A self-consistent Schrödinger–Laplace theory of a sub-nanometer-radius field emitter

Source abstract

Conventional analytical models of field emission (FE) rely on the assumption of plane waves for the electron wave functions and a Cartesian symmetry in the Schrödinger equation, which fail to adequately describe nanoscale emitters with non-planar surfaces. Here, we propose a theoretical approach accounting for the exact symmetry of both the Schrödinger and Laplace equations that provides a quantitative description of FE from ultra-high-curvature surfaces. The model is validated through comparison with experimental data for a carbon emitter, which consists primarily of graphitic carbon and takes the form of a cylindrical nanowire (CNW) with a radius of 0.65 nm, formed on a paraboloidal post with a curvature radius of 3.5 nm. Solving the Laplace equation for this geometry showed significant emission contributions from both the nanowire and the parabolic base. To calculate the current–voltage ( I – V ) characteristic, we employed the solutions of the Schrödinger equation for the nanoscale emitters with the corresponding symmetries (cylinder and paraboloid). The calculations yielded good quantitative agreement between the theoretical I – V curve and experimental data, including the correct prediction of the slope of the ln( I)− 1/ V curve, which was obtained without the use of fitting parameters. Our work highlights the necessity of such theoretical approaches for nanometer-scale emitters.