Energy efficiency of voltage waveform tailoring for the generation of excited species in RF plasma jets operated in He/N₂ mixtures

Figure 1 shows a sketch of the experimental set-up. The measurements are performed using a reference microplasma “COST-jet” developed within the framework of the European Cooperation in Science and Technology. The plasma volume of the jet of $30\times 1\times 1$ mm³ is confined between two stainless steel electrodes and two quartz plates. The electrode gap is fixed at 1 mm. More detailed information about the plasma source can be found elsewhere [54]. The discharge is operated in helium (5.0 purity) with 0.1 % nitrogen (5.0 purity) admixture. The gas handling system has been described in [43, 38]. The flow rates are fixed at 1000 sccm for helium and at 1 sccm for nitrogen. The gas pressure inside the COST-jet is 1.02($\pm 0.02$)$\times 10^{5}$ Pa.

The stability and accuracy conditions of the PIC/MCC scheme dictate to use a very small, $\Delta t_{\rm e}=4.5\times 10^{-14}$s time step for the electrons under the conditions of their highly collisional transport. This ensures that the collision probability in a time step, $P=1-\exp(-\nu\Delta t)$ ($\nu$ being the collision frequency), stays in the order of $\cong 0.1$. Ions are traced with larger time steps in order to optimise the simulation time, for He⁺, He${}_{2}^{+}$, and N${}_{2}^{+}$ ions, these are $\Delta t_{\rm He^{+}}=10\,\Delta t_{\rm e}$ and $\Delta t_{\rm He_{2}^{+}}=\Delta t_{\rm N_{2}^{+}}=100\,\Delta t_{\rm e}$.

We use a buffer gas temperature of $T_{\rm g}$ = 300 K and keep the pressure at $p$ = 1 bar. The probability of the reflection of the electrons at the electrode surfaces is set to $\alpha=$ 0.6 [43, 75, 76]. The ion-induced secondary electron emission coefficients are set to $\gamma_{\rm He^{+}}=$ 0.3, $\gamma_{\rm He_{2}^{+}}=$ 0.2, and $\gamma_{\rm N_{2}^{+}}=$ 0.1 [43, 38]. Further details of the discharge model and its computational implementation can be found in [68, 39, 43].

For the COST-jets in the present He/N₂ mixture, the operating (peak-to-peak) voltages are limited to $\sim 750-800$ V. Above these voltages the discharge switches into a ”constricted” mode where an unstable filament can be observed between the electrodes. This mode is characterized by high powers and it is undesired as it leads to damage of the jet via excessive gas heating.

In order to compare the measured power with 1D simulation results, the calculated mean power densities over one RF period are multiplied by the discharge volume of the jet ($\sim 30$ mm³). The computed power is a sum of two components that represent power absorption by electrons and ions. Both are computed as $jE$, where where $j$ is the current density of the given particle and $E$ is the electric field. The current density is obtained as $j=\pm enu$, where $e$ is the elementary charge, $n$ is the particle density and $u$ is the mean velocity. Despite the simplified approach, very good agreement of the experimental and computational results is generally found for all conditions studied here. For the single frequency case at $f_{0}=13.56$ MHz, the dissipated power increases linearly with the peak-to-peak voltage until around 600 V. At such low powers the discharge is mainly operated in the $\Omega$-mode, see e.g. [43, 79, 57]. In this mode, a strong electric field is generated inside the bulk at the times of high current within each RF period, since the conductivity is low due to a high number of electron-neutral collisions. Thus, electron power absorption and ionization/excitation mainly occur inside the plasma bulk at the times of maximum current within the RF period.

At higher voltages, in Penning mixtures, e.g. He/N₂(O₂), the discharge is operated in the “Penning”-mode [43, 80]. It is based on two main ionization/excitation pathways, which are both attributed to Penning ionization, a direct and an indirect one [43]. The direct channel corresponds to ionization/excitation caused by electrons directly generated by Penning ionization inside the sheaths. These electrons are then accelerated towards the bulk by the sheath electric field and can cause ionization/excitation. The second channel is indirect: positive ions, created via the Penning ionization process inside the sheaths, are accelerated towards the adjacent electrode by the sheath electric field and, upon impact, induce the emission of secondary electrons. These electrons are also accelerated towards the bulk by the sheath electric field and can cause ionization/excitation inside the sheath.

