## Abstract

The orthodox emission hypothesis is a set of physical and mathematical assumptions that permit well-specified analysis of measured current–voltage data relating to field electron emission (FE). If these assumptions are not adequately satisfied, then widely used FE data-analysis methods can generate spurious values for emitter characterization parameters, particularly field enhancement factors (FEFs). This paper describes development of a simple quantitative test for whether FE data are incompatible with the hypothesis. It applies to any geometrical emitter shape and any Fowler–Nordheim (FN) plot type, and involves extracting the related range of scaled-barrier-field values (*f* values). By analysing historical data, this paper identifies ‘apparently reasonable’ and ‘clearly unreasonable’ experimental ranges for *f*. The historical data, taken between 1926 and 1972, are internally self-consistent. This test is then applied to 19 post-1975 datasets, mainly for various carbon and semiconductor nanostructures. Some extracted *f*-value ranges (including many carbon results) are apparently reasonable, some are clearly unreasonable. It is shown that if extracted *f* values are ‘unreasonably high’, then FEF values extracted by literature methods are spuriously large. New materials and published FN plots that generate particularly high FEF values require testing, and improved data-analysis theory is needed for non-orthodox emitters. A spreadsheet for implementing the test is provided.

## 1. Introduction

This paper discusses a simple, well-specified quantitative test that can indicate whether current–voltage characteristics measured in field electron emission (FE) experiments are incompatible with a set of physical and mathematical assumptions widely used in the period 1960–1990, and refined here to become the ‘orthodox emission hypothesis’ described below. If incompatibility is demonstrated in any specific case, then existing, widely used, theoretical data-analysis methods—based on Fowler–Nordheim (FN) plots [1]—are invalid for that emitter, and may generate spurious values for emitter characterization parameters.

A simple quantitative test of this type has long been needed. For many years, the only simple test for consistency of FE data with theory has been the qualitative one of whether an experimental FN plot is effectively straight. For traditional single-tip field emitters (STFEs), Dyke & Trolan [2] developed a better test, involving emitter-profile measurement by electron microscopy. However, this is not practicable for everyday use on STFEs, and is not practicable at all for modern large-area field emitters (LAFEs).

The past 15 years have seen wide interest in developing LAFEs for electronic device applications [3–6]. The author's perception—shared with colleagues—has been that some reported values for the important LAFE characterization parameter ‘field enhancement factor’ (FEF) are implausibly high, if regarded as values for true electrostatic FEFs. One aim of this work has been a test that can determine whether the FEF value extracted from a given experimental dataset is unreliable.

The author has used earlier versions of the test [7,8], and the existence of the improved version discussed here was reported at a recent conference [9]. This paper sets out the background thinking in a more careful and definitive manner, and then discusses new results found by applying the improved test to various experimental FN plots, selected from the literature to cover a range of materials and emitter geometrical structures.

This paper's structure is as follows. Section 2 presents background theory, §3 discusses the formulation of test criteria, §4 applies the test to 19 post-1975 datasets, §5 discusses the mathematical link between test failure and the extraction of spurious FEF values and finally §6 provides a summary and general discussion. Details concerning test application to experimental datasets, and a copy of the spreadsheet used to apply the test, are provided as electronic supplementary material.

In accordance with the mainstream field emission convention, this paper treats fields, current densities and related quantities as positive, even though they would be negative in classical electromagnetism, and uses the symbol *F* for the negative of electrostatic field as classically defined.

## 2. Background theory

### (a) Current-density and current equations

*FN tunnelling* is electron tunnelling though an exact or approximately triangular barrier. *Deep tunnelling* is tunnelling well below the top of the barrier, in a regime where the expression of Landau & Lifschitz [10] for transmission probability is a valid approximation [11]. *Cold field electron emission* (CFE) is the statistical emission regime in which (i) the electrons in the emitting region are effectively in local thermodynamic equilibrium and (ii) emission occurs mainly by deep FN tunnelling from states close in energy to the local emitter Fermi level.

*FN-type equations* are a large family of approximate equations that have been derived to describe CFE from a metal conduction band, provided that the radius of the emitting region is ‘not too small’ (greater than around 10 nm, say). Simple forms of FN-type equation are also widely used as empirical fitting equations (i.e. data-analysis equations) for CFE from small-apex-radius emitters and from many non-metallic materials.

