  We analyze the paramagnetic state of a two-band Hubbard model with finite
Hund's coupling close to integer filling at $n=2$ in two spacial dimensions.
Previously, a Mott metal-insulator transition was established at $n=2$ with a
coexistence region of a metallic and a bad metal state in the vicinity of that
integer filling. The coexistence region ends at a critical point beyond which a
charge instability persists. Here we investigate the transition into negative
electronic compressibility states for an extended filling range close to $n=2$
within a slave boson setup. We analyze the separate contributions from the
(fermionic) quasiparticles and the (bosonic) multiparticle incoherent
background and find that the total compressibility depends on a subtle
interplay between the quasiparticle excitations and collective fields.
Implementing a Blume-Emery-Griffiths model approach for the slave bosons, which
mimics the bosonic fields by Ising-like pseudospins, we suggest a feedback
mechanism between these fields and the fermionic degrees of freedom. We argue
that the negative compressibility can be sustained for heterostructures of such
strongly correlated planes and results in a large capacitance of these
structures. The strong density dependence of these capacitances allows to tune
them through small electronic density variations. Moreover, by resistive
switching from a Mott insulating state to a metallic state through short
electric pulses, transitions between fairly different capacitances are within
reach.
