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We analyze the two-orbital (with $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ orbitals)analogue of the $t$-$J$-$U$ model as applied to the description of thecopper-oxide monolayer deposited on the Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$substrate (CuO$_2$/BSCCO). We show, that an $extended$ $s$-$wave$superconducting dome appears in the heavily hole overdoped regime of the model,with dominant contribution to the pairing coming from the $d_{3z^2-r^2}$orbitals. Also, the lower critical doping for the appearance of the SC stateconcurs with the Lifshitz Transition after which the hole-like Fermi pocketsare created around the $M$ points in the Brillouin zone. The obtained resultsare in accord with the recent experimental result [cf. Y. Zhong et al., Sci.Bull. 61, 1239 (2016)]. An analogous two-band description is also analyzed inthe context of the Ba$_2$CuO$_{4-y}$ bulk compound, where the heavilyhole-overdoped situation is believed to be reached [cf. W. M. Li et al. PNAS116, 12156 (2019)]. As we show, a two dome structure can be realized in suchsystem with the $d$- and $s$-$wave$ symmetries of the gap corresponding to thelow hole-doping and heavily hole overdoped cases, respectively.

Superconducting dome with $extended$ $s$-$wave$ pairing symmetry in the heavily hole-overdoped copper-oxide planes