An FN-type equation is described as *technically complete* if it contains formal correction factors that are defined in such a way that all physical effects relevant to the dependent variable in use are encompassed within the equation. For CFE through a Schottky–Nordheim (SN) barrier [12,13], a technically complete expression for the characteristic local emission current density *J*_{C} can be written, in terms of the local work function *ϕ* and a characteristic local barrier field *F*_{C}, via the linked equations
2.1and
2.2where *a* and *b* are the usual FN constants (table 1), *ν*^{SN}_{F} is the *barrier-form correction factor* for the SN barrier, *λ*_{C} is the relevant *local pre-exponential correction factor* and *J*^{SN}_{k} is a parameter introduced here and called the *kernel current density for the SN barrier*.

Fields, current densities and related parameters vary with position on an emitter surface. The so-called *characteristic* values relate to some specific position (such as the emitter apex) that is considered characteristic of the emitter. The correction factor *λ*_{C} formally takes into account various effects known to influence the emission process, but for which no detailed theory is included in the equation. These effects include the use of more accurate tunnelling theory, correct integration over emitter electron states, the influence of emitter temperature, the use of atomic-level wave functions and the use of accurate electron band theory [14]. The details of these effects are of limited relevance to the present discussion.

Equations (2.1) and (2.2), taken together, provide a characteristic value of the local emission current density given by eqn (2.4) in Forbes [7]. The merit of using the linked form here is that all theoretical uncertainties associated with prediction of *J*_{C} for the SN barrier have been accumulated into the parameter *λ*_{C}: hence, *J*^{SN}_{k} can be calculated exactly.

For the sake of notational simplicity, we now omit the superscript ‘SN’, leaving it to be understood that relevant parameters relate to the SN barrier, unless otherwise indicated.

For a LAFE, a technically complete expression for the macroscopic (i.e. emitter-average) current density *J*_{M} can then be written [7] as
2.3where *α*_{M} is the *area efficiency of emission*, and *λ*_{M} is the relevant *macroscopic pre-exponential correction factor*. For both STFEs and LAFEs, the current *i*_{d} emitted by the device can be written as
2.4where *A*_{f} is a *formal area* introduced here and defined by *A*_{f}≡*i*/*J*_{k}. The formal similarities between equations (2.2), (2.3) and (2.4) ensure that the discussion below is applicable to both STFEs and LAFEs, and to both currents and current densities.

### (b) Universal variables and Fowler–Nordheim plots

In CFE discussions, theoretical results and experimental data can be presented in many different ways, involving different choices of independent variable (e.g. barrier field, *F*_{C}; macroscopic field, *F*_{M}; device voltage, *V* _{d}; or measured voltage, *V* _{m}) and of dependent variable (e.g. characteristic local current density, *J*_{C}; macroscopic current density, *J*_{M}; device current *i*_{d}; or measured current, *i*_{m}). The need to distinguish between device parameters (*i*_{d}, *V* _{d}) and measured parameters (*i*_{m}, *V* _{m}) arises because measurement circuits may, in principle, have resistance in series and in parallel with the field emitter (figure 1). In practice, parallel resistance can usually be made effectively infinite by improved system design, but, in some cases, it may be impossible to eliminate the series resistance.

To discuss theoretical issues applying to all pairs of variables, it is useful to introduce a *universal independent variable X* (which represents *F*_{C}, *F*_{M}, *V* _{d}, *V* _{m} or any other suitable variable), and a *universal dependent variable Y* (which represents *J*_{C}, *J*_{M}, *i*_{d}, *i*_{m} or any other suitable variable), and to define a *universal auxiliary constant c*_{X} by
2.5All the FN-type equations can then be seen as variants of the *universal FN-type equation*
2.6Here, *C*_{YX} is a parameter whose form depends on the choice of *X* and *Y* (and on various other factors), *ν*_{F} is the barrier-form correction factor for the chosen barrier (which can be of arbitrary well-behaved form) and *B*_{X} is a parameter (denoted by *S** in earlier papers) defined by
2.7

In FN coordinates of type versus 1/*X*], equation (2.6) generates a function *L*(*X*^{−1}) given by
2.8The slope *S*_{YX} of the FN plot of equation (2.8) can be written as
2.9where the overall *slope correction factor σ*_{YX} depends on various individual contributing factors, sometimes including the choice of *X* and *Y* , and is given by
2.10

