= Sommerfeld expansion
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{title2=$\int_0^\infty f(E)p_F(E)\,dE=\int_0^\mu f(E)\,dE+\frac{\pi^2}{6}(k_BT)^2f'(\mu)+\cdots$}
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The <Sommerfeld expansion> approximates integrals weighted by the <Fermi-Dirac distribution> when $k_BT\ll\mu$ and $f$ is smooth near $\mu$. Applying it to the number integral at fixed particle number gives $\mu-E_F=-(\pi^2/6)(k_BT)^2g'(E_F)/g(E_F)+\cdots$. 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.
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