Field electron microscopy and related basics

As already indicated, the early experimental work on field electron emission (1910–1920)^[7] was driven by Lilienfeld's desire to develop miniaturized X-ray tubes for medical applications. However, it was too early for this technology to succeed.

After Fowler–Nordheim theoretical work in 1928, a major advance came with the development in 1937 by Erwin W. Mueller of the spherical-geometry field electron microscope (FEM)^[23] (also called the "field emission microscope"). In this instrument, the electron emitter is a sharply pointed wire, of apex radius r. This is placed, in a vacuum enclosure, opposite an image detector (originally a phosphor screen), at a distance R from it. The microscope screen shows a projection image of the distribution of current-density J across the emitter apex, with magnification approximately (R/r), typically 10⁵ to 10⁶. In FEM studies the apex radius is typically 100 nm to 1 μm. The tip of the pointed wire, when referred to as a physical object, has been called a "field emitter", a "tip", or (recently) a "Mueller emitter".

When the emitter surface is clean, this FEM image is characteristic of: (a) the material from which the emitter is made: (b) the orientation of the material relative to the needle/wire axis; and (c) to some extent, the shape of the emitter endform. In the FEM image, dark areas correspond to regions where the local work function φ is relatively high and/or the local barrier field F is relatively low, so J is relatively low; the light areas correspond to regions where φ is relatively low and/or F is relatively high, so J is relatively high. This is as predicted by the exponent of Fowler–Nordheim-type equations [see eq. (30) below].

The adsorption of layers of gas atoms (such as oxygen) onto the emitter surface, or part of it, can create surface electric dipoles that change the local work function of this part of the surface. This affects the FEM image; also, the change of work-function can be measured using a Fowler–Nordheim plot (see below). Thus, the FEM became an early observational tool of surface science.^[24][25] For example, in the 1960s, FEM results contributed significantly to discussions on heterogeneous catalysis.^[26] FEM has also been used for studies of surface-atom diffusion. However, FEM has now been almost completely superseded by newer surface-science techniques.

A consequence of FEM development, and subsequent experimentation, was that it became possible to identify (from FEM image inspection) when an emitter was "clean", and hence exhibiting its clean-surface work-function as established by other techniques. This was important in experiments designed to test the validity of the standard Fowler–Nordheim-type equation.^[27][28] These experiments deduced a value of voltage-to-barrier-field conversion factor β from a Fowler–Nordheim plot (see below), assuming the clean-surface φ–value for tungsten, and compared this with values derived from electron-microscope observations of emitter shape and electrostatic modeling. Agreement to within about 10% was achieved. Only very recently^[29] has it been possible to do the comparison the other way round, by bringing a well-prepared probe so close to a well-prepared surface that approximate parallel-plate geometry can be assumed and the conversion factor can be taken as 1/W, where W is the measured probe-to emitter separation. Analysis of the resulting Fowler–Nordheim plot yields a work-function value close to the independently known work-function of the emitter.

Field electron spectroscopy (electron energy analysis)

Energy distribution measurements of field-emitted electrons were first reported in 1939.^[30] In 1959 it was realized theoretically by Young,^[31] and confirmed experimentally by Young and Mueller^[32] that the quantity measured in spherical geometry was the distribution of the total energy of the emitted electron (its "total energy distribution"). This is because, in spherical geometry, the electrons move in such a fashion that angular momentum about a point in the emitter is very nearly conserved. Hence any kinetic energy that, at emission, is in a direction parallel to the emitter surface gets converted into energy associated with the radial direction of motion. So what gets measured in an energy analyzer is the total energy at emission.

With the development of sensitive electron energy analyzers in the 1960s, it became possible to measure fine details of the total energy distribution. These reflect fine details of the surface physics, and the technique of Field Electron Spectroscopy flourished for a while, before being superseded by newer surface-science techniques.^[33][34]

Field electron emitters as electron-gun sources

Schottky-emitter electron source of an Electron microscope

To achieve high-resolution in electron microscopes and other electron beam instruments (such as those used for electron beam lithography), it is helpful to start with an electron source that is small, optically bright and stable. Sources based on the geometry of a Mueller emitter qualify well on the first two criteria. The first electron microscope (EM) observation of an individual atom was made by Crewe, Wall and Langmore in 1970,^[35] using a scanning electron microscope equipped with an early field emission gun.

