User:Tenzzor/Electronic band structure

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Density of states

The density of states function, ${\text{DOS}}(E)$, is defined as the number of electronic states per unit energy for electron energies between $E$ and $E+dE$. It is represented as ${\text{DOS}}(E)={\frac {dN}{dE}}$ where $N=N(E)$ is number of states occupied at energy $E$. The energy $E=E_{F}$ is the highest occupied energy level, a.k.a. Fermi Energy.

The derivation of $DOS(E)$ for the particle in 3D box model at 0 K can be used as an approximation for electrons in the metal. The energy of a state at ${\boldsymbol {n}}=(n_{x},n_{y},n_{z})$ is given by $E_{\boldsymbol {n}}={\frac {h^{2}}{8m_{e}L^{2}}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right)$, where $h$ is Planck’s constant, $m_{e}$ is the mass of electron, and $L$ is the box length. The number of states with $E\leq E_{F}$is given by twice the volume of a positive octant of the sphere with radius $n$:

$N(E)=2\times {\frac {1}{8}}\times {\frac {4}{3}}\pi |{\boldsymbol {n}}|^{3}={\frac {8\pi }{3}}\left({\frac {2m_{e}EL^{2}}{h^{2}}}\right)^{3/2}={\frac {8\pi }{3}}\left({\frac {2m_{e}E}{h^{2}}}\right)^{3/2}V$,

where $V=L^{3}$ volume of the box. Using the definition of density of states:

$DOS(E)={\frac {dN}{dE}}=4\left({\frac {2m_{e}}{h^{2}}}\right)^{3/2}E^{1/2}V={\frac {V}{2\pi ^{2}}}\left({\frac {2m_{e}}{\hbar }}\right)^{3/2}E^{1/2}V$

The function can also be defined as, $g(E)={\frac {DOS(E)}{V}}$, or density of states per unit volume. Thus, the $g(E)$ at 0K is proportional to $E^{1/2}$. At finite temperatures, $g(E)$ is modulated by Fermi-Dirac distribution.

The density of states function is important for calculations of effects based on band theory. In Fermi's Golden Rule, a calculation for the rate of optical absorption, it provides both the number of excitable electrons and the number of final states for an electron. It appears in calculations of electrical conductivity where it provides the number of mobile states, and in computing electron scattering rates where it provides the number of final states after scattering.

For energies inside a band gap, g(E) = 0.