  Hybrid systems of superconductors and magnets display several intriguing
properties, both from a fundamental physics point of view and with practical
applications. Promising applications in superconducting spintronics motivate a
search for systems where superconductivity can survive larger in-plane critical
magnetic fields than the conventional limit. The Chandrasekhar-Clogston (CC)
limit applies to thin-film conventional superconductors with in-plane magnetic
fields such that orbital effects may be ignored. For a magnetic field strength
comparable to the superconducting gap at zero temperature and zero field, a
spin-split normal state attains lower free energy than the superconducting
state. A multiband superconductor with a flat band placed just below the Fermi
surface has been shown to surpass the CC limit using weak-coupling theory.
Since the dimensionless coupling determining the critical temperature scales
with the density of states, it is natural to anticipate corrections from
strong-coupling theory in flat-band systems, owing to the large density of
states of the flat bands. We derive Eliashberg equations and the free energy
for a multiband superconductor in a magnetic field. First, we show that the CC
limit can be exceeded by a small amount in one-band strong-coupling
superconductors due to self-energy renormalization of the magnetic field. Next,
we consider a two-band system with one flat band and find that the CC limit can
be exceeded by a large amount also in strong-coupling theory, even when
including hybridization between bands that intersect.