From the 1950s onwards, extensive effort has been devoted to the development of field emission sources for use in electron guns.^[36][37][38] [e.g., DD53] Methods have been developed for generating on-axis beams, either by field-induced emitter build-up, or by selective deposition of a low-work-function adsorbate (usually Zirconium oxide – ZrO) into the flat apex of a (100) oriented Tungsten emitter.^[39]

Sources that operate at room temperature have the disadvantage that they rapidly become covered with adsorbate molecules that arrive from the vacuum system walls, and the emitter has to be cleaned from time to time by "flashing" to high temperature. Nowadays, it is more common to use Mueller-emitter-based sources that are operated at elevated temperatures, either in the Schottky emission regime or in the so-called temperature-field intermediate regime. Many modern high-resolution electron microscopes and electron beam instruments use some form of Mueller-emitter-based electron source. Currently, attempts are being made to develop carbon nanotubes (CNTs) as electron-gun field emission sources.^[40][41]

The use of field emission sources in electron optical instruments has involved the development of appropriate theories of charged particle optics,^[37][42] and the development of related modeling. Various shape models have been tried for Mueller emitters; the best seems to be the "Sphere on Orthogonal Cone" (SOC) model introduced by Dyke, Trolan. Dolan and Barnes in 1953.^[43] Important simulations, involving trajectory tracing using the SOC emitter model, were made by Wiesener and Everhart.^[44][45][46] Nowadays, the facility to simulate field emission from Mueller emitters is often incorporated into the commercial electron-optics programmes used to design electron beam instruments. The design of efficient modern field-emission electron guns requires highly specialized expertise.

Atomically sharp emitters

Nowadays it is possible to prepare very sharp emitters, including emitters that end in a single atom. In this case, electron emission comes from an area about twice the crystallographic size of a single atom. This was demonstrated by comparing FEM and field ion microscope (FIM) images of the emitter.^[47] Single-atom-apex Mueller emitters also have relevance to the scanning probe microscopy and helium scanning ion microscopy (He SIM).^[48] Techniques for preparing them have been under investigation for many years.^[47][49] A related important recent advance has been the development (for use in the He SIM) of an automated technique for restoring a three-atom ("trimer") apex to its original state, if the trimer breaks up.^[48]

Large-area field emission sources: vacuum nanoelectronics

Vacuum breakdown and electrical discharge phenomena

As already indicated, it is now thought that the earliest manifestations of field electron emission were the electrical discharges it caused. After Fowler–Nordheim work, it was understood that CFE was one of the possible primary underlying causes of vacuum breakdown and electrical discharge phenomena. (The detailed mechanisms and pathways involved can be very complicated, and there is no single universal cause)^[60] Where vacuum breakdown is known to be caused by electron emission from a cathode, then the original thinking was that the mechanism was CFE from small conducting needle-like surface protrusions. Procedures were (and are) used to round and smooth the surfaces of electrodes that might generate unwanted field electron emission currents. However the work of Latham and others^[51] showed that emission could also be associated with the presence of semiconducting inclusions in smooth surfaces. The physics of how the emission is generated is still not fully understood, but suspicion exists that so-called "triple-junction effects" may be involved. Further information may be found in Latham's book^[51] and in the on-line bibliography.^[60]

Internal electron transfer in electronic devices

In some electronic devices, electron transfer from one material to another, or (in the case of sloping bands) from one band to another ("Zener tunneling"), takes place by a field-induced tunneling process that can be regarded as a form of Fowler–Nordheim tunneling. For example, Rhoderick's book discusses the theory relevant to metal–semiconductor contacts.^[61]

Introduction

The next part of this article deals with the basic theory of cold field electron emission from bulk metals. This is best treated in four main stages, involving theory associated with: (1) derivation of a formula for "escape probability", by considering electron tunneling through a rounded triangular barrier; (2) an integration over internal electron states to obtain the "total energy distribution"; (3) a second integration, to obtain the emission current density as a function of local barrier field and local work function; (4) conversion of this to a formula for current as a function of applied voltage. The modified equations needed for large-area emitters, and issues of experimental data analysis, are dealt with separately.

Fowler–Nordheim tunneling is the wave-mechanical tunneling of an electron through an exact or rounded triangular barrier. Two basic situations are recognized: (1) when the electron is initially in a localized state; (2) when the electron is initially not strongly localized, and is best represented by a travelling wave. Emission from a bulk metal conduction band is a situation of the second type, and discussion here relates to this case. It is also assumed that the barrier is one-dimensional (i.e., has no lateral structure), and has no fine-scale structure that causes "scattering" or "resonance" effects. To keep this explanation of Fowler–Nordheim tunneling relatively simple, these assumptions are needed; but the atomic structure of matter is in effect being disregarded.

