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Layered organic superconductors of the BEDT family are model systems for theinterplay of the Mott transition with superconductivity, magnetic order andfrustration. Recent experimental studies on a hole-doped version of BEDTcompounds reveal an enhancement of superconductivity and a rapid crossoverbetween two different conducting phases above the superconducting dome. One ofthese phases is a Fermi liquid, the other not. Using plaquette cellulardynamical mean field theory with state of the art continuous-time quantum MonteCarlo calculations, we study this problem with the two-dimensional Hubbardmodel on the anisotropic triangular lattice. Phase diagrams as a function oftemperature $T$ and interaction strength $U/t$ are obtained for anisotropyparameters $t'=0.4t$, $t'=0.8t$ and various fillings. As for cuprates, we find,at finite doping, a first-order transition between two normal-state phases. For$T$ above the critical point of the first-order transition, there is a Widomline where crossovers occur. The results are in broad agreement withexperiment. This suggests that for compounds with intermediate to highfrustration, very light-doping should reveal the first-order transition andassociated crossovers. These crossovers could leave traces in thesuperconducting phase. We also predict that destroying the superconductingphase by a magnetic field should reveal the first-order transition. Finally, wepredict that electron-doping should also lead to an increased range of $U/t$for superconductivity but with a reduced maximum $T_c$. This work also clearlyshows that the superconducting dome here is tied to the Mott transition and itscontinuation as a transition separating pseudogap phase from correlated metalin doped compounds, as in the cuprates. Contrary to heavy fermions for example,the maximum $T_c$ is definitely not attached to an antiferromagnetic quantumcritical point.

Superconducting dome in doped quasi-2d organic Mott insulators: a paradigm for strongly-correlated superconductivity