xmpl3d16, 16th 'example' data file for CPO3D

Extended Schottky emission, spherical cathode.

Please see general information on field emission.

This example is of course artificial -real Schottky emission cathodes are sharp points mounted usually on cones. When simulating real cathodes it is important to follow the advice on setting up the ray paramerters. Please also see general information on field emission.

The cathode and anode have radii of 1m and 1cm respectively, the cathode-anode voltage difference is 1000V, the work function of the cathode is 2.7eV, and the temperature is 1800K. A modification of the Richardson-Dushman equation is used, as given by Hawkes and Kasper. The initial velocity components are randomised. The current that should be given by the program is 0.3130 mA, and in fact is 0.3127 mA.

The following data were obtained when the memory and speed of PC's was much more limited than at present, so the available number of segments was small and the requested inaccuracies were fairly high to give a quick demonstration.

In this example the geometry and segment subdivision is the same as in file test3d12.dat, which deals with conventional field emission. Please see that file for important information about the choice of step lengths.

The field at the cathode is 1.0001E7 V/cm and the value of kT is 0.15506 eV, corresponding to T = 1800 K.

Equation 44.42 of Hawkes and Kasper (Principles of Electron Optics, Academic Press, 1989) is used in this program to calculate the current density when the 'extended Schottky emission' option is called. For the present parameters the potential barrier is lowered by delta_w = 1.200 eV, the quantity q is 0.5173 and so the current density is 3.985*10^5 mA/mm^2 (that is, 3.985*10^4 A/cm^2). This can be compared with the more accurate current density calculated numerically by A B El-Kareh, J C Wolfe and J E Wolfe, J Appl Phys, vol 48, 4749 (1977), and reported by J E Wolfe, J Vac Sci Tech, vol 16, 1704 (1979). Reading from Fig 3 of the latter paper, the current density is 9*10^4 A/cm^2, which is more than twice as large as that given by equation 4.42 of Hawkes and Kasper. The agreement would be better for smaller fields or higher temperatures.

The current emitted by the present cathode (from the minimum sector, which is 1/16 of the total area) should be 0.3130 mA (according to equation 44.42). In fact in the present example it is 0.3127 mA.

The current is sensitive to the magnitude of the field at the surface of the cathode -a change of 1% in the field gives a change of 5% in the current, for the present example (and so the sensitivity is smaller than for field emission -see the note to file test3d12.dat). Also it should be noted that the most difficult field evaluation for CPO3DS is the field near to the surface of an electrode, and that the calculated value of such a field is often significantly less accurate than for a field or potential at a point further away from a surface. The field is therefore calculated with extra care, as described for field emission in file test3d12.dat.

It can be noticed in the output data file that there is a randomised component in the energies. The transverse velocity components are given randomised values corresponding to the energy kT, while the longitudinal component corresponds to the higher energy kt*u, where u = pq/sin(pq), pq = pi*q, and q is the quantity mentioned above and defined by equation 44.43 of Hawkes and Kasper. The factor u is therefore 1.628.