Motive energy

For an electron, the one-dimensional Schrödinger equation can be written in the form

${\displaystyle {\frac {\hbar ^{2}}{2m}}{\frac {\mathrm {d} ^{2}\Psi (x)}{\mathrm {d} x^{2}}}=\left[U(x)-E_{\mathrm {n} }\right]\Psi (x)=M(x)\Psi (x),}$

(1)

where Ψ(x) is the electron wave-function, expressed as a function of distance x measured from the emitter's electrical surface,^[62] ħ is the reduced Planck constant, m is the electron mass, U(x) is the electron potential energy, E_n is the total electron energy associated with motion in the x-direction, and M(x) = [U(x) − E_n] is called the electron motive energy.^[63] M(x) can be interpreted as the negative of the electron kinetic energy associated with the motion of a hypothetical classical point electron in the x-direction, and is positive in the barrier.

The shape of a tunneling barrier is determined by how M(x) varies with position in the region where M(x) > 0. Two models have special status in field emission theory: the exact triangular (ET) barrier and the Schottky–Nordheim (SN) barrier.^[64][65] These are given by equations (2) and (3), respectively:

${\displaystyle M^{\mathrm {ET} }(x)=h-eFx}$

(2)

${\displaystyle M^{\rm {SN}}(x)=h-eFx-e^{2}/(16\pi \varepsilon _{0}x),}$

(3)

Here h is the zero-field height (or unreduced height) of the barrier, e is the elementary positive charge, F is the barrier field, and ε₀ is the electric constant. By convention, F is taken as positive, even though the classical electrostatic field would be negative. The SN equation uses the classical image potential energy to represent the physical effect "correlation and exchange".

Escape probability

For an electron approaching a given barrier from the inside, the probability of escape (or "transmission coefficient" or "penetration coefficient") is a function of h and F, and is denoted by D(h,F). The primary aim of tunneling theory is to calculate D(h,F). For physically realistic barrier models, such as the Schottky–Nordheim barrier, the Schrödinger equation cannot be solved exactly in any simple way. The following so-called "semi-classical" approach can be used. A parameter G(h,F) can be defined by the JWKB (Jeffreys-Wentzel-Kramers-Brillouin) integral:^[66]

${\displaystyle G(h,F)=g\int M^{1/2}{\mbox{d}}x,}$

(4)

where the integral is taken across the barrier (i.e., across the region where M > 0), and the parameter g is a universal constant given by

${\displaystyle g\,=2{\sqrt {2m}}/\hbar \approx 10.24624\;{\rm {eV}}^{-1/2}\;{\rm {nm}}^{-1}.}$

(5)

Forbes has re-arranged a result proved by Fröman and Fröman, to show that, formally – in a one-dimensional treatment – the exact solution for D can be written^[67]

${\displaystyle \,D={\frac {P\mathrm {e} ^{-G}}{1+P\mathrm {e} ^{-G}}},}$

(6)

where the tunneling pre-factor P can in principle be evaluated by complicated iterative integrations along a path in complex space.^[67][68] In the CFE regime we have (by definition) G ≫ 1. Also, for simple models P ≈ 1. So eq. (6) reduces to the so-called simple JWKB formula:

${\displaystyle D\approx P\mathrm {e} ^{-G}\approx \mathrm {e} ^{-G}.}$

(7)

For the exact triangular barrier, putting eq. (2) into eq. (4) yields G^ET = bh^3/2/F, where

${\displaystyle b={\frac {2g}{3e}}={\frac {4{\sqrt {2m}}}{3e\hbar }}\approx 6.830890\;{\mathrm {eV} }^{-3/2}\;\mathrm {V} \;{\mathrm {nm} }^{-1}.}$

(8)

This parameter b is a universal constant sometimes called the second Fowler–Nordheim constant. For barriers of other shapes, we write

${\displaystyle G(h,F)=\nu (h,F)G^{\mathrm {ET} }=\nu (h,F)bh^{3/2}/F,}$

(9)

where ν(h,F) is a correction factor that in general has to be determined by numerical integration, using eq. (4).

Correction factor for the Schottky–Nordheim barrier

Schottky–Nordheim barrier for Fowler–Nordheim field emission (and enhanced thermionic emission)

The Schottky–Nordheim barrier, which is the barrier model used in deriving the standard Fowler–Nordheim-type equation,^[69] is a special case. In this case, it is known that the correction factor ${\displaystyle {\it {\nu }}}$ is a function of a single variable f_h, defined by f_h = F/F_h, where F_h is the field necessary to reduce the height of a Schottky–Nordheim barrier from h to 0. This field is given by

${\displaystyle \,F_{h}=(4\pi \epsilon _{0}/e^{3})h^{2}=(0.6944617\;\mathrm {V} \;{\mathrm {nm} }^{-1})(h/{\rm {eV}})^{2}.}$

(10)

The parameter f_h runs from 0 to 1, and may be called the scaled barrier field, for a Schottky–Nordheim barrier of zero-field height h.

