for the net loss (or production) of the species which can change with the plasma conditions.

The fit of the model to the available experimental data and the calculation of the probabilities of the surface reactions (adjustable parameters) is done with a modified Levenberg– Marquardt algorithm [37]. The number of fitted parameters is 8 (table 2) and the number of experimental data is 12. It was not necessary to define different sticking probabilities for each SFx radical in order to predict the available experimental data. The same was true for the recombination probabilities of SFx radicals with adsorbed F. However, it was necessary to differentiate the probabilities of fluorination reactions (S5)–(S7).

The surface production of S2F10 and the dissociative chemisorption of SFx radicals on the surface [58] were included in an intermediate version of the set of surface reactions. Even if these reactions were not found necessary to fit the available experimental data and were not added in the final set, we cannot neglect the possibility that they do take place.

5. Model results and discussion

5.1. Model results and comparison with the experimental data

The comparison of the model results with the available experimental data is shown in figures 5 and 6. The values of the coefficients of the gas phase reactions are included in table 1 (the values for the Druyvesteyn EEDF are used) and the probabilities for the surface reactions, resulting from the fit to the experimental data, are included in table 2.

In figure 5(a), the pressure rise versus power is shown. The model result is close to the measured pressure rise which links to the degree of dissociation of the parent gas. It is an ‘absolute value’ measurement which can be directly (without conversion, as e.g. in the case of the actinometric ratios) compared with model results and used for the validation of the model. The calculated pressure rise is greater if the deposition of SFx radicals at the walls is not taken into account, and it is much greater if no surface reactions are taken into account.

The model also captures the increasing trend and the absolute values of F density versus power, as shown in figure 5(b). It seems that the experimental data are closer to the model results when the value of KF (equation (1)) is35% greater than 0.56. This value (1.35 × 0.56 = 0.76) is within the reported error limits (a factor of two) for KF [42]. The calculated F density is slightly lower when the deposition of SFx radicals at the walls is not taken into account, and it is greater if no surface reactions are taken into account.

The model also well describes the pressure rise versus pOFF, as shown in figure 6(a). This decrease in pressure rise is due to the loss (deposition) of SFx radicals at the walls ((S32)– (S40)). If no deposition occurs, then the pressure rise does not decrease as shown in figure 6(a). The calculated pressure is much greater if no surface reactions are taken into account.

In order to calculate the deposition rate of SFx radicals or the growth rate of the fluoro-sulfur film on the reactor

Figure 5. Comparison of the experimental data with the model

results versus power. pOFF is 0.921 Pa and Tgas is 315 K. (a) Pressure rise. (b) Density of F; the experimental data are coming from

actinometry. Two values for KF of (1) were used (0.56 and 0.76). The results for three configurations of the model are shown: the basic configuration (including all surface reactions of table 2), one with no surface reactions and one with no deposition of SFx radicals.

walls, the density of the film is necessary; there are no pertinent available data. However, in order to make an order of magnitude estimation of the deposition rate predicted by the model, we consider that the density of this film is that of sulfur (2.07 g cm−3) and calculate the fluxes of F and S atoms deposited on the surface. Under the conditions of figure 6, the deposition rate of the fluoro-sulfur film is calculated to be about 60 nm min−1. Note that the density of the fluoro-sulfur film is expected to be greater than that of sulfur as the size of the F atom is smaller than the size of the S atom. Thus, the deposition rate is a fraction of 60 nm min−1.

