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 ...
and
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 phases which arise from electromagnetic forces between atoms. More generally, the sub ...
, the density of states (DOS) of a system describes the number of modes per unit frequency 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), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
, 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 d ...
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 m ...
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, a g ...
, like a
spectral density
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
. 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 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 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 (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
in the dimension of the propagation. The result of the number of states in a
band
Band or BAND may refer to:
Places
*Bánd, a village in Hungary
*Band, Iran, a village in Urmia County, West Azerbaijan Province, Iran
* Band, Mureș, a commune in Romania
*Band-e Majid Khan, a village in Bukan County, West Azerbaijan Province, I ...
is also useful for predicting the conduction properties. For example, in a one dimensional crystalline structure an odd number 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 per atom results in a half-filled top band; there are free electrons at the
Fermi level 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 which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...
.
Depending on the quantum mechanical system, the density of states can be calculated for
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,
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, so they always ...
s, or
phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
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), ...
''k''. 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 ''E'' and ''k'' 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
neutronium
Neutronium (sometimes shortened to neutrium, also referred to as neutrite) is a hypothetical substance composed purely of neutrons. The word was coined by scientist Andreas von Antropoff in 1926 (before the 1932 discovery of the neutron) for the ...
in
neutron star
A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. w ...
s and
free electron gases in metals (examples of
degenerate matter
Degenerate matter is a highly dense state of fermionic matter in which the Pauli exclusion principle exerts significant pressure in addition to, or in lieu of, thermal pressure. The description applies to matter composed of electrons, protons, neu ...
and 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 sp ...
), 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 distance, Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative f ...
. Less familiar systems, like
two-dimensional electron gas
A two-dimensional electron gas (2DEG) is a scientific model in solid-state physics. It is an electron gas that is free to move in two dimensions, but tightly confined in the third. This tight confinement leads to quantized energy levels for motion ...
es (2DEG) in
graphite
Graphite () is a crystalline form of the element carbon. It consists of stacked layers of graphene. Graphite occurs naturally and is the most stable form of carbon under standard conditions. Synthetic and natural graphite are consumed on large ...
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 exh ...
system in
MOSFET
The metal–oxide–semiconductor field-effect transistor (MOSFET, MOS-FET, or MOS FET) is a type of field-effect transistor (FET), most commonly fabricated by the controlled oxidation of silicon. It has an insulated gate, the voltage of which d ...
type devices, have a 2-dimensional Euclidean topology. Even less familiar are
carbon nanotubes
A scanning tunneling microscopy image of a single-walled carbon nanotube
Rotating single-walled zigzag carbon nanotube
A carbon nanotube (CNT) is a tube made of carbon with diameters typically measured in nanometers.
''Single-wall carbon nan ...
, the
quantum wire In mesoscopic physics, a quantum wire is an electrically conducting wire in which quantum effects influence the transport properties. Usually such effects appear in the dimension of nanometers, so they are also referred to as nanowires.
Quantum e ...
and
Luttinger liquid
A Luttinger liquid, or Tomonaga–Luttinger liquid, is a theoretical model describing interacting electrons (or other fermions) in a one-dimensional conductor (e.g. quantum wires such as carbon nanotubes). Such a model is necessary as the commonl ...
with their 1-dimensional topologies. Systems with 1D and 2D topologies are likely to become more common, assuming developments in
nanotechnology
Nanotechnology, also shortened to nanotech, is the use of matter on an atomic, molecular, and supramolecular scale for industrial purposes. The earliest, widespread description of nanotechnology referred to the particular technological goal o ...
and
materials science proceed.
Definition
The density of states related to volume ''V'' and ''N'' 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 ...
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
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 continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
s,
glass
Glass is a non-crystalline, often transparent, amorphous solid that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most often formed by rapid cooling (quenching) of ...
es and
amorphous solid
In condensed matter physics and materials science, an amorphous solid (or non-crystalline solid, glassy solid) is a solid that lacks the long-range order that is characteristic of a crystal.
Etymology
The term comes from the Greek ''a'' ("wit ...
s are examples of a symmetric system whose
dispersion relations
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 d ...
have a rotational symmetry.
Measurements on powders or polycrystalline samples require evaluation and calculation functions and integrals over the whole
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
, most often a
Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice ...
, 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.
[
] The Brillouin zone of the
face-centered cubic lattice
In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.
There are three main varieties o ...
(FCC) in the figure on the right has the 48-fold symmetry of the
point group ''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. Another solid form of carbon known as graphite is the Chemical stability, chemically stable form of car ...
,
silicon
Silicon is a chemical element with the symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic luster, and is a tetravalent metalloid and semiconductor. It is a member of group 14 in the periodic tab ...
and
platinum
Platinum is a chemical element with the symbol Pt and atomic number 78. It is a dense, malleable, ductile, highly unreactive, precious, silverish-white transition metal. Its name originates from Spanish , a diminutive of "silver".
Platinu ...
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
image:tetrahedron.jpg, 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that c ...
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 property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
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 boundaries.RIWD. "Re ...
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 Energy
−1Volume
−1 , in a two dimensional system, the units of DOS is Energy
−1Area
−1 , in a one dimensional system, the units of DOS is Energy
−1Length
−1. The referenced volume is the volume of ''k''-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 d ...
that relates ''E'' to ''k''. An example of a 3-dimensional ''k''-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 ''N'' allowed states at a certain ''k'' 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 ''k'', with respect to ''k''. The volume, area or length in 3, 2 or 1-dimensional spherical ''k''-spaces are expressed by
:
for a n-dimensional ''k''-space with the topologically determined constants
:
for linear, disk and spherical symmetrical shaped functions in 1, 2 and 3-dimensional Euclidean ''k''-spaces respectively.
According to this scheme, the density of wave vector states ''N'' is, through differentiating
with respect to ''k'', 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 ''k'' 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 ''s'' 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