Indirect band gap

An indirect band gap separates a valence band maximum and a conduction band minimum at different values of crystal momentum. A near-edge radiative transition then usually needs a phonon or another momentum-transfer process in addition to the photon, whose momentum is small on the Brillouin zone scale. A direct band gap has the extrema at the same crystal momentum and more readily supports efficient band-edge light emission. A material can have direct transitions at higher energy while its smallest gap is indirect.

For $-V_0(\cos Qx+\cos Qy)$, two-dimensional bands are sums of one-dimensional bands. At first order, the first-band maximum is $E_Q/2-V_0$ at the corner, while the second-band minimum is $E_Q/4+V_0/2$ at an edge centre. Hence ${\Delta }_{\mathrm{ind}}\simeq 3V_0/2-E_Q/4$, giving a rough onset $V_0\sim E_Q/6$. Higher plane waves shift this estimate; insulating filling must be checked separately.