  The Migdal effect inside detectors provides a new possibility of probing the
sub-GeV dark matter (DM) particles. While there has been well-established
methods treating the Migdal effect in isolated atoms, a coherent and complete
description of the valence electrons in semiconductor is still absent. The
bremstrahlung-like approach is a promising attempt, but it turns invalid for DM
masses below a few tens of MeV. In this paper, we lay out a framework where
phonon is chosen as an effective degree of freedom to describe the Migdal
effect in semiconductors. In this picture, a valence electron is excited to the
conduction state via exchange of a virtual phonon, accompanied by a
multi-phonon process triggered by an incident DM particle. Under the incoherent
approximation, it turns out that this approach can effectively push the
sensitivities of the semiconductor targets further down to the MeV DM mass
region.
