Extension of the general thermal field equation for nanosized emitters

Abstract

During the previous decade, K.L. Jensen et. al. developed a general analytical model that successfully describes electron emission from metals both in the field and thermionic regimes, as well as in the transition region. In that development, the standard image corrected triangular potential barrier was used. This barrier model is valid only for planar surfaces and therefore cannot be used in general for modern nanometric emitters. In a recent publication the authors showed that the standard Fowler-Nordheim theory can be generalized for highly curved emitters if a quadratic term is included to the potential model. In this paper we extend this generalization for high temperatures and include both the thermal and intermediate regimes. This is achieved by applying the general method developed by Jensen to the quadratic barrier model of our previous publication. We obtain results that are in good agreement with fully numerical calculations for radii R > 4nm, while our calculated current density differs by a factor up to 27 from the one predicted by the Jensen's standard General-Thermal-Field (GTF) equation. Our extended GTF equation has application to modern sharp electron sources, beam simulation models and vacuum breakdown theory.

Ab initio calculation of field emission from metal surfaces with atomic-scale defects

“…It has already been shown that the FN equation is valid only for surfaces with radii of curvature larger than ≈ 20 nm 31,32 . Although the behaviour of non-planar emitters has been studied extensively [33][34][35][36][37][38] , these works relied on analytical approximations for the surface potential barrier, the assumption of a smooth, well defined surface for the emitter and a semi-classical approach for the transmission probability.…”

A general computational method for electron emission and thermal effects in field emitting nanotips

“…It is the assumptions on the shape of V (x) that limit the validity of the FN and GTF equations and in some cases, even their recent extensions of ref. [12,13]. In the standard theories based on the SN barrier, the electrostatic potential is approximated to be linear V (x) = F x where F is the local field at the emitting point.…”

“…The latter approximation is in principle valid only for planar surfaces, and practically valid for emitter radii of curvature R grater than 15-20nm. In the recent extensions of [12,13], a quadratic curvature-dependent term was added to the linear one, giving the form…”

“…It is important to define whether the tip is "sharp" (nanoscale) or "blunt" (macroscopic scale). If the tip is "sharp", in some cases even the approximations suggested in [12,13] may not be sufficient.…”

Why the Electron Energy Balance During the Thermo-Field Emission From Refractory Metal Micro-Protrusions Can Trigger a Thermal Instability ?

“…Relying on the ever-increasing computing power, several numerical models have followed one another since the '80s. They brought insights into the heating evolution in time [28,29,30] along with more accurate considerations of the underlying quantum dynamics [31,32,33]. Concerning the contribution of the Nottingham effect to the emitter self-heating, modeling works have shown that it was predominant over the resistive heating at low current density, while the situation reverses at higher density [34,35].…”