A common use of FN plots is to estimate the parameter *c*_{X}. Let the slope of the line fitted to the FN plot be *S*^{fit}. In the tangent approach, as described by Forbes *et al*. [15], this fitted line is taken as parallel to the tangent to the theoretical plot (2.8) at some specific (but unknown) value *X*^{−1}_{t} of the horizontal-axis variable *X*^{−1}. The corresponding predicted slope is −*σ*_{t}*B*_{X}, where . Hence, on identifying *S*^{fit} with the predicted slope, and by using some predicted or estimated value for *σ*_{t}, an extracted value for *c*_{X} is obtained from
2.11A common application identifies the independent variable *X* as the apparent macroscopic field, and identifies as the corresponding FEF.

To obtain a predicted value for *σ*_{t}, it is customary to disregard the first and last terms on the r.h.s. of equation (2.10), and to assume that no series-resistance-related effects occur. Forbes *et al*. [15] call this the *basic approximation*, denote the corresponding *basic slope correction factor* by *σ*_{B}, and show that equation (2.10) then reduces to
2.12If, in addition, it is assumed that tunnelling takes place through an SN barrier, then this is the orthodox approximation specified below, and *σ*_{B} becomes given by the mathematical SN-barrier function *s* (see below), and *σ*_{t} by *s*_{t}. However, modern FE literature often makes the *elementary approximation* of taking *σ*_{t}=1.

### (c) The orthodox emission hypothesis

The *orthodox emission hypothesis* is a set of physical and mathematical assumptions that allow a well-defined data-analysis method to be applied to measured current–voltage data. These assumptions are as follows:

— the voltage difference between the emitting regions and a surrounding counter-electrode (all parts of which are at the same voltage) can be treated as uniform across the emitting surface and equal to the measured voltage

*V*_{m};— the measured current

*i*_{m}can be treated as equal to the device current*i*_{d}, and as controlled solely by CFE at the emitter/vacuum interface;— emission can be treated as if it involves deep tunnelling through an SN barrier, with the device current

*i*_{d}described by a related FN-type equation in which the only quantities that depend (directly or indirectly) on the measured voltage*V*_{m}are the independent variable in the equation and the barrier-form correction factor; and— the emitter local work function is constant (and constant across the emitting surface), and has its assumed value.

An implication of these assumptions is that the parameters *λ*_{C}, *λ*_{M}, *A*_{f}, *ϕ* and *c*_{X} used earlier can all be treated as constants.

Real physical emission situations are never ‘exactly orthodox’, but some real situations are expected to be ‘very nearly orthodox’. In particular, emission from a stably mounted, metal, STFE of moderate-to-large apex radius, with a good conducting path to the high-voltage supply, and emitting under stable vacuum conditions, is expected to be very nearly orthodox. Some other emitters can perhaps be described as ‘nearly orthodox’. However, the earlier-mentioned assumptions exclude many complications that can occur in real emission situations.

Exclusions include significant effects resulting from: voltage drop in the measuring circuit, or other forms of ‘saturation’; leakage currents; patch fields; field-emitted vacuum space charge; current-induced changes in emitter temperature; field penetration and band-bending; strong field fall-off; quantum confinement associated with small-apex-radius emitters [16]; and field-related changes in emitter geometry or emission area or local work function.

The specific term ‘orthodox emission’ was introduced in [7], but much of the underlying thinking is considerably older. What is given above is a more careful statement of the assumptions involved in ‘orthodox’ FN-plot-analysis procedures.

From 1960 to 1990, nearly all literature analyses of measured CFE current–voltage data, in practice, assumed the orthodox emission hypothesis. However, as noted earlier, if emission is not physically orthodox, then orthodox data-analysis methods may generate spurious values for extracted physical parameters (see §5). Thus, a test for identifying non-orthodox experimental behaviour will be useful.

In the past 20 years, especially with work on LAFEs, the elementary approximation of taking *σ*_{t}=1 has often been used for FN-plot analysis. However, if emission is not physically orthodox, then elementary data-analysis methods are also likely to generate spurious values for extracted physical parameters. Thus, a test for lack of physical orthodoxy is also relevant for the elementary approximation.

### (d) Use of scaling in the orthodox approximation

The arguments below make extensive use of a parameter *f* introduced some years ago [17,18], and called ‘the scaled barrier field for an SN barrier of zero-field height *ϕ*’. This subsection sets out existing theory (see appendix in [7]), and extends it slightly.

