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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 D(E) = N(E)/V , where N(E)\delta E is the number of states in the system of volume V whose energies lie in the range from E to E+\delta E. 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: : D(E) = \frac \, \sum _^N \delta (E - E(_i)). Because the smallest allowed change of momentum k for a particle in a box of dimension d and length L is (\Delta k)^d = (\tfrac)^d , the volume-related density of states for continuous energy levels is obtained in the limit L \to \infty as : D(E) := \int_ \cdot \delta (E - E(\mathbf)), Here, d is the spatial dimension of the considered system and \mathbf the wave vector. For isotropic one-dimensional systems with parabolic energy dispersion, the density of states isD_(E)=\tfrac(\tfrac)^. In two dimensions the density of states is a constant D_=\tfrac, while in three dimensions it becomes D_(E)=\tfrac(2mE)^. Equivalently, the density of states can also be understood as the derivative of the microcanonical partition function Z_m (E) (that is, the total number of states with energy less than E) with respect to the energy: : D(E) = \frac \cdot \frac. The number of states with energy E' (degree of degeneracy) is given by: : g\left(E'\right) = \lim _ \int _^ D(E) \mathrm E = \lim _ D\left(E'\right) \Delta E, 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 ''Oh'' 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 ''Th''. The HCP structure has the 12-fold prismatic dihedral symmetry of the point group ''D3h''. 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 \Omega_ 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 : \Omega_n(k) = c_n k^n for a n-dimensional ''k''-space with the topologically determined constants : c_1 = 2,\ c_2 = \pi,\ c_3 = \frac 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 \Omega_ with respect to ''k'', expressed by :N_n(k) = \frac = n\; c_n\; k^ The 1, 2 and 3-dimensional density of wave vector states for a line, disk, or sphere are explicitly written as : \begin N_1(k) &= 2 \\ N_2(k) &= 2 \pi k \\ N_3(k) &= 4 \pi k^2 \end One state is large enough to contain particles having wavelength λ. The wavelength is related to ''k'' through the relationship. : k = \frac 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 ''s/V_k'', 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 s = 1. ''Vk'' 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 E inside an interval , E+dE/math>. The general form of DOS of a system is given as :D_n\left(E\right) = \frac The scheme sketched so far ''only'' applies to ''monotonically rising'' and ''spherically symmetric'' dispersion relations. In general the dispersion relation E(k) is not spherically symmetric and in many cases it isn't continuously rising either. To express ''D'' as a function of ''E'' the inverse of the dispersion relation E(k) has to be substituted into the expression of \Omega_n(k) as a function of ''k'' to get the expression of \Omega_n(E) as a function of the energy. If the dispersion relation is not spherically symmetric or continuously rising and can't be inverted easily then in most cases the DOS has to be calculated numerically. More detailed derivations are available.


Dispersion relations

The dispersion relation for electrons in a solid is given by the
electronic band structure In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
. The
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 ...
of a particle depends on the magnitude and direction of 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'', the properties of the particle and the environment in which the particle is moving. For example, the kinetic energy of an
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 ...
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 sp ...
is given by : E = E_0 + \frac \ , where ''m'' is the
electron mass The electron mass (symbol: ''m''e) is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about or about , which has an energy-equivalent of ...
. The dispersion relation is a spherically symmetric parabola and it is continuously rising so the DOS can be calculated easily. For longitudinal
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 in a string of atoms the dispersion relation of the kinetic energy in a 1-dimensional ''k''-space, as shown in Figure 2, is given by :E = 2 \hbar \omega_0 \left, \sin\left(\frac\right)\ where \omega_0 = \sqrt is the oscillator frequency, m the mass of the atoms, k_ the inter-atomic force constant and a inter-atomic spacing. For small values of k \ll \pi / a the dispersion relation is rather linear: :E = \hbar \omega_0 ka When k \approx \pi / a the energy is :E = 2 \hbar \omega_0 \left, \cos\left(\frac\right)\ With the transformation q = k - \pi/a and small q this relation can be transformed to :E = 2 \hbar \omega_0 \left - \left(\frac\right)^2\right


