  The effective interaction between electrons in two-dimensional materials can
be modified by their environment, enabling control of electronic correlations
and phases. Here, we study the dependence of electronic correlations in twisted
bilayer graphene (tBLG) on the separation to the metallic gate(s) in two device
configurations. Using an atomistic tight-binding model, we determine the
Hubbard parameters of the flat bands as a function of gate separation, taking
into account the screening from the metallic gate(s), the dielectric spacer
layers and the tBLG itself. We determine the critical gate separation at which
the Hubbard parameters become smaller than the critical value required for a
transition from a correlated insulator state to a (semi-)metallic phase. We
show how this critical gate separation depends on twist angle, doping and the
device configuration. These calculations may help rationalise the reported
differences between recent measurements of tBLG's phase diagram and suggests
that correlated insulator states can be screened out in devices with thin
dielectric layers.
