## Abstract

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

## 1. General introduction

Nearly 80 years ago, Fowler and Nordheim (FN) published in these Proceedings their seminal paper (Fowler & Nordheim 1928) on cold field electron emission (CFE) from metal surfaces. Their paper has shaped much subsequent work. The name ‘Fowler–Nordheim tunnelling’ is now used for any field-induced electron tunnelling through a roughly triangular (in practice, always rounded) barrier. The main modern contexts are: (i) vacuum breakdown in high-voltage apparatus of all kinds, where one needs to prevent electron emission from asperities, (ii) cold-cathode electron sources—their many applications include bright point sources (for high-resolution electron microscopes and other machines), X-ray generators, electronic displays and space vehicle neutralizers, and (iii) internal electron transfer in some electronic devices.

The original 1928 equation used an unrealistic barrier model, which seriously underpredicts CFE current densities. Various modified equations have been introduced, which we call ‘FN-type equations’. Data analysis has often used the so-called ‘standard FN-type equation’ (2.10), derived from Murphy & Good's (1956) work (MG). However, the MG paper is not easy to follow, confusion abounds in the literature, and this ‘standard CFE theory’ has gained a reputation for being obscure and difficult. Perhaps as a result, some recent experimental research papers use simplified FN-type equations, as found in undergraduate textbooks. These also significantly underpredict CFE current densities, typically by a factor of order 100, and give scope for error.

The standard FN-type equation uses a mathematical function *v*, well known in field emission, and normally evaluated as a function of the Nordheim parameter *y* defined by (2.15). Forbes (2006) reported a good simple approximation for *v*(*y*). However, it seemed better to take *v* as a function of a new parameter ‘scaled barrier field’ (equal to *y*^{2} in standard theory), and then write *v* as a function of barrier field *F*. This gave, for the first time, a simple, reliable algebraic approximation for the exponent in the standard FN-type equation. This makes the equation's behaviour easier to investigate, but background theory needs presenting differently.

Specific aims here are to show the approximation's mathematical origin, and use it to gain fuller understanding of FN plots. But, more important, we present a major reformulation of standard CFE theory itself. This seems timely, for two main reasons.

First, FN plots are the commonest tool used to analyse experimental CFE data. Elementary CFE theory, sometimes employed, can explain slopes. But, as §6 shows, standard CFE theory is probably the simplest theory able to clarify their detailed behaviour. Users of FN plots need to be able to understand standard theory; improved formulation should help.

Second, the standard FN-type equation was derived for free-electron metals with planar surfaces and has well-known deficiencies, including limited applicability to atomically sharp emitters. Reformulation makes it easier to generalize standard theory to treat more realistic tunnelling barriers.

In standard theory, clearer conceptual distinctions are needed between purely mathematical aspects and physical aspects. To replace the use of *y*, we introduce two related theoretical descriptions, one mathematical and the other physical, each (in principle) with its own names, symbols and definitions. The former involves mathematics that applies primarily to the Schottky–Nordheim (SN) tunnelling barrier (Schottky 1914; Nordheim 1928); the latter involves physical definitions and equations that apply (or are easily generalized to apply) to many different barriers. There are also physical equations specific to the SN barrier. Both descriptions are valid in their own right. For the SN barrier they interact; other barriers have their own specific mathematical analyses.

Thus, the SN barrier mathematical description uses a variable *l*′ defined in the context of elliptic function theory (by (4.7)); but the physical description uses the scaled barrier field *f*_{h} defined using physical fields (by (2.12)). The SN barrier analysis has *l*′=*f*_{h}, but other barriers normally do not.

Similarly, the mathematical description uses (for example) the ‘principal field emission elliptic function’ *v*(*l*′) given by (4.14), but the physical description uses the ‘tunnelling exponent correction factor’ *ν*(*f*_{h}) defined by (2.5). For the SN barrier, but normally not for other barriers, *ν*=*v*(*f*_{h}).

All this is normal scientific practice when using mathematical functions, with *v*(*l*′) behaving like cos(*θ*). But standard CFE theory has been different. The same symbol has been used for both the physical variable and the related mathematical function, and the name of the symbol representing a function (e.g. ‘vee’) has been used as the name of the function. This unfamiliar convention and resulting lack of clarity seem to have impeded both wider understanding and theoretical development.

In writing this paper, a particular problem has been to decide whether the letters *r, s, t, u, v* and *w* should represent mathematical functions (i.e. be part of the mathematical description, and normally applicable only to the SN barrier), or should represent general physical quantities applicable to many different barriers. Current practice usually treats *t, u* and *v* as mathematical functions, but *r* and *s* more as physical parameters. The solution adopted here is to treat all as mathematical functions. This makes notation more uniform but means that, especially in §2, we need separate symbols (Greek letters are used) for the quantities in the physical description. For simplicity, here, we introduce only *ν* (‘nu’) and the correction factor *τ* defined by (2.6); others will be needed in future work.

