## Abstract

In this paper, we derive analytically from first principles a generalized Fowler–Nordheim (FN) type equation that takes into account the curvature of a nanoscopic emitter and is generally applicable to any emitter shape provided that the emitter is a good conductor and no field-dependent changes in emitter geometry occur. The traditional FN equation is shown to be a limiting case of our equation in the limit of emitters of large radii of curvature R. Experimental confirmation of the validity of our equation is given by the data of three different groups. Upon applying our equation to experimental FN plots complying with the above limitations, one may deduce (i) R and (ii) standard field emission parameters—e.g. enhancement factor—with better accuracy than by using the FN equation.

## 1. Introduction

The Fowler–Nordheim (FN) equation originally constructed in the 1920s [1,2] for planar surfaces is still being used for the analysis of experimental data [3–5] despite the fact that modern emitters have radii of curvature in the region of 1–20 nm and consequently can no longer be considered as planar. The inadequacy of the planar FN theory has been amply demonstrated since the 1990s [6–9]. In particular, Cutler *et al.* [6–8] have shown—using the exact current integral WKB theory—that for sharp nanosized emitters the FN plot is not linear and the current may differ by orders of magnitude from the planar FN theory using in the theory the local field at the apex. Furthermore, extracted values for the local electric field at the emitting surface or the emission area itself may be quite wrong. Similar results have been obtained by Fursey & Glazanov [9] and more recently by Fischer *et al.* [10]. It is worth pointing out that Forbes [11] has devised a test of ‘FN orthodoxy’ and applied it to the available data in the literature. He concluded that nearly half the data sample he examined failed his test.

There have been algebraic works so far that tackle emission from specific geometry emitters [12–14] but no equation is available so far that is applicable to an arbitrarily sharp nanoscopic tips (*R*<20 nm) in both the near- and far-field modes and which has been derived from first principles. More specifically in Edgcombe [12] and Edgcombe *et al.* [13], the sphere on a cone model is used which is valid in the far-field mode (large anode-tip distance *d*). The results contain functions of two variables that have to be evaluated numerically and thus not very easy to use. Another notable work is the hyperboloid by Zuber *et al.* [14] in which the emission from a single surface element dS is considered to be FN type. In this work, we derive analytically from first principles a generalized FN equation that (i) is applicable to an arbitrary potential (or surface) and (ii) contains as a correction a new function of only one variable, the well-known variable *y* of the standard FN theory [15]. This new function is given by the authors by a simple and sufficiently accurate formula which makes the equation straight and easy to use. Finally, we can obtain from our equations the local field at the tip and the radius of curvature of the emitting surface. A computer program extracting these quantities is freely offered^{1} and has been tested by the available data in the literature.

## 2. The potential: a general form

The main idea behind this work is that for emitters with nanosize radii a proper generalization of the barrier potential is the inclusion of a second-order term to the usual image rounded linear one. Next, we show that an expression for the coefficient of this term can be found for an arbitrary shape surface and then a Taylor expansion using the Leibnitz integral formula leads to the explicit formula for the current density at the apex.

Let the potential be 0 at the cathode and *V* at the anode. We assume rotational symmetry of the problem. Given that the emitting surface is smooth, we define its radius of curvature at the apex by *R*. The corresponding osculating (inscribed) circle has its centre on the *z*-axis at distance *R* from the apex as shown in figure 1. Consider a system of spherical coordinates (*r*, *θ*, *φ*) centred at that point. An arbitrary emitting surface can be described in that system by the equation *r*=*f*(*θ*) with *f*(*θ*) being an even function of *θ*. Given that, its Mclaurin series has only even powers and hence
*θ*^{2} terms above owing to the existence of the osculating circle.

We now write the Laplace equation in that spherical coordinate system (*φ*-derivatives are neglected)
*Φ* is the electrostatic potential. Taking advantage of the fact that on the emitting surface the electrostatic potential is constant, we may prove that at the apex (*r*=*R*, *θ*=0)
*r*=*R*, *θ*=0) becomes
*F* is the local electric field at the apex. We have proved this relation for a spherical surface and an arbitrary potential in our work on the scaling properties of near field scanning electron microscopy [16]. Here we have generalized it to a general surface. This is vital for our claim of the general applicability of the resulting equation.

By *r*-differentiating (2.2), we observe that
*Φ* along the vertical axis (*θ*=0) around the apex (we use the variable *z*=*r*−*R*). Then by virtue of (2.4) and (2.5)
*F*=*O*(1/*R*), *R*→0 and hence the pre-factor *FR* in (2.6) remains *O*(1) as *R* diminishes.