We did not experimentally measure the deposition rate of the fluoro-sulfur film and we are not aware of a relative study. However, Meyyappan [28] mentioned that the reactor walls after SF6 plasma are likely to be significantly coated with products containing sulfur, i.e. fluoro-sulfur products. Sulfur deposit was observed at the base plate of the reactor by Tang and Hess [58] during tungsten etching by SF6 plasma. The formation of a fluoro-sulfur film on the reactor walls during SF6 plasma is consistent with the observation of the peak of F atom (703.7 nm) during an O2 plasma following an SF6 plasma process. Adsorbed sulfur products were detected even on Si surfaces (and not wall surfaces) etched by SF6 plasma by x-ray photoelectron spectroscopy (XPS) [59]. The thickness of the fluoro-sulfur film was estimated to be greater in cases of lower ion energy. The growth of a film on the reactor walls is also

Figure 6. Comparison of the experimental data with the model

results versus pOFF. Power is 2000 W and Tgas is 315 K. (a) Pressure rise. (b) Density of F atoms; the experimental data are coming from

actinometry. Two values for KF of (1) were used (0.56 and 0.76). The results for three configurations of the model are shown: the basic configuration (including all surface reactions of table 2), one with no surface reactions and one with no deposition of SFx radicals.

consistent with XPS measurements of Cunge et al [32]. They measured the composition of the film formed on the walls in a SF6/O2 plasma and found a decrease in the Al (wall material) fraction and an increase in the F fraction of the film as process time increases, indicating that the reactor walls are gradually covered, although no discussion is made about sulfur, probably due to the O2 content of the feed which forms volatile products with sulfur.

It cannot be excluded that a fraction of the deposition rate corresponds to growth of particles in the gas phase. Particles can be suspended in the gas phase of low pressure SF6 [60–62] and other plasmas [62–65]. These particles may originate by nucleation sites from the deposited film on the reactor walls; they can be ablated from the film after film fracture under the stress (thermal or by ion bombardment) induced by plasma [66] and form suspended nucleation sites in the gas phase. These particles had been shown [67] to be negatively charged and trapped in the plasma/sheath boundaries. Thus, the growth process can continue onto the surface of the suspended particles. The existence of Al, which is a material of the reactor wall, in the composition [60, 61] of the particles formed in SF6 plasmas signifies that the walls may aid in particle formation. The presence of Al at the centre of the particles measured by Auger depth profiling [60] reinforces this hypothesis, i.e. that nucleation sites can originate from the deposited film on the reactor walls.

Besides the deposition of SFx radicals and the possible subsequent particle growth, other potential reasons for the

decrease in the pressure rise versus pOFF were theoretically investigated. First, the increasing formation of polymerized molecules (Sx Fy ) could cause a decrease in pressure rise versus pOFF. However, the exact mechanisms of formation and consumption of Sx Fy molecules (e.g. S2F10) are not known and pertinent data are not available. The formation of the stable dimer S2F10 was investigated by adding reactions of formation and consumption of S2F10 in an intermediate version of the model and it was found that it could not cause the decrease in pressure rise versus pOFF; the rate coefficients for the electron impact dissociation of S2F10 were estimated relatively to those of SF6 following the relative rates in the work of Van Brunt et al [30]. Second, a decrease in the gas temperature versus pOFF could explain the decrease in the pressure rise. However, in order to meet the pressure rise at, for example, 3.5 Pa, a gas temperature of less than 100 K is necessary, which is not reasonable. Third, errors in the coefficients of the gas phase reactions (e.g. errors in the coefficients of neutral recombinations, or to the cross sections for neutral dissociation of the parent gas) may be responsible. Nevertheless, we decided to use the available data from the literature for the gas phase reactions; only surface reaction probabilities are allowed to be adjusted to the experimental data.

Summarizing the former reasoning, even if (a) the deposition of SFx radicals and the possible subsequent particle growth in the gas phase is enough to predict the experimental data, and (b) the inclusion of S2F10 in the set was not enough to solely predict the experimental data, the existence of dimers [68, 69] or polymers in the gas phase cannot be excluded. They are not considered in the set of species as there are simply not enough data concerning the electron impact cross sections for them.

In figure 6(b), the density of F atoms versus pOFF is shown. The model well describes the experimental data. As in the case of the results versus power, the experimental data are closer to the model results when the value of KF is 35% greater than 0.56. The calculated F density is lower when the deposition of SFx radicals at the walls is not taken into account, and it is greater if no surface reactions are taken into account.

