Editing Field electron emission (section)

== Fowler–Nordheim tunneling ==
'''Fowler–Nordheim tunneling''' is the [[quantum tunnelling|wave-mechanical tunneling]] of an electron through an exact or rounded triangular barrier. Depending on the material's structure, the electron may be initially [[surface states|localized to the surface]] or delocalized into the bulk and best represented by a [[travelling wave]]. Emission from a metal [[conduction band]] is a situation of the second type, which is the only case treated here. It is also assumed that the barrier is one-dimensional (i.e., has no lateral structure), and has no fine-scale structure that causes "[[scattering]]" or "resonance" effects. These assumptions serve primarily to simplify the theory; but the atomic structure of matter is in effect being disregarded.

The treatment has four main stages:

# Derivation of a formula for [[transmission coefficient|escape probability]], by considering electron tunneling through a rounded triangular barrier; 
# Integration over internal electron states to obtain the total energy distribution; 
# A second integration, to obtain the emission current density as a function of local barrier field and local work function; 
# Conversion of the local work function to a formula for current as a function of applied voltage. 

The modified equations needed for large-area emitters, and issues of experimental data analysis, are dealt with separately.

=== Motive energy ===
For an electron, the one-dimensional [[Schrödinger equation]] can be written in the form

{{NumBlk|:|<math>\frac{\hbar^2}{2 m} \frac{\mathrm{d}^2 \Psi(x)}{\mathrm{d}x^2} = \left[U(x)-E_{\mathrm{n}}\right]\Psi(x) = M(x)\Psi(x), </math>|{{EquationRef|1}}}}

where Ψ(''x'') is the electron [[wave function|wave-function]], expressed as a function of distance ''x'' measured from the emitter's electrical surface,<ref name=F99>{{cite journal|doi=10.1016/S0304-3991(99)00098-4|title=The electrical surface as centroid of the surface-induced charge|year=1999|last1=Forbes|first1=R|journal=Ultramicroscopy|volume=79|issue=1–4|pages=25–34}}</ref> ''ħ'' is the [[reduced Planck constant]], ''m'' is the electron mass, ''U''(''x'') is the [[potential energy|electron potential energy]], ''E''<sub>n</sub> is the [[total energy|total electron energy]] associated with motion in the ''x''-direction, and ''M''(''x'') {{nowrap| {{=}} [''U''(''x'') &minus; ''E''<sub>n</sub>]}} is called the electron motive energy.<ref name=HN49>{{cite journal|doi=10.1103/RevModPhys.21.185|title=Thermionic Emission|year=1949|last1=Herring|first1=Conyers|journal=Reviews of Modern Physics|volume=21|pages=185–270|last2=Nichols|first2=M.|bibcode=1949RvMP...21..185H|issue=2}}</ref> ''M''(''x'') can be interpreted as the negative of the electron kinetic energy associated with the motion of a hypothetical classical point electron in the ''x''-direction, and is positive in the barrier.

The shape of a tunneling barrier is determined by how ''M''(''x'') varies with position in the region where {{nowrap|''M''(''x'') > 0}}. Two models have special status in field emission theory: the ''exact triangular (ET) barrier'', given in ({{EquationNote|2}}); and  the ''Schottky&ndash;Nordheim (SN) barrier'', given in ({{EquationNote|3}}).<ref>{{cite journal|author=W. Schottky|journal=Phys. Z.|volume=15|year=1914|page=872}}</ref><ref name=n28b>{{cite journal|author=L.W. Nordheim|journal=[[Proceedings of the Royal Society A]] | volume=121|year=1928|pages=626–639|doi=10.1098/rspa.1928.0222|title=The Effect of the Image Force on the Emission and Reflexion of Electrons by Metals|bibcode = 1928RSPSA.121..626N|issue=788 |doi-access=free}}</ref> 

{{NumBlk|:|<math>M^{\mathrm{ET}}(x) = h - eFx </math>|{{EquationRef|2}}}}
{{NumBlk|:|<math>M^{\rm{SN} }(x) = h - eFx -\frac{e^2}{16\pi\varepsilon_0 x}, </math>|{{EquationRef|3}}}}

Here ''h'' is the zero-field height (or ''unreduced height'') of the barrier, ''e'' is the [[elementary charge|elementary positive charge]], ''F'' is the barrier field, and ''ε''<sub>0</sub> is the [[vacuum permittivity|electric constant]]. By convention, ''F'' is taken as positive, even though the [[electric field|classical electrostatic field]] would be negative.  The SN equation uses the classical image potential energy to represent the physical effect "correlation and exchange".

