Solid State: Crystals, Bloch's Theorem & Bands
Periodic potentials, Bloch waves, and the origin of energy bands.
In a crystal, ions sit on a periodic lattice with lattice vectors $\mathbf R$. Electrons see a periodic potential $V(\mathbf r + \mathbf R) = V(\mathbf r)$. By Bloch's theorem, energy eigenstates have the form
$$\psi_{n,\mathbf k}(\mathbf r) = e^{i\mathbf k \cdot \mathbf r}\, u_{n,\mathbf k}(\mathbf r),$$with $u_{n,\mathbf k}$ periodic. The crystal momentum $\mathbf k$ is defined modulo the reciprocal lattice and conventionally taken in the first Brillouin zone.
For each band $n$, $E_n(\mathbf k)$ is a smooth function — a band. A simple tight-binding 1D chain (hopping amplitude $t$, lattice spacing $a$):
$$E(k) = -2 t \cos(k a).$$Insulators have a completely full lowest band + a finite gap to the next; metals have a partially filled band. Semiconductors: insulators with a small gap (Si ~ 1.1 eV).