In figure 7, outputs of the model versus power at constant pOFF are presented. SF6, F, F2 and SF4 are the dominant neutral species, as shown in figure 7(a) where the densities of all neutral species are shown. The low density of SF3 for the majority of conditions shows that the potential products of SF3 by single or multi-step dissociation, i.e. SF2, SF and S, do not significantly affect the results of the model, especially the pressure rise. SF+5 is the dominant positive ion and is followed by SF+3 , as shown in figure 7(b). The dominance of SF+5 [54, 70] and SF+3 [47, 52, 71] over the rest of the positive ions has been reported by experimental studies for SF6 and SF6/Ar plasmas under several conditions. As the cross sections for dissociative ionization of SF6 to SF+5 , and as a consequence the corresponding rate coefficient, are much greater than those for all other dissociative ionizations of SF6, SF+5 should rationally be the dominant positive ion. The dominancy of SF+3 reported in [47, 52, 71] may either imply a high degree of dissociation of SF6, which is not true in our case as the pressure rise (linking

Figure 7. Model results versus power. pOFF is 0.921 Pa and Tgas is 315 K. (a) Densities of neutral species. (b) Densities of charged species and electrons. The density of F2+ is below 1016 m−3 and is not shown. (c) Ratio of densities of negative ions to electrons and electron temperature. The ratio decreases from 45 at 50 W to 1 at 3500 W. Electron temperature for Druyvesteyn (used for all calculations) and Maxwellian EEDF is shown.

to the dissociation of the parent gas) is rather low, or that ion– molecule reactions (causing change in the ion composition) are more significant [51, 52]. In fact, the density of SF+3 is found to be very sensitive to the value of the coefficient of ion–molecule reaction (G50); if this coefficient was 10 times greater, then the density of SF+3 would have been greater than that of SF+5 . The rate of (G50) would also be higher at higher pressure, i.e. at higher SF6 density. Generally, the composition of the positive ions is possibly affected by other (than (G50)) ion–molecule reactions [50, 52, 54]. For example, the density of SF+4 is expected to be lower than predicted due to the ion– molecule reaction SF6 + SF+4 → SF5 + SF+5 [53, 54]. The latter reaction was not included in the set of table 1 as its coefficient was not available. Another alternative to justify the dominance of SF+3 in some experimental studies would be to

consider that SF3 radical is a major product of electron impact neutral dissociation of SF6 [16, 20].

The dominant negative ion is F− (figure 7(b)) and the ratio of the densities of negative ions to electrons decreases hyperbolically (scaling to 1/power) from 45 at very low power to 1 at 3500 W, as shown in figure 7(c).

The electron temperature is shown in figure 7(c) for both Druyvesteyn (used for all calculations presented in this work) and Maxwellian EEDF. The trend is the same for both EEDFs: the electron temperature decreases versus power and the difference between the two curves is 0.7 eV, i.e. very close to the difference shown in figure 3, which is almost 0.8 eV.

In figure 8 outputs of the model versus pOFF at constant power are presented. The conclusions for the dominant neutral species and ions (positive and negative) are similar to those for figure 7. As shown in figure 8(c), the ratio of densities of negative ions to electrons increases almost linearly with pOFF, which is consistent with the measurements of Chabert et al [18]. Finally, the electron temperature increases versus pOFF (figure 8(c)); as in the case versus power, the difference between the curves corresponding to Druyvesteyn and Maxwellian EEDF (0.75 to 0.80 eV) is very close to 0.8 eV.

We repeated the runs of figures 7 and 8 by considering a Maxwellian EEDF; only electron temperature results are shown here. Indeed, even though there is a difference in the electron temperature between the cases of Druyvesteyn and Maxwellian EEDF, the differences in the neutral and ion compositions as well as in the pressure rise are not important. This means that the model results are not so sensitive to the differences between the rate coefficients resulting from a Maxwellian and a Druyvesteyn EEDF.

