Electron Field Emission from Low-energy Bands in 2-Dimensional Materials

Conventionally, field emission has been thought to be dominated by the conduction band electrons near the Fermi surface. For vertical emission from 2D materials, this leads to a linear scaling of ln(J/F^β) ∝ 1/F with β=1, which is distinct from the classical Fowler-Nordheim law of β=2. In this work, we show that electrons from the low-energy valence bands can contribute significantly to the field emission current, giving rise to a highly nonlinear scaling when the electric field strength exceeds some threshold F_c. We present a comprehensive formalism for 2D materials with a generic band structure over a wide range of parameters. We find that the valence electron field emission will outweigh the conduction band at F>F_c even when the Fermi energy is inside the conduction band. The threshold field can be readily tuned by varying the Fermi energy and the tunneling barrier, making systems incorporating with 2D materials an ideal platform to realize prominent valence band electron emission. We compute the values of F_c for some common 2D materials, which are well below breakdown field. Our model can be used  to characterize the field emission measurement at different operating conditions and provide important parameters of 2D material-interface such as barrier height and electron injection time. The regime of space charge limited field emission is also identified at higher field.