arXiv subjects

Richard G. Forbes

Publications and source records attributed to Richard G. Forbes.

At least 37 records · Page 2Linked to original sources

arXiv2020

Modeling the Field Emission Enhancement Factor for Capped Carbon Nanotubes using the Induced Electron Density

In many field electron emission experiments on single-walled carbon nanotubes (SWCNTs), the SWCNT stands on one of two well-separated parallel plane plates, with a macroscopic field FM applied between them. For any given location "L" on the SWCNT surface, a field enhancement factor (FEF) is defined as $F_{\rm{L}}$/$F_{\rm{M}}$, where $F_{\rm{L}}$ is a local field defined at "L". The best emission measurements from small-radii capped SWCNTs exhibit characteristic FEFs that are constant (i.e., independent of $F_{\rm{M}}$). This paper discusses how to retrieve this result in quantum-mechanical (as opposed to classical electrostatic) calculations. Density functional theory (DFT) is used to analyze the properties of two short, floating SWCNTS, capped at both ends, namely a (6,6) and a (10,0) structure. Both have effectively the same height ($\sim 5.46$ nm) and radius ($\sim 0.42$ nm). It is found that apex values of local induced FEF are similar for the two SWCNTs, are independent of $F_{\rm{M}}$, and are similar to FEF-values found from classical conductor models. It is suggested that these induced-FEF values relate to the SWCNT longitudinal system polarizabilities, which are presumed similar. The DFT calculations also generate "real", as opposed to ``induced", potential-energy (PE) barriers for the two SWCNTs, for FM-values from 3 V/$μ$m to 2 V/nm. PE profiles along the SWCNT axis and along a parallel ``observation line" through one of the topmost atoms are similar. At low macroscopic fields the details of barrier shape differ for the two SWCNT types. Even for $F_{\rm{M}}=0$, there are distinct PE structures present at the emitter apex (different for the two SWCNTs); this suggests the presence of structure-specific chemically induced charge transfers and related patch-field distributions.

cond-mat.mes-hall↗

arXiv2020

Implementation of the orthodoxy test as a validity check on experimental field emission data

In field electron emission (FE) studies, it is important to check and analyse the quality and validity of experimental current-voltage data, which is usually plotted in one of a small number of standard forms. These include the so-called Fowler-Nordheim (FN), Millikan-Lauritsen (ML) and Murphy-Good (MG) plots. The Field Emission Orthodoxy Test is a simple quantitative test that aims to check for the reasonableness of the values of the parameter "scaled field" that can be extracted from these plots. This is done in order to establish whether characterization parameters extracted from the plot will be reliable or, alternative, likely to be spurious. This paper summarises the theory behind the orthodoxy test, for each of the plot forms, and confirms that it is easy to apply it to the newly developed MG plot. A simple web tool has been developed that extracts scaled-field values from any of these three plot forms, and tests for lack of field emission orthodoxy.

physics.app-ph↗

arXiv2020

Applying the field emission orthodoxy test to Murphy-Good plots

In field electron emission (FE) studies, it is important to check and analyse the quality and validity of results experimentally obtained from samples, using suitably plotted current-voltage [I(V)] measurements. For the traditional plotting method, the Fowler-Nordheim (FN) plot, there exists a so-called "orthodoxy test" that can be applied to the FN plot, in order to check whether the FE device/system generating the results is "ideal". If it is not ideal, then emitter characterization parameters deduced from the FN plot are likely to be spurious. A new form of FE I(V) data plot, the so-called "Murphy-Good (MG) plot" has recently been introduced (R.G. Forbes, Roy. Soc. Open Sci. 6 (2019) 190912. This aims to improve the precision with which characterization-parameter values (particularly values of formal emission area) can be extracted from FE I(V) data. The present paper compares this new plotting form with the older FN and Millikan-Lauritsen (ML) forms, and makes an independent assessment of the consistency with which slope (and hence scaled-field) estimates can be extracted from a MG plot. It is shown that, by using a revised formula for the extraction of scaled-field values, the existing orthodoxy test can be applied to Murphy-Good plots. The development is reported of a prototype web tool that can apply the orthodoxy test to all three forms of FE data plot (ML, MG and FN).

