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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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  • $\begingroup$ To add to this answer (which is correct), to relax an electron's momentum with on the length scales comparable to the Fermi momentum $k_f$ with photons, you would need to relax via release of x-rays ($10^6$meV) which is much higher than $kT$~25meV! Optical light (and even ultraviolet) do not have enough momentum $\endgroup$
    – KF Gauss
    Feb 4, 2019 at 0:58

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