FN-type Equation Research Articles

Application of fractional Fowler-Nordheim law on field electron emission from tungsten and lanthanum hexaboride nano-scale emitters

ABSTRACT The Murphy-Good Fowler-Nordheim equation was derived for planar and smooth conducting surfaces based on the Sommerfeld free-electron theory. It is applicable to a wide range of micro-emitters. The analyses of experimental field emission data from nano-scaled emitters showed deviation from the theoretical predictions. This has either been noticed in the mismatch of the current–voltage characteristics and/or in the FN-plot behaviour. Such deviation is attributed to the non-planar character of the emitting surface and the shape of the surface potential barrier through which the emitting electrons are tunnelling. Recently, Zubair et al. derived a FN-type equation that takes into account the reduced dimensionality of the emission surface and the non-parabolic energy dispersion [Zubair et al., IEEE Transactions on Electron Devices 65(6) 2018]. In this model, the constants of the conventional FN equation become two variables that depend on the fractional parameter expressing the surface complexity. In the present work, the fractional law is applied to field electron emission from clean tungsten (W) nano-emitter and lanthanum hexaboride (LaB6) nano-protrusions. This has been accomplished in the whole range of the fractional parameter in order to find the value that reproduces a better fit with the experimental data. The application requires determining the geometrical parameter of the emitter and the field emission setup. Results show that the experimental data can easily be reproduced at certain values of the fractional parameter. These ‘certain’ values depend on the geometrical parameters and thus the method used to calculate the electric field value at the emitter surface.

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 nonorthodox emission situations.

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 103 and 109. 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.

Thin-slab model for field electron emission

The elementary Fowler–Nordheim-type (FN-type) equation is often used to describe cold field electron emission (CFE) from carbon nanotubes (CNTs) including closed single-walled CNTs of small apex radius. FN-type equations were originally derived to describe CFE from the conduction band of a bulk metal with a flat surface and use a “flat thick-slab” potential-energy model. This article identifies many theoretical difficulties with using FN-type equations to describe CFE from CNTs. The most serious arise because a closed CNT apex is one-atom thick and can be sharply curved. This article explores a flat thin-slab model for CFE and derives an emission current equation. The model is parametrized using graphene data. The main conclusions are that (a) the occupied parts of electronic subbands probably do not overlap so CFE would come from a single subband; (b) an electron at the Fermi level has large momentum and kinetic energy parallel to the slab surface; (c) a clear conceptual distinction must be made between local thermodynamic work function (ϕ) and the zero-field height of the tunneling barrier for a Fermi-level electron; (d) for the thin-slab model, this zero-field barrier height is equal (at 0 K) to ϕ+ξf, where ξf is the occupied width of the subband at 0 K; and (e) the emission onset field, for a current density of 1000 A/m2, is estimated as around 9 V/nm. The relevance of these conclusions to emission from closed single-walled CNTs is considered. The predicted onset field for a thin-slab model is significantly higher than that predicted by the elementary FN-type equation (around 3.5 V/nm), so attempts should be made to measure onset field accurately for CFE from closed CNTs. In practice, with closed CNTs, it might be more favorable for emission to occur from quasiatomic or quasimolecular localized states rather than a subband; in this case, the question of how these states are supplied arises. More generally, this work draws attention to the need, with thin emitters, to consider issues relating to how the direction of electron motion prior to emission affects the mechanism of emission and the emission probability.The elementary Fowler–Nordheim-type (FN-type) equation is often used to describe cold field electron emission (CFE) from carbon nanotubes (CNTs) including closed single-walled CNTs of small apex radius. FN-type equations were originally derived to describe CFE from the conduction band of a bulk metal with a flat surface and use a “flat thick-slab” potential-energy model. This article identifies many theoretical difficulties with using FN-type equations to describe CFE from CNTs. The most serious arise because a closed CNT apex is one-atom thick and can be sharply curved. This article explores a flat thin-slab model for CFE and derives an emission current equation. The model is parametrized using graphene data. The main conclusions are that (a) the occupied parts of electronic subbands probably do not overlap so CFE would come from a single subband; (b) an electron at the Fermi level has large momentum and kinetic energy parallel to the slab surface; (c) a clear conceptual distinction must be made between ...

