Effect of gas properties on the dynamics of the electrical slope asymmetry effect in capacitive plasmas: comparison of Ar, H₂ and CF₄

Going beyond classic dual-frequency capacitive plasmas between two parallel electrodes [1–6], discharges excited by tailored voltage waveforms (TVWs), comprising a fundamental signal frequency in the radio frequency (RF) domain combined with higher harmonics with controlled phases and amplitudes, have been investigated in recent years [7–15]. Amplitude asymmetry of the waveform, i.e. the difference between its maximum and its minimum value, and the associated dynamics occurring in these plasmas have previously been studied for the separate control of ion energy and flux due to the electrical asymmetry effect (EAE) [16–19].

Recently, a new type of temporally asymmetric TVW was proposed, called 'sawtooth' waveforms [20–22], comprising a fast rise and a slow fall, or vice versa. These waveforms present a slope asymmetry but no amplitude asymmetry. In argon this was shown to induce different sheath motions in front of the two electrodes, leading to strong ionization close to the electrode where the sheath expansion is fast, with only weak ionization close to the electrode where the sheath expansion is slow [22]. However, in molecular, electronegative plasmas significantly different physics phenomena can occur. For example, ionization mechanisms, other than sheath expansion, have been observed [23–25]. This letter compares sawtooth-excited discharges operated in gases with distinctly different properties, namely argon (Ar, electropositive), hydrogen (H₂, light molecular gas), and carbon tetrafluoride (CF₄, electronegative). For each gas, experimental data are compared to specific particle-in-cell (PIC) simulations. For Ar we used the code developed by Lafleur [26], for H₂ that of the Bari University group [27, 28], and for CF₄ the code from the Wigner Research Centre for Physics [18].

The experimental setup is displayed in figure 1. A reactor comprising 20 cm diameter parallel electrodes with an inter-electrode gap of 2.5 cm was made geometrically symmetric by adding a thick Teflon ring (inner diameter 10 cm), fitted with a 2.5  ×  2.5  ×  10 cm borosilicate glass window to allow optical access. The voltage waveform is generated as presented in detail in [8], without any impedance matching. An impedance-matched generator has been proposed by Franek et al [29], but was not employed here as a high coupled power was not required. After an iterative waveform optimization (using the method of Patterson et al [30]) the desired waveform (shown in blue for a sawtooth with n  =  5 frequencies) can be generated at the powered electrode. The gate of the intensified CCD camera (Andor iStar) used for phase resolved optical emission spectroscopy (PROES) measurements is synchronized to the RF generation system via a delay generator. A 2 ns gate was used, providing 91 images within one RF-cycle for a fundamental frequency of 5.5 MHz. The respective emission lines (750.4 nm for Ar, 656 nm (H-alpha) for H₂, and 703.7 nm (F atom) for CF₄,) were selected using interference filters. In all cases, the electron impact excitation rate was deduced from the measured emission by deconvoluting the radiative lifetime of the emitting state [31].

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Since in any experiment the voltage waveforms delivered have a finite frequency bandwidth, the voltage waveforms used are approximations of an ideal sawtooth, where a waveform consisting of a slow decrease and an infinitely fast increase is referred to as a 'sawtooth-down'. The approximation is performed by truncating the Fourier series for a perfect sawtooth waveform after n harmonics according to equation (1) (with a positive sign for sawtooth-down),

$$\begin{eqnarray}&&{{V}_{\text{ac}}}(t)=\pm {{V}_{0}}\underset{k=1}{\overset{n}{\mathop \sum}}\,\frac{1}{k}\sin (k\omega t),\end{eqnarray} \tag{ 1 }$$

where V_ac is the voltage applied to the powered electrode, ω is the fundamental angular frequency, and V₀ is adjusted to give the desired peak-to-peak voltage V_PP. Alternatively, 'sawtooth-up' waveforms (equation (1), minus sign) can be used. In a geometrically symmetric reactor this signal inversion produces a reversal of the roles of each electrode. The experiments were carried out at pressures of 400 mTorr, 900 mTorr, and 600 mTorr, for Ar, H₂, and CF_4, respectively. The exact value of the pressure has a negligible effect on the qualitative features shown hereafter. For example, it was shown in [21] that only a 5% increase of the ion flux ratio was obtained by increasing the pressure from 400 mTorr to 800 mTorr in Ar. The peak-to-peak voltage V_PP was set to 250 V, 350 V, and 400 V for Ar, H₂, and CF₄ respectively.

Figure 2 compares the experimental results and simulations for the three gases using a sawtooth-down waveform with n  =  5 (corresponding to the waveform displayed in figure 1). The results for Ar, H₂, and CF₄ are displayed from top to bottom. The x-axis represents position (with the powered electrode at x  =  0 cm and the grounded electrode at x  =  2.5 cm), while the y-axis represents time, and spans one fundamental RF cycle. The first column shows the excitation rate, K_exc, of the respective observed emitting states, as derived from PROES measurements. The next three columns display simulation results. The second column shows the excitation rate, K_exc, while the third shows the electric field distribution, E(x, t) divided by V_PP, outside of the sheath regions (which are shown in black). The same scale is used for the three gases, so that the normalized electric field values can be directly compared. The fourth column shows the electron power absorption, S_e. The sheath edge position was determined using Brinkmann's criterion [32], and the sheath edge position is shown as a white line in the second and fourth columns. For all three gases the simulations show fast sheath expansion and slow sheath contraction in front of the grounded electrode, and slow sheath expansion with fast sheath contraction in front of the powered electrode, as expected from the sawtooth-down waveform used.