=== Escape probability ===
For an electron approaching a given barrier from the inside, the ''probability of escape'' (or "[[transmission coefficient]]" or "penetration coefficient") is a function of ''h'' and ''F'', and is denoted by {{nowrap|''D''(''h'', ''F'')}}. The primary aim of tunneling theory is to calculate {{nowrap|''D''(''h'', ''F'')}}. For physically realistic barrier models, such as the Schottky–Nordheim barrier, the Schrödinger equation cannot be solved exactly in any simple way. The following so-called "semi-classical" approach can be used. A parameter {{nowrap|''G''(''h'', ''F'')}} can be defined by the [[WKB|JWKB (Jeffreys-Wentzel-Kramers-Brillouin)]] integral:<ref>{{cite journal|author=H. Jeffreys|journal=Proceedings of the London Mathematical Society |volume=23|year=1924|pages=428–436| doi = 10.1112/plms/s2-23.1.428|title=On Certain Approximate Solutions of Lineae Differential Equations of the Second Order}}</ref>

{{NumBlk|:|<math>G(h, F) = g\int M^{1/2}\mbox{d}x, </math>|{{EquationRef|4}}}}

where the integral is taken across the barrier (i.e., across the region where ''M''&nbsp;>&nbsp;0), and the parameter ''g'' is a universal constant given by

{{NumBlk|:|<math> g \,= 2\sqrt{2m}/\hbar \approx 10.24624 \; {\rm{eV}}^{-1/2}\; {\rm{nm}}^{-1}. </math>|{{EquationRef|5}}}}

Forbes has re-arranged a result proved by Fröman and Fröman, to show that, formally – in a one-dimensional treatment – the exact solution for ''D'' can be written<ref name=F08c>{{cite journal|doi=10.1063/1.2937077|title=On the need for a tunneling pre-factor in Fowler–Nordheim tunneling theory|year=2008|last1=Forbes|first1=Richard G.|journal=Journal of Applied Physics|volume=103|bibcode = 2008JAP...103k4911F|issue=11 |pages=114911–114911–8|url=http://epubs.surrey.ac.uk/22/1/fulltext.pdf}}</ref>

{{NumBlk|:|<math>\,D = \frac{P\mathrm{e}^{-G}}{1 + P\mathrm{e}^{-G}}, </math>|{{EquationRef|6}}}}

where the ''tunneling pre-factor'' ''P'' can in principle be evaluated by complicated iterative integrations along a path in [[Complex Hilbert space|complex space]], but is ≈&nbsp;1 for simple models.<ref name=F08c/><ref>H. Fröman and P.O. Fröman, "JWKB approximation: contributions to the theory" (North-Holland, Amsterdam, 1965).</ref> In the CFE regime we have (by definition) ''G''&nbsp;≫&nbsp;1. So eq. (6) reduces to the so-called simple [[WKB|JWKB]] formula:

{{NumBlk|:|<math>D\approx P \mathrm{e}^{-G} \approx \mathrm{e}^{-G}. </math>|{{EquationRef|7}}}}

For the exact triangular barrier, putting eq.&nbsp;({{EquationNote|2}}) into eq.&nbsp;({{EquationNote|4}}) yields {{nowrap|1=''G''<sup>ET</sup> = ''bh''<sup>3/2</sup>/''F''}}, where

{{NumBlk|:|<math> b = \frac{2g}{3e} = \frac{4\sqrt{2 m}}{3e\hbar} \approx 6.830890 \; {\mathrm{eV}}^{-3/2} \; \mathrm{V} \; {\mathrm{nm}}^{-1}. </math>|{{EquationRef|8}}}}

This parameter ''b'' is a universal constant sometimes called the ''second Fowler&ndash;Nordheim constant''. For barriers of other shapes, we write

{{NumBlk|:|<math>G(h, F) = \nu(h, F) G^{\mathrm{ET}} = \nu(h, F)b h^{3/2}/F, </math>|{{EquationRef|9}}}}

where {{nowrap|''ν''(''h'', ''F'')}} is a correction factor that determined by [[numerical integration]] of eq.&nbsp;({{EquationNote|4}}).

