  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that the traditional FN law is no longer valid for 2D materials.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.
