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ID: past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2019/section-b/4/c/solution

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

Codex  0 2026-10-18

Let

ϵ_{B} = ℏ^{2} g^{2} / (8 m)

and take

V_{0} > 0

as the coupling magnitude. The dispersion becomes

E_{\pm} (q) = ϵ_{B} + \frac{ℏ ^{2} q ^{2}}{2 m} \pm V_{0}^{2} + \frac{ℏ ^{4} g ^{2} q ^{2}}{4 m ^{2}} .

Expanding the square root at the boundary gives

E_{\pm} (q) = ϵ_{B} \pm V_{0} + \frac{ℏ ^{2} q ^{2}}{2 m} (1 \pm \frac{2 ϵ _{B}}{V _{0}}) + O (q^{4}) .

Using

1/ m^{*} = ℏ^{- 2} d^{2} E / d q^{2}

, the band-edge effective mass in a weak periodic potential is

\frac{m _{\pm}^{*}}{m} = \frac{1}{1 \pm 2 ϵ _{B} / V _{0}} ≃ \pm \frac{V _{0}}{2 ϵ _{B}} (V_{0} ≪ ϵ_{B}) .

The 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

ℏ^{2} g ∣ q ∣/ (2 m) ≪ V_{0}

; its validity interval shrinks with the gap.

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