{"history":[],"article":{"id":"38abe2d3-be3f-4dd5-8607-737cd05626d6","title":"Field electron emission","slug":"field-electron-emission","content":":::summary\n:::\n\n# \n**Field electron emission**, also known as **field emission** (**FE**) and electron **field emission**, is emission of [electrons](https://bliptext.com/articles/electron) induced by an [electrostatic field](https://bliptext.com/articles/electrostatic-field).\nThe most common context is field emission from a [solid](https://bliptext.com/articles/solid) surface into a [vacuum](https://bliptext.com/articles/vacuum).\nHowever, field emission can take place from solid or [liquid](https://bliptext.com/articles/liquid) surfaces, into a vacuum, a [fluid](https://bliptext.com/articles/fluid) (e.g. [air](https://bliptext.com/articles/air)), or any [non-conducting](https://bliptext.com/articles/non-conducting) or weakly conducting [dielectric](https://bliptext.com/articles/dielectric).\nThe field-induced promotion of electrons from the [valence](https://bliptext.com/articles/valence-chemistry) to [conduction band](https://bliptext.com/articles/conduction-band) of [semiconductors](https://bliptext.com/articles/semiconductors) (the [Zener effect](https://bliptext.com/articles/zener-effect)) can also be regarded as a form of field emission.\nThe terminology is historical because related phenomena of surface photoeffect, [thermionic emission](https://bliptext.com/articles/thermionic-emission) (or [Richardson–Dushman effect](https://bliptext.com/articles/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.\nWhen the term field emission is used without qualifiers, it typically means \"cold emission\".\nField emission in pure metals occurs in high [electric fields](https://bliptext.com/articles/electric-field): the gradients are typically higher than 1 gigavolt per metre and strongly dependent upon the [work function](https://bliptext.com/articles/work-function).\nWhile 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](https://bliptext.com/articles/electrical-breakdown) phenomena, which engineers work to prevent.\nExamples of applications for surface field emission include the construction of bright electron sources for high-resolution [electron microscopes](https://bliptext.com/articles/electron-microscope) or the discharge of induced charges from [spacecraft](https://bliptext.com/articles/spacecraft).\nDevices that eliminate induced charges are termed [charge-neutralizers](https://bliptext.com/articles/charge-neutralizer).\nField emission was explained by [quantum tunneling](https://bliptext.com/articles/quantum-tunneling) of electrons in the late 1920s.\nThis was one of the triumphs of the nascent [quantum mechanics](https://bliptext.com/articles/quantum-mechanics).\nThe theory of field emission from bulk metals was proposed by [Ralph H. Fowler](https://bliptext.com/articles/ralph-h-fowler) and [Lothar Wolfgang Nordheim](https://bliptext.com/articles/lothar-wolfgang-nordheim).\nA family of approximate equations, [Fowler–Nordheim equations](https://bliptext.com/articles/), is named after them.\nStrictly, Fowler–Nordheim equations apply only to field emission from bulk metals and (with suitable modification) to other bulk [crystalline solids](https://bliptext.com/articles/crystalline-solid), but they are often used – as a rough approximation – to describe field emission from other materials.\n\n# Terminology and conventions\nField electron emission, field-induced electron emission, field emission and electron field emission are general names for this experimental phenomenon and its theory.\nThe first name is used here.\n[Fowler–Nordheim tunneling](https://bliptext.com/articles/) is the wave-mechanical tunneling of electrons through a rounded triangular barrier created at the surface of an electron conductor by applying a very high electric field.\nIndividual electrons can escape by Fowler–Nordheim tunneling from many materials in various different circumstances.\n[Cold field electron emission](https://bliptext.com/articles/) (CFE) is the name given to a particular statistical emission regime, in which the electrons in the emitter are initially in internal [thermodynamic equilibrium](https://bliptext.com/articles/thermodynamic-equilibrium), and in which most emitted electrons escape by Fowler–Nordheim tunneling from electron states close to the emitter [Fermi level](https://bliptext.com/articles/fermi-level).\n(By contrast, in the [Schottky emission](https://bliptext.com/articles/thermionic-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.\n[Fowler–Nordheim-type equations](https://bliptext.com/articles/) are a family of approximate equations derived to describe CFE from the internal electron states in bulk metals.\nThe different members of the family represent different degrees of approximation to reality.\nApproximate equations are necessary because, for physically realistic models of the tunneling barrier, it is in principle mathematically impossible to solve the [Schrödinger equation](https://bliptext.com/articles/schr-dinger-equation) exactly in any simple way.\nThere is no theoretical reason to believe that Fowler–Nordheim-type equations validly describe field emission from materials other than bulk crystalline solids.\nFor metals, the CFE regime extends to well above room temperature.\nThere are other electron emission regimes (such as \"[thermal electron emission](https://bliptext.com/articles/thermionic-emission)\" and \"[Schottky emission](https://bliptext.com/articles/thermionic-emission)\") that require significant external heating of the emitter.\nThere 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.\nA 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.\nCare is necessary because 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.\nHistorically, 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\".\nEquations in this article are written using the [International System of Quantities](https://bliptext.com/articles/international-system-of-quantities) (ISQ).\nThis is the modern (post-1970s) international system, based around the rationalized-meter-kilogram-second (rmks) system of equations, which is used to define SI units.\nOlder field emission literature (and papers that directly copy equations from old literature) often write some equations using an older equation system that does not use the quantity ε0.\nIn this article, all such equations have been converted to modern international form.\nFor clarity, this should always be done.\nSince 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), values of most universal constants are given here in units that involve eV, V and nm.\nIncreasingly, this is normal practice in field emission research.\nHowever, all equations here are ISQ-compatible equations and remain dimensionally consistent, as is required by the modern international system.\nTo indicate their status, numerical values of universal constants are given to seven significant figures.\nValues are derived using the 2006 values of the fundamental constants.\n\n# Early history of field electron emission\nField electron emission has a long, complicated and messy history.\nThis section covers the early history, up to the derivation of the original Fowler–Nordheim-type equation in 1928.\nIn retrospect, it seems likely that the electrical discharges reported by J.H. Winkler in 1744 were started by CFE from his wire electrode.\nHowever, meaningful investigations had to wait until after [J.J. Thomson](https://bliptext.com/articles/j-j-thomson)'s identification of the electron in 1897, and until after it was understood – from [thermal emission](https://bliptext.com/articles/thermionic-emission) and [photo-emission](https://bliptext.com/articles/photoemission) work – that electrons could be emitted from inside metals (rather than from [surface-adsorbed gas molecules](https://bliptext.com/articles/adsorption)), and that – in the absence of applied fields – electrons escaping from metals had to overcome a work function barrier.\nIt was suspected at least as early as 1913 that field-induced emission was a separate physical effect.\nHowever, only after vacuum and specimen cleaning techniques had significantly improved, did this become well established.\n[Lilienfeld](https://bliptext.com/articles/julius-edgar-lilienfeld) (who was primarily interested in electron sources for medical [X-ray](https://bliptext.com/articles/x-ray) applications) published in 1922 the first clear account in English of the experimental phenomenology of the effect he had called \"autoelectronic emission\".\nHe had worked on this topic, in Leipzig, since about 1910.\nKleint describes this and other early work.\nAfter 1922, experimental interest increased, particularly in the groups led by [Millikan](https://bliptext.com/articles/robert-andrews-millikan) at the California Institute of Technology (Caltech) in [Pasadena, California](https://bliptext.com/articles/pasadena-california), and by Gossling at the [General Electric Company](https://bliptext.com/articles/general-electric-company) in London.\nAttempts to understand autoelectronic emission included plotting experimental current–voltage (i–V) data in different ways, to look for a straight-line relationship.\nCurrent increased with voltage more rapidly than linearly, but plots of type log(i) vs. V were not straight.\n[Walter H. Schottky](https://bliptext.com/articles/walter-h-schottky) suggested in 1923 that the effect might be due to thermally induced emission over a field-reduced barrier.