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

Marcelino, Edgar; Assis, Thiago A. de; Castilho, C.M.C. de

doi:10.1116/1.4989428

Cited by 15 publications

(11 citation statements)

References 28 publications

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“…Indeed, SC is analytically proved to be valid under these limits in the case of a two-stage field emitter consisting of a rectangular protrusion placed on the center and the top of another rectangular protrusion on a line [28]. Latter, SC was also analytically proved for the case in which a triangular protrusion is placed on the center and the top of a rectangular one on a line, under the same limits [20], a result pointing out that the lack of self-similarity does not affect the validity of SC under these conditions. Surprisingly, different surveys based in many different methods have also verified the validity of SC at some significant region beyond the aforementioned situation [20,23,24,[27][28][29]32].…”

Section: Introductionmentioning

confidence: 92%

“…In particular, Single Tip Field emitters (STFE) [20][21][22][23][24][25][26] are of greater interest. This happens because strong electric fields are required to extract electrons from surfaces, usually from the order of a few volts per nanometer.…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless, STFEs with large aspect-ratios may involve some limitations due to the lack of mechanical stability. This problem may be solved by considering multi-stage structures, since these are able to provide mechanical stability together with significant increasing of the FEF, leading to intense theoretical investigation [20,23,24,[27][28][29]32]. Some times multi-stage emitters appear even at unexpected situations.…”

Section: Introductionmentioning

confidence: 99%

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Analytical proof of Schottky Conjecture for multi-stage field emitters

Marcelino

2019

Preprint

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Schottky Conjecture is analytically proved for multi-stage field emitters consisting on the superposition of rectangular or trapezoidal protrusions on a line under some specific limit. The case in which a triangular protrusion is present on the top of each emitter is also considered as an extension of the model. The results presented here are obtained via Schwarz-Christoffel conformal mapping and reinforce the validity of Schottky Conjecture when each protrusion is much larger than the ones above it, even when an arbitrary number of stages is considered. Moreover, it is showed that it is not necessary to require self-similarity between each of the stages in order to ensure the validity of the conjecture under the appropriate limits.

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Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analytical proof of Schottky Conjecture for multi-stage field emitters

Marcelino

2019

Preprint

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“…We ob-tain a suitable conformal mapping to perform a firstprinciples evaluation of the FEF in the vicinity of the tip of the nanostructure, where the charge density is maximal. Since actual nanostructures are modeled by possibly curved unidimensional lines, standard conformal mapping techniques [36][37][38][39][40][41][42][43], as the Schwarz-Christoffel transformation (SCT) [44,45], are not suitable to solve the problem. Therefore, we consider the Loewner's equation (LE) approach [46] to obtain the desired conformal mappings.…”

mentioning

confidence: 99%

“…The vertical slit can be viewed as the limit of an infinity line with an isosceles triangular protrusion of height L and half-width a, when a tends to zero. The evaluation of γ near the apex (z = iL) of this system [42]…”

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First-Principles Analysis of Nanoelectromechanical Systems Using the Loewner Equation

et al. 2019

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The Loewner equation (LE) is used to obtain conformal mappings that lead to exact and analytical expressions for several electrostatic properties of realistic quasi-unidimensional nanoelectromechanical systems (NEMS). The LE approach also embraces curved geometries, impossible to be addressed by traditional methods such as the Schwarz-Christoffel transformation, often used in this scenario. Among the possible applications of the formalism, we show that it allows for an exact evaluation of the field enhancement factor (FEF) close to the apex of different emitters. Despite its key role in the demodulation process for radio-receiver nano-devices, actual FEF values have been mostly obtained via numerical and/or phenomenological approaches. This work extends the already huge universe of applications of the LE and provides an analytical method to evaluate the FEF, even for curved emitters. Furthermore, our results provide a signature of the varying emitted current's response due to the nanostructure oscillation, justifying its role in the demodulation process of radio-frequency.The study of nanoelectromechanical systems (NEMS) [1,2] currently attracts great attention of the scientific community, not only due to the interesting theoretical aspects involved [3][4][5][6], but also as a result of the enormous number of potential applications that can be derived among the many issues related to the field. Some examples include quantum nanomechanical resonators [7], single-molecule detection [8], chemical, mass and thermal sensing [9-13], integrated circuits [14], high-frequency signal sources/generation [14-16] and field emitting nanotubes operating similarly to diode detectors [17]. Most of these applications involve the oscillation of nanotubes [18], or other similarly shaped structures, with lateral dimensions around a few nanometers [1]. Besides that, NEMSs formed by field emission (FE) diode-like nanodetectors [19,20] must present a large aspect ratio.Under the action of an external macroscopic electrostatic field, E 0 , the local field close to the apex of a NEMS is largely enhanced, which is measured by the field enhancement factor (FEF), reaching typical values ∼ 10 2 − 10 3 . This makes carbon nanotubes (CNTs) [18,21] suitable for producing related technological applications [2,17,[22][23][24]. Indeed, by field-induced emission of electrons, it was possible to control the resonance vibrations of CNTs with 40 µm height and radii in the range between 10 and 20 nm (aspect ratio ∼ 10 3 ), when the tip anode is a few millimeters far away from the nanotube apex, under ultra high vacuum conditions [20]. For such conditions, the nanotube apex-FEF, which is evaluated at a well characterized distance from the uppermost atom as we will define latter, is expected to depend only on the geometry [25,26]. These limits, which are valid for technological purposes, will be taken into account here.When a CNT is excited by a Lorentz force, the spatial and temporal variations of its apex-FEF during oscillation becomes a key parameter f...

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Interplay between morphological and shielding effects in field emission via Schwarz-Christoffel transformation

Marcelino

Assis

Castilho

2018

Journal of Applied Physics

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It is well known that sufficiently strong electrostatic fields are able to change the morphology of Large Area Field Emitters (LAFEs). This phenomenon affects the electrostatic interactions between adjacent sites on a LAFE during field emission and may lead to several consequences, such as: the emitter's degradation, diffusion of absorbed particles on the emitter's surface, deflection due to electrostatic forces, and mechanical stress. These consequences are undesirable for technological applications, since they may significantly affect the macroscopic current density on the LAFE. Despite the technological importance, these processes are not completely understood yet. Moreover, the electrostatic effects due to the proximity between emitters on a LAFE may compete with the morphological ones. The balance between these effects may lead to a non trivial behavior in the apex-Field Enhancement Factor (FEF). The present work intends to study the interplay between proximity and morphological effects by studying a model amenable for an analytical treatment. In order to do that, a conducting system under an external electrostatic field, with a profile limited by two mirror-reflected triangular protrusions on an infinite line, is considered. The FEF near the apex of each emitter is obtained as a function of their shape and the distance between them via a Schwarz-Christoffel transformation. Our results suggest that a tradeoff between morphological and proximity effects on a LAFE may provide an explanation for the observed reduction of the local FEF and its variation at small distances between the emitter sites.

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

Cited by 15 publications

References 28 publications

Analytical proof of Schottky Conjecture for multi-stage field emitters

Analytical proof of Schottky Conjecture for multi-stage field emitters

First-Principles Analysis of Nanoelectromechanical Systems Using the Loewner Equation

Interplay between morphological and shielding effects in field emission via Schwarz-Christoffel transformation

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