  Although the direct or indirect nature of the bandgap transition is an
essential parameter of semiconductors for optoelectronic applications, the
understanding why some of the conventional semiconductors have direct or
indirect bandgaps remains ambiguous. In this Letter, we revealed that the
existence of the occupied cation d bands is a prime element in determining the
directness of the bandgap of semiconductors through the s-d and p-d couplings,
which push the conduction band energy levels at the X- and L-valley up, but
leaves the {\Gamma}-valley conduction state unchanged. This unified theory
unambiguously explains why Diamond, Si, Ge, and Al-containing group III-V
semiconductors, which do not have active occupied d bands, have indirect
bandgaps and remaining common semiconductors, except GaP, have direct bandgaps.
Besides s-d and p-d couplings, bond length and electronegativity of anions are
two remaining factors regulating the energy ordering of the {\Gamma}-, X-, and
L-valley of the conduction band, and are responsible for the anomalous bandgap
behaviors in GaN, GaP, and GaAs that have direct, indirect, and direct
bandgaps, respectively, despite the fact that N, P, and As are in ascending
order of the atomic number. This understanding will shed light on the design of
new direct bandgap light-emitting materials.
