  Modulation of electronic properties of materials by electric fields is
central to the operation of modern semiconductor devices, providing access to
complex electronic behaviors and greater freedom in tuning the energy bands of
materials. Here, we explore one-dimensional superlattices induced by a
confining electrostatic potential in monolayer MoS$_2$, a prototypical
two-dimensional semiconductor. Using first-principles calculations, we show
that periodic potentials applied to monolayer MoS$_2$ induce electrostatic
superlattices in which the response is dominated by structural distortions
relative to purely electronic effects. These structural distortions reduce the
intrinsic band gap of the monolayer substantially while also polarizing the
monolayer through piezoelectric coupling, resulting in spatial separation of
charge carriers as well as Stark shifts that produce dispersive minibands.
Importantly, these minibands inherit the valley-selective magnetic properties
of monolayer MoS$_2$, enabling fine control over spin-valley coupling in
MoS$_2$ and similar transition-metal dichalcogenides.