Equation (2.6) is the key to understanding the approximations used in field emission. The tunnelling current is primarily determined by the electrostatic potential within the forbidden region (again typically 1–2 nm). Thus, when *R*>50 nm taking into account up to the linear term of (2.6) (as the field emission theory has done for many years) is sufficient. However, when *R*<20 nm neglecting the quadratic term results in approximately 10% error in *Φ* which is very important in the calculation of tunnelling currents and when *R*<5–6 nm, we have found that even the cubic term cannot be neglected. Furthermore, for very small *R* (1–2 nm), the electron paths are no longer straight in the tunnelling region [17]. In this paper, we will keep up to the quadratic term which will be adequate for radii *R*>5 nm as will be shown below.

Keeping up to the quadratic term in (2.6), adding the image interaction for a sphere and the work function *W*, we obtain the potential energy barrier ‘seen’ by an electron at the Fermi level approaching the emitter surface at right angles
*B*=*e*^{2}/16*πε*_{0}≈0.36 eVnm.

Figure 2 tests the accuracy of our approximation by comparing the values given by equation (2.7) for an ellipsoidal tip to nearly exact values obtained by detailed numerical computation based on the same method as used in Kyritsakis & Xanthakis [18] for the same tip. The ellipsoidal geometry was chosen as it is a harder test. A field of *F*=5 V nm^{−1} and a work function of 4.5 eV have been used. It can clearly be seen that inclusion of the second-order term is sufficient for radii of curvature down to 5 nm for standard values of *W* and *F*. As *R* decreases more terms of the expansion are needed to describe the potential barrier.

## 3. The transmission coefficient

Using now the well-known WKB integral formula for the transmission coefficient, we get for the Gamow exponent *G*

Then we find
*Ξ*(*x*,*y*) is dimensionless. The variable *x*=*φ*/*eFR* is a metric of the magnitude of the forbidden region length *L* in comparison with *R*. We note that for *R*=10 nm and typical values of *W* (=4.5 V) and *F* (=5 V nm^{−1}), we get *x*<0.1, so *x* can in fact be an expansion parameter for *Ξ*(*x*,*y*).

Indeed, *Ξ*(*x*,*y*) can now be approximated linearly in *x* by the Taylor expansion
*Ξ*/∂*x*, we may use the Leibniz integral rule and given that the integrand is zero at the integral limits the result for *G* is
*ζ*_{1} and *ζ*_{2} are the zeros of 1−*ζ*−*y*^{2}/4*ζ*.

We define
*v*(*y*) is the standard correction for the image-corrected linear barrier which is widely analysed and used in the FN theory literature [15,19]. *ω*(*y*) is a new function which we shall call the nonlinear potential correction function. This function has an analytic expression which is rather complicated. The function *ω*(*y*) can also be expanded in exact series but it still remains very complicated. The simplest and accurate enough way to give *ω*(*y*) is an approximation equivalent to the one given for *v*(*y*) by Forbes [20; 21]

## 4. The current density

Finally, the field emission current can now be found by the relation [22]
*Z*_{S}=4*πem*/*h*^{3}≈1.62×10^{5}(nA/(eV)^{2}nm^{2}) known as the Sommerfeld current constant [23]. Combining the two above equations and after some manipulations, we derive our final FN type equation
*t*, *ψ* have been given again by approximations similar to (3.6). All *y*-functions are given also in tabular form. We have also introduced the constants *a*=*Z*_{S}*e*^{2}/*g*^{2} and *b*=2*g*/3*e* known in the literature as first and second FN constants correspondingly.

It is evident that in the limit (*φ*/*eFR*)→0, equation (4.2) reduces to the standard Murphy-Good (MG) [15] form of FN-type equation. In particular, the factors *t*(*y*) and *v*(*y*) are those that appear in the MG formulation and the terms involving *ψ*(*y*) and *ω*(*y*) are the corrections needed for sharp emitters, with both terms proportional to the small parameter *φ*/*eFR*.

## 5. Comparison with experiment-extraction of R

The validity of our theory is tested in figure 3 where we plot the current density *J* in the form of FN plots (dashed lines) for various radii *R* and at a constant *φ*=4.5 eV together with nearly exact FN plots calculated using the nearly exact potentials of figure 2 and the first principles WKB integral formula (solid lines). The calculations refer to an ellipsoidal tip of eccentricity *ε*=0.99. It is evident that very good results are obtained for *R*≥10 nm, satisfactory accuracy is obtained for 5 nm≤*R*<10 nm and the accuracy is not satisfactory for *R*<5 nm. It should be noted of course that the error depends on the ratio *φ*/*eFR*. As a summary, we can state that a reliable FN plot can be obtained for the standard range of *F* values 3–10 V nm^{−1}, when *φ*=3–5 eV and *R*>5 nm. For smaller *φ*, the accuracy obviously increases. Furthermore, both the accuracy and usefulness of our method will be validated below when we compare it to experimental results and extract the curvature of a real tip for which scanning electron microscope (SEM) or transmission electron microscope (TEM) pictures exist.

