Past exam of the quantum condensed matter course of the University of Cambridge / 2019 / Section B / 4 / c / Solution

Let ${ϵ}_B={\hslash }^2{g}^2/(8m)$ and take $V_0>0$ as the coupling magnitude. The dispersion becomesExpanding the square root at the boundary givesUsing $1/m^{\ast }={\hslash }^{-2}{d}^{2}E/d{q}^2$, the band-edge effective mass in a weak periodic potential isThe requested expression is the leading weak-potential limit. The upper band has positive band curvature and small positive effective mass; the lower band has negative curvature at its maximum and can instead be described by positive-mass electron holes. The small-$q$ expansion additionally requires ${\hslash }^{2}g|q|/(2m)\ll V_0$; its validity interval shrinks with the gap.