  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.
