Explain how the energy band structure for electrons determines the conductivity properties of crystalline materials.

A semiconductor has a conduction band with a lower edge $E_{c}$ and a valence band with an upper edge $E_{v}$. Assuming that the density of states for electrons in the conduction band is

$$\rho_{c}(E)=B_{c}\left(E-E_{c}\right)^{\frac{1}{2}}, \quad E>E_{c}$$

and in the valence band is

$$\rho_{v}(E)=B_{v}\left(E_{v}-E\right)^{\frac{1}{2}}, \quad E<E_{v}$$

where $B_{c}$ and $B_{v}$ are constants characteristic of the semiconductor, explain why at low temperatures the chemical potential for electrons lies close to the mid-point of the gap between the two bands.

Describe what is meant by the doping of a semiconductor and explain the distinction between $n$-type and $p$-type semiconductors, and discuss the low temperature limit of the chemical potential in both cases. Show that, whatever the degree and type of doping,

$$n_{e} n_{p}=B_{c} B_{v}[\Gamma(3 / 2)]^{2}(k T)^{3} e^{-\left(E_{c}-E_{v}\right) / k T}$$

where $n_{e}$ is the density of electrons in the conduction band and $n_{p}$ is the density of holes in the valence band.