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Semiconductor Devices: Physics and Technology, 3rd Edition by Ming-Kwei Lee, Simon M. Sze

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images Appendix H

Derivation of the Density of States in a Semiconductor

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/cm3).

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

images

where nx is an integer. The wavelength can be expressed by de Broglie hypothesis:

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where h is the Planck's constant and px is the momentum in the x-direction. Substituting Eq. 2 into Eq. 1 gives

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The incremental momentum dpx required for a unity increase in nx is

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

The volume dpx dpy dpz in the ...

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