Field electron emission

Field electron emission, also known as field-induced electron emission, field emission and electron field emission, is the emission of electrons from a material placed in an electrostatic field. The most common context is field emission from a solid surface into a vacuum. However, field emission can take place from solid or liquid surfaces, into a vacuum, a fluid, or any non-conducting or weakly conducting dielectric. The field-induced promotion of electrons from the valence to conduction band of semiconductors can also be regarded as a form of field emission.
Field emission in pure metals occurs in high electric fields: the gradients are typically higher than 1 gigavolt per metre and strongly dependent upon the work function. While electron sources based on field emission have a number of applications, field emission is most commonly an undesirable primary source of vacuum breakdown and electrical discharge phenomena, which engineers work to prevent. Examples of applications for surface field emission include the construction of bright electron sources for high-resolution electron microscopes or the discharge of induced charges from spacecraft. Devices that eliminate induced charges are termed charge-neutralizers.
Historically, the phenomenon of field electron emission has been known by a variety of names, including "the aeona effect", "autoelectronic emission", "cold emission", "cold cathode emission", "field emission", "field electron emission" and "electron field emission". In some contexts, the name "field emission" is applied to the field-induced emission of ions, rather than electrons, and because in some theoretical contexts "field emission" is used as a general name covering both field electron emission and field ion emission.
Field emission was explained by quantum tunneling of electrons in the late 1920s. This was one of the triumphs of the nascent quantum mechanics. The theory of field emission from bulk metals was proposed by Ralph H. Fowler and Lothar Wolfgang Nordheim. A family of approximate equations, Fowler–Nordheim equations, is named after them. Strictly, Fowler–Nordheim equations apply only to field emission from bulk metals and to other bulk crystalline solids, but they are often used – as a rough approximation – to describe field emission from other materials.
The related phenomena of surface photoeffect, thermionic emission and "cold electronic emission", i.e. the emission of electrons in strong static electric fields, were discovered and studied independently from the 1880s to 1930s. In the modern context, cold field electron emission 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. Many solid and liquid materials can emit electrons in a CFE regime if an electric field of an appropriate size is applied. When the term field emission is used without qualifiers, it typically means "cold emission".
For metals, the CFE regime extends to well above room temperature. There are other electron emission regimes 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 emission if most of the electrons escape by tunneling, but strictly it is not CFE, and is not accurately described by a Fowler–Nordheim-type equation.

Terminology and conventions

Equations in this article are written using the modern International System of Quantities. Older field emission literature often work with Gaussian units such that they omit the physical constant ε₀. In this article, all such equations have been converted to modern international form.
Since work function is normally given with the unit electronvolt, and for fields it is often convenient to use the unit volt per nanometer ; increasingly, this is normal practice in field emission research. , Numerical values of universal constants given here are written in units derived from eV, V and nm, and calculated to seven significant figures using the 2006 values of the fundamental constants.

Early history of field electron emission

In retrospect, it seems likely that the electrical discharges reported by J.H. Winkler in 1744 were started by CFE from his wire electrode. However, meaningful investigations had to wait until after J.J. Thomson's identification of the electron in 1897, and until after it was understood – from thermal emission and photo-emission work – that electrons could be emitted from inside metals, 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 published in 1922 the first clear account in English of the experimental phenomenology of the effect he had called "autoelectronic emission". He had worked on this topic, in Leipzig, since about 1910.
After 1922, experimental interest increased, particularly in the groups led by Millikan at the California Institute of Technology in Pasadena, California, and by Gossling at the General Electric Company in London. Attempts to understand autoelectronic emission included plotting experimental current–voltage data in different ways, to look for a straight-line relationship. Current increased superlinearly with voltage, but plots of type log 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 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 found that plots of log vs. 1/V yielded good straight lines. This result was published by Millikan and Lauritsen in early 1928. Theoretical explanation and the original Fowler–Nordheim-type equation came shortly thereafter.
Oppenheimer had predicted that the field-induced tunneling of electrons from atoms would have this i dependence, had found this dependence in the published experimental field emission results of Millikan and Eyring, and proposed that CFE was due to field-induced tunneling of electrons from atomic-like orbitals in surface metal atoms. An alternative Fowler–Nordheim theory proposed field-induced tunneling from free-electron-type states in what we would now call a metal conduction band, with the electron states occupied in accordance with Fermi–Dirac statistics. The Fowler-Nordheim theory explained both the Millikan–Lauritsen finding and the very weak dependence of current on temperature.
Oppenheimer had mathematical details of his theory seriously incorrect. There was also a small numerical error in the final equation given by Fowler–Nordheim theory for CFE current density, corrected in a 1929 paper. If the barrier field in Fowler–Nordheim 1928 theory is exactly proportional to the applied voltage, and if the emission area is independent of voltage, then the Fowler–Nordheim 1928 theory predicts that plots of log vs. 1/V should be exact straight lines. However, contemporary experimental techniques could not distinguish between the Fowler–Nordheim theoretical result and the Millikan–Lauritsen experimental result.
The physics literature often presents Fowler and Nordheim's work as a proof of electron tunneling, as predicted by wave-mechanics. Whilst this is correct, wave-mechanics was largely accepted by 1928. Instead, the Fowler–Nordheim paper was more revolutionary in establishing modern electron band theory. Prior to 1928 it had been hypothesized that two types of electrons, "thermions" and "conduction electrons", existed in metals, and that thermally emitted electron currents were due to the emission of thermions, but that field-emitted currents were due to the emission of conduction electrons, only in 1927 did Sommerfeld argue that Fermi–Dirac statistics applied to the behavior of electrons in metals. The Fowler–Nordheim 1928 work suggested that thermions did not need to exist as a separate class of internal electrons: electrons could come from a single band occupied in accordance with Fermi–Dirac statistics, but would be emitted in statistically different ways under different conditions of temperature and applied field. The success of Fowler–Nordheim theory did much to support the correctness of Sommerfeld's ideas.
In particular, the original Fowler–Nordheim-type equation was one of the first to incorporate the statistical-mechanical consequences of the existence of electron spin into the theory of an experimental condensed-matter effect. The Fowler–Nordheim paper also established the physical basis for a unified treatment of field-induced and thermally induced electron emission.
The ideas of Oppenheimer, Fowler and Nordheim were also an important stimulus to the development, by George Gamow, and Ronald W. Gurney and Edward Condon, later in 1928, of the theory of the radioactive decay of nuclei.

Practical applications: past and present

Field electron microscopy and related basics

As already indicated, the early experimental work on field electron emission was driven by Lilienfeld's desire to develop miniaturized X-ray tubes for medical applications. However, it was too early for this technology to succeed.
After Fowler–Nordheim theoretical work in 1928, a major advance came with the development in 1937 by Erwin W. Mueller of the spherical-geometry field electron microscope . 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, 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, 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 a "Mueller emitter".
When the emitter surface is clean, this FEM image is characteristic of:

The material from which the emitter is made.
The orientation of the material relative to the needle/wire axis; and
To some extent, the shape of the emitter endform.

In the FEM image, dark areas correspond to regions where the local work function φ is relatively high and/or the local barrier field F is relatively low, so J is relatively low; the light areas correspond to regions where φ is relatively low and/or F is relatively high, so J is relatively high. This is as predicted by the exponent of Fowler–Nordheim-type equations .
The adsorption of layers of gas atoms 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. 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 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, 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.