5.2. The contribution of the reactions

In order to investigate the most important (gas phase or surface) reactions of the model, their relative contributions to the production and consumption of neutral and charged species are calculated under specific conditions (0.921 Pa, 2000 W). The results for SF6, SF5, SF4 and F are shown in figure 9. In particular, the seven most important reactions for each species are shown. SF6 is mainly consumed by the neutral dissociation reaction to SF5 and F (figure 9(a)). The major production reaction is the gas phase recombination of SF5 with F. The major sinks of SF5 are the gas phase recombination of SF5 with F and SF5. SF5 is mainly produced by the neutral dissociation of SF6 and the recombination of SF4 with adsorbed F (figure 9(b)). The latter reaction and the neutral dissociation of SF4 are the major sinks of SF4 (figure 9(c)). SF4 is mainly produced by the gas phase recombination reactions of SF3 with F and SF5 with SF5. F is mainly produced by the neutral dissociations of F2, SF6, and SF4 (figure 9(d)). The major sinks of F are the adsorption on the surface, surface recombination with adsorbed F, and the gas phase recombination with SF5. Concerning the major charged species (not shown in figure 9), SF+5 is mainly produced by the dissociative ionization of SF6, and is consumed by recombinations with the negative ions, while F− is mainly produced by dissociative attachment to SF6, and is consumed by recombinations with the positive ions.

Figure 8. Model results versus pOFF. Power is 2000 W and Tgas is 315 K. (a) Densities of neutral species. (b) Densities of charged species and electrons. The density of F2+ is below 1016 m−3 and is not shown. (c) Ratio of densities of negative ions to electrons and electron temperature. The ratio increases from 0.03 at 0.1 Pa to 25 at 4.5 Pa. Electron temperature for Druyvesteyn (used for all calculations) and Maxwellian EEDF is shown.

The gas phase reactions between neutral species are found to be important sources or sinks of the neutral species consistently with [24, 26]. Surface reactions are also found to be significant; at least one surface reaction is included in the set of the most important reactions for each species. Both of them play an important role in the discharge as they keep the degree of dissociation of the parent gas and the pressure rise low.

5.3. The effective sticking coefficients of neutral species

The effective sticking or surface loss coefficient of a species i on a surface (SE,i) signifies the net consumption (or production) of the species on the surface. It is defined as the ratio of the flux of the species consumed net on the surface over the flux of the

Figure 9. Contributions of the reactions to the production and consumption of (a) SF6, (b) SF5, (c) SF4 and (d) F. The power is

2000 W, pOFF is 0.921 Pa and Tgas is 315 K. The x-axis shows the production (positive x-axis) and the consumption (negative x-axis)

rate in m−3 s−1 multiplied by 10−21. The rates of the seven most important reactions for each species are shown.

species reaching the surface. The sign of the effective sticking coefficient signifies if the species is produced or consumed on the surface. A positive (negative) sign implies net surface consumption (production) of the species.

In order to calculate the effective sticking coefficient of a species, the rates of all surface reactions this species joins should be taken into account: for example, the net consumption of F comes from the sum of reactions (S1) and (S5)–(S8) minus the sum of reactions (S16)–(S19) in table 2.

The effective sticking coefficients depend on the densities of all species in the gas phase which change as the operational parameters (e.g. power) change. In figure 10 the effective sticking coefficients of F, F2, SF3, SF4, SF5 and SF6 are shown versus power. The effective sticking coefficients of F, SF4 and SF3 are positive, implying that there is a net consumption of these species on the surface. The effective sticking coefficients of F2 and SF6 are negative as expected, as there is no mechanism for surface consumption of F2 and SF6 in the set of the reactions in table 2. Finally, SF5 is consumed below 150 W and is produced at higher power at the walls: the rate of production of SF5 at the surface (after 150 W) overcomes the rate of consumption. This is consistent with figure 9(b) where the most important surface reaction for SF5 is a production reaction. The opposite is true for SF4 (figure 9(c)).

The effective sticking coefficient of F varies from 0.1 to 0.15 versus power (figure 10). It is in the same range versus pressure (not shown in this work). Finally, the varying values of the effective sticking coefficients of SFx radicals versus