The simulations allow to observe the effects of the surface processes on the discharge characteristics by turning these processes (the ion-induced secondary electron emission and the elastic reflection of electrons from the electrodes) on and off. As figure 3(a) shows, in the absence of surface processes the increase of the absorbed power remains linear with the peak-to-peak voltage even beyond 600 V. The observations also imply that the direct Penning ionization is not strong enough to cause the non-linear increase of the dissipated power observed experimentally at 700 - 800 V. At these conditions in the simulation, excitation and ionization occur mainly within the plasma bulk, corresponding to the $\Omega$-mode.

The behavior of the absorbed power with the peak-to-peak value of the driving voltage shows a characteristic change when the surface processes are included in the simulation. The coefficients characterising these processes were adjusted to get good agreement between the computational results and the experimental findings for ”Jet 2” at $f_{0}=13.56$ MHz and $N=1$. The electron reflection probability at the electrodes was set this way to $\alpha=$ 0.6 and the ion-induced secondary electron emission coefficients were set to $\gamma_{\rm He^{+}}=$ 0.3, $\gamma_{\rm He_{2}^{+}}=$ 0.2, and $\gamma_{\rm N_{2}^{+}}=$ 0.1. If the surface effects are taken into account via these coefficients, no convergence of the simulations can be achieved at peak-to-peak voltages above 750 V, which corresponds to the ”constricted”-mode in the experiment mentioned above. At lower voltages an excellent agreement is reached between the simulation results and the experimental data.

The effect of increasing the driving frequency on the dissipated power in single frequency discharges is depicted in figure 3(b) for different peak-to-peak driving voltages. The frequency is chosen to be $f_{0}=54.24$ MHz, which corresponds to the highest harmonic used in this work to construct “valleys”-waveforms. In contrast to $f_{0}=13.56$ MHz, much higher power densities are dissipated in the high frequency discharge at low voltages. At 54.24 MHz the discharge cannot be operated at low powers comparable to those observed in the 13.56 MHz single frequency scenario. Already near the breakdown voltage of about 325 V, more than 1.5 W power is dissipated. Similar frequency effects were previously observed in a $\mu$APPJ by Chirokov et al.  [58] for a frequency range of $13.56-40.68$ MHz. At $\gtrsim 500$ V peak-to-peak voltage a hot filamentary discharge at the tip of the jet and a very narrow and bright emission close to the electrodes are observed. Simulations indicate, that due to the increasing number of secondary electrons there is a strong avalanche formation leading to the sheath collapse/breakdown. In this regime, the power consumption is drastically increased, which causes thermal damage of the COST-jet in the experiment.

The experimental data are again well reproduced by the PIC/MCC simulations, the small quantitative differences between the computational and the experimental results might be attributed to temperature and/or multidimensional effects, which are not captured by the 1D simulations.

Figure 3 (c) compares experimental and simulation results for “valleys”-waveforms constructed from a fundamental frequency $f_{0}=13.56$ MHz and $N=4$ consecutive harmonics. By switching the driving voltage waveform from “valleys” to “peaks” the absolute value of the dissipated power remains the same, therefore we limit our following discussion to “valleys”-waveforms. In the single-frequency cases, adjusting the peak-to-peak driving voltage within the limits of a stable discharge allows to change the power consumption by a factor of $\sim 6$ for $f_{0}=13.56$ MHz and $\sim 3$ for $f_{0}=54.24$ MHz. In case of the “valleys”-waveforms this factor is around 10, i.e. a better power range control can be realized via VWT compared to single frequency operation. The range of the absorbed power is between those obtained with single frequency excitation at 13.56 MHz and 54.24 MHz. Similar to the previous cases a hot unstable filamentary discharge at the tip of the jet and a very narrow bright emission close to the powered electrode are observed, at $\gtrsim 550$ V peak-to-peak driving voltage.

In conclusion, by changing the base frequency and applying tailored voltage waveforms the power range of stable operation of the plasma source can be significantly extended. In the following section, we will reveal how the observed power dissipation in the COST-jet under single and VWT conditions results in the formation of helium metastables.

Helium atoms have two, singlet He-I 2¹S₁ and triplet He-I 2³S₁, long-lived metastable states. In our plasma, the population of triplet metastables is expected to prevail over the singlet metastables due to an efficient conversion channel via superelastic collisions of the singlet metastables with thermal electrons  [77, 81]: $\textnormal{He}(2^{1}\textnormal{S}_{1})+\textnormal{e}^{-}\rightarrow\textnormal{He}(2^{3}\textnormal{S}_{1})+\textnormal{e}^{-}+0.79\textnormal{ eV}.$ As this spin changing reaction is not included in the simulations, figure 4 [(d)-(i)] shows the sum of the densities of metastable species obtained from the simulation that is believed to be comparable to the measured He-I 2³S₁ triplet metastable density.

