  Electron doping is an excellent tuning knob to explore different phases of
matter in two-dimensional (2D) materials. For example, tuning the Fermi level
at a van Hove singularity in twisted bilayer graphene can enhance
electron-electron interactions and induce a diverse range of correlated phases.
Here, using a single-particle picture, we study the electronic reconstruction
of the band edges of a 2D semiconductor, monolayer MoS2, on a hexagonal moire
potential induced by another MoS2 monolayer. We find that such system
transitions from a honeycomb to a hexagonal symmetry when the Fermi level is
tuned from the conduction to the valence side. We also study the system under
magnetic fields, and construct the Hofstadter's butterfly in the electron- and
hole-doped side. Our findings are confirmed by simulating the conductance
across a large-scale two-terminal device. We conclude that this duality is a
general property that MoS2 and other transition-metal-dichalcogenides exhibit
under non-symmetric superlattice potentials.