In summary, this paper aims to reduce confusion, make standard CFE theory more complete and more accessible, and permanently change the way that basic CFE theory is discussed. Analysis is from basic principles, but relies on results in Forbes (1999*b*) and in Forbes (2004).

The paper's structure is as follows: §2 presents additional background; §3 derives revised definitions for the main functions used in standard CFE theory; §4 discusses mathematical expressions for *v*(*l*′); §5 establishes simple algebraic approximations for the other functions; §6 presents new results relating to FN plots; §7 discusses implications; and appendix A presents the exact series expansion for *v*(*l*′).

## 2. Theoretical background

The following notations are used: let *e* denote the elementary positive charge; *m*_{e} the electron mass in free space; *h*_{P} Planck's constant; *ϵ*_{0} the electric constant; and *ϕ* the local work function of the emitting surface. The field at the emitter surface is denoted by *F* and called the ‘barrier field’, and the emission current density is denoted by *J*; in CFE theory, these positive quantities are the negative of the like-named quantities used in conventional electrostatics. This field *F* determines the tunnelling barrier. ‘Unreduced barrier height’ *h* is the height of a tunnelling barrier when *F*=0. Universal constants are evaluated using the 2002 values of the fundamental constants and given to seven significant figures.

CFE theory involves, first, calculation of the escape probability *D* for an electron approaching the emitter surface in a given internal electronic state, and then summation over all occupied states to give *J*. Many different levels of theoretical approximation exist. The basic ‘free-electron’ CFE treatments discussed here: (i) ignore atomic structure and assume a Sommerfeld free-electron model, (ii) assume that the electron distribution is in thermodynamic equilibrium and obeys Fermi–Dirac statistics, (iii) take temperature as zero, and (iv) assume a flat planar emitter surface, of constant uniform local work function *ϕ* (i.e. d*ϕ*/d*F*=0), with a uniform electric field *F* outside.

Choices are then needed about modelling the tunnelling barrier and about theoretical method. The Schrödinger equation can be solved exactly for FN's original triangular barrier, but has no ordinary analytical solutions for most model barriers of physical interest, including the Schottky–Nordheim barrier used in standard CFE theory. So, normally, some approximate method must be used. Usually this is a JWKB-type approximation (Jeffreys 1925, also see Fröman & Fröman 1965) or the closely related Miller & Good (1953) approximation. There are several different JWKB-type formulae, applicable to barriers of different kinds.

For ‘strong’ barriers, the escape probability *D* can be written (Landau & Lifschitz 1958) as(2.1)where *G* is the so-called ‘JWKB exponent’ defined by(2.2)and *P* is a ‘tunnelling pre-factor’. Here, *g*_{e} [≡4*π*(2*m*_{e})^{1/2}/*h*_{P}≈10.24624 eV^{−1/2}nm^{−1}] is the JWKB constant for an electron, and *z* is the distance measured from the emitter's electrical surface (Lang & Kohn 1973; Forbes 1999*c*). The function *M*(*z*) defines the barrier, with integration performed between the classic turning points, i.e. the zeros of *M*(*z*). By definition, a strong barrier has *G* sufficiently large that exp[−*G*]≪1.

The pre-factor *P* is included in (2.1) for conceptual completeness. Normally, it is tacitly assumed that *P* differs from unity by a presumed unimportant multiplying factor (between 1/5 and 5, say), and is slowly varying with barrier height *h* in comparison with exp[−*G*]. So normal practice sets *P*=1 and uses the so-called ‘simple JWKB approximation’,(2.3)The Miller–Good approximation also reduces to (2.3) when *G* is large.

For the elementary triangular barrier of height *h* and slope −*eF*, *M*=*h*−*eFz*; (2.2) then yields the quantity *G*^{el} given by(2.4)where the ‘second Fowler–Nordheim constant’ *b*≡2*g*_{e}/3*e*≡(8*π*/3)(2*m*_{e})^{1/2}/*eh*_{P}≈6.830890 eV^{−3/2} V nm^{−1}. For other barriers, a tunnelling exponent correction factor *ν* is defined generally by(2.5)and a ‘decay-rate correction factor’ *τ* is defined generally by(2.6)where partial derivatives are taken at constant barrier field *F*. It is easily shown that(2.7)For the elementary triangular barrier, the correction factors *ν* and *τ* are both unity.

