Local density of states
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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 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), 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: 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 = (/)^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 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 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 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 ...
''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. 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 ''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 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 \Omega_ 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 \Omega_n(k) = c_n k^n for a -dimensional -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 -spaces respectively. According to this scheme, the density of wave vector states is, through differentiating \Omega_ with respect to , 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 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 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 + \mathrm 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 form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
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 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 ...
in 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 ...
is given by E = E_0 + \frac \ , where ''m'' is the
electron mass In particle physics, the electron mass (symbol: ) 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 ...
. The dispersion relation is a spherically symmetric parabola and it is continuously rising so the DOS can be calculated easily. For longitudinal
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 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_\text the inter-atomic force constant and a inter-atomic spacing. For small values of k \ll \pi / a the dispersion relation is 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 \approx 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 ...
because it interrelates two wave properties and it is
isotropic In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...
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)^ , Accordingly, the volume of n-dimensional -space containing wave vectors smaller than 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)^ , 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 \\ ex D_2\left(E\right) &= \frac \\ ex D_3\left(E\right) &= 2\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 particles that can ...
s, acoustic phonons, 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 \\ ex D_2\left(E\right) &= 2\pi\frac \\ ex D_3\left(E\right) &= 4 \pi \frac \end


Distribution functions

The density of states plays an important role in the kinetic theory of solids. The product of the density of states and the probability distribution function 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: 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 subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s are particles which obey the Pauli exclusion principle (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 Chemical specie, 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 potent ...
(also denoted as EF and called 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 ...
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: 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 half odd-intege ...
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 energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accoun ...
per unit volume u, the number of particles N,
specific heat capacity In thermodynamics, the specific heat capacity (symbol ) of a substance is the amount of heat that must be added to one unit of mass of the substance in order to cause an increase of one unit in temperature. It is also referred to as massic heat ...
c, and
thermal conductivity The thermal conductivity of a material is a measure of its ability to heat conduction, conduct heat. It is commonly denoted by k, \lambda, or \kappa and is measured in W·m−1·K−1. Heat transfer occurs at a lower rate in materials of low ...
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)\,\mathrm dE \\ ex N &= V \int f(E)\, g(E)\,\mathrm dE \\ ex c &= \frac \int E\, f(E)\, g(E) \,\mathrm dE \\ ex k &= \frac\frac \int E f(E)\, g(E)\, \nu(E)\, \Lambda(E)\,\mathrm dE \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 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. In nanostructured media the concept of local density 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. 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 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 denotes the -th update step. The simulation finishes when the modification factor is less than a certain threshold, for instance The Wang and Landau algorithm has some advantages over other common algorithms such as multicanonical simulations and parallel tempering. 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 relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
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 relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
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 (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.


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 A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch electrical signals and electric power, power. It is one of the basic building blocks of modern electronics. It is composed of semicondu ...
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 optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
and
photonics Photonics is a branch of optics that involves the application of generation, detection, and manipulation of light in the form of photons through emission, transmission, modulation, signal processing, switching, amplification, and sensing. E ...
, 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. Different photonic structures have different LDOS behaviors with different consequences for spontaneous emission. In photonic crystals, near-zero LDOS are expected, inhibiting 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, and 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