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Renewing the Mainstream Theory of Field and Thermal Electron Emission

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Modern Developments in Vacuum Electron Sources

Part of the book series: Topics in Applied Physics ((TAP,volume 135))

Abstract

Mainstream field electron emission (FE) theory—the theory normally used by FE experimentalists—employs a Sommerfeld-type free-electron model to describe FE from a metal emitter with a smooth planar surface of very large extent. This chapter reviews the present state of mainstream FE theory, noting aspects of the history of FE and thermal electron emission theory. It sets out ways of improving the theory’s presentation, with the ultimate aim of making it easier to reliably compare theory and experiment. This includes distinguishing between (a) emission theory and (b) device/system theory (which deals with field emitter behaviour in electrical circuits), and between ideal and non-ideal device behaviours. The main focus is the emission theory. Transmission regimes and emission current density regimes are discussed. With FE, a method of classifying different FE equations is outlined. With theories that assume tunnelling through a Schottky-Nordheim (SN) (“planar-image-rounded”) barrier, a careful distinction is needed between the barrier form correction factor ν (“nu”) and the special mathematical function v (“vee”). This function v is presented as dependent on the Gauss variable x. The pure mathematics of v(x) is summarised, and reasons are given for preferring the use of x over the older convention of using the Nordheim parameter y [=+√x]. It is shown how the mathematics of v(x) is applied to wave-mechanical transmission theory for basic Laurent-form barriers (which include the SN barrier). A brief overview of FE device/system theory defines and discusses different auxiliary parameters currently in use, outlines a preferred method for characterising ideal devices when using FN plots and notes difficulties in characterising non-ideal devices. The chapter concludes by listing some of the future tasks involved in upgrading FE science.

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Notes

  1. 1.

    Prof. Marshall Stoneham (now deceased), a former President of the UK Institute of Physics, thought that some of the most difficult unsolved problems in theoretical physics were in field electron emission (private communication to the author, 2001).

  2. 2.

    Reference [55] contains a typographical error, in that the numerical value in his equation (5) should be 6.97 × 106, as given correctly in Table 9.1 later in the paper. The modern value is slightly lower, as can be shown from (9.12).

  3. 3.

    A well-behaved barrier has no regions inside it where M(z) is negative.

  4. 4.

    ‘Corpuscle’ was J.J. Thomson’s original term for what we now call the electron.

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Authors and Affiliations

  1. Advanced Technology Institute & Department of Electrical and Electronic Engineering, University of Surrey, Guildford, GU2 7XH, UK

    Richard G. Forbes

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  1. Richard G. Forbes
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Editors and Affiliations

  1. Aachen, Nordrhein-Westfalen, Germany

    Georg Gaertner

  2. FMB/IFQ, Otto von Guericke University Magdeburg, Magdeburg, Germany

    Wolfram Knapp

  3. Advanced Technology Institute and Department of Electronic Engineering, University of Surrey, Guildford, Surrey, UK

    Richard G. Forbes

Appendices

Appendix 9.1. Fundamental and Universal Constants Used in Field Emission Theory

Table 9.4 presents values of the fundamental constants used in FE theory, both in SI units and in the ‘field emission customary units’ in which they are often given. Both sets of units are defined in the context of ISQ-based equations, and the normal rules of quantity algebra apply to their values. These customary units take the electronvolt, rather than the joule, as the unit of energy, and normally measure field in V/nm and charge in units equal to the elementary positive charge. Their use greatly simplifies basic calculations when energies are measured in eV and fields in V/nm, as is often the case. The numerical values of the constants, measured in both sets of units, are based on the values of the fundamental constants given in October 2019 on the website of the US National Institute of Standards and Technology (NIST) (see http://physics.nist.gov/constants). These values incorporate the SI system changes made in May 2019.

Table 9.4 The May 2019 values of the electronvolt (eV) and fundamental constants used in emission physics, given in SI units (either exactly or to 11 significant figures), and in field emission customary units (to 7 significant figures). Asterisks indicate constants where (since May 2019) the value in SI units is specified exactly
Full size table

Table 9.5 gives the values, in FE customary units, of many universal constants, and combinations thereof, that are used in field emission theory. Where necessary, the tabulated values take the May 2019 changes into account. The third column in Table 9.5 is interesting, because it shows that several of the universal constants used in field emission are related in a relatively simple way to the constants κe and zS, which are very basic universal constants that appear centrally in the Schrödinger equation and in statistical mechanics, respectively.

Table 9.5 Some fundamental constants used in field emission and related theory. Values are given in field emission customary units
Full size table

Appendix 9.2. High-Precision Formulae for v(x) and u(x)

This Appendix provides formulae for estimating high-precision values of the FE special mathematical functions v(x) and u(x) [≡ –dv/dx] (and hence of all the FE special mathematical functions). The parameter x is the Gauss variable. The two functions are estimated by the following series, derived from those given in [86] by replacing the symbol l' by the symbol x now preferred, and by slightly adjusting the form of the resulting series for v(x) (without changing its numerical predictions)

$$\text{v}(x) \, \cong \, (1 - x)\left( {1 + \sum\limits_{i = 1}^{4} {p_{i} x^{i} } } \right) + x\ln x\sum\limits_{i = 1}^{4} {q_{i} x^{i - 1} }$$
(9.86)
$$\text{u}(x) \, \cong \, \text{u}_{1} - (1 - x)\sum\limits_{i = 0}^{5} {s_{i} x^{i} } - \ln x\sum\limits_{i = 0}^{4} {t_{i} x^{i} }$$
(9.87)

Values of the constant coefficients pi, qi, si and ti are shown in Table 9.6.

Table 9.6 Numerical constants for use in connection with (9.86) and (9.87)
Full size table

It can be seen that at the values x = 0, 1, formula (9.86) generates the correct values v(0) = 1, v(1) = 0, and that at x = 1, formula (9.87) generates the correct value u(1) = u1 = 3π/8√2.

Equation (9.86) mimics the form of the lower-order terms in the (infinite) exact series expansion for v(x) [57], but the coefficients in Table 9.6 have been determined by numerical fitting to exact expressions for v(x) and u(x) (in term of complete elliptic integrals) evaluated by the computer algebra package MAPLE™. In the range 0 ≤ x ≤ 1, v(x) takes values lying in the range 1 ≥ v(x) ≥ 0, and the maximum error associated with formulae (9.86) and (9.87) is less than 8×10−10 [57]. In Murphy-Good-type FE equations, these formulae are applied by setting x = fC.

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Forbes, R.G. (2020). Renewing the Mainstream Theory of Field and Thermal Electron Emission. In: Gaertner, G., Knapp, W., Forbes, R.G. (eds) Modern Developments in Vacuum Electron Sources. Topics in Applied Physics, vol 135. Springer, Cham. https://doi.org/10.1007/978-3-030-47291-7_9

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