physics.app-ph↗

arXiv2019

Restoring observed classical behavior of the carbon nanotube field emission enhancement factor from the electronic structure

Experimental Fowler-Nordheim plots taken from orthodoxly behaving carbon nanotube (CNT) field electron emitters are known to be linear. This shows that, for such emitters, there exists a characteristic field enhancement factor (FEF) that is constant for a range of applied voltages and applied macroscopic fields $F_\text{M}$. A constant FEF of this kind can be evaluated for classical CNT emitter models by finite-element and other methods, but (apparently contrary to experiment) several past quantum-mechanical (QM) CNT calculations find FEF-values that vary with $F_\text{M}$. A common feature of most such calculations is that they focus only on deriving the CNT real-charge distributions. Here we report on calculations that use density functional theory (DFT) to derive real-charge distributions, and then use these to generate the related induced-charge distributions and related fields and FEFs. We have analysed three carbon nanostructures involving CNT-like nanoprotrusions of various lengths, and have also simulated geometrically equivalent classical emitter models, using finite-element methods. We find that when the DFT-generated local induced FEFs (LIFEFs) are used, the resulting values are effectively independent of macroscopic field, and behave in the same qualitative manner as the classical FEF-values. Further, there is fair to good quantitative agreement between a characteristic FEF determined classically and the equivalent characteristic LIFEF generated via DFT approaches. Although many issues of detail remain to be explored, this appears to be a significant step forwards in linking classical and QM theories of CNT electrostatics. It also shows clearly that, for ideal CNTs, the known experimental constancy of the FEF value for a range of macroscopic fields can also be found in appropriately developed QM theory.

cond-mat.mes-hall↗

arXiv2019

On the quantum mechanics of how an ideal carbon nanotube field emitter can exhibit a constant field enhancement factor

Measurements of current-voltage characteristics from ideal carbon nanotube (CNT) field electron emitters of small apex radius have shown that these emitters can exhibit a linear Fowler-Nordheim (FN) plot [e.g., Dean and Chalamala, Appl. Phys. Lett., 76, 375, 2000]. From such a plot, a constant (voltage-independent) characteristic field enhancement factor (FEF) can be deduced. Over fifteen years later, this experimental result has not yet been convincingly retrieved from first-principles electronic structure calculations, or more generally from quantum mechanics (QM). On the contrary, several QM calculations have deduced that the characteristic FEF should be a function of the macroscopic field applied to the CNT. This apparent contradiction between experiment and QM theory has been an unexplained feature of CNT emission science, and has raised doubts about the ability of existing QM models to satisfactorily describe experimental CNT emission behavior. In this work we demonstrate, by means of a density functional theory analysis of single-walled CNTs "floating" in an applied macroscopic field, the following significant result. This is that agreement between experiment, classical-conductor CNT models and QM calculations can be achieved if the latter are used to calculate (from the "real" total-charge-density distributions initially obtained) the distributions of $\textit{induced}$ charge-density, induced local fields and induced local FEFs. The present work confirms, more reliably and in significantly greater detail than in earlier work on a different system, that this finding applies to the common "post-on-a-conducing plane" situation of CNT field electron emission. This finding also brings out various further theoretical questions that need to be explored.

cond-mat.mes-hall↗

arXiv2019

Comments on the continuing widespread and unnecessary use of a defective emission equation in field emission related literature

