Generalized High-Energy Thermionic Electron Injection at Graphene Interface

Graphene thermionic electron emission across a 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 the 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, Φ_B^{( c )}≈3.5 eV, beyond which the Dirac cone approximation crosses over from underestimation to overestimation. In the high-temperature thermionic emission regime at the 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 provides an improved theoretical avenue for the accurate analysis, modeling, and design of graphene-based thermionic energy devices.