3.1 Modeling of electron-molecule interactions with BOLSIG+
3.1.2 Influence of additional parameters on the elec- elec-tron-molecule interactionselec-tron-molecule interactions
3.1.1.1 Dependence of EEDF and rates on the cross sections The results of the calculations based on the cross section data sets (Kushner and Zhang [66], Voloshinet al.[80], see Figure 3.2) are shown comparatively in Figure 3.4 in dependence on the mean electron energy.
• Qualitatively, a similar dependence of the rate coefficients on the mean electron energy is observed. CF3 is throughout the major product from neutral feed gas dissociation (top line in both graphs), followed byCHF2 and CF. CF2 is also produced, but the rate constant is at least some orders of magnitude lower than the one of CF. Further fragmentation is obtained from ionization dissociation and electron-attachment dissociation (bottom line in both figures).
• Quantitatively, differences between the two data sets are observed due to the different interpretations of low-lying cross section features. The neutral dissociation data show bad agreement even after the removal of the 11 eV dissociation add-on. In case the additional attachment process around 11.5 eV in Kushner’s data (hollow squares in the figure) is discarded, the calculated rate coefficients for the formation of ions show good agreement.
The electronic system is affected by the differences in the cross sections as well. Figure 3.5 shows the mean electron energies and electron mobilities vs. reduced electric field. However, the effect is less than 10% and turned out to be negligible. Such, the overall agreement between the two different cross section sets is fairly good when the ionization rates, the dependence of mean energy on E/N, and the mobility are considered. Contrary, the neutral dissociation rates show much larger deviations. As the quantum-mechanical calculations give a better theoretical background to the data published by Voloshinet al., these cross sections were adopted to the present work.
3.1.2 Influence of additional parameters on the
Figure 3.5: Top: Simulated mean energy hi versus the reduced electric field, E/N, for conditions as noted in Table 3.1, based on different cross section data sets (Kushner and Zhang [66] / Voloshinet al.[80]). Bottom: Simulated electron mobilityµ×N versus the reduced electric field versus the reduced electric field, E/N.
Figure 3.6: Neutral dissociation cross sections of Voloshinet al. [80]).
For this study, the cross section data published in [80] were adopted.
The neutral dissociation cross sections are shown in detail in Figure 3.6.
3.1.2.1 Influence of plasma parameters
So far, only positive ions have been regarded. But it is known that the den-sity of negative ions in pure rf discharges ofCHF3can exceed the electron density by about one order of magnitude [83], and therefore, the degree of ionization can be about one order of magnitude larger than expected from electron density measurements. In BOLSIG+, this can be accounted for by setting the degree of ionization separately from the electron density. Upon the variation of the plasma density (nplasma =ne+n++n−) as shown in Figure 3.7, an increase in weight in the beginning as well as in the tail of the EEDF is observed in case that the plasma density is increased. The enhancement of the ion density, resulting from an increase in the degree of ionization, α, at unchanged electron density, causes the same increase.
Such, the ion density is the major reason for this change in the EEDF com-pared to standard conditions. This result is comparable to simulations in argon in [7]. Contrary, the separate increment of the electron density ne
only by one order of magnitude does not show any difference compared to the standard conditions. The explanation for such behavior is an increase of the electron-electron collisions upon an increasing degree of ionization, repopulating the high-energy tail of the EEDF like observed in fluid-model
Figure 3.7: Electron energy distribution functionf()for several different condi-tions for E/N= 4.28 Td. Variation parameters: plasma density, degree of ioniza-tion, electron density, reduced excitation frequency. Standard stands for standard parameters as listed in Table 3.1, inc. and dec. stand for increase and decrease of the variable by an order of magnitude, respectively.
simulations of argon discharges [7]. Finally, the reduced frequency ω/n affects the EEDF as well, again in agreement with the literature for simu-lations of argon discharges [7]. In this case, only the low-energy part of the EEDF is enhanced: it is known that the EEDF shifts towards a Maxwellian for very high frequencies [5].
Figure 3.8: hi and µ×N versus the reduced electric field, E/N. Variation of plasma density, degree of ionization, electron density, reduced angular frequency.
As an input for further discharge simulations, the electron mobilities are of importance as well. With the same parameter set of Table 3.1 as a basis and the previous parameter variations, the mean electron energieshiand electron mobilitiesµewere calculated in dependence of the reduced electric field,E/Nand are plotted in Figure 3.8. The variation of the plasma
param-eters has hardly any effect on the dependence ofhion theE/N as shown in the top graph. This can be attributed to the comparably low charge carrier density. Hardly any change in µe can be seen in the lower graph when ne and the α are varied, except again in the low Townsend regime, where differences of about ±30% are observed whenω/N is increased by one order of magnitude. This may be related to the shifting of the EEDF towards an Maxwellian distribution when the frequency is increased.
3.1.2.2 Influence of gas composition
Figure 3.9: Electron energy distribution functionf()for several different condi-tions for E/N= 4.28 Td in a CHF3/Ar gas mixture. Variation parameter: per-centage of trifluoromethane.
As can be seen in Figure 3.9, the gas composition also has an influence on the EEDF, which is exemplary shown for E/N = 4.28 Td: In the series from pure argon to pure CHF3, f() is enhanced for < 3eV as well as for > 11.2eV, while the intermediate population decreases. The change must be attributed to the different cross sections of argon, where there total inelastic cross section is zero below 11.55 eV, and trifluoromethane, which exhibits low energy vibrations and attachment processes.
The dependence ofhion the reduced electric field, as shown in Figure 3.10, top, shows a decrease if the partial pressure of CHF3 is increased.
This is due to the overall higher energy consumption by fragmentation.
Figure 3.10: hiandµ×N in dependence of E/N (oscillating field) for various gas compositions.
The gas composition also has a slight influence on the mobility in the range of about 15% for concentrations between 0-20%, especially in the low field region, as shown in Figure 3.10, bottom. It becomes flat upon increasing the concentration ofCHF3. This is related to the inelastic cross sections of both molecules.