Semiconductors are materials where the chemical potential lies in a band gap and the band gap is less than about 3 eV. To calculate the properties of a semiconductor, first the crystal structure must be determined. This can be done by x-ray diffraction. Once the crystal structure is known, a band structure calculation can be performed to determine the electron dispersion relation. The band structure calculation is done for different lattice constants to check that the calculation that gives the minimum of the energy has the lattice constant that is measured in a diffraction experiment. From the change in energy as a function of lattice constant, an effective spring constant between the atoms can be determined and this can be used to calculate the phonon dispersion relation and the phonon density of states. The phonon density of states can be used to calculate the phonon contribution to the thermodynamic properties. At room temperature, the phonon contribution to the thermodynamic properties usually dominates over the electron contribution of semiconductors. An example of these calculations for silicon can be found here.

Only the top of the valence band and bottom of the conduction band are important for many electronic properties. The minimum of the conduction band can be fit by a paraboloid. For an appropriate choice of coordinate axes, the minimum of the conduction band can be approximated as,

\begin{equation} E_{c}=E_g+\frac{\hbar^2}{2m_{ex}}\left(k_x-k_{0x}\right)^2+\frac{\hbar^2}{2m_{ey}}\left(k_y-k_{0y}\right)^2+\frac{\hbar^2}{2m_{ez}}\left(k_z-k_{0z}\right)^2, \end{equation}

where $(k_{0x},k_{0y},k_{0z})$ is the position of the minimum and $m_{ex}$, $m_{ey}$, and $m_{ez}$ are the effective masses of the electrons. At maximum of the valence band can similarily described by

\begin{equation} E_{v}=-\frac{\hbar^2}{2m_{hx}}\left(k_x-k_{0x}\right)^2-\frac{\hbar^2}{2m_{hy}}\left(k_y-k_{0y}\right)^2-\frac{\hbar^2}{2m_{hz}}\left(k_z-k_{0z}\right)^2, \end{equation}

where $m_{hx}$, $m_{hy}$, and $m_{hz}$ are the effective masses of the holes.

Because the dispersion relation is parabolic near the top of the valence band and the bottom of the conduction band, the electron density of states grows like the square root of energy near the band edges like it does in the free electron model. Near the bottom of the conduction band the electron density of states is proportional to $\sqrt{E-E_c}$ and near the top of the valence band the electron density of states is proportional to $\sqrt{E_v-E}$. Using the Boltzmann approximation, it is possible to calculate the concentration of electrons in the conduction band $n = N_c\exp\left(\frac{\mu-E_c}{k_BT}\right)$ and holes in the valence band $p = N_v\exp\left(\frac{E_v-\mu}{k_BT}\right)$. Here $N_c$ is the effective density of states in the conduction band and $N_v$ is the effective density of states in the valence band. Plots of results of the the Boltzmann approximations are available here. From these formulas it is possible to the intrinsic carrier concentration $n_i$, and the chemical potential for an intrinsic semiconductor.

	$N_c$	$N_v$	$E_g$
Si (300 K)	2.78 × 1025 m-3	9.84 × 1024 m-3	1.12 eV
Ge (300 K)	1.04 × 1025 m-3	6.0 × 1024 m-3	0.66 eV
GaAs (300 K)	4.45 × 1023 m-3	7.72 × 1024 m-3	1.424 eV
4H-SiC (300 K)	1.7 × 1025 m-3	2.5 × 1025 m-3	3.3 eV

The conductivity of semiconductors can be modified dramatically by doping. For an $n$-type semiconductor in the extrinsic regime, the concentration of electrons in the conduction band is the concentration of donors $n=N_D$. For a $p$-type semiconductor in the extrinsic regime, the concentration of holes in the valence band is the concentration of acceptors $p=N_A$. These conditions can be used to calculate the chemical potential of a doped semiconductor. The minority carrier concentrations are then determined by the law of mass action $np=n_i^2$. Outside the extrinsic regime, the carrier concentrations of doped semicnductors can be determined numerically.