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
{\displaystyle D(E)=N(E)/V}
, where
N
(
E
)
d
E
{\displaystyle N(E)\delta E}
is the number of states in the system of volume
V
{\displaystyle V}
whose energies lie in the range from
E
{\displaystyle E}
to
E
+
d
E
{\displaystyle E+\delta E}
. 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. The density of states is directly related to the dispersion relations 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 systems however, such as atoms or molecules in the gas phase, the density distribution is discrete, like a spectral density. Local variations, most often due to distortions of the original system, are often referred to as local densities of states (LDOSs).
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 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 electrons 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.
Depending on the quantum mechanical system, the density of states can be calculated for electrons, photons, or phonons, and can be given as a function of either energy or the wave vector 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 in neutron stars and free electron gases in metals (examples of degenerate matter and a Fermi gas), have a 3-dimensional Euclidean topology. Less familiar systems, like two-dimensional electron gases (2DEG) in graphite layers and the quantum Hall effect system in MOSFET type devices, have a 2-dimensional Euclidean topology. Even less familiar are carbon nanotubes, 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 and materials science proceed.
The density of states related to volume V and N countable energy levels is defined as:
For isotropic one-dimensional systems with parabolic energy dispersion, the density of states is
D
1
D
(
E
)
=
1
2
p
ℏ
(
2
m
E
)
1
/
2
{\textstyle D_{1D}(E)={\tfrac {1}{2\pi \hbar }}({\tfrac {2m}{E}})^{1/2}}
. In two dimensions the density of states is a constant
D
2
D
=
m
2
p
ℏ
2
{\displaystyle D_{2D}={\tfrac {m}{2\pi \hbar ^{2}}}}
, while in three dimensions it becomes
D
3
D
(
E
)
=
m
2
p
2
ℏ
3
(
2
m
E
)
1
/
2
{\displaystyle D_{3D}(E)={\tfrac {m}{2\pi ^{2}\hbar ^{3}}}(2mE)^{1/2}}
.
Equivalently, the density of states can also be understood as the derivative of the microcanonical partition function
Z
m
(
E
)
{\displaystyle Z_{m}(E)}
(that is, the total number of states with energy less than
E
{\displaystyle E}
) with respect to the energy:
The number of states with energy
E
′
{\displaystyle E'}
(degree of degeneracy) is given by:
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 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. 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 (FCC) in the figure on the right has the 48-fold symmetry of the point group Oh with full octahedral symmetry. 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, silicon and platinum 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 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. 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.
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]−1[Volume]−1, in a two dimensional system, the units of DOS is [Energy]−1[Area]−1, in a one dimensional system, the units of DOS is [Energy]−1[Length]−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 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.
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. This procedure is done by differentiating the whole k-space volume
O
n
,
k
{\displaystyle \Omega _{n,k}}
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
According to this scheme, the density of wave vector states N is, through differentiating
O
n
,
k
{\displaystyle \Omega _{n,k}}
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 l. The wavelength is related to k through the relationship.
In a quantum system the length of l 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
{\displaystyle 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
{\displaystyle 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.
To finish the calculation for DOS find the number of states per unit sample volume at an energy
E
{\displaystyle E}
inside an interval
[
E
,
E
+
d
E
]
{\displaystyle [E,E+\mathrm {d} E]}
. The general form of DOS of a system is given as
The dispersion relation for electrons in a solid is given by the electronic band structure.
The kinetic energy of a particle depends on the magnitude and direction of the wave vector k, the properties of the particle and the environment in which the particle is moving. For example, the kinetic energy of an electron in a Fermi gas is given by
where m is the electron mass. 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
When
k
≈
p
/
a
{\displaystyle k\approx \pi /a}
the energy is
With the transformation
q
=
k
−
p
/
a
{\displaystyle q=k-\pi /a}
and small
q
{\displaystyle q}
this relation can be transformed to
The two examples mentioned here can be expressed like
This expression is a kind of dispersion relation because it interrelates two wave properties and it is isotropic 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:
Accordingly, the volume of n-dimensional k-space containing wave vectors smaller than k is:
Substitution of the isotropic energy relation gives the volume of occupied states
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,
D
n
(
E
)
{\displaystyle D_{n}\left(E\right)}
, for electrons in a n-dimensional systems is
for
E
>
E
0
{\displaystyle E>E_{0}}
, with
D
(
E
)
=
0
{\displaystyle D(E)=0}
for
E
<
E
0
{\displaystyle E
0
{\displaystyle x>0}
the expression is
In fact, we can generalise the local density of states further to
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.
LDOS can be used to gain profit into a solid-state device. For example, the figure on the right illustrates LDOS of a transistor 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.
In optics and photonics, 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 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.