The shape of the density profile for the single frequency scenarios ($N=1$) is spatially symmetric as a result of the symmetric spatio-temporal electron impact excitation dynamics [43]. At high peak-to-peak driving voltages, the discharge is operated in the Penning-mode where metastables are mainly generated inside the sheaths and close to the electrodes. The main destruction mechanism of the metastables is Penning ionization of nitrogen molecules and the reaction time of this process is much shorter than the characteristic diffusion time for the metastables. Thus, the profile of the measured densities along the gap should be similar to that of the electron impact excitation rate, for details see also Korolov et al. [40].

The increase of the (single) driving frequency from 13.56 MHz to 54.12 MHz leads to a higher metastable density (from $1.8\times 10^{11}$ cm^-3 at 700 V$\rm{pp}$ and 13.56 MHz to $4.6\times 10^{12}$ cm^-3 at 500 V$\rm{pp}$ and 54.12 MHz ). At the same time the plasma density also increases which leads to a shorter sheath width. Consequently, the metastables are generated closer to the electrodes. The PIC/MCC simulations show that the highest electron/ion density, which can be reached for stable operational conditions, is $1.7\times 10^{11}$ cm^-3 at $f_{0}=13.56$ MHz ($N=1$, 700 V$\rm{pp}$) and $2\times 10^{12}$ cm^-3 at $f_{0}=54.12$ MHz ($N=1$, 500 V$\rm{pp}$).

By increasing the number of harmonics, $N$, i.e. by using Voltage Waveform Tailoring, the spatio-temporal electron impact excitation dynamics can be tailored. If “valleys”-waveforms are used to drive the discharge, these dynamics exhibit a strong spatial and temporal asymmetry which enhanced as a function of $N$. The electron power absorption is confined to a small region in both space and time close to the grounded electrode during the local sheath collapse, which breaks the spatial symmetry of the density profile. This effect (about which more detailed information can be found elsewhere [39, 38, 42]) significantly enhances the production of the helium metastables close to the grounded electrode. Figure 4 shows that our simulation accurately describes the COST-jet under such discharge conditions including the generation of metastables. For $f_{0}=13.56$ MHz ($N=4$) and 500 V peak-to-peak driving voltage, the peak electron density is found from simulations to be $0.95\times 10^{12}$ cm^-3, which is approximately two times lower than for the single frequency case where only the highest harmonic ($f_{0}=54.12$ MHz, 500 V$\rm{pp}$) of the tailored voltage waveform is used.

Figure 5 compares the space- and time-averaged and the peak helium metastable densities obtained from the PIC/MCC simulations and experimentally for different driving waveforms and the corresponding dissipated powers. The peak densities obtained from the simulations are calculated from the ”smoothed” density profiles to allow for a direct comparison with the experimental data, see figure 4 and the discussion above. We find excellent agreement between the experimental and the simulation data. The highest peak and mean metastable densities, which can be reached for the single-frequency case at $f_{0}=13.56$ MHz, are below $2\times 10^{11}$ cm^-3. To achieve a higher generation of helium metastables, the base frequency needs be increased or the driving voltage waveform shape must be modified. While both approaches yield higher metastable densities, the energy efficiency is significantly different for the high single frequency case ($f_{0}=54.12$ MHz) and the Voltage Waveform Tailoring case ($N=4$). To reach high mean densities around $3\times 10^{11}$ cm^-3 in the experiment, 2.8 times more dissipated power is needed for the high single frequency case compared to the ”valleys”-waveform case. The difference between the dissipated power in these two scenarios is even larger for the peak densities. To reach peak metastable densities of $1\times 10^{12}$ cm^-3 a factor of $\sim 4$ higher dissipated power is required in the single high frequency compared to the VWT case. This observation allows us to conclude that VWT is a more energy efficient way to control and generate metastable densities. The energy efficiency of VWT to generate high metastable densities is higher by a factor of 3 - 4 compared to single frequency operation. A detailed description and understanding of this effect is provided in the next section. We note that, according to figure 5, an extrapolation of the 13.56 MHz single frequency data to higher dissipated powers approximately yields the 54.12 MHz single frequency results, i.e. all single frequency results for the helium metastable density as a function of the dissipated power lie approximately on a single curve. The same is expected for the VWT results, i.e. increasing the fundamental driving frequency is expected to result in higher dissipated powers that yield higher metastable densities according to an extrapolation of the corresponding curve in figure 5. This, however, could not be tested in the frame of this work, but remains a topic for future investigations.

