  When combined with graphene, hexagonal boron nitride (h-BN) is an ideal
substrate and gate dielectric with which to build metalh-BN|graphene
field-effect devices. We use first-principles density functional theory (DFT)
calculations for Cu|h-BN|graphene stacks to study how the graphene doping
depends on the thickness of the h-BN layer and on a potential difference
applied between Cu and graphene. We develop an analytical model that describes
the doping very well, allowing us to identify the key parameters that govern
the device behaviour. A predicted intrinsic doping of graphene is particularly
prominent for ultrathin h-BN layers and should be observable in experiment. It
is dominated by novel interface terms that we evaluate from DFT calculations
for the individual materials and for interfaces between h-BN and Cu or
graphene.
