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 FL/FM, where FL 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 FM). 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 (∼5.46 nm) and radius (∼0.42 nm). It is found that apex values of local induced FEF are similar for the two SWCNTs, are independent of FM, and are similar to FEF values found from classical conductor models. It is suggested that these induced-FEF values are related 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/µm 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 FM = 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.

Comment on 'Metallic nanowire-graphene hybrid nanostructures for highly flexible field emission devices'.

Comments are made on theoretical aspects of a recent paper on cold field electron emission (CFE) from a large area field emitter (LAFE), in Arif et al (2011 Nanotechnology 22 355709). (1) Anomalies in the extraction of characterization parameters from the published graphical data are corrected. (2) Quantitative tests show that the data are not compatible with the hypothesis that the measured current is controlled by an orthodox CFE process. Hence, it may not be possible to interpret the extracted slope characterization parameter as a simple electrostatic field enhancement factor. (3) The Fowler-Nordheim-type equation used is defective, because it is missing an important pre-exponential correction factor, and would over-predict LAFE-average current density by a very large factor.

Extraction of emission parameters for large-area field emitters, using a technically complete Fowler-Nordheim-type equation.

In papers on cold field electron emission from large-area field emitters (LAFEs), it has become widespread practice to publish a misleading Fowler-Nordheim-type (FN-type) equation. This equation over-predicts the LAFE-average current density by a large highly variable factor thought to usually lie between 10(3) and 10(9). This equation, although often referenced to FN's 1928 paper, is a simplified equation used in undergraduate teaching, does not apply unmodified to LAFEs and does not appear in the 1928 paper. Technological LAFE papers often do not cite any theoretical work more recent than 1928, and often do not comment on the discrepancy between theory and experiment. This usage has occurred widely, in several high-profile American and UK applied-science journals (including Nanotechnology), and in various other places. It does not inhibit practical LAFE development, but can give a misleading impression of potential LAFE performance to non-experts. This paper shows how the misleading equation can be replaced by a conceptually complete FN-type equation that uses three high-level correction factors. One of these, or a combination of two of them, may be useful as an additional measure of LAFE quality; this paper describes a method for estimating factor values using experimental data and discusses when it can be used. Suggestions are made for improved engineering practice in reporting LAFE results. Some of these should help to prevent situations arising whereby an equation appearing in high-profile applied-science journals is used to support statements that an engineering regulatory body might deem to involve professional negligence.

Field emitter electrostatics: a review with special emphasis on modern high-precision finite-element modelling.

This review of the quantitative electrostatics of field emitters, covering analytical, numerical and 'fitted formula' approaches, is thought the first of its kind in the 100 years of the subject. The review relates chiefly to situations where emitters operate in an electronically ideal manner, and zero-current electrostatics is applicable. Terminology is carefully described and is 'polarity independent', so that the review applies to both field electron and field ion emitters. It also applies more generally to charged, pointed electron-conductors-which exhibit the 'electrostatic lightning-rod effect', but are poorly discussed in general electricity and magnetism literature. Modern electron-conductor electrostatics is an application of the chemical thermodynamics and statistical mechanics of electrons. In related theory, the primary role of classical electrostatic potentials (rather than fields) becomes apparent. Space and time limitations have meant that the review cannot be comprehensive in both detail and scope. Rather, it focuses chiefly on the electrostatics of two common basic emitter forms: the needle-shaped emitters used in traditional projection technologies; and the post-shaped emitters often used in modelling large-area multi-emitter electron sources. In the post-on-plane context, we consider in detail both the electrostatics of the single post and the interaction between two identical posts that occurs as a result of electrostatic depolarization (often called 'screening' or 'shielding'). Core to the review are discussions of the 'minimum domain dimensions' method for implementing effective finite-element-method electrostatic simulations, and of the variant of this that leads to very precise estimates of dimensionless field enhancement factors (error typically less than 0.001% in simple situations where analytical comparisons exist). Brief outline discussions, and some core references, are given for each of many 'related considerations' that are relevant to the electrostatic situations, methods and results described. Many areas of field emitter electrostatics are suggested where further research and/or separate mini-reviews would probably be useful.

Field emission: calculations supporting a new methodology of comparing theory with experiment.

This paper provides a demonstration-of-concept of a new methodology for comparing field electron emission (FE) theory and experiment. It uses the parameter κ in the mathematical equation I m = CV m κ exp[-B/V m] (where B and C are weakly varying or constants) that is taken to describe how measured current I m depends on measured voltage V m for electronically ideal FE systems (i.e. systems that (i) have constant configuration during voltage application and (ii) have I m(V m) given by the emission physics alone). Experimental parameter values (κ m) are used to compare two alternative FE theories, for which allowable (but different) κ ranges have been established. At present, contributions to the 'total theoretical κ' made by voltage dependence of notional emission area are not well known: simulations reported here provide data about four commonly investigated emitter shapes. The methodology is then applied to compare 1928/1929 Fowler-Nordheim (FN) FE theory and 1956 Murphy-Good (MG) FE theory. It is theoretically certain that the 1956 theory is 'better physics' than the 1928/1929 theory. As in previous attempts to reach known correct theoretical conclusions by experimentally based argument, the new methodology tends to favour MG FE theory, but is formally indecisive at this stage. Further progress needs better methods of establishing error limits and of measuring κ m.

