To study the effects of varying the dielectric wall capacitance on the self-bias, two additional CFD-plasma simulations ofPRare performed with0.5mm and1.5mm thick discharge cham- ber walls. To distinguish between the three CFD-plasma simulations, the default geometry with a1mm thick wall is designated asPR-10, while the two other geometries are designated asPR-05 and PR-15. Only the external radius of the discharge chamber wall is modified in thePR-05 andPR-15 simulation meshes, so as to preserveAgnd/Apwr (2.3), (2.4).
The capacitance of a cylindrical dielectric capacitor is given by:
C = 2πε0εrL
ln Rr (5.8)
whereε0 is the vacuum permittivity,εr is the relative permittivity of the dielectric,L is the
length of the cylinder, andRandrare its external and internal radii respectively. Hence, the
capacitances of the section of theεr= 9.6 Al2O3 wall immediately in front of theL= 5mm powered electrode in PR-05, PR-10, and PR-15 are: Cwall = 12.5pF, 6.9pF, and 5.0pF respectively, which equates to an impedance of|Zwall|=|−1/ωCwall|= 0.94kΩ,1.7kΩ, and 2.4kΩ respectively. For comparison, the capacitance of the blocking capacitor used in the PRRF circuit (Section2.1.4) isCblock = 100pF.
and εr = 1 to give Cs,pwr0 ≈3.2pF. This calculation uses the external and internal radii of the powered sheath in PR-10 (Figure 4.7) with r = 2.1mm and r = 0.9mm, but remains approximately valid in PR-05 and PR-15 where the powered sheath widths are only very slightly different. Due to the larger size of the grounded sheath, it may be approximated by a parallel plate capacitor with an area equivalent to the internal grounded surface of the structure of the plenum plus the surface of the end wall facing the downstream region. The width of the grounded sheath at the front wall of the plenum is approximatelyd= 1.94mm, soCs,gnd0 =ε0Agnd/d≈229pF, with a slight variation of.1pF between the three geometries due to different internal radii of the rear (S-P) and end (S-D1) walls as a result of the modified external radius of the discharge chamber wall. The collisional sheath capacitance equation
Cs,sym≈0.76·ε0Asym/dis not able to be used forPRas it is only valid for symmetric plasma systems [154,156].
The Cs,pwr0 and C 0
s,gnd values quoted above are not necessarily accurate in reality, espe- cially in the τi τRF regime where the plasma sheaths become resistive instead of capa-
citive. Instead, it is more appropriate to quantify their impedances, which in theτi∼ τRF
regime is some combination of resistanceR and capacitive reactance XC = −1/ωC. Since R ∝ d/A uses the same geometrical parameters as XC ∝ d/A, the effective impedance is therefore |Z| =
q
R2+XC2 ∝ |XC|. Hence, the powered and grounded sheath imped-
ances are Z 0 s,pwr ≈ ζ ·3.7kΩ and Z 0 s,gnd
≈ ζ·51Ω respectively, where ζ is an unknown proportionality constant.
Figures 5.2 and 5.3 give a closer look at the electric potential Φ− and Φ+ during the negative (τ−) and positive (τ+) peaks of the RF cycle respectively, averaged over the10 RF cycles in the final solution. Φ− and Φ+ from PR-05 are represented byred lines, PR-10 by green lines, and PR-15 by blue lines. The horizontal axes are labelled according to the plasma and RF circuit features, tracing a path from the plasma bulk (aqua), through the powered sheath (magenta), dielectric wall (colour coded according to each geometry), to the powered electrode (brown). Note that the vertical axes are at different scales in the two figures. Despite significant changes in the dielectric wall capacitance Cwall, the
minima and maxima plasma potentialΦ−p and Φ+p profiles remain largely similar across the three geometries. Figures 5.2 and 5.3 also demonstrate the alternately positive (Φ+
p and Φ−p −Φ−wall) and clamping (Φp− and Φ+p −Φ+wall) behaviour of the plasma potential and the potential difference across the powered sheath, as well as the Φp(t) > Φwall(t) behaviour mentioned in Section 5.1. The maxima, minima, peak-to-peak, and cycle average values of Φp(t) for the three geometries are summarised in Table 5.1. The values for PR-05 tend to be slightly higher than the other two geometries, but overall the results are very close.
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Figure 5.2: Electric potentialΦ+ during the positive peak of the RF cycle through different regions of PR-05(blue line),PR-10(green line), andPR-15(red line). The plasma potential Φ+p profile remain largely similar across the three geometries. The blue, green, and red bars on the bottom of the plot denotes the thickness of the dielectric wall in the respective geometries.
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Figure 5.3: Electric potentialΦ− during the negative peak of the RF cycle through differ- ent regions of PR-05 (blue line), PR-10 (green line), and PR-15 (red line). The plasma potentialΦ−p profile remain largely similar across the three geometries, but the gradi- ent ∇rΦ−p in the powered sheath and the electric potential at the plasma-facing surface of the discharge chamber wall Φ−wall are slightly different due to the different dielectric wall capacitancesCwall.
The most conspicuous differences between the three geometries are visible in Figure5.3: namely the gradient of the plasma potential ∇rΦ−p in the powered sheath and the electric potential at the plasma-facing surface of the discharge chamber wall during the negative peak of the RF cycle. These differences are in response to the different extraneous impedances in the RF circuit, in this case the capacitance Cwall of the dielectric discharge chamber wall,
described earlier in Section5.1. Ions falling through the powered sheath at the negative peak of the RF cycle are accelerated to very high velocities by the radial electric fieldEr =−∇rΦ−p, therefore differences in the gradient of Φ−p or the magnitude of the potential drop in the powered sheath are expected to shift the position of the high energy peak of the IED in the powered sheath. SinceΦ+p is unaffected by the change in the dielectric wall capacitance, the low energy peak of the IED in the powered sheath is expected to remain constant. As the dimensions of the PR discharge chamber are too small to admit a RFEA or other invasive probes, the only way to gain valuable insight into the ion dynamics in the powered sheath is through CFD-plasma simulations.
Table 5.1: Plasma potential[V] Geometry Φ−p Φ+p Φ + p −Φ − p Φp PR-05 12.8 +0−0..22 64.1 +1.9 −3.3 51.3 +2.1 −3.6 28.8 +0.4 −0.3 PR-10 12.0 +0−0..37 60.9 +4.3 −4.1 48.9 +4.4 −3.4 27.0 +0.4 −0.9 PR-15 12.3 +0−0..56 60.2 +3.9 −2.1 47.8 +3.6 −2.5 26.8 +0.6 −0.9