Dielectric wall surface potential

Figure5.4plots the electric potential on the plasma-facing surface of the dielectric discharge chamber wall Φ_wall(t) at the coordinates (z, r) = (−9.9,2.1)inPR-05 (blue line), PR-10 (green line), and PR-15 (red line). Also plotted on the same scale is the electric potential in the powered electrodeΦpwr(t) (magenta line). The supplied RF waveform is the same for all three geometries. Only four of the total10 RF cycles of the final solution are shown for clarity. The coloured lines denote the mean of each waveform over all 10 RF cycles. Figure 5.4 clearly shows the asymmetric response ofΦ_wall(t) to the different discharge chamber wall impedances. TheseΦwall(t) profiles suggest for the powered sheath an IED with a constant low energy peak given the constant maximaΦ+_wall, and a high energy peak that is shifted depending on the value of the minimaΦ−_wall.

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Figure 5.4: Electric potential on the plasma-facing surface of the dielectric discharge chamber wall Φ_wall(t) in PR-05 (blue line), PR-10 (green line), and PR-15 (red line), and the electric potential in the powered electrode Φ_pwr(t) (magenta line). Coloured lines denote the respective mean values. Φ_wall(t) responds asymmetrically to the different discharge chamber wall impedances, affecting only the negative peaks and minimaΦ−_wall.

For clearer comparison of the shapes of theΦ_wall(t)waveforms, Figure5.5plotsΦ0_wall(t) = Φwall(t)−Φwall, or the original waveform relative to its mean, alongsideΦpwr(t)(magenta line). The profiles have a similar shape acrossPR-05 (blue line), PR-10 (green line), and PR-15 (red line). Φ0_wall(t) mostly preserves the sinusoidal waveform of Φ_pwr(t), but is asymmetric about its mean, with a diminished trailing edge at each positive peak [145]. The peak-to-peak voltages are greatest inPR-05 and least inPR-15.

Table 5.2 lists the maxima, minima, peak-to-peak, and mean values of Φwall(t) for each geometry, including the degree of variation in each parameter. Φ−_wall in the second column of Table5.2quantifies the asymmetric response exhibited by the negative peaks in theΦ_wall(t) waveforms to different extraneous impedances in the RF circuit, in this case the dielectric wall capacitance, described earlier in Section 5.1 and visible in Figures 5.3 and 5.4. In PR, the affected parameter is Φ_wall(t) instead of Φ_pwr(t) because the plasma is in contact with the discharge chamber wall instead of the powered electrode. Essentially, the dielectric wall functions as a blocking capacitor with capacitance C_wall (Section 5.3.2) instead of the Cblock= 100pF blocking capacitor which is positioned before the powered electrode in thePR RF circuit (Figure2.4). Since the self-bias manifests inΦwall(t)after the powered electrode, Φ_pwr(t) maintains the electric potential of the supplied RF waveform.

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Figure 5.5: When plottingΦwall(t) relative to its mean,Φ 0

wall(t) across the three geometries mostly preserve the sinusoidal waveform ofΦ_pwr(t), but are asymmetric with a diminished trailing edge at each positive peak.

Table 5.2: Discharge chamber wall electric potential[V] Geometry Φ−_wall Φ+_wall

Φ + wall−Φ − wall Φwall PR-05 −472.4 −+21..04 55.6 +1−3..29 528.0 +2−5..73 −200.1 +1−0..59 PR-10 −458.1 +1−1..09 52.7 +4.3 −3.8 510.8 +3.9 −2.6 −193.0 +0.9 −1.8 PR-15 −441.3 +2−1..58 52.2 +3.8 −2.2 493.5 +3.8 −3.6 −184.7 +2.0 −1.5

Of particular interest is the alignment of the positive peaks ofΦ_wall(t)for all three geomet- ries in Figure5.4. The positive peaks have an average maximum ofDΦ+_wall

E

= 53.5V (Table

5.2) with a standard deviation of σ = 2.7V, and τ₊/τ− = 14/46 (out of 60 time-steps) is approximately constant. SinceA_gnd/A_pwris unchanged, (5.5) implies thatc_s,in_i,pwr/c_s,in_i,gnd

is approximately constant in each of the three geometries. While it is possible to verify this by integrating over all the plasma-facing powered (section of the discharge chamber wall) and grounded electrode surfaces over each RF cycle with the present CFD-plasma simula- tion results, it is a nontrivial task given the size of the dataset and the presence of other non-electrode surfaces inPR.

