User:Lagerbaer/DOS

In physics, the density of states (DOS) of a system is a measure for the number of accessible quantum states in a given energy interval. It is introduced by replacing a discrete set of closely neighbored energy eigenstates with a continuum of states. This idea is analogous to the common mass density of a material which measures how much mass is found in a given volume of the material and replaces a discrete set of point masses. The density of states finds widespread use in statistical physics and condensed matter physics where systems generally exhibit many states in close proximity.

Explanation
Waves, or wave-like particles, can only exist within quantum mechanical (QM) systems if the properties of the system allow the wave to exist. In some systems, the interatomic spacing and the atomic charge of the material allows only electrons of certain wavelengths to exist. In other systems, the crystalline structure of the material allows waves to propagate in one direction, while suppressing wave propagation in another direction. Waves in a QM system have specific wavelengths and can propagate in specific directions, and each wave occupies a different mode, or state. Because many of these states have the same wavelength, and therefore share the same energy, there may be many states available at certain energy levels, while no states are available at other energy levels. For example, the density of states for electrons in a semiconductor is shown in red in Fig. 2. For electrons at the conduction band edge, very few states are available for the electron to occupy. As the electron increases in energy, the electron density of states increases and more states become available for occupation. However, because there are no states available for electrons to occupy within the bandgap, electrons at the conduction band edge must lose at least $$E_g$$ of energy in order to transition to another available mode. The density of states can be calculated for electron, photon, or phonon in QM systems. The DOS is usually represented by one of the symbols g, ρ, D, n, or N, and can be given as a function of either energy or wavevector k. To convert between energy and wavevector, the specific relation between E and k must be known. For example, the formula for electrons in free space is
 * $$E = \frac{(\hbar k)^2}{2m} \ ,$$

and for photons in free space the formula is
 * $$E = \hbar c k \ ,$$

where c is the speed of light in free space, $$\hbar = h/2\pi$$ is the reduced Planck's constant and m is the electron mass.

Derivation
The density of states is dependent upon the dimensional limits of the object itself. The role dimensions play is evident from the units of DOS (Energy-1Volume-1). In the limit that the system is 2 dimensional a volume becomes an area and in the limit of 1 dimension it becomes a length. It is important to note that the volume being referenced 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 specific k-space is given in Fig. 1. It can be seen that the dimensionality of the system itself will confine 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 is done by dividing the whole k-space volume Vk at an arbitrary k, by a volume increment dΩ (area for 2D, length for 1D) in k-space that contains one state. One state is large enough to contain particles having wavelength λ. The wavelength is related to k through the relationship.
 * $$k = \frac{2\pi}{\lambda}$$

In a quantum system the length of λ will depend on a characteristic spacing of the system L that is confining the particles. For example, a 3D crystal of length L has a k-space length increment of $$dk_i = 2\pi/L$$ that gives a $$d\Omega = (2\pi/L)^3$$.Note that L3 is the volume V of the crystal. Finally, N is given by the expression
 * $$N = \frac{sV_k}{d\Omega}$$

Here $$s$$ 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 containing all states whose wavevectors are smaller than a certain k serving as a parameter. To finish the calculation for DOS find the number of states per unit sample volume at an energy $$E$$ inside an interval $$[E, E+dE]$$. The general form of DOS of a system with volume $$V$$ is given as
 * $$g\left(E\right) = \frac{1}{V}\frac{dN}{dE}$$

More detailed derivations are available.



Example: parabolic E versus k
In the case of a parabolic relation, such as applies to free electrons, or to electrons in a solid with an isotropic parabolic band structure, the wavevector is related to the energy as:


 * $$ k = \frac{1}{\hbar} \sqrt{2m\left(E-E_0\right)} \, $$

where E0 is the energy at the bottom of the band from which the parabolic approximation to the energy begins.

Accordingly, the volume of k-space containing wavevectors smaller than k is:


 * $$V_k = \int_0^k \ dk \ \int_0^{2\pi} \ d \theta \ k \ \int_0^{\pi} \ d \phi\ k \sin \phi =\frac {4\pi}{3} k^3 = \frac{4\pi}{3 \hbar^3}\left(2m \left(E-E_0\right)\right)^{3/2}\ ,$$

resulting in a density of states for electrons (s = 2 for spin) increasing as the square root of the energy:


 * $$g\left(E\right) = \frac{1}{V}\frac{dN}{dE} =2\frac{1}{(2\pi)^3}\frac{4\pi}{3 \hbar^3} \frac {d}{dE}\left(2m\left(E-E_0\right)\right)^{3/2} = \frac {(2m)^{3/2}}{2\pi^2 \hbar^3} \left(E-E_0\right)^{1/2} \ .$$

Density of states and distribution functions
The DOS is often combined with a probability distribution that gives the likelihood of occupation of a particular state. The product of the DOS 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 DOS can give rise to physical properties.



Fermi-Dirac: The Fermi-Dirac probability distribution function, Fig. 2, is used to find the probability that a fermion occupies a specific quantum state in a system at thermal equilibrium. Fermions are particles which obey the Pauli Exclusion Principle (e.g. electrons, protons, neutrons). The distribution function can be written as
 * $$f_{\mathrm{FD}}\left(E\right) = \frac{1}{\exp\left(\frac{E-\mu}{k_BT}\right)+1}$$

$$\mu$$ is the chemical potential, $$k_B$$ is the Boltzmann constant, and $$T$$ is temperature. Fig. 2 illustrates how the product of the Fermi-Dirac distribution function and the three dimensional DOS for a semiconductor can give insight to physical properties such as carrier concentration and Energy band gaps.

Bose-Einstein: 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. Bosons are particles which do not obey the Pauli Exclusion Principle (e.g. phonons and photons). The distribution function can be written as
 * $$f_{\mathrm{BE}}\left(E\right) = \frac{1}{\exp\left(\frac{E-\mu}{k_BT}\right)-1}$$

From these two distributions it is possible to calculate properties such as the internal energy $$U$$, the density of particles $$n$$, specific heat capacity $$C$$, and thermal conductivity $$k$$. The relationships between these properties and the product of the DOS and the probability distribution are
 * $$U = \int E f\left(E\right)g\left(E\right)\,dE$$
 * $$n = \int f(E)g(E)\,dE$$
 * $$C = \frac{\partial}{\partial T}\int E f\left(E\right)g\left(E\right)\,dE$$
 * $$k = \frac{1}{d}\frac{\partial}{\partial T}\int E f\left(E\right)g\left(E\right)\nu\left(E\right)\Lambda\left(E\right)\,dE$$

$$d$$ is dimensionality, $$\nu$$ is sound velocity and $$\Lambda$$ is mean free path.

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. The DOS for all 3 dimensions are produced on the same graph in Fig. 3

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 with certain wavelengths, causing the creation of a photonic bandgap. Other structures can inhibit the propagation of light in certain directions, creating photonic waveguides. These devices are known as photonic crystals.