Field electron emission (FE) has relevance in many different technological contexts. However, many related technological papers use a physically defective elementary FE equation for local emission current density (LECD). This equation takes the tunneling barrier as exactly triangular, as in the original FE theory of 90 years ago. More than 60 years ago, it was shown that the so-called Schottky-Nordheim (SN) barrier, which includes an image-potential-energy term (that models exchange-and-correlation effects) is better physics. For a metal-like emitter with work-function 4.5 eV, the SN-barrier-related Murphy-Good FE equation predicts LECD values that are higher than the elementary equation values by a large factor, often between around 250 and around 500. By failing to mention/apply this 60-year-old established science, or to inform readers of the large errors associated with the elementary equation, many papers (aided by defective reviewing) spread a new kind of "pathological science", and create a modern research-integrity problem. The present paper aims to enhance author and reviewer awareness by summarizing relevant aspects of FE theory, by explicitly identifying the misjudgment in the original 1928 Fowler-Nordheim paper, by explicitly calculating the size of the resulting error, and by showing in detail why most FE theoreticians regard the 1950s modifications as better physics. Suggestions are made, about nomenclature and about citation practice, that may help to diminish misunderstandings.

cond-mat.mes-hall↗

arXiv2019

Field emission: Why converting LAFE voltages to macroscopic fields before making a Fowler-Nordheim plot has often led to spurious characterization results

An important parameter used to characterize large-area field electron emitters (LAFEs) is the characteristic apex field enhancement factor γ_C. This parameter is normally extracted from the slope of a Fowler-Nordheim (FN) plot. Several years ago, the development of an "orthodoxy test" allowed a sample of 19 published FN plots relating to LAFEs to be tested, and it was found that about 40% of the related papers were reporting spuriously high values for γ_C. In technological papers relating to LAFE characterization, common practice is to pre-convert the measured voltage into an (apparent) value of macroscopic field before making and analyzing a FN plot. This paper suggests that the cause of the "spurious-FEF-value" problem is the widespread use of a pre-conversion equation that is defective (for example, not compatible with ordinary electrical circuit theory) when it is applied to so-called "non-ideal" field emission devices/systems. Many real devices/ systems are non-ideal. The author argues that FN plots should be made using raw experimental current-voltage data, and that an orthodoxy test should be applied to the resulting FN plot before any more-detailed analysis, and that (in view of growing concerns over the reliability of published "scientific" results) reviewers should scrutinize field emission materials-characterization papers with enhanced care.

physics.app-ph↗

arXiv2019

The Murphy-Good plot: a better method of analysing field emission data

Measured field electron emission (FE) current-voltage Im(Vm) data are traditionally analysed via Fowler-Nordheim (FN) plots, as ln{Im/(Vm)**2} vs 1/Vm. These have been used since 1929, because in 1928 FN predicted they would be linear. In the 1950s, a mistake in FN's thinking was found. Corrected theory by Murphy and Good (MG) made theoretical FN plots slightly curved. This causes difficulties when attempting to extract precise values of emission characterization parameters from straight lines fitted to experimental FN plots. Improved mathematical understanding, from 2006 onwards, has now enabled a new FE data-plot form, the "Murphy-Good plot". This plots ln{Im/(Vm)**(2-(η/6)} vs 1/Vm, where η depends only on local work function. Modern ("21st century") MG theory predicts that a theoretical MG plot should be "almost exactly" straight. This makes precise extraction of well-defined characterization parameters from ideal I_m(V_m) data much easier. This article gives the theory needed to extract characterization parameters from MG plots, setting it within the framework of wider difficulties in interpreting FE Im(Vm) data (among them, use of the "planar emission approximation"). Careful use of MG plots could also help remedy other problems in FE technological literature. It is argued MG plots should now supersede FN plots.

cond-mat.mes-hall↗

arXiv2018

Numerical testing by a transfer-matrix technique of Simmons' equation for the local current density in metal-vacuum-metal junctions

We test the consistency with which Simmons' model can predict the local current density obtained for flat metal-vacuum-metal junctions. The image potential energy used in Simmons' original papers had a missing factor of 1/2. Besides this technical issue, Simmons' model relies on a mean-barrier approximation for electron transmission through the potential-energy barrier between the metals. In order to test Simmons' expression for the local current density when the correct image potential energy is included, we compare the results of this expression with those provided by a transfer-matrix technique. This technique is known to provide numerically exact solutions of Schrodinger's equation for this barrier model. We also consider the current densities provided by a numerical integration of the transmission probability obtained with the WKB approximation and Simmons' mean-barrier approximation. The comparison between these different models shows that Simmons' expression for the local current density actually provides results that are in good agreement with those provided by the transfer-matrix technique, for a range of conditions of practical interest. We show that Simmons' model provides good results in the linear and field-emission regimes of current density versus voltage plots. It loses its applicability when the top of the potential-energy barrier drops below the Fermi level of the emitting metal.