In this section, we investigate the effects of different driving voltage waveforms on the electron and ion power absorption dynamics to explain why VWT allows to generate helium metastables in a more energy efficient way as compared to classical single frequency operations of $\mu$APPJs. Based on PIC/MCC simulation data, figure 6 shows the fraction of the total power dissipated to the electrons and to the ions in the COST-jet as a function of the total dissipated power. Our findings are that

• •

  in contrast to low pressure capacitively coupled plasmas, where a large fraction of the total power is dissipated to the ions [82, 83, 84, 85], at atmospheric pressure the power is mostly dissipated to electrons under the conditions studied here;

• •

  the power dissipated to the ions is, however, not negligibly small and varies from 7.5% up to 25% for the conditions studied. There are numerous fluid, global or electrical approaches of atmospheric pressure radio-frequency capacities discharges [86, 87, 44, 55, 88]. These models are usually limited to low powers, where the discharge is operated in the $\Omega$-mode, or/and they neglect mobility-driven ion fluxes towards the electrodes. According to our results, the simplification that all the input power is absorbed by the electrons only may lead to significant deviations of the obtained plasma parameters from the measured ones. Thus, it is recommended to always consider in the models the power absorbed by ions.

In our previous studies, the He metastable source was shown to be directly correlated with the spatial distribution of the electron impact excitation rate from the helium ground state into other high-lying excited He levels [40]. One of these is the He-I (3s)³S₁ level that has an excitation energy of 22.7 eV and emits radiation with a wavelength of 706.5 nm which is often chosen in experiments to characterize the plasma by Phase Resolved Optical Emission Spectroscopy (PROES) [43, 38]. Figure 7(a)-(c) shows spatio-temporal distributions of the computed electron impact excitation rate from the ground state into the He-I (3s)³S₁-state for different waveforms. The peak-to-peak driving voltages are chosen to be close to the maximum values that allow generating stable discharges in the experiments and in the simulations.

According to the findings of the previous section, the single frequency scenario ($N$ = 1) is characterized by a spatially symmetric pattern of the metastable densities. This is caused by the symmetric spatio-temporal electron impact excitation dynamics shown in figure 7(a) and (b): Two excitation maxima with the same intensity are observed inside the expanded sheath at each electrode at an equal distance from the adjacent electrode within each RF period. Under these conditions, the discharges operate in the Penning-mode. With increasing frequency, the position of the excitation maxima shift towards the electrodes due to the increased plasma density and the decreased sheath width. The positions of these peaks, as expected, are at similar distances from the adjacent electrode as the positions of the maxima of the metastable atom density depicted in figure 4. By increasing the number of consecutive harmonics to $N=4$ to generate the “valleys” driving voltage waveform, the dynamics of the boundary sheaths within the fundamental RF period can be customized and the symmetry of the spatio-temporal excitation dynamics can be broken in space and time [39, 38, 40, 80]. Such a waveform, depicted in figure 7(l), induces a short and fast sheath collapse at the grounded electrode, while at the powered electrode the sheath collapses for a long fraction of the fundamental RF period (figure 3(c)). In order to compensate the continuous positive ion flux to the grounded electrode within one fundamental RF period, a high electron current has to be driven to this electrode during the short sheath collapse phase. To induce such an electron current a strong electric field is generated, while the sheath is collapsed [90]. This field can accelerate electrons to energies high enough to cause strong excitation/ionization including metastable generation in the vicinity of the grounded electrode. Therefore, in the experiment and in the simulation, see figure 4[(c),(f) and (i)], the peak metastable density is found to be strongly enhanced close to this electrode for this tailored driving voltage waveform.