Comment on "Design and circuit simulation of nanoscale vacuum channel transistors" by J. Xu, Y. Qin, Y. Shi, Y. Yang and X. Zhang, Nanoscale Adv., 2020, 2, 3582.

These comments aim to correct some apparent weaknesses in the theory of field electron emission given in a recent paper about nanoscale vacuum channel transistors, and to improve the presentation of this theory. In particular, it is argued that a "simplified" formula stated in the paper should not be used, because this formula is known to under-predict emission current densities by a large factor (typically around 300 for an emitting surface with local work function 4.5 eV). Thus, the "simplified" formula may significantly under-predict the practical performance of a nanoscale vacuum channel transistor.

Comment on "Advanced field emission measurement techniques for research on modern cold cathode materials and their applications for transmission-type x-ray sources" [Rev. Sci. Instrum. 91, 083906 (2020)].

This Comment suggests that technological field electron emission (FE) papers, such as the paper under discussion [P. Serbun et al., Rev. Sci. Instrum. 91, 083906 (2020)], should use FE theory based on the 1956 work of Murphy and Good (MG), rather than a simplified version of FE theory based on the original 1928 work of Fowler and Nordheim (FN). The use of the 1928 theory is common practice in the technological FE literature, but the MG treatment is known to be better physics than the FN treatment, which contains identifiable errors. The MG treatment predicts significantly higher emission current densities and currents for emitters than does the FN treatment. From the viewpoint of the research and development of electron sources, it is counterproductive (and unhelpful for non-experts) for the technological FE literature to use theory that undervalues the performance of field electron emitters.

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

Measured field electron emission (FE) current-voltage I m(V m) data are traditionally analysed via Fowler-Nordheim (FN) plots, as ln { I m / V m 2 } versus 1/V m. 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 plot has the form ln ⁡ { I m / V m ( 2 - η / 6 ) } versus 1/V m, where η ≅ 9.836239 (eV/ϕ)1/2 and ϕ is the local work function. Modern (twenty-first 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 I m(V m) data (among them, use of 'smooth planar emitter methodology'). Careful use of MG plots could also help remedy other problems in FE technological literature. It is suggested that MG plots should replace FN plots.

Appendix to "Coulomb interactions in Ga LMIS" by Radlicka and Lencová.

This appendix comments on the input data and simulation results of Radlicka and Lencová (RL), reported in the preceding paper. These simulations relate to the effects of space-charge on the full-width at half-maximum (FWHM) of the ion energy distribution from a gallium liquid metal ion source (LMIS), and on its virtual source size. Corrections are necessary to the input data, relating to LMIS shape, that were supplied by the present author. It is clear from RL's work that their simulations, when applied to the corrected shape data, would generate very good agreement between experimental results and numerical simulations concerning the FWHM. This work confirms that an inappropriate hydrodynamic condition has been used in past modelling programmes for LMIS shapes and current/voltage characteristics; consequently, some quantitative results from these old programmes must be considered unreliable in detail, especially at low emission currents, although qualitative trends have been correctly exhibited.

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

Recent analyses of the apex field enhancement factor (FEF) for many forms of field emitter 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 [Formula: see text] (c being the separation and h is the height of the emitters) and large separations ([Formula: see text]) has not been predicted by the existing formulas in field emission literature, for post-like emitters of any shape. In this work, we use first principles to derive a simple two-parameter formula for fractional reduction that can be useful for experimentalists for modeling and interpreting the FEFs for small clusters of emitters or arrays at separations of interest. For the structures tested, the agreement between numerical and analytical data is ∼1%.

Field electron and ion emission from charged surfaces: a strategic historical review of theoretical concepts.

The field-electron (FE) and field-ion techniques directly observe and measure atomic-level surface processes that occur in very high electric fields. In theoretical terms, the high fields put large additional terms into Hamiltonians and free energies, and significantly modify many aspects of the surface physics and chemistry, as compared with the field-free situation. This paper presents a strategic review of the fundamental science of some of these high-field surface effects and processes, as developed in the context of the field electron and ion emission techniques. It outlines the main theoretical concepts developed, notes some twists of scientific history, and suggests useful contributions made to mainstream science. Topics covered are basic aspects of FE emission, surface field ionisation, localised field adsorption, charged surfaces theory, field-ion image contrast theory and associated imaging-gas kinetics, field evaporation, and aspects of the thermodynamics of charged surfaces. Despite many years of effort, important aspects of the theory remain incomplete. Some theoretical challenges are noted.

Some comments on models for field enhancement.

To estimate the apex field-enhancement factor gamma(a)associated with a pointed protrusion on a flat planar surface, the simple physical models of a 'floating sphere at emitter-plane potential' and a 'hemisphere on a post' are often discussed. The corresponding mathematical expressions have the form: gamma(a)=m+h/rho, where rho is the sphere or hemisphere radius, h is its 'height above the emitter plane', and m is a constant variously taken as 0, 2 or 3. Recent numerical simulations for the 'hemisphere on a post' model, reported elsewhere by two of us (CJE and GV) and by Kokkaris, Modinos and Xanthakis, have shown that all of these simple formulae significantly overpredict gamma(a) if h/rho is large. This article first reexamines the basis of these simple formulae and confirms that they are less secure than is sometimes thought. The formulae reported elsewhere as fits to the numerical results are then quoted and compared with the simple formulae, and with the known exact analytical result for the 'hemi-ellipsoid on a plane' model. Discrepancies can be rationalised. Some general conclusions are drawn.