dimensions the density of states is a constant while in three dimensions it becomes
Equivalently, the density of states can also be understood as the derivative of the microcanonical partition function
(that is, the total number of states with energy less than
) with respect to the energy:
The number of states with energy
(degree of degeneracy) is given by:
where the last equality only applies when the mean value theorem for integrals is valid.
Symmetry
There is a large variety of systems and types of states for which DOS calculations can be done.
Some condensed matter systems possess a
structural
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
symmetry
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
on the microscopic scale which can be exploited to simplify calculation of their densities of states. In spherically symmetric systems, the integrals of functions are one-dimensional because all variables in the calculation depend only on the radial parameter of the dispersion relation.
Fluid
In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
s,
glass
Glass is an amorphous (non-crystalline solid, non-crystalline) solid. Because it is often transparency and translucency, transparent and chemically inert, glass has found widespread practical, technological, and decorative use in window pane ...
es and
amorphous solid
In condensed matter physics and materials science, an amorphous solid (or non-crystalline solid) is a solid that lacks the long-range order that is a characteristic of a crystal. The terms "glass" and "glassy solid" are sometimes used synonymousl ...
s are examples of a symmetric system whose
dispersion relations have a rotational symmetry.

Measurements on powders or polycrystalline samples require evaluation and calculation functions and integrals over the whole
domain, most often a
Brillouin zone
In mathematics and solid state physics, the first Brillouin zone (named after Léon Brillouin) is a uniquely defined primitive cell in reciprocal space
Reciprocal lattice is a concept associated with solids with translational symmetry whic ...
, of the dispersion relations of the system of interest. Sometimes the symmetry of the system is high, which causes the shape of the functions describing the dispersion relations of the system to appear many times over the whole domain of the dispersion relation. In such cases the effort to calculate the DOS can be reduced by a great amount when the calculation is limited to a reduced zone or
fundamental domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each ...
.
The Brillouin zone of the
face-centered cubic lattice (FCC) in the figure on the right has the 48-fold symmetry of the
point group
In geometry, a point group is a group (mathematics), mathematical group of symmetry operations (isometry, isometries in a Euclidean space) that have a Fixed point (mathematics), fixed point in common. The Origin (mathematics), coordinate origin o ...
''O
h'' with full
octahedral symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
. This configuration means that the integration over the whole domain of the Brillouin zone can be reduced to a 48-th part of the whole Brillouin zone. As a
crystal structure periodic table shows, there are many elements with a FCC crystal structure, like
diamond
Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Diamond is tasteless, odourless, strong, brittle solid, colourless in pure form, a poor conductor of e ...
,
silicon
Silicon is a chemical element; it has symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic lustre, and is a tetravalent metalloid (sometimes considered a non-metal) and semiconductor. It is a membe ...
and
platinum
Platinum is a chemical element; it has Symbol (chemistry), symbol Pt and atomic number 78. It is a density, dense, malleable, ductility, ductile, highly unreactive, precious metal, precious, silverish-white transition metal. Its name origina ...
and their Brillouin zones and dispersion relations have this 48-fold symmetry. Two other familiar crystal structures are the body-centered cubic lattice (BCC) and hexagonal closed packed structures (HCP) with cubic and hexagonal lattices, respectively. The BCC structure has the 24-fold
pyritohedral symmetry of the point group ''T
h''. The HCP structure has the 12-fold
prismatic dihedral symmetry of the point group ''D
3h''. A complete list of symmetry properties of a point group can be found in
point group character tables.
In general it is easier to calculate a DOS when the symmetry of the system is higher and the number of topological dimensions of the dispersion relation is lower. The DOS of dispersion relations with rotational symmetry can often be calculated analytically. This result is fortunate, since many materials of practical interest, such as steel and silicon, have high symmetry.
In
anisotropic
Anisotropy () is the structural property of non-uniformity in different directions, as opposed to isotropy. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit ver ...
condensed matter systems such as a
single crystal
In materials science, a single crystal (or single-crystal solid or monocrystalline solid) is a material in which the crystal lattice of the entire sample is continuous and unbroken to the edges of the sample, with no Grain boundary, grain bound ...
of a compound, the density of states could be different in one crystallographic direction than in another. These causes the anisotropic density of states to be more difficult to visualize, and might require methods such as calculating the DOS for particular points or directions only, or calculating the projected density of states (PDOS) to a particular crystal orientation.
''k''-space topologies

The density of states is dependent upon the dimensional limits of the object itself. In a system described by three orthogonal parameters (3 Dimension), the units of DOS is in a two dimensional system, the units of DOS is in a one dimensional system, the units of DOS is The referenced volume is the volume of -space; the space enclosed by the
constant energy surface of the system derived through a
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the ...
that relates to . An example of a 3-dimensional -space is given in Fig. 1. It can be seen that the dimensionality of the system confines the momentum of particles inside the system.
Density of wave vector states (sphere)
The calculation for DOS starts by counting the allowed states at a certain that are contained within inside the volume of the system. This procedure is done by differentiating the whole k-space volume
in n-dimensions at an arbitrary , with respect to . The volume, area or length in 3, 2 or 1-dimensional spherical -spaces are expressed by
for a -dimensional -space with the topologically determined constants
for linear, disk and spherical symmetrical shaped functions in 1, 2 and 3-dimensional Euclidean -spaces respectively.
According to this scheme, the density of wave vector states is, through differentiating
with respect to , expressed by
The 1, 2 and 3-dimensional density of wave vector states for a line, disk, or sphere are explicitly written as
One state is large enough to contain particles having wavelength λ. The wavelength is related to through the relationship.
In a quantum system the length of λ will depend on a characteristic spacing of the system L that is confining the particles. Finally the density of states ''N'' is multiplied by a factor
, where is a constant degeneracy factor that accounts for internal degrees of freedom due to such physical phenomena as spin or polarization. If no such phenomenon is present then
. ''V
k'' is the volume in k-space whose wavevectors are smaller than the smallest possible wavevectors decided by the characteristic spacing of the system.
Density of energy states
To finish the calculation for DOS find the number of states per unit sample volume at an energy
inside an interval