Past exam of the quantum condensed matter course of the University of Cambridge / 2021 / Section B / 5 / a / 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 givesHere $g$ is the total spin-summed density of states for the sample. For a three-dimensional parabolic band, $g(E_F)=3N/(2E_F)$, yieldingThe linear law concerns the electronic contribution; the low-temperature lattice heat capacity usually adds a $T^3$ term.