Field emission: Applying the “magic emitter” validity test to a recent paper, and related research-literature integrity issues

This paper forms part of a wider project to put the topic of field electron emission (FE) onto an improved scientific basis. The wider project concentrates on basic aspects of FE theory and FE data analysis and is envisaged to have five main strands: (1) encouraging use of modern (“21st century”) planar emission physics and of modern treatments of the FE special mathematical function “v” (e.g., Ref. 1); (2) encouraging the view² that the interpretation of FE current–voltage [I(V)] characteristics is, in principle, a specialized form of electrical engineering and setting improved methods of I(V) data analysis (e.g., Refs. 3 and 4) in this context; (3) attempting to remove many entrenched mistakes and misunderstandings, particularly about data analysis, from the FE technological literature; (4) attempting to determine reliable methods of making comparisons between theory and experiment^5,6 (the recent work^7,8 of Ayari and colleagues links into this); and (5) exploring how to “get atoms into FE theory” (as recently discussed, for example, by Lepetit),⁹ and exploring how to formulate the related quantum mechanics in an exactly correct manner.

Other aspects of FE theory are also important; in particular, emission theory for sharply curved and atomically sharp emitters, the effects of temperature and/or photon impact on emission, emission theory for nonmetals such as semiconductors or carbon-based materials, and emission theory for two-dimensional materials. But the author's view is that the first priority is to get the basic science of FE into better order.

This paper addresses the third strand above, namely, the removal of entrenched mistakes and misunderstandings from the FE technological literature.

Large-area electron emitters based on FE have many actual and potential technological applications. In related materials-development activities, it is important that the current–voltage characteristics of such emitters be analyzed correctly and that reliable values of emitter characterization parameters—in particular, dimensionless field enhancement factors (FEFs)—be reported. However, many papers reporting on materials developments of this kind make mistakes in the current–voltage data-analysis procedure, and consequently report spurious FEF values. Mistakes are also commonly made in emission theory stated in such papers. Clearly, journal editorial and reviewing procedures have been unable to detect these weaknesses.

The present paper aims to draw attention to and correct weaknesses of this kind. This will be done by using a recently published paper¹⁰ to exemplify these difficulties. It is emphasized that the materials research in Ref. 10 is of high quality: the problems are with the emission theory, data analysis, and the resulting reported FEF value. Correction of these problems adds value to the paper, and, probably, provides the research funders with better value for money.

Reference 10 has been chosen because it is a recent paper that provides a particularly clear example of the weaknesses discussed below. However, weaknesses of this kind appear to have been widespread in the FE materials-development literature for the last 15 years or more. As a demonstrator of their occurrence, I have submitted to the editor of JVSTB a list of 20 papers, published in the last two years or so, in which weaknesses of the kind to be discussed have been detected. Further details of how these have been selected are given in the  Appendix, but there has seemed no useful merit in publishing this list of papers.

It is shown below that, in many cases, the “spurious FEF value” problem can be rapidly and easily uncovered, in the immediate-presubmission or review stages of a paper, by applying, to reported experimental and characterization data, a newly developed validity test² provisionally called the “magic emitter test.” The present discussion thus provides an illustration of the usage of this test.

A much fuller discussion of one of the main issues involved (whether an FE system is “electronically ideal”) can be found in Ref. 2, and a discussion of some less-critical issues appears in an earlier paper.¹¹ 

Most FE materials-development papers (including the paper under discussion and the others recently published) analyze FE current–voltage [I(V)] experimental data using: (a) a methodology—the Fowler–Nordheim (FN) plot—originally developed¹² in 1929 and (b) a modern simplified version of the FE equation that was originally developed by Fowler and Nordheim¹³ in 1928 and clarified by Stern et al.¹² in 1929. This simplified version omits a prefactor of order unity that appears in the original 1928/29 equation and is usually an acceptable mathematical approximation of the 1928/29 FE theory.

Underlying this widely used methodology is the unstated assumption that analysis of the data is a problem involving only the emission physics and the zero-current system electrostatics. An emitter for which this assumption is adequately valid has been termed electronically ideal (see Ref. 2). The metal emitters under consideration in 1929 could usually be treated as electronically ideal, but this is not necessarily true for modern emitters fabricated from nonmetallic materials. In such cases, it may often be necessary to take into account wider electrical-engineering aspects of the complete emission and measurement system (see Ref. 2). Twelve different possible causes (“system complications”) that can make FE systems not electronically ideal have been identified in Ref. 2. In any particular nonideal case, more than one of these might be operating. If an FE system is not electronically ideal, then it is not scientifically valid to apply the 1929 methodology: results derived by using it inappropriately—in particular, the values of dimensionless field enhancement factors (FEFs)—may be spurious.

