Schottky’s conjecture, field emitters, and the point charge model

This work presents an accurate numerical study of the electrostatics of a system formed by individual nanostructures mounted on support substrate tips, which provides a theoretical prototype for applications in field electron emission or for the construction of tips in probe microscopy that requires high resolution. The aim is describe the conditions to produce structures mechanically robust with desirable field enhancement factor (FEF). We modeled a substrate tip with a height h1, radius r1 and characteristic FEF γ1, and a top nanostructure with a height h2, radius r2 < r1 and FEF γ2, for both hemispheres on post-like structures. The nanostructure mounted on the support substrate tip then has a characteristic FEF, γC . Defining the relative difference ηR = (γC − γ1)/(γ3 − γ1), where γ3 corresponds to the reference FEF for a hemisphere of the post structure with a radius r3 = r2 and height h3 = h1 + h2, our results show, from a numerical solution of Laplace's equation using a finite element scheme, a scaling ηR = f (u ≡ λθ −1 ), where λ ≡ h2/h1 and θ = r1/r2. Given a characteristic variable uc, for u uc, we found a power law ηR ∼ u κ , with κ ≈ 0.55. For u uc, ηR → 1, which led to conditions where γC → γ3. As a consequence of scale invariance, it is possible to derive a simple expression for γC and to predict the conditions needed to produce related systems with a desirable FEF that are robust owing to the presence of the substrate tip. Finally, we discuss the validity of Schottky's conjecture (SC) for these systems, showing that, while to obey SC is indicative of scale invariance, the opposite is not necessarily true. This result suggests that a careful analysis must be performed before attributing SC as an origin of giant FEF in experiments.Producing nanostructures that allow one to amplify the applied electric field in their vicinity and which are mechanically stable remains an engineering challenge. This can be observed already a long time ago in the pioneer work by Gomer who discuss a method for growing metal whiskers in a modified field emission tube [1]. In fact, the issue of mechanical stability requires a solution for the degradation and failure of nanostructures that occurs during field electron emission at or near the substrate emitter contact [2] and for the self-mechanical oscillations that occur during field electron emission measurements [3,4] or from electrostatic interactions [5]. In particular, a method to study the self-oscillations of a nanostructure mounted on a macroscopic frame requires using a laser beam to excite the sample; subsequently, a second laser beam is then used to register the amplitude of vibrations at a certain point from the object [6].Applications of these nanostructures mounted on tip devices include carbon nanotubes (CNTs) mounted on a support tip, which can be used as an electron source in a high-resolution electron beam. The latter acquires properties such as a stable emitted current and high brightness [7]. Moreover, due to screening effects [8], there is a tendency to...

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“…As already pointed, this feature has been found analytically in Ref. [15], but for conical field emitters using point charge model.…”

Section: 55supporting

confidence: 65%

“…Finally, with the aim to compare our results with those of Refs. [14] and [15], we show in Fig. 5 the behavior of γ C /(γ 1 γ 2 ) as a function of parameter h 2 /r 1 , for specific values of h 1 /r 1 and h 2 /r 2 .…”

Section: 55mentioning

confidence: 99%

Mechanically stable nanostructures with desirable characteristic field enhancement factors: a response from scale invariance in electrostatics

Assis

Dall’Agnol

2016

Nanotechnology

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“…SC became a paradigm of a good approximation in order to explain large FEFs obtained in CFE experiments, nevertheless it is well known that this conjecture is not rigorously true and there is still some lack of theoretical results studying its validity. Previous results suggest that SC is valid when the dimensions of the upper-stage structure are much smaller than the ones from the lower-stage [12,13,18,19]. Furthermore, results using a point charge model suggest that SC remains valid even for some situations in which the two stages have dimensions of the same order of magnitude [12].…”

Section: Introductionmentioning

confidence: 99%

“…A simple example consists on a metal single tip field emitter (STFE). Alternatively one could consider a two-stage-structure which, depending on the dimensions of each of the stages, produces a high local field enhancement factor (FEF), as compared with the single-stage case [10][11][12][13]. All these aspects have motivated the development of field emitter designs featuring these properties [14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

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Unexpected validity of Schottky's conjecture for two-stage field emitters: A response via Schwarz–Christoffel transformation

Marcelino

Assis

Castilho

2017

Journal of Vacuum Science &Amp; Technology B, Nanotechnology and Microelectronics: Materials, Processing, Measurement, and Phen

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The electric field in the vicinity of the top of an emitter with a profile consisting of a triangular protrusion on an infinite line is analytically obtained when this system is under an external uniform electric field. The same problem is also studied when the profile features a two-stage system, consisting of a triangular protrusion centered on the top of a rectangular one on a line. These problems are approached by using a Schwarz-Christoffel conformal mapping and the validity of Schottky's conjecture (SC) is discussed. We provide an analytical proof of SC when the dimensions of the upper-stage structure are much smaller than the lower-stage ones, for large enough aspect ratios and considering that the field enhancement factor (FEF) of the rectangular structure is evaluated on the center of the top of the structure, while the FEF of the triangular stage is evaluated near the upper corner of the protrusion. The numerical solution of our exact equations shows that SC may remain valid even when both stages feature dimensions from the same order of magnitude, reinforcing the validity of SC for multi-stage field emitters.

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A Thermal-Field-Photoemission Model and Its Application

Jensen

2020

Modern Developments in Vacuum Electron Sources

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Schottky’s conjecture, field emitters, and the point charge model

Cited by 28 publications

References 23 publications

Mechanically stable nanostructures with desirable characteristic field enhancement factors: a response from scale invariance in electrostatics

Mechanically stable nanostructures with desirable characteristic field enhancement factors: a response from scale invariance in electrostatics

Unexpected validity of Schottky's conjecture for two-stage field emitters: A response via Schwarz–Christoffel transformation

A Thermal-Field-Photoemission Model and Its Application

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