Let's take the indirect semiconductors Si, Ge & Diamond. All these semiconductors are indirect, meaning that the maximum of the valence band is not directly under the minimum of the conduction band. This is how we can explain that the band gap for Si is $1.12$ eV, whereas the average required energy to cerate an electron-hole pair is $3.65$ eV (basically, the energy difference goes into phonons, heat).
But then, for the direct semiconductor $\rm CdTe$ (Cadmium-Tellur), why is the required energy for the creation of an $e/h$ pair $4.43$ eV, whereas the band gap is $1.44$ eV!? Very similar numbers also hold for GaAs (reference: Kolanoski-Wermes "Particle Detectors. Fundamentals and Applications. 2020. p. 261").
I thought that for direct semiconductors, we don't have phonon excitations.. But okay, I was probably wrong here.