Semiconductor Devices: Physics and Technology, 3rd Edition

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

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 x-direction. Substituting Eq. 2 into Eq. 1 gives

The incremental momentum dp_x required for a unity increase in n_x is

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

The volume dp_x dp_y dp_z in the ...