Past exam of the quantum condensed matter course of the University of Cambridge / 2021 / Section B / 5 / a

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Solution

For a degenerate Fermi gas, the Pauli exclusion principle restricts thermally changed occupations to an energy shell of width

k_{B} T

around the Fermi energy. The number of active Electrons is of order

g (E_{F}) k_{B} T \sim N k_{B} T / E_{F}

; each carries an excitation energy of order

k_{B} T

. Thus the thermal energy grows as

N (k_{B} T)^{2} / E_{F}

and the electronic heat capacity is proportional to

T

.

More precisely, the fixed-particle-number Sommerfeld expansion gives

U (T) - U (0) = \frac{π ^{2}}{6} g (E_{F}) (k_{B} T)^{2} + O (T^{4}), C_{e} = \frac{π ^{2}}{3} g (E_{F}) k_{B}^{2} T + O (T^{3}) .

Here

g

is the total spin-summed density of states for the sample. For a three-dimensional parabolic band,

g (E_{F}) = 3 N / (2 E_{F})

, yielding

C_{e} = A \frac{N k _{B}^{2} T}{E _{F}}, A = \frac{π ^{2}}{2}, k_{B} T ≪ E_{F} .

The linear law concerns the electronic contribution; the low-temperature lattice heat capacity usually adds a

T^{3}

term.

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