## Abstract

In field electron emission theory, evaluating the transmission coefficient *D*^{ET} for an exact triangular (ET) potential energy barrier is a paradigm problem. This paper derives a compact, exact, general analytical expression for *D*^{ET}, by means of an Airy function approach that uses a reflected barrier and puts the origin of coordinates at the electron’s outer classical turning point. This approach has simpler mathematics than previous treatments. The expression derived applies to both tunnelling and ‘flyover’ (wave-mechanical transmission over the barrier), and is easily evaluated by computer algebra. The outcome is a unified theory of transmission across the ET barrier. In different ranges of relevant physical parameters, the expression yields different approximate formulae. For some ranges, no simple physical dependences exist. Ranges of validity for the most relevant formulae (including the Fowler–Nordheim 1928 formula for *D*^{ET}) are explored, and a regime diagram constructed. Previous treatments are assessed and some discrepancies noted. Further approximations involved in deriving the Fowler–Nordheim 1928 equation for current density are stated. To assist testing of numerical procedures, benchmark values of *D*^{ET} are stated to six significant figures. This work may be helpful background for research into transmission across barriers for which no exact analytical theory yet exists.

## 1. Introduction

### (a) Overview

This paper is one of a series that aims to clarify and extend the basic theory of electron emission. In it, we re-examine wave-mechanical electron transmission from right to left across the ‘exact triangular (ET) barrier’ described in §1*d*. We derive a compact general theoretical expression for the transmission coefficient, *D*^{ET} (i.e. the probability that an emitter electron approaching the ET barrier will escape), evaluate accurate values for *D*^{ET} using the computer algebra package MAPLE and discuss transmission regimes. The derived formula applies both to tunnelling through the barrier and to escape over it. Thus, the outcome is a unified theory of transmission *across* the ET barrier, with various special cases. The relationship to previous treatments is noted.

This work needs a name for wave-mechanical transmission over a barrier. The literature sometimes uses ‘ballistic emission’. The term ‘flyover’ is preferred here.

### (b) Previous treatments

Determining the ET-barrier transmission coefficient *D*^{ET} is a paradigm problem in electron emission, with three older generations of mathematical treatment. Fowler & Nordheim (1928) (FN) used a Hankel function and other Bessel functions in their work on cold field electron emission (CFE). Their analysis is complicated (and, for many people, obscure). For tunnelling, Rokhlenko (2011) has recently improved this approach.

Gadzuk & Plummer (1973) outlined a simpler treatment based on the Airy functions Ai and Bi as introduced by Jeffreys (1928, 1942) (see Olver 2010 for their properties). A third generation of treatments was initiated by Jensen & Ganguly (1993), who published—without a fully detailed derivation—a revised formula for *D*^{ET} in terms of Ai and Bi (their eqn (34)).

We have found mathematical discrepancies in the pre-1993 treatments (see electronic supplementary material, ESM3). FN’s approximate formula for *D*^{ET} for deep tunnelling, used in deducing their current-density equation, is correct. However, in neither treatment is the published unapproximated mathematics suitable for general use and the published treatments are not satisfactory mathematical derivations of the approximate formula. Further, neither paper quantifies the approximate formula’s range of validity.

Over the last 20 years, Jensen has developed effective numerical methods for solving transmission problems, based on modified Airy functions ‘Zi’ and ‘Di’ (Jensen & Ganguly 1993; Jensen 2001, 2003, 2007). Although some small discrepancies have been detected (see electronic supplementary material, ESM3), these publications contain between them formulae equivalent to several of those derived below. These agreements help confirm the correctness of the formulae concerned.

In our view, the Jensen derivations are more complicated than would be needed if the objective were solely an exact formula for *D*^{ET}, rather than his objective of developing formulae suitable for numerical modelling. The impression is also given in Jensen (2007) (see discussion immediately after eqn (242)) that slightly different formulae might be needed for tunnelling and flyover. In reality, this appears to be a notation issue, related to the arguments of Airy functions, which does not arise with the convention used in the present paper (see electronic supplementary material, ESM7).

Our original research motives are noted in §1*e*. The difficulties with the pre-1993 treatments emerged during our work. The 1928 Fowler–Nordheim paper is the seminal treatment of field-induced tunnelling from a travelling-wave state inside the emitter to a travelling-wave state outside it. This paper has been highly influential in CFE theory and related technology, and in tunnelling theory generally. The scientific record needs to show, transparently and as simply as possible, how the original FN-type equation for CFE current density can be derived from basic principles of wave mechanics and statistical mechanics. In our view, it currently does not; also, this should be part of a complete discussion of transmission across the ET barrier.