\nIf so, then plots of log(i) vs. √V should be straight, but they were not.\nNor is Schottky's explanation compatible with the experimental observation of only very weak temperature dependence in CFE – a point initially overlooked.\nA breakthrough came when [C.C. Lauritsen](https://bliptext.com/articles/charles-christian-lauritsen)\n(and [J. Robert Oppenheimer](https://bliptext.com/articles/j-robert-oppenheimer) independently ) found that plots of log(i) vs. 1/V yielded good straight lines.\nThis result, published by Millikan and Lauritsen in early 1928, was known to [Fowler](https://bliptext.com/articles/ralph-h-fowler) and [Nordheim](https://bliptext.com/articles/lothar-wolfgang-nordheim).\nOppenheimer had predicted that the field-induced [tunneling](https://bliptext.com/articles/quantum-tunnelling) 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](https://bliptext.com/articles/quantum-tunnelling) of electrons [from atomic-like orbitals](https://bliptext.com/articles/atomic-orbital) in surface metal atoms.\nAn alternative Fowler–Nordheim theory explained both the Millikan–Lauritsen finding and the very weak dependence of current on temperature.\nFowler–Nordheim theory predicted both to be consequences if CFE were due to field-induced tunneling from [free-electron-type states](https://bliptext.com/articles/free-electron-model) in what we would now call a metal [conduction band](https://bliptext.com/articles/electronic-band-structure), with the electron states occupied in accordance with [Fermi–Dirac statistics](https://bliptext.com/articles/fermi-dirac-statistics).\nOppenheimer had mathematical details of his theory seriously incorrect.\nThere was also a small numerical error in the final equation given by Fowler–Nordheim theory for CFE [current density](https://bliptext.com/articles/current-density): this was corrected in the 1929 paper of.\nStrictly, 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 the form (log(i/V2) vs. 1/V) should be exact straight lines.\nHowever, contemporary experimental techniques were not good enough to distinguish between the Fowler–Nordheim theoretical result and the Millikan–Lauritsen experimental result.\nThus, by 1928 basic physical understanding of the origin of CFE from bulk metals had been achieved, and the original Fowler–Nordheim-type equation had been derived.\nThe literature often presents Fowler–Nordheim work as a proof of the existence of [electron tunneling](https://bliptext.com/articles/quantum-tunnelling), as predicted by wave-mechanics.\nWhilst this is correct, the validity of wave-mechanics was largely accepted by 1928.\nThe more important role of the Fowler–Nordheim paper was that it was a convincing argument from experiment that [Fermi–Dirac statistics](https://bliptext.com/articles/fermi-dirac-statistics) applied to the behavior of electrons in metals, as suggested by [Sommerfeld](https://bliptext.com/articles/arnold-sommerfeld) in 1927.\nThe success of Fowler–Nordheim theory did much to support the correctness of Sommerfeld's ideas, and greatly helped to establish modern [electron band theory](https://bliptext.com/articles/electronic-band-structure).\nIn particular, the original Fowler–Nordheim-type equation was one of the first to incorporate the [statistical-mechanical](https://bliptext.com/articles/statistical-mechanics) consequences of the existence of [electron spin](https://bliptext.com/articles/electron-spin) into the theory of an experimental condensed-matter effect.\nThe Fowler–Nordheim paper also established the physical basis for a unified treatment of field-induced and [thermally induced electron emission](https://bliptext.com/articles/thermionic-emission).\nPrior 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.\nThe 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](https://bliptext.com/articles/electronic-band-structure) occupied in accordance with Fermi–Dirac statistics, but would be emitted in statistically different ways under different conditions of temperature and applied field.\nThe ideas of [Oppenheimer](https://bliptext.com/articles/j-robert-oppenheimer), Fowler and Nordheim were also an important stimulus to the development, by [George Gamow](https://bliptext.com/articles/george-gamow), and [Ronald W. Gurney](https://bliptext.com/articles/ronald-w-gurney) and [Edward Condon](https://bliptext.com/articles/edward-condon), later in 1928, of the theory of the [radioactive decay](https://bliptext.com/articles/radioactive-decay) of nuclei (by [alpha particle](https://bliptext.com/articles/alpha-particle) tunneling).\n\n# Practical applications: past and present\n\n## Field electron microscopy and related basics\nAs already indicated, the early experimental work on field electron emission (1910–1920) was driven by [Lilienfeld's](https://bliptext.com/articles/julius-edgar-lilienfeld) desire to develop miniaturized [X-ray](https://bliptext.com/articles/x-ray) tubes for medical applications.\nHowever, it was too early for this technology to succeed.\nAfter Fowler–Nordheim theoretical work in 1928, a major advance came with the development in 1937 by [Erwin W. Mueller](https://bliptext.com/articles/erwin-wilhelm-mueller) of the spherical-geometry [field electron microscope](https://bliptext.com/articles/field-emission-microscope) (FEM) (also called the \"field emission microscope\").\nIn this instrument, the electron emitter is a sharply pointed wire, of apex radius r.\nThis is placed, in a vacuum enclosure, opposite an image detector (originally a phosphor screen), at a distance R from it.\nThe 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.\nIn FEM studies the apex radius is typically 100 nm to 1 μm.\nThe 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\".\nWhen 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.\nIn 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.\nThis is as predicted by the exponent of Fowler–Nordheim-type equations [see eq.\n(30) below].\nThe [adsorption](https://bliptext.com/articles/adsorption) of layers of gas atoms (such as oxygen) onto the emitter surface, or part of it, can create surface [electric dipoles](https://bliptext.com/articles/electric-dipole) that change the local work function of this part of the surface.\nThis affects the FEM image; also, the change of work-function can be measured using a Fowler–Nordheim plot (see below).\nThus, the FEM became an early observational tool of [surface science](https://bliptext.com/articles/surface-science).\nFor example, in the 1960s, FEM results contributed significantly to discussions on [heterogeneous catalysis](https://bliptext.com/articles/heterogeneous-catalysis).\nFEM has also been used for studies of [surface-atom diffusion](https://bliptext.com/articles/surface-diffusion).\nHowever, FEM has now been almost completely superseded by newer surface-science techniques.\nA 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.\nThis was important in experiments designed to test the validity of the standard Fowler–Nordheim-type equation.\nThese 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](https://bliptext.com/articles/transmission-electron-microscope) observations of emitter shape and electrostatic modeling.\nAgreement to within about 10% was achieved.\nOnly 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.\nAnalysis of the resulting Fowler–Nordheim plot yields a work-function value close to the independently known work-function of the emitter.\n\n## Field electron spectroscopy (electron energy analysis)\nEnergy distribution measurements of field-emitted electrons were first reported in 1939.\nIn 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\").\nThis is because, in spherical geometry, the electrons move in such a fashion that [angular momentum](https://bliptext.com/articles/angular-momentum) about a point in the emitter is very nearly conserved.\nHence any [kinetic energy](https://bliptext.com/articles/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.\nSo what gets measured in an energy analyzer is the [total energy](https://bliptext.com/articles/total-energy) at emission.\nWith the development of sensitive electron energy analyzers in the 1960s, it became possible to measure fine details of the total energy distribution.\nThese reflect fine details of the [surface physics](https://bliptext.com/articles/surface-physics), and the technique of Field Electron Spectroscopy flourished for a while, before being superseded by newer surface-science techniques.\n\n## Field electron emitters as electron-gun sources\nTo achieve high-resolution in [electron microscopes](https://bliptext.com/articles/transmission-electron-microscope) and other electron beam instruments (such as those used for [electron beam lithography](https://bliptext.com/articles/electron-beam-lithography)), it is helpful to start with an electron source that is small, optically bright and stable.\nSources based on the geometry of a Mueller emitter qualify well on the first two criteria.\nThe first electron microscope (EM) observation of an individual atom was made by Crewe, Wall and Langmore in 1970, using a [scanning electron microscope](https://bliptext.com/articles/scanning-electron-microscope) equipped with an early field emission gun.\nFrom the 1950s onwards, extensive effort has been devoted to the development of field emission sources for use in [electron guns](https://bliptext.com/articles/electron-gun).