 Solution

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2014/paper-2/section-c/c11/i/solution

∣ k_{x} ∣, ∣ k_{y} ∣ \leq π / a

. For one atom per primitive cell and two spin states, a monovalent free-electron gas has

n_{2 D} = \frac{1}{a ^{2}} = \frac{k _{F}^{2}}{2 π}, k_{F} a = 2 π .

Its 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.

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