### (c) The present treatment

The treatment here differs in various respects from previous ones. The ET-barrier mathematics is simpler if the barrier is taken to slope downwards to the left and the origin of coordinates is suitably chosen, if energy-like parameters are used in formulae, and if the common Airy functions Ai and Bi are used. (Modern computer algebra packages readily calculate Ai and Bi and associated dependences on physical parameters: this is a strong reason for using these functions.) We directly develop (by wave-matching) a single general formula, in a form well suited for evaluation by computer algebra. Our emphasis is on the mathematical physics of transmission-coefficient regimes, and complements Jensen’s work. Our treatment makes it clear that a single universal formula exists for *D*^{ET}, applicable to both tunnelling and flyover, and that separate derivations are not needed for tunnelling and flyover.

The rest of §1 discusses our barrier model, methodology and original motives. Section 2 derives an exact analytical formula for *D*^{ET}. Section 3 illustrates results. Section 4 considers the deep-tunnelling limit, re-derives the FN approximate formula for *D*^{ET} and discusses its reliability. Section 5 considers other special cases. Section 6 treats ranges of validity systematically, via regime diagrams. Section 7 records benchmark *D*^{ET} values. Section 8 draws conclusions.

In what follows, *e* denotes the elementary positive charge, *m* the electron mass in free space, *h*_{P} Planck’s constant and . To keep equations simple, various auxiliary constants and parameters are defined. The electronic supplementary material comprises notes on the following: acronyms (ESM1); emission constants and parameters (ESM2); discrepancies in previous treatments (ESM3); and various mathematical and other details (ESM4–ESM10). In particular, note ESM6 indicates how the original FN-type current-density equation is derived, and records limitations on using FN-type equations, and note ESM7 sets out the non-customary notation we use when asymptotically expanding the Airy functions for real values of their arguments.

As usual in electron emission theory, fields, currents and related quantities are treated as positive. Apart from this, quantities and equations follow the current International System of Quantities (ISQ) (BIPM 2006), but universal constants are given (to seven significant figures) in the ISQ-compatible units customary in field emission; these simplify calculations when energies are given in electronvolts and electric fields in volts per nanometre.

### (d) The exact triangular barrier

The emitter’s atomic structure is neglected and a smooth, flat planar surface is assumed. The outwards direction normal to the surface is named the ‘forwards direction’, and related energy components are called ‘forwards energies’. The electron potential energy (PE) is uniform parallel to the surface, and its variation in the forwards direction is given by the one-dimensional barrier model shown in figure 1.

This ET barrier is laterally inverted when compared with those commonly drawn. The emitter is represented by the usual Sommerfeld-type PE box, but electrons approach the barrier from the right, and an electron emitted into the vacuum is represented by a wave moving to the left. Figure 1*a* depicts tunnelling, and figure 1*b* the flyover. The ‘vacuum-side’ region to the left of the PE step is ‘region V’ and the ‘emitter-side’ region to its right is ‘region E’.

*F* denotes barrier field, and *χ* the local inner PE of the emitting face. *χ* is assumed independent of *F*. The forwards kinetic energy (KE) of an electron (in region E) that is approaching the PE step is called the ‘approach KE’ and denoted by *W*. The parameter *H*(=*χ*−*W*) is assumed independent of *F*, but—for consistency with the usage for other barrier shapes—is called the ‘zero-field height’ of the barrier. When *W*>*χ*, as in figure 1*b*, then *H* is negative.

The ‘transmission energy’ *w* is defined as the energy difference between the approach KE *W* and the approach KE for transmission at the level of the top of the barrier. Positive *w*-values indicate flyover and negative *w*-values tunnelling. For the ET barrier (though not for barriers where the field pulls the top of the barrier down), *w*=−*H*=*W*−*χ*. Tunnelling theory normally uses the parameter *H*. However, *w* is the better variable for discussing transmission more generally. Hence, some relationships below are given in alternative forms, one involving *H*, the other *w*.

Distance measured towards the right is denoted by *X*. In figure 1*a*, the origin of *X* is to the left of the PE step, and the step is at position *X*=*L*=*H*/*eF*=−*w*/*eF*. *L* is the ‘matching distance’. When *w*≤0, *L* is the width of the tunnelling barrier. Physically, the origin of *X* is at the electron’s ‘outer classical turning point’, in region V. At a turning point, a charged particle modelled as an electrified massive point using classical mechanics has zero forwards KE. If the particle is modelled as a wave using wave mechanics, the wave function *ψ* changes from oscillatory to quasi-exponential at the turning point (because ∂^{2}*ψ*/∂*X*^{2} changes sign there).

In flyover, as shown in figure 1*b*, the origin of *X* is to the right of the PE step. (If the emitter were absent and the field continued towards the right, this origin would be the electron’s classical turning point.) The step position is again given by *X*=*L*=−*w*/*eF*, but *w* is now positive and *L* negative. Other parameters are as in figure 1*a*.

The ET barrier is not a physically realistic model for the actual surface barrier experienced by escaping electrons. The Schottky–Nordheim (SN) barrier (Schottky 1914; Nordheim 1928), which contains an image-PE term, is a better physical model, certainly for metals. However, the Schrödinger equation for the SN barrier cannot be solved exactly in terms of the established functions of mathematical physics. The ET barrier has the marked advantage that an exact analytical solution to the related Schrödinger equation exists. It has the disadvantage that—although trends can be found—quantitative predictions of experimental quantities such as current densities are not accurate. This means that some mathematical results in this paper cannot be directly supported by reference to experiment, and underlines the scientific need for transparent mathematical proofs.