cond-mat.mes-hall↗

arXiv2018

Physics-based derivation of a formula for the mutual depolarization of two post-like field emitters

Recent analyses of the field enhancement factor (FEF) from multiple emitters have revealed that the depolarization effect is more persistent with respect to the separation between the emitters than originally assumed. It has been shown that, at sufficiently large separations, the fractional reduction of the FEF decays with the inverse cube power of separation, rather than exponentially. The behavior of the fractional reduction of the FEF encompassing both the range of technological interest $0<c/h\lesssim5$ ($c$ being the separation and $h$ is the height of the emitters) and $c\rightarrow\infty$, has not been predicted by the existing formulas in field emission literature, for post-like emitters of any shape. In this letter, we use first principles to derive a simple two-parameter formula for fractional reduction that can be of interest for experimentalists to modeling and interpret the FEF from small clusters of emitters or arrays in small and large separations. For the structures tested, the agreement between numerical and analytical data is $\sim1\%$.

cond-mat.mes-hall↗

arXiv2018

Field electron emission theory (October 2016),v2

This document provides an updated account of material originally presented in two field electron emission (FE) tutorial lectures given at the 2016 Young Researchers' School in Vacuum Micro/Nano Electronics, held in Saint-Petersburg in October 2016. The aim of the tutorial lectures was to set out modern versions of some of the basics of mainstream FE theory. This paper indicates, in some depth, the scope, structure and content of the tutorials, and also where some of the related published material can be found.

cond-mat.mes-hall↗

arXiv2017

Field emission: the theoretical link between voltage loss, reduction in field enhancement factor, and Fowler-Nordheim-plot saturation

With a large-area field electron emitter, when an individual post-like emitter is sufficiently resistive, and current through it sufficiently large, then voltage loss occurs along it. This Letter provides a simple analytical and conceptual demonstration that this voltage loss is directly and inextricably linked to a reduction in the field enhancement factor (FEF) at the post apex. A formula relating apex-FEF reduction to this voltage loss was obtained in the paper by E. Minoux, O. Groening, K. B. K. Teo, S. H. Dalal, L. Gangloff, J.-P. Schnell, L. Hudanski, I. Y. Y. Bu., P. Vincent, P. Legagneux, G. A. J. Amaratunga, and W. I. Milne [Nano Lett. 5, 2135 (2005)], by fitting to numerical results from a Laplace solver. This Letter derives the same formula analytically, by using a "floating sphere" model. The analytical proof brings out the underlying physics more clearly, and shows that the effect is a general phenomenon, related to reduction in the magnitude of the surface charge in the most protruding parts of an emitter. Voltage-dependent FEF-reduction is one cause of "saturation" in Fowler-Nordheim plots. Another is a voltage-divider effect, due to measurement-circuit resistance. An integrated theory of both effects is presented. Both together, or either by itself, can cause saturation. Experimentally, if saturation occurs but voltage loss is small (< 20 V, say), then saturation is more probably due to FEF-reduction than voltage division. In this case, existing treatments of electrostatic interaction ("shielding") between closely spaced emitters may need modification. Other putative causes of saturation exist, so the present theory is a partial story. Its extension seems possible, and could lead to a more general physical understanding of the causes of FN-plot saturation.

