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Illustrating field emission theory by using Lauritsen plots of transmission probability and barrier strength

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 lnD can be used to illustrate how basic aspects of FN tunnelling theory are influenced by the mathematical form of the tunnelling 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.


[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 lnD can be used to illustrate how basic aspects of FN tunnelling theory are influenced by the mathematical form of the tunnelling 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.

I. INTRODUCTION
Fowler-Nordheim-type (FN-type) equations are a family of approximate equations that describe cold field electron emission (CFE) from a metal conduction band, for emitters that are "not too sharp" (tip radius of order 10 nm, or greater). FN-type equations are also used as empirical fitting equations for all types of CFE, but the interpretation of extracted results may be problematic if the emission situation is not orthodox [1][2][3] .
The FN-type equations most suited to discuss basic theory give the local emission current density J L in terms of the local work-function φ and the local barrier field F L . As is well known, for a planar emitter, a FN plot of type [ln{J L /F L 2 } vs F L -1 ] is predicted to be nearly straight; for this reason, FN plots are widely used to represent and interpret experimental CFE data. One can also make theoretical plots of the form [lnD F vs F L -1 ] and/or [-G F vs F L -1 ], where D F and G F are the tunneling probability and barrier strength defined below. This technical note suggests that such plots can be a useful (and perhaps under-appreciated) way of illustrating theoretical effects that relate specifically to the barrier form and the tunneling process, rather than partly to the summation over electron states (and, for currents, the integration over the emitting surface) used to derive FN-type equations.
The note's structure is as follows. Section II provides background theory; Section III illustrates the use of Lauritsen plots relating to tunneling probability and barrier strength, by describing five applications; and Section IV provides discussion. The usual electron emission convention is followed that fields, current densities and related quantities are treated as positive, even though negative in classical electromagnetism. In particular, the basic symbol F denotes a positive quantity that is the negative of electrostatic field as used in classical electromagnetism.
To help discussion, this note introduces a special name for the class of data plots where the horizontal axis shows the reciprocal of either local barrier field F L or a parameter proportional to F L .
This plotting method was first used by  , and was the critical breakthrough that led Fowler where D F is the tunnelling probability for a Fermi-level emitter electron moving "forwards" (normal to the emitter surface). This electron is said to be "in state F", and sees a barrier of zero-field height φ.
The subscript "F" is used to label quantities associated with this electron state or the related tunnelling barrier. Z F is the "effective supply" for state F (i.e., the effective incident current density, for electrons approaching the emitter surface from the inside), and is found by summation over all occupied electron states.
Form (1) was originally introduced, in Refs 10 and 11, for free-electron models. For such models Z F is easily calculated, in particular for planar or large-radius emitters 10 , by the evaluation of appropriate integrals. However, because both D F and J L can be defined at all emitter surface positions, form (1) in fact applies at any surface position on an emitter of any shape, made from any material.
The form of Z F will depend on the circumstances, and may sometimes be very difficult to calculate.
A technically complete FN-type equation for the emission current i can then be written where J C is a characteristic value of J L , and A n is the related notional emission area 12 . As where g e [≈ 10.24624 eV -1/2 nm -1 ] is a universal constant called 14,15 the "JWKB constant for an electron". M(z) is the motive energy that determines electron motion, and is given by M=U-E n , where (in the one-dimensional Schrödinger equation) U is the potential energy and E n is the total-energy component in the z-direction. The integral is taken "across the barrier", i.e., over the range of z where M(z)≥0. For tunneling, G can also be identified with the quantity 2|K| used by Fröman and Fröman 16 .
In past literature, G has been called the "Gamow exponent", the "WKB exponent" and the "JWKB exponent". We now prefer the physically more descriptive name "barrier strength"; thus G F is the barrier strength for state F. For consistency, we call Eq. (3) the "barrier-strength integral".
For an arbitrary, well-behaved, "general barrier" (GB), G F and Z F can be written in the forms where a and b are the usual universal FN constants 15  is the effective-electron-supply term used 10 in the elementary equation. ν F GB is a correction factor related to the mathematical form of the general barrier, and λ L GB is a local pre-exponential correction factor related to electron supply and summation over electron states; these factors correct the exponent and pre-exponential, respectively, of the elementary equation.
For the general barrier, writing Eq. (1) in so-called FN coordinates, yields The term ln P F GB is usually in the range {-1< ln P F GB <1} and is slowly varying; thus, barrier effects on FN plots are mainly due to the barrier-strength term −G F GB .

