Density of states

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In statistical and condensed matter physics, the density of states (DOS) of a system describes the number of states at each energy level that are available to be occupied. A high DOS at a specific energy level means that there are many states available for occupation. A DOS of zero means that no states can be occupied at that energy level.

Contents

[edit] 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 Eg 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.

[edit] 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.

Figure 1: Spherical constant energy surface in k-space for electrons in a three-dimensional crystalline material with isotropic effective mass. Any wavevector \color{BrickRed}{ \boldsymbol{\vec{\mathrm{k}}}} with tip on the sphere corresponds to the same energy value: E = (ħ k)2/(2m).

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 dki = 2π / L that gives a dΩ = (2π / 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.[1][2]

Volume element in spherical coordinates; dΩ = r2sinφ dr dθ dφ.

[edit] 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:[3]

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} \ .

[edit] Density of States and Distribution Functions

Main article: Probability of occupation

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.

Figure 2: The DOS, probability distribution, and their product shown for a semiconductor. The DOS is shown in red, the Fermi-Dirac distribution is shown in black, and their product is shown in blue. The valence band and the conduction band are labeled on the y-axis.

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}

μ is the chemical potential, kB 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, ν is sound velocity and Λ is mean free path.

[edit] Applications

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

[edit] Quantization

Figure 3: Density of states for electrons in bulk semiconductors (3D; in blue), quantum wells (2D; red), quantum wires (1D; green) and quantum dots (0D; black).

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

[edit] 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), causing the creation of a photonic bandgap: 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.

[edit] Notes and in-line references

  1. ^ Sample density of states calculation
  2. ^ Another density of states calculation
  3. ^ Charles Kittel (1996). Introduction to Solid State Physics (7th Edition ed.). Wiley. Equation (37), p. 216. ISBN 0471111813. 

[edit] Further reading

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

[edit] See also

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