As already indicated, large-area field electron sources have some interesting actual and potential technological applications. Field enhancement factors are one of the parameters used to characterize the newly developed emitting materials for such sources, and it is important for research-and-development purposes to have accurate FEF values reported in the literature. But it is the author's belief that many FEF values published in the FE technological literature are, in fact, spurious. A small survey,¹⁴ conducted about ten years ago, provisionally suggested that up to about 40% of published FEF values might be spurious. A more recent literature examination is reported above.

In order to deal with this situation, the present author has introduced the idea of a validity check (also called a validity test). This is an engineering type test that analyses the measured current–voltage data in order to establish whether the measured data show that the system is not (or may not be) electronically ideal. One of the original validity checks was the so-called orthodoxy test.¹⁴ Recently,² I have proposed a new validity check, the “magic emitter test.” This can be seen as a simplified version of the orthodoxy test that can be rapidly applied to the published data when information is provided both about the range of macroscopic-field magnitudes (F_M) apparently used in the experiments and about the derived value of the apparent characteristic FEF. [This is denoted here by (γ_MC)^app, but often by β in the literature.]

For example, in Ref. 10, for their sample S3, the lowest reported value (see Fig. 6) of 1/F_M is about 0.17 μm/V, which means that F_M(max) is about 5.9 V/μm. The reported value of (γ_MC)^app is 14 594, which yields the value F_C^app(max)= 8.6 × 10⁴ V/μm = 86 V/nm.

It is well established that if the field applied to a field electron emitter is gradually increased in magnitude, then eventually, the emitter will self-destruct, though precisely how this happens may depend on particular circumstances. A parameter of interest is the “reference local-field magnitude” F_R(ϕ) at which—for an emitter of a characteristic local work function ϕ—the top of a Schottky–Nordheim tunneling barrier is pulled down to the emitter Fermi level. Sample S3 is stated to have a true work function of 3.25 eV; the corresponding reference-field magnitude is about¹⁵ 7.3 V/nm. The usual thinking is that when the barrier top is at the Fermi level, then the electrons certainly pour out of the emitter in such numbers that the resulting current density creates heating effects that rapidly destroy the emitter. However, older investigations¹⁶ of FE in a traditional field electron microscope configuration suggest that (in that context) the highest safe continuous-current situation corresponds (for a 4.50 eV emitter) to a field magnitude significantly lower than F_R(ϕ), namely, 0.34 F_R(ϕ) (see Ref. 15). For a ϕ = 3.25 eV emitter, this would indicate a “highest safe” local surface-field magnitude of around 2 V/nm.

The original logic behind the “magic emitter” test was that if the value of F_C^app(max) found from Eq. (1) is significantly greater than the range of surface-field-magnitude values where the emitter is expected to self-destruct, as it is for sample S3, then authors apparently have a “magic emitter” immune to normal self-destruction processes. In reality, of course, authors in this position have failed to carry out a validity check, have applied 1929 data-analysis methodology in circumstances where it is not scientifically valid to do so, and are reporting spurious (unreasonably high) FEF values.

My revised thinking is that (for consistency), it is better to link the decision criterion in the magic emitter test directly to the related criterion used in the orthodoxy text. This states that the extracted FEF value is almost certainly spurious if the derived value of F_C^app(max) is greater than f_ub(ϕ)⋅F_R(ϕ), where f_ub(ϕ) is a work function-dependent parameter originally tabulated in Table II of Ref. 14 and retabulated in Table I here. For ϕ = 4.50 eV, we have f_ub = 0.75 (which is the value usually quoted), but for ϕ = 3.25 eV, the slightly higher value of f_ub = 0.88 should be used. Sample S3 has a derived value of F_C^app(max) equal to 86 V/nm, which is equivalent to about 12 F_R(ϕ), so—as already concluded—the extracted FEF value is clearly “almost certainly spurious.”

When the derived characteristic local-field magnitude is close to f_ub(ϕ)⋅F_R(ϕ) (either above or below this value), then more careful investigation is needed.

Part of this discrepancy is presumably a side-effect of the weakness discussed above, but certainly, there is also another partial cause. Equation (2) above is, in fact, an equation for local emission current density (LECD). Because field enhancement is included, this equation is being applied in the high field region near the apex of a spike-like emitter. Thus, what Eq. (2) actually predicts (using 1920s-style theory) is a characteristic LECD J_C at or near the emitter apex. By contrast, the symbol “J” on the vertical axis of Fig. 6(a) in Ref. 10 represents the macroscopic (or “footprint average”) current density J_M. This J_M is expected to be much lower than J_C. For electronically ideal emitters, the ratio J_M/J_C could be as low as 10⁻⁷ or lower: however, actual values of this ratio are likely to be very variable as between different materials and different fabrication methodologies and are not well known.

