Tight-Binding Band Structure

The simplest model of a solid: one orbital per site, an on-site energy, and a hopping amplitude $t$ to neighbours. Bloch's theorem turns it into a band $E(k)=\varepsilon_0-2t\cos ka$ for the 1D chain (width $4t$), the curvature at the bottom fixing the effective mass $m^*=\hbar^2/2ta^2$. Dimerize the chain (alternating $t_1,t_2$) and a gap $2|t_1-t_2|$ opens at the zone boundary, the SSH insulator. On the 2D square lattice $E=-2t(\cos k_xa+\cos k_ya)$ has a van Hove saddle and a Fermi surface that you can fill by dragging $E_F$.