  Possible field-temperature superconducting (SC) phase diagrams in two-band
quasi-two-dimensional materials with a strong paramagnetic pair-breaking (PPB)
are considered theoretically. Attention is paid to the case under a magnetic
field $perpendicular$ to the SC layers and to essential differences from the
counterpart in the ordinary single-band material. It is found by examining the
$H_{c2}(T)$ curve and the vortex lattices to be realized close to $H_{c2}$ that
a PPB-induced SC phase with a spatial modulation parallel to the field tends to
occur more easily than in the single-band case, and that a crisscrossing vortex
lattice proposed previously can occur in place of the conventional
Fulde-Ferrell-Larkin-Ovchinnikov state within a parameter range. The relevance
of the obtained results to FeSe and other iron-based superconductors is
discussed.
