A band's energy obeys two facts already in hand: it is unchanged if the electron's…

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

A band's energy E(k)E(k) obeys two facts already in hand: it is unchanged if the electron's wave is reversed, E(k)=E(−k)E(k)=E(-k) (reversing kk only swaps which wave is forward and which is reflected, so nothing physical changes), and it repeats with the reciprocal lattice vector, E(k)=E(k+2π/a)E(k)=E(k+2\pi/a) (Bloch's theorem, stated earlier in this chapter). Assuming only that E(k)E(k) has a definite slope at the boundary, and nothing else about its shape, use these two facts to show that dE/dk=0\mathrm{d}E/\mathrm{d}k=0 at the first zone boundary, k=π/ak=\pi/a.

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