^{1,a)}and Jonathan H. B. Deane

^{2}

### Abstract

The Fowler–Nordheim-type (FN-type) equation developed by Murphy and Good [Phys. Rev.102, 1464 (1956)] contains in its exponent a correction function , best called the “principal Schottky–Nordheim (SN) barrier function.” In the last 50 years a large number of different approximations have been developed for . This article reformulates these approximations to be functions of the scaled barrier field rather than the Nordheim parameter , and then compares them with the approximations and 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 . However, particularly with the first formula, some older approximations are superior over limited ranges of 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 (rather than ) 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.

I. BACKGROUND

II. FORBES 2006 AND RELATED APPROXIMATIONS

III. COMPARISON WITH OLDER APPROXIMATIONS

IV. ANALYSIS OF FOWLER–NORDHEIM PLOTS

A. Introduction and background theory

B. FN-plot analysis: General

C. Tangent method

D. Substitution and linearization methods

E. Chord method

F. Commentary

V. SUMMARY AND DISCUSSION

## Figures

Graphical comparison of various approximations for the principal Schottky–Nordheim barrier function , over the range , classified according the maximum magnitude of error involved: (a) ; (b) ; (c) . In all cases, the vertical axis represents the absolute difference between the approximate result and the exact result.

Graphical comparison of various approximations for the principal Schottky–Nordheim barrier function , over the range , classified according the maximum magnitude of error involved: (a) ; (b) ; (c) . In all cases, the vertical axis represents the absolute difference between the approximate result and the exact result.

(Color online) Schematic illustrating how, in the tangent method, a formula relating the quantities and is derived. The symbol denotes the term . The line is the tangent (taken at point ) to the theoretical curve . (For clarity, the curvature of is much exaggerated). Point marks the voltage at which the correction factor in the chosen FN-type equation becomes zero. At this voltage, the height of the tunneling barrier and the exponent in the chosen FN-type equation become zero, and the value of predicted by the FN-type equation is . Line represents the theoretical calculation of (i.e., for ), using the chosen FN-type equation.

(Color online) Schematic illustrating how, in the tangent method, a formula relating the quantities and is derived. The symbol denotes the term . The line is the tangent (taken at point ) to the theoretical curve . (For clarity, the curvature of is much exaggerated). Point marks the voltage at which the correction factor in the chosen FN-type equation becomes zero. At this voltage, the height of the tunneling barrier and the exponent in the chosen FN-type equation become zero, and the value of predicted by the FN-type equation is . Line represents the theoretical calculation of (i.e., for ), using the chosen FN-type equation.

## Tables

To show, for the formulas listed, the maximum magnitudes of the absolute errors (Abs) and relative errors (Rel) in the ranges of indicated.

To show, for the formulas listed, the maximum magnitudes of the absolute errors (Abs) and relative errors (Rel) in the ranges of indicated.

Values of the Schottky–Nordheim barrier functions (for definitions, see Ref. 4), tabulated as functions of or . [“nd” is not usefully defined for .]

Values of the Schottky–Nordheim barrier functions (for definitions, see Ref. 4), tabulated as functions of or . [“nd” is not usefully defined for .]

Comparison of the Charbonnier and Martin (CM) and Spindt *et al.* (SBHW) approximations with the corresponding exact results for the substitution method as used in FN-plot theory. The correction factor implied for the chord method (for ) is .

Comparison of the Charbonnier and Martin (CM) and Spindt *et al.* (SBHW) approximations with the corresponding exact results for the substitution method as used in FN-plot theory. The correction factor implied for the chord method (for ) is .

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