For an SN barrier of zero-field height *ϕ*, one can define a *reference field F*_{R} that reduces the barrier height to zero. For this SN barrier, the *scaled barrier field f* corresponding to barrier field *F* is
2.13where *c* is the *Schottky constant* (table 1). The value *f*_{C} corresponds to the characteristic barrier field *F*_{C} in equation (2.2). When deriving equation (2.2), it can be shown that *ν*^{SN}_{F} is given by the mathematical value *v*(*f*_{C}) obtained by substituting *l*′=*f*_{C} into the mathematical function *v*(*l*′) that Forbes & Deane [18] call the *principal SN barrier function*.

Work-function-dependent parameters *η*(*ϕ*) and *θ*(*ϕ*) can be defined by
2.14and
2.15where *a* and *b* are the FN constants, as before. The characteristic kernel current density *J*_{k} can then be written exactly in scaled form as
2.16Values of *η*(*ϕ*) and *θ*(*ϕ*) are shown in table 2 for a range of work-function values.

To make equations physically explicit, the subscript ‘C’ is used above to label characteristic values. For the sake of notational simplicity, the subscript ‘C’ is now dropped, it being understood that relevant parameters are characteristic values.

The slope *S*_{f} of an FN plot of type versus *f*^{−1}] is
2.17where *s*(*f*) is the mathematical function (expressed here as a function of *f*) that is the slope correction function for the SN barrier [18]. Because the orthodox emission hypothesis treats *λ*_{C}, *λ*_{M} and *A*_{f} as constants, and requires that *i*_{m}=*i*_{d}, equation (2.17) also gives the predicted slopes of FN plots that use *f* as the independent variable and either *J*_{C}, *J*_{M} or *i*_{m} as the dependent variable.

### (e) Extraction of *f* values

With the orthodox emission hypothesis, all sensible ‘independent’ variables are linearly related to each other, and consequently *f* can be used as a scaled value of the universal variable *X*. Thus,
2.18where *X*_{R} is the reference value of *X* at which the barrier height becomes zero. If a given dataset is plotted as a function of *X*^{−1} rather than *f*^{−1}, then the slopes of the two plots are related by
2.19It follows, using *X*_{R}=*X*/*f* and equation (2.17), that
2.20In the FN-plot analysis, this applies in particular to the ‘fitting point’ *X*_{t} at which the tangent to the theoretically predicted plot is parallel to the fitted line. Hence,
2.21where *s*_{t}≡*s*(*f*_{t}), and *X* and *f* are values related by equation (2.18) for a specific value of *X*_{R}. It follows that the value *f*^{extr} that relates to the experimental value (*X*^{−1})^{expt} read from the horizontal (1/*X*) axis of an FN plot is
2.22In orthodox data-analysis theory (data analysis based on the orthodox emission hypothesis), this result applies for all appropriate choices of independent and dependent variables. It guarantees that the test for lack of orthodoxy constructed below works for any physically relevant form of FN plot.

A value is now needed for *s*_{t}. A simple good approximation exists for *s*(*f*), namely *s*(*f*)≈1−*f*/6 [18]. Re-analysis [19] of the experimental results of Dyke & Trolan [2] for their emitter X89 showed that their ‘safe DC working voltage range’ corresponded to the *f*-value range 0.20<*f*<0.34. The related range for *s*(*f*) is 0.967>*s*>0.943. Hence, to an adequate approximation, because *f*_{t} is near the middle of the stated range, one may take *s*_{t}≈0.95. The Dyke & Trolan results for emitter X89 (which was a very carefully investigated tungsten STFE) are used here because this was the main set of results used in the 1950s tests of the applicability of FN theory to CFE data.

Obviously, there is a question as to the accuracy of the extracted *f* values derived from equation (2.22). The largest source of possible error is expected to result from any difference between the actual work-function value experienced by tunnelling electrons and the value (*ϕ*) assumed in deriving the value of *η* in equation (2.22). Because *η*∼*ϕ*^{−1/2}, an error of 0.5 eV in a work function expected to be 4.5 eV would generate errors of around 6%; errors of this size would not normally have any significant effect on the outcome of the test described below.