Now the experimentally observable quantity in field emission is the current and not the current density. In order to calculate the current, one has to calculate the current density and perform the corresponding surface integral which is usually written
*J*—as in (4.2)—the magnitude of the current density along the *z*-axis, that is, the maximum one. With ** J**, we denote the vector current density at an arbitrary point of the emitting surface. The effective area

*A*

_{eff}is defined by the above formula and is a very useful quantity used widely in the literature [12,13]. The calculation of the surface integral cannot be done analytically. Extensive numerical calculations that are partly shown in figure 4 indicate that on a logarithmic scale

*A*

_{eff}is a constant with respect to the electric field

*F*for values of

*F*in the typical range of field emission experiments. Therefore, any analysis of any FN plot will be dictated to a very good accuracy by

*J*as shown again in figure 4. Furthermore,

*A*

_{eff}can be combined into a product with the parameter

*σ*that gives the correction to the supply function of materials that deviate from free electron metals [24]. Nevertheless, a set of analytic formulae that give approximately the effective area is given in [25]. Note, however, that—as far as analysing the FN plot—it is sufficient to take

*A*

_{eff}constant.

In figure 5, we plot the FN curves of three different experimental groups [3,4,26] and (Cabrera H, Zanin DA, De Pietro LG, Ramsperger U, Vindigni A, Pescia D. 2014, private communication) together with our theoretical fits to them. The degree of reproduction of experimental results is more than satisfactory. From the traditional fit using the FN equation an experimentalist would obtain the voltage to local field conversion factor *β* (*F*=*β*V) and the product *σA*_{eff}. Using our generalized FN equation (equation (4.3)) we obtain the above two parameters with even greater accuracy and the radius of curvature of the emitter. In particular, the data of H Cabrera, DA Zanin, LG De Pietro, U Ramsperger, A Vindigni, D Pescia (2014, private communication) are accompanied by a SEM picture of the tip on which the experimental group has fitted a hyperboloid of apex radius of curvature *R*∼4 nm. Our extracted radius 4.2 nm is in very good agreement with that. From the data of fig. 12 in [4], we obtain a radius of curvature of 10.3 nm, 4.7 nm higher than the average radius of the field emission array measured from their SEM pictures. Also the extremely small pre-factor *σA*_{eff} seems reasonable given that the emitting material is *N*-doped silicon and much fewer electron states contribute to the current than in a metal. The data in fig. 6b of [26] are not accompanied by SEM or TEM pictures but the extracted radius 6.5 nm falls in the range of *R* values of the tips used by this group, i.e. 4–10 nm [27]. Finally, the data in fig. 3 of [3] are not supported by SEM or TEM pictures of the emitters. We note also that the enhancement factor *β*_{exp} estimated in [26] without using the FN equation and our extracted *β* are in excellent agreement (*β*=0.99*β*_{exp}). If the traditional FN equation were to be used there would be an answer *β*=0.79*β*_{exp}. For our calculations we have used the work function values given by the corresponding experimental papers *φ*=4.05 eV for Si in [4], *φ*=4.35 eV for Mo in [3] and *φ*=4.5 eV for *φ* in H Cabrera, DA Zanin, LG De Pietro, U Ramsperger, A Vindigni, D Pescia (2014, private communication) [26]. In the above analysis, we have assumed that the curvature in the FN plots is primarily due to emitter curvature and other factors such as series resistance and supply saturation play a minor role.

In the limiting case where the curvature is small, Burges *et al.* [28] and later Forbes [29] have introduced the notion of a slope correction factor to the traditional FN equation but so far there is no systematic way of obtaining such a factor. From our theory we obtain—ignoring the *F* variation in the pre-exponential factor
*F* range will suffice. Of course the information on the tip radius *R* will be lost.

## 6. Conclusion

In conclusion, we have derived algebraically a generalization of the FN equation appropriate for curved emitters with radii down to 4–5 nm (equation (4.2)). If it is applied to experimental data the radius of curvature of the emitter may be deduced provided that other factors such as series resistance do not play a significant role in the observed FN plot curvature. Standard parameters that are usually extracted from the FN equation can be deduced with even better accuracy.

## Data accessibility

There are no primary data in this article.

## Authors' contributions

This is a theoretical paper. Both authors contributed equally to conception, mathematical analysis and execution.

## Funding statement

There was no funding assisting this research.

## Competing interests

There are no competing interests.

## Acknowledgements

We are grateful to Prof. D. Pescia for I-V data prior to publication.

## Appendix A: proof of equation (2.3)

Given that the potential is constant at the emitting surface *r*=*f*(*θ*), it is obvious that on this surface

Expanding the above for *n*=1 we get
*n*=2
*θ*=0), from equation (2.1) it is *f*′′(0)=*f*′(0)=0. Thus, if we take (A 2) and (A 3) at this point, all terms multiplied by the derivatives of *f*(*θ*) vanish and we are left with ∂*Φ*/∂*θ*=0 from (A 2) and ∂^{2}*Φ*/∂*θ*^{2}=0 from (A 3). Hence (2.3) is proved.

## Footnotes

- Received October 18, 2014.
- Accepted December 4, 2014.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.