A. The barrier strengths of the exact triangular and Schottky-Nordheim barriers
The first application compares the strengths of the two most commonly used barrier models: the exact triangular (ET) barrier, which has M ET = φ-eF L z; and the Schottky-Nordheim (SN) barrier, which has M SN = φ-eF L z-e 2 /16πε 0 z, where e is the elementary positive charge, ε 0 is the electric constant, and z is distance measured from the emitter's electrical surface. This comparison has been made before 17 , but is presented differently here. It uses the scaled barrier field f and the parameter η In scaled form, the barrier strength G F ET for an ET barrier of height φ is The barrier strength G F SN for an SN barrier of zero-field height φ can be written where the simple good expansion 17 v( f ) ≈ 1− f + 1 6 f ln f has been used for the principal SN barrier function 17 v(f). Hence the difference Δ(−G F ) between the strengths of the SN and ET barriers is In Fig. 1, the straight line ET shows how -G F ET varies with f -1 , and the slightly curved line SN shows this for -G F SN . Point "R" is the reference point (1,0) at which curve SN starts, and line PL is a straight line drawn parallel to line ET, a vertical distance η above it. Line PL passes through point R.
For these values, Table I (6)) be written formally as with where λ L SN is the local pre-exponential correction factor for the SN barrier. A merit of the new function r 2012 is that it allows the equation for the tangent to Eq. (11) to be written where r is distance from the sphere centre, and the barrier field F L is defined as the field at r=R.
Edgcombe 19 has evaluated the related barrier-strength integral. In terms of a dimensionless parameter υ ("upsilon") (his x U ), called here "Edgcombe's parameter" and defined by his result can be written where ν F CL is the barrier-form correction factor for the Coulomb barrier.
This Coulomb-barrier tunneling problem was first addressed by Gamow 20  Taylor expansion of G F CL about υ=0, using MAPLE, yields (after some algebraic manipulation) where the symbol O(υ 4 ) stands for terms of order υ 4 and higher. In the range 0≤υ≤0.5, this expansion has an accuracy of better than 4%. As far as we are aware, this is the first time that this particular series expansion has been reported.
Obviously, in the limit of very large model radius R (very small υ) this result goes over into the result (ν F ET =1) for the exact triangular barrier, which represents the electrostatic component of the electron potential energy outside a planar emitter.
In Eq. (17), the leading "unity" would generate a straight line (representing -G F ET ) when -G F is plotted against υ or (for constant φ and R) against Unlike the case of the SN barrier ( Fig. 1), where (a) the exact plot diverges upwards from line "PL" (and hence from line "ET") and (b) curvature of the exact plot is so small as to be hardly detectable, the plot curvature associated with a Coulomb barrier becomes noticeable when the model radius R gets below a value between 20 and 50 nm. Another difference between the two cases is that, for the SN barrier, the curvature gets greater towards the left-hand side of the plot (the high-field, low-barrier-strength side), whereas for the Coulomb barrier the curvature gets greater towards the right-hand side (the high-barrier-strength side).
We cut off Lauritsen barrier-strength plots for G F values greater than 30, on the grounds that, in most experiments, the corresponding currents would be too small to detect.
These results provide useful qualitative understanding of effects likely to occur with small-radius real emitters, but caution is needed in applying them quantitatively. This is because the sphericalemitter model tends to lose its validity as R becomes smaller than about 10 to 20 nm, due (a) to the influence of the emitter shank on the barrier form; and (b) to the possible onset of quantum confinement effects 21 . For the more realistic emitter shapes, with a shank and quasi-spherical endregion, the potential distribution associated with a sphere is no longer an adequate approximation to the solution of Laplace's equation for the more realistic emitter, when R drops below about 10 to 20 nm. This point has previously been made by Edgcombe 22 .
Another point is that formula (17) performs quantitatively well only if F L >2φ/eR; however, this is often not a practical difficulty. For φ= 4.5 eV and R=10 nm, this condition implies that F L should be greater than about 1 V/nm. Thus, for emitter radii where the model is physically adequate, formula 11 (17) should usually be mathematically adequate for F L -values of practical interest.