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Looking at the excitation rate, one can see that an excellent agreement is found between experiments and simulations for all three gas chemistries, despite significantly different excitation features between them. In Ar, excitation is strongly localized close to the grounded electrode, and occurs during the fast sheath expansion (as indicated by the simulations in the second column). Only very weak excitation is observed close to the powered electrode during the (much slower) sheath expansion. In contrast, for H₂ two excitation peaks of similar amplitude occur almost simultaneously, the first one close to the grounded electrode where the sheath is expanding rapidly (sheath expansion heating) and a second peak close to the powered electrode where the sheath is contracting rapidly (due to a reversed electric field). Finally, in CF₄, weak excitation occurs close to the grounded electrode during the fast expansion, followed by a strong peak very close to the powered electrode during the fast sheath contraction. In order to understand the difference in the excitation features between the gas chemistries, both the electric field and the electron power absorption were analyzed.

In Ar, no significant electric field is observed within the bulk plasma outside of the sheath regions (the green color indicates zero field). In contrast, in H₂, a strong positive electric field occurs close to the powered electrode during the fast sheath contraction, corresponding to an electric field reversal. It should be noted that, although smaller, a field reversal also occurs at the grounded electrode when the sheath collapses. These field reversals occur because the motion of electrons is hindered by collisions (which are much more significant in H₂, a molecular gas, compared to Ar), and therefore a significant (reversed) electric field is necessary to accelerate the electrons to the electrodes and balance the escaping ion current [33]. The intensity of these field reversals therefore only depends on how fast the sheath contracts. When the sheath at the powered electrode is contracting rapidly the electron thermal flux is too low to ensure current balance, and a high, reversed, electric field is necessary. In contrast, the sheath contraction at the grounded electrode is slow, and therefore a weaker reversed electric field is sufficient to ensure current balance.

In CF₄, a very strong electric field (much stronger than in H₂) is observed during the fast sheath contraction. This electric field builds up because of the low electron density (because the majority of electrons have been converted to heavy F⁻ ions by dissociative attachment to CF₄), leading to low conductivity. In this discharge, the negative ion density (not shown), is particularly high close to the electrode where fast sheath contraction occurs (in this case the powered electrode), hence the electron density and the conductivity are particularly low. In contrast, close to the opposite (grounded) electrode the negative ion density is significantly lower and the electron density is higher, and therefore the conductivity is higher. As a consequence, a strong electric field is only observed close to the powered electrode, where fast sheath contraction occurs. Furthermore, on approaching the powered electrode, this positive electric field (corresponding to field reversal) is followed by a negative electric field, therefore creating an electric potential well, which acts as a trap for the electrons at one moment in time and limits their flux to the electrode.

The electron power absorption dynamics (shown in the fourth column) is clearly strongly affected by the gas properties. In all cases (for these relatively high gas pressure plasmas) the maximum in the electron power absorption occurs close to where the highest excitation rate is observed. This is because the electron mean free path is short compared to the discharge length. In Ar strong electron power absorption occurs close to the grounded electrode during the fast sheath expansion, while much lower electron power absorption occurs near the powered electrode during the slow sheath expansion. This is consistent with the hard wall model of Godyak [34, 35], according to which, the faster the sheath expansion rate, the greater the electron energy gain. In addition, because there is no significant electric field in the plasma bulk, and because secondary electron emission from the electrodes plays no significant role under these conditions, there is no other source of energy gain for the electrons. The discharge therefore operates in α-mode. In H₂, the situation is more complex: strong electron power absorption again occurs close to the grounded electrode during the fast sheath expansion, but is immediately followed by a second electron power absorption peak of similar amplitude, close to the powered electrode, due to the electric field reversal in this region. These two power absorption peaks are therefore the cause for the two excitation peaks.

In CF₄, the situation is again different. While some weak electron power absorption can be observed close to the grounded electrode during the fast sheath expansion, the largest power absorption peak is observed close to the powered electrode during the fast sheath contraction. Indeed, the large reversed electric field during this period strongly accelerates electrons from the plasma bulk towards the powered electrode. This leads to significant power absorption, accelerating electrons up to a mean energy of about 10 eV at this point. Because the electric fields in this phase can be split into drift and ambipolar components, this heating mode has been named the 'drift-ambipolar' (DA) mode [24]. This power absorption peak again coincides with a strong excitation peak. The latter is further strengthened by the formation of a potential well, which traps the energetic electrons and reduces their loss to the electrode, hence increasing the probability for these electrons to excite the neutral gas. In addition, this electron confinement increases the probability of electron attachment, and therefore the generation of negative ions. This further decreases the conductivity, and therefore increases the reversed electric field, forming a positive feedback loop.

