  Vertical heterostructures of van der Waals materials enable new pathways to
tune charge and energy transport characteristics in nanoscale systems. We
propose that graphene Schottky junctions can host a special kind of
photoresponse which is characterized by strongly coupled heat and charge flows
that run vertically out of the graphene plane. This regime can be accessed when
vertical energy transport mediated by thermionic emission of hot carriers
overwhelms electron-lattice cooling as well as lateral diffusive energy
transport. As such, the power pumped into the system is efficiently extracted
across the entire graphene active area via thermionic emission of hot carriers
into a semiconductor material. Experimental signatures of this regime include a
large and tunable internal responsivity ${\cal R}$ with a non-monotonic
temperature dependence. In particular, ${\cal R}$ peaks at electronic
temperatures on the order of the Schottky potential $\phi$ and has a large
upper limit ${\cal R} \le e/\phi$ ($e/\phi=10\,{\rm A/W}$ when $\phi =
100\,{\rm meV}$). Our proposal opens up new approaches for engineering the
photoresponse in optically-active graphene heterostructures.
