CPO2D -30th example file, enhancement factor for a 'hemisphere on post'.

The 'post' has a height h = 1 micron (ie 0.01mm) and a radius a = 1nm. To create a field an anode is placed at an arbitrary 9 micron away, with a potential of 0.01V, giving a 'far field' of 1V/mm.

The field enhancement factor depends only on h/a

The ray output file contains the field at 10 points at distances of 0.05 to 0.5 nm from the surface of the cap.

Results, fields in V/mm:

 r z er ez etotal 0.000E+00 5.000E-08 0.000000E+00 -5.464995E+02 5.464995E+02 0.000E+00 1.000E-07 0.000000E+00 -4.984520E+02 4.984520E+02 0.000E+00 1.500E-07 0.000000E+00 -4.567765E+02 4.567765E+02 0.000E+00 2.000E-07 0.000000E+00 -4.203965E+02 4.203965E+02 0.000E+00 2.500E-07 0.000000E+00 -3.884485E+02 3.884485E+02 0.000E+00 3.000E-07 0.000000E+00 -3.602371E+02 3.602371E+02 0.000E+00 3.500E-07 0.000000E+00 -3.351977E+02 3.351977E+02 0.000E+00 4.000E-07 0.000000E+00 -3.128681E+02 3.128681E+02 0.000E+00 4.500E-07 0.000000E+00 -2.928671E+02 2.928671E+02 0.000E+00 5.000E-07 0.000000E+00 -2.748781E+02 2.748781E+02

Analysis of the results:

Extrapolate field to z=0 by quartic extrapolation of E(z)*(z+1e-6)**2: gives E(a) = 602.29. Therefore enhancement factor, gamma = 602.29

Vary N = number of segments, dz = z_anode - z_baseplate, r = radius of bridge to ensure that inaccuracy is less than about 0.01 percent. All dimensions in mm, radius a of cap = 1E-6mm

Vary length h of post:

 h l/a N dz r gamma Edgcombe Fit difference% .003 3001 1650 .06 .03 1644.99 1643.6 1643.29 0.10 .001 1001 750 .01 .01 602.82 597.5 603.33 0.09 .0003 301 950 .01 .01 202.21 200.5 202.88 0.33 .0001 101 750 .01 .01 76.994 76.37 76.76 0.30 .00003 31 950 .01 .01 28.405 28.18 28.43 0.07

Here l = h + a, the total length of the nanotube.

The empirical fit is to 1.0782*(l/a + 4.7)**0.9152

The column labelled 'Edgcombe' are those given by C J Edgcombe and U Valdre, Solid State Electronics 45 (2001) 857-863. These authors use the fit

1.125*(l/a + 2)**0.91

Dependence of field strength on angle (ie angle to axis):

 theta Er fit difference% 0 602.84 601.65 0.2 10 601.54 600.25 0.2 20 597.51 596.14 0.2 30 590.61 589.24 0.2 40 580.54 579.33 0.2 50 566.78 566.12 0.1 60 548.45 549.20 0.1 70 523.98 527.28 0.6 80 489.68 497.15 1.5 90 428.29 422.59 1.3

Fit is 422.6 + 179.1*sqrt(cos(theta))