D. Breakdown of the planar-emitter deep-tunneling approximations
For both the ET and SN barriers, the well-known formulas for tunneling probability D F are approximations valid in the deep-tunneling transmission regime, i.e., when the barrier strength is positive and sufficiently large.
For the ET barrier, there exists an exact general formula for D F that is known to be mathematically correct 14,23 . This has the mathematical form 14 where A, B, A' and B' are values of the Airy functions and their derivatives, evaluated at a defined (field-dependent) value of their argument, and ω F is a parameter that depends on the barrier field F L and on the Fermi energy (i.e., the kinetic energy K F of an electron in state F). Equation (18) where FN's tunneling pre-factor P F FN is defined by the term in square brackets in Eq. (19). This formula is an asymptotic approximation to Eq.  there exists a semi-classical approximation formula for transmission probability D F that is considered to be mathematically reliable over most (perhaps all) fields of practical interest to CFE. This was derived from the work of Fröman and Fröman 16 and can be put in the mathematical form For most barriers, exact analytical expressions for P F in terms of the established functions of mathematical physics are not known and probably do not exist. Mayer has developed 25-27 numerical procedures able to calculate P F . However, to evaluate transmission probability D F SN for the SN barrier, it is more common to use the Kemble approximation 28,29 (later derived in a different mathematical way by Miller and Good 30 ). In effect, this sets P F =1 in Eq. (20) and (for the SN barrier) takes For the SN barrier, wave-mechanical flyover (transmission over the barrier) corresponds to the scaled-barrier-field range 1<f≤∞. The principal SN barrier function v(f) is continuous through the value f=1, and remains well-defined 32,33 for f>1 (although the approximation given earlier for v(f) increasingly loses its accuracy above f=1, and performs very poorly above about f=4). As f increases above 1, v(f) becomes increasingly negative 32 . It follows that the barrier strength G F SN remains mathematically well-defined for f>1, but becomes increasingly negative as f increases. In the high-

IV. Discussion
This technical note has given five illustrations of the use of Lauritsen plots, mainly of the natural logarithm of the transmission probability D or of (the negative of) the barrier strength G. We emphasize that these plots are not FN plots as conventionally understood, because they do not explicitly involve a current or a current density.

15
To make these Lauritsen plots directly relevant to FN-type equations, we have used plots that relate to barriers with zero-field height equal to the local work function φ. However, these Lauritsen plots could equally well be used to illustrate the characteristics of barriers of any (positive) height.
These Lauritsen plots illustrate, in a convenient way, the direct effects of barrier form (i.e., the form of M(z)) on barrier strength and transmission probability. Via Eqs (1) and (2), they also indicate how barrier and transmission effects influence field dependences in emission current density and current. (In simple models, FN plots are parallel to the corresponding Lauritsen plot of lnD F ; in more realistic models, additional curvature will often be introduced.) For practical field electron emission, the most interesting results are probably those related to plot curvature for a spherical emitter. The Coulomb-barrier results here clearly illustrate that one possible cause of curvature in conventional FN plots is a barrier-form effect: if the field fall-off across the width of a real emitter barrier is sufficiently large, then a corresponding conventional FN plot is expected to be curved, particularly so at the low-field (high barrier strength) side of the plot.
Because an algebraic series expansion has been found for the barrier-form correction factor ν F CL for the Coulomb barrier, there seems a reasonable chance that future work may be able to develop satisfactory algebraic expressions for the related slope and (new-type) intercept correction factors, although (due to the onset of shank effects) these would not be accurate for practical emitters of apex radius less than around 20 nm.
More generally, comparisons can be made between a Lauritsen plot of ln{D F } and the corresponding FN plot (for example, a plot involving ln{J M /F M 2 }, where F M is macroscopic field and J M is macroscopic current density). Differences in shape between the plots would suggest/confirm that factors other than transmission probability were affecting the experimental plot shape, and that additional investigation might be helpful. Detailed discussion of applications of this kind is beyond the scope of the present paper.
In conclusion, we believe that the illustrations above of the uses of Lauritsen plots of lnD and -G and have demonstrated the merits of these plots in discussions of CFE theory. We also believe that other forms of Lauritsen plot, such as those used to display the properties of the SN barrier functions in Figs (2c) and (2d) of Ref. 17, will prove useful in future work. A long-term potential advantage is