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ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2018/paper-2/section-c/c11/a/solution

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Past exam of the physics A course of the University of Cambridge / 2018 / Paper 2 / Section C / C11 / a / Solution by

Codex  0 2026-10-18

A Bloch state in a crystal experiences a periodic lattice potential. Its band dispersion relation

ϵ (k)

encodes the lattice response, so an effective mass allows a wavepacket's acceleration under a weak applied force to be described without explicitly following the microscopic periodic forces.

In one dimension the wavepacket group velocity and semiclassical equation for crystal momentum are

v_{g} = \frac{1}{ℏ} \frac{d ϵ}{d k}, ℏ \dot{k} = F .

(1)

Differentiating the first equation with respect to time gives

a = \frac{1}{ℏ} \frac{d ^{2} ϵ}{d k ^{2}} \dot{k} = \frac{F}{ℏ ^{2}} \frac{d ^{2} ϵ}{d k ^{2}} .

(2)

Defining

F = m^{*} a

yields

m^{*} = ℏ^{2} (\frac{d ^{2} ϵ}{d k ^{2}})^{- 1} .

(3)

This band effective mass is controlled by band curvature, not the slope. A band minimum has positive

m^{*}

; a band maximum has negative Electron

m^{*}

and is often more conveniently described with positive-mass electron holes. In three dimensions the inverse effective mass is generally a tensor,

(m^{*- 1})_{ij} = ℏ^{- 2} \partial_{k_{i}} \partial_{k_{j}} ϵ

. The scalar formula assumes motion along a specified direction in an isolated band, weak fields and negligible interband transitions; the bare Electron mass has not physically changed.

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