\n[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](https://bliptext.com/articles/adsorption) (usually [Zirconium oxide](https://bliptext.com/articles/zirconium-oxide) – ZrO) into the flat apex of a [(100) oriented](https://bliptext.com/articles/miller-index) [Tungsten](https://bliptext.com/articles/tungsten) emitter.\nSources that operate at room temperature have the disadvantage that they rapidly become covered with adsorbate [molecules](https://bliptext.com/articles/molecule) that arrive from the [vacuum](https://bliptext.com/articles/vacuum) system walls, and the emitter has to be cleaned from time to time by \"flashing\" to high temperature.\nNowadays, it is more common to use Mueller-emitter-based sources that are operated at elevated temperatures, either in the [Schottky emission](https://bliptext.com/articles/thermionic-emission) regime or in the so-called temperature-field intermediate regime.\nMany modern high-resolution electron microscopes and electron beam instruments use some form of Mueller-emitter-based electron source.\nCurrently, attempts are being made to develop [carbon nanotubes](https://bliptext.com/articles/carbon-nanotubes) (CNTs) as electron-gun field emission sources.\nThe 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.\nVarious shape models have been tried for Mueller emitters; the best seems to be the \"Sphere on Orthogonal Cone\" (SOC) model introduced by Dyke, Trolan.\nDolan and Barnes in 1953.\nImportant simulations, involving trajectory tracing using the SOC emitter model, were made by Wiesener and Everhart.\nNowadays, the facility to simulate field emission from Mueller emitters is often incorporated into the commercial electron-optics programmes used to design electron beam instruments.\nThe design of efficient modern field-emission electron guns requires highly specialized expertise.\n# [Schottky-emitter electron source of an Electron microscope | upload.wikimedia.org/wikipedia/commons/0/0c/Schottky-Emitter///01.jpg]\n\n## Atomically sharp emitters\nNowadays it is possible to prepare very sharp emitters, including emitters that end in a single atom.\nIn this case, electron emission comes from an area about twice the crystallographic size of a single atom.\nThis was demonstrated by comparing FEM and [field ion microscope](https://bliptext.com/articles/field-ion-microscope) (FIM) images of the emitter.\nSingle-atom-apex Mueller emitters also have relevance to the [scanning probe microscopy](https://bliptext.com/articles/scanning-probe-microscopy) and [helium scanning ion microscopy](https://bliptext.com/articles/scanning-helium-ion-microscope) (He SIM).\nTechniques for preparing them have been under investigation for many years.\nA 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.\n\n## Large-area field emission sources: vacuum nanoelectronics\n\n### Materials aspects\nLarge-area field emission sources have been of interest since the 1970s.\nIn these devices, a high density of individual field emission sites is created on a substrate (originally silicon).\nThis research area became known, first as \"vacuum microelectronics\", now as \"vacuum nanoelectronics\".\nOne of the original two device types, the \"[Spindt array](https://bliptext.com/articles/spindt-tip)\", used [silicon-integrated-circuit (IC)](https://bliptext.com/articles/integrated-circuit) fabrication techniques to make regular arrays in which [molybdenum](https://bliptext.com/articles/molybdenum) cones were deposited in small cylindrical voids in an oxide film, with the void covered by a counterelectrode with a central circular aperture.\nThis overall geometry has also been used with [carbon nanotubes](https://bliptext.com/articles/carbon-nanotubes) grown in the void.\nThe other original device type was the \"Latham emitter\".\nThese 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.\nThe device field-emits because its microstructure/nanostructure has field-enhancing properties.\nThis material had a potential production advantage, in that it could be deposited as an \"ink\", so IC fabrication techniques were not needed.\nHowever, in practice, uniformly reliable devices proved difficult to fabricate.\nResearch advanced to look for other materials that could be deposited/grown as thin films with suitable field-enhancing properties.\nIn 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.\nIf 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\nThe parameter γ is called the \"field enhancement factor\" and is basically determined by the object's shape.\nSince 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.\nHence, for a given value of W, the lower the applied voltage V at which significant emission occurs.\nFor 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](https://bliptext.com/articles/diamond-like-carbon).\nHowever, interest subsequently lessened, partly due to the arrival of [CNT](https://bliptext.com/articles/carbon-nanotube) 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](https://bliptext.com/articles/chemical-vapor-deposition): this suggested that [quality control](https://bliptext.com/articles/quality-control) of an industrial-scale production process might be problematic.\nThe introduction of CNT field emitters, both in \"mat\" form and in \"grown array\" forms, was a significant step forward.\nExtensive research has been undertaken into both their physical characteristics and possible technological applications.\nFor field emission, an advantage of CNTs is that, due to their shape, with its high [aspect ratio](https://bliptext.com/articles/aspect-ratio), they are \"natural field-enhancing objects\".\nIn 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.\nA particular aim is to develop \"high-γ\" nanostructures with a sufficiently high density of individual emission sites.\nThin films of nanotubes in form of nanotube webs are also used for development of field emission electrodes.\nIt is shown that by fine-tuning the fabrication parameters, these webs can achieve an optimum density of individual emission sites.\nDouble-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.\nCommon 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.\nThus, an important industrial requirement is \"robustness in poor vacuum conditions\"; this needs to be taken into account in research on new emitter materials.\nAt 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.\n\n### Applications\nThe development of large-area field emission sources was originally driven by the wish to create new, more efficient, forms of [electronic information display](https://bliptext.com/articles/flat-panel-display).\nThese are known as \"[field-emission displays](https://bliptext.com/articles/field-emission-display)\" or \"nano-emissive displays\".\nAlthough 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].\nOther proposed applications of large-area field emission sources include [microwave](https://bliptext.com/articles/microwave) generation, space-vehicle neutralization, [X-ray generation](https://bliptext.com/articles/x-ray-generation), and (for array sources) multiple [e-beam lithography](https://bliptext.com/articles/electron-beam-lithography).\nThere are also recent attempts to develop large-area emitters on flexible substrates, in line with wider trends towards \"[plastic electronics](https://bliptext.com/articles/plastic-electronics)\".\nThe development of such applications is the mission of vacuum nanoelectronics.\nHowever, field emitters work best in conditions of good ultrahigh vacuum.\nTheir most successful applications to date (FEM, FES and EM guns) have occurred in these conditions.\nThe 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.\n\n## Vacuum breakdown and electrical discharge phenomena\nAs already indicated, it is now thought that the earliest manifestations of field electron emission were the electrical discharges it caused.\nAfter Fowler–Nordheim work, it was understood that CFE was one of the possible primary underlying causes of vacuum breakdown and electrical discharge phenomena.\n(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.\nProcedures were (and are) used to round and smooth the surfaces of electrodes that might generate unwanted field electron emission currents.\nHowever the work of Latham and others showed that emission could also be associated with the presence of semiconducting inclusions in smooth surfaces.\nThe physics of how the emission is generated is still not fully understood, but suspicion exists that so-called \"triple-junction effects\" may be involved.\nFurther information may be found in Latham's book and in the on-line bibliography.\n\n## Internal electron transfer in electronic devices\nIn some electronic devices, electron transfer from one material to another, or (in the case of sloping bands) from one band to another (\"[Zener tunneling](https://bliptext.com/articles/zener-diode)\"), takes place by a field-induced tunneling process that can be regarded as a form of Fowler–Nordheim tunneling.\nFor example, [Rhoderick's](https://bliptext.com/articles/emlyn-rhoderick) book discusses the theory relevant to [metal–semiconductor contacts](https://bliptext.com/articles/metal-semiconductor-junction).\n\n# Fowler–Nordheim tunneling\n\n## Introduction\nThe next part of this article deals with the basic theory of cold field electron emission from bulk metals.