  We develop analytical models of optical-field electron tunneling from the
edge and surface of two-dimensional (2D) materials, including the effects of
reduced dimensionality, non-parabolic energy dispersion, band anisotropy,
quasi-time dependent tunneling and emission dynamics indueced by the laser
field. We discover a universal scaling between the tunneling current density
$J$ and the laser electric field $F$: In($J/|F|^{\beta})\propto1/|F|$ with
$\beta = 3 / 2$ in the edge emission and $\beta = 1$ in the vertical surface
emission, which both are distinctive from the traditional Fowler-Nordheim (FN)
model of $\beta = 2$. The current density exhibits an unexpected high-field
saturation effect due to the reduced dimensionality of 2D materials, which is
completely different from the space-charge saturation commonly observed in
traditional bulk materials. Our results reveal the dc bias as an efficient
method in modulating the optical-field tunneling sub-optical-cycle emission
characteristics. Importantly, our model is in excellent agreement with a recent
experiment on graphene. Our findings offer a theoretical foundation for the
understanding of optical-field tunneling emission from the 2D material system,
which is useful for the development of 2D-material based optoelectronics and
vacuum nanoelectronics.