### (e) Original motives

Before difficulties with the pre-1993 treatments were found, this research had two motives. The first was to investigate the mathematical accuracy of FN’s approximate formula (18) for the transmission coefficient for deep-tunnelling, by comparing this with their fuller formula (given at the top of their p. 178) and with the equivalent form found by Forbes (2008) (eqn (18)).

A second motive concerned SN-barrier transmission. The SN barrier was used by Murphy & Good (1956) in their seminal paper on CFE theory. However, there are unresolved fundamental problems relating to the mathematical correctness of their approach (which is based on the underlying work by Miller & Good 1953), especially in relation to flyover. In addition, discrepancies have been found between analytical and numerical treatments of SN-barrier transmission (Bahm *et al*. 2008). Deeper discussion of these problems is not appropriate here. Our thinking was that, before further examination of SN-barrier problems, it would be helpful to understand ET-barrier transmission better, as the mode changes from tunnelling to flyover, and as *F* becomes small.

### (f) Basic methodology

FN’s solution of the Schrödinger equation in region V used the Hankel function, . Because this has a branch point at the origin, they needed to consider how to ‘connect’ solutions valid on opposite sides of the outer classical turning point. Gadzuk & Plummer (1973) indicated that it is simpler to use the Airy functions, Ai(*z*) and Bi(*z*), where *z* is the complex number, *x*+*iy*. Ai(*z*) and Bi(*z*) are ‘entire’ functions, i.e. analytical and well-defined over the whole complex plane; thus, there is no ‘connection’ problem.

To understand our approach, think in terms of a ‘mathematical space’ used to describe Airy functions, and a ‘model space’ used to draw the ET barrier. On the real axis of the mathematical space, the origin is a transition point, with the Airy functions oscillatory to its left and quasi-exponential to its right. All Airy function reference sources use this convention.

The ET-barrier model normally used in field electron emission takes the barrier as sloping downwards to the right, and puts the distance–coordinate origin at the PE step. This model needs Airy function combinations that represent a wave travelling to the right. To convert the mathematical space to this model space, one needs to: (i) expand the real axis in a nonlinear way (see below); (ii) reflect the real axis about the origin; and (iii) shift the origin from the outer classical turning point to the PE step.

In contrast, our model space has the barrier sloping downwards to the left and the origin at the outer classical turning point. Thus, only the first transformation above is needed. This makes the Airy function mathematics (in particular, the algebra of the arguments) much easier to apply. Obviously, after a formula has been derived for *D*^{ET}, it does not matter which way round the barrier is then drawn or if a different origin is chosen.

Physically, with our model, the need is: (i) to use a linear combination of Airy functions that corresponds, for large negative distances, to a wave travelling to the left; and (ii) at the position (*L*) of the PE step, to match the resulting function with a combination of travelling waves of the forms e^{ikX} and e^{−ikX}, using continuity of the wave function and its derivative.

## 2. Development of exact expressions for transmission coefficient

Theory is initially developed in the familiar context of tunnelling. In any particular case, take the electron PE *U*_{e} to be zero at the outer classical turning point for its motion (shown as the open circle in figure 1*a*). This means the electron’s total forwards energy *E*_{n} must also be assigned the value zero. *U*_{e} is then given by *U*_{e}=*eFX* in region V and *U*_{e}=−*W* in region E, and the one-dimensional Schrödinger equation [d^{2}*ψ*/d*X*^{2}−*κ*^{2}(*E*_{n}−*U*_{e})*ψ*=0] becomes
2.1
and
2.2
where *ψ*(*X*) is a one-dimensional wave function (see electronic supplementary material, ESM4), and *κ* is the universal constant
2.3
In region E, the solutions have the form *C*_{±}e^{±ikX}, where *k*=*κW*^{1/2} and *C*_{+} and *C*_{−} are complex amplitudes (see electronic supplementary material, ESM4). In a one-dimensional model, the related probability current *Π* is given by
2.4
where *ψ** is the complex conjugate of *ψ*. For *ψ*=*C*_{−}e^{−ikX}, this yields . Thus, a wave travelling to the left in region E, and carrying a probability current of unit magnitude *Π*_{u} (=1 s^{−1}, in SI units), is represented by (see electronic supplementary material, ESM4)
2.5
where e^{iγ} is a phase factor that can be chosen arbitrarily. The wave function *ψ*^{ucn}_{E,l} is said to be ‘unit current-normalized’.