\nThis is best treated in four main stages, involving theory associated with: (1) derivation of a formula for \"[escape probability](https://bliptext.com/articles/transmission-coefficient)\", 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.\nThe modified equations needed for large-area emitters, and issues of experimental data analysis, are dealt with separately.\nFowler–Nordheim tunneling is the [wave-mechanical tunneling](https://bliptext.com/articles/quantum-tunnelling) of an electron through an exact or rounded triangular barrier.\nTwo basic situations are recognized: (1) when the electron is initially in a [localized state](https://bliptext.com/articles/surface-states); (2) when the electron is initially not strongly localized, and is best represented by a [travelling wave](https://bliptext.com/articles/travelling-wave).\nEmission from a bulk metal [conduction band](https://bliptext.com/articles/conduction-band) is a situation of the second type, and discussion here relates to this case.\nIt is also assumed that the barrier is one-dimensional (i.e., has no lateral structure), and has no fine-scale structure that causes \"[scattering](https://bliptext.com/articles/scattering)\" or \"resonance\" effects.\nTo keep this explanation of Fowler–Nordheim tunneling relatively simple, these assumptions are needed; but the atomic structure of matter is in effect being disregarded.\n\n## Motive energy\nFor an electron, the one-dimensional [Schrödinger equation](https://bliptext.com/articles/schr-dinger-equation) can be written in the form\nwhere Ψ(x) is the electron [wave-function](https://bliptext.com/articles/wave-function), expressed as a function of distance x measured from the emitter's electrical surface, ħ is the [reduced Planck constant](https://bliptext.com/articles/reduced-planck-constant), m is the electron mass, U(x) is the [electron potential energy](https://bliptext.com/articles/potential-energy), En is the [total electron energy](https://bliptext.com/articles/total-energy) associated with motion in the x-direction, and M(x) = [U(x) − En] is called the electron motive energy.\nM(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.\nThe shape of a tunneling barrier is determined by how M(x) varies with position in the region where M(x) > 0.\nTwo models have special status in field emission theory: the exact triangular (ET) barrier and the Schottky–Nordheim (SN) barrier.\nThese are given by equations (2) and (3), respectively:\nHere h is the zero-field height (or unreduced height) of the barrier, e is the [elementary positive charge](https://bliptext.com/articles/elementary-charge), F is the barrier field, and ε0 is the [electric constant](https://bliptext.com/articles/vacuum-permittivity).\nBy convention, F is taken as positive, even though the [classical electrostatic field](https://bliptext.com/articles/electric-field) would be negative.\nThe SN equation uses the classical image potential energy to represent the physical effect \"correlation and exchange\".\n\n## Escape probability\nFor an electron approaching a given barrier from the inside, the probability of escape (or \"[transmission coefficient](https://bliptext.com/articles/transmission-coefficient)\" or \"penetration coefficient\") is a function of h and F, and is denoted by D(h, F).\nThe primary aim of tunneling theory is to calculate D(h, F).\nFor physically realistic barrier models, such as the Schottky–Nordheim barrier, the Schrödinger equation cannot be solved exactly in any simple way.\nThe following so-called \"semi-classical\" approach can be used.\nA parameter G(h, F) can be defined by the [JWKB (Jeffreys-Wentzel-Kramers-Brillouin)](https://bliptext.com/articles/wkb) integral:\nwhere 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\nForbes 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\nwhere the tunneling pre-factor P can in principle be evaluated by complicated iterative integrations along a path in [complex space](https://bliptext.com/articles/complex-hilbert-space).\nIn the CFE regime we have (by definition) G ≫ 1.\nAlso, for simple models P ≈ 1.\nSo eq.\n(6) reduces to the so-called simple [JWKB](https://bliptext.com/articles/wkb) formula:\nFor the exact triangular barrier, putting eq.\n(2) into eq.\n(4) yields GET = bh3/2/F, where\nThis parameter b is a universal constant sometimes called the second Fowler–Nordheim constant.\nFor barriers of other shapes, we write\nwhere ν(h, F) is a correction factor that in general has to be determined by [numerical integration](https://bliptext.com/articles/numerical-integration), using eq.\n(4).\n\n## Correction factor for the Schottky–Nordheim barrier\nThe Schottky–Nordheim barrier, which is the barrier model used in deriving the standard Fowler–Nordheim-type equation, is a special case.\nIn this case, it is known that the correction factor \\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.\nThis field is given by\nThe 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.\nFor the Schottky–Nordheim barrier, ν(h, F) is given by the particular value ν(fh) of a function ν(ℓ).\nThe latter is a function of mathematical physics in its own right and has been called the principal Schottky–Nordheim barrier function.\nAn explicit series expansion for ν(ℓ) is derived in a 2008 paper by J. Deane.\nThe following good simple approximation for ν(fh) has been found:\n# [Schottky–Nordheim barrier for Fowler–Nordheim field emission (and enhanced thermionic emission) | upload.wikimedia.org/wikipedia/commons/5/53/Sn///barrier.svg]\n\n## Decay width\nThe decay width (in energy), dh, measures how fast the escape probability D decreases as the barrier height h increases; dh is defined by:\nWhen h increases by dh then the escape probability D decreases by a factor close to e ( ≈ 2.718282).\nFor an elementary model, based on the exact triangular barrier, where we put ν = 1 and P ≈ 1, we get\nThe decay width dh derived from the more general expression (12) differs from this by a \"decay-width correction factor\" λd, so:\nUsually, the correction factor can be approximated as unity.\nThe decay-width dF for a barrier with h equal to the local work-function φ is of special interest.\nNumerically this is given by:\nFor metals, the value of dF is typically of order 0.2 eV, but varies with barrier-field F.\n\n## Comments\nA historical note is necessary.\nThe idea that the Schottky–Nordheim barrier needed a correction factor, as in eq.\n(9), was introduced by Nordheim in 1928, but his mathematical analysis of the factor was incorrect.\nA new (correct) function was introduced by Burgess, [Kroemer](https://bliptext.com/articles/herbert-kroemer) and Houston in 1953, and its mathematics was developed further by Murphy and Good in 1956.\nThis 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\".\nOnly recently (2006 to 2008) has it been realized that, mathematically, it is much better to use the variable ℓ ( = y2).\nAnd 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](https://bliptext.com/articles/hypergeometric-differential-equation)).\nAlso, approximation (11) has been found only recently.\nApproximation (11) outperforms, and will presumably eventually displace, all older approximations of equivalent complexity.\nThese recent developments, and their implications, will probably have a significant impact on field emission research in due course.\nThe following summary brings these results together.\nFor tunneling well below the top of a well-behaved barrier of reasonable height, the escape probability D(h, F) is given formally by:\nwhere ν(h, F) is a correction factor that in general has to be found by numerical integration.\nFor 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.\nThe 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.\nThe 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.\nThe approach described here was originally developed to describe Fowler–Nordheim tunneling from smooth, classically flat, planar emitting surfaces.\nIt is adequate for smooth, classical curved surfaces of radii down to about 10 to 20 nm.\nIt 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.\nWhen 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.\nAs 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.\nTaking atomic structure properly into account is a very difficult problem, and only limited progress has been made.\nHowever, 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.\n(15), by amounts that cannot easily be estimated at present.\nAll 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](https://bliptext.com/articles/travelling-wave).\nThe 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.\n\n# Total-energy distribution\nThe 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](https://bliptext.com/articles/surface-physics) and for the field emission sources used in electron beam instruments such as [electron microscopes](https://bliptext.com/articles/electron-microscopes).\nIn the latter case, the \"width\" (in energy) of the distribution influences how finely the beam can be focused.\nThe theoretical explanation here follows the approach of Forbes.\nIf ε denotes the total electron energy relative to the emitter Fermi level, and Kp denotes the [kinetic energy](https://bliptext.com/articles/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\nTwo 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.\nWhen the emitter Fermi level is used as the reference zero level, both ε and εn can be either positive or negative.