Semiconductors have a band gap $E_g < \text{ 3 eV}$, that separates the valence band and the conduction band.
You should be able to describe the difference between a direct band gap semiconductor and an indirect band gap semiconductor. You should be able to explain why a direct band gap is required for light-emitting diodes, and why a direct band gap is not required for a solar cell.
Know what an effective mass is and be able to describe what holes are.
Be able to sketch the electron density of states near the band gap for one-dimensional, two-dimensional, and three-dimensional semiconductors.
You should be able to describe the Boltzmann approximation used to calculate the density of electrons in the conduction band or the concentration of holes in the valence band.
As temperature $T \rightarrow 0$, the chemical potential of an intrinsic semiconductor moves to the middle of the band gap $\frac{E_v+E_c}{2}$. As temperature increases, the chemical potential shifts towards the band with the lower of $D_c$ and $D_v$ but it typically is not far from the middle of the band gap at room temperature.
The law of mass action states $np = n_i^2$ where $n_i$ is the intrinsic carrier concentration $n_i = \frac{\sqrt{\pi D_vD_c}}{2}(k_BT)^{3/2}\exp{\left(\frac{-E_g}{2k_BT}\right)}$.
The electron contribution to the internal energy density is the sum of the energies of the electrons in the conduction band minus the sum of the energies of the holes in the valence band, $u(T) \approx u(T=0) + \int\limits_{E_c}^{\infty}ED(E)f(E)dE- \int\limits_{-\infty}^{E_v}ED(E)(1-f(E))dE$.
Other thermodynamic properties such as the specific heat, the entropy, and the Helmholtz free energy can be derived from the internal energy density.
Donors add occupied states in the band gap just below the conduction band, and acceptors add empty states in the band gap just above the valence band. Donors and acceptors are chosen so that for $N_d \gt N_a$, $n = N_d - N_a$ and $p = n_i^2/n$, while for $N_a \gt N_d$, $p = N_a - N_d$ and $n = n_i^2/p$.
Doped semiconductors are electrically neutral so the concentrations of electrons plus the ionized acceptors must equal the concentrations of the holes plus the ionized donors, $n+N_a^- = p+N_d^+,$.
The charge neutrality condition can be used to determine the chemical potential. It is closer to the conduction band for $n$-type semiconductors, and closer to the valence band for $p$-type semiconductors.
By solving the charge neutrality condition numerically for different temperatures, three temperature regimes can be identified. In the low-temperature freeze-out regime, the concentration of mobile carriers decreases rapidly with temperature. In the extrinsic regime, the concentration of mobile carriers remains about constant. In the intrinsic regime, the concentration of mobile carriers increases with temperature as the electrons are thermally excited across the band gap.
In the extrinsic regime, the thermodynamic properties of the charge carriers are the same as for an ideal gas.