## 3. Development of a test for lack of field emission orthodoxy

### (a) Basic thinking

For any given emitter work function *ϕ*, the CFE regime occupies a region of the coordinate space that has barrier field *F* on the horizontal axis and temperature *T* on the vertical axis [20]. Alternatively (for given *ϕ*), one can depict the regime in a coordinate space involving *f* and *T*, as illustrated in figure 1, which is derived specifically for tunnelling through an SN barrier. The precise boundary of the CFE regime depends on the details of the FN-type equation and boundary condition used, but the general shape of the region is always similar to that in figure 2.

FN-type equations describe CFE. Figure 2 shows qualitatively that, for any given emitter temperature, an FN-type equation based on the SN barrier is valid only for a restricted range of *f* values. Consequently, if an orthodox data-analysis procedure based on equation (2.22), with *s*_{t}=0.95, yields an extracted *f* value outside the range of values relevant to the CFE regime, then the emission situation cannot be physically orthodox. This basic thinking suggests that tests for lack of field emission orthodoxy can be based on specifying an ‘experimentally reasonable’ range of *f* values for physically orthodox emission.

In reality, the *f* values corresponding to practically useful emission current densities may lie somewhat inside the regime boundary shown in figure 2. At sufficiently low *f* values, owing to signal-to-noise issues in measuring instruments, the resulting emission currents may be too low to be measured. At sufficiently high *f* values, the local current density may (for many emitter materials) be high enough to cause significant emitter heating, and consequent emitter failure (or, alternatively, be high enough to induce vacuum breakdown and emitter failure by some other mechanism). Thus, the question of ‘what is a reasonable range of practical *f* values for physically orthodox emission?’ is better decided by reference to actual past experimental data.

### (b) Formulation of test criteria

The emitters corresponding most closely to the physical assumptions of the orthodox emission hypothesis are the traditional, metal, STFEs of relatively large apex radius, used in arrangements with a low-resistance path to the high-voltage supply. This means that one can appeal to the large body of experimental work in such emission situations, mainly carried out in the 1925–1975 period.

The author's initial choice [7] was to base an orthodoxy test on the re-analysed Dyke & Trolan [2] data described above, which yielded the ‘safe DC working range’ 0.20<*f*<0.34, and to use a test that required the midpoint of the extracted *f* values to lie in the smaller range 0.22<*f*<0.32. In this original test, a formula for *s*_{t} was used that made its value consistent with the experimental data. However, this formula yielded unphysical *s*_{t} values for physically non-orthodox emission data. The revised approach, subsequently used [8,9], of approximating *s*_{t} as 0.95 makes the *s*_{t} value used consistent with tunnelling theory, and is considered simpler and better.

It was also decided (in response to reviewer's comments in the preparation of [8]) that a better orthodoxy test—for a linear FN plot—would be to require the whole of the extracted range of *f* values to lie within a specified ‘reasonable’ range. In [8], the Dyke & Trolan range 0.20<*f*<0.34 was used as the ‘reasonable’ range.

Later, as indicated in [9], a more extensive investigation of past experimental results was undertaken. For some of the best-known CFE experiments between 1926 and 1972, extracted *f*-value ranges are shown in table 3. In all cases, the work function is assumed to be 4.50 eV.

Dataset 1 is that used earlier, and is also shown as fig. 13 in Gomer's [34] well-known textbook. Dataset 6 is also shown as fig. 8 in Good & Müller's [35] review article. Dataset 12, taken in wire-and-concentric-cylinder geometry [36], is the current–voltage (*i*_{m}−*V* _{m}) dataset with which Lauritsen first discovered linearity between and 1/*V* _{m} [31,37,38]; this form of plot, and directly related forms involving other independent and dependent variables, are called here ‘Millikan–Lauritsen (ML) plots’. Dataset 16, also taken in wire-and-concentric-cylinder geometry, is the Millikan & Eyring [33] dataset subsequently used by Stern *et al*. [1] to deduce that experimental FN plots are more nearly linear than ML plots. The results of Rother [39] are excluded because, for the tip–anode spacing tested in his Tafel 5 (0.015 mm), an FN re-plot of his results does not yield a straight line.

Further information about the experimental datasets used to produce table 3, and related references, and details of the related spreadsheet calculations are provided as electronic supplementary material. Several of the datasets in table 3 were, in fact, published as ML plots rather than FN plots: the necessary formula for converting ML-plot slope to FN-plot slope is given in the supplementary material.