This failure to properly distinguish between local and macroscopic current densities appears to be widespread in the FE technological literature, and the author has detected many other papers that include large undiscussed apparent discrepancies between theory and experiment.

A further difficulty with Eq. (2) above, and with the related equation in the paper under discussion, is that these equations represent versions of the FE theory that was developed in the late 1920s. In the 1950s, serious errors were confirmed¹⁷ in all earlier FE theories, in that exchange-and-correlation effects (normally modeled as image-force effects) had not been taken into account in a physically and mathematically correct fashion. This resulted in a 1956 reformulation of the FE theory by Murphy and Good¹⁸ (MG). The consequences in principle of this reformulation have been discussed elsewhere.¹¹ 

In the case under discussion, the main effect of this reformulated theory is to put into the exponent of Eq. (2), a correction factor that (for a ϕ = 3.25 eV emitter) has values 0.8 down to 0.4, over the orthodoxy-test pass range of scaled field (f) between 0.175 and 0.525. This results in predictions of local current density that are larger, by a factor from around 1150 down to around 350, than those provided by simplified 1920s-style equations.

Here, the first and second Fowler–Nordheim constants are represented by the lower-case letters a and b, rather than by A and B; v_F and t_F are appropriate particular values (appropriate to a Schottky–Nordheim (SN) tunneling barrier defined by ϕ and F_C) of well-known special mathematical functions “v” and “t” (see any of Refs. 1 and 18–20); and α is a parameter, best called an area efficiency, that is a measure of how much of the footprint area of a large-area field electron source is actually emitting electrons. If required, the surface-field magnitude F_C can be expressed in terms of a macroscopic-field magnitude by using an appropriate field enhancement factor.

For clarity, it is strongly recommended that Eq. (4) be called the Murphy–Good FE equation or (perhaps better—see Ref. 3) the Extended Murphy–Good FE equation, in order to distinguish it from the 1928/29 Fowler–Nordheim FE equation. There is some tendency in the literature for theoreticians and some experimentalists to apply the name “Fowler–Nordheim equation” to the 1956 Murphy–Good FE equation. The present author finds this nomenclature convention very unhelpful and suspects that it may be a partial cause of the literature confusion between the 1928/29 and 1956 FE equations (which, as just indicated, are significantly different in their numerical predictions).

It needs to be made clear that these comments should be seen, not as a specific criticism of the authors of the paper under discussion or of the reviewing system operated by Applied Surface Science, but rather as comments on the scientific state of FE technological literature as a whole, as exemplified by the paper under discussion. It is the present author's perception that similar weaknesses can be found in many papers (perhaps hundreds of papers) and the reviewing systems of many journals.

Sociological discussions of scientific-community behavior sometimes use the term “pathological” to describe situations where individual scientists or communities of scientists believe things and/or carry out procedures that scientists outside the community do not accept as valid. (The term “pathological” derives from the paper by Langmuir and Hall,²² but its meaning has become extended over time.)

A characteristic associated with pathological scientific communities is that members, normally, mainly cite work performed by other members of the community and do not cite relevant work by scientists outside the community. The paper¹⁰ under discussion, in fact, illustrates this: no citation is given to any modern paper on FE theory or to any of the 15 or so FE research handbooks/textbooks that have been published since 1960, all of which discuss Murphy–Good FE theory (for a list see Ref. 1).

This pathological literature situation has been ongoing in FE technological contexts for around 15 years. The research integrity of this research area is now significantly damaged, and the peer review system in this research area is also significantly damaged. Attempts by individual scientists over the last ten years to improve the situation by using the normal correction processes of science have had only limited effect.

The phenomenon of field electron emission is the basis for many established and potential technologies (some very credible, although some very speculative). It is not helpful, and arguably, it is not in the public interest, that apparently relevant published scientific literature contain large amounts of defective and misleading information. This situation is particularly unhelpful for nonexperts, especially those interested in medical and defense applications, as well as new FE graduate students. The present author believes that, in individual countries such as the UK and USA, there is a growing case for some form of parliamentary, congressional, or other official investigation, with a focus on how to safeguard the integrity of scientific literature published in that country. This should probably take place in the context of a wider investigation into research integrity and the difficulties of ensuring it. The situation in FE could, perhaps, be a useful example of possible wider problems of this general kind, because the issues in the FE case are very clearly defined.

