  For two decades, two-dimensional carbon species, including graphene, have
been the core of research in pursuing next-generation logic applications beyond
the silicon technology. Yet the opening of a gap in a controllable range of
doping, whilst keeping high conductance outside of this gapped state, has
remained a grand challenge in them thus far. Here we show that, by bringing
Bernal-stacked bilayer graphene in contact with an anti-ferromagnetic insulator
CrOCl, a strong insulating behavior is observed in a wide range of positive
total electron doping $n_\mathrm{tot}$ and effective displacement field
$D_\mathrm{eff}$ at low temperatures. Transport measurements further prove that
such an insulating phase can be well described by the picture of an inter-layer
excitonic state in bilayer graphene owing to electron-hole interactions. The
consequential over 1 $\mathrm{G\Omega}$ excitonic insulator can be readily
killed by tuning $D_\mathrm{eff}$ and/or $n_\mathrm{tot}$, and the system
recovers to a high mobility graphene with a sheet resistance of less than 100
$\mathrm{\Omega}$. It thus yields transistors with "ON-OFF" ratios reaching
10$^{7}$, and a CMOS-like graphene logic inverter is demonstrated. Our findings
of the robust insulating phase in bilayer graphene may be a leap forward to
fertilize the future carbon computing.
