  Van der Waals semiconductor heterostructures could be a platform to harness
hot photoexcited carriers in the next generation of optoelectronic and
photovoltaic devices. The internal quantum efficiency of hot-carrier devices is
determined by the relation between photocarrier extraction and thermalization
rates. Using \textit{ab-initio} methods we show that the photocarrier
thermalization time in single-layer transition metal dichalcogenides strongly
depends on the peculiarities of the phonon spectrum and the electronic
spin-orbit coupling. In detail, the lifted spin degeneracy in the valence band
suppresses the hole scattering on acoustic phonons, slowing down the
thermalization of holes by one order of magnitude as compared to electrons.
Moreover, the hole thermalization time behaves differently in MoS$_2$ and
WSe$_2$ because spin-orbit interactions differ in these seemingly similar
materials. We predict that the internal quantum efficiency of a tunneling van
der Waals semiconductor heterostructure depends qualitatively on whether
MoS$_2$ or WSe$_2$ is used.