=== Correction factor for the Schottky–Nordheim barrier ===
[[File:Sn barrier.svg|thumb|Schottky–Nordheim barrier for Fowler–Nordheim field emission (and [[Schottky effect|enhanced thermionic emission]])]]
The Schottky–Nordheim barrier, which is the barrier model used in deriving the standard Fowler–Nordheim-type equation,<ref name=fd07>{{cite journal|doi=10.1098/rspa.2007.0030|title=Reformulation of the standard theory of Fowler–Nordheim tunnelling and cold field electron emission|year=2007|last1=Forbes|first1=Richard G.|last2=Deane|first2=Jonathan H.B.|journal=[[Proceedings of the Royal Society A]]|volume=463|pages=2907–2927|bibcode = 2007RSPSA.463.2907F|issue=2087 |s2cid=121328308}}</ref> is a special case. In this case, it is known that the correction factor <math> \it{\nu} </math> is a function of a single variable ''f<sub>h</sub>'', defined by ''f<sub>h</sub>''&nbsp;=&nbsp;''F''/''F<sub>h</sub>'', where ''F<sub>h</sub>'' is the field necessary to reduce the height of a Schottky&ndash;Nordheim barrier from ''h'' to 0. This field is given by

{{NumBlk|:|<math> \, F_h = (4\pi \epsilon_0/e^3) h^2 = (0.6944617 \; \mathrm{V}\; {\mathrm{nm}}^{-1})(h/{\rm{eV}})^2. </math>|{{EquationRef|10}}}}

The parameter ''f<sub>h</sub>'' runs from 0 to 1, and may be called the ''scaled barrier field'', for a Schottky–Nordheim barrier of zero-field height ''h''.

For the Schottky&ndash;Nordheim barrier, {{nowrap|''ν''(''h'', ''F'')}} is given by the particular value ''ν''(''f<sub>h</sub>'') of a function ''ν''(''ℓ''{{prime}}). The latter is a function of mathematical physics in its own right with explicit series expansion<ref name="DF08">{{cite journal |last1=Deane |first1=Jonathan H B |last2=Forbes |first2=Richard G |year=2008 |title=The formal derivation of an exact series expansion for the principal Schottky–Nordheim barrier function, using the Gauss hypergeometric differential equation |journal=Journal of Physics A: Mathematical and Theoretical |volume=41 |issue=39 |page=395301 |bibcode=2008JPhA...41M5301D |doi=10.1088/1751-8113/41/39/395301 |s2cid=122711134}}</ref> and has been called the ''principal Schottky&ndash;Nordheim barrier function''. The following good simple approximation for ''ν''(''f<sub>h</sub>'') has been found:<ref name=fd07/>

{{NumBlk|:|<math>v(f_h) \approx 1 - f_h + \tfrac{1}{6} f_h\ln f_h. </math>|{{EquationRef|11}}}}

===Decay width===
The ''decay width'' (in energy), ''d<sub>h</sub>'', measures how fast the escape probability ''D'' decreases as the barrier height ''h'' increases; ''d<sub>h</sub>'' is defined by:

{{NumBlk|:|<math>\frac{1}{d_h} = -\frac{\mathrm{d}(\ln D)}{\mathrm{d}h}. </math>|{{EquationRef|12}}}}

When ''h'' increases by ''d<sub>h</sub>'' then the escape probability ''D'' decreases by a factor close to e (&nbsp;≈&nbsp;2.718282). For an elementary model, based on the exact triangular barrier, where we put ''ν''&nbsp;=&nbsp;1 and ''P''&nbsp;≈&nbsp;1, we get
: <math>d_h^{\mathrm{(el)}} = \frac{2F}{3b\sqrt{h}} = \frac{e F}{g \sqrt{h}}. </math>

The decay width ''d<sub>h</sub>'' derived from the more general expression ({{EquationNote|12}}) differs from this by a "decay-width correction factor" ''λ<sub>d</sub>'', so:

{{NumBlk|:|<math>d_h=  \lambda_d d_h^{\mathrm{(el)}} = \frac{\lambda_d e F}{g \sqrt{h}}. </math>|{{EquationRef|13}}}}

Usually, the correction factor can be approximated as unity.