\nEnergy analysis experiments have been made on field emitters since the 1930s.\nHowever, 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).\nThis is also true (or nearly true) when the emission comes from a small field enhancing protrusion on an otherwise flat surface.\nTo see how the total energy distribution can be calculated within the framework of a [Sommerfeld free-electron-type model](https://bliptext.com/articles/free-electron-model), look at the P-T energy-space diagram (P-T=\"parallel-total\").\nThis shows the \"parallel kinetic energy\" Kp on the horizontal axis and the total energy ε on the vertical axis.\nAn electron inside the bulk metal usually has values of Kp and ε that lie within the lightly shaded area.\nIt can be shown that each element dεdKp of this energy space makes a contribution to the electron current density incident on the inside of the emitter boundary.\nHere, zS is the universal constant (called here the [Sommerfeld](https://bliptext.com/articles/arnold-sommerfeld) supply density):\nand is the [Fermi–Dirac distribution function](https://bliptext.com/articles/fermi-dirac-statistics):\nwhere T is [thermodynamic temperature](https://bliptext.com/articles/thermodynamic-temperature) and kB is the [Boltzmann constant](https://bliptext.com/articles/boltzmann-constant).\nThis element of incident current density sees a barrier of height h given by:\nThe corresponding escape probability is D(h, F): this may be expanded (approximately) in the form\nwhere DF is the escape probability for a barrier of unreduced height equal to the local work-function φ.\nHence, the element dεdKp makes a contribution to the emission current density, and the total contribution made by incident electrons with energies in the elementary range dε is thus\nwhere the integral is in principle taken along the strip shown in the diagram, but can in practice be extended to ∞ when the decay-width dF is very much less than the [Fermi energy](https://bliptext.com/articles/fermi-energy) KF (which is always the case for a metal).\nThe outcome of the integration can be written:\nwhere and are values appropriate to a barrier of unreduced height h equal to the local work function φ, and is defined by this equation.\nFor a given emitter, with a given field applied to it, is independent of F, so eq.\n(21) shows that the shape of the distribution (as ε increases from a negative value well below the Fermi level) is a rising exponential, multiplied by the FD distribution function.\nThis generates the familiar distribution shape first predicted by Young.\nAt low temperatures, goes sharply from 1 to 0 in the vicinity of the Fermi level, and the [FWHM](https://bliptext.com/articles/full-width-at-half-maximum) of the distribution is given by:\nThe fact that experimental CFE total energy distributions have this basic shape is a good experimental confirmation that electrons in metals obey [Fermi–Dirac statistics](https://bliptext.com/articles/fermi-dirac-statistics).\n# [P-T energy-space diagram, showing the region in P-T energy space where traveling-wave electron states exist. | upload.wikimedia.org/wikipedia/commons/3/35/P-T///Energy///Space///%28cropped%29.jpg]\n\n# Cold field electron emission\n\n## Fowler–Nordheim-type equations\n\n### Introduction\nFowler–Nordheim-type equations, in the J–F form, are (approximate) theoretical equations derived to describe the local current density J emitted from the internal electron states in the conduction band of a bulk metal.\nThe emission current density (ECD) J for some small uniform region of an emitting surface is usually expressed as a function J(φ, F) of the local work-function φ and the local barrier field F that characterize the small region.\nFor sharply curved surfaces, J may also depend on the parameter(s) used to describe the surface curvature.\nOwing to the physical assumptions made in the original derivation, the term Fowler–Nordheim-type equation has long been used only for equations that describe the ECD at zero temperature.\nHowever, it is better to allow this name to include the slightly modified equations (discussed below) that are valid for finite temperatures within the CFE emission regime.\n\n### Zero-temperature form\nCurrent density is best measured in A/m2.\nThe total current density emitted from a small uniform region can be obtained by integrating the total energy distribution j(ε) with respect to total electron energy ε.\nAt zero temperature, the [Fermi–Dirac distribution function](https://bliptext.com/articles/fermi-dirac-statistics) fFD = 1 for ε < 0, and fFD = 0 for ε > 0.\nSo the ECD at 0 K, J0, is given from eq.\n(18) by\nwhere is the effective supply for state F, and is defined by this equation.\nStrictly, the lower limit of the integral should be −KF, where KF is the [Fermi energy](https://bliptext.com/articles/fermi-energy); but if dF is very much less than KF (which is always the case for a metal) then no significant contribution to the integral comes from energies below KF, and it can formally be extended to –∞.\nResult (23) can be given a simple and useful physical interpretation by referring to Fig. 1. The electron state at point \"F\" on the diagram (\"state F\") is the \"forwards moving state at the Fermi level\" (i.e., it describes a Fermi-level electron moving normal to and towards the emitter surface).\nAt 0 K, an electron in this state sees a barrier of unreduced height φ, and has an escape probability DF that is higher than that for any other occupied electron state.\nSo it is convenient to write J0 as ZFDF, where the \"effective supply\" ZF is the current density that would have to be carried by state F inside the metal if all of the emission came out of state F.\nIn practice, the current density mainly comes out of a group of states close in energy to state F, most of which lie within the heavily shaded area in the energy-space diagram.\nSince, for a [free-electron model](https://bliptext.com/articles/free-electron-model), the contribution to the current density is directly proportional to the area in energy space (with the Sommerfeld supply density zS as the constant of proportionality), it is useful to think of the ECD as drawn from electron states in an area of size dF2 (measured in eV2) in the energy-space diagram.\nThat is, it is useful to think of the ECD as drawn from states in the heavily shaded area in Fig. 1. (This approximation gets slowly worse as temperature increases.)\nZF can also be written in the form:\nwhere the universal constant a, sometimes called the First Fowler–Nordheim Constant, is given by\nThis shows clearly that the pre-exponential factor aφ−1F2, that appears in Fowler–Nordheim-type equations, relates to the effective supply of electrons to the emitter surface, in a free-electron model.\n\n### Non-zero temperatures\nTo obtain a result valid for non-zero temperature, we note from eq.\n(23) that zSdFDF = J0/dF. So when eq.\n(21) is integrated at non-zero temperature, then – on making this substitution, and inserting the explicit form of the Fermi–Dirac distribution function – the ECD J can be written in the form:\nwhere λT is a temperature correction factor given by the integral.\nThe integral can be transformed, by writing and, and then u = \\exp(x), into the standard result:\nThis is valid for w > 1 (i.e., dF/kBT > 1).\nHence – for temperatures such that kBT < dF:\nwhere the expansion is valid only if (πkBT / dF) ≪ 1.\nAn example value (for φ = 4.5 eV, F = 5 V/nm, T = 300 K) is λT = 1.024.\nNormal thinking has been that, in the CFE regime, λT is always small in comparison with other uncertainties, and that it is usually unnecessary to explicitly include it in formulae for the current density at room temperature.\nThe emission regimes for metals are, in practice, defined, by the ranges of barrier field F and temperature T for which a given family of emission equations is mathematically adequate.\nWhen the barrier field F is high enough for the CFE regime to be operating for metal emission at 0 K, then the condition kBT < dF provides a formal upper bound (in temperature) to the CFE emission regime.\nHowever, it has been argued that (due to approximations made elsewhere in the derivation) the condition kBT < 0.7dF is a better working limit: this corresponds to a λT-value of around 1.09, and (for the example case) an upper temperature limit on the CFE regime of around 1770 K. This limit is a function of barrier field.\nNote that result (28) here applies for a barrier of any shape (though dF will be different for different barriers).\n\n### Physically complete Fowler–Nordheim-type equation\nResult (23) also leads to some understanding of what happens when atomic-level effects are taken into account, and the [band-structure](https://bliptext.com/articles/band-structure) is no longer free-electron like.\nDue to the presence of the atomic ion-cores, the surface barrier, and also the electron [wave-functions](https://bliptext.com/articles/wave-function) at the surface, will be different.\nThis will affect the values of the correction factor \\nu, the prefactor P, and (to a limited extent) the correction factor λd. These changes will, in turn, affect the values of the parameter DF and (to a limited extent) the parameter dF. For a real metal, the supply density will vary with position in energy space, and the value at point \"F\" may be different from the Sommerfeld supply density.\nWe can take account of this effect by introducing an electronic-band-structure correction factor λB into eq.