Field electron emission

Field (electron) emission (FE) is an experimental phenomenon involving the electric-field-induced emission of electrons from the surface of a condensed material (either solid or liquid), into vacuum or into another material. This second material may be a gas, a liquid, or a non-metallic solid with low electrical conductivity. FE occurs at surface locations where the local surface electric field is particularly high. To generate significant amounts of emission, fields of 1 volt per nanometre (1 V/nm, or 1 000 000 000 volts per metre) or more are required. The exact field needed depends both on the nature of the materials involved and on the amount of electron current you want to generate. No external stimulation (in particular, no heating) is needed. FE is a physical effect in its own right.

Electron sources based on FE have (or potentially have) many practical contexts, ranging from bright electron sources for high-resolution electron microscopes to charge-neutralizers for spacecraft. FE is also one of the possible primary causes of vacuum breakdown and electrical discharge phenomena. In some contexts one wants to maximize/optimize FE currents; in others one wants to suppress FE entirely. Billion-dollar industries have been engaged in both endeavours.

The scientific explanation of field electron emission is that it is due to the wave-mechanical tunneling of electrons out of the condensed material. (This is also called quantum-mechanical tunneling, but the wave aspects of electron behaviour, rather than the quantisation aspects, are more important for FE.) Because electrons have wave-like behaviour, they can tunnel through a surface barrier that has been made sufficiently thin by the applied electric field.

Field electron emission was the first major physical effect to be firmly identified as due to tunneling, in the early days of quantum mechanics. This was done by Fowler and Nordheim [1], who gave the first basically correct physical explanation of field emission from bulk metals in early 1928. A family of (approximate) equations, "Fowler-Nordheim-type equations", is named after them. Strictly, these 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 FE from other materials.

In some respects, field electron emission is a paradigm example of what physicists mean by tunneling. Unfortunately, simple solvable models of the tunneling barrier lead to equations (including the original 1928 FN-type equation) that get predictions of emission current density too low by a factor of 100 or more. If one inserts a more realistic barrier model into the simplest form of the Schroedinger equation, then an awkward mathematical problem arises over the resulting differential equation: it is mathematically impossible in principle to solve this equation exactly in terms of the usual functions of mathematical physics, or in any simple way. To get even an approximate solution, it is necessary to use special approximate methods known in physics as "semi-classical" or "quasi-classical" methods. Worse, a mathematical error was made in the original application of these methods to FE, and even the corrected theory that was put in place in the 1950s has been formally incomplete until very recently. A consequence of these (and other) difficulties has been a heritage of misunderstanding and disinformation that still persists in some current FE research literature. An aim of this page should be to present a basic account of FE "for the 21st century and beyond" that is free from these confusions.

Some basics facts and terminology

Field (electron) emission and field-induced electron emission are general names for this experimental phenomenon and its theory.

Fowler-Nordheim (FN) 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 FN tunneling from many materials in various different circumstances. It is very difficult for experimental field-induced tunneling not to be FN tunneling.

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 of the emitted electrons escape by FN tunneling from electron states close to the emitter Fermi level. [By contrast, in the Schottky emission regime, most of the electrons escape over the top of a field-reduced barrier, from states well above 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.

Fowler-Nordheim-type (FN-type) equations are a family of (approximate) equations derived to describe CFE from the internal electron states in bulk metals. The different members of this family represent different degrees of approximation to reality. Approximate equations are necessary because, for physically realistic models of the tunnelling barrier, it is mathematically impossible in principle to solve the Schroedinger equation exactly in any simple way. There is no theoretical reason to believe that FN-type equations validly describe FE 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") which 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 tunnelling, but strictly it is not CFE, and is not accurately described by a FN-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", "field emission", "field electron emission" and "electron field emission".

