  A key strategy for the direct detection of sub-GeV dark matter is to search
for small ionization signals. These can arise from dark matter-electron
scattering or when the dark matter-nucleus scattering process is accompanied by
a "Migdal" electron. We show that the theoretical descriptions of both
processes are closely related, which allows for a principal mapping between
dark matter-electron and dark matter-nucleus scattering rates once the dark
matter interactions with matter are specified. We explore this parametric
relationship for noble-liquid targets and, for the first time, provide an
estimate of the "Migdal" ionization rate in semiconductors that is based on
evaluating a crystal form factor that accounts for the semiconductor band
structure. We also present new dark-matter-nucleus scattering limits down to
dark matter masses of 500 keV using published data from XENON10, XENON100, and
a SENSEI prototype Skipper-CCD. For a dark photon mediator, the dark
matter-electron scattering rates dominate over the Migdal rates for dark matter
masses below 100 MeV. We also provide projections for proposed experiments with
xenon and silicon targets.
