Collisionless electron heating by capacitive radio-frequency plasma sheaths 2 and Lieberman[2, 3, 4], where the electrons moving towards the sheath ar

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1 Collisionless electron heating by capacitive radio-frequency plasma sheaths G. Gozadinosyx, D. Vendery, M.M. Turnery and M.A. Liebermanz yplasma Research Laboratory, School of Physical Sciences Dublin City University, Dublin 9, Ireland zdepartment of Electrical Engineering and Computer Science, University of California, Berkeley, CA 94720,USA Abstract. Low pressure capacitive rf plasmas can be maintained chiefly by collisionless heating in the rf modulated sheaths adjacent to the electrodes. Theoretical models dealing with this mechanism are often based on a `hard wall' approximation where the electrons are considered to collide elastically with the oscillating sheath edge. The power transfer is then calculated by averaging forward and reverse power fluxes over an rf period. There are however several drawbacks in this approach: the models are sensitive to assumptions regarding the incident electron distribution, transit time effects in the sheath electric field are neglected, electron loss is not considered and current conservation is not satisfied. In order to examine the validity of the theoretical models, we use a Monte Carlo approach to study electron interactions with model and selfconsistent fields providing modifications that can lead to a more consistent treatment of the electron dynamics inside the sheath. Of particular importance is the presence of a small field behind the moving electron sheath edge which maintains quasi-neutrality between the electron sheath position, and the bulk plasma. In addition a semi-infinite particle-in-cell (PIC) simulation is used to investigate in detail sheath dynamics. The errors that the `hard wall' approximation gives are calculated, and power deposition scalings with current drive, frequency and electron temperature are provided. Our results indicate that collisionless heating cannot be attributed to the stochastic heating mechanism based on the `hard wall' approximation and that in contrast electron inertia plays a dominant role as far as collisionless heating is concerned. 1. Introduction The comprehension of plasma sheaths and electron dynamics near the sheath vicinity is fundamental for the understanding of low-pressure radio-frequency glow discharges. Of particular importance is the heating mechanism associated with the sheaths which may sustain the plasma at low pressures where collisions are rare and explain the heating mode transition observed experimentally by Godyak and Piejak in [1]. Models based on a `hard wall' approximation were proposed in the 1960's and further developed by Godyak x Electronic address (internet):gg@physics.dcu.ie

2 Collisionless electron heating by capacitive radio-frequency plasma sheaths 2 and Lieberman[2, 3, 4], where the electrons moving towards the sheath are considered to collide elastically with the sheath edge and bounce back. According to this `Fermi acceleration' argument[5], while the sheath expands or contracts the electrons are being heated or cooled respectively. Using simplified models [6, 7] it has been shown that provided there is sufficient phase-space randomization in the bulk plasma, this process leads to net heating (usually referred to as stochastic heating). A thorough review of the application of Fermi acceleration argument to plasma discharges has appeared recently in Lieberman and Godyak [8]. A drawback of this approach, as has already been pointed out [9], is that since electron inertia is neglected, transit time effects in the sheath electric field are ignored. Wendt and Hitchon [10] using a Monte-Carlo simulation and assuming a sinusoidal temporal and linear spatial variation of the sheath electric field explored the effect of power loss due to electron escape to the electrode. Surendra and Dalvie [11] noticed that current conservation is not satisfied in the theoretical models based on the hard wall approximation (hereafter denoted by HWA"). In addition, there are currently two other approaches on the heating mechanism discussed, one due to Kaganovich, Aliev, Tsendin and Schlüter [12, 13] where the inhomogeneity of the rf field in combination with the nonlocal electron dynamics is used to explain the heating mode transition, and another due to Surendra and Turner [11, 14, 15, 16] where the idea that the compression and decompression of the electron cloud near the sheath vicinity isresponsiblefortheobserved heating is put forth. In what follows, we will only deal with stochastic heating based on the Fermi acceleration mechanism, attempting to clarify whether and why HWA might fail using results obtained from analytical models as well as self-consistent simulations. In section 2, we use Lieberman's analytical model for the collisionless rf sheath as a representative model based on the HWA in order to discuss the implications of HWA when current is conserved. In the section that follows, we use a semi-infinite particle-in-cell (PIC) simulation to obtain the sheath field self-consistently. We provide measurements showing that the use of HWA has limited applicability. In addition, we obtain scaling laws using the current density, the frequency and the electron temperature as independent parameters. Although our results show reasonable agreement with the theoretical predictions of models based on the HWA, detailed calculation from the PIC simulations show that the underlying heating mechanism is not the one assumed. Finally, we note the presence of plasma oscillations that occur due to a quasi-neutrality failure near the sheath vicinity and might beresponsible for the observed heating. 2. Monte-Carlo Simulation In order to demonstrate the effects of neglecting electron inertia and field details we will use Lieberman's model [2] for a current-driven collisionless rf sheath. The structure of the sheath is shown in figure 1. Ions respond to the time-averaged field and are accelerated towards the electrode, whereas electrons are assumed to be in Boltzmann