Theories of field and thermionic electron emissions from carbon nanotubes

Taking into account the effect of the low-energy band structure of carbon nanotubes (CN), we develop the theories of CN field and thermionic emissions. We give the analytic field and thermionic emission equations for both metal and semiconducting CNs. These theories modify the conventional Fowler–Nordheim (FN) and Murphy–Good (MG) theories. For large-diameter CNs and high fields, the field-emission equation reduces the FN-type field-emission equation. For small-diameter CNs and low fields, the field-emission equation goes beyond the FN-type behavior, which provides a possible way to understand the non-FN behavior observed in experimental results. Based on these theories, we give the electron-emission phase diagram on the field, thermionic, and intermediate emissions in the field-temperature space, whose boundaries have a slight shift to the corresponding boundaries of MG’s theory. These differences come from the energy-band structure difference between CN and conventional emitters. This theory provides an understanding of field and thermionic emissions for nanoscale materials.

Comparison of approximations for the principal Schottky–Nordheim barrier function v(f), and comments on Fowler–Nordheim plots

The Fowler–Nordheim-type (FN-type) equation developed by Murphy and Good [Phys. Rev. 102, 1464 (1956)] contains in its exponent a correction function v, best called the “principal Schottky–Nordheim (SN) barrier function.” In the last 50 years a large number of different approximations have been developed for v. This article reformulates these approximations to be functions of the scaled barrier field f rather than the Nordheim parameter y, and then compares them with the approximations v(f)≈1−f+(1/6)f ln f and v(f)≈1−f+0.1715f ln f recently reported. It is confirmed, as expected, that these new formulas outperform older approximations of equivalent complexity when the comparison is made over the range 0≤f≤1. However, particularly with the first formula, some older approximations are superior over limited ranges of f because they have been designed to perform well over these limited ranges. The article also presents an updated and fuller account of the theory of FN-plot analysis and an updated tabulation of values of the SN-barrier functions (also called field emission elliptic functions). Both updates use f (rather than y) as the main auxiliary variable. The limitations of the existing FN-plot theory are clarified; this reinforces the case for the development of higher quality measurements of current-voltage characteristics and of new methods of data analysis that may be able to supersede the use of FN plots.

On the need for a tunneling pre-factor in Fowler–Nordheim tunneling theory

This paper argues that a tunneling prefactor should appear in expressions for the tunneling probability D relevant to cold field electron emission (CFE) and in Fowler–Nordheim (FN) type equations. Except in the case of the “ideally smooth” parabolic barrier, a prefactor is always present for barriers where D can be found by exact solution of the Schrödinger equation. A review of the Jeffreys–Wentzel–Kramers–Brillouin (JWKB) approach to solving the Schrödinger equation shows that tunneling barriers should be classified according to whether they are weak or strong and ideally smooth or not: there are four different JWKB-type formulas, depending on the nature of the barrier. CFE tunneling barriers are not ideally smooth but since the 1950s have usually been analyzed using the JWKB formulas for ideally smooth barriers. These analyses, and the standard Murphy–Good FN-type equation, seem mathematically and physically incomplete. The FN-type equations currently used to describe CFE should be revised to explicitly include a tunneling prefactor. Some implications are explored.

Physics of generalized Fowler-Nordheim-type equations

Cold field electron emission (CFE) from metals is described by a large family of approximate equations called Fowler-Nordheim-type (FN-type) equations. This article discusses FN-type equations that give emission current density in terms of local work function and barrier field. Starting from the widely used standard FN-type equation [E. L. Murphy and R. H. Good, Phys. Rev. 102, 1464 (1956)], which is a hybrid physical/mathematical equation applicable to CFE from flat free-electron-metal surfaces at 0K, this article builds up a generalized “physical FN-type equation” in which physical correction factors are used to represent effects not included in standard CFE theory. The derivation starts by separating mathematical and physical aspects of standard theory, making use of a new independent physical variable (the scaled barrier field) and a physical “barrier-shape correction factor” that applies to any well-behaved barrier. Correction of long-standing error in applying Jeffries-Wentzel-Kramers-Brillouin-type approximations to CFE theory introduces a tunneling prefactor and a related small correction factor associated with integration over occupied electron states. Further correction factors represent effects due to non-free-electron band structures and emitter temperature. The outcome is an equation that provides a physical framework for FN-type equations that apply to CFE from a single metal-like conduction band. This equation is conceptually more complete than the equations normally used. Estimates are presented for the sizes of correction factors.