Why in excitation in indirect bandgap materials photon absorption is always before phonon absorption? Why is it always photon->phonon? Why can't it be phonon->photon?
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 A: This is a second order process. The photon and the phonon are simultaneously absorbed. You may just as well draw the arrows in a different order. Now, if the first arrow extended all the way to the green line, followed by the second arrow to the conduction band minimum, then the order would be important. Then we would interpret this as absorption of a photon, followed by thermalization due to electron-phonon interaction. In the case of phonon-assisted absorption, however, the first arrow stops inside the band gap, where there are no states, and thus cannot be considered independently as a transition.
A similar situation arises with, e.g., two-photon absorption. Two photons are simultaneously absorbed, although the process is sometimes displayed as the first photon taking to an intermediate virtual state, followed by the second photon taking into the conduction band. This should be contrasted with the case where two photons are sequentially absorbed, in which case two real excited states are involved.