For the Schottky–Nordheim barrier, ν(h,F) is given by the particular value ν(f_h) of a function ν(ℓ′). The latter is a function of mathematical physics in its own right and has been called the principal Schottky–Nordheim barrier function. An explicit series expansion for ν(ℓ′) is derived in a 2008 paper by J. Deane.^[70] The following good simple approximation for ν(f_h) has been found:^[69]

${\displaystyle v(f_{h})\approx 1-f_{h}+{\tfrac {1}{6}}f_{h}\ln f_{h}.}$

(11)

Decay width

The decay width (in energy), d_h, measures how fast the escape probability D decreases as the barrier height h increases; d_h is defined by:

${\displaystyle {\frac {1}{d_{h}}}=-{\frac {\mathrm {d} (\ln D)}{\mathrm {d} h}}.}$

(12)

When h increases by d_h then the escape probability D decreases by a factor close to e ( ≈ 2.718282). For an elementary model, based on the exact triangular barrier, where we put ν = 1 and P ≈ 1, we get

${\displaystyle d_{h}^{\mathrm {(el)} }={\frac {2F}{3b{\sqrt {h}}}}={\frac {eF}{g{\sqrt {h}}}}.}$

The decay width d_h derived from the more general expression (12) differs from this by a "decay-width correction factor" λ_d, so:

${\displaystyle d_{h}=\lambda _{d}d_{h}^{\mathrm {(el)} }={\frac {\lambda _{d}eF}{g{\sqrt {h}}}}.}$

(13)

Usually, the correction factor can be approximated as unity.

The decay-width d_F for a barrier with h equal to the local work-function φ is of special interest. Numerically this is given by:

${\displaystyle d_{\mathrm {F} }={\frac {\lambda _{d}eF}{g{\sqrt {\phi }}}}\approx {\frac {eF}{g{\sqrt {\phi }}}}\approx 0.09759678\;\mathrm {eV} \,\cdot {\sqrt {\frac {1\ \mathrm {eV} }{\phi }}}\cdot {\frac {F}{1\ \mathrm {V} \ \mathrm {nm} ^{-1}}}.}$

(14)

For metals, the value of d_F is typically of order 0.2 eV, but varies with barrier-field F.

Comments

A historical note is necessary. The idea that the Schottky–Nordheim barrier needed a correction factor, as in eq. (9), was introduced by Nordheim in 1928,^[65] but his mathematical analysis of the factor was incorrect. A new (correct) function was introduced by Burgess, Kroemer and Houston^[71] in 1953, and its mathematics was developed further by Murphy and Good in 1956.^[72] This corrected function, sometimes known as a "special field emission elliptic function", was expressed as a function of a mathematical variable y known as the "Nordheim parameter". Only recently (2006 to 2008) has it been realized that, mathematically, it is much better to use the variable ℓ′ ( = y²). And only recently has it been possible to complete the definition of ν(ℓ′) by developing and proving the validity of an exact series expansion for this function (by starting from known special-case solutions of the Gauss hypergeometric differential equation). Also, approximation (11) has been found only recently. Approximation (11) outperforms, and will presumably eventually displace, all older approximations of equivalent complexity. These recent developments, and their implications, will probably have a significant impact on field emission research in due course.

The following summary brings these results together. For tunneling well below the top of a well-behaved barrier of reasonable height, the escape probability D(h,F) is given formally by:

${\displaystyle D(h,F)\approx P\exp \left[-{\frac {\nu (h,F)bh^{3/2}}{F}}\right],}$

(15)

where ν(h,F) is a correction factor that in general has to be found by numerical integration. For the special case of a Schottky–Nordheim barrier, an analytical result exists and ν(h,F) is given by ν(f_h), as discussed above; approximation (11) for ν(f_h) is more than sufficient for all technological purposes. The pre-factor P is also in principle a function of h and (maybe) F, but for the simple physical models discussed here it is usually satisfactory to make the approximation P = 1. The exact triangular barrier is a special case where the Schrödinger equation can be solved exactly, as was done by Fowler and Nordheim;^[1] for this physically unrealistic case, ν(f_h) = 1, and an analytical approximation for P exists.

The approach described here was originally developed to describe Fowler–Nordheim tunneling from smooth, classically flat, planar emitting surfaces. It is adequate for smooth, classical curved surfaces of radii down to about 10 to 20 nm. It can be adapted to surfaces of sharper radius, but quantities such as ν and D then become significant functions of the parameter(s) used to describe the surface curvature. When the emitter is so sharp that atomic-level detail cannot be neglected, and/or the tunneling barrier is thicker than the emitter-apex dimensions, then a more sophisticated approach is desirable.