cond-mat.mes-hall↗

arXiv2016

Physical electrostatics of small field emitter arrays/clusters

This paper improves understanding of electrostatic influences on apex field enhancement factors (AFEFs) for small field emitter arrays. Using the "floating sphere at emitter-plate potential" (FSEPP) model, it re-examines the electrostatics and mathematics of three simple systems of identical post-like emitters. For the isolated emitter, various approaches are noted. On need consider only the effects of sphere charges and (for separated emitters) image charges. For the 2-emitter system, formulas are found for "charge-blunting" and "neighbour-field" effects, for widely spaced and "sufficiently closely spaced" emitters. Mutual charge-blunting is always dominant, with a related (negative) fractional AFEF-change δ_two. For sufficiently small emitter spacing c, |δ_two| varies as 1/c; for large spacing, |δ_two| decreases as 1/c^3. In a 3-emitter linear array, differential charge-blunting and differential neighbor-field effects occur, but the former are dominant, and cause the "exposed" outer emitters to have higher AFEF (γ_0) than the central emitter (γ_1). Formulas are found for the exposure ratio Ξ=γ_0/γ_1, for large and for sufficiently small separations. The FSEPP model for an isolated emitter has accuracy around 30%. Line-charge models (LCMs) are an alternative, but an apparent difficulty with recent LCM models is identified. Better descriptions of array electrostatics may involve developing good fitting equations for AFEFs derived from accurate numerical solution of Laplace's equation, perhaps with equation form(s) guided qualitatively by FSEPP-model results. In existing fitting formulas, the AFEF-reduction decreases exponentially as c increases, which differs from FSEPP-model formulas. FSEPP models might provide a useful guide to the qualitative behaviour of small field emitter clusters larger than those investigated.

cond-mat.mes-hall↗

arXiv2016

Fowler-Nordheim Plot Analysis: a Progress Report

The commonest method of characterizing a cold field electron emitter is to measure its current-voltage characteristics, and the commonest method of analysing these characteristics is by means of a Fowler-Nordheim (FN) plot. This tutorial/review-type paper outlines a more systematic method of setting out the Fowler-Nordheim-type theory of cold field electron emission, and brings together and summarises the current state of work by the authors on developing the theory and methodology of FN plot analysis. This has turned out to be far more complicated than originally expected. Emphasis is placed in this paper on: (a) the interpretation of FN-plot slopes, which is currently both easier and of more experimental interest than the analysis of FN-plot intercepts; and (b) preliminary explorations into developing methodology for interpreting current-voltage characteristics when there is series resistance in the conduction path from the high-voltage generator to the emitter's emitting regions. This work reinforces our view that FN-plot analysis is best carried out on the raw measured current-voltage data, without pre-conversion into another data format, particularly if series resistance is present in the measuring circuit. Relevant formulae are given for extracting field-enhancement-factor values from such an analysis.

cond-mat.mes-hall↗

arXiv2015

ISQ derivation ("derivation in SI units") of a formula for the electrostatic field ionization rate-constant for a hydrogenic atom in its ground electronic state

Technical applications using electrostatic field ionization (ESFI) need a formula for the rate-constant K_e for the free-space ESFI of a hydrogenic atom in its ground electronic state. This formula must indicate the dependence on ionization energy I. Existing formulae were derived using atomic units. However, many scientists work with ESFI as an important technical process, but are not familiar with the Gaussian or atomic units systems. In the 1970s, the present International System of Quantities (ISQ), which includes the equations behind SI units, was designated as the main system for university teaching and the communication of scientific equations. 40 years on, ISQ derivations of ESFI rate-constant formulae are still not easily found, but are now needed. This tutorial paper gives a detailed ISQ derivation of a formula for K_e. The derivation is closely modelled on the Landau and Lifshitz (LL) approach used in their 1958 work (in atomic units) on hydrogen atom ESFI. The ISQ derivation confirms that, for hydrogenic atoms, ionization energy appears in the pre-exponential as I^(5/2), and defines a universal "field ionization constant". It also shows how the ISQ formula relates to the Gurney and Condon "attempt frequency" form for tunnelling rate-constants, and an ISQ formula is given for the motive energy in the related JWKB integral. The ISQ derivation uses a motive-energy transformation analogous to a transformation used by LL. The need for this transformation in ESFI theory raises questions as to the correctness of theoretical treatments of field electron emission from non-planar emitters, which do not make this transformation.

