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A key strategy for the direct detection of sub-GeV dark matter is to searchfor small ionization signals. These can arise from dark matter-electronscattering or when the dark matter-nucleus scattering process is accompanied bya "Migdal" electron. We show that the theoretical descriptions of bothprocesses are closely related, which allows for a principal mapping betweendark matter-electron and dark matter-nucleus scattering rates once the darkmatter interactions with matter are specified. We explore this parametricrelationship for noble-liquid targets and, for the first time, provide anestimate of the "Migdal" ionization rate in semiconductors that is based onevaluating a crystal form factor that accounts for the semiconductor bandstructure. We also present new dark-matter-nucleus scattering limits down todark matter masses of 500 keV using published data from XENON10, XENON100, anda SENSEI prototype Skipper-CCD. For a dark photon mediator, the darkmatter-electron scattering rates dominate over the Migdal rates for dark mattermasses below 100 MeV. We also provide projections for proposed experiments withxenon and silicon targets.

On the relation between Migdal effect and dark matter-electron scattering in isolated atoms and semiconductors