 Solution

ID: past-exam-of-the-condensed-matter-physics-course-of-the-university-of-oxford/2012/4/b/solution

A

A / (2 π)^{2}

. Include two electron spin states and fill a disk of radius

k_{F}

at zero temperature. Then

N = 2 \frac{A}{( 2 π ) ^{2}} π k_{F}^{2}, n_{s} = N / A = \frac{k _{F}^{2}}{2 π} .

E = ℏ^{2} k^{2} / (2 m_{e})

, with its minimum chosen as zero, so

k_{F} = 2 π n_{s}, E_{F} = \frac{π ℏ ^{2} n _{s}}{m _{e}} .

These are the Fermi wavevector and Fermi energy for a spin-degenerate two-dimensional free electron gas. An imposed spin polarization or additional valley degeneracy would change the state count.

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