The incident wave *ψ*^{ucn}_{E,l} is partly reflected at the PE step. The reflected wave function *ψ*_{E,r} is written as the product of a unit current-normalized wave function *ψ*^{ucn}_{E,r} and a complex number *c*_{E,r}, thus:
2.6
To solve the Schrödinger equation in region V, a new (dimensionless) variable *x* is defined by
2.7
where *k*_{A} is defined by formula (2.7). Unlike some earlier treatments (which have a ‘plus’ sign in their Airy-type equations), this generates the Airy equation in its standard mathematical form
2.8
The functions Ai(*x*) and Bi(*x*) are linearly independent solutions of equation (2.8). In the notation system used here, the real variable *x* is used as the argument of the Airy functions and can take positive, zero and negative values (see electronic supplementary material, ESM7). For large negative *x*, these functions take the asymptotic forms (see electronic supplementary material, ESM7 and eqns (9.7.1), (9.7.9) and (9.7.11) in Olver 2010):
2.9
and
2.10
where *δ*,*δ*^{−1/4} and *δ*^{3/2} are positive quantities defined by taking *δ*=−*x*. The function *ψ*_{V,l}(*x*)=*C*_{A}[Ai(*x*)−iBi(*x*)] (where *C*_{A} is a complex amplitude) thus represents a wave travelling in the positive *δ* direction, i.e. to the left in figure 1.

By converting *ψ*_{V,l}(*x*) back to the variable *X* and applying formula (2.4), it can be shown (see electronic supplementary material, ESM4*c*) that for this wave the probability current *Π* is negative (thus confirming that it is moving to the left), and . Thus, a wave function *ψ*_{V,l} (representing a wave travelling to the left in region V) can be written as the product of a unit current-normalized wave function *ψ*^{ucn}_{V,l} and the complex number *c*_{V,l}:
2.11
This carries a probability current of magnitude |*c*_{V,l}|^{2}*Π*_{u}.

In a one-dimensional theory, the transmission coefficient (i.e. transmission probability) *D* is defined as the ratio of the transmitted probability current to the incident probability current; hence
2.12
To obtain *c*_{V,l}, wave-matching is done at the PE step, i.e. at *X*=*L*=*H*/*eF*=−*w*/*eF*. This corresponds to
2.13a
where *c*_{κ} is the universal constant
2.13b
and eV^{−1}(V nm^{−1})^{2/3}.

Matching *ψ*-values at *X*=*L* yield
2.14
By choosing *γ*=*kL*, using the notation *A*≡Ai(*k*_{A}*L*), *B*≡Bi(*k*_{A}*L*), rearranging initial factors and defining a dimensionless parameter *ω* by *ω*=*k*/*k*_{A}, equation (2.14) is simplified to
2.15
where *ω*^{1/2} is positive. On matching values of d*ψ*/d*X* at *X*=*L*, equivalent simplifications yield
2.16
where *A*′ is the value of d Ai(*x*)/d*x* at *x*=*x*_{L}=*k*_{A}*L*, and *B*′ is defined similarly. Hence:
2.17
and
2.18
The Wronskian function (*AB*′−*A*′*B*)=1/*π* (Olver 2010). Thus, equation (2.18) reduces to
2.19a
From the definitions of *k*, *k*_{A} and *c*_{κ} above, the positive real parameter *ω* is given by:
2.19b

Equation (2.19) is equivalent to eqn (34) in Jensen & Ganguly (1993) and eqn (242) in Jensen (2007). For use below, note that defining *ω*_{0}=*c*_{κ}*F*^{−1/3}*χ*^{1/2} yields the exact result
2.20
Subject to the limitation *W*>0 (which implies *w*>−*χ*), the mathematics of wave-matching as set out above is valid for all *L* and *x*_{L}, whether positive, negative or zero. Thus, formulae (2.17) and (2.19) also apply to flyover. In all cases, these formulae can be evaluated precisely by computer algebra packages.

For ET-barrier transmission there are three independent physical variables, best chosen as inner PE *χ*, transmission energy *w* and field *F* (though other choices are possible). These can be used to define various dimensionless variables. Since Airy function values depend only on the dimensionless variable *x*_{L}, result (2.19) illustrates that two (and only two) suitably chosen dimensionless variables are needed to obtain values for *D*^{ET}. Result (2.19) uses the pair {*x*_{L},*ω*}. Suitable pairs can be chosen in other ways, with different pairs useful in different contexts.

## 3. Illustrative behaviour

In CFE, most electrons escape from states near the emitter Fermi level, and face a barrier of zero-field height *H* close to the local work-function *ϕ* of the emitting face. For tungsten, *ϕ* is around 4.5 eV, and *χ* can be approximated as around 15 eV (see electronic supplementary material, ESM5); hence, *W*≈10.5 eV. For tungsten at room temperature, a typical barrier field is around 5 V nm^{−1}. We take these as ‘illustrative values for a refractory metal emitter’.

