Past exam of the quantum condensed matter course of the University of Cambridge / 2008 / Section B / B2 / a / Solution

The tight-binding approximation starts with localized atomic orbitals. Their overlap allows hopping, broadening each atomic energy into a band. For example, a single orbital on a simple cubic lattice with nearest-neighbor hopping gives $E(\mathbf{k})=E_0-2t[\cos (k_{x}a)+\cos (k_{y}a)+\cos (k_{z}a)]$. Small overlap means a narrow band; orbital symmetry and hopping signs determine its extrema and dispersion.

The nearly-free electron model starts instead with free-electron parabolas, folded into the Brillouin zone, and treats the periodic crystal potential as a perturbation. Near a Bragg plane, states differing by a reciprocal vector become degenerate and mix, opening a gap approximately $2|V_{\mathbf{G}}|$. Away from such degeneracies, the energy remains close to the free-electron dispersion. These are complementary limits of the same Bloch-band framework, not different counts of electronic states. Both can give metals or insulators depending on filling and band overlap, but their natural basis functions and useful expansion parameters differ.