3 Collisionless electron heating by capacitive radio-frequency plasma sheaths 3 equilibrium with the instantaneous field, i.e., electron inertia is neglected. Due to ion flux conservation, the ion density n i (x) falls as they move from the bulk to the electrode, whereas the electron density n e (t; x) is assumed to be equal to the ion density for x < s(t) and zero for x > s(t), where s(t) is the position of the electron sheath edge, measured from the bulk. The electron sheath edge oscillates nonlinearly between the bulk plasma and the maximum sheath thickness s m, denoted in figure 1 as x = 0 and x = s m respectively. In Lieberman's treatment of the collisionless heating effect, individual electron trajectories are assumed to be unaffected by the electric field (the quasi-neutrality field") that necessarily exists between the bulk plasma and the instantaneous sheath edge. As we will show, this is not consistent with conservation of current in this region. The quasi-neutrality field, although small compared to the sheath field, causes a potential difference which is of the order of the electron temperature and therefore affects the incident distribution that arrives at the sheath edge. Lieberman includes the quasi-neutrality field in his model only implicitly via his assumption of Boltzmann equilibrium for the electron fluid. This field has been analytically explored in [17] but the resulting equations were not solved. However, a first-order time-independent approximation can be obtained if we assume that electrons are in Boltzmann equilibrium with the quasi-neutrality field so that dn e (x + dx) =n e (x)e dv Te ; (1) where T e is the electron temperature measured in volts. From quasi-neutrality n e = n i, and taking the potential at the bulk to be zero, we can solve for the potential V (x) and differentiate to obtain the field strength! d E qn (x) = T e ψlog n i(x) : (2) dx n 0 In order to establish our argument on the necessity of the quasi-neutrality field we present a direct comparison of results obtained from Monte-Carlo simulations of the original model and a modification of it, with the quasi-neutrality field present. The numerical simulation used is based on following individual electron trajectories as they interact with the model sheath electric field. A similar approach, using a simpler model, can be found in [10]. The loading process (particles entering the sheath from the bulk plasma) assumes a one-dimensional time-dependent drifting Maxwellian flux. At a random phase a particle is given a velocity from the flux using a Monte-Carlo approach. The equations of motion for the particle are then numerically integrated with the appropriate boundary conditions inside the sheath region, using a Runge-Kutta integration scheme. The electric field strength is interpolated using dense tabulated data for different phases of the rf cycle and different positions in the sheath. While the particle remains inside the sheath region, its velocity and position are allocated to grids. By averaging over a large number of particles (' 10 5 ) and properly rescaling these grids we can obtain all standard diagnostics.

