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ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2022/paper-2/section-a-ii/16/ii/solution

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Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2022 / Paper 2 / Section A(ii) / 16 / ii / Solution by

Codex  0 2026-10-20

For an in-plane parabolic band

E (k) = E_{0} + ℏ^{2} k^{2} / (2 m^{*})

, one state occupies area

(2 π)^{2} / A

in two-dimensional wave-vector space. Including spin degeneracy two and the single GaAs conduction-band valley,

\frac{N ( E )}{A} = 2 \frac{π k ^{2}}{( 2 π ) ^{2}} = \frac{k ^{2}}{2 π} = \frac{m ^{*}}{π ℏ ^{2}} (E - E_{0}) .

Differentiation gives the constant two-dimensional density of states

D_{2 D} (E) = \frac{m ^{*}}{π ℏ ^{2}} Θ (E - E_{0}) = \frac{4 π m ^{*}}{h ^{2}} Θ (E - E_{0}) .

Using the supplied

h

and effective mass gives

D_{2 D} = 1.75 \times 1 0^{36} J^{- 1} m^{- 2} = 2.80 \times 1 0^{13} e V^{- 1} c m^{- 2} .

This counts available one-electron states, not the number actually occupied at arbitrary temperature. Occupation requires the Fermi-Dirac distribution; each additional subband would add another step above its own threshold.

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