  We theoretically investigate gate-defined graphene superlattices with broken
inversion symmetry as a platform for realizing tunable valley dependent
transport. Our analysis is motivated by recent experiments [C. Forsythe et al.,
Nat. Nanotechnol. 13, 566 (2018)] wherein gate-tunable superlattice potentials
have been induced on graphene by nanostructuring a dielectric in the
graphene/patterneddielectric/gate structure. We demonstrate how the electronic
tight-binding structure of the superlattice system resembles a gapped Dirac
model with associated valley dependent transport using an unfolding procedure.
In this manner we obtain the valley Hall conductivities from the Berry
curvature distribution in the superlattice Brillouin zone, and demonstrate the
tunability of this conductivity by the superlattice potential. Finally, we
calculate the valley Hall angle relating the transverse valley current and
longitudinal charge current and demonstrate the robustness of the valley
currents against irregularities in the patterned dielectric.