4 Collisionless electron heating by capacitive radio-frequency plasma sheaths 4 As can be seen in figure 2, gross distortion of the electron density profile occurs in the absence of the quasi-neutrality field, leading to violation of current conservation. However, due to the non self-consistent character of the model, absolute current conservation cannot be achieved even though the inclusion of the quasi-neutrality field improves the density and current profiles. The importance of current conservation in any sheath model becomes apparent when one tries to calculate power deposition to the plasma due to stochastic heating based on HWA. In [2] Lieberman finds that the instantaneous power dissipated into the plasma is given by S stoc (t) = 2mZ 1 u s u s (u u s ) 2 f s (u; t) du (3) where u s is the sheath speed and f s the electron velocity distribution function at the sheath boundary at time t. Furthermore, taking the electron distribution at the bulk to be of the form f 0 (u; t) =g 0 (u u e;0 ) where g 0 is the Maxwellian distribution and u e;0 (t) the time varying electron average velocity (drift) due to the oscillating rf field at time t on the bulk plasma, f s is approximated by f s = n s n 0 g 0 (u u e;0 ); (4) so as to take account of the effect of the quasi-neutrality electric field. Averaging (3) over the rf period the power transfer per unit area is found to be positive. However the result is inconsistent since the drift velocity on the sheath edge u e;s should be used. That velocity can be found by electron flux conservation n i u e;s = n 0 u e;0 and when used in combination with equations (3) and (4) it yields S stoc (t) = 2mZ 1 0 n s n 0 u s u 0 2 g(u 0 ) du; (5) where u 0 = u u e;s. From current conservation u s = u e;s and using n s u s = n 0 ~u 0 sin(!t) we get after carrying out the integration S stoc (t) = T e n 0 ~u 0 sin(!t): (6) Averaging (6) over time we get zero, implying that if current is conserved there can be no net heating. Therefore, in order for heating due to the proposed mechanism to occur, either deviations from HWA must be present or the incident electron flux comes from a distribution which is not a drifting Maxwellian (In fact, any distribution which is symmetrical about its drift velocity will give a flux that yields zero heating). We will explore these matters further in the following sections. 3. PIC Simulation The previous results emphasize the need for a self-consistent treatment of the problem. With this in mind, we created a semi-infinite, collisionless, one-dimensional PIC simulation [18, 19, 20, 21, 22], which differs from the usual approach in that not the whole of the plasma is modeled. This type of simulation has been presented before by Surendra and Vender in [23]. Since we are only interested in the interaction of

5 Collisionless electron heating by capacitive radio-frequency plasma sheaths 5 electrons with the sheath, we assume an infinite uniform bulk plasma on the left side and take the potential to be zero on that boundary, while on the right side a perfectly absorbing electrode is assumed. Ions are loaded at the left boundary from a warm beam distribution of room temperature centered at the Bohm velocity u B =(T e =m i ) 1=2 where T e is the electron temperature measured in volts. Although experimental and theoretical research [24, 25] indicates that ions have quite different distributions near the sheath than the one assumed, wehave tested our results with different ion temperatures without noticing significant discrepancies. This is due to the fact that the sheath structure is not sensitive to the ion distribution. Electrons are loaded at the left boundary from a single-temperature (2:57 ev) Maxwellian distribution in such a way that a given sinusoidal current is driven into the simulation area. Secondary electron emission from the electrode is not considered. For the results presented here, the bulk density is kept constant at n e;0 = n i;0 = 1: cm 2 while the rf frequency has the typical value of 13:56MHz unless otherwise stated. The rest of the implementation is conventional: a one-dimensional explicit particle mover is utilized, the time step and cell size are chosen to satisfy the usual stability and accuracy criteria and a model gas of helium for the sake of speed has been used (argon has been tested as well without qualitatively changing the results). This type of PIC simulation is of theoretical interest as there is no coupling of the sheath with the bulk, and theoretical models that study the sheath independently can be easily tested. In addition, the small simulation area enables us to have an improved resolution for all diagnostics. A typical sheath field obtained by this simulation, with the quasi-neutrality field mentioned in the previous section clearly seen, is shown in figure Power deposition into the plasma It is interesting at this point to compare the power deposition as calculated by the PIC simulation with the values predicted by Lieberman's model. The average power per unit area deposited into the plasma by the oscillating sheaths can be calculated directly from the PIC simulation by Z sm Z T P = 1 J e Edt`dx (7) T 0 0 where E is the electric field and J e is the electron current density. Since this model does not account for electron loss, in order for the comparison to be made we have to exclude the contribution to J e E by the electrons which are being lost at the electrode: every escaping electron has contributed 1 2 m eu 2 i 1 2 m eu 2 f of energy to J e E, where u i is its initial velocity when it enters the simulation area and u f is its velocity when it exits. Since the potential is zero at the bulk side, this is equal to q e V (t) where V (t) is the instantaneous potential of the electrode. Averaging this quantity over many rf cycles, we can obtain P loss (t), the power lost due to electron loss as a function of phase. In figure 4we plot the power flux to the plasma as a function of the current drive. Our results agree well with Lieberman's calculation. In figure 5 we compare J e E