As noted at the beginning, the effects of the atomic structure of materials are disregarded in the relatively simple treatments of field electron emission discussed here. Taking atomic structure properly into account is a very difficult problem, and only limited progress has been made.^[33] However, it seems probable that the main influences on the theory of Fowler–Nordheim tunneling will (in effect) be to change the values of P and ν in eq. (15), by amounts that cannot easily be estimated at present.

All these remarks apply in principle to Fowler Nordheim tunneling from any conductor where (before tunneling) the electrons may be treated as in travelling-wave states. The approach may be adapted to apply (approximately) to situations where the electrons are initially in localized states at or very close inside the emitting surface, but this is beyond the scope of this article.

Fowler–Nordheim-type equations

CFE theoretical equations

The preceding section explains how to derive Fowler–Nordheim-type equations. Strictly, these equations apply only to CFE from bulk metals. The ideas in the following sections apply to CFE more generally, but eq. (30) will be used to illustrate them.

For CFE, basic theoretical treatments provide a relationship between the local emission current density J and the local barrier field F, at a local position on the emitting surface. Experiments measure the emission current i from some defined part of the emission surface, as a function of the voltage V applied to some counter-electrode. To relate these variables to J and F, auxiliary equations are used.

The voltage-to-barrier-field conversion factor β is defined by:

${\displaystyle F=\;\beta V,}$

(31)

The value of F varies from position to position on an emitter surface, and the value of β varies correspondingly.

For a metal emitter, the β−value for a given position will be constant (independent of voltage) under the following conditions: (1) the apparatus is a "diode" arrangement, where the only electrodes present are the emitter and a set of "surroundings", all parts of which are at the same voltage; (2) no significant field-emitted vacuum space-charge (FEVSC) is present (this will be true except at very high emission current densities, around 10⁹ A/m² or higher^[27][80]); (3) no significant "patch fields" exist,^[63] as a result of non-uniformities in local work-function (this is normally assumed to be true, but may not be in some circumstances). For non-metals, the physical effects called "field penetration" and "band bending" [M084] can make β a function of applied voltage, although – surprisingly – there are few studies of this effect.

The emission current density J varies from position to position across the emitter surface. The total emission current i from a defined part of the emitter is obtained by integrating J across this part. To obtain a simple equation for i(V), the following procedure is used. A reference point "r" is selected within this part of the emitter surface (often the point at which the current density is highest), and the current density at this reference point is denoted by J_r. A parameter A_r, called the notional emission area (with respect to point "r"), is then defined by:

${\displaystyle i=A_{\mathrm {r} }J_{\mathrm {r} }=\int J\mathrm {d} A,}$

(32)

where the integral is taken across the part of the emitter of interest.

This parameter A_r was introduced into CFE theory by Stern, Gossling and Fowler in 1929 (who called it a "weighted mean area").^[16] For practical emitters, the emission current density used in Fowler–Nordheim-type equations is always the current density at some reference point (though this is usually not stated). Long-established convention denotes this reference current density by the simple symbol J, and the corresponding local field and conversion factor by the simple symbols F and β, without the subscript "r" used above; in what follows, this convention is used.

The notional emission area A_r will often be a function of the reference local field (and hence voltage),^[30] and in some circumstances might be a significant function of temperature.

Because A_r has a mathematical definition, it does not necessarily correspond to the area from which emission is observed to occur from a single-point emitter in a field electron (emission) microscope. With a large-area emitter, which contains many individual emission sites, A_r will nearly always be very very much less than the "macroscopic" geometrical area (A_M) of the emitter as observed visually (see below).

Incorporating these auxiliary equations into eq. (30a) yields

${\displaystyle i=\;A_{\mathrm {r} }a{\phi ^{-1}}{\beta }^{2}V^{2}\exp[-v(f)\;b\phi ^{3/2}/\beta V],}$

(33)

This is the simplified standard Fowler–Nordheim-type equation, in i-V form. The corresponding "physically complete" equation is obtained by multiplying by λ_ZP_F.

Modified equations for large-area emitters

The equations in the preceding section apply to all field emitters operating in the CFE regime. However, further developments are useful for large-area emitters that contain many individual emission sites.