The decay-width ''d''<sub>F</sub> for a barrier with ''h'' equal to the local work-function ''φ'' is of special interest. Numerically this is given by:

{{NumBlk|:|<math> d_{\mathrm{F}}= \frac{\lambda_d e F}{g \sqrt{\phi}}  \approx \frac{e F}{g \sqrt{\phi}} \approx 0.09759678 \; \mathrm{eV} \, \cdot \sqrt{\frac{1\ \mathrm{eV}}{\phi}} \cdot \frac{F}{1\ \mathrm{V}\ \mathrm{nm}^{-1}}. </math>|{{EquationRef|14}}}}

For metals, the value of ''d''<sub>F</sub> is typically of order 0.2&nbsp;eV, but varies with barrier-field ''F''.

=== Comments ===
A historical note is necessary. The idea that the Schottky–Nordheim barrier needed a correction factor, as in eq. ({{EquationNote|9}}), was introduced by Nordheim in 1928,<ref name=n28b/> but his mathematical analysis of the factor was incorrect. A new (correct) function was introduced by Burgess, [[Herbert Kroemer|Kroemer]] and Houston<ref name=BKH53>{{cite journal|doi=10.1103/PhysRev.90.515|title=Corrected Values of Fowler–Nordheim Field Emission Functions v(y) and s(y)|year=1953|last1=Burgess|first1=R. E.|last2=Houston|first2=J. M.|journal=Physical Review|volume=90|page=515|last3=Houston|first3=J.|bibcode = 1953PhRv...90..515B|issue=4 }}</ref> in 1953, and its mathematics was developed further by Murphy and Good in 1956.<ref name=MG56>{{cite journal|doi=10.1103/PhysRev.102.1464|title=Thermionic Emission, Field Emission, and the Transition Region|year=1956|last1=Murphy|first1=E. L.|last2=Good|first2=R. H.|journal=Physical Review|volume=102|pages=1464–1473|bibcode = 1956PhRv..102.1464M|issue=6 }}</ref> This corrected function, sometimes known as a "special field emission elliptic function", was expressed as a function of a mathematical variable ''y'' known as the "Nordheim parameter". Only recently (2006 to 2008) has it been realized that, mathematically, it is much better to use the variable ''ℓ''{{prime}} {{nowrap|1=( = ''y''<sup>2</sup>)}}. And only recently has it been possible to complete the definition of ''ν''(''ℓ''{{prime}}) 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 ({{EquationNote|11}}) has been found only recently. Approximation ({{EquationNote|11}}) outperforms, and will presumably eventually displace, all older approximations of equivalent complexity. These recent developments, and their implications, will probably have a significant impact on field emission research in due course.

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

{{NumBlk|:|<math>D(h, F) \approx P\exp\left[-\frac{\nu(h, F) bh^{3/2}}{F}\right], </math>|{{EquationRef|15}}}}

where {{nowrap|''ν''(''h'', ''F'')}} is a correction factor that in general has to be found by numerical integration. For the special case of a Schottky–Nordheim barrier, an analytical result exists and {{nowrap|''ν''(''h'', ''F'')}} is given by ''ν''(''f<sub>h</sub>''), as discussed above; approximation (11) for ''ν''(''f<sub>h</sub>'') 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''&nbsp;=&nbsp;1. The exact triangular barrier is a special case where the Schrödinger equation can be solved exactly, as was done by Fowler and Nordheim;<ref name="Fowler1928"/> for this physically unrealistic case, ''ν''(''f<sub>h</sub>'')&nbsp;=&nbsp;1, and an analytical approximation for ''P'' exists.

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

As noted at the beginning, the effects of the atomic structure of materials are disregarded in the relatively simple treatments of field electron emission discussed here. Taking atomic structure properly into account is a very difficult problem, and only limited progress has been made.<ref name=mo84/> However, it seems probable that the main influences on the theory of Fowler–Nordheim tunneling will (in effect) be to change the values of ''P'' and ''ν'' in eq. (15), by amounts that cannot easily be estimated at present.

All these remarks apply in principle to Fowler&nbsp;Nordheim tunneling from any conductor where (before tunneling) the electrons may be treated as in [[travelling wave|travelling-wave states]]. The approach may be adapted to apply (approximately) to situations where the electrons are initially in localized states at or very close inside the emitting surface, but this is beyond the scope of this article.