6 Collisionless electron heating by capacitive radio-frequency plasma sheaths 6 scaling with frequency for a constant rf voltage as predicted theoretically by stochastic heating theories (J e E /! 2 ) with the results obtained from simulation experiments. In order to keep the voltage across the sheath constant in these simulations we varied the current drive until the voltage was stabilized at the value of V rf = (400 ± 5) V p p. The predicted scaling does not agree with the results obtained. It also has to be noted, as reported previously [14, 15], that increasing the frequency leads to an injection into the plasma of a fast current due to a pressure wave developing as the electron fluid is compressed and decompressed by the sheath. This current imposes a small electric field on the bulk side of the plasma which leads to a negative J e E in the bulk. Therefore, the integration over J e E is performed from where the sheath begins. Performing the integration for the whole simulation area only makes the disagreement with the theoretical results more profound. Finally, when the electron temperature is used as a scaling parameter (see figure 6) there is also a fair agreement between theory and simulation results (J e E / T 1=2 e ). The rf voltage amplitude is kept almost constant (' 2000V p p ). Note however the importance of electron thermal loss to the electrode. Following our arguments in the previous sections, although the above scalings of power deposition show a fair agreement with the theoretical predictions, this can only happen if one of the two important assumptions of the underlying theory breaks down: that is, either the HWA or the Maxwellian nature of the incident electron fluxes to the sheath. It is therefore of interest to examine the error that the HWA would give when applied on the self-consistent field obtained by the PIC simulation. The procedure we followed is to consider that the instantaneous sheath edge at time t is found where the displacement current has dropped to 10% of the value it had at the electrode. We smooth the sheath position with an FFT filter to avoid anomalies caused by the plasma oscillations previously mentioned and differentiate to obtain the sheath velocity at time t. A typical case of this process is shown in figure 7. Although this procedure seems arbitrary, it gives us an accurate estimate of the sheath position and velocity. The results obtained hereafter are not prone to important changes when different criteria (drop percentage) for the sheath edge are being used. A good test for the validity of our criterion would be to check whether the average electron velocity at the sheath edge matches the sheath speed, implying current continuity. This is shown in figure 8 where the two velocities are very close to each other except for the phases where the sheath is near the electrode. This emphasizes the point that conservation of current implies that the electrons will arrive at the sheath edge with an average velocity equal to that of the sheath and therefore will be reflected without a change in their speed. The disagreement of the two velocities at phases where the sheath is nearly collapsed occurs because at those phases the magnitude of the electric field is small and therefore electron current in the sheath region, as well as electron loss, become significant.

7 Collisionless electron heating by capacitive radio-frequency plasma sheaths 7 Using the Monte-Carlo code we described previously with the self-consistent field as a model field, we can calculate the relative error of the HWA ff = u r 2u s + u i 2u s u i ; (8) where u i is the incidentvelocity of the electron coming from the bulk, u r is the velocityof the reflected electron and u s is the velocity of the sheath edge at time t = 1 2 (t 1+t 2 ) where t 1 and t 2 are the times when the electron entered and exited the sheath respectively. The result averaged over many electrons as a function of the rf phase is shown in figure 9. From the graph it is clear that the errors remain of the order of 10% for most of the rf period. They are the result of neglecting electron inertia and the time dependence of the quasi-neutrality field. During the retraction phase of the sheath, the errors can grow large due to the rapid movement of the sheath and the presence of a field reversal. Electrons lost to the electrode or reflected by the quasi-neutrality field are not included in the calculation, but should be considered in general. It remains to investigate the nature of the incident and reflected distributions of the electrons interacting with the sheath edge. In figure 10 the electron distribution function sampled over many rf cycles (' 500) near the instantaneous sheath edge at various phases (see the caption on figure 11) is shown along with the drifting Maxwellian assumed by the theory. We notice that during the initial and final parts of the cycle the distribution departs from being Maxwellian. The gross distortion at the end of the collapse phase is not of big interest since it is due to the field reversal and the electrons concerned escape and do not otherwise contribute to heating. From what we have shown so far it is apparent that there exist deviations from both the HWA and the assumption that the electron distribution is a drifting Maxwellian at the sheath edge. These deviations leave open the possibility for heating to appear due to the Fermi acceleration mechanism. Therefore, one has to try to evaluate directly whether heating does occur due to that mechanism or not. Our approach is the following: knowing the instantaneous sheath position from the method we described previously, we collect a set of electron distribution functions on the instantaneous sheath edge at a hundred evenly distributed time intervals in the rf cycle. From these one can evaluate (3) directly and hence calculate the heating which can be attributed to the mechanism proposed. A typical result is shown in figure 12 along with the actual heating rate calculated from J e E P loss, and the theoretical prediction. Notice that the three curves not only are quite different from each other, but the one corresponding to the direct calculation of (3) integrates to almost null. We repeated the above procedure for a range of different parameters and found the result to remain invariable: the average power dissipated is always much smaller than J e E P loss and averages to almost zero. Therefore, electron heating through a hard-wall type of interaction with the sheaths cannot be the main mechanism involved in collisionless heating.

