4.4 RF Plasma Transport Models
4.4.1 Isothermal Diffusion Solution
4.4.2.2 Ion Saturation Current
∂¯nr
∂τ = ¯Kgrn¯gn¯e+ ¯Kmrn¯mn¯e+ K¯prn¯e+ ¯Apr
n¯p
−
K¯rm+ ¯Krp+ ¯Krg+ ¯Kri
n¯e+ ¯Arg+ ¯Def f.
¯
nr (4.42)
∂¯np
∂τ = ¯Kgpn¯gn¯e+ ¯Kmpn¯mn¯e+ ¯Krp¯nrn¯e+ ¯νr¯ne
−
K¯pm+ ¯Kpr+ ¯Kpg + ¯Kpi
¯ne+ ¯Apm+ ¯Apr+ ¯Def f.
¯
np. (4.43) As in Sec. 3.3, ¯ng =ntotal/n0−(¯ne+ ¯nm+ ¯nr+ ¯np).
The numerical algorithm used to advance these equations forward in time is described in detail in Appendix G. A low, non-zero initial electron density (1014–1015m−3) was specified in order to ensure numerical stability of the energy equation, and the RF power was ramped up from 0 to its maximum value over a period of 3 µs. The initial electron temperature was chosen to be Te ∼ 2 eV, but much higher or lower values could be used without any noticeable effect on the simulation results at t 1 µs. The boundary conditions on ¯Pe
at the edges of the simulation domain (see Fig. 4.16) were derived from the requirement
∂Te/∂z¯= 0 for zero heat flux at the boundaries (see Eq. 4.19)—refer to Appendix G for details. No boundary conditions were necessary for nm, nr, and np because there are no spatial derivatives in Eqs. 4.41, 4.42, and 4.43.
0 50 100 150 200 250 0
5 10 15 20 25 30 35 40
Time (µs)
Ion saturation current (mA)
Ti= 0.025 eV Ti= 0.05 eV Ti= 0.1 eV
z = −5 cm
z = −4 cm z = −6 cm z = −7 cm
Figure 4.17: Isat.(z, t) calculated by the 1D time-dependent model for three different ion temperatures. The gas pressure was pAr = 60 mTorr, and a 10.5 cm long RF power deposition region was assumed, with total absorbed powerPRF = 1190 W.
The temperature of the neutral atoms and ions was left as a free parameter, which was adjusted within the plausible range 0.025 – 0.25 eV for the best fit to the data. Changing Ti = Tg affected the ambipolar diffusion rate through the dependence of the ion-neutral collision frequency νin on vT i (see Eq. 4.29) and also altered the cross sections for re- absorption of line emission (see Eq. E.7). The overall impact of varyingTion the simulation results was modest—in general, with higher Ti the predictedIsat. rose more slowly at each location, and the ultimate quasi-steady state density downstream of the antenna was lower, as illustrated in Fig. 4.17.
The best-fit model results forpAr = 30 mTorr are shown overlaid on the Langmuir probe data in Fig. 4.18. RF power absorption was assumed to occur only within the inner 8.5 cm of the 10.5 cm-long antenna. A simpler assumption would have been to set the length of the power absorption region equal to the antenna length, as in Sec. 3.3. However, in the data shown in Fig. 4.18 there is a delay of several µs between the initial density rise at z =−7 cm and the time at which the density begins to rise atz =−6 cm, implying that plasma had to be transported to z=−6 cm rather than being created there initially (the front edge of the antenna was located at z = −5.9 cm, so z = −6 cm was just inside the
0 50 100 150 200 250
−5 0 5 10 15 20 25 30 35
Time (µs)
Ion saturation current (mA)
z = −5 cm z = −4 cm
z = −3 cm z = −6 cm
30 mTorr z = −7 cm
0 50 100 150 200 250
−5 0 5 10 15 20 25 30 35
Time (µs)
Ion saturation current (mA)
z = −5 cm z = −4 cm
z = −3 cm z = −7 cm
z = −6 cm
30 mTorr
Figure 4.18: ModelIsat.predictions (dashed curves) overlaid on Langmuir probe data (solid curves) for unmagnetized discharges at pAr = 30 mTorr. The RF power deposition region was assumed to extend from z=−15.4 cm toz=−6.9 cm (the boundaries of the antenna in the experiment were at z = −16.4 cm and z = −5.9 cm). The best fit between the model and data was achieved with Ti = 0.1 eV and PRF = 1600 W, with the time- dependence of the RF power input after the initial ramp-up period modeled as PRF(t) = PRF,peak[1−(t−tpeak)/(700µs)] up until t = 200 µs, at which time the RF power was instantaneously switched off. t= 0µs for the model was set to coincide with the timing of the initial fast density rise atz=−7 cm in the experiment, which occurred att≈16µs on the plots. In the left panel, the raw model results are shown, while in the right panel, the downstream model curves (z =−6 cm to z=−3 cm) have been shifted in time to match the measured timing of the initial density rise at each location in order to demonstrate the excellent agreement in the calculated and measured ∂Isat./∂t (the model curves at z = −6 cm through z = −4 cm are barely visible on this figure because they precisely overlap with the data).