The first row of figure 7 shows the spatio-temporal distribution of the electron impact excitation rate from the ground state into the He-I (3s)³S₁-state, which is highly correlated with the generation of helium metastables, while the second and the third rows show the power densities dissipated to electrons and ions, respectively. Analyzing these plots allows to understand why VWT, i.e. the use of valleys driving voltage waveforms, is a more energy efficient way to generate the metastables as compared to single frequency operation. Based on a comparison of the first and the second rows of figure 7, we can state that in both single-frequency cases only a small fraction of the total power dissipated to electrons results in the generation of helium metastables, while a much larger fraction is used for this in the VWT scenario. Generally, most power is dissipated to electrons inside the plasma bulk, where the electron density is high, so that the power dissipated per electron can be low and insufficient to accelerate electrons above the threshold energy to generate helium metastables by electron impact (ca. 19.8 eV) despite the high total electron power absorption in the bulk. This is true for both single frequency cases as documented by the fact that there is almost no excitation in the bulk, although most power is dissipated to electrons there. In fact, most excitation and helium metastable generation occurs inside the sheaths in the single frequency scenarios, although only little power is dissipated to electrons inside the sheaths. This is caused by the fact that the electron density inside the sheaths is extremely low so that the energy gain per electron is high, although the local total electron power absorption is low. Such electrons inside the sheaths are either directly generated by Penning ionization or indirectly by secondary electron emission from the electrodes due to the impact of positive ions generated by Penning ionization. Such electrons are accelerated by the high sheath electric field into the bulk and gain enough energy to cause excitation and generation of metastables.

In principle, these same mechanisms are present in the low and high single frequency cases. The only difference is the fact that the RF period is shorter in the 54.12 MHz compared to the 13.56 MHz case and, thus, the sheaths oscillate more frequently per unit time in the high frequency case. Thus, more power is dissipated in the high frequency case and higher metastable densities can be produced, but the energy efficiency of this process is not improved (see figure 5). Overall, in terms of helium metastable generation, the electron power absorption is inefficient under single frequency operation as most power is dissipated to electrons in the bulk in a way that each electron gains little energy, which is subsequently lost by electron-neutral collisions and is too low to generate metastables.

This is very different in case of VWT, i.e. valleys driving voltage waveforms, as shown in the third column of figure 7. Due to the specific sheath dynamics induced by this particular waveform most power is now dissipated to electrons within a narrow domain of space and time within the fundamental RF period. Thus, a large fraction of the total power is now dissipated to a small group of electrons located at the corresponding spatial position. This position is located outside the sheath, where the electron density is relatively high. Correspondingly, a high number of electrons gains enough energy to generate helium metastables. This explains why a given power dissipated to electrons results in higher metastable densities in case of the valleys driving voltage waveform as compared to single frequency operation. Thus, the energy efficiency of helium metastable generation is higher for VWT compared to the classical single frequency operation of $\mu$APPJs.

Figures 6 and  7 (third row) show that the power dissipated to ions is not negligibly small. The electric field inside the sheaths is much stronger than the electric field inside the bulk for all presented cases. This strong sheath electric field accelerates ions towards the electrodes, as shown in figure 8, which leads to an enhancement of the ion current density inside the sheath and, thus, to an increase of the dissipated power. With increasing applied voltage the ion density and the electric field inside the sheath increase leading to an enhancement of the total power dissipated to ions, see figure 6. In low-pressure RF discharged the ions are usually assumed not to follow the RF electric field due to their high mass. Instead, they only react to the time averaged electric field. At atmospheric pressure, this assumption is not correct, since the ion-neutral collision frequency is high, i.e. $\sim 1.4\times 10^{10}$ s^-1 (according to an estimation based on the Langevin cross-section [91]), which is much higher than the driving frequencies used in this work. The average time between such collisions is around 74 picoseconds and the ion energy relaxation time is assumed to be around 1 ns [92]. Thus, the ions can almost instantaneously follow the change of the electric field. The effect can be clearly seen in figure 8, where the spatio-temporal velocity distribution of the dominant ion, N${}_{2}^{+}$, obtained from the simulations is plotted for the different driving voltage waveforms.

Generally, the dynamics of the power dissipation to electrons and ions is important and should be taken into account in models used to describe such plasmas. Models that use only the time averaged dissipated power as an input parameter, according to figure 7, need to take the time dependence of the power dissipation into account. For instance, in the case of the single-frequency waveforms, the difference between the mean and the instantaneous peak power values within one RF period is less than a factor of two, while for the “valleys”-waveform used in this paper this factor is more than 4. Therefore, using time-averaged power values as an input for, e.g., global models [93, 47, 94], can lead to serious miscalculations of the plasma parameters. This is distinctly visible in figure 5: For the same dissipated power of approximately 2 W, the mean value of the helium metastable densities for single frequency operation is a factor of three lower than for the “valleys” waveform.