In short, sawtooth waveforms induce a strong asymmetry in the excitation rate in both the Ar and CF₄ cases. However, the location of maximal excitation in CF₄ is opposite that of Ar, due to the change in the dominant electron power absorption mechanism. In the case of H₂, any asymmetry is cancelled out as both mechanisms occur almost simultaneously.

From the location of the electron excitation (figure 2, first and second columns) one would expect differences in the ion fluxes to the grounded and powered electrodes. Figure 3(a) shows the ratio β, of the ion flux to the powered electrode divided by that to the grounded electrode, for Ar (black squares), H₂ (red circles), and CF₄ (blue triangles) discharges, obtained from simulations, as a function of the number of applied harmonics n. This ratio exceeds unity for CF₄, because the (phase-integrated) excitation and ionization maxima are localized close to the powered electrode. In contrast, the ratio is below one for Ar, because the majority of the excitation and ionization occurs near the grounded electrode. It is worth noting that the asymmetry in the ion flux obtained for CF₄ is not only opposite compared to Ar, it is also significantly larger. As one can see, an ion flux ratio of about five can be obtained in a geometrically symmetric reactor with CF₄, while this drops to 0.6 (corresponding to an ion flux 1.6 times larger at the grounded electrode) in the case of Ar under these conditions. Finally, for H₂ the ion flux ratio remains close to unity (it is always between 0.9 and 1), since no clear asymmetry is observed in the spatial profile of the excitation dynamics.

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In capacitively coupled plasmas, a dc self-bias voltage establishes such as to equalize the period-averaged electron and ion fluxes to the electrodes. Figure 3(b) shows the dc self-bias voltage, η, normalized to the peak-to-peak voltage V_PP, obtained from simulations (full symbols), and experiments (empty symbols), showing excellent agreement (with the exception of n  =  2 in CF₄, for reasons not yet known). In CF₄, η is always positive, whereas it is always negative in Ar and H₂. It is worth noting that a strong correlation exists between the ion flux ratio, β, and η. The sign of η will be positive if β  >  1, and negative if β  <  1. Its amplitude is proportional to the difference of β from unity. This link can be understood by considering the expression for η given by Heil et al [1] applied to the case of sawtooth waveforms (which have no amplitude asymmetry):

$$\begin{eqnarray}&&\eta =-\frac{{{V}_{\text{PP}}}}{2}\times \frac{1-\varepsilon}{1+\varepsilon}\end{eqnarray} \tag{ 2 }$$

where ε is the symmetry parameter, which can be approximated by the ratio of the mean ion densities in the sheath at the powered and grounded electrode. Because the mean ion densities and the ion fluxes are strongly linked, we can expect a relationship between ε and β. Assuming a power law, ε  =  β^a (because we should have ε  =  1 when β  =  1, and ε  →  1/ε when β  →  1/β), where a is a fitting parameter, and replacing ε by its expression as a function of β in equation (2), it is possible to obtain a relationship between the normalized dc self-bias voltage and the ion flux ratio. Figure 3(c) shows η/V_PP as a function of β, for the three gases and for different values of n. Also shown in black is the best fit obtained by replacing ε by its expression as a function of β in equation (2). We find a  =  0.8, indicating that the ion flux ratio is always larger than the symmetry parameter. The agreement is excellent for all three gases and for all conditions, suggesting that this relationship may have general applicability.

In conclusion, we have observed that the charged particle dynamics occurring in plasmas of gases with different properties leads to fundamental differences in the spatial asymmetry of discharges excited with the same sawtooth waveform. The spatiotemporal distribution of the dominant electron energy gain mechanism is identified as the main cause for these asymmetries. When the discharge is in α-mode (Ar), most of the ionization occurs close to the electrode where the fast sheath expansion occurs, whereas in a strongly electro-negative gas where the DA mode dominates (CF₄) ionization is predominantly close to the electrode where fast sheath contraction occurs. Hydrogen discharges were found to operate in a mixed mode, where significant ionization is simultaneously observed near both sheath edges when they contract or expand quickly. The net effect for such H₂ discharges is an absence of asymmetry in the spatial profile of the time-averaged ionization. Finally, despite these differences, a power-law relationship is observed between the ion flux ratio and the normalized dc self-bias, and a single value of exponent fits the data for all three gas chemistries studied herein.

The authors would like to thank F Farci and P Bulkin for their help with the reactor used in this study. Work described in this paper was made possible by financial support from the ANR projects APOCALYPSO (ANR-13-PRGE-0003-01), CleanGRAPH (ANR-13-BS09-0019), the UK EPSRC (EP/K018388/1 & EP/H003797/1), the York-Paris Low Temperature Plasma Collaborative Research Centre, and the Hungarian Fund for Scientific Research (OTKA K 105476). B Bruneau is also supported by the Doctoral School of Ecole Polytechnique (EDX grant, Ecole Doctorale de l'X). P Diomede and D J Economou are grateful to the Department of Energy, Office of Fusion Energy Science, contract DE-SC0001939 for financial support.