Setting up limits to final cathode parameters

In cathode iterations the user is asked to specify the final cathode parameters (either Richardson’s constant or the current per unit area), as well as the initial values..

The final values have two uses. The first is to obtain the fastest possible convergence of the iterations, without affecting the final current. The second is to provide, if necessary, a physical limit to the current, caused by example by a temperature-limited cathode.

In general the initial Richardson’s constant should be chosen to be significantly lower than the final value, to give a gentle start to the cathode iterations. If using current densities, the initial value should be significantly lower than the final value, for the same reason.

The user has to establish the best values of the final maximum current density or Richardson’s constant empirically -see experimenting with parameters.

The value of the final temperature or final maximum current density for the later iterations is usually not critical in CPO2DS, nor in CPO3DS if kT is zero, provided that it is higher than the final space-charge limited value, which for a planar is given by Child’s Law

0.002334*V**1.5/d**2,

in mA/mm**2, if d is in mm (for a spherical diode, see the 'bench-test' files test3d10.dat and test2d11.dat).

But in CPO3DS with a non-zero kT great care must be taken. Langmuir’s relationships is used, with corrections for any curvature that the cathode might have. If the final temperature or final maximum current density is set too large a dense space-charge cloud builds up in front of the cathode. This cloud creates a potential minimum which becomes the effective cathode -called the ‘virtual cathode. Most of the emitted electrons are reflected by the potential minimum and are returned to the cathode. The depth of the minimum can be larger than kT/e.

In these circumstances the convergence of the iterations can be slow. The results will probably not reproduce those obtained in real systems because in practice the temperature of the cathode is almost always reduced until it is near the temperature that would give a temperature-limited cathode current.

On the other hand if the final temperature or maximum current density is set too low the cathode current will be effectively temperature-limited.

If it is known that the current is space-charge limited then the maximum current density that you enter (either directly, or indirectly using Richardson’s constant) should be larger than the cathode current density (mA/mm**2) that you expect to obtain finally. But to avoid unnecessary oscillations it should not be much larger. 50% larger is usually suitable, and 400% should be regarded as an upper limit when kT is non-zero -see also the note on the cathode about not using high values, and the footnotes of the example files xmpl3d11 and xmpl3d21.

As mentioned above this is the second maximum current density (or the second Richardson’s constant) that has to be specified, and so would be redundant except that it enables the user to start the iterations gently by making the first value smaller than the value that applies for the later iterations.

Start by doing a quick approximate simulation to establish the approximate asymptotic final values of the parameters of the cathode. Then use these values as the initial values. When kT is non-zero try to set the maximum current emitted for the cathode surface at approximately 1.5 times the value of the space-charge limited current -see the footnotes of the example files xmpl3d11 and xmpl3d21. A series of runs may be needed to establish the most appropriate values to use.