For such emitters, the notional emission area will nearly always be very very much less than the apparent "macroscopic" geometrical area (A_M) of the physical emitter as observed visually. A dimensionless parameter α_r, the area efficiency of emission, can be defined by

${\displaystyle A_{\mathrm {r} }=\;\alpha _{\mathrm {r} }A_{\mathrm {M} }.}$

(34)

Also, a "macroscopic" (or "mean") emission current density J_M (averaged over the geometrical area A_M of the emitter) can be defined, and related to the reference current density J_r used above, by

${\displaystyle J_{\mathrm {M} }=\;i/A_{\mathrm {M} }=\alpha _{\mathrm {r} }(i/A_{\mathrm {r} })=\alpha _{\mathrm {r} }J_{\mathrm {r} }.}$

(35)

This leads to the following "large-area versions" of the simplified standard Fowler–Nordheim-type equation:

${\displaystyle J_{\mathrm {M} }=\alpha _{\mathrm {r} }a{\phi ^{-1}}F^{2}\exp[-v(f)\;b\phi ^{3/2}/F],}$

(36)

${\displaystyle i=\;\alpha _{\mathrm {r} }A_{\mathrm {M} }a{\phi ^{-1}}{\beta }^{2}V^{2}\exp[-v(f)\;b\phi ^{3/2}/\beta V],}$

(37)

Both these equations contain the area efficiency of emission α_r. For any given emitter this parameter has a value that is usually not well known. In general, α_r varies greatly as between different emitter materials, and as between different specimens of the same material prepared and processed in different ways. Values in the range 10⁻¹⁰ to 10⁻⁶ appear to be likely, and values outside this range may be possible.

The presence of α_r in eq. (36) accounts for the difference between the macroscopic current densities often cited in the literature (typically 10 A/m² for many forms of large-area emitter other than Spindt arrays^[50]) and the local current densities at the actual emission sites, which can vary widely but which are thought to be generally of the order of 10⁹ A/m², or possibly slightly less.

A significant part of the technological literature on large-area emitters fails to make clear distinctions between local and macroscopic current densities, or between notional emission area A_r and macroscopic area A_M, and/or omits the parameter α_r from cited equations. Care is necessary in order to avoid errors of interpretation.

It is also sometimes convenient to split the conversion factor β_r into a "macroscopic part" that relates to the overall geometry of the emitter and its surroundings, and a "local part" that relates to the ability of the very-local structure of the emitter surface to enhance the electric field. This is usually done by defining a "macroscopic field" F_M that is the field that would be present at the emitting site in the absence of the local structure that causes enhancement. This field F_M is related to the applied voltage by a "voltage-to-macroscopic-field conversion factor" β_M defined by:

${\displaystyle F_{\mathrm {M} }=\;\beta _{\mathrm {M} }V.}$

(38)

In the common case of a system comprising two parallel plates, separated by a distance W, with emitting nanostructures created on one of them, β_M = 1/W.

A "field enhancement factor" γ is then defined and related to the values of β_r and β_M by

${\displaystyle \gamma =\;F_{\mathrm {r} }/F_{\mathrm {M} }=\beta _{\mathrm {r} }/\beta _{\mathrm {M} }.}$

(39)

With eq. (31), this generates the following formulae:

${\displaystyle F=\;\gamma F_{\mathrm {M} }=\beta V;}$

(40)

${\displaystyle \beta =\;\beta _{\mathrm {M} }\gamma$ ;}

(41)

where, in accordance with the usual convention, the suffix "r" has now been dropped from parameters relating to the reference point. Formulae exist for the estimation of γ, using classical electrostatics, for a variety of emitter shapes, in particular the "hemisphere on a post".^[81]

Equation (40) implies that versions of Fowler–Nordheim-type equations can be written where either F or βV is everywhere replaced by ${\displaystyle \gamma F_{\mathrm {M} }}$. This is often done in technological applications where the primary interest is in the field enhancing properties of the local emitter nanostructure. However, in some past work, failure to make a clear distinction between barrier field F and macroscopic field F_M has caused confusion or error.

More generally, the aims in technological development of large-area field emitters are to enhance the uniformity of emission by increasing the value of the area efficiency of emission α_r, and to reduce the "onset" voltage at which significant emission occurs, by increasing the value of β. Eq. (41) shows that this can be done in two ways: either by trying to develop "high-γ" nanostructures, or by changing the overall geometry of the system so that β_M is increased. Various trade-offs and constraints exist.

In practice, although the definition of macroscopic field used above is the commonest one, other (differently defined) types of macroscopic field and field enhancement factor are used in the literature, particularly in connection with the use of probes to investigate the i-V characteristics of individual emitters.^[82]

In technological contexts field-emission data are often plotted using (a particular definition of) F_M or 1/F_M as the x-coordinate. However, for scientific analysis it usually better not to pre-manipulate the experimental data, but to plot the raw measured i-V data directly. Values of technological parameters such as (the various forms of) γ can then be obtained from the fitted parameters of the i-V data plot (see below), using the relevant definitions.

