Field electron emission, also known as field-induced electron emission, field emission (FE) and electron field emission, is the emission of electrons from a material placed in an electrostatic field. The most common context is field emission from a solid surface into a vacuum. However, field emission can take place from solid or liquid surfaces, into a vacuum, a fluid (e.g. air), or any non-conducting or weakly conducting dielectric. The field-induced promotion of electrons from the valence to conduction band of semiconductors (the Zener effect) can also be regarded as a form of field emission. Field emission in pure metals occurs in high electric fields: the gradients are typically higher than 1 gigavolt per metre and strongly dependent upon the work function. While electron sources based on field emission have a number of applications, field emission is most commonly an undesirable primary source of vacuum breakdown and electrical discharge phenomena, which engineers work to prevent. Examples of applications for surface field emission include the construction of bright electron sources for high-resolution electron microscopes or the discharge of induced charges from spacecraft. Devices that eliminate induced charges are termed charge-neutralizers. Historically, the phenomenon of field electron emission has been known by a variety of names, including "the aeona effect", "autoelectronic emission", "cold emission", "cold cathode emission", "field emission", "field electron emission" and "electron field emission". In some contexts (e.g. spacecraft engineering), the name "field emission" is applied to the field-induced emission of ions (field ion emission), rather than electrons, and because in some theoretical contexts "field emission" is used as a general name covering both field electron emission and field ion emission. Field emission was explained by quantum tunneling of electrons in the late 1920s. This was one of the triumphs of the nascent quantum mechanics. The theory of field emission from bulk metals was proposed by Ralph H. Fowler and Lothar Wolfgang Nordheim. A family of approximate equations, Fowler–Nordheim equations, is named after them. Strictly, Fowler–Nordheim equations apply only to field emission from bulk metals and (with suitable modification) to other bulk crystalline solids, but they are often used – as a rough approximation – to describe field emission from other materials. The related phenomena of surface photoeffect, thermionic emission (or Richardson–Dushman effect) and "cold electronic emission", i.e. the emission of electrons in strong static (or quasi-static) electric fields, were discovered and studied independently from the 1880s to 1930s. In the modern context, cold field electron emission (CFE) is the name given to a particular statistical emission regime, in which the electrons in the emitter are initially in internal thermodynamic equilibrium, and in which most emitted electrons escape by Fowler–Nordheim tunneling from electron states close to the emitter Fermi level. (By contrast, in the Schottky emission regime, most electrons escape over the top of a field-reduced barrier, from states well above the Fermi level.) Many solid and liquid materials can emit electrons in a CFE regime if an electric field of an appropriate size is applied. When the term field emission is used without qualifiers, it typically means "cold emission". For metals, the CFE regime extends to well above room temperature. There are other electron emission regimes (such as "thermal electron emission" and "Schottky emission") that require significant external heating of the emitter. There are also emission regimes where the internal electrons are not in thermodynamic equilibrium and the emission current is, partly or completely, determined by the supply of electrons to the emitting region. A non-equilibrium emission process of this kind may be called field (electron) emission if most of the electrons escape by tunneling, but strictly it is not CFE, and is not accurately described by a Fowler–Nordheim-type equation.
Terminology and conventions Equations in this article are written using the modern International System of Quantities (ISQ). Older field emission literature (and papers that directly copy equations from old literature) often work with Gaussian units such that they omit the physical constant ε0. In this article, all such equations have been converted to modern international form. Since work function is normally given with the unit electronvolt (eV), and for fields it is often convenient to use the unit volt per nanometer (V/nm); increasingly, this is normal practice in field emission research. , Numerical values of universal constants given here are written in units derived from eV, V and nm, and calculated to seven significant figures using the 2006 values of the fundamental constants.
