Earlier in this chapter the gap was built at the first zone boundary,

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Free electrons, energy bands and electronic structure · Free electrons, energy bands and electronic structure

Condensed Matter Physics, Ch Electrons in a periodic potential: Bloch's theorem and the opening of a gap

Problem

Earlier in this chapter the gap Eg=2∣UG∣E_g=2|U_G| was built at the first zone boundary, k=π/ak=\pi/a, from the single cosine U(x)=2UGcos⁡(2πx/a)U(x)=2U_G\cos(2\pi x/a). It was also stated, without being shown, that this same potential opens no gap to this order at the second boundary, k=2π/ak=2\pi/a. Show it: form the two standing waves at k=2π/ak=2\pi/a, ψ+∝cos⁡(2πx/a)\psi_+\propto\cos(2\pi x/a) and ψ−∝sin⁡(2πx/a)\psi_-\propto\sin(2\pi x/a), and show that ⟨U⟩+=⟨U⟩−=0\langle U\rangle_+=\langle U\rangle_-=0, so the two standing waves carry the same energy and no gap opens there.

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