quant-ph↗

arXiv2012

Improved approach to Fowler-Nordheim plot analysis

This article introduces an improved approach to Fowler-Nordheim (FN) plot analysis, based on a new type of intercept correction factor. This factor is more cleanly defined than the factor previously used. General enabling theory is given that applies to any type of FN plot of data that can be fitted using a FN-type equation. Practical use is limited to emission situations where slope correction factors can be reliably predicted. By making a series of well-defined assumptions and approximations, it is shown how the general formulas reduce to provide an improved theory of orthodox FN-plot data analysis. This applies to situations where the circuit current is fully controlled by the emitter characteristics, and tunneling can be treated as taking place through a Schottky-Nordheim (SN) barrier. For orthodox emission, good working formulas make numerical evaluation of the slope correction factor and the new intercept correction factor quick and straightforward. A numerical illustration, using simulated emission data, shows how to use this improved approach to derive values for parameters in the full FN-type equation for the SN barrier. Good self-consistency is demonstrated. The general enabling formulas also pave the way for research aimed at developing analogous data-analysis procedures for non-orthodox emission situations.

cond-mat.mes-hall↗

arXiv2012

Illustrating field emission theory by using Lauritsen plots of transmission probability and barrier strength

This technical note relates to the theory of cold field electron emission (CFE). It starts by suggesting that, to emphasize common properties in relation to CFE theory, the term 'Lauritsen plot' could be used to describe all graphical plots made with the reciprocal of barrier field (or the reciprocal of a quantity proportional to barrier field) on the horizontal axis. It then argues that Lauritsen plots related to barrier strength (G) and transmission probability (D) could play a useful role in discussion of CFE theory. Such plots would supplement conventional Fowler-Nordheim (FN) plots. All these plots would be regarded as particular types of Lauritsen plot. The Lauritsen plots of -G and lnD can be used to illustrate how basic aspects of FN tunnelling theory are influenced by the mathematical form of the tunnelling barrier. These, in turn, influence local emission current density and emission current. Illustrative applications used in this note relate to the well-known exact triangular and Schottky-Nordheim barriers, and to the Coulomb barrier (i.e., the electrostatic component of the electron potential energy barrier outside a model spherical emitter). For the Coulomb barrier, a good analytical series approximation has been found for the barrier-form correction factor; this can be used to predict the existence (and to some extent the properties) of related curvature in FN plots.

cond-mat.mes-hall↗

arXiv2012

Influence of barrier form on Fowler-Nordheim plot analysis

Recent research has described an improved method of Fowler-Nordheim plot analysis, based on the definition and evaluation of a slope correction factor and a new form of intercept correction factor. In this improved approach there exists a basic approximation that neglects certain terms in the general theory, and focuses on the influence of the form of the tunneling barrier on the values of basic slope (σB) and intercept (ρB) correction factors. Simple formulae exist that allow these to be evaluated numerically for a barrier of arbitrary well-behaved form. This paper makes an initial exploration of the effects of barrier form on FN plot analysis. For a planar emitter, two models for the correlation-and-exchange (C and E) potential energy (PE) are used. For the Schottky-Nordheim barrier, it is shown that numerical and analytical approaches generate equivalent results. This agreement supports the validity of the numerical methods used. Comparisons with results for the Cutler-Gibbons barrier show that small differences in the assumed C and E PE make little difference to values of σB and ρB. Schottky's planar image PE has then been used, in conjunction with the electrostatic PE variation associated with a spherical emitter model, to explore the influence of apex radius r_a on correction-factor values, for values of r_a greater than 20 nm. Both σB and ρB increase significantly as r_a decreases, especially ρB. At low values of barrier field F, σB depends approximately linearly on 1/F, with a slope that depends on r_a. Suggestions are made for how the exploratory work described in this paper might be extended.

cond-mat.mes-hall↗