Choice of method also occurs when summing tunnelling current contributions from the internal electron states. Forbes (2004) summed over states on a spherical constant-total-energy surface, and then integrated with respect to total electron energy. This approach resembles that used for non-free-electron band structures (e.g. Gadzuk & Plummer 1973; Modinos 1984). It can be applied to any well-behaved barrier model, and leads to the current-density equation(2.8)where the ‘first Fowler–Nordheim constant’ *a*≡*e*^{3}/8*πh*_{P}≈1.541434×10^{−6}A eV V^{−2}, and *ν*_{F} and *τ*_{F} are the values of *ν* and *τ* that apply to a barrier of unreduced height *h* equal to the local work function *ϕ*. For clarity, later, equations of this kind are called ‘curve equations’. The emission current *I* is then given by *I*=*AJ*, where *A* is a notional emission area that is often field dependent.

The subscript ‘F’ on a quantity shows that it applies to the particular barrier encountered by a Fermi-level electron that is moving ‘forwards’ (i.e. towards and normal to the emitting surface). *ν*_{F} and *τ*_{F} enter the theory because deriving (2.8) involves Taylor expansion of the JWKB exponent *G* (=*νG*^{el}) about the Fermi level.

The symbols *ν*, *ν*_{F}, *τ* and *τ*_{F} represent general correction factors, and appear in equations that apply to any well-behaved barrier model. Detailed analyses require a specific barrier model, and the general quantities in (2.5)–(2.8) must then be replaced by correction factors specific to this model. Mathematical evaluations (by computer if necessary) must then be performed for these specific correction factors. Since all factors specific to a barrier model can be derived from the ‘*ν*-like’ factor in the specific version of (2.5), detailed analysis concentrates on this factor.

The Schottky–Nordheim (SN) barrier is defined (for *z* greater than some minimum value *z*_{min}) by(2.9)Putting (2.9) in (2.2) leads to a specific correction factor *ν*^{SN}. Burgess *et al*. (1953) showed that *ν*^{SN} is given by a mathematical function *v* that they specified1 and MG subsequently used. *v* is best understood as a function of mathematical physics in its own right, albeit a very specialized one, and can be called the principal field emission elliptic function (or, better, the ‘principal SN barrier function’). A function *t* is then derived via a specific version of (2.7), namely (3.1*b*). The final outcome is the so-called standard FN-type curve equation (for the emission current *I*^{st} predicted by standard theory),(2.10)where *v*_{F} and *t*_{F} are the values of *v* and *t* that apply to the barrier encountered by a forward-moving Fermi-level electron (when this barrier is modelled as a SN barrier).

The mathematical functions *v* and *t*, with *s*, *u* and *w* below, are known collectively as the ‘special field emission elliptic functions’.2 They depend on a single mathematical variable, hitherto taken as the Nordheim parameter *y* (see (2.15)). All can be derived from *v* and its derivative (see Forbes (1999*b*) for past definitions using *y*).

These special elliptic functions can be evaluated accurately by various methods, and (except for *w*) the results are tabulated (Burgess *et al*. 1953; Good & Müller 1956; Miller 1966; Forbes & Jensen 2001). *v* can also be expressed in terms of the complete elliptic integrals *K* and *E* (see MG and Forbes (1999*b*),3 for definitions using *y*). However, a need exists for simple, reliable, algebraic approximations for these functions, especially *v*. Jensen & Ganguly (1993) and Jensen (2001) have derived formulae for *v*(*y*), but these are complex and awkward to use in further analysis. Fitting procedures have generated simple empirical formulae, for example the Spindt *et al*. (1976) approximation, but these do not represent *v* accurately over the whole range 0≤*y*≤1.

As already noted, Forbes (2006) reported a simple good approximation for *v*, but argued that *y*^{2} is the natural variable to use. In mathematics, it seems best to put *l*′≡*y*^{2}, call *l*′ a ‘complementary elliptic variable’4 and write(2.11)The discovery that *v*(*l*′) satisfies a simple-looking differential equation in *l*′, (4.13), supports using *l*′ as the independent variable. If *y* is used, the resulting differential equation is more complex.

For the real physical situation, we can define a scaled barrier field *f*_{h} by(2.12)where *F*_{h} is the real field that reduces barrier height from *h* to zero. For the Schottky–Nordheim (SN) barrier, Schottky (1914) showed that a field *F* lowers the barrier by an energy *Δ*_{S}=*cF*^{1/2}, where *c*=(*e*^{3}/4*πϵ*_{0})^{1/2}≈1.199985 eV V^{−1/2} nm^{1/2}. So, for the SN barrier (but not for other barriers), we estimate *F*_{h} and *f*_{h} by(2.13)(2.14)The Nordheim parameter *y* (Nordheim 1928) is defined as(2.15)

In the JWKB integral for the SN barrier, the term (*e*^{3}/4*πϵ*_{0})*Fh*^{−2} has previously been replaced with *y*^{2}. Here, it is replaced with *l*′ (see §4). Equation (2.14) shows that the mathematical variable *l*′ can be identified with the SN barrier parameter , and so has a physical interpretation. Probably, the concept of scaled barrier field will prove more readily understandable than the Nordheim parameter has been. These are further advantages of using *l*′ as the independent mathematical variable.

