Approximation Technique
5.1 LOS Approximation Method
5.1.2 Comparison of Results with the Full Trajectory Analysis
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Figure 5.2: Beamlet-shell contribution to the CEX-flux distribution: Approximations. Contribu- tions from the same beamlet shell from the example in Section 4.7.2 were computed using different simplifying approximations. The line-of-sight (LOS) only approximation assumes the CEX ions scatter directly to the target point and does not take account of the change in potential ΔΦ. The energy-corrected LOS approximation also assumes the CEX ions scatter directly to the target point, but accounting for the change in potential between the scattering and target points is also made.
The energy- and angle-corrected LOS approximation adjusts all LOS scattering angles such that the maximum ion energy matches that from the distribution obtained by full trajectory analysis (from Figure 4.26).
scattering angle solution branch.
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Figure 5.3: Sputtering-rate-distribution contribution comparison. Contributions, Y(E) dΓk/dr, from the same beamlet shell from the example in Section 4.7.2 were computed using different simpli- fying approximations. The expected sputtering is obtained from the flux distributions in Figure 5.2.
mate distribution is significantly less. When the scattering angle is corrected to match the maximum energy, the approximation yields a distribution (dashed line) very similar to the actual distribution.
At the lowest energies, we can do little in the way of comparison, for as we found in our analysis in Chapter 4, even the complete-trajectory computation method is inaccurate at these energies. Since the majority of the CEX ions populate the low-energy region, as anticipated, integration of each of these functions yields vastly different results for the total density contribution from this particular beamlet shell.
Our ultimate goal is to predict the amount of sputtering expected. For each distribution resulting from the different approximations, the expected sputtered flux from ions originating from this shell is shown in Figure 5.3. As we found for the ion flux distribution, the sputtering rate expected using only LOS trajectories is very different from the rate predicted by the complete trajectory computation. Even enforcing energy conservation is insufficient to produce a distribution similar to the full-trajectory analysis. On the other hand, the sputtering rate distribution obtained using both the energy and scattering-angle corrections to the LOS approximation is very similar to the distribution computed using complete-trajectory analysis, however it still predicts a lower rate.
To obtain a better idea of how well the energy- and angle-corrected LOS approximation does in duplicating the results obtained using the full-trajectory analysis, the sputtering rate contributions from three entire NSTAR beamlets were computed using both methods, and compared. The results for each of the beamlets are shown in Figures 5.4 through 5.6. The contributions to the CEX- ion flux distribution from each beamlet are shown at the top of each figure. The inset depicts a representative inclination scattering-angle solution surface for one of the shells in the beamlet. The lower-left plot in each figure shows the contribution to the flux distribution (left axis), integrated over the beamlet-shell radii, and the sputtered flux distribution (right axis), also integrated over the beamlet-shell radii. The solid lines correspond to the solutions found based on the full-trajectory analysis and the dashed lines correspond to the solutions found using the energy- and angle-corrected LOS approximation. The lower-right plot in each figure shows the contribution to the sputtering rate (left axis), integrated over CEX-ion energy, as a function of the beamlet-shell radii, and the cumulative integral of the sputtering rate (right axis), integrated over the beamlet-shell radii. Again, the solid lines correspond to the solutions found based on the full-trajectory analysis, and the dashed lines correspond to the solutions found using the energy- and angle-corrected LOS approximation.
The total contributions to the sputter rate, Υk, due to the approximating method in the first and second examples shown (Figures 5.4 and 5.5) are less than that predicted using the full-trajectory analysis by approximately 10%. The beamlet in the first example originated from the hole located at (x, y)k = (−0.111,5.96) cm, which is nearly on the axis transverse to the axis on which the target point was located (S(˜x) = (60,0,3)), and at approximately a third of the grid radius from the center of the grid (Rg = 15 cm). The beamlet in the second example originated from the hole located at (x, y)k= (5.994,0) cm, which is on the same axis as the target point, and also approximately a third of the grid radius from the center of the grid.
It is interesting to note from the second example the presence of the “valley” in the CEX-ion flux distribution at energies around 30 eV for shell radii less than approximately 4 cm. As the shell radii increase, the valley disappears, indicating a merging of two separate populations (“high”- and
“low”-energy). The reason for this is apparent from the scattering-angle solutions shown in the
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Figure 5.4: Full-beamlet approximation comparison: Hole (x, y)k = (−0.111,5.96) cm. Top: The contribution to the CEX-ion energy distribution, as a function of energy and beamlet shell radius, computed using the full-trajectory analysis. The inset displays a characteristic inclination scattering- angle solution surface. Lower-left: The total beamlet contribution to the (1) CEX-ion energy distribution (left axis), integrated over beamlet-shell radii, and (2) sputtering rate (right axis), as a function of CEX-ion energy. Lower-right: The total beamlet contribution to the sputtering rate (1) as a function of beamlet-shell radius (left axis), and (2) integrated (cumulative) over beamlet-shell radius (right axis). Solid lines indicate results obtained using the full-trajectory analysis. Dashed lines indicate results obtained using the energy- and angle-corrected LOS approximation.
