Generalized Fowler-Nordheim field-induced vertical electron emission model for two-dimensional materials

Current theoretical description of field-induced electron emission remains mostly bounded by the classic Fowler-Nordheim (FN) framework developed nearly one century ago. For the emerging class of two-dimensional (2D) materials, many basic assumptions of FN model become invalid due to their reduced dimensionality and exotic electronic properties. In this work, we develop analytical and semi-analytical models of field-induced vertical electron emission from the surface of 2D materials by explicitly taking into account the reduced dimensionality, non-parabolic energy spectrum, non-conserving in-plane electron momentum, finite-temperature and space-charge-limited effects. We show that the traditional FN law is no longer valid for 2D materials. The modified vertical field emission model developed here provides better agreement with experimental results. Intriguingly, a new high-field regime of \emph{saturated surface field emission} emerges due to the reduced dimensionality of 2D materials. A remarkable consequence of this saturated field emission effect is the absence of space-charge-limited current normally expected at high field in three-dimensional bulk material.