Early History

Field 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 Winkler [6] in 1744 were started by CFE from his wire electrode. However, meaningful investigations had to wait until after J.J. Thomson's identification [7] of the electron in 1897, and until after it was understood – from thermal emission [8] and photo-emission [9] 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 [10]. But 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 [11] the first clear account in English of the experimental phenomenology of the effect he had called "autoelectronic emission". Kleint describes this and other early work [12-14].

After 1922, experimental interest increased, particularly in the groups led by Millikan at the California Insitute in Pasadena (e.g., Ref. 15), and by Gossling at the General Electric Company in London (see Ref. 16). 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 [15]. Schottky [17] had suggested in 1923 that the effect might be due to thermally-induced emission over a field-reduced barrier. If so, then plots of type [log{i} vs $V^{1/2}$] should be straight; but they were not [15]. (Nor is Schottky's explanation compatible with the experimental observation of only very weak temperature dependence in CFE [11] – a point initially overlooked [10].)

A breakthrough came when Lauritsen (see Ref. 18) (and Oppenheimer independently [19]) found that plots of type [log{i} vs 1/V] yielded good straight lines. This result, published by Millikan and Lauritsen (ML) [18] in early 1928, was known to Fowler and Nordheim.

Oppenheimer had predicted [19] that the field-induced tunnelling of electrons from atoms (the effect we now call field ionization) would have this i(V) dependence, had found this dependence in the published experimental FE results of Millikan and Eyring [15], and proposed that CFE was due to field-induced tunneling of electrons from atomic-like orbitals in surface metal atoms. FN's alternative theory [1] explained both the ML finding and the very weak dependence of current on temperature. FN predicted both to be consequences if CFE were due to field-induced tunnelling 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 (FD) statistics.

In fact, Oppenheimer (although right in principle about the theory of field ionization) had mathematical details of his theory seriously incorrect (see Ref. 20). There was also a small numerical error in the final equation given by FN for CFE current density: this was corrected in the 1929 paper of Stern, Gossling and Fowler [21].

Strictly, if the barrier field in FN's 1928 theory is exactly proportional to the applied voltage, and if the emission area is independent of voltage, then the FN 1928 theory predicts that plots of the form [log{$i/V^{2}$} vs 1/V] should be exact straight lines. But – as FN comment – contemporary experimental techniques were not good enough to distinguish between the FN theoretical result and the ML experimental result.

Thus, by 1928 basic physical understanding of the origin of CFE from bulk metals had been achieved, and the original FN-type equation had been derived.

The literature often presents FN's 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 FN paper was that it was the first convincing argument from experiment that Fermi-Dirac statistics applied to the behaviour of electrons in metals, as suggested by Summerfeld [22] in 1927. The success of FN's theory did much to confirm the correctness of Sommerfeld's ideas, and greatly helped to establish modern electron band theory (see Ref. 23). In particular, the original FN-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 FN paper also established the physical basis for an unified treatment of field-induced and thermally-induced electron emission (see Ref. 23). Prior to 1928 it had been hypothesised 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 FN 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 FD 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 Gurney and Condon [24, 25], later in 1928, of the theory of the radioactive decay of nuclei (by alpha-particle tunnelling). (See Ref. 26 for an historical account).

Fowler-Nordheim tunneling

Fowler-Nordheim (FN) 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 localised state; (2) when the electron is initially not strongly localised, 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 FN tunneling relatively simple, these assumptions are needed; but the atomic structure of matter is in effect being disregarded.

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

$(h_{\rm {P}}^{2}/8\pi ^{2}m){\mbox{d}}^{\rm {2}}{\it {{\Psi }/{\mbox{d}}x^{\rm {2}}=\left(U-E_{\rm {n}}\right)\Psi =M\Psi ,..........{\rm {(1)}}}}$

where ${\it {{\Psi }(x)}}$ is the electron wave-function, expressed as a function of distance x measured from the emitter's electrical surface [27], $h_{\rm {P}}$ is Planck's constant, m is the electron mass, U(x) is the electron potential energy, $E_{\rm {n}}$ is the total electron energy associated with motion in the x-direction, and M(x) [=U(x)–$E_{\rm {n}}$] is called the electron motive energy [28]. M(x) is the negative of the electron kinetic energy associated with motion in the x-direction, and is positive in the barrier.