When *h* is the local work function *ϕ*, then (2.13) yields the ‘critical SN barrier field’ () at which the SN barrier for a forward-moving Fermi-level electron vanishes (Schottky 1923). The corresponding scaled SN barrier field is(2.16)So we reach the Forbes (2006) result (but in more careful notation),(2.17)

All these things make it desirable to reformulate standard CFE theory to be in terms of the mathematical variable *l*′ and its physical partner the scaled barrier field. For simplicity in this paper, we now mostly drop the label ‘SN’ and just use *f*: it is always clear from context whether or is meant. Since in standard theory *l*′=*f*, there is some choice as to which is used in formulae. We use *l*′ in strictly mathematical contexts, *f* when the formula relates more to experiment.

## 3. Parameters for Fowler–Nordheim analysis

This section creates *f*-based definitions for the mathematical functions used in standard CFE theory. They could equally well be presented using *l*′, but using *f* will make them easier to generalize in future work. Apart from (3.1*b*), the definitions here are specific mathematical versions of general physical relationships valid for any well-behaved barrier model.

In standard theory, *ν* is given by *v*, *τ* by *t*. So from (2.7), and then the SN barrier formula (2.14),(3.1a)

(3.1b)

The functions *r, s, u* and *w* relate to Fowler–Nordheim plots. We first convert tunnelling exponents to be in terms of 1/*f*. Define a dimensionless parameter *η* by(3.2)where *F*_{ϕ} is the critical field at which a tunnelling barrier of unreduced height *ϕ* vanishes. This leads to the general result(3.3)So, in standard theory, from (2.10),(3.4)Equation (3.4) is said to be ‘in FN coordinates of type [ln{*I*/*F*^{2}} versus 1/*f*]’. Our notation for logarithms follows the rule (ISO 1992) that placing an expression in curly brackets means ‘take the numerical value of this expression, when evaluated in the specified units’. Elsewhere these brackets are used normally. In SI units, (*I*/*F*^{2}) and (*Aaϕ*^{−1}) would be in A V^{−2} m^{2}. Since both *F* and *f* appear, (3.4) is a ‘mixed’ FN plot form; but this form is useful for discussing basic theory.

Equation (3.4) is a ‘curve equation, expressed in FN coordinates’. If any field dependence in *ϕ*, or *A* is ignored, then its slope with respect to 1/*f* at any point may be written as −*sη*, where *s* is the standard slope correction function5 introduced by Houston (1952). Letting *x*≡1/*f*, we have(3.5)

Forbes (1999*a*) argued that the most convenient theoretical model for an experimental FN plot is the *tangent* to the chosen FN-type curve equation, when this curve equation is expressed in FN coordinates. In standard theory, again ignoring any field dependence in *ϕ*, or *A*, a suitable form for this ‘FN-type tangent equation’ is(3.6)where ln{*rAaϕ*^{−1}} is the intercept the tangent makes with the ln{*I*/*F*^{2}} axis. The ‘standard intercept correction function’ denoted here by *r* is the function *r*_{N} introduced by Forbes (1999*a*). Both *r* and *s* vary along the curve.

Figure 3*b* shows plots of the quantity Δ*Y*^{st} [=ln{*I*^{st}/*F*^{2}}−ln{*Aaϕ*^{−1}}] against 1/*f*. For any specific value 1/*f*_{P}, Δ*Y*^{st}(*f*_{P}) can be found either from the curve equation via line V or from the tangent equation via line T. Subtracting (3.4) from (3.6), and using (3.5), yields(3.7)In terms of *f*, the function *u* in Forbes (1999*b*) is *u*(*f*)≡−d*v*_{F}/d*f*. Hence,(3.8)

(3.9)

Finally, a new function *w* is introduced to describe the curvature of a FN plot made against 1/*f*,(3.10)For example, *w*=0.02 means the FN plot slope changes by 2% when 1/*f* changes by 1.

Clearly, all these functions can be obtained from *v*_{F}(*f*) and its first two derivatives.

