**3-D Density of States**

For a three dimensional (3-D) structure such as a bulk semiconductor, to calculate the electron and hole concentrations in the conduction and valence bands, respectively, we need to know the density of states, that is, the number of allowed energy states per unit energy per unit volume (i.e., in the unit of number of states/eV/cm^{3}).

When electrons move back and forth along the *x*-direction in a semiconductor material, the movements can be described by standing-wave oscillations. The wavelength *λ* of a standing wave is related to the length of the semiconductor *L* by

where *n _{x}* is an integer. The wavelength can be expressed by de Broglie hypothesis:

where *h* is the Planck's constant and *p _{x}* is the momentum in the

The incremental momentum *dp _{x}* required for a unity increase in

For a three-dimensional cube of side *L*, we have

The volume *dp _{x} dp_{y} dp_{z}* in the ...

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