8 Collisionless electron heating by capacitive radio-frequency plasma sheaths 8 5. Discussion - Conclusions Although a revision of collisionless rf heating theory seems necessary, it is not very clear in which direction this should be attempted. Animated videos of electron phasespace trajectories obtained from our simulation suggest that there is a powerful heating mechanism associated with a traveling plasma wave: When the sheath is near the end of its expansion phase, quasi-neutrality breaks down due to an overshoot of hot electrons into the bulk plasma. This causes the plasma near the sheath vicinity to ring at the plasma frequency (see figure 13). These oscillations are similar to others that have been reported before [9, 26] but a thorough analysis of their nature has not yet been given. The electrons interact strongly with the oscillating potential and on average gain energy through the well-known Landau damping mechanism. These oscillations appear to be enhanced at high frequencies and low temperatures, which agrees with the observation that the transferred power to the plasma drops as we increase temperature. An approach based on quasi-linear theory which attempts to link stochastic heating with collisionless power dissipation through a Landau damping like mechanism has appeared in [12]. This is one of several attempts to link the capacitive collisionless heating effects with acoustic or plasma wave phenomena (see also [11, 14, 15, 16]). None of these efforts has yet produced a generally satisfactory treatment, but this may be a fruitful direction for future research. Summarizing, using analytic and self-consistent models we have investigated electron dynamics in the sheath region of an rf capacitive discharge. Our results indicate that the presence of a small field in front of the sheath edge which will preserve quasi-neutrality isimportant, and its exclusion from models which attempt to describe the sheath dynamics leads to a violation of current conservation. A time-independent approximation of this field has been provided but a more realistic solution requires a self-consistent treatment of the whole sheath. In addition, we have shown that the HWA (although it can be used for the derivation of models that attempt to investigate analytically the global dynamics of the rf sheath), fails to provide an understanding of the mechanisms governing collisionless heating in capacitive rf discharges: when the sheath is moving slowly, HWA could be applied but as we have shown a consistent calculation of the power induced to the plasma should give no net heating in this case. In contrast, the applicability of the HWA turns out to be very limited when the sheath is moving fast (high voltages and driving frequencies) due to the fact that electron inertia and transit time effects are neglected. The scalings we performed with a current drive and electron temperature as parameters showed fair agreement with the existing theory, but not the scaling using the driving frequency as a parameter. This implies, for example, that the existing theories of collisionless heating in capacitive discharges may not be reliable in the operating regime of the dual frequency" capacitive sources which are now being investigated. In conclusion, our results indicate that collisionless heating remains an open question and that electron inertia, transient effects and non-local behaviour have indeed

