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 R with a non-monotonic temperature dependence. In particular, R peaks at electronic temperatures on the order of the Schottky potential φ and has a large upper limit R ≤ e/φ (e/φ = 10 A/W when φ = 100 meV). Our proposal opens up new approaches for engineering the photoresponse in optically-active graphene heterostructures.Keywords: graphene; hot carriers; Schottky junction; photocurrent Vertical heterostructures comprising layers of van der Waals (vdW) materials have recently emerged as a platform for designer electronic interfaces [1]. Of special interest are heterostructures which feature tunable interlayer transport characteristics, as exemplified by g/X Schottky junctions [2-10]; here 'g' denotes graphene, and X is a semiconductor material, such as Si, MoS 2 or WSe 2 . These junctions are characterized by Schottky barriers φ that span two orders of magnitude φ ≈ 0.01−1 eV and exhibit in situ control through applied bias or by using gate potentials [7][8][9][10][11]. The wide range of φ achievable across the g/X interface, combined with the unique graphene photoresponse mediated by long-lived hot carriers (elevated electronic temperatures, T g , different from those of the lattice, T 0 [12-17]), make graphene Schottky junctions a prime target for accessing novel vertical energy transport regimes [18].Here we show that specially designed graphene Schottky junctions can host an enhanced thermionicdominated photoresponse driven by strongly coupled charge and energy currents. Such photoresponse proceeds, as illustrated in Fig. 1a, via the thermionic emission of graphene hot carriers with energy larger than the Schottky barrier. At steady state, an equal number of cold carriers are injected at the Fermi surface through an ohmic contact, giving a net flow of heat J ⊥ q out of the graphene electronic system balancing the energy pumped into the system. Strikingly, thermionic emission yields strong heat transport running vertically out of the hot electron system, which dominates over more conventional electronic cooling channels, e.g. electron-lattice cooling. Indeed, we find that J ⊥ q can be significant in graphene (see Fig. 1c) when k B T g ≈ φ/2, dominating over acoustic and optical phonon cooling [12,13] in pristine graphene Sch...