The shape of a tunneling barrier is determined by how M(x) varies with position, in the region where M(x)>0. Two models have special status in FE theory. The exact triangular (ET) barrier is

$\,M^{\rm {ET}}(x)=h-eFx,.........{\rm {(2)}}$

where h is the zero-field height (or unreduced height) of the barrier, e is the elementary positive charge, and F is the barrier field. By convention, F is taken as positive, even though the classical electrostatic field would be negative. The Schottky-Nordheim (SN) barrier [29,30] is

$M^{\rm {SN}}(x)=h-eFx-e^{2}/(16\pi \varepsilon _{0}x),.........{\rm {(3)}}$

where ${\it {\epsilon }}_{0}$ is the electric constant (formerly called the "permittivity of free space"). This uses the classical image potential energy to represent the physical effect "correlation and exchange".

For an electron approaching a given barrier from the inside, the probability of escape (or "transmission coefficient" or "pentration coefficient") is a function of h and F, and is denoted by D(h,F). The primary aim of tunnelling theory is to calculate D(h,F). For physically realistic barrier models, such as the SN barrier, the Schroedinger equation cannot be solved exactly in terms of the usual functions of mathematical physics. The following so-called "semi-classical" approach can be used. A parameter G(h,F) can be defined by the JWKB (Jeffreys-Wentzel-Kramers-Brillouin) integral [31]:

$G(h,F)=g\int M^{1/2}{\mbox{d}}x...........(4)$

The parameter g [$=4\pi {(2m)}^{1/2}/h_{\rm {P}}$] is an universal constant of value 10.24624 ${\rm {eV}}^{-1/2}$ ${\rm {nm}}^{-1},$ and the integral is taken across the barrier (i.e., across the region where M>0). Forbes [32] has re-arranged a result proved by Fröman and Fröman [33], to show that – formally – in a one-dimensional treatment – the exact solution for D can be written

$\,D=P{\mbox{e}}^{-G}/(1+P{\mbox{e}}^{-G}),..........(5)$

where the tunnelling pre-factor P can in principle be evaluated by complicated iterative integrations along a path in complex space [31]. In the CFE regime we have (by definition) G>>1. Also, for simple models P~1. So eq. (5) reduces to the so-called simple JWKB formula:

$D\approx P{\mbox{e}}^{-G}\approx \exp[-G]...........(6)$

For the exact triangular barrier, putting eq. (2) into eq. (4) yields $G^{\rm {ET}}=bh^{3/2}/F$, where b = 2g/3e = 6.830890 ${\rm {eV}}^{-3/2}$ V ${\rm {nm}}^{-1}$. This parameter b is an universal constant sometimes called the Second Fowler-Nordheim Constant. For barriers of other shapes, we write:

$\,G(h,F)={\it {{\nu }(h,F)G^{\rm {ET}}={\it {{\nu }(h,F)bh^{\rm {3/2}}/F,..........{\rm {(7)}}}}}}$

where ${\it {\nu }}$ ("nu") is a correction factor that in general has to be determined by numerical integration, using eq. (4).

The Schottky-Nordheim barrier, which is the barrier model used in deriving the standard FN-type equation, is a special case. In this case, it is known that the correction factor ${\it {\nu }}$ is a function of a single variable $f_{h}$, defined by $f_{h}=F/F_{h}$, where $F_{h}$ is the field necessary to reduce the height of a SN barrier from h to 0. This field is given by: $F_{h}=(4\pi \epsilon _{0}/e^{3})h^{2}$ = (0.6944617 V ${\rm {nm}}^{-1}$)•${(h/{\rm {{eV})}}}^{2}$. It follows that $f_{h}$ runs from 0 to 1. The parameter $f_{h}$ may be called the scaled barrier field, for a barrier of zero-field height h.

