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.