## 4. Expressions for the mathematical function *v*(*l*′)

### (a) ODE in the complementary elliptic variable l′

We now write *v* as a function of the purely mathematical complementary elliptic variable *l*′, and derive the ordinary differential equation (ODE) that *v*(*l*′) satisfies. Following MG, Forbes (1999*b*) showed how to express *v*(*y*) in terms of the complete elliptic integrals *K* and *E*. His equation numbers are here prefixed F. We can replace (F12) by defining(4.1)Comparison with (2.14) shows *l*′=. Hence, for the SN barrier (but not for other barriers), *l*′ has a physical interpretation as the scaled barrier field. Equation (4.1) also means we can treat *y* in Forbes (1999*b*) as a convenient notation for . Noting *b*=(8*π*/3)(2*m*_{e})^{1/2}/*eh*_{P}, we can use (F17*a*) to provide definitions convenient for numerical integration,(4.2)

(4.3)

In MG and Forbes (1999*b*), the transformation applied to (F24*b*) yields the following results. *K* and *E* are defined in terms of the elliptic parameter *m* as used by Abramowitz & Stegun (1965 (AS)),(4.4)(4.5)If we put, as in (F26*b*),(4.6)then, noting (F26*a*) and (F29),(4.7)(4.8)(4.9)(4.10)Equation (4.7) establishes the link between *l*′ and elliptic function theory. From (F27), (4.6), and then (4.10),(4.11)

(4.12)

(4.13)

Thus, *v*(*l*′) obeys the ODE (4.13). Note that no factors in *l*′^{1/2} appear. This ODE appears to be new both in elliptic function theory and in mathematical physics. It is simpler in form than the ODEs associated with *K* and *E* (Cayley 1876).

### (b) Exact series expansion

Two independent solutions for ODE (4.13) have been found using the method of Frobenius to develop series expansions. The boundary conditions on *v* and d*v*/d*l*′, as *l*′→0, then determine an exact series expansion for *v*(*l*′). The first few terms are(4.14)Recurrence relations for the coefficients are given in appendix A. Series (4.14) was found earlier using Maple (Forbes 2006).

The derivation of the recurrence relations is lengthy and is presented separately elsewhere (Deane *et al.* 2007). It shows that the ln *l*′ terms are an intrinsic part of the expansion, and that terms involving non-integral powers of *l*′ are not needed. Appendix A contains a shorter alternative derivation; this directly confirms the lower order terms but does not bring out the underlying mathematics.

Evaluating coefficients to five decimal places, we obtain(4.15)(4.16)Form (4.16) is explicitly exact at *l*′=0 and 1 and has good convergence.

### (c) Approximations and numerical evaluations

The two simplest ways to obtain high accuracy values for *v*(*l*′) are to evaluate (4.8) or (4.14) using a mathematical package, or to integrate (2.2) or (4.2) numerically. We have checked that different methods give the same numerical result to at least 12 decimal places.

For some applications, including spreadsheet calculations, approximation formulae are useful. For a degree-*j* formula always exact at *l*′=0 and 1, we write(4.17)Best-fit values for the coefficients *p*_{i} and *q*_{i} were chosen by least squares minimization of numerical approximations to , where *v*_{E} is the exact value as determined numerically. Choosing *j*=4 yields formulae with absolute error |*ϵ*|≤8×10^{−10}, without using error spreading techniques. Table 1 shows coefficient values. This performance exceeds, by a factor of approximately 25, the Hastings (1955) result |*ϵ*|≤2×10^{−8} for *E*(*m*), which did use error-spreading techniques. As the universal constants are known only to about 1 part in 10^{7}, this precision is far more than needed physically.

Similarly, a formula for d*v/*d*l*′, with |*ϵ*|≤7×10^{−10}, uses(4.18)where , *s*_{0}=[*u*_{1}−(9/8)ln 2], *t*_{0}=3/16, and the other coefficients come from minimization of absolute errors. This formula ‘goes to infinity in the correct way’ as *l*′→0 and is exact at *l*′=1. As compared with (4.17), extra coefficients are needed to achieve similar precision. A spreadsheet using these formulae to calculate the standard theory functions is available from RGF.

For analytical explorations and preliminary data analysis, a very simple formula is best. As already reported, (2.11) has a relative accuracy of 0.33% or better, over the whole range 0≤*l*′≤1. This outperforms earlier approximations of equivalent complexity, due to Andreev (1952), Charbonnier & Martin (1962), Dobretsov & Gomoyunova (1966), Miller (1966), Beilis (1971), Spindt *et al*. (1976), Eupper (1980) (quoted by Hawkes & Kasper (1989)) and Miller (1980). The reason is that, unlike the other formulae, (2.11) resembles the low-order terms in the exact expansion. However, some formulae do outperform (2.11) over limited ranges, because they were optimized to perform well there.

For comparison with (2.11), we searched numerically for the best approximations of form(4.19)where *q* is an adjustable constant. For 0≤*l*′≤1, least squares minimization of absolute and relative errors leads to *q*-values of 0.1715 and 0.1691, respectively, so *q*=1/6(≈0.1667) is close to optimum for both. Figure 1 compares all three formulae. For *q*=1/6, the largest absolute error *ϵ* in *v* is 0.0024 and occurs near *l*′=0.19; the largest relative error is 0.33% and occurs near *l*′=0.3.