\n(23). Modinos has discussed how this factor might be calculated: he estimates that it is most likely to be between 0.1 and 1; it might lie outside these limits but is most unlikely to lie outside the range 0.01 < λB < 10.\nBy defining an overall supply correction factor λZ equal to λT λB λd2, and combining equations above, we reach the so-called physically complete Fowler–Nordheim-type equation:\nwhere [= (φ, F)] is the exponent correction factor for a barrier of unreduced height φ.\nThis is the most general equation of the Fowler–Nordheim type.\nOther equations in the family are obtained by substituting specific expressions for the three correction factors, PF and λZ it contains.\nThe so-called elementary Fowler–Nordheim-type equation, that appears in undergraduate textbook discussions of field emission, is obtained by putting λZ → 1, PF → 1, → 1; this does not yield good quantitative predictions because it makes the barrier stronger than it is in physical reality.\nThe so-called standard Fowler–Nordheim-type equation, originally developed by Murphy and Good, and much used in past literature, is obtained by putting λZ → tF−2, PF → 1, → vF, where vF is v(f), where f is the value of fh obtained by putting h = φ, and tF is a related parameter (of value close to unity).\nWithin the more complete theory described here, the factor tF−2 is a component part of the correction factor λd2 [see, and note that λd2 is denoted by λD there].\nThere is no significant value in continuing the separate identification of tF−2.\nProbably, in the present state of knowledge, the best approximation for simple Fowler–Nordheim-type equation based modeling of CFE from metals is obtained by putting λZ → 1, PF → 1, → v(f).\nThis re-generates the Fowler–Nordheim-type equation used by Dyke and Dolan in 1956, and can be called the \"simplified standard Fowler–Nordheim-type equation\".\n\n### Recommended form for simple Fowler–Nordheim-type calculations\nExplicitly, this recommended simplified standard Fowler–Nordheim-type equation, and associated formulae, are:\nwhere Fφ here is the field needed to reduce to zero a Schottky–Nordheim barrier of unreduced height equal to the local work-function φ, and f is the scaled barrier field for a Schottky–Nordheim barrier of unreduced height φ.\n[This quantity f could have been written more exactly as fφSN, but it makes this Fowler–Nordheim-type equation look less cluttered if the convention is adopted that simple f means the quantity denoted by fφSN in, eq.\n(2.16).] For the example case (φ = 4.5 eV, F = 5 V/nm), f ≈ 0.36 and v(f) ≈ 0.58; practical ranges for these parameters are discussed further in.\nNote that the variable f (the scaled barrier field) is not the same as the variable y (the Nordheim parameter) extensively used in past field emission literature, and that \"v(f)\" does NOT have the same mathematical meaning and values as the quantity \"v(y)\" that appears in field emission literature.\nIn the context of the revised theory described here, formulae for v(y), and tables of values for v(y) should be disregarded, or treated as values of v(f1/2).\nIf more exact values for v(f) are required, then provides formulae that give values for v(f) to an absolute mathematical accuracy of better than 8×10−10.\nHowever, approximation formula (30c) above, which yields values correct to within an absolute mathematical accuracy of better 0.0025, should gives values sufficiently accurate for all technological purposes.\n\n### Comments\nA historical note on methods of deriving Fowler–Nordheim-type equations is necessary.\nThere are several possible approaches to deriving these equations, using [free-electron theory](https://bliptext.com/articles/free-electron-model).\nThe approach used here was introduced by Forbes in 2004 and may be described as \"integrating via the total energy distribution, using the parallel kinetic energy Kp as the first variable of integration\".\nBasically, it is a free-electron equivalent of the Modinos procedure (in a more advanced quantum-mechanical treatment) of \"integrating over the surface Brillouin zone\".\nBy contrast, the free-electron treatments of CFE by Young in 1959, Gadzuk and Plummer in 1973 and Modinos in 1984, also integrate via the total energy distribution, but use the normal energy εn (or a related quantity) as the first variable of integration.\nThere is also an older approach, based on a seminal paper by Nordheim in 1928, that formulates the problem differently and then uses first Kp and then εn (or a related quantity) as the variables of integration: this is known as \"integrating via the normal-energy distribution\".\nThis approach continues to be used by some authors.\nAlthough it has some advantages, particularly when discussing resonance phenomena, it requires integration of the Fermi–Dirac distribution function in the first stage of integration: for non-free-electron-like electronic band-structures this can lead to very complex and error-prone mathematics (as in the work of Stratton on [semiconductors](https://bliptext.com/articles/semiconductors)).\nFurther, integrating via the normal-energy distribution does not generate experimentally measured electron energy distributions.\nIn general, the approach used here seems easier to understand, and leads to simpler mathematics.\nIt is also closer in principle to the more sophisticated approaches used when dealing with real bul**k** crystalline solids, where the first step is either to integrate contributions to the ECD over [constant energy surfaces](https://bliptext.com/articles/constant-energy-surface) in a [wave-vector](https://bliptext.com/articles/wave-vector) space (**k**-space), or to integrate contributions over the relevant surface Brillouin zone.\nThe Forbes approach is equivalent either to integrating over a spherical surface in **k**-space, using the variable Kp to define a ring-li**k**e integration element that has cylindrical symmetry about an axis in a direction normal to the emitting surface, or to integrating over an (extended) surface Brillouin zone using circular-ring elements.\n\n## CFE theoretical equations\nThe preceding section explains how to derive Fowler–Nordheim-type equations.\nStrictly, these equations apply only to CFE from bulk metals.\nThe ideas in the following sections apply to CFE more generally, but eq.\n(30) will be used to illustrate them.\nFor 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.\nExperiments 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.\nTo relate these variables to J and F, auxiliary equations are used.\nThe voltage-to-barrier-field conversion factor β is defined by:\nThe value of F varies from position to position on an emitter surface, and the value of β varies correspondingly.\nFor 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](https://bliptext.com/articles/space-charge) (FEVSC) is present (this will be true except at very high emission current densities, around 109 A/m2 or higher ); (3) no significant \"patch fields\" exist, as a result of non-uniformities in [local work-function](https://bliptext.com/articles/work-function) (this is normally assumed to be true, but may not be in some circumstances).\nFor non-metals, the physical effects called \"field penetration\" and \"[band bending](https://bliptext.com/articles/band-bending)\" [M084] can make β a function of applied voltage, although – surprisingly – there are few studies of this effect.\nThe emission current density J varies from position to position across the emitter surface.\nThe total emission current i from a defined part of the emitter is obtained by integrating J across this part.\nTo obtain a simple equation for i(V), the following procedure is used.\nA 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 Jr. A parameter Ar, called the notional emission area (with respect to point \"r\"), is then defined by:\nwhere the integral is taken across the part of the emitter of interest.\nThis parameter Ar was introduced into CFE theory by Stern, Gossling and Fowler in 1929 (who called it a \"weighted mean area\").\nFor 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).\nLong-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.\nThe notional emission area Ar will often be a function of the reference local field (and hence voltage), and in some circumstances might be a significant function of temperature.\nBecause Ar 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](https://bliptext.com/articles/field-emission-microscope).\nWith a large-area emitter, which contains many individual emission sites, Ar will nearly always be very very much less than the \"macroscopic\" geometrical area (AM) of the emitter as observed visually (see below).\nIncorporating these auxiliary equations into eq.\n(30a) yields\nThis is the simplified standard Fowler–Nordheim-type equation, in i–V form.\nThe corresponding \"physically complete\" equation is obtained by multiplying by λZPF.\n\n## Modified equations for large-area emitters\nThe equations in the preceding section apply to all field emitters operating in the CFE regime.\nHowever, further developments are useful for large-area emitters that contain many individual emission sites.\nFor such emitters, the notional emission area will nearly always be very very much less than the apparent \"macroscopic\" geometrical area (AM) of the physical emitter as observed visually.