Figure 2 shows how the transmission coefficient *D*^{ET} varies with *w*(=−*H*), for *χ*=15 eV and for barrier fields *F* of 0.1, 1, 5 and 10 V nm^{−1}. Features of interest are: (i) the well-known exponential fall-off when *w*≪0; (ii) *D*^{ET} is noticeably different from unity for *w*-values as high as 10 eV; (iii) the curve increases monotonically, with no oscillations on the positive-*w* side; (iv) the coefficient for transmission at *w*=0 varies with field (also see figure 4); and (v) as *F* becomes small, the curve approaches that for transmission at a rectangular step, namely *D*^{ET}=0 for *w*≤0, with *D*^{ET} rising from 0 to 1 as *w* increases from *w*=0. This diagram is similar to fig. 24 of Jensen (2007), which is calculated using his approximate eqn (250).

## 4. The ‘deep tunnelling’ limit

### (a) Asymptotic expansions for

On the real axis of the mathematical space, there are three ways of writing the Airy functions and their derivatives with respect to *x* as series expansions in the variable *x*: (i) as asymptotic expansions valid for large positive *x*; (ii) as Maclaurin expansions (most useful near *x*=0); and (iii) as asymptotic expansions valid for large negative *x*. Thus, different approximate formulae for *D*^{ET} exist for different ranges of *x*_{L}, and hence of *F* and *w*.

For sufficiently large positive *x*_{L} (i.e. large negative *w*, large positive *H*), the asymptotic expressions for *B* and *B*′ (see electronic supplementary material, ESM7 and equations (9.7.7) and (9.7.8) in Olver 2010) can be written as
4.1
and
4.2
where
4.3
and |*x*_{L}|^{−1/4}, |*x*_{L}|^{1/4}, |*x*_{L}|^{3/2} and *ξ*_{L} are all positive real numbers.

The modulus signs have been introduced for reasons of overall notational consistency, because this paper uses *x* and *x*_{L} to denote real variables with positive, zero, and negative values, whereas the variables that appear in the asymptotic expansions of Airy functions on the real axis are always positive. As discussed in the electronic supplementary material, ESM7, the asymptotic expansions given in Olver (2010) and other handbooks are—for good reasons—customarily expressed in a different (but potentially confusing) way.

In order to relate (in a later paper) the analysis here to discussions of transmission theory elsewhere, it is necessary to arrange that the parameter *G*^{ET} becomes negative when *w*>0. This is done by the formal definition
4.4a
where *b* is a universal constant, sometimes called the second FN constant, defined by
4.4b
In addition, a positive quantity *u* is formally defined, using equations (2.13a) and (2.19b), by
4.5
The present paper uses *G*^{ET} only in the context of tunnelling. For tunnelling, *x*_{L}>0 and *G*^{ET}>0, and it is clearer to use *H* rather than *w*; thus we have the simpler formulae
4.6a
and
4.6b
Also, for notational simplicity, we drop ‘ET’ and use *G* (rather than *G*^{ET}) in some equations and discussion that follow. In terms of *u* and *G*, equations (2.19), (4.1) and (4.2) yield
4.7
and
4.8
The asymptotic series expansions for *πωA*^{2} and *πω*^{−1}*A*^{′2} can be put into similar forms (see eqns (9.7.5) and (9.7.6) in Olver 2010), but with each containing the factor e^{−G} (rather than e^{G}) and a modified power series.

In this deep-tunnelling asymptotic formulation, *D*^{ET} again depends only on two dimensionless variables; *G*^{ET} and *u* is a convenient pair.

### (b) Fowler and Nordheim’s approximate formula

In the Airy function approach, FN’s approximate formula for *D*^{ET} can be derived as follows. When e^{2G} is large, *A*^{2} and *A*^{′2} may be neglected in comparison with *B*^{2} and *B*^{′2}. This reduces equation (2.19) to
4.9
If only the leading term in each of expansions (4.7) and (4.8) is taken, then equation (4.9) becomes
4.10
Since *u*>0, the value of (1/4){*u*^{−1}+*u*} can never be less than 1/2. Thus, the term (1/2)e^{−G} may be neglected if *G* is sufficiently large. In this case, equation (4.10) yields the formulae
4.11a
with
4.11b
This is FN’s approximate formula for *D*^{ET}; denoted here by *D*^{FNa}, and *P*^{FN} is the FN transmission pre-factor. The illustrative values *χ*=15 eV, *H*=4.5 eV, yield *u*≈0.655, *P*^{FN}≈1.83. The value *χ*=10 eV (which is more realistic for some metals) yields *u*≈0.905, *P*^{FN}≈2.00.

If *H*≪*χ*, then equation (4.11b) reduces mathematically to
4.12

### (c) Mathematical reliability of the Fowler and Nordheim approximate formula

For parameter values typical in CFE, *G* is the main influence on the validity of the assumptions above. If *G*>∼3, *A*^{2} and *A*^{′2} are less than *B*^{2} and *B*^{′2} by a factor of order 400 or more and may be neglected. If higher terms in expansions (4.7) and (4.8) are used, then equation (4.10) is replaced by
4.13
The term (*P*^{FN})^{−1} yields the FN approximate formula, the others constitute a correction.

