One may estimate the effective emission area from Fowler-Nordheim equation that reads as, I = 1.5 × 10−6 (A/φ) (βV/d)2 exp (10.4/√φ) exp (−6.44 × 109φ3/2 d/βV),

given A is the effective emitting area in m2, φ the work function in electronvolts, d the cathode-to-anode spacing in meters and β the field enhancement factor in dimensionless units that determines the virtual electric field at the apex of the emitter tip such that E = β(V/d). Thus, if one has the work function of the emitting material and field enhancement factor obtained from F-N plot, [ln(I) vs. 1/V] of a given measurement, the effective emitting area is readily estimated from the above equation.

may be these articles can help you:

1. Hailong Hu, Dian Zhang, Yuhui Liu, Weiming Yu, Tailiang Guo. Highly enhanced

field emission from CuO nanowire arrays by coating of carbon nanotube network films. // Vacuum. 2015. 115. P. 70-74.

2. Xiaomeng Song, Jun Chen. Non-crystallization and enhancement of field emission of cupric oxidenanowires induced by low-energy Ar ion bombardment. //  Applied Surface Science 329 (2015) 94–103.

One may estimate the effective emission area from Fowler-Nordheim equation that reads as, I = 1.5 × 10−6 (A/φ) (βV/d)2 exp (10.4/√φ) exp (−6.44 × 109φ3/2 d/βV),

given A is the effective emitting area in m2, φ the work function in electronvolts, d the cathode-to-anode spacing in meters and β the field enhancement factor in dimensionless units that determines the virtual electric field at the apex of the emitter tip such that E = β(V/d). Thus, if one has the work function of the emitting material and field enhancement factor obtained from F-N plot, [ln(I) vs. 1/V] of a given measurement, the effective emitting area is readily estimated from the above equation.

The problem of estimating the effective area (A) in field emission is quite complex. A detailed summary of your possible methods is found here [1]. However, in nanoscale field emitters the problem gets even more difficult as the F-N equation is not precise [2] (it is constructed for planar emitters). Furthermore, any possible calculation of the current via standard methods has an uncertainty pre-factor of about 4 orders of mgnitude [2] which makes it impossible to adequately determine A using only experimental I-V data.

I think that what you need is detailed theoretical calculations. I have developed various computational tools that could be helpful. If you are interested, please send me a PM to discuss with more detail your problem and possible solutions.

[1] R.G. Forbes, J. Vac. Sci. Technol. B 17, 526 (1999); http://dx.doi.org/10.1116/1.590588

[2] A. Kyritsakis and J.P. Xanthakis, Proc. R. Soc. A 471: 20140811 (2015)

http://dx.doi.org/10.1098/rspa.2014.0811 

[3]A. Modinos, Solid-State Electron. 45, 809–816. (2001)

http://dx.doi.org/10.1016/S0038-1101(00)00218-5

Thank you all for your reply. This is exactly what I was looking for.

All answers presented to this question are reasonable and correct. But why don't you try measuring the emitter size using SEM? You can also count the number of emitters per unit area of your substrate. The emission current is that contributed by all possible activated sites under certain applied voltage. The effective emission area is the total sum of areas of individual emitters in the nanostructured film. You have to figure out how to calculate the contribution of each individual emitter to the total emission current. 

Dear Erich,

Some answers to your question have been published in :

Scanning anode field emission microscopy analysis for studies of planar cathodes

V. Semet, R. Mouton and Vu Thien Binh

J. Vac. Sci. Technol. B 23(2) Mar/Apr 2005; pp 671-675.

Best Regards.

A traveller in the remote reaches of a distant land asked a local inhabitant the best way to get back to the country's capital. The answer was:  "Well, sir, if I was you, I wouldn't start from here". Unfortunately, the best answer to Erich's question may be rather similar.  What Erich asks to know is not really the effective emission area, but the "effective active macroscopic area A" under the sphere. (The effective emission area under the sphere will probably be less than this by a factor of 1 000 000 or more). The paper by Semet et al. shows (correctly) that working out A––and its possible variation with voltage and with sphere-film separation––is a VERY complicated business. These sphere-on-plane systems are quite good for investigating film variability across the surface of a film, but not so good for making accurate quantitative estimates of (average) film characteristics, which we presumably also wish to know. The best experimental answer to measuring these is to construct or borrow a separate rig that has a planar anode that can be set a known distance away from the cathode, and use a planar substrate onto which a cathode of KNOWN macroscopic area has been deposited (measure it by a microscope technique if needed). This approach removes nearly all the complications described by Semet et al.

However, a second set of complications then comes into play. The usual types of Fowler-Nordheim plot analysis theory (and the simpler approximate versions often used) only work  properly if the emission situation is effectively what I describe as "orthodox" [1].  Some film emitters are orthodox; some are not. Known reasons for non-orthodoxy include series resistance in the measuring circuit (e.g., at the point where a CNT joins the substrate), and Maxwell stress that pulls carbon fibres upright and makes their field enhancement factors voltage-dependent. Ref. [1] describes an orthodoxy test, and one can attempt to apply it on results from the sphere/plane system (though the results may be more difficult to interpret than in a planar system).  If your film turns out to be non-orthodox, then very probably it is currently impossible to determine effective emission area accurately, because we don't yet have the necessary scientific knowledge. Instead, simply looking at the emission pattern on a phosphor-coated planar anode may give you some guide as to film quality. if orthodox, the usual FN-plot analysis methods apply [provided that the dominant emitters are not too small (apex radius greater than about 10-20 nm). Feel free to contact me directly by e-mail if you wish to go into more detail. 

[1] R.G. Forbes, Proc. R. Soc. Lond. A 469 (2013) 20130271.

Dear Erich,

I'm sending you an article of mine specifically about the effective area of emission from capped and open single CNTs in a "pseudo" array. I hope it is useful for you. This article is still being considered for publication.

Yours sincerely, Fernando.