A Modern Theory for the Emission of Electrons from Ultrasharp Nanometric Metal Tips in Intense Electric Fields – Application to the Lateral Resolution of the Near-Field-Emission Scanning Electron Microscope
Creators: name: Chatziafratis, Andreas
Publisher: name: Zenodo
Year: 2020
Subjects: subject: exact electron eigenstates and Whittaker wavefunctions in atomically sharp nanoemitters
subject: three-dimensional Schrödinger equation in parabolic coordinates
subject: Jeffreys-Wentzel-Kramers-Brillouin (WKB) approximation method collapse
subject: lateral resolution interpretation for the Near-Field Emission Scanning Electron Microscope (NFE-SEM)
subject: nonlinear nanoscale Fowler–Nordheim plots and slope correction factors for experimental emitter characterization
subject: low-energy electrons and field electron emission fron nanoscale metal tips
subject: three-dimensional quantum tunneling and curved electron paths
subject: electron supply function and parallel velocity in extremely curved nano-emitters
subject: ab initio mathematical modelling and
subject: quantum confinement and extreme quantization effects
subject: Applied Mathematics
subject: FOS: Mathematics
subject: FOS: Mathematics
subject: Quantum Physics
subject: FOS: Physical sciences
subject: FOS: Physical sciences
subject: Nanotechnology
subject: FOS: Nanotechnology
subject: FOS: Nanotechnology
subject: Electron Microscopy

description: Summary: Progress in field emission theory from its original Fowler–Nordheim (FN) form is focused on the transmission coefficient (or penetration factor) of quantum tunneling through a potential barrier. For the incident supply of electrons impinging on the internal surface of emitter tips, advanced theories still assume the constant value due to a supply of plane-waves states. However, it has been observed in the literature that for emitting tips of apical radius of curvature R in the range 1–5nm, which are achievable with modern fabrication techniques, this is highly questionable. To address this issue, we assumed such a sharp quantum well (standing for metal or carbon nanotubes within the Sommerfeld model) in rotationally parabolic coordinates and exactly solved Schrödinger’s equation, which gives rise to the Whittaker wavefunctions, hence obtaining the exact eigenstates of the system. We then assumed a nearby sample (anode), solved exactly Laplace’s equation to find the electric field in three-dimensional (3D) space and used a hybrid three-dimensional Jeffreys-Wentzel-Kramers-Brillouin (JWKB) method with a parabolic image charge to calculate the emitted current of such nanoscale systems, which are employed nowadays in state-of-the-art nanotechnologies. Significant differences from the usual Cartesian geometry are noticed and the relevant field emission quantities are shown to exhibit a non-FN behavior. Highlights of our findings include: (1) Both the normally incident and parallel electron fluxes are functions of the angle to the emitter axis and affect the emission angle. (2) The WKB approximation fails for this system. (3) The eigenfunctions of the nanoemitter form a continuum only in one dimension while complete discretization occurs in the other two directions. (4) the parallel, to tip surface, electron velocity vanishes at the emitter’s apex which partly explains the recent measurements of the beam spot-size, and therefore the imaging precision or resolution, of the newly developed Near Field Emission Scanning Electron Microscope (NFESEM). (5) Competing effects are found, between locally enhanced fields and current densities, as the tip radius decreases down to 1nm: The electric field increases but the total supply function decreases so that an optimum radius must exist. (6) Transmitted current density of such an emitter is a function of angle of emission θ and R. (7) The angular dependence is different from that of the classical model of a hemisphere on a post. (8) The resulting Fowler-Nordheim plots appear to be non-straight which impacts parameters extraction in laboratory studies.