For the illustrative value *u*=0.655, the three correction terms shown contribute less than 10 per cent to the denominator when *G*>∼2.8. At this *G*-value, the term (1/2)e^{−G} contributes most. The term in *G*^{−1} becomes dominant when *G*>∼4.7 (equivalent to *F*<∼17 V nm^{−1}). In reality—although both results are qualitatively suggestive—equation (4.13) is not accurate for such *G*-values, because higher powers of *G*^{−1} are being neglected, and because the expansion itself is asymptotic. Fortunately, corresponding field values are above those of practical interest for a *ϕ*=4.5 eV emitter.

The progressive breakdown of the FN approximate formula (4.11) as *G*^{ET} diminishes below about 5 is clearly shown in figure 3*a*. The dashed line represents the approximate formula, evaluated for *ϕ*=4.5 eV and a range of fields, but plotted as [ versus −*G*^{ET}]; the continuous line represents the exact result. For very large fields (very small values of *G*^{ET}), the exact curve goes though a maximum; this effect is discussed further in §5.

For *F*=5 V nm^{−1}, the following apply: *G*∼13; the *G*^{−1} term is the dominant correction term; and the predicted correction to the FN approximate result is about 0.17 per cent. This is confirmed by exact calculations using equation (2.19). Thus, for bulk tungsten emitters operating under typical conditions (and for emitters operating at similar current densities), the FN approximate formula is *mathematically* reliable, and the *G*^{−1} term provides a reasonable (but not precise) measure of the formula’s accuracy.

In these conditions, equation (4.12) is in error by about 20 per cent. Since formula (4.12) is an approximation for equation (4.11), and is valid in only part of the parameter space where the latter is valid, it is better to use equation (4.11).

Deep tunnelling is one of the main transmission-coefficient regimes (see §6). In each main regime, one seeks a ‘good working formula’ with a good trade-off between simplicity and accuracy. This section has confirmed that, for deep tunnelling, the FN approximate formula for *D*^{ET} is a good working formula. Its range of validity is discussed further in §6. For completeness, electronic supplementary material, ESM6 indicates how the original FN equation for CFE current density can be derived from equation (4.11).

## 5. Other special cases

### (a) Zero transmission energy

Transmission at the barrier peak has *x*_{L}=0, *w*=0, *W*=*χ*. Let the corresponding transmission coefficient be , and let the Airy functions and derivatives have values *A*_{0}, *B*_{0}, *A*′_{0} and *B*′_{0} there. Values taken from eqns (9.2.3) to (9.2.6) in Olver (2010) are given in the table in electronic supplementary material, ESM2*c*.

For *w*=0, equation (2.19) generates the mathematically *exact* formulae
5.1a
5.1b
and
5.1c
is a single-peaked function of *F*, with a maximum at
5.2

Figure 4 plots against *F*^{1/3}, for *χ*=15 eV. The maximum (about 0.93) occurs at *F*≈770 V nm^{−1}.

For *χ*=15 eV, *F*=5 V nm^{−1}, the terms in equation (5.1a) have the values 0.5, 1.18 and 0.054, respectively. Thus, for fields of practical interest, the term in *F*^{1/3} can be neglected. For fields less than about 0.02 V nm^{−1}, the term in *F*^{−1/3} dominates, and . Hence, *D*^{ET}_{0} goes smoothly to zero as *F*→0, as expected physically. This result is interesting, because the mathematical theory normally used to describe SN-barrier transmission seems to break down as *F*→0.

A formula exists for the limit, namely . This is a good approximation only at incredibly high fields (10^{6} V nm^{−1} or more) and is of theoretical interest only—it shows (surprisingly) that *D*^{ET}_{0} ultimately goes smoothly to zero as .

Formula (5.3) is best considered as describing a special ‘zero-*w*’ sub-regime within the barrier-top regime described next.

### (b) The barrier-top regime

The Airy functions and their derivatives have Maclaurin series expansions valid for all *x* (see Olver 2010), but these converge quickly only if |*x*| is small. Hence, they yield acceptable transmission formulae only in the ‘barrier-top regime’, where |*x*_{L}| is sufficiently less than 1, i.e. when (|*w*|/*F*^{2/3}) is sufficiently small. These formulae describe ‘shallow tunnelling’ when *w*<0 and ‘low flyover’ when *w*>0.

Putting the Maclaurin expansions into equation (2.19), and using low-order terms, gives the formulae (see electronic supplementary material, ESM8):
5.3a
5.3b
5.3c
and
5.3d
As *A*′_{0} is negative, *c*_{1} is positive.

Only the term involving contains a positive power of *F*; thus, this term dominates when *F* is extremely large, and leads to the formula (of theoretical interest only):
5.4
This result, and the unexpected behaviour at very high fields, were first found by Rokhlenko (2011).