The use of *q*=1/6 in (4.19) is not an analytical result but a good approximation that is convenient for its algebraic simplicity, and also fit for purpose. The important thing is that (4.19), with *q*=1/6, performs sufficiently well over the whole range 0*≤l*′≤1 that we can trust it to reliably model the mathematical behaviour of the exact function *v*(*l*′).

## 5. Explicit approximate expressions for the special elliptic functions

Using (4.19), earlier definitions yield(5.1)(5.2)(5.3)(5.4)(5.5)With (5.4), (4.11) has been used to replace d^{2}*v*/d*l*′^{2}. Numerics below take *q* as 1/6.

Figure 2 shows exact values for *s*, *t*, *u*, *v* and *w*, and absolute errors *ϵ* when using these formulae. All are shown as functions of both *l*′ (0≤*l*′≤1) and *x* (≡1/*l*′) (1≤*x*≤10). Note that *u* rises steeply as *l*′ falls below approximately 0.2 and goes to infinity as ln(1/*l*′), as *l*′→0. For *s*, *t*, *v* and *w*, maximum values of |*ϵ*| are 0.0035, 0.0042, 0.0024 and 0.00009, respectively; maximum magnitudes of relative errors (found separately) are 0.37, 0.39, 0.33 and 0.33%, respectively. For *u*, over the range 0.2≤*l*′≤1, the corresponding figures are 0.0043 and 0.45%; errors get progressively worse as *l*′ falls below 0.2 and become serious below 0.1, but this is not important because *l*′-values of 0.1 or lower are rarely of practical interest.

The established accuracy in predicting *v*(*l*′) is thus reflected in the accuracy in predicting *s*(*l*′), *t*(*l*′) and *w*(*l*′) for 0≤*l*′≤1, and *u*(*l*′) for *l*′≳0.12. In these ranges, the approximations can be used reliably in most algebraic manipulations (though use in exponents needs caution).

## 6. Applications

We now put *l*′=*f* and investigate the standard FN-type equation and associated theoretical FN plots. For simplicity, this section drops the suffix from *v*_{F}, *t*_{F} and *ν*_{F}. ‘Scaled’ forms, with the exponent written as *η*/*f*, are used because they are more general. In standard theory,(6.1)*η* varies slowly with *ϕ*. The range 2.7<(*ϕ*/eV)<6 is 6>*η*>4. The typical value *ϕ=*4.5 eV is *η*=4.64. From (3.3) and (4.19) (with *l*′ replaced with *f*), note that(6.2)

### (a) Predicted straight-line semi-logarithmic plot

From (2.10), (6.2) and *f*=*F*/*F*_{ϕ}, the emission current *I*^{st} predicted by standard theory is(6.3)The expanded form of *t*^{−2}(*f*) is difficult to manipulate. Since *t*≈1 and has weak field dependence, it is left unexpanded. In the first bracket in (6.3), the only assumed field dependence is in *t*, and this may be ignored. So, in principle, an exact straight-line plot is given by ln{*I*/*F*^{(2−qη)}} versus 1/*F*, not ln{*I*/*F*^{2}} versus 1/*F*. This is a new prediction; typically, (2−*qη*) is approximately 1.2.

However, real emitters often have field dependence in the emission area (e.g. Abbott & Henderson 1939), which opposes the −*qη* term. Field dependence in *ϕ*, and hence *η* (e.g. Jensen 1999), might also exist. For real emitters, the success of the conventional experimental FN plot could be partly due to mutual cancellation of opposing effects. Experiments to measure the true power of *F* in FN-type equation pre-exponentials would be of considerable interest, but very difficult. Change from using the traditional FN plot is not justified.

### (b) Relationship between standard and elementary CFE theory

Related to (2.10), there is an elementary FN-type equation obtained by replacing *v*_{F} and *t*_{F} with 1. In scaled form, the predicted emission current *I*^{el} is(6.4)From (2.10), (6.4) and (6.2), since *t*^{−2}≈1,(6.5)Typically, *f* is approximately 0.15–0.45, *η* approximately 4.6, *q* close to 1/6 and *qη* approximately 3/4, so (6.5) shows that typically the current density and current predicted by standard theory are roughly 100 times greater than the related elementary theory predictions. The ratio increases as *f* decreases.

Using *t*_{1} to denote *t*(*f*=1), (2.10), (6.2) and (6.4) give(6.6)To show behaviour on the semi-logarithmic FN plot, define(6.7)The elementary FN equation (6.4) then takes either of the forms,(6.8)The elementary FN plot intersects the *y*-axis at *Y*_{0}; this value *Y*_{0} serves as the reference zero for Δ*Y*. Figure 3 has the *y*-axes labelled in this way.