antenna).
There are four main properties of the experimental Isat. curves that a model should ideally reproduce: the time delay before the density begins to rise at each axial location, the rate of increase∂Isat./∂tduring the density rise, the final quasi-steady state axialIsat.(z) profile, and the rate of decrease of Isat. after RF power turn-off. As seen in Fig. 4.18, our model performed admirably by the second and third measures, and reasonably well by the fourth. The quasi-steady stateIsat.(z) profile matched the data extremely closely, and the rate at whichIsat.rose at each location was predicted nearly perfectly as well. In the region where the plasma density was high, the predicted rate ofIsat. decline in the afterglow was too slow, but excellent agreement with the measuredIsat. decay rate was obtained at some downstream locations.
0 50 100 150 200 250
−5 0 5 10 15 20 25 30 35 40 45
Time (µs)
Ion saturation current (mA)
z = −6 cm
z = −3 cm
60 mTorr z = −7 cm
z = −4 cm z = −5 cm
0 50 100 150 200 250
−5 0 5 10 15 20 25 30 35 40 45
Time (µs)
Ion saturation current (mA)
60 mTorr z = −7 cm
z = −6 cm
z = −5 cm z = −4 cm
z = −3 cm
Figure 4.19: ModelIsat.predictions (dashed curves) overlaid on Langmuir probe data (solid curves) for unmagnetized discharges atpAr = 60 mTorr. The left panel shows the raw model results, while in the right panel the downstream model curves (z=−6 cm toz =−3 cm) have been shifted in time to match the measured timing of the initial density rise at each location. The RF power deposition region was assumed to extend from z = −15.4 cm to z=−6.9 cm, and the best fit between the model and data was achieved withTi= 0.05 eV and PRF = 1400 W.
There was, however, one major discrepancy between the model and data, illustrated in the left panel of Fig. 4.18: in the simulation, the plasma density both within the antenna and downstream began to rise soon after the RF power was turned on, while in reality there was a delay of ∼5–10µs before Isat. began to rise at each successive axial location. Only if the model Isat.(t) curves are artificially translated in time, as in the right panel of Fig.
4.18, is the excellent agreement in∂Isat./∂tat each location demonstrated. The reason for the immediate downstream density rise in the model can be seen from the detailed results presented in Sec. 4.4.2.3. At the beginning of the discharge, the plasma density is low, and the high level of RF power input causesTeto become high inside the antenna. The resulting gradient inTeleads to a large axial heat flux (see Eqs. 4.20 and 4.28) that quickly raises the temperature downstream (see Fig. 4.22). WithTe∼5 eV, the rate coefficient for ionization out of the ground state isKgi∼2×10−15m3/s (see Table 3.1). Then withng ≈1021m−3, the ionization rate is Kging ∼ 2×106 s−1. This is much larger that the radial loss rate νloss = 2hRcs/R ∼ 2×105 s−1, so Eq. 4.29 approximately reduces to ∂ne∂t ≈ Kgingne, implying thatnegrows exponentially with a time constant10 of∼500 ns until the electrons
10This argument neglects stepwise ionization out of excited states, so the actual e-folding time would be even shorter.