Isotropic dispersion relations

The two examples mentioned here can be expressed like :E = E_0 + c_k k^p This expression is a kind of
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 ...
because it interrelates two wave properties and it is
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
because only the length and not the direction of the wave vector appears in the expression. The magnitude of the wave vector is related to the energy as: :k = \left(\frac\right)^\frac \ , Accordingly, the volume of n-dimensional ''k''-space containing wave vectors smaller than ''k'' is: :\Omega_n(k) = c_n k^n Substitution of the isotropic energy relation gives the volume of occupied states :\Omega_n(E) = \frac\left(E - E_0\right)^\frac\ , Differentiating this volume with respect to the energy gives an expression for the DOS of the isotropic dispersion relation :D_n\left(E\right) = \frac \Omega_n(E) = \frac \left(E - E_0\right)^


Parabolic dispersion

In the case of a parabolic dispersion relation (''p'' = 2), such as applies to free electrons in a Fermi gas, the resulting density of states, D_n\left(E\right), for electrons in a n-dimensional systems is :\begin D_1\left(E\right) &= \frac \\ D_2\left(E\right) &= \frac \\ D_3\left(E\right) &= \pi \sqrt \ . \end for E > E_0, with D(E) = 0 for E < E_0. In 1-dimensional systems the DOS diverges at the bottom of the band as E drops to E_0. In 2-dimensional systems the DOS turns out to be independent of E. Finally for 3-dimensional systems the DOS rises as the square root of the energy. Including the prefactor ''s/V_k'', the expression for the 3D DOS is :N(E) = \frac \left(\frac\right)^\frac\sqrt, where V is the total volume, and N(E-E_0) includes the 2-fold spin degeneracy.


Linear dispersion

In the case of a linear relation (''p'' = 1), such as applies to
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,
acoustic phonon In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanical ...
s, or to some special kinds of electronic bands in a solid, the DOS in 1, 2 and 3 dimensional systems is related to the energy as: :\begin D_1\left(E\right) &= \frac \\ D_2\left(E\right) &= \frac\left(E - E_0\right) \\ D_3\left(E\right) &= \frac\left(E - E_0\right)^2 \end


Distribution functions

The density of states plays an important role in the
kinetic theory of solids In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanical ...
. The product of the density of states and the
probability distribution function Probability distribution function may refer to: * Probability distribution * Cumulative distribution function * Probability mass function * Probability density function In probability theory, a probability density function (PDF), or density ...
is the number of occupied states per unit volume at a given energy for a system in thermal equilibrium. This value is widely used to investigate various physical properties of matter. The following are examples, using two common distribution functions, of how applying a distribution function to the density of states can give rise to physical properties.
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 Fermi–Dirac probability distribution function, Fig. 4, is used to find the probability that a fermion occupies a specific quantum state in a system at thermal equilibrium.
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 are particles which 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 ...
(e.g. electrons, protons, neutrons). The distribution function can be written as : f_(E) = \frac. \mu is the
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 ...
(also denoted as EF and called the Fermi level when ''T''=0), k_\mathrm is the Boltzmann constant, and T is temperature. Fig. 4 illustrates how the product of the Fermi-Dirac distribution function and the three-dimensional density of states for a semiconductor can give insight to physical properties such as carrier concentration and Energy band gaps.
Bose–Einstein statistics In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic e ...
: The Bose–Einstein probability distribution function is used to find the probability that a boson occupies a specific quantum state in a system at thermal equilibrium.
Boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s are particles which do not obey the Pauli exclusion principle (e.g. phonons and photons). The distribution function can be written as :f_(E) = \frac From these two distributions it is possible to calculate properties such as the
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
U, the number of particles N,
specific heat capacity In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
C, and
thermal conductivity The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
k. The relationships between these properties and the product of the density of states and the probability distribution, denoting the density of states by g(E) instead of D(E), are given by :\begin U &= \int E\, f(E)\, g(E)\,E \\ N &= \int f(E)\, g(E)\,E \\ C &= \frac \int E\, f(E)\, g(E) \,E \\ k &= \frac\frac \int E f(E)\, g(E)\, \nu(E)\, \Lambda(E)\,E \end d is dimensionality, \nu is sound velocity and \Lambda is
mean free path In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...
.