Some final points need to be made. First, it is not being claimed here that Murphy–Good FE theory is “correct physics:” clearly it is not, because it is a theory of FE from metals with smooth planar surfaces, and also disregards the effects of atomic structure. What is being claimed here is that MG FE theory is “obviously better physics” than 1920s-style FE physics and is “probably good enough” for data analysis in many or most modern technological contexts.

When emitters are sharply curved, then this may not be true, and more careful discussion is in principle needed. If an FE system containing sharply curved emitters is “seriously nonideal” then the validity checks discussed above should detect this correctly. However, with electronically ideal FE systems containing sharply curved emitters, the possibility exists that the present set of validity checks may return a “fail” result. This would be because Murphy–Good FE theory and the related “planar emission approximation” (on both of which the theory of the present checks is based) are “not adequately applicable” to sharply curved emitters. When the primary objective of applying a validity check is to detect or prevent the publication of spurious FEF values, then a result of this kind would have the status of a “false negative.” As far as the primary objective is concerned, limited numbers of false negatives (although undesirable) are not of great significance for the primary objective.

Kyritsakis and colleagues have developed an emission theory²³ for earthed spherical emitters of arbitrary radius somewhat greater than atomic radii, have shown how to employ this theory to analyze current–voltage data, using numerical multilinear regression techniques, and have developed a relevant open-access computer package.²⁴ What is currently lacking (and is now needed) is a validity check customized to apply to sharply curved emitters. Developing such a check looks not entirely straightforward, because the emitter apex radius cannot be assumed known in advance. Thus, there is a need for research and simulation related to developing a validity check of this kind.

Obviously, there is a need for a criterion of what is meant by “sharply curved.” No consensus on this matter yet exists, and further discussion (and simulation) is needed. The present author currently takes the view that a provisional working criterion is “an emitter is sharply curved if its apex radius of curvature is less than roughly 50 nm,” but this figure could be subjected to change in light of detailed calculations and discussion.

Last, I emphasize again that this is a community problem affecting (primarily) the FE technology community interested in materials’ development for large-area field electron sources. It seems that hundreds of authors and reviewers, and very many journal editors, have been “hoaxed” by the related FE literature over the last 15 years or so. It would be inappropriate (and unfair) to attribute significant fault to the particular authors and journals used here to exemplify a very much wider problem. They, like many others, have been “hoaxed.”

In summary, the main points being made in this paper are the following:

• - There are several significant weaknesses in data analysis in the paper¹⁰ under discussion and (obviously) in the related peer review process. Where practicable, the present paper corrects these weaknesses.

• - The author's perception is that similar weaknesses (and others) appear in many papers (perhaps in hundreds of papers in the last 15 years or so) in the FE technological literature relating to materials’ development for large-area field electron sources. Research integrity and the peer review process in this specialist area both seem significantly damaged.

• - The “magic emitter” validity test appears to be effective in detecting situations where an FE system is not electronically ideal (and hence, reported field enhancement factor values may be unreliable).

• - There seems a growing case for some sort of “official” external investigation into the situation described, as part of a wider investigation into research integrity in scientific literature and the difficulties of ensuring it.

As a contribution toward reducing the difficulties described above, I have placed on ResearchGate a Technical Report.²⁵ This provides background theory and guidelines for researchers intending to use Murphy–Good FE theory to validity check and (where appropriate) analyze experimental FE current–voltage data.

The authors have no conflicts to disclose.

Richard G. Forbes: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Project administration (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available within the article.

As noted earlier, I have submitted to the editor of JVSTB a list of 20 papers, published in the last two years or so, that fail the “magic emitter” validity test. This has been done in order to demonstrate the existence of a research integrity problem in FE literature relating to large-area field electron emitters. It is not an attempt to assess the full scale of the problem, which (as discussed below) must be much larger.

These papers are drawn from the set of FE papers reported to me, near daily, by ResearchGate and GoogleScholar: I always now check these papers for theoretical and data-analysis problems. In the time-period (approximately two years) corresponding to the 20 validity test failures, I have identified 37 papers with problems over the definition of current density and 43 papers that use obsolete 1920s-style equations.

These papers usually reach me because I have been cited in the paper. Since there must be many FE papers to which this does not apply, and since the situation has been ongoing for many years (probably around 15 years), it seems reasonable to conclude that, probably, hundreds of published papers suffer from one or more of the weaknesses under discussion in the present paper.