Fixed-chemical-potential Sommerfeld density increase

For a three-dimensional parabolic band and fixed positive $\mu -E_c$, the Sommerfeld expansion gives $n(T)=n(0)[1+{\pi }^2(k_{B}T/(\mu -E_c){)}^2/8+O(T^4)]$. This increase differs from the usual fixed-density calculation, in which the chemical potential decreases with temperature. The exact density is proportional to $T^{3/2}{F}_{1/2}[(\mu -E_c)/(k_{B}T)]$.

The temperature derivative of the Fermi occupation has opposite signs below and above $\mu$. Pairing energies $\mu \pm x$ leaves a positive contribution because the density of states increases with energy. The unpaired higher-energy tail is also positive. Hence the density increases at fixed $\mu$ for an unbounded three-dimensional parabolic band.