Applications

The density of states appears in many areas of physics, and helps to explain a number of quantum mechanical phenomena.


Quantization

Calculating the density of states for small structures shows that the distribution of electrons changes as dimensionality is reduced. For quantum wires, the DOS for certain energies actually becomes higher than the DOS for bulk semiconductors, and for
quantum dots Quantum dots (QDs) are semiconductor particles a few nanometres in size, having optical and electronic properties that differ from those of larger particles as a result of quantum mechanics. They are a central topic in nanotechnology. When the ...
the electrons become quantized to certain energies.


Photonic crystals

The photon density of states can be manipulated by using periodic structures with length scales on the order of the wavelength of light. Some structures can completely inhibit the propagation of light of certain colors (energies), creating a photonic band gap: the DOS is zero for those photon energies. Other structures can inhibit the propagation of light only in certain directions to create mirrors, waveguides, and cavities. Such periodic structures are known as
photonic crystals A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of natural crystals gives rise to X-ray diffraction and that the atomic ...
. In nanostructured media the concept of
local density of states In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states ...
(LDOS) is often more relevant than that of DOS, as the DOS varies considerably from point to point.


Computational calculation

Interesting systems are in general complex, for instance compounds, biomolecules, polymers, etc. Because of the complexity of these systems the analytical calculation of the density of states is in most of the cases impossible. Computer simulations offer a set of algorithms to evaluate the density of states with a high accuracy. One of these algorithms is called the
Wang and Landau algorithm The Wang and Landau algorithm, proposed by Fugao Wang and David P. Landau, is a Monte Carlo method designed to estimate the density of states of a system. The method performs a non-Markovian random walk to build the density of states by quickly vi ...
. Within the Wang and Landau scheme any previous knowledge of the density of states is required. One proceeds as follows: the cost function (for example the energy) of the system is discretized. Each time the bin ''i'' is reached one updates a histogram for the density of states, g(i), by : g(i) \rightarrow g(i) + f where ''f'' is called the modification factor. As soon as each bin in the histogram is visited a certain number of times (10-15), the modification factor is reduced by some criterion, for instance, : f_ \rightarrow \frac f_ where ''n'' denotes the ''n''-th update step. The simulation finishes when the modification factor is less than a certain threshold, for instance f_n < 10^ . The Wang and Landau algorithm has some advantages over other common algorithms such as multicanonical simulations and
parallel tempering Parallel tempering in physics and statistics, is a computer simulation method typically used to find the lowest free energy state of a system of many interacting particles at low temperature. That is, the one expected to be observed in reality. ...
. For example, the density of states is obtained as the main product of the simulation. Additionally, Wang and Landau simulations are completely independent of the temperature. This feature allows to compute the density of states of systems with very rough energy landscape such as proteins. Mathematically the density of states is formulated in terms of a tower of covering maps.


Local density of states

An important feature of the definition of the DOS is that it can be extended to any system. One of its properties are the translationally invariability which means that the density of the states is
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
and it's the same at each point of the system. But this is just a particular case and the LDOS gives a wider description with a
heterogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
density of states through the system.


Concept

Local density of states (LDOS) describes a space-resolved density of states. In materials science, for example, this term is useful when interpreting the data from a
scanning tunneling microscope A scanning tunneling microscope (STM) is a type of microscope used for imaging surfaces at the atomic level. Its development in 1981 earned its inventors, Gerd Binnig and Heinrich Rohrer, then at IBM Zürich, the Nobel Prize in Physics in 1986. ...
(STM), since this method is capable of imaging electron densities of states with atomic resolution. According to crystal structure, this quantity can be predicted by computational methods, as for example with
density functional theory Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
.


