Fowler-Nordheim equation.


(Taken from the footnotes of the test data files test2d13.dat and test3d12.dat).

The Fowler-Nordheim equation has been taken from R. H. Good and E. W. Muller, Handbuch der Physik, vol 21 (1956) and P. W. Hawkes and E. Kasper, Principles of Electron Optics (Academic Press, 1989). Distances are in mm, the field F is in V/mm, and the current density j is in mA/mm**2. The work function of the material of the cathode is P eV.


y = 0.0012000*sqrt(F)/P,

t(y) = 1 + 0.1107*y**1.33,

v(y) = 1 - y**1.69 + 0.0028*sin(2*pi*y),

(the 3rd term being a new empirical correction term)

c = 6.8309E6*P**1.5*v/F,

d = 9.7596E-8*F/(sqrt(P)*t),

p = kT/d,


 j = 0.0015414*(F/t)**2*(1/P)*exp(-c)*(pi*p)/sin(pi*p).


This is valid for y < 1 and p < 0.7. To quote Hawkes and Kasper ''For larger values of p no satisfactory simple theory is known''.

Note that for cpo2ds kT and p are zero and so the factor (pi*p)/sin(pi*p) = 1.


The most probable energy, ei, of the electrons is given by

 ei = -(P + d).

(In fact d is the energy gained over a distance lambda/(4*pi), where lambda is the de Broglie wavelength at the energy eP.)


Note that the energy ei is negative and that it corresponds to an energy slightly below the Fermi energy of the cathode (in the present example d is 0.176 eV). This energy is used as the initial energy for the ray tracings.


The field emission electrons have an energy spread of width 2.44*d, but in CPO2DS this is not taken into account. For a more careful treatment, including thermal effects, use CPO3DS.


It can be noticed in the output data file that there is a randomised component in the energies. This is present because as explained in the relevant note the velocity components are randomised even if kT is zero. The energy spread is characterised by the acceleration energy d given above -see R H Good and E W Muller, Handbuch der Physik, Vol 21, 176 (1956) and P W Hawkes and E Kasper, Principles of Electron Optics (Academic Press, 1989). In effect, d replaces kT in the usual velocity distributions, even when T = 0.


An 'artificial' option is also available (in CPO3DS) in which the electrons start with fixed energy (chosen by the user), normal to the surface.