Universal model for electron thermal-field emission from two-dimensional semimetals

We present the theory of out-of-plane (or vertical) electron thermal-field emission from two-dimensional (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 other and bear no special signature in electron emission characteristics. Our findings represent the quantum extension of the universal semiclassical thermionic emission scaling law in 2D materials and provide theoretical foundations for the understanding of electron emission from cathode and charge interface transport for the design of 2D-material-based vacuum nanoelectronics.