Abstract
Vertical field emission for two-dimensional (2D) materials was normally assumed to have dominant emission by the conduction band electrons near to the Fermi surface with a linear scaling of ln(J/Fβ) ∝ 1/F and β = 1, which is different from the classical Fowler–Nordheim law having β = 2. In this paper, we show that field emission from a low-energy valence band can contribute significantly when the field (F) is above a threshold value, Fc (below the breakdown field), giving rise to an abrupt change in the linear scaling. We present a comprehensive formalism for 2D materials with a generic band structure over a wide range of parameters. It is found that the valence electron field emission will outweigh the conduction band at F > Fc 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. The values of Fc are calculated for some common 2D materials. When the model is used to characterize the measurement of field emission under different operating conditions, the characterization can provide important parameters of the 2D material-interface, such as barrier height and electron injection time. The regime of space charge limited field emission is also identified at higher fields.