The concept of the density of states is important in solid state physics and semiconductor physics.* The density of states function describes the number of allowed energy states (called eigenstates) per unit energy per unit volume in a crystal. Determining the allowed energy states requires difficult quantum mechanical calculations. However, precise energy bandstructure information isn't necessary for semiconductor applications, as I will discuss.
* I consider solid state physics a field in physics and semiconductor physics a field in engineering.
The eigenstates of crystals form allowed-energy bands separated by forbidden-energy band gaps. In the case of semiconductors, only the energy states directly above or directly below a band gap are important for engineering purposes. This is because in semiconductors, conduction band electrons are those electrons in energy states right above the band gap, and valence band holes are the unfilled energy states directly below the band gap.
Since only the energy states near the band gap are important in semiconductor physics, general formulas for the density of states can be written. These formulas are only valid at the edges of the conduction band and the valence band and only valid if the semiconductor extends many nanometers in all three dimensions. There are different formulas for 2d and 1d, and new devices such as quantum wells and quantum wires can exploit the new properties.
Density of states of electrons in the conduction band and holes in the valence band
gc(E) = 8πmnsqrt(2mn(E - Ec))/h3 for E greater than Ec
gv(E) = 8πmpsqrt(2mp(Ev - E))/h3 for E less than Ev
where gc(E) is the density of states in the conduction band, gv(E) is the density of states in the valence band, mn is the effective mass of an electron, mp is the effective mass of a hole, and h is Planck's constant.
Of course, knowledge of the effective masses of the carriers in a semiconductor requires either quantum mechanics or empirical data, so these formulas aren't magical. Nevertheless, effective masses are fairly easy to extract from experiments. If the densities of states are multiplied by the Fermi function, which describes the probability of a given energy state being filled, and the products are integrated, then the actual concentrations of electrons and holes in the semiconductor can be obtained.