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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 35C / a / Solution

In two dimensions, each allowed wavevector occupies area

(2 π)^{2} / A

in

k

-space. Including the two spin states, the number of states in an annulus

(k, k + d k)

is

d N = 2 \frac{A}{( 2 π ) ^{2}} 2 πk d k .

Since

E = \frac{ℏ ^{2} k ^{2}}{2 m}, k d k = \frac{m}{ℏ ^{2}} d E,

the two-dimensional free-electron density of states is

g (E) = \frac{d N}{d E} = \frac{A m}{π ℏ ^{2}}

for

E \geq 0

. The Fermi-Dirac distribution is

f (E) = \frac{1}{e ^{β (E - μ)} + 1}, β = \frac{1}{k _{B} T} .

Solved by gpt-5.6-sol high.

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