Nonetheless, Figures 5.6 and 5.7 are provided for reference. Figure 5.6 plots the cycle average ion density ni ( lines) and the cycle average radial ion drift velocity ur,i ( lines) radially along z = −9.9mm under the powered electrode in the discharge chamber,

over the 10 RF cycles in the final solution, in PR-05 (blue), PR-10 (green), and PR-15 (red). n_i peaks on the z-axis, and falls sharply in the powered sheath as u_r,i increases to ur,i≈30km s

−1 _{due to the steep potential drop from}

Φ_p to Φ_wall across the powered sheath (Figure4.8). Figure5.7plots the cycle average axial ion densityni ( lines) and the cycle average axial ion drift velocityu_z,i ( lines) along thez-axis atr= 0mm using the same colours for each geometry. The position of the powered electrode is shown by thebrownbar at the top. n_i peaks under the powered electrode at the coordinates (z, r) = (−9.9,0) in all three geometries, with similarly shaped profiles featuring a strong central gamma mode peak and two shoulder alpha mode plateaus. The peak values are n_i = 7.010×1017m−3, 5.373×1017m−3, and 4.271×1017m−3 for PR-05, PR-10, and PR-15 respectively. The left edge of the plot atz=−30mm is the grounded electrode, where ions impact onto the front wall of the plenum at uz,i ≈ −5.0km s

−1_{. Along the}

z-axis, ions are on average moving

away from the central n_i peak, which is expected since the plasma potential Φ_p is most positive at the peak. Figures 5.6 and 5.7 show a consistent u_r,in_i,pwr/u_z,in_i,gnd across the

three geometries in the powered sheath and front wall grounded sheath respectively, which is a reasonable indication of similar behaviour for the plasma sheaths at all of the grounded electrode surfaces in general. Regardless, the high degree of alignment of the positive peaks and the preservation of a constantτ₊/τ− in Figure 5.4is sufficient evidence to prove it so.

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Figure 5.6: Radial profile alongz=−9.9mm of the cycle average ion densityn_i ( lines)

and the cycle average radial ion drift velocity ur,i ( lines) in the discharge chamber of PR-05(blue),PR-10 (green), andPR-15 (red). The internal radius of the discharge chamber is kept constant at2.1mm, and only the external radius is changed inPR-05 andPR-15.

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Figure 5.7: Axial profile of the cycle average ion densityni ( lines) and the cycle average axial ion drift velocityu_z,i( lines) along thez-axis forPR-05 (blue),PR-10 (green), and PR-15 (red). n_i peaks at (z, r) = (−9.9,0) in all three geometries, with similarly shaped profiles featuring a strong central gamma mode peak and two shoulder alpha mode plateaus. The final assumptions to be verified before calculating the self-bias voltage Vbias are the ion transit timeτ_iand the variation in the ion drift velocity∆u_r,iin the powered sheath. The

radial transit time is calculated by performing a fourth order Runge-Kutta analysis on u_r,i

to obtain the cycle average radial drift position of the ionr_i(t)as a function of time (Section

4.6.2). The ion transit times across the powered sheath are: τi= 134ns= 1.8·τRFinPR-05,

τ_i= 103ns= 1.4·τ_RF in PR-10, and τ_i= 86ns = 1.2·τ_RF in PR-15. Given thatτ_i ∼τ_RF,

∆u_r,i is expected to be somewhat similar across the three geometries, at ∆u_r,i =±29.9 %,

±27.1 %, and ±24.6 % respectively. Since ∆u_r,i is significant in all three geometries, the

assumptions made for theτiτRF regime are reasonably valid for PR.