Modified equations for nanometrically sharp emitters

Most of the theoretical derivations in the field emission theory are done under the assumption that the barrier takes the Schottky–Nordheim form eq. (3). However, this barrier form is not valid for emitters with radii of curvature ${\displaystyle R}$ comparable to the length of the tunnelling barrier. The latter depends on the work function and the field, but in cases of practical interest, the SN barrier approximation can be considered valid for emitters with radii ${\displaystyle R>20{\text{nm}}}$, as explained in the next paragraph.

The main assumption of the SN barrier approximation is that the electrostatic potential term takes the linear form ${\displaystyle \Phi =Fx}$ in the tunnelling region. The latter has been proved to hold only if ${\displaystyle x\ll R}$.^[83] Therefore, if the tunnelling region has a length ${\displaystyle L}$, ${\displaystyle x<L}$ for all ${\displaystyle x}$ that determines the tunnelling process; thus if ${\displaystyle L\ll R}$ eq. (1) holds and the SN barrier approximation is valid. If the tunnelling probability is high enough to produce measurable field emission, L does not exceed 1-2 nm. Hence, the SN barrier is valid for emitters with radii of the order of some tens of nm.

However, modern emitters are much sharper than this, with radii that of the order of a few nm. Therefore, the standard FN equation, or any version of it that assumes the SN barrier, leads to significant errors for such sharp emitters. This has been both shown theoretically^[84][85] and confirmed experimentally.^[86]

The above problem was tackled by Kyritsakis and Xanthakis,^[83] who generalized the SN barrier by including the electrostatic effects of the emitter curvature. The general barrier form for an emitter with radius of average curvature ${\displaystyle R}$ (inverse of the average of the two principal curvatures) can be asymptotically expanded as^[87]

${\displaystyle M^{KX}(x)=h-eFx\left[1-{\frac {x}{R}}+O\left({\frac {x}{R}}\right)^{2}\right]-{\frac {e^{2}}{16\pi \epsilon _{0}x}}\left[1-{\frac {x}{2R}}+O\left({\frac {x}{R}}\right)^{2}\right]}$.

(43)

After neglecting all ${\displaystyle O(x/R)^{2}}$ terms, and employing the JWKB approximation (4) for this barrier, the Gamow exponent takes a form that generalizes eq. (5)

${\displaystyle G(h,F,R)={\frac {bh^{3/2}}{F}}\left(v(f)+\omega (f){\frac {h}{eFR}}\right)}$

(44)

where ${\displaystyle f}$ is defined by (30d), ${\displaystyle v(f)}$ is given by (30c) and ${\displaystyle \omega (f)}$ is a new function that can be approximated in a similar manner as (30c) (there are typographical mistakes in ref.,^[83] corrected here):

${\displaystyle \omega (f)\approx {\frac {4}{5}}-{\frac {7}{40}}f+{\frac {1}{200}}\log(f).}$

(45)

Given the expression for the Gamow exponent as a function of the field-free barrier height ${\displaystyle h}$, the emitted current density for cold field emission can be obtained from eq. (23). It yields

${\displaystyle J=a{\frac {F^{2}}{\phi }}\left({\frac {1}{\lambda _{d}(f)}}+{\frac {\phi }{eFR}}\psi (f)\right)^{-2}\exp \left[-{\frac {b\phi ^{3/2}}{F}}\left(v(f)+\omega (f){\frac {\phi }{eFR}}\right)\right]}$

(46)

where the functions ${\displaystyle \lambda _{d}(f)}$ and ${\displaystyle \psi (f)}$ are defined as

${\displaystyle {\frac {1}{\lambda _{d}(f)}}\equiv v(f)-{\frac {4}{3}}f{\frac {\partial v}{\partial f}}\approx 1+{\frac {f}{9}}-{\frac {f\log(f)}{22}}}$

(47a)

and

${\displaystyle \psi (f)\equiv {\frac {5}{3}}\omega (f)-{\frac {4}{3}}f{\frac {\partial \omega }{\partial f}}\approx {\frac {4}{3}}-{\frac {f}{15}}-{\frac {f\log(f)}{1200}}}$

(47b)

In equation (46), for completeness purposes, ${\displaystyle \lambda _{d}}$ is not approximated by unity as in (29) and (30a), although for most practical cases it is a very good approximation. Apart from this, equations (43), (44) and (46) coincide with the corresponding ones of the standard Fowler–Nordheim theory (3), (9), and (30a), in the limit ${\displaystyle R\to \infty }$; this is expected since the former equations generalise the latter.

