  We use first-principles calculations to investigate the band structure
evolution of W$X_2$/Mo$X_2$ ($X$ = S, Se) heterobilayers under a perpendicular
electric field. We characterize the extent to which the type-II band alignment
in these compounds can be tuned or inverted electrostatically. Our results
demonstrate two effects of stacking configuration. First, different stackings
produce different net dipole moments, resulting in band offset variations that
are larger than 0.1 eV. Second, based on symmetry constraints that depend on
stacking, a perpendicular electric field may hybridize W$X_2$ and Mo$X_2$ bands
that cross at the Brillouin Zone corner $K$. Our results suggest that external
electric fields can be used to tune the physics of intralayer and interlayer
excitons in heterobilayers of transition metal dichalcogenides.