\nA dimensionless parameter αr, the area efficiency of emission, can be defined by\nAlso, a \"macroscopic\" (or \"mean\") emission current density JM (averaged over the geometrical area AM of the emitter) can be defined, and related to the reference current density Jr used above, by\nThis leads to the following \"large-area versions\" of the simplified standard Fowler–Nordheim-type equation:\nBoth these equations contain the area efficiency of emission αr. For any given emitter this parameter has a value that is usually not well known.\nIn general, αr varies greatly as between different emitter materials, and as between different specimens of the same material prepared and processed in different ways.\nValues in the range 10−10 to 10−6 appear to be likely, and values outside this range may be possible.\nThe presence of αr in eq.\n(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](https://bliptext.com/articles/spindt-tip) ) 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.\nA 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 Ar and macroscopic area AM, and/or omits the parameter αr from cited equations.\nCare is necessary in order to avoid errors of interpretation.\nIt is also sometimes convenient to split the conversion factor βr 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.\nThis is usually done by defining a \"macroscopic field\" FM that is the field that would be present at the emitting site in the absence of the local structure that causes enhancement.\nThis field FM is related to the applied voltage by a \"voltage-to-macroscopic-field conversion factor\" βM defined by:\nIn the common case of a system comprising two parallel plates, separated by a distance W, with emitting nanostructures created on one of them, βM = 1/W.\nA \"field enhancement factor\" γ is then defined and related to the values of βr and βM by\nWith eq.\n(31), this generates the following formulae:\nwhere, in accordance with the usual convention, the suffix \"r\" has now been dropped from parameters relating to the reference point.\nFormulae exist for the estimation of γ, using [classical electrostatics](https://bliptext.com/articles/electrostatics), for a variety of emitter shapes, in particular the \"hemisphere on a post\".\nEquation (40) implies that versions of Fowler–Nordheim-type equations can be written where either F or βV is everywhere replaced by.\nThis is often done in technological applications where the primary interest is in the field enhancing properties of the local emitter nanostructure.\nHowever, in some past work, failure to make a clear distinction between barrier field F and macroscopic field FM has caused confusion or error.\nMore 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 αr, and to reduce the \"onset\" voltage at which significant emission occurs, by increasing the value of β.\nEq.\n(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 βM is increased.\nVarious trade-offs and constraints exist.\nIn 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.\nIn technological contexts field-emission data are often plotted using (a particular definition of) FM or 1/FM as the x-coordinate.\nHowever, for scientific analysis it usually better not to pre-manipulate the experimental data, but to plot the raw measured i–V data directly.\nValues 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.\n\n## Modified equations for nanometrically sharp emitters\nMost of the theoretical derivations in the field emission theory are done under the assumption that the barrier takes the Schottky–Nordheim form eq.\n(3). However, this barrier form is not valid for emitters with radii of curvature R comparable to the length of the tunnelling barrier.\nThe 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 R > 20 nm, as explained in the next paragraph.\nThe main assumption of the SN barrier approximation is that the electrostatic potential term takes the linear form \\Phi = Fx in the tunnelling region.\nThe latter has been proved to hold only if x \\ll R. Therefore, if the tunnelling region has a length L, x 0.\nTwo models have special status in field emission theory: the exact triangular (ET) barrier and the Schottky–Nordheim (SN) barrier.\nThese are given by equations (2) and (3), respectively:\nHere h is the zero-field height (or unreduced height) of the barrier, e is the [elementary positive charge](https://bliptext.com/articles/elementary-charge), F is the barrier field, and ε0 is the [electric constant](https://bliptext.com/articles/vacuum-permittivity).\nBy convention, F is taken as positive, even though the [classical electrostatic field](https://bliptext.com/articles/electric-field) would be negative.\nThe SN equation uses the classical image potential energy to represent the physical effect \"correlation and exchange\".\n\n## Escape probability\nFor an electron approaching a given barrier from the inside, the probability of escape (or \"[transmission coefficient](https://bliptext.com/articles/transmission-coefficient)\" or \"penetration coefficient\") is a function of h and F, and is denoted by D(h, F).\nThe primary aim of tunneling theory is to calculate D(h, F).\nFor physically realistic barrier models, such as the Schottky–Nordheim barrier, the Schrödinger equation cannot be solved exactly in any simple way.\nThe following so-called \"semi-classical\" approach can be used.\nA parameter G(h, F) can be defined by the [JWKB (Jeffreys-Wentzel-Kramers-Brillouin)](https://bliptext.com/articles/wkb) integral:\nwhere 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\nForbes 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\nwhere the tunneling pre-factor P can in principle be evaluated by complicated iterative integrations along a path in [complex space](https://bliptext.com/articles/complex-hilbert-space).\nIn the CFE regime we have (by definition) G ≫ 1.\nAlso, for simple models P ≈ 1.\nSo eq.\n(6) reduces to the so-called simple [JWKB](https://bliptext.com/articles/wkb) formula:\nFor the exact triangular barrier, putting eq.\n(2) into eq.\n(4) yields GET = bh3/2/F, where\nThis parameter b is a universal constant sometimes called the second Fowler–Nordheim constant.\nFor barriers of other shapes, we write\nwhere ν(h, F) is a correction factor that in general has to be determined by [numerical integration](https://bliptext.com/articles/numerical-integration), using eq.\n(4).\n\n## Correction factor for the Schottky–Nordheim barrier\nThe Schottky–Nordheim barrier, which is the barrier model used in deriving the standard Fowler–Nordheim-type equation, is a special case.\nIn this case, it is known that the correction factor \\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.\nThis field is given by\nThe 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.\nFor the Schottky–Nordheim barrier, ν(h, F) is given by the particular value ν(fh) of a function ν(ℓ).\nThe latter is a function of mathematical physics in its own right and has been called the principal Schottky–Nordheim barrier function.\nAn explicit series expansion for ν(ℓ) is derived in a 2008 paper by J. Deane.\nThe following good simple approximation for ν(fh) has been found:\n# [Schottky–Nordheim barrier for Fowler–Nordheim field emission (and enhanced thermionic emission) | upload.wikimedia.org/wikipedia/commons/5/53/Sn///barrier.svg]\n\n## Decay width\nThe decay width (in energy), dh, measures how fast the escape probability D decreases as the barrier height h increases; dh is defined by:\nWhen h increases by dh then the escape probability D decreases by a factor close to e ( ≈ 2.718282).\nFor an elementary model, based on the exact triangular barrier, where we put ν = 1 and P ≈ 1, we get\nThe decay width dh derived from the more general expression (12) differs from this by a \"decay-width correction factor\" λd, so:\nUsually, the correction factor can be approximated as unity.\nThe decay-width dF for a barrier with h equal to the local work-function φ is of special interest.\nNumerically this is given by:\nFor metals, the value of dF is typically of order 0.2 eV, but varies with barrier-field F.\n\n## Comments\nA historical note is necessary.\nThe idea that the Schottky–Nordheim barrier needed a correction factor, as in eq.\n(9), was introduced by Nordheim in 1928, but his mathematical analysis of the factor was incorrect.\nA new (correct) function was introduced by Burgess, [Kroemer](https://bliptext.com/articles/herbert-kroemer) and Houston in 1953, and its mathematics was developed further by Murphy and Good in 1956.\nThis 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\".\nOnly recently (2006 to 2008) has it been realized that, mathematically, it is much better to use the variable ℓ ( = y2).\nAnd 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](https://bliptext.com/articles/hypergeometric-differential-equation)).\nAlso, approximation (11) has been found only recently.\nApproximation (11) outperforms, and will presumably eventually displace, all older approximations of equivalent complexity.\nThese recent developments, and their implications, will probably have a significant impact on field emission research in due course.\nThe following summary brings these results together.\nFor tunneling well below the top of a well-behaved barrier of reasonable height, the escape probability D(h, F) is given formally by:\nwhere ν(h, F) is a correction factor that in general has to be found by numerical integration.\nFor 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.