0 50 100 150 200 250
−10 0 10 20 30 40 50 60
Time (µs)
Ion saturation current (mA)
z = −7 cm
z = −6 cm
z = −5 cm z = −4 cm
z = −3 cm
120 mTorr
0 50 100 150 200 250
−10 0 10 20 30 40 50 60
Time (µs)
Ion saturation current (mA)
120 mTorr z = −7 cm
z = −6 cm
z = −4 cm z = −3 cm z = −5 cm
Figure 4.20: ModelIsat.predictions (dashed curves) overlaid on Langmuir probe data (solid curves) for unmagnetized discharges at pAr = 120 mTorr. The left panel shows the raw model results, while in the right panel the downstream model curves (z = −6 cm to z =
−3 cm) have been shifted in time to match the measured timing of the initial density rise at each location. The RF power deposition region was assumed to extend fromz=−15.4 cm to z=−6.9 cm, and the best fit between the model and data was achieved withTi= 0.025 eV and PRF = 1230 W.
have cooled enough to reduce the ionization rate.
This sequence of events probably did not occur in the experiments because the electron heat fluxqezcould not actually be high in regions where there was little or no plasma present.
The Braginskii heat flux expression (Eq. 4.20) is independent ofne, but it was derived under the assumption that the electron-ion collision mean free path was much smaller than the characteristic length scale for the problem, and thus it is not valid in the region out in front of the expanding plasma. However, it is not clear what expression for the heat flux should be used instead of Eq. 4.20 in this region; more work is needed to determine whether this aspect of RF plasma transport can be accurately modeled within a two-fluid framework.
The times at which the density began to rise at each location in the experiment may be used to infer an “expansion velocity” vexp. for the RF plasma. We find vexp. ≈ 1380 m/s at 30 mTorr (Fig. 4.18), vexp. ≈1210 m/s at 60 mTorr (Fig. 4.19), vexp. ≈850 m/s at 120 mTorr (Fig. 4.20), and vexp.≈780 m/s at 240 mTorr (Fig. 4.21). If the expansion velocity were proportional to the ambipolar diffusion coefficient, it would approximately scale with p−1Ar (see Eq. 4.12), but this was not the case, probably because the expansion involved a combination of both diffusion and ionization of pre-existing neutral gas. However, the
0 50 100 150 200 250
−10 0 10 20 30 40 50 60 70 80
Time (µs)
Ion saturation current (mA) z = −3 cm
z = −7 cm 240 mTorr
z = −5 cm z = −6 cm
z = −4 cm
0 50 100 150 200 250
−10 0 10 20 30 40 50 60 70 80
Time (µs)
Ion saturation current (mA)
z = −7 cm
z = −6 cm
z = −5 cm
z = −4 cm z = −3 cm
240 mTorr
Figure 4.21: ModelIsat.predictions (dashed curves) overlaid on Langmuir probe data (solid curves) for unmagnetized discharges at pAr = 240 mTorr. The left panel shows the raw model results, while in the right panel the downstream model curves (z = −6 cm to z =
−3 cm) have been shifted in time to match the measured timing of the initial density rise at each location. The RF power deposition region was assumed to extend fromz=−15.4 cm to z=−6.9 cm, and the best fit between the model and data was achieved withTi= 0.025 eV and PRF = 1100 W.
details of the physics at the expansion front are not yet understood.
Consider now the calculated and measured Isat.(z, t) at pAr = 60 mTorr, shown in Fig.