Clearly, many experiments used slightly lower onset conditions than were used for emitter X89; hence, a revised test lower limit will be set at *f*^{extr}=0.15. Dataset 8 is the only one with a still lower onset (0.11). This could be because Abbott & Henderson [27], who were attempting to check the linearity of FN plots over as wide a field range as practicable, used an electrometer for their low-current measurements, rather than the galvanometer used for their higher-current measurements.

For steady emission, datasets taken in the 1950s and later terminate (on the high-*f* side) at 0.35 or less. However, the older datasets go up as far as *f*=0.59. This difference could be due to increased later awareness that emitting high steady current density can cause premature emitter destruction. Another factor may be that two datasets involving high *f* values (numbers 7 and 11) were parts of experimental investigations of the effects of temperature on emission, where it would be natural to use relatively high field values in the room-temperature datasets. Given the historical spread, a revised test upper limit for modern results is set, somewhat arbitrarily, at *f*^{extr}=0.45.

Another table feature, where pulsed emission has been used to explore emission at higher *f* values, is the apparent variability in the *f* value at which nonlinearity is detected. This could partly be due to difficulties in quantifying the onset of nonlinearity; alternatively, the most obvious physical explanation lies in geometrical differences in space-charge effects.

In the 90 years of practical CFE since the first review in English [40] of what was then called autoelectronic emission, table 3 appears to be the first published attempt to make a systematic *quantitative* comparison of historical CFE experimental results. The overall picture is one is of general consistency (apart from Rother's work [39], which was probably influenced by gas-discharge effects). This consistency is perhaps slightly surprising, given the general state of vacuum techniques in the 1920s and 1930s; the weak dependence of *η* on work function may be part of the reason.

Because of the difficulties in setting precise test limits before gaining extensive experience in its practical use, this range-ends test currently needs to be treated as an ‘engineering triage’ test: it sorts experimental results into the three categories: ‘apparently reasonable’, ‘clearly unreasonable’ and ‘undecided—more detailed investigation required’. The limits proposed above are the limits of the central ‘apparently reasonable’ range (for an emitter with *ϕ*=4.50 eV). As boundaries for the ‘clearly unreasonable’ range, the author proposes *f*^{extr}<0.10 and *f*^{extr}>0.75, the former because it is slightly lower than any observed onset value in table 3, the latter because it is somewhat higher than any *f* value used in the pulsed-emission experiments listed.

These proposed limits may need adjustment in the light of experience and/or for certain classes of emitter. For example, it is arguable that test boundaries may need to be lowered if highly sensitive measuring techniques are used, and may need to be raised for emitters (such as carbon emitters) that have very high melting points. They may also need to be raised for very sharp emitters with good heat dissipation properties—including liquid–metal field electron emitters—that might be able to sustain current densities corresponding to *f* values closer to unity.

### (c) Tests for other work-function values

For orthodoxly behaving materials with work-function values different from 4.50 eV, one can define the boundaries of the ‘apparently reasonable’ range in the form *f*_{low}≤*f*^{extr}≤*f*_{up}, and the boundaries of the ‘clearly unreasonable’ range in the forms *f*^{extr}<*f*_{lb} and *f*^{extr}>*f*_{ub}. All these boundaries will vary with *ϕ*. It seems probable that the effects determining the boundaries relate strongly to the emitter's characteristic local emission current density *J*_{C}; this, in turn, relates to the kernel current density *J*_{k}. For *ϕ*=4.50 eV, the boundary values for *J*_{k} (namely *J*_{lb}, *J*_{low}, *J*_{up} and *J*_{ub}) are derived from equation (2.16) as, respectively, 3.02×10^{−6}, 2.56×10^{1}, 8.79×10^{10} and 1.02×10^{13} A m^{−2}. For other *ϕ* values, the *f* values corresponding to these *J*_{k} values can be found by trial-and-error methods based on equation (2.16). These values are shown in table 2, and can be used to provide orthodoxy tests for materials with work functions different from 4.50 eV.

## 4. Illustrative applications to post-1975 datasets

This modified test has now been applied to selected examples of various emitting materials. Table 4 shows the material, the work function assumed and extracted *f* values. Values outside the ‘apparently reasonable’ range are marked with a single asterisk, ‘clearly unreasonable’ values with a double asterisk. Where given by the original authors, the extracted ‘apparent field enhancement factor’ (parameter *β* in equation (5.1)) is also shown. Details of table preparation are provided as electronic supplementary material. Brief notes on individual tests follow.