Early history of field electron emission In retrospect, it seems likely that the electrical discharges reported by J.H. Winkler in 1744 were started by CFE from his wire electrode. However, meaningful investigations had to wait until after J.J. Thomson's identification of the electron in 1897, and until after it was understood – from thermal emission and photo-emission work – that electrons could be emitted from inside metals (rather than from surface-adsorbed gas molecules), and that – in the absence of applied fields – electrons escaping from metals had to overcome a work function barrier. It was suspected at least as early as 1913 that field-induced emission was a separate physical effect. However, only after vacuum and specimen cleaning techniques had significantly improved, did this become well established. Lilienfeld (who was primarily interested in electron sources for medical X-ray applications) published in 1922 the first clear account in English of the experimental phenomenology of the effect he had called "autoelectronic emission". He had worked on this topic, in Leipzig, since about 1910. After 1922, experimental interest increased, particularly in the groups led by Millikan at the California Institute of Technology (Caltech) in Pasadena, California, and by Gossling at the General Electric Company in London. Attempts to understand autoelectronic emission included plotting experimental current–voltage (i–V) data in different ways, to look for a straight-line relationship. Current increased superlinearly with voltage, but plots of type log(i) vs. V were not straight. Walter H. Schottky suggested in 1923 that the effect might be due to thermally induced emission over a field-reduced barrier. If so, then plots of log(i) vs. √V should be straight, but they were not. Nor is Schottky's explanation compatible with the experimental observation of only very weak temperature dependence in CFE – a point initially overlooked. A breakthrough came when C.C. Lauritsen (and J. Robert Oppenheimer independently) found that plots of log(i) vs. 1/V yielded good straight lines. This result was published by Millikan and Lauritsen in early 1928. Theoretical explanation and the original Fowler–Nordheim-type equation came shortly thereafter. Oppenheimer had predicted that the field-induced tunneling of electrons from atoms (the effect now called field ionization) would have this i(V) dependence, had found this dependence in the published experimental field emission results of Millikan and Eyring, and proposed that CFE was due to field-induced tunneling of electrons from atomic-like orbitals in surface metal atoms. An alternative Fowler–Nordheim theory proposed field-induced tunneling from free-electron-type states in what we would now call a metal conduction band, with the electron states occupied in accordance with Fermi–Dirac statistics. The Fowler-Nordheim theory explained both the Millikan–Lauritsen finding and the very weak dependence of current on temperature. Oppenheimer had mathematical details of his theory seriously incorrect. There was also a small numerical error in the final equation given by Fowler–Nordheim theory for CFE current density, corrected in a 1929 paper. If the barrier field in Fowler–Nordheim 1928 theory is exactly proportional to the applied voltage, and if the emission area is independent of voltage, then the Fowler–Nordheim 1928 theory predicts that plots of log(i/V2) vs. 1/V should be exact straight lines. However, contemporary experimental techniques could not distinguish between the Fowler–Nordheim theoretical result and the Millikan–Lauritsen experimental result. The physics literature often presents Fowler and Nordheim's work as a proof of electron tunneling, as predicted by wave-mechanics. Whilst this is correct, wave-mechanics was largely accepted by 1928. Instead, the Fowler–Nordheim paper was more revolutionary in establishing modern electron band theory. Prior to 1928 it had been hypothesized that two types of electrons, "thermions" and "conduction electrons", existed in metals, and that thermally emitted electron currents were due to the emission of thermions, but that field-emitted currents were due to the emission of conduction electrons, only in 1927 did Sommerfeld argue that Fermi–Dirac statistics applied to the behavior of electrons in metals. The Fowler–Nordheim 1928 work suggested that thermions did not need to exist as a separate class of internal electrons: electrons could come from a single band occupied in accordance with Fermi–Dirac statistics, but would be emitted in statistically different ways under different conditions of temperature and applied field. The success of Fowler–Nordheim theory did much to support the correctness of Sommerfeld's ideas. In particular, the original Fowler–Nordheim-type equation was one of the first to incorporate the statistical-mechanical consequences of the existence of electron spin into the theory of an experimental condensed-matter effect. The Fowler–Nordheim paper also established the physical basis for a unified treatment of field-induced and thermally induced electron emission. The ideas of Oppenheimer, Fowler and Nordheim were also an important stimulus to the development, by George Gamow, and Ronald W. Gurney and Edward Condon, later in 1928, of the theory of the radioactive decay of nuclei (by alpha particle tunneling).
