Abstract
We develop analytical models of optical-field-driven electron tunneling from the edge and surface of free-standing two-dimensional (2D) materials. We discover a universal scaling between the tunneling current density (J) and the electric field near the barrier (F): In(J/|F|beta) beta 1/|F| with beta values of (3)/(2) and 1 for edge emission and vertical surface emission, respectively. At ultrahigh values of F, the current density exhibits an unexpected high-field saturation effect due to the reduced dimensionality of the 2D material, which is absent in the traditional bulk material. Our calculation reveals the dc bias as an efficient method for modulating the optical-field tunneling subcycle emission characteristics. Importantly, our model is in excellent agreement with a recent experiment on graphene. Our results offer a useful framework for understanding optical-field tunneling emission from 2D materials, which are helpful for the development of optoelectronics and emerging petahertz vacuum nanoelectronics.