  We examine the full static non-linear optical response of uniaxially strained
transition metal dichalcogenide monolayers doped with a finite carrier density
in the conduction band, in the presence of disorder. We find that the customary
shift current is suppressed, yet we identify a strong, valley-dependent
non-reciprocal response, which we term a \textit{unidirectional
valley-contrasting photo-current} (UVCP). This DC current originates from the
combined effect of strain and Kramers symmetry breaking by trigonal warping,
while the contributions due to individual valleys can be separated by
introducing an energy offset between them by means of a magnetization. This
latter fact enables one to monitor inter-valley transitions. The UVCP is
proportional to the mobility and is enhanced by the excitonic Coulomb
interaction and inter-valley scattering, as well as by a top gate bias. We
discuss detection strategies in state-of-the-art experiments.