Figure 3*a* and (6.6) show, more clearly than earlier discussions, how a standard FN plot (curve S) is derived from the related elementary plot (line E). First, line L is drawn parallel to line E, above it by [*η*−2 ln *t*_{1}]. Point B is then marked on L at 1/*f*=1. (For 1/*f*<1, the Schottky–Nordheim barrier is below the Fermi level, so curve S starts from 1/*f*=1.) As 1/*f* increases, S lies increasingly above L, by [*qη*ln(1/*f*)−2 ln (*t*/*t*_{1})]; for large 1/*f,* the slopes tend to become equal.

If we ignore small terms in *t*, the shift from line E to line L relates to the e^{η} term in (6.5) and the shift from line L to curve S relates to the (1/*f*)^{qη} term. Figure 3*a* shows that the e^{η} term has the larger effect, for *f-*values of practical interest.

Experimental data points lie in a small range of *f-*values (typically in part of 0.15<*f*<0.45), and often seem to lie on a straight line (the ‘experimental FN plot’). As already noted, the most convenient theoretical model (for a line fitted by linear regression to the data points) is the *tangent* to S, taken at a value of 1/*f* among the data points. The intercept of this tangent with the ln{*I*/*F*^{2}} axis is the theoretical prediction of the regression line intercept. In standard CFE theory, the slope and intercept of the tangent relate to those of line E via the mathematical correction functions *s* and *r*. A tangent taken at 1/*f*=5 is shown as line T_{5}.

The shape of S makes *s* and *r* vary with 1/*f*. At 1/*f*=1, (5.5) shows that *r* has the value , typically approximately 40. As 1/*f* increases and the tangent point P moves to the right along S, three effects occur: P moves downwards; P becomes increasingly distant from line E; and the magnitude of the slope of S increases. The result is to increase *r*. Equation (5.5) shows, perhaps counterintuitively, that this increase is due, almost exclusively, to the increasing difference between S and L, effectively by the factor (1/*f*)^{qη}. For example, for 1/*f*=5, (1/*f*)^{qη} is typically approximately 3.3 and *r* is typically approximately 140. As is well known, if one attempts to extract emission area from an experimental FN plot by putting the *Y*-intercept equal to ln{*Aaϕ*^{−1}} rather than ln{*rAaϕ*^{−1}}, overestimation by a factor *r* occurs.

This analysis provides a clearer picture of how the FN plot works. It is also a reminder of the disadvantages of using elementary rather than standard theory: predicted currents will be too low and extracted areas too high, typically by factors of 100 or more.

### (c) The underlying mathematics of the standard FN plot

The underlying mathematics of the standard FN plot deserves comment. In (6.6), ignore the terms in *t,* write *x*≡1/*f*, and use (6.7) to give(6.9)Figure 2*c* shows clearly that *v*(*x*) varies quite sharply with *x*; but figure 3*a* shows that line S, i.e. Δ*Y*^{st}(*x*), is almost straight. So a vital mathematical question is: ‘why does (6.9) generate a straight line?’ At first sight, this behaviour is counter-intuitive. Its mathematical origin is as follows.

From (4.19), with *x*≡1/*l*′, ln *x*≡−ln *l*′,(6.10)(6.11)Equation (6.11) is another equation for S. Clearly, the reason for its near-linearity is the form of (6.10). The terms in *x*^{−1} generate a constant term (*η*) and the slowly varying term *qη* ln *x*; the ‘1’ simply generates the linear term −*ηx*.

### (d) The implications of FN plot linearity

This argument also operates in reverse. A legitimate question is: ‘what does observed linearity in a FN plot mean?’ The argument above implies that, if an experimental FN plot of type [ln{*I*/*E*^{2}} versus 1/*E*], where *E* is any of *f*, *F*, voltage *V* or macroscopic field *F*_{M}, is effectively linear, then, to predict this, the tunnelling exponent correction factor *ν* must have the form *ν*(*E*) =*B*+*CE* (or reduce to it in the relevant range of *E*), where *B* and *C* are constants or slowly varying with *E*.

Thus, FN plot curvature relates to ∂^{2}*ν*/∂*E*^{2}. Edgcombe & de Jonge (2006) reach an equivalent conclusion. Observed linearity implies small ∂^{2}*ν*/∂*E*^{2}. For standard theory, this is confirmed qualitatively by figure 2*a*, which shows that *v*(*l*′) is nearly linear, and quantitatively by the small values of the curvature function *w*. At *l*′=1/3 (equivalent to *J≈*10^{9} A m^{−2} for a *ϕ*=4.5 eV emitter), *w*(1/3)*≈*0.02.

