  A recent scientific debate has arisen: Which processes underlie the actual
ground of the valley Hall effect (VHE) in two-dimensional materials? The
original VHE emerges in samples with ballistic transport of electrons due to
the anomalous velocity terms resulting from the Berry phase effect. In
disordered samples though, alternative mechanisms associated with electron
scattering off impurities have been suggested: (i) asymmetric electron
scattering, called skew scattering, and (ii) a shift of the electron wave
packet in real space, called a side jump. It has been claimed that the side
jump not only contributes to the VHE but fully offsets the anomalous terms
regardless of the drag force for fundamental reasons and, thus, the side-jump
together with skew scattering become the dominant mechanisms. However, this
claim is based on equilibrium theories without any external valley-selective
optical pumping, which makes the results fundamentally interesting but
incomplete and impracticable. We develop in this paper a microscopic theory of
the photoinduced VHE using the Keldysh nonequilibrium diagrammatic technique
and find that the asymmetric skew scattering mechanism is dominant in the
vicinity of the interband absorption edge. This allows us to explain the
operation of optical transistors based on the VHE.
