Sommerfeld expansion (${\int }_0^{\infty }f(E)p_F(E)dE={\int }_0^{\mu }f(E)dE+\frac{{\pi }^2}{6}(k_{B}T{)}^2{f}^{\prime }(\mu )+\cdots$)

The Sommerfeld expansion approximates integrals weighted by the Fermi-Dirac distribution when $k_{B}T\ll \mu$ and $f$ is smooth near $\mu$. Applying it to the number integral at fixed particle number gives $\mu -E_F=-({\pi }^2/6)(k_{B}T{)}^2{g}^{\prime }(E_F)/g(E_F)+\cdots$. The sign depends on the slope of the density of states. For an unbounded parabolic band, $g(E)\propto E^{d/2-1}$, yielding opposite quadratic shifts in one and three dimensions. In two dimensions the leading algebraic shift vanishes, but the finite band-bottom cutoff still gives an exponentially small negative shift.

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.