For the SN barrier, $\nu (h,F)$ is given by the particular value v($f_{h}$) of a mathematical function v($l'$) discussed in Refs 3 and 34. This mathematical function v($l'$) has been called the Principal Schottky-Nordheim Barrier Function, and is a function of mathematical physics in its own right: an explicit series expansion for it is derived in Ref. 34. The following good simple approximation for v($f_{h}$) has been found (see Ref. 3):

$v(f_{h})\approx 1-f_{h}+(1/6)f_{h}\ln f_{h}...........{\rm {(8)}}$

The decay width (in energy), $d_{h}$, measures how fast the escape probability D decreases as the barrier height h increases. $d_{h}$ is defined by:

${d_{h}}^{-1}=-{\rm {{dln}{\it {{D}/{\rm {{d}{\it {{h}...........{\rm {(9)}}}}}}}}}}$

When h increases by $d_{h}$ then the escape probability D decreases by a factor close to e [≈2.718282], where e is the base of natural logarithms. For an elementary model, based on the exact triangular barrier, where we put $\nu =1$ and P≈1, we get $d_{h}{\rm {{(el)}=(2/3{\it {{b}){h^{-{\rm {1/2}}}}F=eg^{\mathrm {-1} }{h^{-\mathrm {1/2} }}F}}}}$. The decay width $d_{h}$ derived from the more general expression (9) differs from this by a correction factor written as ${\sqrt {\lambda _{D}}}$, so we have:

$d_{h}={\lambda _{D}}^{1/2}d_{h}\mathrm {(el)} ={\lambda _{D}}^{1/2}eg^{-1}{h^{-\mathrm {1/2} }}F..........{\rm {(10).}}$

Usually, the correction factor can be approximated as unity.

The decay-width $d_{\mathrm {F} }$ for a barrier with h equal to the local work-function ${\mathit {\phi }}$ is of special interest. Numerically this is given by:

$d_{\mathrm {F} }={\lambda _{D}}^{1/2}eg^{-1}{\phi ^{-\mathrm {1/2} }}F\approx eg^{-1}{\phi ^{-\mathrm {1/2} }}F\approx 0.09759678\;\mathrm {eV} \,\times {(\mathrm {eV} /\phi )}^{1/2}(F/{\rm {{V}\,{nm}^{-1})...........{\rm {(11).}}}}$

For metals, the value of $d_{\mathrm {F} }$ is typically of order 0.2 eV, but varies with barrier-field F.

A historical note is necessary. The idea that the SN barrier needed a correction factor, as in eq. (7), was introduced by Nordheim [29] in 1928, but his mathematical analysis of the factor was incorrect. A new (correct) function was introduced by Burgess, Kroemer and Houston [35] in 1953, and its mathematics was developed further by Murphy and Good [2] in 1956. This corrected function, sometimes known as a "special field emission elliptic function", was expressed as a function of a mathematical variable y known as the "Nordheim parameter". Only recently (2006 to 2008) has it been realised that, mathematically, it is much better to use the variable $l'$ [$=y^{2}$]. And only recently has it been possible to complete the definition of v($l'$) by developing and proving the validity of an exact series expansion for this function (by starting from known special-case solutions of the Gauss hypergeometric differential equation). Also, approximation (8) has been found only recently. Approximation (8) outperforms, and will presumably eventually displace, all older approximations of equivalent complexity. These recent developments, and their implications, will probably have a significant impact on field emission research in due course.

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

$D(h,F)\approx P\exp \left[-\nu (h,F)bh^{3/2}/F\right],.........(12)$

where $\nu (h,F)$ is a correction factor that in general has to be found by numerical integration. For the special case of a SN barrier, an analytical result exists and $\nu (h,F)$ is given by v($f_{h}$), as discussed above; approximation (8) for v($f_{h}$) is more than sufficient for all technological purposes. The pre-factor P is also in principle a function of h and (maybe) F, but for the simple physical models discussed here it is usually satisfactory to make the approximation P=1. The exact triangular barrier is a special case where the Schroedinger equation can be solved exactly, as was done by FN [1]; for this physically unrealistic case, $\nu (h,F)$ =1, and an analytical approximation for P exists (see Ref. 1).

