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.
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