Electrical-Equivalent van der Waals Gap for Two-Dimensional Bilayers

Vertical stacks of two-dimensional (2D) materials separated by a van der Waals gap and held together by van der Waals forces are immensely promising for a plethora of nanotechnological applications. Charge control in these stacks may be modeled with use of either a simple electrostatics approach or a detailed atomistic one. In this paper, we compare these approaches for a gated 2D transition-metal dichalcogenide bilayer and show that recently reported electrostatics-based models of this system give large errors in band energy compared with atomistic (density-functional-theory) simulations. These errors are due to the tails of the ionic potentials that reduce the electrical-equivalent van der Waals gap between the 2D layers, and can be corrected by use of the reduced gap in the electrostatic model. For a physical van der Waals gap (defined as the chalcogen-to-chalcogen distance) of 3 angstrom in a 2D bilayer, the electrical-equivalent gap is less than 1 angstrom. For the example of band-to-band-tunneling-based ultra-low-power transistors, this is seen to lead to errors of several hundred millivolts or more in the threshold voltage estimated from electrostatics.