Finally, note that the above analysis is asymptotic in the limit ${\displaystyle L\ll R}$, similarly to the standard Fowler–Nordheim theory using the SN barrier. However, the addition of the quadratic terms renders it significantly more accurate for emitters with radii of curvature in the range ~5–20 nm. For sharper emitters there is no general approximation for the current density. In order to obtain the current density, one has to calculate the electrostatic potential and evaluate the JWKB integral numerically. For this purpose, scientific computing software has been developed (see e.g. GETELEC^[88]).

Empirical CFE i–V equation

At the present stage of CFE theory development, it is important to make a distinction between theoretical CFE equations and an empirical CFE equation. The former are derived from condensed matter physics (albeit in contexts where their detailed development is difficult). An empirical CFE equation, on the other hand, simply attempts to represent the actual experimental form of the dependence of current i on voltage V.

In the 1920s, empirical equations were used to find the power of V that appeared in the exponent of a semi-logarithmic equation assumed to describe experimental CFE results. In 1928, theory and experiment were brought together to show that (except, possibly, for very sharp emitters) this power is V⁻¹. It has recently been suggested that CFE experiments should now be carried out to try to find the power (κ) of V in the pre-exponential of the following empirical CFE equation:^[89]

${\displaystyle i=\;CV^{\kappa }\exp[-B/V],}$

(48)

where B, C and κ are treated as constants.

From eq. (42) it is readily shown that

${\displaystyle -\mathrm {d} \ln i/\mathrm {d} (1/V)=\;\kappa V+B,}$

(49)

In the 1920s, experimental techniques could not distinguish between the results κ = 0 (assumed by Millikan and Laurtisen)^[13] and κ = 2 (predicted by the original Fowler–Nordheim-type equation).^[1] However, it should now be possible to make reasonably accurate measurements of dlni/d(1/V) (if necessary by using lock-in amplifier/phase-sensitive detection techniques and computer-controlled equipment), and to derive κ from the slope of an appropriate data plot.^[50]

Following the discovery of approximation (30b), it is now very clear that – even for CFE from bulk metals – the value κ=2 is not expected. This can be shown as follows. Using eq. (30c) above, a dimensionless parameter η may be defined by

${\displaystyle \eta =b\phi ^{3/2}/F_{\phi }=\;(be^{3}/4\pi \epsilon _{0}){\phi }^{-1/2}\approx 9.836239\;\;(\mathrm {eV} /\phi )^{1/2}.}$

(50)

For φ = 4.50 eV, this parameter has the value η = 4.64. Since f = F/F_φ and v(f) is given by eq (30b), the exponent in the simplified standard Fowler–Nordheim-type equation (30) can be written in an alternative form and then expanded as follows:^[69]

${\displaystyle \exp[-v(f)\;b{\phi }^{3/2}/F]\;=\;\exp[-v(f)\;\eta /f]\;\approx \;{\mathrm {e} }^{\eta }f^{-\eta /6}\exp[-\eta /f]\;=\;{\mathrm {e} }^{\eta }f^{-\eta /6}\exp[-b{\phi }^{3/2}/F].}$

(51)

Provided that the conversion factor β is independent of voltage, the parameter f has the alternative definition f = V/V_φ, where V_φ is the voltage needed, in a particular experimental system, to reduce the height of a Schottky–Nordheim barrier from φ to zero. Thus, it is clear that the factor v(f) in the exponent of the theoretical equation (30) gives rise to additional V-dependence in the pre-exponential of the empirical equation. Thus, (for effects due to the Schottky–Nordheim barrier, and for an emitter with φ=4.5 eV) we obtain the prediction:

${\displaystyle \kappa \approx 2-\eta /6=2-0.77=1.23.}$

(52)

Since there may also be voltage dependence in other factors in a Fowler–Nordheim-type equation, in particular in the notional emission area^[30] A_r and in the local work-function, it is not necessarily expected that κ for CFE from a metal of local work-function 4.5 eV should have the value κ = 1.23, but there is certainly no reason to expect that it will have the original Fowler–Nordheim value κ = 2.^[90]

A first experimental test of this proposal has been carried out by Kirk, who used a slightly more complex form of data analysis to find a value 1.36 for his parameter κ. His parameter κ is very similar to, but not quite the same as, the parameter κ used here, but nevertheless his results do appear to confirm the potential usefulness of this form of analysis.^[91]

Use of the empirical CFE equation (42), and the measurement of κ, may be of particular use for non-metals. Strictly, Fowler–Nordheim-type equations apply only to emission from the conduction band of bulk crystalline solids. However, empirical equations of form (42) should apply to all materials (though, conceivably, modification might be needed for very sharp emitters). It seems very likely that one way in which CFE equations for newer materials may differ from Fowler–Nordheim-type equations is that these CFE equations may have a different power of F (or V) in their pre-exponentials. Measurements of κ might provide some experimental indication of this.