\nThe 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.\nThe 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.\nThe approach described here was originally developed to describe Fowler–Nordheim tunneling from smooth, classically flat, planar emitting surfaces.\nIt is adequate for smooth, classical curved surfaces of radii down to about 10 to 20 nm.\nIt 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.\nWhen 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.\nAs 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.\nTaking atomic structure properly into account is a very difficult problem, and only limited progress has been made.\nHowever, 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.\n(15), by amounts that cannot easily be estimated at present.\nAll 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](https://bliptext.com/articles/travelling-wave).\nThe 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.\n\n# Total-energy distribution\nThe 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](https://bliptext.com/articles/surface-physics) and for the field emission sources used in electron beam instruments such as [electron microscopes](https://bliptext.com/articles/electron-microscopes).\nIn the latter case, the \"width\" (in energy) of the distribution influences how finely the beam can be focused.\nThe theoretical explanation here follows the approach of Forbes.\nIf ε denotes the total electron energy relative to the emitter Fermi level, and Kp denotes the [kinetic energy](https://bliptext.com/articles/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\nTwo 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.\nWhen the emitter Fermi level is used as the reference zero level, both ε and εn can be either positive or negative.\nEnergy analysis experiments have been made on field emitters since the 1930s.\nHowever, 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).\nThis is also true (or nearly true) when the emission comes from a small field enhancing protrusion on an otherwise flat surface.\nTo see how the total energy distribution can be calculated within the framework of a [Sommerfeld free-electron-type model](https://bliptext.com/articles/free-electron-model), look at the P-T energy-space diagram (P-T=\"parallel-total\").\nThis shows the \"parallel kinetic energy\" Kp on the horizontal axis and the total energy ε on the vertical axis.\nAn electron inside the bulk metal usually has values of Kp and ε that lie within the lightly shaded area.\nIt can be shown that each element dεdKp of this energy space makes a contribution to the electron current density incident on the inside of the emitter boundary.\nHere, zS is the universal constant (called here the [Sommerfeld](https://bliptext.com/articles/arnold-sommerfeld) supply density):\nand is the [Fermi–Dirac distribution function](https://bliptext.com/articles/fermi-dirac-statistics):\nwhere T is [thermodynamic temperature](https://bliptext.com/articles/thermodynamic-temperature) and kB is the [Boltzmann constant](https://bliptext.com/articles/boltzmann-constant).\nThis element of incident current density sees a barrier of height h given by:\nThe corresponding escape probability is D(h, F): this may be expanded (approximately) in the form\nwhere DF is the escape probability for a barrier of unreduced height equal to the local work-function φ.\nHence, the element dεdKp makes a contribution to the emission current density, and the total contribution made by incident electrons with energies in the elementary range dε is thus\nwhere the integral is in principle taken along the strip shown in the diagram, but can in practice be extended to ∞ when the decay-width dF is very much less than the [Fermi energy](https://bliptext.com/articles/fermi-energy) KF (which is always the case for a metal).\nThe outcome of the integration can be written:\nwhere and are values appropriate to a barrier of unreduced height h equal to the local work function φ, and is defined by this equation.\nFor a given emitter, with a given field applied to it, is independent of F, so eq.\n(21) shows that the shape of the distribution (as ε increases from a negative value well below the Fermi level) is a rising exponential, multiplied by the FD distribution function.\nThis generates the familiar distribution shape first predicted by Young.\nAt low temperatures, goes sharply from 1 to 0 in the vicinity of the Fermi level, and the [FWHM](https://bliptext.com/articles/full-width-at-half-maximum) of the distribution is given by:\nThe fact that experimental CFE total energy distributions have this basic shape is a good experimental confirmation that electrons in metals obey [Fermi–Dirac statistics](https://bliptext.com/articles/fermi-dirac-statistics).\n# [P-T energy-space diagram, showing the region in P-T energy space where traveling-wave electron states exist. | upload.wikimedia.org/wikipedia/commons/3/35/P-T///Energy///Space///%28cropped%29.jpg]\n\n# Cold field electron emission\n\n## Fowler–Nordheim-type equations\n\n### Introduction\nFowler–Nordheim-type equations, in the J–F form, are (approximate) theoretical equations derived to describe the local current density J emitted from the internal electron states in the conduction band of a bulk metal.\nThe emission current density (ECD) J for some small uniform region of an emitting surface is usually expressed as a function J(φ, F) of the local work-function φ and the local barrier field F that characterize the small region.\nFor sharply curved surfaces, J may also depend on the parameter(s) used to describe the surface curvature.\nOwing to the physical assumptions made in the original derivation, the term Fowler–Nordheim-type equation has long been used only for equations that describe the ECD at zero temperature.\nHowever, it is better to allow this name to include the slightly modified equations (discussed below) that are valid for finite temperatures within the CFE emission regime.\n\n### Zero-temperature form\nCurrent density is best measured in A/m2.\nThe total current density emitted from a small uniform region can be obtained by integrating the total energy distribution j(ε) with respect to total electron energy ε.\nAt zero temperature, the [Fermi–Dirac distribution function](https://bliptext.com/articles/fermi-dirac-statistics) fFD = 1 for ε 0.\nSo the ECD at 0 K, J0, is given from eq.\n(18) by\nwhere is the effective supply for state F, and is defined by this equation.\nStrictly, the lower limit of the integral should be −KF, where KF is the [Fermi energy](https://bliptext.com/articles/fermi-energy); but if dF is very much less than KF (which is always the case for a metal) then no significant contribution to the integral comes from energies below KF, and it can formally be extended to –∞.\nResult (23) can be given a simple and useful physical interpretation by referring to Fig. 1. The electron state at point \"F\" on the diagram (\"state F\") is the \"forwards moving state at the Fermi level\" (i.e., it describes a Fermi-level electron moving normal to and towards the emitter surface).\nAt 0 K, an electron in this state sees a barrier of unreduced height φ, and has an escape probability DF that is higher than that for any other occupied electron state.\nSo it is convenient to write J0 as ZFDF, where the \"effective supply\" ZF is the current density that would have to be carried by state F inside the metal if all of the emission came out of state F.\nIn practice, the current density mainly comes out of a group of states close in energy to state F, most of which lie within the heavily shaded area in the energy-space diagram.\nSince, for a [free-electron model](https://bliptext.com/articles/free-electron-model), the contribution to the current density is directly proportional to the area in energy space (with the Sommerfeld supply density zS as the constant of proportionality), it is useful to think of the ECD as drawn from electron states in an area of size dF2 (measured in eV2) in the energy-space diagram.\nThat is, it is useful to think of the ECD as drawn from states in the heavily shaded area in Fig. 1. (This approximation gets slowly worse as temperature increases.)\nZF can also be written in the form:\nwhere the universal constant a, sometimes called the First Fowler–Nordheim Constant, is given by\nThis shows clearly that the pre-exponential factor aφ−1F2, that appears in Fowler–Nordheim-type equations, relates to the effective supply of electrons to the emitter surface, in a free-electron model.\n\n### Non-zero temperatures\nTo obtain a result valid for non-zero temperature, we note from eq.\n(23) that zSdFDF = J0/dF. So when eq.\n(21) is integrated at non-zero temperature, then – on making this substitution, and inserting the explicit form of the Fermi–Dirac distribution function – the ECD J can be written in the form:\nwhere λT is a temperature correction factor given by the integral.\nThe integral can be transformed, by writing and, and then u = \\exp(x), into the standard result:\nThis is valid for w > 1 (i.e., dF/kBT > 1).\nHence – for temperatures such that kBT 20 nm, as explained in the next paragraph.\nThe main assumption of the SN barrier approximation is that the electrostatic potential term takes the linear form \\Phi = Fx in the tunnelling region.\nThe latter has been proved to hold only if x \\ll R. Therefore, if the tunnelling region has a length L, x

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