Editing Field electron emission (section)

=== Modified equations for nanometrically sharp emitters ===
Most of the theoretical derivations in the field emission theory are done under the assumption that the barrier takes the Schottky–Nordheim form eq. (3). However, this barrier form is not valid for emitters with radii of curvature ''R'' comparable to the length of the tunnelling barrier. The latter depends on the work function and the field, but in cases of practical interest, the SN barrier approximation can be considered valid for emitters with radii {{nowrap|''R'' > 20 nm}}, as explained in the next paragraph.

The main assumption of the SN barrier approximation is that the electrostatic potential term takes the linear form <math>\Phi = Fx</math> in the tunnelling region. The latter has been proved to hold only if <math>x \ll R</math>.<ref name=KX>{{cite journal|doi=10.1098/rspa.2014.0811|title=Derivation of a generalized Fowler–Nordheim equation for nanoscopic field-emitters|year=2015|last1=Kyritsakis|first1=A.|last2=Xanthakis|first2=J. P. |journal=Proceedings of the Royal Society A|volume=471|issue=2174|page=20140811|bibcode=2015RSPSA.47140811K|doi-access=free}}</ref> Therefore, if the tunnelling region has a length <math>L</math>, <math>x<L</math> for all <math>x</math> that determines the tunnelling process; thus if <math>L \ll R</math> eq. (1) holds and the SN barrier approximation is valid. If the tunnelling probability is high enough to produce measurable field emission, L does not exceed 1–2&nbsp;nm. Hence, the SN barrier is valid for emitters with radii of the order of some tens of nm.

However, modern emitters are much sharper than this, with radii that of the order of a few nm. Therefore, the standard FN equation, or any version of it that assumes the SN barrier, leads to significant errors for such sharp emitters. This has been both shown theoretically<ref>{{cite journal|doi=10.1063/1.106257|title=Derivation of a generalized Fowler–Nordheim equation for nanoscopic field-emitters|year=1991|last1=He|first1=J.|last2=Cutler|first2=P. H.|journal=Applied Physics Letters|volume=59|issue=13|page=1644|bibcode=1991ApPhL..59.1644H}}</ref><ref>{{cite journal|doi=10.1116/1.589929|title=Deviations from the Fowler–Nordheim theory and peculiarities of field electron emission from small-scale objects|year=1998|last1=Fursey|first1=G. N.|last2=Glazanov|first2=D. V.|journal=Journal of Vacuum Science and Technology B|volume=16|issue=2|page=910|bibcode=1998JVSTB..16..910F}}</ref> and confirmed experimentally.<ref>{{cite journal|doi=10.1103/PhysRevB.87.115436|title=Scale invariance of a diodelike tunnel junction|year=2013|last1=Cabrera|first1=H.|journal=Physical Review B|volume=87|issue=11|page=115436|display-authors=etal|arxiv=1303.4985|bibcode=2013PhRvB..87k5436C|s2cid=118361236}}</ref>

The above problem was tackled by Kyritsakis and Xanthakis,<ref name=KX/> who generalized the SN barrier by including the electrostatic effects of the emitter curvature. The general barrier form for an emitter with radius of average curvature <math>R</math> (inverse of the average of the two principal curvatures) can be [[asymptotic expansion|asymptotically expanded]] as<ref>{{Cite journal |last=Kyritsakis |first=Andreas |date=2023-03-21 |title=General form of the tunneling barrier for nanometrically sharp electron emitters |url=https://doi.org/10.1063/5.0144608 |journal=Journal of Applied Physics |volume=133 |issue=11 |pages=113302 |doi=10.1063/5.0144608 |arxiv=2207.06263 |bibcode=2023JAP...133k3302K |s2cid=256390628 |issn=0021-8979}}</ref> {{NumBlk|:|<math> M^{KX} (x) = h - eFx \left[ 1-\frac{x}{R} + O \left(\frac{x}{R} \right)^2 \right] - \frac{e^2}{16\pi \epsilon_0 x} \left[ 1-\frac{x}{2R} + O \left(\frac{x}{R} \right)^2 \right] </math>.|{{EquationRef|43}}}}

After neglecting all <math>O(x/R)^2</math> terms, and employing the [[WKB approximation|JWKB approximation]] (4) for this barrier, the Gamow exponent takes a form that generalizes eq. (5) 
{{NumBlk|:|<math> G(h,F,R) = \frac{b h^{3/2}}{F} \left(v(f) + \omega(f)\frac{h}{eFR} \right) </math>|{{EquationRef|44}}}}
where <math>f</math> is defined by (30d), <math>v(f)</math> is given by (30c) and <math>\omega(f)</math> is a new function that can be approximated in a similar manner as (30c) (there are typographical mistakes in ref.,<ref name="KX" /> corrected here): 
{{NumBlk|:|<math>\omega(f) \approx \frac{4}{5} - \frac{7}{40} f  + \frac{1}{200} \log(f) . </math>|{{EquationRef|45}}}}

Given the expression for the Gamow exponent as a function of the field-free barrier height <math>h</math>, the emitted current density for cold field emission can be obtained from eq. (23). It yields 
{{NumBlk|:|<math> J = a \frac{F^2}{\phi} \left(\frac{1}{\lambda_d(f)} + \frac{\phi}{eFR} \psi(f) \right)^{-2} \exp\left[-\frac{b \phi^{3/2}}{F} \left(v(f) + \omega(f)\frac{\phi}{eFR} \right) \right] </math> |{{EquationRef|46}}}}
where the functions <math>\lambda_d(f)</math> and <math>\psi(f)</math> are defined as
{{NumBlk|:|<math>\frac{1}{\lambda_d(f)} \equiv v(f) - \frac{4}{3}f \frac{\partial v}{\partial f} \approx 1 + \frac{f}{9} - \frac{f \log(f)}{22} </math>|{{EquationRef|47a}}}}
and 
{{NumBlk|:|<math>\psi(f) \equiv \frac{5}{3} \omega(f) - \frac{4}{3}f\frac{\partial\omega}{\partial f} \approx \frac{4}{3} - \frac{f}{15} - \frac{f \log(f)}{1200} </math>|{{EquationRef|47b}}}}
In equation (46), for completeness purposes, ''λ''<sub>''d''</sub> is not approximated by unity as in (29) and (30a), although for most practical cases it is a very good approximation. Apart from this, equations (43), (44) and (46) coincide with the corresponding ones of the standard Fowler–Nordheim theory (3), (9), and (30a), in the limit {{nowrap|''R'' → ∞}}; this is expected since the former equations generalise the latter.

Finally, note that the above analysis is asymptotic in the limit {{nowrap|''L'' ≪ ''R''}}, similarly to the standard Fowler–Nordheim theory using the SN barrier. However, the addition of the quadratic terms renders it significantly more accurate for emitters with radii of curvature in the range ~&nbsp;5–20&nbsp;nm. For sharper emitters there is no general approximation for the current density. In order to obtain the current density, one has to calculate the electrostatic potential and evaluate the [[WKB approximation|JWKB integral]] numerically. For this purpose, scientific computing software has been developed (see e.g. [https://getelec.org GETELEC]<ref>{{cite journal|doi=10.1016/j.commatsci.2016.11.010|title=A general computational method for electron emission and thermal effects in field emitting nanotips|year=2017|last1=Kyritsakis|first1=A.|last2=Djurabekova|first2=F. |journal=Computational Materials Science|volume=128|page=15|arxiv=1609.02364|s2cid=11369516}}</ref>).