Practical applications: past and present
Field electron microscopy and related basics As already indicated, the early experimental work on field electron emission (1910–1920) 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) (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:
The material from which the emitter is made. The orientation of the material relative to the needle/wire axis; and 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. For example, in the 1960s, FEM results contributed significantly to discussions on heterogeneous catalysis. 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. 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 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. In 1959 it was realized theoretically by Young, and confirmed experimentally by Young and Mueller 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.
Field electron emitters as electron-gun sources
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 Albert Crewe, J. Wall and J. Langmore in 1970, using a scanning transmission 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. [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. Sources that operate at room temperature have the disadvantage that they can 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 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. Most modern high-resolution electron microscopes and electron beam instruments use some form of field emission electron source. Currently, attempts are being made to develop carbon nanotubes (CNTs) as electron-gun field emission sources. The use of field emission sources in electron optical instruments has involved the development of appropriate theories of charged particle optics, 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. Important simulations, involving trajectory tracing using the SOC emitter model, were made by Wiesener and Everhart. 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. Single-atom-apex Mueller emitters also have relevance to the scanning probe microscopy and helium scanning ion microscopy (He SIM). Techniques for preparing them have been under investigation for many years. 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.
Large-area field emission sources: vacuum nanoelectronics
Materials aspects Large-area field emission sources have been of interest since the 1970s. In these devices, a high density of individual field emission sites is created on a substrate (originally silicon). This research area became known, first as "vacuum microelectronics", now as "vacuum nanoelectronics". One of the original two device types, the "Spindt array", used silicon-integrated-circuit (IC) fabrication techniques to make regular arrays in which molybdenum cones were deposited in small cylindrical voids in an oxide film, with the void covered by a counterelectrode with a central circular aperture. This overall geometry has also been used with carbon nanotubes grown in the void. The other original device type was the "Latham emitter". These were MIMIV (metal-insulator-metal-insulator-vacuum) – or, more generally, CDCDV (conductor-dielectric-conductor-dielectric-vacuum) – devices that contained conducting particulates in a dielectric film. The device field-emits because its microstructure/nanostructure has field-enhancing properties. This material had a potential production advantage, in that it could be deposited as an "ink", so IC fabrication techniques were not needed. However, in practice, uniformly reliable devices proved difficult to fabricate. Research advanced to look for other materials that could be deposited/grown as thin films with suitable field-enhancing properties. In a parallel-plate arrangement, the "macroscopic" field FM between the plates is given by FM = V/W, where W is the plate separation and V is the applied voltage. If a sharp object is created on one plate, then the local field F at its apex is greater than FM and can be related to FM by
F = γ F M . {\displaystyle F=\gamma F_{\mathrm {M} }.}
The parameter γ is called the "field enhancement factor" and is basically determined by the object's shape. Since field emission characteristics are determined by the local field F, then the higher the γ-value of the object, then the lower the value of FM at which significant emission occurs. Hence, for a given value of W, the lower the applied voltage V at which significant emission occurs. For a roughly ten year-period from the mid-1990s, there was great interest in field emission from plasma-deposited films of amorphous and "diamond-like" carbon. However, interest subsequently lessened, partly due to the arrival of CNT emitters, and partly because evidence emerged that the emission sites might be associated with particulate carbon objects created in an unknown way during the deposition process: this suggested that quality control of an industrial-scale production process might be problematic. The introduction of CNT field emitters, both in "mat" form and in "grown array" forms, was a significant step forward. Extensive research has been undertaken into both their physical characteristics and possible technological applications. For field emission, an advantage of CNTs is that, due to their shape, with its high aspect ratio, they are "natural field-enhancing objects". In recent years there has also been massive growth in interest in the development of other forms of thin-film emitter, both those based on other carbon forms (such as "carbon nanowalls") and on various forms of wide-band-gap semiconductor. A particular aim is to develop "high-γ" nanostructures with a sufficiently high density of individual emission sites. Thin films of nanotubes in form of nanotube webs are also used for development of field emission electrodes. It is shown that by fine-tuning the fabrication parameters, these webs can achieve an optimum density of individual emission sites. Double-layered electrodes made by deposition of two layers of these webs with perpendicular alignment towards each other are shown to be able to lower the turn-on electric field (electric field required for achieving an emission current of 10 μA/cm2) down to 0.3 V/μm and provide a stable field emission performance. Common problems with all field-emission devices, particularly those that operate in "industrial vacuum conditions" is that the emission performance can be degraded by the adsorption of gas atoms arriving from elsewhere in the system, and the emitter shape can be in principle be modified deleteriously by a variety of unwanted subsidiary processes, such as bombardment by ions created by the impact of emitted electrons onto gas-phase atoms and/or onto the surface of counter-electrodes. Thus, an important industrial requirement is "robustness in poor vacuum conditions"; this needs to be taken into account in research on new emitter materials. At the time of writing, the most promising forms of large-area field emission source (certainly in terms of achieved average emission current density) seem to be Spindt arrays and the various forms of source based on CNTs.
