  There are currently several existing and proposed experiments designed to
probe sub-GeV dark matter (DM) using electron ionization in various materials.
The projected signal rates for these experiments assume that this ionization
yield arises only from DM scattering directly off electron targets, ignoring
secondary ionization contributions from DM scattering off nuclear targets. We
investigate the validity of this assumption and show that if sub-GeV DM couples
with comparable strength to both protons and electrons, as would be the case
for a dark photon mediator, the ionization signal from atomic scattering via
the Migdal effect scales with the atomic number $Z$ and 3-momentum transfer
$\mathbf{q}$ as $Z^2 \mathbf{q}^2$. The result is that the Migdal effect is
always subdominant to electron scattering when the mediator is light, but that
Migdal-induced ionization can dominate over electron scattering for heavy
mediators and DM masses in the hundreds of MeV range. We put these two
ionization processes on identical theoretical footing, address some theoretical
uncertainties in the choice of atomic wavefunctions used to compute rates, and
discuss the implications for DM scenarios where the Migdal process dominates,
including for XENON10, XENON100, and the recent XENON1T results on light DM
scattering.