Test 16 relates to a Spindt array [41]. The upper end is outside the apparently reasonable range. This could be because the emitters were being operated at particularly high current density, or because the individual emitters have very small tip radius, or a mixture of both.

Tests 17 and 18 relate to emitters developed for use as electron microscope sources. Allowing for the low work function of the zirconium-coated tungsten emitter, both *f*-value ranges are apparently reasonable.

Test 19 relates to a hybrid nanostructure that uses relatively closely spaced gold nanowires grown on a graphene film. The whole extracted range is ‘clearly unreasonable’. This result has already been reported [8,42], and is not fully understood. Possible explanations include field-dependent changes in emitter geometry and/or changes in collective electrostatic screening effects.

Tests 20 and 21 relate to what Li *et al*. [43] call a ‘flexible SnO_{2} nanoshuttle’. The related FN plot has two quasi-linear sections. For both sections, the whole extracted range is ‘clearly unreasonable’; this could be an effect related to field-dependent geometry.

Tests 22–26 relate to various forms of carbon field emitters. Apart from two end values, the whole extracted range is apparently reasonable in every case. Carbon FE is not necessarily well described by FN-type equations, so this result is slightly unexpected. It may indicate that these carbon emitters (and their supporting structures) have adequate conductivity, and that carbon barrier behaviour is sufficiently close to that of an SN barrier.

Test 24 is particularly interesting because it relates to a large-scale electrical engineering device, namely a gas-discharge tube surge protector (used, for example, to protect telecommunication circuits from lightning strikes). Žumer *et al*. [44] consider that gas breakdown is initiated/stimulated by CFE from the edge of graphite platelets used in the device; obviously, the test result here supports this proposition.

Tests 27 and 28 relate to a carbon nanotube (CNT) mat on a silicon substrate. The FN plot has two sections, with the high field one obviously a ‘saturated’ regime. The two tests between them clearly pick out the high-*f* regime as non-orthodox.

Tests 29 and 30 relate to two different CNT-in-matrix composites. The first is apparently reasonable, the other is not. A point of interest is that the FN plot used for test 28 has only a single quasi-linear section; there is no ‘unsaturated’ section.

Tests 31–34 relate to four different types of semiconductor nanostructures. For tests 32–34, the FN plots are chosen because they have the three highest ‘field enhancement factors’ listed in table 1 in the review by Zhai *et al*. [6]. As can be seen, one test result is apparently reasonable, but the other three are non-orthodox. This means that one of the four high reported values of ‘apparent field enhancement factor’ seems valid, but the other three seem spurious (see §5).

The comments provided here are not intended, at this point, as definitive scientific conclusions. Rather, the tests and comments are intended as ‘proof of concept’ and as a ‘demonstration of usefulness’. They illustrate the types of issues that applying the test might raise or illuminate.

## 5. Non-orthodoxy and spuriously large values for field enhancement factors

A direct mathematical link exists between non-orthodoxy and spuriously large values of apparent FEF. In FE literature, a formula written here as
5.1has been widely used to derive an experimental value for an ‘apparent FEF’ denoted here by *β*. This parameter *β* is often interpreted as a true electrostatic FEF. However, if the magnitude |*S*^{fit}| of an experimental FN-plot slope is significantly lower than the value that would be predicted for orthodox emission from the emitter concerned, then this anomalously low value of |*S*^{fit}| will generate both anomalously high extracted *f* values, via equation (2.22), and a spuriously large apparent FEF value via equation (5.1).

Thus, failure of the orthodoxy test means that the apparent FEF derived from equation (5.1) will be larger than the true electrostatic FEF. As discussed in §4, table 4 shows that some (but certainly not all) reported high FEF values appear to be spuriously high. For safety of interpretation, all experimental results for which equation (5.1) generates particularly high FEF values need to be checked for lack of orthodoxy.

Mathematical reality is that equation (5.1) is defective, and that the correct mathematical formula for the true electrostatic FEF (denoted here by *γ*) should have the form
5.2where *κ* is a correction factor that can, in some cases, be significantly less than unity. The issue of whether it is possible to reliably predict *κ* values (and, if so, then under what conditions) seems very complicated, and is beyond the scope of this paper.

