Editing Field electron emission (section)

==== Non-zero temperatures ====
To obtain a result valid for non-zero temperature, we note from eq. (23) that ''z''<sub>S</sub>''d''<sub>F</sub>''D''<sub>F</sub> = ''J''<sub>0</sub>/''d''<sub>F</sub>. So when eq. (21) is integrated at non-zero temperature, then – on making this substitution, and inserting the explicit form of the Fermi–Dirac distribution function – the ECD ''J'' can be written in the form:

{{NumBlk|:|<math> J=J_0 \int_{-\infty}^{\infty} \frac{\exp(\epsilon / d_{\mathrm{F}})}{1 + \exp [(\epsilon/d_{\mathrm{F}})(d_{\mathrm{F}}/k_{\mathrm{B}} T)]} \mathrm{d}(\epsilon/ d_{\mathrm{F}}) = \lambda_T J_0 ,</math>|{{EquationRef|26}}}}

where ''λ''<sub>''T''</sub> is a temperature correction factor given by the integral. The integral can be transformed, by writing <math> w = d_{\mathrm{F}}/k_{\mathrm{B}}T </math> and <math> x=\epsilon/d_{\mathrm{F}} </math>, and then <math> u = \exp(x) </math>, into the standard result:<ref>{{cite book|author=Gradshteyn and Rhyzhik|title=Tables of Integrals, Series and Products|year=1980|publisher=Academic, New York|bibcode=1980tisp.book.....G }} see formula 3.241 (2), with ''μ''=1</ref>

{{NumBlk|:|<math> \int_{-\infty}^{\infty} \frac{ {\mathrm{e}}^x } { 1+ {\mathrm{e}}^{wx} } \mathrm{d}x = \int_{0}^{\infty} \frac{\mathrm{d}u}{1+u^w} = \frac{\pi} {w\sin(\pi/w)}. </math>|{{EquationRef|27}}}}

This is valid for {{nowrap|''w'' > 1}} (i.e., {{nowrap|''d''<sub>F</sub>/''k''<sub>B</sub>''T'' > 1}}). Hence – for temperatures such that {{nowrap|''k''<sub>B</sub>''T'' < ''d''<sub>F</sub>}}:

{{NumBlk|:|<math> \lambda_T = \frac{\pi k_{\mathrm{B}} T/d_{\mathrm{F}} }{ \sin(\pi k_{\mathrm{B}} T / d_{\mathrm{F}})} \approx 1 + \frac{1}{6} \left( {\frac{\pi k_{\mathrm{B}} T}{ d_{\mathrm{F}}}} \right) ^2, </math>|{{EquationRef|28}}}}

where the expansion is valid only if ({{nowrap|π''k''<sub>B</sub>''T'' / ''d''<sub>F</sub>) ≪ 1}}. An example value (for {{nowrap|1=''φ'' = 4.5 eV}}, {{nowrap|1=''F'' = 5 V/nm}}, {{nowrap|1=''T'' = 300 K}}) is {{nowrap|1=''λ''<sub>''T''</sub> = 1.024}}. Normal thinking has been that, in the CFE regime, ''λ''<sub>''T''</sub> is always small in comparison with other uncertainties, and that it is usually unnecessary to explicitly include it in formulae for the current density at room temperature.

The emission regimes for metals are, in practice, defined, by the ranges of barrier field ''F'' and temperature ''T'' for which a given family of emission equations is mathematically adequate. When the barrier field ''F'' is high enough for the CFE regime to be operating for metal emission at 0&nbsp;K, then the condition {{nowrap|''k''<sub>B</sub>''T'' < ''d''<sub>F</sub>}} provides a formal upper bound (in temperature) to the CFE emission regime. However, it has been argued that (due to approximations made elsewhere in the derivation) the condition {{nowrap|''k''<sub>B</sub>''T'' < 0.7''d''<sub>F</sub>}} is a better working limit: this corresponds to a ''λ''<sub>''T''</sub>-value of around 1.09, and (for the example case) an upper temperature limit on the CFE regime of around 1770&nbsp;K. This limit is a function of barrier field.<ref name=mo84/><ref name=MG56/>

Note that result (28) here applies for a barrier of any shape (though ''d''<sub>F</sub> will be different for different barriers).