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

As noted at the beginning, the effects of the atomic structure of materials are disregarded in the relatively simple treatments of field electron emission discussed here. Taking atomic structure properly into account is a very difficult problem, and only limited progress has been made (for example, see Ref. 5). However, it seems probable that the main effects on the theory of FN tunneling will be to change the values of P and $\nu$ in eq. (9), by amounts that cannot easily be estimated at present..

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

Total-Energy Distribution (TED)

The energy distribution of the emitted electrons is important both for scientific experiments that use the emitted electron energy distribution to probe aspects of the emitter surface physics [] and for the field electron emission sources used in electron beam instruments such as electron microscopes []. In the latter case, the "width" (in energy) of the distribution determines how finely the beam can be focused.

The theoretical explanation here follows the approach of Forbes []. If $\epsilon$ denotes the total electron energy relative to the emitter Fermi level, and $K_{\mathrm {p} }$ denotes the kinetic energy of the electron parallel to the emitter surface, then the electron's "normal energy" ${\it {\epsilon }}_{\mathrm {n} }$ (sometimes called the "forwards energy") is defined by

$\;\epsilon _{\mathrm {n} }=\epsilon -K_{\mathrm {p} }...........(13)$.

Two types of theoretical energy distribution are recognised: the "normal-energy distribution (NED)", which shows how the energy ${\it {\epsilon }}_{\mathrm {n} }$ is distributed; and the "total-energy distribution (TED)", which shows how the total energy ${\it {\epsilon }}$ is distributed. When the emitter Fermi level is used as the reference zero level, both ${\it {\epsilon }}$ and ${\it {\epsilon }}_{\mathrm {n} }$ can be either positive or negative.

Energy analysis experiments have been made on field emitters since the 1930's. But only in the late 1950's was it realised that these experiments [] always measured the TED, which is now usually denoted by $j({\it {\epsilon }})$. This is also true (or nearly true) when the emission comes from a small field enhancing protrusion on an otherwise flat surface [].

Theory of field emission from metals [AWAITING REVISION - THIS IS 1950s THEORY AND IS NOT FULLY COMPATIBLE WITH PREVIOUS SECTION]

The Fowler–Nordheim theory is generally used in order to quantitatively describe the FE process for metals, which requires calculating the FE current density as a function of the electric field. Since this process is essentially a tunneling process, the tunneling transition probability for the electron to tunnel through the potential barrier and the number of electrons incident on the potential barrier must be found. Integrating these over all energy values gives the desired current density. The assumptions of the Fowler–Nordheim theory are [4]:

• The metal obeys the free electron model of Sommerfeld with Fermi–Dirac statistics.
• The metal surface is planar, reducing the problem to a one-dimensional one. So long as the potential barrier thickness is several orders of magnitude less than the emitter radius, this assumption is justified.
• The potential in the metal, $V_{1}\left(z\right)$, is a constant, $-V_{0}$. The potential barrier outside the metal is entirely due to the image forces, $V_{z}=-e^{2}/4z$; the applied electric field does not affect the electron states in the metal.
• The temperature of the system is $T=0\ K$.

