Richard G. Forbes

Atom probe tomography

Atom probe tomography is a powerful tool for the characterization of the size, morphology and composition of ultrafine features in a variety of materials. With the development of new forms of specimen preparation especially with focused ion beam milling systems, atom probe tomography should be extended to a wider variety of applications in nanotechnology.

Reformulation of the standard theory of Fowler-Nordheim tunnelling and cold field electron emission

This paper presents a major reformulation of the standard theory of Fowler–Nordheim (FN) tunnelling and cold field electron emission (CFE). Mathematical analysis and physical interpretation become easier if the principal field emission elliptic function v is expressed as a function v(l′) of the mathematical variable l′≡y2, where y is the Nordheim parameter. For the Schottky–Nordheim (SN) barrier used in standard CFE theory, l′ is equal to the ‘scaled barrier field’ f, which is the ratio of the electric field that defines a tunnelling barrier to the critical field needed to reduce barrier height to zero. The tunnelling exponent correction factor ν=v(f). This paper separates mathematical and physical descriptions of standard CFE theory, reformulates derivations to be in terms of l′ and f, rather than y, and gives a fuller account of SN barrier mathematics. v(l′) is found to satisfy the ordinary differential equation l′(1−l′)d2v/dl′2=(3/16)v; an exact series solution, defined by recurrence formulae, is reported. Numerical approximation formulae, with absolute error |ε|<8×10−10, are given for v and dv/dl′. The previously reported formula v≈1−l′+(1/6)l′ ln l′ is a good low-order approximation, with |ε|<0.0025. With l′=f, this has been used to create good approximate formulae for the other special CFE elliptic functions, and to investigate a more universal, ‘scaled’, form of FN plot. This yields additional insights and a clearer answer to the question: ‘what does linearity of an experimental FN plot mean?’ FN plot curvature is predicted by a new function w. The new formulation is designed so that it can easily be generalized; thus, our treatment of the SN barrier is a paradigm for other barrier shapes. We urge widespread consideration of this approach.

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 FNs 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.

Simple good approximations for the special elliptic functions in standard Fowler-Nordheim tunneling theory for a Schottky-Nordheim barrier

The discovery is reported of simple, good approximate formulas for special elliptic functions that appear in the standard theory of Fowler-Nordheim (FN) [Proc. R. Soc. London, Ser. A 119, 173 (1914)] tunneling through an image-rounded Schottky-Nordheim [W. Schottky, Z. Phys. 15, 872 (1923); L. W. Nordheim, Proc. R. Soc. London, Ser. A 121, 626 (1928)] barrier and in the standard FN equation. The FN-exponent correction factor v can be written as v(y)≈1−y2+(1∕3)y2lny, where y is the Nordheim parameter. This formula has a respectable mathematical basis, predicts exact values of v(y) to within 0.33% in 0⩽y⩽1, and can be rewritten to give (after nearly 80years) a simple, reliable algebraic formula for the explicit dependence of v on barrier field. Significant consequences are expected.

Refining the application of Fowler-Nordheim theory

Abstract This paper contributes to Fowler–Nordheim (FN) theory, i.e. the theory of cold field emission of electrons from metals, by summarising recent developments in standard FN theory and its application to FN plots. The paper is part of a theoretical `tidy up’ and consolidation at the level of the Murphy and Good 1956 treatment, and is a step towards further developments. It re-states the FN equation, in elementary and generalised forms, using modern conventions and the 1986 values of the fundamental physical constants. It presents a formal theory for interpreting FN plots, using a slope correction factor and an intercept correction factor, and relates these to parameters in the generalised FN equation. It explains the concept of an exact emission-area extraction function, and shows how values of this parameter may be used to extract emission-area estimates from FN plots, in the presence of uncertainty about values of field, work-function and current density. It compares the new procedures with existing procedures based on numerical approximations. It briefly explores possible effects of the recently reported field dependence in local work-function. It presents formulae defining the onset of field-induced ballistic emission, and discusses implications. The standard physical assumptions behind the theory are stated, and some ways in which they can break down are noted.

A generalised theory of standard field ion appearance energies

Abstract By means of arguments based on a thermodynamic cycle, general formulae are derived which express standard field ion appearance energies in terms of atomic parameters and molecular term values, and in terms of thermodynamic parameters. These formulae may be applied to field ionization, field evaporation, or to the field desorption of well-behaved molecules, and apply to a very wide range of desorption mechanisms. Previous theories of energy deficits are reviewed in the light of the general result. The extent of agreement between theory and experiment is assessed, and the need for continued theoretical development is noted. Formulae are provided for converting the variety of energy-deficit parameters encountered in the literature into appearance energies; and the various terminologies are correlated.

Field evaporation theory: a review of basic ideas

Abstract This paper reviews the basic principles of field-evaporation theory, as it applies to pure metals. The approach is primarily conceptual, and part of the aim is to indicate where and how earlier uncertainties have been resolved, and to note some unsolved problems. A particular topic is the physical origin of Mullers evaporation-field formula. A derivation based solely on the energetics of field evaporation is presented in more detail than previously. It is then argued that Mullers formula does not correctly represent the energetics of his image-hump escape mechanism; rather, the success of Mullers formula lies in the fact that it represents quite well the energetics of Gomers charge-exchange escape mechanism. A simplified derivation of the authors activation-energy formula is also given.

Physics of generalized Fowler-Nordheim-type equations

The aim here is to generalise the standard FN-type equation, so that the physical equation where the local emission current density (ECD) J is related to the barrier field F becomes clearer for metals. This is a preliminary need before the above equation can be fully linked with cold field electron emission (CFE) theory for non-metals. Each generalisation step needs stating in detail; this paper deals with the first few.

Use of a spreadsheet for Fowler–Nordheim equation calculations

Field electron emission theory as developed in the 1950s and 1960s was incomplete, in the sense that it contained no formally exact treatment of how to analyse the intercept of a Fowler-Nordheim plot. Such a treatment is readily developed by introducing additional functions broadly similar to the earlier correction factors. Due to hesitations about the correctness of Nordheims assumption, the formal treatment is best presented in terms of generalised correction factors. But, within the framework of calculations based on Nordheims assumption, the treatment involves the definition of three new functions: a further Nordheim-type function, the intercept correction factor, and the exact emission-area extraction function.

New results in the theory of Fowler–Nordheim plots and the modelling of hemi-ellipsoidal emitters

This paper reports further progress in understanding the theory of emission-area extraction from Fowler-Nordheim plots, and reports some useful interim results derived by modelling field electron emission from hemi-ellipsoidal emitters. The mathematical nature of the relationship between a new approach to emission-area extraction, recently proposed, and older approaches is demonstrated. The new approach is extended to cover field dependence in emission area. Preliminary results are reported from an investigation into the effects of making erroneous assumptions about the tunnelling barrier seen by the electron and the absence of field dependence in emission area. If wrong theoretical assumptions are made, then emission area can be overpredicted by a factor of as much as 10 or 20. On the other hand, if correct theoretical assumptions are made, then the extracted emission area is close to an emission area derived directly from the model calculations. The problematical nature of the concept of emission area, when emission area is a function of field, is pointed out.