Applications The development of large-area field emission sources was originally driven by the wish to create new, more efficient, forms of electronic information display. These are known as "field-emission displays" or "nano-emissive displays". Although several prototypes have been demonstrated, the development of such displays into reliable commercial products has been hindered by a variety of industrial production problems not directly related to the source characteristics [En08]. Other proposed applications of large-area field emission sources include microwave generation, space-vehicle neutralization, X-ray generation, and (for array sources) multiple e-beam lithography. There are also recent attempts to develop large-area emitters on flexible substrates, in line with wider trends towards "plastic electronics". The development of such applications is the mission of vacuum nanoelectronics. However, field emitters work best in conditions of good ultrahigh vacuum. Their most successful applications to date (FEM, FES and EM guns) have occurred in these conditions. The sad fact remains that field emitters and industrial vacuum conditions do not go well together, and the related problems of reliably ensuring good "vacuum robustness" of field emission sources used in such conditions still await better solutions (probably cleverer materials solutions) than we currently have.
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 the 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) 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 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 and in the on-line bibliography.
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.
Fowler–Nordheim tunneling Fowler–Nordheim tunneling is the wave-mechanical tunneling of an electron through an exact or rounded triangular barrier. Depending on the material's structure, the electron may be initially localized to the surface or delocalized into the bulk and best represented by a travelling wave. Emission from a metal conduction band is a situation of the second type, which is the only case treated here. 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. These assumptions serve primarily to simplify the theory; but the atomic structure of matter is in effect being disregarded. The treatment has four main stages:
Derivation of a formula for escape probability, by considering electron tunneling through a rounded triangular barrier; Integration over internal electron states to obtain the total energy distribution; A second integration, to obtain the emission current density as a function of local barrier field and local work function; Conversion of the local work function 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.
Motive energy For an electron, the one-dimensional Schrödinger equation can be written in the form
where Ψ(x) is the electron wave-function, expressed as a function of distance x measured from the emitter's electrical surface, ħ is the reduced Planck constant, m is the electron mass, U(x) is the electron potential energy, En is the total electron energy associated with motion in the x-direction, and M(x) = [U(x) − En] is called the electron motive energy. 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, given in (2); and the Schottky–Nordheim (SN) barrier, given in (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:
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
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
where the tunneling pre-factor P can in principle be evaluated by complicated iterative integrations along a path in complex space, but is ≈ 1 for simple models. In the CFE regime we have (by definition) G ≫ 1. So eq. (6) reduces to the so-called simple JWKB formula:
For the exact triangular barrier, putting eq. (2) into eq. (4) yields GET = bh3/2/F, where
This parameter b is a universal constant sometimes called the second Fowler–Nordheim constant. For barriers of other shapes, we write
where ν(h, F) is a correction factor that determined by numerical integration of eq. (4).