Additionally, the model assumes that the electrons in the metal remain at equilibrium, despite the electrons escaping the metal surface. Integrating the product of the flux of electrons incident on the surface potential barrier and the tunneling probability over all electron energies. Define $E_{z}$ to be the z-component of the electron energy:

$E_{z}=E-{\frac {p_{x}^{2}}{2m}}-{\frac {p_{y}^{2}}{2m}}={\frac {p_{z}^{2}}{2m}}+V(z).\qquad (3)$

Let $N\left(E_{z}\right)dE_{z}$ be the number of electrons per unit area per second with the z-component of their energy within $dE_{z}$ of $E_{z}$ incident on the surface potential barrier; and let $D\left(E_{z}\right)$ be the tunneling probability, also known as the transmission coefficient. Thus, the product $D\left(E_{z}\right)N\left(E_{z}\right)dE_{z}$ gives the number of electrons per unit area per second within $dE_{z}$ of $E_{z}$ emitted from the metal surface. Then the current density is

$j=e\int _{-V_{0}}^{\infty }D\left(E_{z}\right)N\left(E_{z}\right)\,dE_{z}.\qquad (4)$

The electron flux incident on the metal surface is

$N\left(E_{z}\right)dE_{z}={\frac {2}{h^{3}}}dE_{z}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }{\frac {dp_{x}dp_{y}}{1+exp\left({\frac {E_{z}-\zeta }{kT}}+{\frac {p_{x}^{2}+p_{y}^{2}}{2mkT}}\right)}}={\frac {4\pi mkT}{h^{3}}}log\left(1+e^{-\left(E_{z}-\zeta \right)/kT}\right)dE_{z},\;(5)$

where $h$ is the Planck constant, $-\zeta$ is the work function, $k$ is the Boltzmann constant, $T$ is the temperature, and $m$ is the electron mass [3]. Using the semiclassical WKB approximation, the transmission coefficient is

$D\left(E_{z}\right)=exp\left[-{\frac {8\pi \left(2m\right)^{1/2}}{3he}}{\frac {|E_{z}|^{3/2}}{F}}\vartheta \left(y\right)\right],\qquad (6)$

where $F$ is the applied electric field. The Nordheim function, $\vartheta \left(y\right)$, is

$\vartheta \left(y\right)=2^{-1/2}\left[1+\left(1-y^{2}\right)^{1/2}\right]^{1/2}\cdot \left[E\left(k\right)-\left\{1-\left(1-y^{2}\right)^{1/2}\right\}\right]K\left(k\right),\qquad (7)$

where $y=\left(e^{3}F\right)^{1/2}/|E_{z}|$. The complete elliptic integrals of the first and second kinds, $E\left(k\right)$ and $K\left(k\right)$, are given by

$E\left(k\right)=\int _{0}^{\pi /2}\left(1-k^{2}sin^{2}\alpha \right)^{-1/2}d\alpha$, and $K\left(k\right)=\int _{0}^{\pi /2}\left(1-k^{2}sin^{2}\alpha \right)^{1/2}d\alpha ,\qquad (8)$

where $k^{2}=2\left(1-y^{2}\right)^{1/2}{\bigg /}\left(1+\left(1-y^{2}\right)^{1/2}\right)$ [4]. Combining equations (5) and (6), the number of electrons within $dE_{z}$ emitted per unit area per second is

$D\left(E_{z}\right)N\left(E_{z}\right)dE_{z}={\frac {4\pi mkT}{h^{3}}}exp\left[-{\frac {8\pi \left(2m\right)^{1/2}}{3he}}{\frac {|E_{z}|^{3/2}}{F}}\vartheta \left(y\right)\right]ln\left(1+e^{-\left(E_{z}-\zeta \right)/kT}\right)dE_{z}.\qquad (9)$

There are a few applicable simplifications for field emission assumptions listed above. Since field-emitted electrons have energies near $E_{z}=\zeta$, approximating the exponent in equation (9) with the first two terms of a power series expansion at $E_{z}=\zeta$ is valid. In this approximation, the exponent reduces to

