The electronic structure of graphene field-effect transistors with a hexagonal boron nitride (h-BN) gate dielectric was studied in terms of the gate electric field and dielectric thickness using the density functional theory combined with the effective screening medium method. The calculation results showed that the carrier penetration into monolayer h-BN depended on the gate voltage that injected electrons or holes into graphene. The critical voltage for hole penetration was lower than that for electron penetration. At a fixed carrier concentration, carrier penetration soon occurred with increasing h-BN thickness for hole doping.