4.19. Once again, the model accurately predicts the ultimate quasi-steady stateIsat.(z) pro- file and the time-dependence of∂Isat./∂tduring the initial density rise at each location and during the afterglow. The corresponding results forpAr = 120 mTorr andpAr = 240 mTorr are shown in Figs. 4.20 and 4.21, respectively. At these higher pressures, although the model matches the measured rise and fall rates ofIsat.well, the predicted quasi-steady state Isat.(z) declines far too quickly moving away from the power deposition region. Therefore, there must be some physical process missing from the model that is relatively unimpor- tant at p . 60 mTorr but becomes critical for achieving a high downstream density at
p&120 mTorr. Many approximations were made in developing the model, so there are nu-
merous possible candidates ranging from axial radiative transport (re-absorption of emitted photons changes the excited state population densities non-locally) to some exotic colli- sional atomic process. The main culprit has yet to be conclusively identified, however.
Interestingly, while the measured downstream density profile was not fit well by the model atpAr ≥120 mTorr, the initial∂Isat./∂tinside the power deposition region at z=−7 cm
0 50 100 150 200 250 0
0.5 1 1.5 2 2.5 3
Time (µs) Te(eV)
z = −7 cm z = −6 cm z = −5 cm z = −4 cm z = −3 cm
60 mTorr
18.5 19 19.5 20 20.5 21
0 1 2 3 4 5 6 7 8 9 10
Time (µs) Te(eV)
z = −7 cm z = −6 cm z = −5 cm z = −4 cm z = −3 cm
60 mTorr
Figure 4.22: Electron temperature vs. time calculated by the 1D numerical discharge model withpAr = 60 mTorr. The same simulation parameters were used as in Fig. 4.19. The right panel zooms in on the first 2.5 µs of the simulation, which was started at t = 18.5 µs to facilitate comparison with the data in Fig. 4.19.
was predicted more accurately in Figs. 4.20 and 4.21 than at lower pressures11.
The best fits between the model and data were achieved by setting Ti = 0.1 eV for the 30 mTorr calculation, Ti = 0.05 eV for the 60 mTorr calculation, and Ti = 0.025 eV for the 120 mTorr and 240 mTorr calculations. It makes sense intuitively that Ti =Tg should have been lower at high pAr, since the ions and neutrals were heated by electric impacts, and the ratio of the number of electrons to the number of heavy particles was lower at high pressure.
The RF power level that was necessary to make the model Isat.(t) fit the Langmuir probe data at z = −7 cm was also pressure-dependent, ranging from PRF = 1600 W at pAr = 30 mTorr toPRF = 1100 W atpAr = 240 mTorr. On the other hand, the measured RF power delivered to the load as a whole at t = 100 µs was PL = 2290 W at 30 mTorr and PL = 2100 W at 240 mTorr. The more drastic pressure dependence in the absorbed powers inferred from the model was probably not physically meaningful, but rather arose because the ion saturation current measured by the probe was depressed at high pressures due to collisional effects (see Sec. D.2 for more discussion). This non-ideal probe behavior was not taken into account when using Eq. D.9 to convert the model results forne(z, t) and
11Note that while the time dependence ofPRF(t) used in the model was adjusted to fit the gradual decay inIsat.at later times due to the decreasing power output from the RF amplifier, there was no fine-tuning to fit the initialIsat. rise rate—the RF power was always ramped up linearly to reach its maximum value 3µs after the start of the simulation.
0 50 100 150 200 250 0
0.5 1 1.5 2 2.5
3x 1018
Time (µs) nm(m−3)
z = −7 cm z = −6 cm z = −5 cm z = −4 cm z = −3 cm
60 mTorr
0 50 100 150 200 250
0 0.5 1 1.5 2 2.5
3x 1018
Time (µs) nr(m−3)
z = −7 cm z = −6 cm z = −5 cm z = −4 cm z = −3 cm
60 mTorr
0 50 100 150 200 250
0 0.5 1 1.5 2 2.5
3x 1018
Time (µs) np(m−3)
z = −7 cm z = −6 cm z = −5 cm z = −4 cm z = −3 cm
60 mTorr
Figure 4.23: Ar I excited state population densities vs. time calculated by the model with pAr = 60 mTorr. The same simulation parameters were used as in Fig. 4.19. The left panel shows the 4s metastable state population density, the center panel shows the 4sresonant state population density, and the right panel shows the 4pstate population density.
Te(z, t) into a predicted ion saturation current.