## 6. Discussion

### (a) Summary

This paper has provided a careful definition of the orthodox emission hypothesis, and—by using the idea of ‘apparently reasonable’ and ‘clearly unreasonable’ ranges of scaled barrier fields—has confirmed that experimental tests for lack of field emission orthodoxy can be developed. Using results from historical CFE experiments, the limits involved in the range-ends test have been refined. Thus, after 90 years of practical CFE, we now have (for the first time) a simple, quantitative easy-to-apply test that indicates when measured current–voltage characteristics are not compatible with FN-type equations and theory. Demonstrated incompatibility means either that the emitting-device behaviour does not conform to an FN-type equation, or that the emitting device behaviour may be conforming (for a given barrier field), but that series resistance or some other system feature is distorting the measured characteristics.

Table 3 shows that historical CFE experiments (on metal emitters of relatively large apex radius) are broadly consistent with the orthodox emission hypothesis. Although this was to be expected, demonstrating this explicitly and quantitatively is helpful for FE science.

The improved test has been applied to various modern emitter types, including many non-metal emitters and many LAFEs. Some show apparently orthodox behaviour, others do not. Two-section ‘kinked’ FN plots are characteristic of ‘saturation’, which is usually due to series resistance in the current path. As might be expected, the high-*f* section of such plots tested non-orthodox, in both cases tested. However, in one case, the low-*f* section also tested non-orthodox. There are also single-section FN plots that tested non-orthodox. So—contrary to belief in the literature—observation of an approximately straight-line FN plot, without a ‘kink’, does not necessarily mean that the emission is physically orthodox.

Not surprisingly, field-dependent geometry and saturation effects appear as likely causes of non-orthodoxy. The finding that most carbon emitters tested, including some CNT-based emitters, have apparently orthodox behaviour was slightly unexpected. This perhaps suggests that when a CNT-based emission situation tests non-orthodox, this may be due to series resistance in the substrate or in the substrate-to-CNT contact.

### (b) General implications and future theoretical development

Non-orthodoxy usually implies that a spurious FEF value will be generated if orthodox or elementary data-analysis formulae are used. Hence, it is arguable that, as a routine part of data analysis, all new forms of field emitter should have their current–voltage characteristics tested for lack of orthodoxy, and that the results of the test should be reported in related publications. A copy of the spreadsheet provided as part of electronic supplementary material could be used to do this.

There is also a case for applying the test to past results where particularly high FEF values have been reported, and to representative sets of past results for the different LAFE types, to see if any systematic behaviour can be detected. One would certainly expect that the physical structure of an emitter, and the resistivities of the materials involved, would affect the extent to which the measured current–voltage characteristics are non-orthodox.

Lack of orthodoxy need not imply poor technological quality—for example, in some applications, the ‘ballasting’ provided by series resistance is a practical advantage. What non-orthodoxy implies is that orthodox and elementary data-analysis methods are not applicable to the measured characteristics, and may generate spurious values for characterization parameters.

Clearly, more sophisticated data-analysis theory is required. Although useful progress has been made in modelling series-resistance effects (e.g. [45] in the literature of FN tunnelling in metal–oxide–silicon structures, and [46,47] in the LAFE literature), this is only one of many potential causes of non-orthodoxy (as shown by the list of excluded complications in §2*c*). The way in which orthodox data-analysis theory needs to be generalized will depend, in part, on the particular cause of non-orthodoxy. The overall task seems likely to be highly complicated, will almost certainly need to proceed in stages, and will be addressed elsewhere in due course.

An interesting question is why a quantitative test of this type has taken so long to be found. Part of the answer may be that it relies heavily on the concept of the scaled barrier field *f*, and the usefulness of *f* in simplifying orthodox CFE theory is a relatively recent realization.

In conclusion, it is believed that the test developed here will prove a useful tool in developing field emission technology. For experimentalists, it should help in assessing the validity of their extracted parameters; for technologists, it may assist with the choice of emitter material; for theoreticians, it should help identify materials and emission situations that need detailed attention. At present, the test is a relatively coarse tool, aimed largely at identifying the existence of problems. Hopefully, as results build up from its use, it may be possible to refine the test to become more informative. More generally, it is considered that the development of this test contributes to making the link between theory and experiment in FE science stronger and more quantitative.

## Acknowledgements

I thank the University of Surrey for provision of facilities.

- Received April 29, 2013.
- Accepted June 28, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.