  In recent years, a growing experimental program has begun to search for
sub-GeV dark matter through its scattering with electrons. An associated
theoretical challenge is to compute the dark matter scattering rate in
experimental targets, and to find materials with large scattering rates. In
this paper we point out that, if dark matter scatters through a mediator that
couples to EM charge, then electromagnetic sum rules place limits on the
achievable scattering rates. These limits serve as a useful sanity check for
calculations, as well as setting a theoretical target for proposed detection
methods. Motivated by this analysis, we explore how conductor-dielectric
heterostructures can result in enhanced scattering rates compared to bulk
conductors, for dark matter masses $\lesssim$ MeV. These effects could be
especially important in computing the scattering rates from thin-film targets,
e.g. superconducting detectors such as SNSPDs, TESs or MKIDs, for which the
scattering rate could be enhanced by orders of magnitude at low enough dark
matter masses, as well as introducing or enhancing directional dependence.
