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

^{2}, Andreas Fischer

^{3,b)}and Marwan S. Mousa

^{3}

### Abstract

This technical note relates to the theory of cold field electron emission (CFE). It starts by suggesting that, to emphasize common properties in relation to CFE theory, the term "Lauritsen plot" could be used to describe all graphical plots made with the reciprocal of barrier field (or the reciprocal of a quantity proportional to barrier field) on the horizontal axis. It then argues that Lauritsen plots related to barrier strength (*G*) and transmission probability (*D*) could play a useful role in discussion of CFE theory. Such plots would supplement conventional Fowler–Nordheim (FN) plots. All these plots would be regarded as particular types of Lauritsen plot. The Lauritsen plots of –*G* and ln*D* can be used to illustrate how basic aspects of FN tunneling theory are influenced by the mathematical form of the tunneling barrier. These, in turn, influence local emission current density and emission current. Illustrative applications used in this note relate to the well-known exact triangular and Schottky–Nordheim barriers, and to the Coulomb barrier (i.e., the electrostatic component of the electron potential energy barrier outside a model spherical emitter). For the Coulomb barrier, a good analytical series approximation has been found for the barrier-form correction factor; this can be used to predict the existence (and to some extent the properties) of related curvature in FN plots.

Andreas Fischer thanks the Alexander von Humboldt foundation for a Feodor Lynen fellowship and Mu'tah University for hospitality.

I. INTRODUCTION

II. BACKGROUND THEORY

III. APPLICATIONS

A. Barrier strengths of the exact triangular and Schottky–Nordheim barriers

B. Slope and intercept correction functions for the SN barrier

C. Plot curvature in the deep tunneling regime

D. Breakdown of the planar-emitter deep-tunneling approximations

E. Comparison of theoretical high-field (low barrier-strength) behaviors

IV. DISCUSSION

### Key Topics

- Electron transfer
- 37.0
- Tunneling
- 21.0
- Field emission
- 20.0
- Current density
- 9.0
- Probability theory
- 5.0

## Figures

Comparison of barrier-strength dependences on inverse scaled barrier field, for the exact triangular (ET) and Schottky–Nordheim (SN) barrier models. Line PL is drawn parallel to line ET, a distance *η* above it. Curve SN starts at the reference point "R," at (1,0).

Comparison of barrier-strength dependences on inverse scaled barrier field, for the exact triangular (ET) and Schottky–Nordheim (SN) barrier models. Line PL is drawn parallel to line ET, a distance *η* above it. Curve SN starts at the reference point "R," at (1,0).

To illustrate the relationships between the SN-barrier correction functions (*f*), *s*(*f*) and *r* _{2012}(*ϕ*,*f*), for the specific values *ϕ* = 4.50 eV (*η* ≈ 4.637), *f* = 0.2. The line T(5) is the tangent to curve SN at point "P," at which *f* ^{−1} = 5. The slopes of lines ET, V, and T(5) are, respectively, −*η*, −*η·* (0.2), and −*η·s*(0.2), and *G* _{F} ^{ET} = 5*η*.

To illustrate the relationships between the SN-barrier correction functions (*f*), *s*(*f*) and *r* _{2012}(*ϕ*,*f*), for the specific values *ϕ* = 4.50 eV (*η* ≈ 4.637), *f* = 0.2. The line T(5) is the tangent to curve SN at point "P," at which *f* ^{−1} = 5. The slopes of lines ET, V, and T(5) are, respectively, −*η*, −*η·* (0.2), and −*η·s*(0.2), and *G* _{F} ^{ET} = 5*η*.

To show how the barrier-form correction factor for the Coulomb barrier (*ν* _{F} ^{CL}) varies with Edgcombe's parameter *υ* ("upsilon"), defined by Eq. (18) .

To show how the barrier-form correction factor for the Coulomb barrier (*ν* _{F} ^{CL}) varies with Edgcombe's parameter *υ* ("upsilon"), defined by Eq. (18) .

To show how, for a Coulomb barrier, the barrier strength for state F varies with inverse barrier field, for the work-function value 4.50 eV, and the emitter radii shown. For sufficiently small model radii, the curvature in the Lauritsen plot is detectable.

To show how, for a Coulomb barrier, the barrier strength for state F varies with inverse barrier field, for the work-function value 4.50 eV, and the emitter radii shown. For sufficiently small model radii, the curvature in the Lauritsen plot is detectable.

To show how the transmission probability *D* _{F} varies with inverse barrier field. (a) and (b) show results for an exact triangular barrier of height 4.50 eV, as predicted by the original Fowler–Nordheim formula (FN) and by an exact treatment (FD). (c) and (d) show results for a Schottky–Nordheim barrier of zero-field height 4.50 eV, as predicted by the usual simple-JWKB treatment (JWKB) and by the Kemble approximation (Kem). For each barrier, the left-hand figure shows the high-field (low *F* _{L} ^{−1}) region in greater detail.

To show how the transmission probability *D* _{F} varies with inverse barrier field. (a) and (b) show results for an exact triangular barrier of height 4.50 eV, as predicted by the original Fowler–Nordheim formula (FN) and by an exact treatment (FD). (c) and (d) show results for a Schottky–Nordheim barrier of zero-field height 4.50 eV, as predicted by the usual simple-JWKB treatment (JWKB) and by the Kemble approximation (Kem). For each barrier, the left-hand figure shows the high-field (low *F* _{L} ^{−1}) region in greater detail.

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