Correction factor for the Schottky–Nordheim barrier
The Schottky–Nordheim barrier, which is the barrier model used in deriving the standard Fowler–Nordheim-type equation, is a special case. In this case, it is known that the correction factor ν {\displaystyle {\it {\nu }}} is a function of a single variable fh, defined by fh = F/Fh, where Fh is the field necessary to reduce the height of a Schottky–Nordheim barrier from h to 0. This field is given by
The parameter fh 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(fh) of a function ν(ℓ′). The latter is a function of mathematical physics in its own right with explicit series expansion and has been called the principal Schottky–Nordheim barrier function. The following good simple approximation for ν(fh) has been found:
Decay width The decay width (in energy), dh, measures how fast the escape probability D decreases as the barrier height h increases; dh is defined by:
When h increases by dh 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
d h ( e l ) = 2 F 3 b h = e F g h . {\displaystyle d_{h}^{\mathrm {(el)} }={\frac {2F}{3b{\sqrt {h}}}}={\frac {eF}{g{\sqrt {h}}}}.}
The decay width dh derived from the more general expression (12) differs from this by a "decay-width correction factor" λd, so:
Usually, the correction factor can be approximated as unity. The decay-width dF for a barrier with h equal to the local work-function φ is of special interest. Numerically this is given by:
For metals, the value of dF 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, but his mathematical analysis of the factor was incorrect. A new (correct) function was introduced by Burgess, Kroemer and Houston in 1953, and its mathematics was developed further by Murphy and Good in 1956. 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:
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(fh), as discussed above; approximation (11) for ν(fh) 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; for this physically unrealistic case, ν(fh) = 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. 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.
Total-energy distribution The energy distribution of the emitted electrons is important both for scientific experiments that use the emitted electron energy distribution to probe aspects of the emitter surface physics and for the field emission sources used in electron beam instruments such as electron microscopes. In the latter case, the "width" (in energy) of the distribution influences how finely the beam can be focused. The theoretical explanation here follows the approach of Forbes. If ε denotes the total electron energy relative to the emitter Fermi level, and Kp denotes the kinetic energy of the electron parallel to the emitter surface, then the electron's normal energy εn (sometimes called its "forwards energy") is defined by
Two types of theoretical energy distribution are recognized: the normal-energy distribution (NED), which shows how the energy εn is distributed immediately after emission (i.e., immediately outside the tunneling barrier); and the total-energy distribution, which shows how the total energy ε is distributed. When the emitter Fermi level is used as the reference zero level, both ε and εn can be either positive or negative. Energy analysis experiments have been made on field emitters since the 1930s. However, only in the late 1950s was it realized (by Young and Mueller [,YM58]) that these experiments always measured the total energy distribution, which is now usually denoted by j(ε). This is also true (or nearly true) when the emission comes from a small field enhancing protrusion on an otherwise flat surface. To see how the total energy distribution can be calculated within the framework of a Sommerfeld free-electron-type model, look at the P-T energy-space diagram (P-T="parallel-total").
This shows the "parallel kinetic energy" Kp on the horizontal axis and the total energy ε on the vertical axis. An electron inside the bulk metal usually has values of Kp and ε that lie within the lightly shaded area. It can be shown that each element dεdKp of this energy space makes a contribution z S f F D d ϵ d K p {\displaystyle z_{\mathrm {S} }f_{\mathrm {FD} }\mathrm {d} {\it {\epsilon }}\mathrm {d} K_{\mathrm {p} }} to the electron current density incident on the inside of the emitter boundary. Here, zS is the universal constant (called here the Sommerfeld supply density):
and f F D {\displaystyle f_{\mathrm {FD} }} is the Fermi–Dirac distribution function:
where T is thermodynamic temperature and kB is the Boltzmann constant. This element of incident current density sees a barrier of height h given by:
The corresponding escape probability is D(h, F): this may be expanded (approximately) in the form
where DF is the escape probability for a barrier of unreduced height equal to the local work-function φ. Hence, the element dεdKp makes a contribution