Past exam of the physics A course of the University of Cambridge / 2014 / Paper 2 / Section C / C11 / i / Solution

The first Brillouin zone is $|k_x|,|k_y|\le \pi /a$. For one atom per primitive cell and two spin states, a monovalent free-electron gas hasIts circular Fermi surface lies inside the zone and encloses half its area: a partially filled band is conducting. Here odd filling obstructs a spin-degenerate band insulator: a monovalent band insulator is impossible in this spin-degenerate one-atom independent-electron model. It would require changed premises, such as unit-cell enlargement or physics beyond the free-electron picture; no insulating free-electron contour should be invented.