Generalized high-energy thermionic electron injection at graphene interface

Graphene thermionic electron emission across high-interface-barrier involves energetic electrons residing far away from the Dirac point where the Dirac cone approximation of the band structure breaks down. Here we construct a full-band model beyond the simple Dirac cone approximation for the thermionic injection of high-energy electrons in graphene. We show that the thermionic emission model based on the Dirac cone approximation is valid only in the graphene/semiconductor Schottky interface operating near room temperature, but breaks down in the cases involving high-energy electrons, such as graphene/vacuum interface or heterojunction in the presence of photon absorption, where the full-band model is required to account for the band structure nonlinearity at high electron energy. We identify a critical barrier height, $\Phi_B^{(\text{c})} \approx 3.5$ eV, beyond which the Dirac cone approximation crosses over from underestimation to overestimation. In the high-temperature thermionic emission regime at graphene/vacuum interface, the Dirac cone approximation severely overestimates the electrical and heat current densities by more than 50\% compared to the more accurate full-band model. The large discrepancies between the two models are demonstrated using a graphene-based thermionic cooler. These findings reveal the fallacy of Dirac cone approximation in the thermionic injection of high-energy electrons in graphene. The full-band model developed here can be readily generalized to other 2D materials, and shall provide an improved theoretical avenue for the accurate analysis, modeling and design of graphene-based thermionic energy devices.