  When the electron density of highly crystalline thin films is tuned by
chemical doping or ionic liq- uid gating, interesting effects appear including
unconventional superconductivity, sizeable spin-orbit coupling, competition
with charge-density waves, and a debated low-temperature metallic state that
seems to avoid the superconducting or insulating fate of standard
two-dimensional electron systems. Some experiments also find a marked tendency
to a negative electronic compressibility. We suggest that this indicates an
inclination for electronic phase separation resulting in a nanoscopic inhomo-
geneity. Although the mild modulation of the inhomogeneous landscape is
compatible with a high electron mobility in the metallic state, this
intrinsically inhomogeneous character is highlighted by the peculiar behaviour
of the metal-to-superconductor transition. Modelling the system with super-
conducting puddles embedded in a metallic matrix, we fit the peculiar
resistance vs. temperature curves of systems like TiSe2, MoS2, and ZrNCl. In
this framework also the low-temperature debated metallic state finds a natural
explanation in terms of the pristine metallic background embedding
non-percolating superconducting clusters. An intrinsically inhomogeneous
character naturally raises the question of the formation mechanism(s). We
propose a mechanism based on the interplay be- tween electrons and the charges
of the gating ionic liquid.
