Carrier compensation and even band filling in a square lattice

With spin degeneracy two, one full square-zone band has density $Q^2/(2{\pi }^2)$. Thus $n_e-n_h=n-Q^2/(2{\pi }^2)$. If $Q=\alpha \sqrt{2\pi n}$, compensation requires $\alpha =\sqrt{\pi }$ and the real-space cell contains two electrons. A global gap is also necessary for an insulator.