Band gap

A band gap is an energy interval between adjacent energy bands containing no allowed bulk state.

A direct band gap compares two energy bands at the same crystal momentum. A global gap instead requires the lowest energy of the upper band to exceed the highest energy of the lower band, allowing the two extrema to occur at different momenta. For example, the periodic toy bands $E_1(k)=A\cos k$ and $E_2(k)=A\cos k+\delta$, with $A>0$ and $0<\delta <2A$, have positive direct separation $\delta$ everywhere but global overlap $2A-\delta$. A positive direct splitting at each Brillouin zone boundary therefore does not alone establish an insulating filled band. At a suitable electron count the overlap can produce both electron and hole pockets.

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.

A band gap whose conduction band minimum and valence band maximum occur at the same crystal momentum, allowing an optical transition without a momentum-balancing phonon.

An avoided crossing occurs when two parameter-dependent eigenvalues approach one another but an interaction prevents equality. Near a two-level crossing, a nonzero off-diagonal matrix element opens a positive minimum gap.