9 Collisionless electron heating by capacitive radio-frequency plasma sheaths 9 a dominant role and are indispensable for the understanding of the heating associated with the rf sheath. Acknowledgments This work is supported by Association EURATOM DCU Contract ERB CT One of the authors (MAL) acknowledges the support of National Science Foundation Grant ECS , California Industries, the Lam Research Corporation, and the State of California UC-SMART programme under Contract Authors GG and MAL are grateful to Prof. R.W. Boswell of the Space Plasma and Plasma Processing Group, Plasma Research Laboratory, Research School of Physical Sciences and Engineering, Australian National University, Canberra, Australia for hosting and supporting their research visit, during which some of this research was performed. 6. References [1] V. A. Godyak and R. B. Piejak. Abnormally low electron temperature and heating-mode transition in a low-pressure rf discharge at MHz. Phys. Rev. Lett., 65(8): , August [2] M. A. Lieberman. Analytical solution for capacitive rf sheath. IEEE Trans. Plasma Sci., 16(6): , December [3] V. A. Godyak and N. Sternberg. Dynamic model of the electrode sheaths in symmetrically driven rf discharges. Phy. Rev. A, 42(4): , August [4] V. A. Godyak. Statistical heating of electrons at an oscillating plasma boundary. Sov. Phys. - Tech. Phys., 16(7): , January [5] E. Fermi. On the Origin of the Cosmic Radiation. Phy. Rev., 75(8): , April [6] C. G. Goedde, A. J. Lichtenberg, and M. A. Lieberman. Self-consistent stochastic electron heating in radio frequency discharges. J. Appl. Phys., 64(9): , November [7] B. P. Wood, M. A. Lieberman, and A. J. Lichtenberg. Stochastic electron heating in a capacitive RF discharge with non-maxwellian and time-varying distributions. IEEE Trans. Plasma Sci., 23(1):89 96, February [8] M. A. Lieberman and V. A. Godyak. From fermi acceleration to collisionless discharge heating. IEEE Trans. Plasma Sci., 26(3): , June [9] D. Vender and R. W. Boswell. Electron-sheath interaction in capacitive radio-frequency plasmas. J. Vac. Sci. Technol. A, 10(4): , Jul/Aug [10] A. E. Wendt and W. N. G. Hitchon. Electron heating by sheaths in radio frequency discharges. J. Appl. Phys., 71(10): , May [11] M. Surendra and M. Dalvie. Moment analysis of rf parallel-plate-discharge simulations using the particle-in-cell with monte carlo collisions technique. Phy. Rev. E, 48(5): , November [12] Y. M. Aliev, I. D. Kaganovich, and H. Schlüter. Collisionless electron heating in rf gas dicharges: i. quasilinear theory. In Uwe Kortshagen and Lev D. Tsendin, editors, Electron Kinetics and Applications of Glow Discharges, volume 367 of NATO ASI Series B. Plenum Press, New York, [13] I. D. Kaganovich and L. D. Tsendin. The space-time-averaging procedure and modeling of the rf discharge, part ii: Model of collisional low-pressure rf discharge. IEEE Trans. Plasma Sci., 20(2):66 75, April [14] M. Surendra and D. B. Graves. Electron accoustic waves in capacitively coupled, low-pressure rf glow discharges. Phys. Rev. Lett., 66(11): , November 1991.

10 Collisionless electron heating by capacitive radio-frequency plasma sheaths 10 [15] M. M. Turner. Pressure heating of electrons in capacitively-coupled rf discharges. Phys. Rev. Lett., 75(7): , August [16] M. M. Turner. Collisionless heating in capacitively-coupled radio frequency discharges. In Uwe Kortshagen and Lev D. Tsendin, editors, Electron Kinetics and Applications of Glow Discharges, volume 367 of NATO ASI Series B. Plenum Press, New York, [17] I. D. Kaganovich and L. D. Tsendin. Low-pressure rf discharge in the free-flight regime. IEEE Trans. Plasma Sci., 20(2):86 92, April [18] C. K. Birdsall and A. B. Langdon. Plasma Physics via Computer Simulation. Adam Hilger, Bristol, [19] R. W. Hockney and J. W. Eastwood. Computer Simulation Using Particles. Adam Hilger, Bristol, [20] C. K. Birdsall. Particle-in-cell charged-particle simulations, plus Monte Carlo collisions with neutral atoms, PIC-MCC. IEEE Trans. Plasma Sci., 19(2):65 85, April [21] V. Vahedi, G. DiPeso, C. K. Birdsall, M. A. Lieberman, and T. D. Rognlien. Capacitive RF discharges modelled by particle-in-cell Monte Carlo simulation. I. analysis of numerical techniques. Plasma Sources Sci. Technol., 2: , [22] V. Vahedi, C. K. Birdsall, M. A. Lieberman, G. DiPeso, and T. D. Rognlien. Capacitive RF discharges modelled by particle-in-cell Monte Carlo simulation. II. Comparison with laboratory measurements of electron energy distribution functions. Plasma Sources Sci. Technol., 2: , [23] M. Surendra and D. Vender. Collisionless electron heating by radio-frequency plasma sheaths. Appl. Phys. Lett., 65(2): , July [24] A. T. Mense G. A. Emmert, R. M. Wieland and J. N. Davidson. Electric sheath and presheath in a collisionless finite ion temperature plasma. Phys. Fluids, 23(4):803, April [25] R. C. Bissell and P. C. Johnson. The solution of the plasma equation in plane parallel geometry with a maxwellian source. Phys. Fluids, 30(3):779, March [26] J. Borovsky. The dynamic sheath: Objects coupling to plasmas on electron-plasma frequency time scales. Phys. Fluids, 31: , 1988.