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Figure 5.5: Full-beamlet approximation comparison: Hole (x, y)k= (5.994,0) cm. Top: The contri- bution to the CEX-ion energy distribution, as a function of energy and beamlet shell radius, com- puted using the full-trajectory analysis. The inset displays a characteristic inclination scattering- angle solution surface. Lower-left: The total beamlet contribution to the (1) CEX-ion energy distribution (left axis), integrated over beamlet-shell radii, and (2) sputtering rate (right axis), as a function of CEX-ion energy. Lower-right: The total beamlet contribution to the sputtering rate (1) as a function of beamlet-shell radius (left axis), and (2) integrated (cumulative) over beamlet-shell radius (right axis). Solid lines indicate results obtained using the full-trajectory analysis. Dashed lines indicate results obtained using the energy- and angle-corrected LOS approximation.
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Figure 5.6: Full-beamlet approximation comparison: Hole (x, y)k= (0,11.92) cm. Top: The contri- bution to the CEX-ion energy distribution, as a function of energy and beamlet shell radius, com- puted using the full-trajectory analysis. The inset displays a characteristic inclination scattering- angle solution surface. Lower-left: The total beamlet contribution to the (1) CEX-ion energy distribution (left axis), integrated over beamlet-shell radii, and (2) sputtering rate (right axis), as a function of CEX-ion energy. Lower-right: The total beamlet contribution to the sputtering rate (1) as a function of beamlet-shell radius (left axis), and (2) integrated (cumulative) over beamlet-shell radius (right axis). Solid lines indicate results obtained using the full-trajectory analysis. Dashed lines indicate results obtained using the energy- and angle-corrected LOS approximation.
top inset. For shell radii less than approximately 4 cm, all points within the beamlet shell have scattering events that contribute CEX ions to the target point. Additionally, all of the contributing points within the shell fall within Case II: there are multiple scattering-angle solutions. For these small shell radii, there are no points where Case III arises: a situation where the multiple solutions merge to a single solution and d ˜A/dΩ→ 0. Thus, there is a range of energy which no CEX ions originating from these beamlet shells close to the grid can have. The eventual merging of the two CEX-ion populations at larger beamlet shell radii is due to the scattering-angle solutions eventually becoming more like that shown previously in Figure 4.24 or the inset of Figure 5.4: the two solution branches merge at some point within the beamlet shell, and some points within the shell make no contribution to the CEX-ion population at the target point.
The total contribution to the sputter rate, Υk, due to the approximating method in the third example (Figure 5.6) is also less than that predicted using the full-trajectory analysis, but only by approximately 1%1. The beamlet in this last example originated from the hole located at (x, y)k= (0,11.92) cm, which is on the axis transverse to the axis on which the target point was located, and at nearly 80% of the grid radius from the center of the grid. Note that the contributions from each of the beamlets in these examples have values ranging over two orders of magnitude. The contribution from the third beamlet is an order of magnitude less than that from the first beamlet, which, in turn, is less by another order of magnitude than the contribution from the second beamlet.
The three examples shown indicate that the approximations made — LOS trajectories corrected for energy conservation with an adjustment to the scattering angles to match the maximum CEX- ion energy — give a good approximation of the results expected from the full-trajectory analysis, with an error within the range of 10%. These approximations reduce the computations required by limiting the number of scattering-angle solutions that need to be found to only a few mesh points, instead of all the mesh points within any beamlet shell. The computational time saved is exceptional,
1The large difference in error between this example and the previous two (10%) is now thought to be attributed to the expected contributing area of the beamlet shell. Figure 5.2 indicates that the LOS approximation reduces the contributing area, since the gradient of the surface is larger for the LOS solutions than for those found using the full-trajectory analysis. If effort was made to match the surface gradients as well, it is expected that the contributing area from the LOS approximation would increase, and yield better agreement with the full analysis. In the case of this third example, the contributing area is already small (see inset of Figure 5.6), so it is expected the differing surface gradients do not have as much impact as for the others. We arrived at this line of thought after the work was completed, and therefore were not able to test it further.
since the majority of the computation during the full-trajectory analysis is spent on locating the scattering-angle solutions using the Gauss-Newton search algorithm of Section 4.6.
The pseudo-code outlining the procedure for adjusting the scattering-angle solutions is presented next. Section 5.2 addresses the case where the grid is not axisymmetric. In such cases, the compu- tational time required can substantially increase, since the electric field cannot be expected to be axisymmetric either.