$-{\frac {8\pi \left(2m\right)^{1/2}}{3he}}{\frac {|E_{z}|^{3/2}}{F}}\vartheta \left(y\right)\approx -c+{\frac {E_{z}-\zeta }{d}};\qquad (10)$

where

${\begin{array}{lcl}c&=&{\frac {8\pi \left(2m\right)^{1/2}}{3he}}{\frac {|E_{z}|^{3/2}}{F}}\vartheta \left(y\right),\\d&=&{\frac {heF}{4\pi \left(2m\phi \right)^{1/2}t\left(\left(e^{3}F\right)^{1/2}/\phi \right)}},\\t\left(y\right)&=&\vartheta \left(y\right)-{\frac {2}{3}}y{\frac {d\vartheta \left(y\right)}{dy}},\end{array}}$

and the work function is $\phi =-\zeta$ [3]. For sufficiently low temperatures, the temperature dependent part of equation (5) reduces as follows:

$kTln\left(1+exp\left(-\left(E_{z}-\zeta \right)/kT\right)\right)={\begin{cases}0,&{\mbox{for }}{E_{z}}>{\zeta }\\{\zeta -E_{z}},&{\mbox{for }}{E_{z}}<{\zeta }\end{cases}}.\qquad (11)$

Upon substituting (11) into equation (9), the following is obtained:

$D\left(E_{z}\right)N\left(E_{z}\right)dE_{z}={\begin{cases}0,&{\mbox{for }}{E_{z}}>{\zeta }\\{{\frac {4\pi m}{h^{3}}}exp\left(-c+{\frac {E_{z}-\zeta }{d}}\right)\left(\zeta -E_{z}\right)},&{\mbox{for }}{E_{z}}<{\zeta }\end{cases}}.\qquad (12)$

Integration of equation (12) will give the current density. Assuming that $-V_{0}\ll \zeta$, the Fermi energy, the lower limit of the integral can be set to $-\infty$. The current density obtained via these assumptions is [3]

${\begin{aligned}j&=e\int _{-\infty }^{\zeta }{\frac {4\pi m}{h^{3}}}\left(\zeta -E_{z}\right)exp\left(-c+{\frac {E_{z}-\zeta }{d}}\right)dE_{z}={\frac {4\pi med^{2}}{h^{3}}}e^{-c}\\&={\frac {e^{3}F^{2}}{8\pi h\phi t^{2}\left(\left(e^{3}F\right)^{1/2}{\big /}\phi \right)}}exp\left({\frac {4\left(2m\right)^{1/2}\phi ^{3/2}}{3heF}}\vartheta \left({\frac {\left(e^{3}F\right)^{1/2}}{\phi }}\right)\right)\\\end{aligned}}.\qquad (13)$

The dependences of the emission current on the work function and the field strength in the above expression match those observed experimentally [4].

Extending the field emission theory from metals

For finite, non-zero temperatures, the Fermi–Dirac distribution that applies to electrons indicates that there will be electrons in the metal with energies greater than the Fermi level [4]. Since the transmission coefficient increases with the particle’s incident energy, these electrons with energies greater than the Fermi level are more likely to tunnel through the potential barrier at the metal surface. The emission current changes only slightly from that at $T=0\ K$ for small temperatures, but in the high temperature limit, which is thermionic emission, the electrons with energies greater than the barrier height constitute the majority of the current].

Using the expansion around the Fermi level in equation (10) in equation (9) and approximating the natural logarithm term for $E_{z}>\zeta$ gives

$\ln \left(1+\exp \left(-\left(E_{z}-\zeta \right)/kT\right)\right)\approx \exp \left(-\left(E_{z}-\zeta \right)/kT\right).\qquad \qquad (14)$

Substituting equations (10) and (14) in equation (9), the expression becomes

$D\left(E_{z}\right)N\left(E_{z}\right)dE_{z}={\frac {4\pi mkT}{h^{3}}}\exp \left[-c+\left(E_{z}-\zeta \right)\left({\frac {1}{d}}-{\frac {1}{kT}}\right)\right].\qquad \qquad (15)$

Integrating equation (15) gives the following expression for the current density for non-zero temperature, valid only for $T\leq 1000\ K$:

