Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES
Lecture 30 : Quantum Mechanical Concepts
  Density of States
Density of states at an energy $ E$ is the number of states per unit volume available per unit enit energy interval with energy between $ E$ and $ E+dE$. This would require counting of states, i.e., enumeration of different values of $ n_x, n_y, n_z$ corresponding to the energy of states within this interval. This is obviously a difficult task. However, given the large dimension of a crystal, the states are very closely packed and and one can essentially treat the $ k-$ values as continuous.
  Equation of constant energy given by eqn. (B) is a sphere in $ n_x-n_y-n_z$ space with a radius $ \sqrt{2mE}L/\pi\hbar$. As the points in this space are separated from the adjacent ones by one unit in each direction, each point effectively occupies a volume $ (2\pi)^3/V$ in the $ k-$ space. Thus a unit volume in $ k-$ space contains $ V/(2\pi)^3$ number of states. As each $ k-$ state can accommodate two electrons (corresponding to two distinct spin states), the number of electrons per unit volume of $ k-$ space is $ V/4\pi^3$.
 
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