Computation of CEX-Ion Flux and Sputtering Rate
11. Output x, v, and h
4.8 Chapter Summary
the sputtering as well.
It took approximately 30 minutes on a modern desktop computer to do the entire computation required to obtain the data contained in Figure 4.30. The large majority of this time was spent executing the root-finding algorithm which determines the scattering-angle solutions. This execution time is impractical, for, to obtain the final expected sputtering rate at a specific point around any particular grid, the computations shown in this example must be repeated for all of the nearly 15,000 holes. Some savings in time could be made by both reducing the number of mesh points (J = 441 in this example), and reducing the number of shells computed in each beamlet (some beamlets may also not require as many shells as others, due to a small contribution). Additionally, since the model is based on the superposition of individual beamlets, the model is amenable to parallel computing methods which, given the required computing resources, could significantly decrease the time required.
All of these time-saving methods make little difference in an optimization problem, as we are attempting to do. Incorporating this model into an optimization routine requires computing the sputtering rate for multiple grids, not only the multiple holes of one grid. In order to implement an optimization routine using this model, we must reduce the computational time required. In Chapter 5 we discuss some time-reducing simplifications made.
trajectories through the plume are calculated using a time-adaptive velocity-Verlet algorithm.
Both the charge-exchange differential cross-section and the sputter yield were obtained from physical measurements, which were parameterized by applying simple curve-fits. The differential cross-section data was from scattering measurements of xenon ions, with an energy of 1100 eV, colliding with stationary (thermal) neutral xenon atoms. The data for the sputter yield was obtained from measurements of xenon ions impinging on a molybdenum surface.
A two-dimensional problem was analyzed from which information was obtained that assists both in developing the model, as well as in interpreting results from the full three-dimensional problem. It was found that, from any small scattering volume, there may be multiple scattering angle solutions, which result in CEX ions passing through the specified target point, associated with any primary ion velocity vector. The presence of multiple solutions imply that there are certain “turn around” angles where the streamtube of ions scattered at these angles gets focussed. Scattering-angle solutions that approach one of these “turn around” angles result in a very large flux through the target point, due to the focussing effect, dA/dΩ→0. A Gauss-Newton root-finding algorithm was developed in order to locate the scattering-angle solutions for any primary-ion velocity vector passing through a specified point in space.
In order to facilitate numerical integration of the ion flux and sputtering rate equations, the idea of a beamlet shell and mesh were introduced. As an implementation example, one beamlet shell within an NSTAR beamlet was modeled. It was demonstrated how all the required quantities are computed for mesh points in the beamlet shell, and from which the numerical integral was computed.
An example was also shown of how the sputtering rate contribution from one beamlet is obtained by integration over multiple beamlet shell radii. The vast majority of CEX ions were found to have energies comparable to the potential difference between the scattering point and the target point ΔΦ. This is in agreement with the assumption made in other models: most CEX ions receive very little energy from the charge-exchange collision process. Though the calculated CEX-ion density was found to be within the range of values computed by other models, it was determined that the current model inadequately treats the lowest-energy ions. Since the low-energy ions are the most
populous, density calculations from this model are unreliable.
Despite the inadequacies of the model in its treatment of low-energy ions, it was found that the majority of the sputtering is due to collisions involving CEX ions that have significantly more energy than ΔΦ. This is a direct result of the sensitivity of the sputter yield to energy. Since the sputtering contribution from the low-energy ions was found to be so small, it was determined that the errors in computing the number of these ions have little impact on the computed sputtering rate. It was found that the highest-energy CEX ions originate from a region very close to the grid surface. These ions are the products of ion-atom encounters involving collision angles smaller than 90◦, and were found to have values in the low 80◦s or lower. This finding is in disagreement with other models, where the the assumption is that collision dynamics play a minor role in sputtering, due to the vast majority of low-energy ions.
Implementation of the model, as presented in this chapter, was found to be a formidable task, due to the large amount of time required. If the computation was to be done for only one grid, parallel computing methods could be used to reduce the required time to a reasonable amount.
However, incorporating the model into an optimization routine, for which computation of multiple grids would be required, is impractical. We compare, in Chapter 5, the results obtained using some simplifications with those obtained using the full computational description discussed in this chapter.