Comparison of approximations for the principal Schottky–Nordheim barrier function v(f), and comments on Fowler–Nordheim plots

Forbes, Richard G.; Deane, Jonathan H. B.

doi:doi:10.1116/1.3363858

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Journal of Vacuum Science & Technology B / Volume 28 / Issue 2 / PAPERS FROM THE 22nd INTERNATIONAL VACUUM NANOELECTRONICS CONFERENCE / Fundamental

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J. Vac. Sci. Technol. B 28, C2A33 (2010); http://dx.doi.org/10.1116/1.3363858 (10 pages)

Comparison of approximations for the principal Schottky–Nordheim barrier function v(f), and comments on Fowler–Nordheim plots

Richard G. Forbes¹ and Jonathan H. B. Deane²

¹Advanced Technology Institute, Faculty of Engineering and Physical Sciences, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom
²Department of Mathematics, Faculty of Engineering and Physical Sciences, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom

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(Published online 31 March 2010)

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The Fowler–Nordheim-type (FN-type) equation developed by Murphy and Good [Phys. Rev. 102, 1464 (1956) ] contains in its exponent a correction function v, best called the “principal Schottky–Nordheim (SN) barrier function.” In the last 50 years a large number of different approximations have been developed for v. This article reformulates these approximations to be functions of the scaled barrier field f rather than the Nordheim parameter y, and then compares them with the approximations v(f) ≈ 1−f+(1/6)f ln f and v(f) ≈ 1−f+0.1715f ln f recently reported. It is confirmed, as expected, that these new formulas outperform older approximations of equivalent complexity when the comparison is made over the range 0 ≤ f ≤ 1. However, particularly with the first formula, some older approximations are superior over limited ranges of f because they have been designed to perform well over these limited ranges. The article also presents an updated and fuller account of the theory of FN-plot analysis and an updated tabulation of values of the SN-barrier functions (also called field emission elliptic functions). Both updates use f (rather than y) as the main auxiliary variable. The limitations of the existing FN-plot theory are clarified; this reinforces the case for the development of higher quality measurements of current-voltage characteristics and of new methods of data analysis that may be able to supersede the use of FN plots.

Article Outline

BACKGROUND
FORBES 2006 AND RELATED APPROXIMATIONS
COMPARISON WITH OLDER APPROXIMATIONS
ANALYSIS OF FOWLER–NORDHEIM PLOTS
1. Introduction and background theory
2. FN-plot analysis: General
3. Tangent method
4. Substitution and linearization methods
5. Chord method
6. Commentary
SUMMARY AND DISCUSSION

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KEYWORDS and PACS

Keywords

electron field emission, tunnelling

PACS

79.70.+q

Field emission, ionization, evaporation, and desorption
73.40.Gk

Tunneling

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History

Received 11 September 2009
Accepted 22 February 2010
Published online 31 March 2010

Digital Object Identifier

http://dx.doi.org/10.1116/1.3363858

PUBLICATION DATA

ISSN

1071-1023 (print)

1520-8567 (online)

Publisher

American Vacuum Society

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