Электронная эмиссия в электрическом поле

Emission of electrons induced by an electrostatic field
Field electron emission, also known as field emission (FE) and electron field emission, is emission of electrons induced by 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. The terminology is historical because 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. When field emission is used without qualifiers it typically means "cold 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 which eliminate induced charges are termed charge neutralizers. 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.

Field electron emission, field induced electron emission, field emission and electron field emission are general names for this experimental phenomenon and its theory. The first name is used here. Fowler–Nordheim tunneling 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. Individual electrons can escape by Fowler–Nordheim tunneling from many materials in various different circumstances. 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. Fowler–Nordheim type equations are a family of approximate equations derived to describe CFE from the internal electron states in bulk metals. The different members of the family represent different degrees of approximation to reality. Approximate equations are necessary because, for physically realistic models of the tunneling barrier, it is mathematically impossible in principle to solve the Schrödinger equation exactly in any simple way. There is no theoretical reason to believe that Fowler–Nordheim type equations validly describe field emission from materials other than bulk crystalline solids. 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. Care 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. 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". Equations in this article are written using the International System of Quantities (ISQ). This 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. Older 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. In this article, all such equations have been converted to modern international form. For clarity, this should always be done. Since work function is normally given in electronvolts (eV), and it is often convenient to measure fields in volts per nanometer (V/nm), values of most universal constants are given here in units involving the eV, V and nm. Increasingly, this is normal practice in field emission research. However, all equations here are ISQ compatible equations and remain dimensionally consistent, as is required by the modern international system. To indicate their status, numerical values of universal constants are given to seven significant figures. Values are derived using the 2006 values of the fundamental constants.

Field electron emission has a long, complicated and messy history. This section covers the early history, up to the derivation of the original Fowler–Nordheim type equation in 1928. 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. Kleint describes this and other early work. 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 with voltage more rapidly than linearly, 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. 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, published by Millikan and Lauritsen in early 1928, was known to Fowler and Nordheim. 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 explained both the Millikan–Lauritsen finding and the very weak dependence of current on temperature. Fowler–Nordheim theory predicted both to be consequences if CFE were due to 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. 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: this was corrected in the 1929 paper of
Strictly, 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. However, contemporary experimental techniques were not good enough to distinguish between the Fowler–Nordheim theoretical result and the Millikan–Lauritsen experimental result. Thus, 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. The literature often presents Fowler–Nordheim work as a proof of the existence of electron tunneling, as predicted by wave mechanics. Whilst this is correct, the validity of wave mechanics was largely accepted by 1928. The more important role of the Fowler–Nordheim paper was that it was a convincing argument from experiment that Fermi–Dirac statistics applied to the behavior of electrons in metals, as suggested by Sommerfeld in 1927. The success of Fowler–Nordheim theory did much to support the correctness of Sommerfeld's ideas, and greatly helped to establish modern electron band theory. 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. 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. 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 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).

As already indicated, the early experimental work on field electron emission (1910–1920) (also called the "field emission microscope"). In this instrument, the electron emitter is a sharply pointed wire, of apex radius r. This is placed, in a vacuum enclosure, opposite an image detector (originally a phosphor screen), at a distance R from it. The microscope screen shows a projection image of the distribution of current density J across the emitter apex, with magnification approximately (R/r), typically 105 to 106. In FEM studies the apex radius is typically 100 nm to 1 μm. The tip of the pointed wire, when referred to as a physical object, has been called a "field emitter", a "tip", or (recently) a "Mueller emitter". When the emitter surface is clean, this FEM image is characteristic of: (a) the material from which the emitter is made: (b) the orientation of the material relative to the needle/wire axis; and (c) to some extent, the shape of the emitter endform. In the FEM image, dark areas correspond to regions where the local work function φ is relatively high and/or the local barrier field F is relatively low, so J is relatively low; the light areas correspond to regions where φ is relatively low and/or F is relatively high, so J is relatively high. This is as predicted by the exponent of Fowler–Nordheim type equations [see eq. (30) below]. The adsorption of layers of gas atoms (such as oxygen) onto the emitter surface, or part of it, can create surface electric dipoles that change the local work function of this part of the surface. This affects the FEM image; also, the change of work function can be measured using a Fowler–Nordheim plot (see below). Thus, the FEM became an early observational tool of surface science. 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.

Energy distribution measurements of field emitted electrons were first reported in 1939. 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.

To achieve high resolution in electron microscopes and other electron beam instruments (such as those used for electron beam lithography), it is helpful to start with an electron source that is small, optically bright and stable. Sources based on the geometry of a Mueller emitter qualify well on the first two criteria. The first electron microscope (EM) observation of an individual atom was made by Crewe, Wall and Langmore in 1970, using a scanning electron microscope equipped with an early field emission gun. From the 1950s onwards, extensive effort has been devoted to the development of field emission sources for use in electron guns. [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 rapidly become covered with adsorbate molecules that arrive from the vacuum system walls, and the emitter has to be cleaned from time to time by "flashing" to high temperature. Nowadays, it is more common to use Mueller emitter based sources that are operated at elevated temperatures, either in the Schottky emission regime or in the so called temperature field intermediate regime. Many modern high resolution electron microscopes and electron beam instruments use some form of Mueller emitter based electron source. Currently, attempts are being made to develop carbon nanotubes (CNTs) as electron gun field emission sources. 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.

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. 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

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, ") 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. 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.

As already indicated, it is now thought that the earliest manifestations of field electron emission were the electrical discharges it caused. After Fowler–Nordheim work, it was understood that CFE was one of the possible primary underlying causes of vacuum breakdown and electrical discharge phenomena. (The detailed mechanisms and pathways involved can be very complicated, and there is no single universal cause) 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

The next part of this article deals with the basic theory of cold field electron emission from bulk metals. This is best treated in four main stages, involving theory associated with: (1) derivation of a formula for "escape probability", by considering electron tunneling through a rounded triangular barrier; (2) an integration over internal electron states to obtain the "total energy distribution"; (3) a second integration, to obtain the emission current density as a function of local barrier field and local work function; (4) conversion of this to a formula for current as a function of applied voltage. The modified equations needed for large area emitters, and issues of experimental data analysis, are dealt with separately. Fowler–Nordheim tunneling is the wave mechanical tunneling of an electron through an exact or rounded triangular barrier. Two basic situations are recognized: (1) when the electron is initially in a localized state; (2) when the electron is initially not strongly localized, and is best represented by a travelling wave. Emission from a bulk metal conduction band is a situation of the second type, and discussion here relates to this case. It is also assumed that the barrier is one dimensional (i. e., has no lateral structure), and has no fine scale structure that causes "scattering" or "resonance" effects. To keep this explanation of Fowler–Nordheim tunneling relatively simple, these assumptions are needed; but the atomic structure of matter is in effect being disregarded.