For fields of practical interest, and when |*w*| is greater than zero but sufficiently small, we can put *W*≈*χ*, and reduce equation (5.3) to
5.5
Hence, a plot of *D*^{ET} versus *w* should be approximately linear near *w*=0, with a field-dependent slope. Figure 2 showed this behaviour. Thus, it is confirmed algebraically that tunnelling goes smoothly over into flyover, as *w* goes from negative to positive. As |*w*| gets larger, the approximations behind equation (5.5) soon deteriorate, as shown in figure 2.

In practice, the terms *T*_{n} pick up dependences on in the Maclaurin series. However, *x*_{L} also affects *ω* via equation (2.20). Considering this (see electronic supplementary material, ESM8), we found a good working formula for the barrier-top regime (for fields of practical interest) by using the sum *T*_{0}+*T*_{1}, but with *W*^{1/2} in term *T*_{1} replaced by (*χ*/*W*^{1/2}). This yields
5.6
In summary, transmission theory in the barrier-top regime is physically and mathematically well-behaved, but mildly complicated analytically. To obtain values of *D*^{ET}, it may usually be simpler to evaluate equation (2.19) directly.

### (c) The ‘high flyover’ limit

The Airy functions and their derivatives also have asymptotic expansions valid for large negative *x*-values (Olver 2010). In the ‘high flyover’ limit, it can be shown (see electronic supplementary material, ESM9) that:
5.7
and
5.8
where *δ*_{L}=−*x*_{L}. Sine and cosine terms in the expansions for *A*, *B*, *A*′ and *B*′ are eliminated when (*A*^{2}+*B*^{2}) and (*A*′^{2}+*B*′^{2}) are formed.

Expressions (5.7) and (5.8) suggest that higher order terms can be neglected if *δ*_{L}≫1. If only the leading terms are used in these expressions, equation (2.19) yields
5.9
This can be re-arranged (without further approximation) to give a good working formula for the high-flyover regime, namely
5.10
This is also the standard result for transmission across a rectangular step of height *χ*=(*W*−*w*) (e.g. Landau & Lifschitz (1958), problem 1 in §23). Jensen (2007) gives an equivalent result as eqn (247). Note that formula (5.10) has no field-dependence.

If the terms in in equations (5.7) and (5.8) were included, this would generate a term in equation (5.9) that behaves as *F*^{2} (provided *δ*_{L}≫1) and hence becomes negligible as *F* is reduced for a given *w*-value (as is needed for physical consistency).

For sufficiently large *w*, equation (5.10) reduces to
5.11
Thus, *D*^{ET}→1 as , as physically expected, and a plot of *D*^{ET} versus −1/*w*^{2} will tend to be straight at sufficiently high *w*, as shown in figure 3*b*.

These results are a reminder that noticeable wave-mechanical reflection effects can occur at least several electron volts above the barrier peak, and are also consistent with the common assumption that the presence of low fields (not large enough to induce significant Schottky lowering of the barrier) will have no effect on emission characteristics for thermal electron emission.

## 6. Transmission-coefficient regimes

In emission theory, a ‘regime’ is a region of parameter space where a specific formula is an adequate approximation. Given a boundary criterion, regimes can be shown on a ‘regime diagram’ such as figure 5. This paper uses formulae (4.11), (5.6) and (5.10) to define three main transmission-coefficient regimes: deep tunnelling (DT); the barrier top (BT); and high flyover (HF). The criterion used is a 10 per cent difference between the exact formula (2.19) and the relevant good working formula, i.e.
6.1
Evaluations of *W*, *ω* and *u* assume *χ*=15 eV .

The details of a regime diagram will be affected by the particular working formulae and boundary criterion used, and by the choice of *χ*-value. Hence, the results here should be taken as illustrative.

For simplicity, regime names have been based on ‘height in energy’. In reality, the best mathematical discriminator is the dimensionless parameter *δ*_{L} given by . Hence, figure 5 shows *F*^{2/3} on the horizontal axis and *w* on the vertical axis. Straight lines radiating from the origin correspond to constant *δ*_{L} (equivalent to constant values of *w*/*F*^{2/3}), and, for *w*<0, to constant *G*^{ET}. The main regimes appear approximately as circular sectors (i.e. ‘pie slices’). In the field range shown in figure 5, the regimes are largely distinct. However, there is a region of overlap where both the DT and BT formulae give results accurate to within 10%, and a second region of overlap where both the BT and HF formulae give results to this accuracy. There are also two regions (shown shaded) where none of the working formulae perform well, and the exact formula (2.19) (or some engineered approximation) has to be used. The ST and LF components of the BT regime are labelled separately, but together constitute a single main regime.

Figure 5 covers the experimentally relevant ranges of *w* and *F*^{2/3} (given that practical emitters melt or explode at fields well below 20 V nm^{−1}). However mathematically, figure 5 is part of a larger regime diagram, shown in the electronic supplementary material, ESM10. Diagram structure is more complicated at higher fields, and may be of theoretical interest.

Regime diagrams can be used to show ranges of validity for formulae used to obtain numerical values. This is less important for the ET barrier, because an exact general formula exists, but figure 5 is useful for showing theoretical structure and the regions where well-defined physical dependences exist. It also shows clearly that varying *F* for constant *w*, or *w* for constant *F*, can cause a regime boundary to be crossed.

