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

Generalized high-energy thermionic electron injection at graphene interface