11 Collisionless electron heating by capacitive radio-frequency plasma sheaths Figures n o Bulk Plasma <n e > n e n i Electrode 0 S(t) Sm Figure 1. Structure of the rf sheath.

12 Collisionless electron heating by capacitive radio-frequency plasma sheaths 12 Figure 2. Electron density profile from the Monte-Carlo simulation without (a) and with (b) the quasi-neutrality field. n e =1: m 3.

13 Collisionless electron heating by capacitive radio-frequency plasma sheaths 13 Figure 3. The quasi-neutrality and sheath electric fields. The field has been clipped at the value of 10 kv=m. The maximum of the field when the sheath is fully expanded is at E ' 1: V=m. I rf I rf =60A=m 2, n e =5: m 3

14 Collisionless electron heating by capacitive radio-frequency plasma sheaths 14 Figure 4. Average J e E scaling with current. The bare solid line corresponds to Lieberman's prediction, + to J e E from the simulation and Π to J e E P loss. n e =1: m 3

15 Collisionless electron heating by capacitive radio-frequency plasma sheaths 15 Figure 5. Average J e E scaling with frequency. The bare solid line corresponds to Lieberman's prediction, + to J e E from the simulation and Π to J e E P loss. V rf =400V p p, n e =1: m 3

16 Collisionless electron heating by capacitive radio-frequency plasma sheaths 16 Figure 6. Average J e E scaling with temperature. The bare solid line corresponds to Lieberman's prediction, + to J e E from the simulation and Π to J e E P loss. V rf =2000V p p, n e =1: m 3

17 Collisionless electron heating by capacitive radio-frequency plasma sheaths 17 Figure 7. Dots indicate the sheath position obtained by the method described in the text. The solid line corresponds to the FFT fitting. I rf = 60 Am 2, n e =5: m 3

18 Collisionless electron heating by capacitive radio-frequency plasma sheaths 18 Figure 8. The average electron velocity on the sheath edge (solid line) and the sheath velocity (dashed line). Parameters same as in figure 7

19 Collisionless electron heating by capacitive radio-frequency plasma sheaths 19 Figure 9. The error given by the HWA. Solid line is for I rf =130Am 2, dotted line for I rf = 140 Am 2 and dashed line for I rf = 150Am 2, n e =1: m 3.

20 Collisionless electron heating by capacitive radio-frequency plasma sheaths 20 Figure 10. Electron velocity distribution functions normalized to unity. Solid lines correspond to the distributions calculated from the simulation at phases and positions indicated in figure 11. Dashed lines correspond to the expected Maxwellian distributions. The average electron velocity under this conditions is very small compared to the thermal velocity. I rf =130Am 2, n e =1: m 3.

21 Collisionless electron heating by capacitive radio-frequency plasma sheaths 21 Figure 11. gathered. The positions and phases where the distributions of figure 10 were

22 Collisionless electron heating by capacitive radio-frequency plasma sheaths 22 Figure 12. The contribution of stochastic heating to J e E as a function of phase. The bare solid line corresponds to Lieberman's prediction of J e E, Π to J e E P loss, and + is a direct evaluation of Equation (3). I rf =130A=m 2, n e =1: m 3

23 Collisionless electron heating by capacitive radio-frequency plasma sheaths 23 Figure 13. The evolution of the failure of quasi-neutrality and the propagation of a plasma wavetowards the bulk(left) is shown during the expansion phase of the sheath. During the contraction phase a similar wave moves to the opposite direction. (a) t =0:2 T rf,(b) t =0:25 T rf, (c) t =0:3 T rf,(d)t =0:4 T rf. Here! rf =2ß =100:0 MHz, I rf =500Am 2