$j\left(T\right)=j\left(0\right){\frac {\pi kT/d}{\sin \left(\pi kT/d\right)}},\qquad \qquad (16)$

where $j\left(0\right)$ is the current density at zero temperature shown in equation (13). Using equation (16) for a field of $4.7\times 10^{7}\ Vcm^{-1}$ and a work function of $4.5\ eV$, the current densities at room temperature and at $1000\ K$ are $j\left(300\ K\right)\approx 1.03j\left(0\right)$ and $j\left(1000\ K\right)\approx 1.5j\left(0\right)$. Treatment of the transition region between field emission and thermionic emission requires a more rigorous analysis, such as that presented by Murphy and Good, or numerical calculations.

The assumption of a planar, or very smooth, surface of the emitter is one of the primary assumptions of the Fowler–Nordheim theory. This assumption is fairly accurate for atomically smooth emitters with a radius of curvature approximately greater than $0.1\ \mu m$, for which the surface potential barrier width is much less than the radius of curvature. However, the planar assumption, which reduces the problem to a one-dimensional one, is not valid for field emitters with a radius of curvature around $1-20\ nm$, which is on the order of the barrier width. A rigorous treatment of such systems requires solving the three-dimensional Schrödinger equation with an asymmetric barrier potential. Additionally, the field at the apex of emitters with small radii of curvature will be larger than those with larger radii of curvature for the same applied bias, with the field inversely proportional to the radius of curvature.

.

Compared to FE from metals, the process of FE from semiconductors is much more complicated. A qualitative theory of the field emission process from semiconductors has yet to be developed, despite the plentiful amount of experimental data. Experimental results have shown that the relationship between $\ln \left(j\right)$ and $\left(1/F\right)$ is nonlinear for semiconductors with low conduction band carrier concentrations. This is in contrast to metals, for which the relationship between $\ln \left(j\right)$ and $\left(1/F\right)$ is linear over a wide range of field strengths.

The Morgulis-Stratton theory adequately describes experimental results for FE from semiconductors for relatively small currents. The theory describes the linear increase in the natural logarithm of the current with the inverse of the applied bias, the lack of photosensitivity, and the constant emission image size. The theory assumes that the electron gas is degenerate due to penetration of the electric field into the emitter near the surface, increasing the free electron concentration near the surface. Additionally, the theory assumes that the tunneling transmission coefficient is small. Calculation of the emission current density follows the method of the Fowler–Nordheim model.

Applications

The field emission microscope (FEM), invented in 1936 by E. W. Müller, is one of the primary applications of the field emission phenomena. The introduction of commercial FEMs enabled more accurate field calculations, which confirmed the validity of the Fowler–Nordheim theory within experimental error and the exponential dependence of the emission current density on $\phi ^{3/2}$. A fluorescent screen anode is placed at a macroscopic distance from the field emitter cathode. The image that appears on the screen is a projection of the emitter apex produced by the impinging field-emitted electrons. Due to the parabolic trajectories of the emitted electrons, the magnification is proportional to the quotient of the anode-cathode distance and the emitter radius of curvature. Due to the work function change induced by adsorption of molecules on surfaces and the sensitivity of the emission current on the work function, the FEM is effective for studying adsorption phenomena such as diffusion on surfaces and adsorption-desorption kinetics.

In addition to the applications of FE to surface science studies, FE is used in vacuum microelectronic devices, which rely on electron transport through vacuum rather than carrier transport in semiconductors. Displays based on field emitter arrays are by far the most common use of vacuum microelectronic devices. These field emission displays generally replace the thermal cathodes of traditional cathode ray tube displays with arrays of field emitters. In addition to applications in display technology, several other vacuum microelectronic devices have been demonstrated, including FE triodes and amplifiers and microwave frequency devices.

References

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See also

• Field emission microscope
• Field emitter array

External links

• Field emission – Fowler–Nordheim tunneling, Principles of Semiconductor Devices, Bart Van Zeghbroeck, 1997