Why do electrons emit phonons when they "relax" into the minimum energy level of the conduction band after getting into it from the valence band by absorbing a photon with an energy higher than their bandgap? Why don't they simply emit a photon with an energy equivalent to the energy of the phonon emitted? In other words, why a phonon and not a photon?
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"To conserve the k-vector." To find a place in the band diagram, an electron should have right k and right energy E. Please see any E-k diagram of the conduction band. Emitting a photon will only lower the energy of the electron with unaltered k value. But if it emits a phonon, both k and E are reduced such that it finds a suitable place in the E-k plot.
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