A general definition

In a local density of states the contribution of each state is weighted by the density of its wave function at the point. N(E) becomes n(E,x) :n(E,x)=\sum_j , \phi_j (x), ^2\delta(E-\varepsilon_j) the factor of , \phi_j (x), ^2 means that each state contributes more in the regions where the density is high. An average over x of this expression will restore the usual formula for a DOS. The LDOS is useful in inhomogeneous systems, where n(E,x) contains more information than n(E) alone. For a one-dimensional system with a wall, the sine waves give :n_(E,x)=\frac\sqrt\sin^2 where k=\sqrt/\hbar. In a three-dimensional system with x>0 the expression is :n_(E,x)=\left(1-\frac\right)n_(E) In fact, we can generalise the local density of states further to :n(E,x,x')=\sum_j \phi_j (x)\phi^*_j (x')\delta(E-\varepsilon_j) this is called the ''spectral function'' and it's a function with each wave function separately in its own variable. In more advanced theory it is connected with the Green's functions and provides a compact representation of some results such as
optical absorption In physics, absorption of electromagnetic radiation is how matter (typically electrons bound in atoms) takes up a photon's energy — and so transforms electromagnetic energy into internal energy of the absorber (for example, thermal energy). ...
.


Solid state devices

LDOS can be used to gain profit into a solid-state device. For example, the figure on the right illustrates LDOS of a
transistor upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink). A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch e ...
as it turns on and off in a ballistic simulation. The LDOS has clear boundary in the source and drain, that corresponds to the location of band edge. In the channel, the DOS is increasing as gate voltage increase and potential barrier goes down.


Optics and photonics

In
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
and
photonics Photonics is a branch of optics that involves the application of generation, detection, and manipulation of light in form of photons through emission, transmission, modulation, signal processing, switching, amplification, and sensing. Though ...
, the concept of local density of states refers to the states that can be occupied by a photon. For light it is usually measured by fluorescence methods, near-field scanning methods or by cathodoluminescence techniques. For different photonic structures, the LDOS have different behaviors and they are controlling spontaneous emission in different ways. In photonic crystals, the near-zero LDOS are expected and they cause inhibition in the spontaneous emission. The LDOS are still in photonic crystals but now they are in the cavity. In this case, the LDOS can be much more enhanced and they are proportional with Purcell enhancements of the spontaneous emission. Similar LDOS enhancement is also expected in plasmonic cavity. However, in disordered photonic nanostructures, the LDOS behave differently. They fluctuate spatially with their statistics are proportional to the scattering strength of the structures. In addition, the relationship with the
mean free path In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...
of the scattering is trivial as the LDOS can be still strongly influenced by the short details of strong disorders in the form of a strong Purcell enhancement of the emission. and finally, for the plasmonic disorder, this effect is much stronger for LDOS fluctuations as it can be observed as a strong near-field localization.


See also


References


Further reading

*Chen, Gang. Nanoscale Energy Transport and Conversion. New York: Oxford, 2005 *Streetman, Ben G. and Sanjay Banerjee. Solid State Electronic Devices. Upper Saddle River, NJ: Prentice Hall, 2000. *Muller, Richard S. and Theodore I. Kamins. Device Electronics for Integrated Circuits. New York: John Wiley and Sons, 2003. *Kittel, Charles and Herbert Kroemer. Thermal Physics. New York: W.H. Freeman and Company, 1980 *Sze, Simon M. Physics of Semiconductor Devices. New York: John Wiley and Sons, 1981


External links


Online lecture:ECE 606 Lecture 8: Density of States
by M. Alam
Scientists shed light on glowing materials
How to measure the Photonic LDOS {{DEFAULTSORT:Density Of States Statistical mechanics Physical quantities Electronic band structures