  We present the theory of out-of-plane (or vertical) electron thermal-field
emission from 2D semimetals. We show that the current-voltage-temperature
characteristic is well-captured by a universal scaling relation applicable for
broad classes of 2D semimetals, including graphene and its few-layer, nodal
point semimetal, Dirac semimetal at the verge of topological phase transition
and nodal line semimetal. Here an important consequence of the universal
emission behavior is revealed: in contrast to the common expectation that band
topology shall manifest differently in the physical observables, band
topologies in two spatial dimension are indistinguishable from each others and
bear no special signature in the electron emission characteristics. Our
findings represent the quantum extension of the universal semiclassical
thermionic emission scaling law in 2D materials, and provide the theoretical
foundations for the understanding of electron emission from cathode and charge
interface transport for the design of 2D-material-based vacuum nanoelectronics.