It would be no theoretical surprise if more realistic barriers had generally similar behaviour, so it is no surprise that FN plots have often been nearly linear. Even for emitters with tip radius of order 1 nm, FN plot curvature can be relatively small, as the work of Edgcombe & de Jonge (2006) brings out. So, where marked curvature occurs in experimental plots, usually some other effect must be operating, such as the presence of vacuum space charge, electron supply limitation inside the emitter, or statistical effects associated with a many-emission-sites electron source.

## 7. Discussion

This work was stimulated by finding a new approximation for *v*, and then realizing that *y*^{2} was better as the independent variable and consequently derivations needed reformulating. It seemed best to separate the mathematical and physical descriptions inherent in standard CFE theory, and generalize the physical description to apply to all well-behaved barriers. This paper has laid some foundations and has given a fuller account of Schottky–Nordheim barrier mathematics. By providing approximation formulae with absolute error |*ϵ*|<8×10^{−10}, we hope to make *v*(*l*′) almost as readily accessible as cos(*θ*).

In our view, JWKB-type approaches to linking CFE current density to barrier field will take the following form in future. There will be general physical quantities and equations that relate to a general physical form of FN-type equation and to FN plots. These quantities will include an independent variable (the scaled barrier field of (2.12)), correction factors for the exponent and pre-exponential of the curve equation, and factors that relate to the intercept, slope and curvature of the FN plot. The exponent correction factor *ν* will be defined by (2.2) and (2.5), and the others by more general, physical, versions of the §3 equations. d*ν*/d*f* will be needed as part of this.

In standard CFE theory, with the SN barrier, the dependence of these physical quantities on the real value of scaled barrier field (2.12) is modelled via the specific mathematical functions discussed earlier. We suggest that, in future, they could be known collectively as the ‘Schottky–Nordheim barrier functions’. Other barriers can be defined by different expressions for *M*(*z*), particularly those associated with sharply curved emitters. For each barrier ‘B’ (different for each model of emitter shape, etc.), new quantities *F*_{ϕ}^{B} and *f*_{ϕ}^{B} would be determined, and the dependent physical quantities modelled by new mathematical functions or (equivalently) value sets generated numerically. *f*_{ϕ}^{B} will again lie in the range 0≤*f*_{ϕ}^{B}≤1, so the treatment of the SN barrier becomes a paradigm for the treatment of more realistic barriers.

There remain, even for standard theory, awkward issues over how to relate the physical quantities discussed to the actual behaviour of experimental FN plots based on measurements of current versus voltage. These include: how to calibrate barrier field precisely; how best to define the notional emission area *A*; the effects of field dependence in *ϕ*, *t*_{F}^{−2} and *A*; nonlinearity in the dependence of barrier field on applied voltage; and how to establish the *f-*value you are operating at. There are also the purely statistical difficulties of fitting to noisy experimental data and the problems of poorly characterized experiments. In standard theory, formula (2.17) should help the investigation of some of these. As in Forbes (1999*a*), the problem is how to extract results and error limits under conditions of physical uncertainty.

Overall, this paper provides renewal of standard CFE theory, and underpinning for future developments. We hope many will find this theoretical approach more complete, more fruitful and easier than some older literature. We strongly urge that clearer distinctions be made between mathematical entities and physical quantities, and that the special field emission elliptic functions be treated as functions of *l*′(=*y*^{2}), rather than *y*, and be thought of as the ‘SN barrier functions’. We also commend the scaled form of FN plot, which exhibits CFE theory in a more universal form.

## Acknowledgments

We wish to acknowledge an initial stimulus provided by Dr C. J. Edgcombe's work on the theory of CFE from curved emitters (e.g. Edgcombe 2005), in particular his use of dimensionless variables. We also thank the referees for their constructive comments.

## Footnotes

↵A function specified by Nordheim (1928) is not a correct mathematical expression for

*ν*^{SN}, owing to a mistake in defining the argument of a complete elliptic integral.↵But a better collective name might be the ‘Schottky–Nordheim barrier functions’.

↵Typographic errors occur in Forbes (1999

*b*): (i) in the definition of*K*(*m*), both brackets should be raised to the power (−1/2) and (ii) the value of coefficient*a*_{4}in eqn (32*c*) should be*a*_{4}=0.01451196212.↵The prime indicates that it is a ‘complementary’ variable, i.e.

*l*′→0 as the elliptic parameter*m*→1.↵But note his calculations of

*s*are in error, owing to the mistake in Nordheim (1928); Burgess*et al*. (1953) gave corrected results.- Received May 13, 2007.
- Accepted July 24, 2007.

- © 2007 The Royal Society