  The physics of quantum many-body systems have been studied using bulk
correlated materials, and recently, moir\'e superlattices formed by atomic
bilayers have appeared as a novel platform in which the carrier concentration
and the band structures are highly tunable. In this brief review, we introduce
an intermediate platform between those systems, namely, a band-filling- and
bandwidth-tunable electric double-layer transistor based on a real organic Mott
insulator $\kappa$-(BEDT-TTF)$_{2}$Cu[N(CN)$_{2}$]Cl. In the proximity of the
bandwidth-control Mott transition at half filling, both electron and hole
doping induced superconductivity (with almost identical transition
temperatures) in the same sample. The normal state under electric double-layer
doping exhibited non-Fermi liquid behaviors as in many correlated materials.
The doping levels for the superconductivity and the non-Fermi liquid behaviors
were highly doping-asymmetric. Model calculations based on the anisotropic
triangular lattice explained many phenomena and the doping asymmetry, implying
the importance of the noninteracting band structure (particularly the flat part
of the band).
