Semiconductors occupy the regime between metals and insulators, where the electronic structure is governed by a finite band gap $E_g$ and by the detailed symmetry of the crystal potential. Their usefulness comes from the fact that carrier density, mobility, optical response, and transport can be tuned by doping, strain, dimensionality, and external fields.

For a periodic potential $V(\mathbf{r}+\mathbf{R})=V(\mathbf{r})$, eigenstates take the Bloch form $\psi_{n\mathbf{k}}(\mathbf{r})=u_{n\mathbf{k}}(\mathbf{r})e^{i\mathbf{k}\cdot\mathbf{r}}$. Band gaps arise from Bragg scattering at zone boundaries, and the low-energy physics near extrema can often be approximated by effective masses $m^*$ defined through the band curvature.

Near an extremum $\mathbf{k}_0$, the dispersion can be expanded as $E(\mathbf{k})\approx E(\mathbf{k}_0)+\frac{\hbar^2}{2}(\mathbf{k}-\mathbf{k}_0)^T m^{*-1}(\mathbf{k}-\mathbf{k}_0)$. The $k\cdot p$ approach systematically captures interband coupling, nonparabolicity, spin-orbit effects, and valley physics.