## 7. Benchmark values

With physically realistic barrier shapes, it is rarely possible to obtain an exact analytical expression for *D* in terms of the established functions of mathematical physics. One then needs to consider how to validate numerical solution methods. As tests of adequacy, numerical procedures ought to be able to reproduce faithfully the exact results for barrier shapes that can be solved analytically. To assist such testing, table 1 lists accurate (‘benchmark’) values of *D*^{ET}, precise to six significant figures, for selected values of *F* and *w* (using *χ*=15 eV).

## 8. Discussion

### (a) The nature and use of the solutions

A feature of the ET barrier problem is that the exact universal formula (2.19) generates significantly different approximate formulae in different ranges of field *F* and transmission energy *w*. The basic reasons are as follows. The Airy functions have unique definitions in terms of integrals (Jeffreys 1928; Olver 2010) but three different forms of series expansion. A particular expansion generates a simple, physically useful approximate formula only if the lowest order term or terms are quantitatively dominant. Hence, only in a limited range of *δ*_{L}(=−*x*_{L}) is a particular expansion both mathematically valid and physically useful. The form of formula (2.19) (which results from its derivation by wave-matching) adds to the number of special cases.

These approximate formulae are useful for showing functional dependences in specific parameter ranges. However, to calculate reliable numerical values using them, their ranges of validity need to be known. In practice, the general expression (2.19) may be more convenient—provided one has access to software that is able to evaluate Airy functions precisely, either a computer algebra package or a specific code (see NIST 2010). There is a sense in which modern computer algebra packages have modified the nature of analytical evaluation in science and engineering, and evaluating *D*^{ET} is a good example. However, in working simulations where embedded code is needed that is ‘sufficiently accurate for purpose’, there remains a role for specially engineered and widely applicable approximations, such as eqn (250) in Jensen (2007).

### (b) Summary of achievement

Because transmission across the ET barrier has been an incompletely analysed paradigm problem, this paper’s aim has been an examination that is comprehensive, correct, transparent, subject to the peer review process of a regular scientific journal, and (hopefully) able to bring closure to an 80-year-old problem. In places we have identified flaws in the existing treatments, in places our work has served to confirm formulae first derived by others, and in places our work has generated new formulae that fill gaps.

The outcome, we believe, is an account of the ET barrier problem that is coherent and well-structured, and is derived via transparent physical and mathematical arguments. To develop this, we have used the ‘reflected’ barrier model of figure 1, and the ordinary Airy functions. We find it a strength that some findings have already been independently derived by slightly different mathematical methods, namely via Jensen’s modified Airy functions Zi and Di, and via Rokhlenko’s use of Hankel and other Bessel functions (see electronic supplementary material, ESM3*e*). However, we think that using the ordinary Airy functions, and energies rather than wave numbers, provides the most transparent approach.

### (c) Wider relevance

An important point, which emerges more clearly from our treatment than from the pre-1993 ones, is that the well-known FN formula is an asymptotic approximation, valid only in the deep-tunnelling transmission regime. However, our most instructive results relate to the barrier-top regime. It has been shown that the ET barrier has three main transmission regimes, not two. This identification of the barrier-top regime as separate is not clearly made in the existing literature. We have found that the transmission energy *w* is a useful variable, that exact solutions exist for *w*=0, and that (for *w*=0) the transmission coefficient *D*^{ET} goes smoothly to zero both at very low and at extremely high fields. Within the regime, there is a smooth physical and mathematical transition from shallow tunnelling to low flyover; and both are described by a single set of expansion formulae.

Within this regime (or close to it), *D*^{ET} changes rapidly with *w* (figure 2). Since this means that the lower energy side of the electron normal-energy distribution will be relatively narrow, it should come as no theoretical surprise that the so-called Schottky electron emitter, often used for high-resolution electron microscopes, operates in or close to the barrier-top regime (Swanson & Schwind 2009). (Obviously, details are complicated because the measured distribution is the total energy distribution and because the ET barrier is not a good surface model).

An important question is to what extent the richness of theoretical behaviour found for the ET barrier is specifically associated with the sharply peaked nature of the barrier, to what extent it also exists for the SN barrier and other field-influenced smoothly varying barrier shapes. This issue seems primarily mathematical in nature and needs further research.

In summary, this analysis has put the theory of the paradigm model of field-influenced barrier transmission onto a secure, transparent and reasonably complete mathematical and scientific basis. We hope this will help stimulate fresh thinking about transmission theory for field-influenced barriers that are better physical models than the ET barrier, but for which no convincing exact analytical theory yet exists.

## Acknowledgements

Research support was provided by the University of Surrey. We thank the six reviewers for helpful comments. Electronic supplementary material is downloadable with this article or available directly from one of us (R.G.F).

- Received January 10, 2011.
- Accepted April 18, 2011.

- This journal is © 2011 The Royal Society