Description
Motivated by the recent observations of small Fermi energies
and comparatively large superconducting gaps, present also
on bands not crossing the Fermi energy incipient bands in
iron-based superconductors, we analyse the doping evolution
of superconductivity in a four-band model across the Lifshitz
transition including BCS-BEC crossover effects on the shallow
bands. Similar to the BCS case we find that with hole doping
the phase difference between superconducting order
parameters of the hole bands changes from $0$ to $\pi$
through an intermediate $s+is$ state breaking time-reversal
symmetry. The transition however occurs in the region where
electron bands are incipient and chemical potential
renormalization in the superconducting state leads to a
significant broadening of the $s+is$ region. Our results are
quite actual given that the signatures of the s+is state were
found recently by means of muSR experiments in Ba122
doped with K. Furthermore, we present the qualitative
features of the $s+is$ state that can be observed in scanning
tunnelling microscopy (STM) experiments using the recently
proposed method.
