Density of states

It is mathematically represented as a distribution by a probability density function, and it is generally an average over the space and time domains of the various states occupied 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.

Some condensed matter systems possess a structural symmetry 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.

Fluids, glasses and amorphous solids 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, of the dispersion relations of the system of interest.

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.

In anisotropic condensed matter systems such as a single crystal of a compound, the density of states could be different in one crystallographic direction than in another.

The calculation for DOS starts by counting the N allowed states at a certain k that are contained within [k, k + dk] inside the volume of the system.

for linear, disk and spherical symmetrical shaped functions in 1, 2 and 3-dimensional Euclidean k-spaces respectively.

, where s is a constant degeneracy factor that accounts for internal degrees of freedom due to such physical phenomena as spin or polarization.

To finish the calculation for DOS find the number of states per unit sample volume at an 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.

The dispersion relation is a spherically symmetric parabola and it is continuously rising so the DOS can be calculated easily.

For longitudinal phonons 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

Differentiating this volume with respect to the energy gives an expression for the DOS of the isotropic dispersion relation

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,

4, is used to find the probability that a fermion occupies a specific quantum state in a system at thermal equilibrium.

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.

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

Calculating the density of states for small structures shows that the distribution of electrons changes as dimensionality is reduced.

The photon density of states can be manipulated by using periodic structures with length scales on the order of the wavelength of light.

Computer simulations offer a set of algorithms to evaluate the density of states with a high accuracy.

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,

This feature allows to compute the density of states of systems with very rough energy landscape such as proteins.

One of its properties are the translationally invariability which means that the density of the states is homogeneous and it's the same at each point of the system.

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.

In more advanced theory it is connected with the Green's functions and provides a compact representation of some results such as optical absorption.

[15] 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.

First Brillouin zone of the FCC lattice , a truncated octahedron , showing symmetry labels for high symmetry lines and points
Octahedron.
Figure 1: Spherical surface in k -space for electrons in three dimensions.
Figure 2: Monatomic chain phonon dispersion relation
Figure 3: Free-electron DOS in 3-dimensional k-space
Figure 4: The Fermi-Dirac probability distribution, density of states, and their product for a semiconductor. The lower green lobe depicts hole energy, and thus uses as the distribution function.
Space resolved local density of states. A sequence of images with varying gate bias in a nanowire MOSFET at drain bias {{{1}}} Notice the confined energy levels as they move with increasing gate bias.