4 Modeling of a capacitive RF discharge

Transcription

1 4 Modeling of a capacitive discharge 4.1 PIC MCC model for capacitive discharge Capacitive radio frequency () discharges are very popular, both in laboratory research for the production of low-temperature plasmas, and industry, where they are commonly used for thin film deposition and surface etching [Lieberman, 1994; Raizer, 1995; Bogaerts, 2002]. The typical capacitive discharge consists of two parallel electrodes, placed in a vacuum vessel. The electrodes are powered with voltage from a power source. The working gas, fed into a system, gets ionized by electrons, accelerated in the electric field, producing the weekly-ionized plasma with an ionization degree of about The typical distance between the electrodes is of order 1-10 cm. The driving voltage is usually about V with a frequency between 1 and 100 MHz. The pressure of the working gas, depending on application of discharge, varies in the range of Pa. In order to enhance the transfer of knowledge and insight gained in discharge studies and make easier the comparisons between results obtained in different experiments, the Gaseous Electronics Conference Radio-Frequency Reference Cell (GEC) was developed in 1988 [Hargis, 1994]. Now this standardized set-up is used by a large number of experimental groups, working with capacitive discharges. In Fig 4.1 we present the scheme of one such set-up, which is used in the Laser and Plasma Physics Group, Bochum University for the investigation of plasma-chemistry processes in low temperature plasmas [Bush, 1999; Bush, 2001; Möller, 2003]. In this set-up the disk electrodes with spacing d = 4 cm are powered with voltage with frequency MHz. power input in the system varies in the range of W. Mixtures of methane and oxygen at pressures Pa are normally used as the working gas. For the typical operation parameters a plasma with density n e ~ cm -3 and electron temperature T e ~ 3 ev is obtained in the discharge.

2 Although the experimental set-up for capacitive discharge seems to be rather simple, the discharge itself is inherently complex. In such discharges the physics of a non-equilibrium non-stationary plasma is combined with the complexity of reactive plasma processes, including the surface interaction, which makes the modeling of such systems a real challenge. Despite the numerous experimental and theoretical studies performed on capacitive discharges (see, for example, [Raizer, 1995] and references contained therein), the understanding of its behavior is still far from complete. The appropriate model for capacitive discharges should be able to resolve the dynamics of a non-maxwellian bounded plasma in a varying field coupled with the kinetics of chemical processes between plasma species. The Particle-in-Cell model with Monte Carlo Collisions (see Chapter 2) meets these requirements. Particle models were recently successfully applied for modeling of discharges in helium [Surendra, 1990], hydrogen [Vender, 1992] and argon [Vahedi, 1993a; Turner, 1993] and proved to be a promising tool for simulation of such plasmas, providing insight into discharge parameters which are difficult to measure experimentally. Figure 4.1 Capacitively coupled plasma reactor. Figure from [Möller, 2003]. We applied the 2d3v PIC-MCC model, described in Chapter 2, to a capacitive discharge in a methane-hydrogen mix, similar to one described in [Bush, 1999], 68

3 making special emphasis on accurate treatment of the relevant electron-neutral collisions and their influence on the electron energy distribution. In the simulations the initial electron density and temperature were chosen as ne 0 = cm -3 and Te 0 = 20 ev respectively. The mix of CH 4 and H 2 was used as a background gas. The 14-3 gas temperature, T n = 500 K, and densities, n CH4 = 7 10 cm, n H = cm, were chosen close to those used in [Busch 1999], the total pressure of the background gas was p = Pa (0.085 Torr). A rectangular domain with the length d = Ymax = 128λD 0 = 4.25 cm and the width X max = 8λ D 0 = 0.19 cm was used. In the Y direction at positions of the electrodes Y = 0 and Y = Y max the absorbing wall boundary conditions were applied. The potential at Y = Y max was fixed at zero, corresponding to the grounded electrode. At the position of the powered electrode at Y = 0 the potential was assumed to oscillate harmonically according to applied voltage: ϕ ( ) 0, t = U sin( ω t) with ω 2π = MHz. In order to reach equilibrium discharge conditions, the amplitude of applied voltage U was automatically adjusted using the feedback control (Chapter 2.9). At the boundaries in the X direction a periodic boundary condition was applied. As neutral species densities are much larger than densities of charged species, the neutral species were treated as background with fixed density and temperature. Only the dynamics of charged particles was followed. In order to obtain an accurate electron energy distribution, the comprehensive list of electron-neutral reactions for CH 4 and H 2 was added in the model, including the rotational, vibrational and electronic excitation as well as dissociation and ionization collisions and the elastic scattering. Cross-sections for these collisional processes were collected from the compilation used in [Bush, 1999]. In Appendix B we present the energy dependent plots of cross-sections for electron-neutral collisions used in the model. In calculations a grid size x = λ 0 2 = cm and time step 11 = 0.2 = s was used. The number of computational particles per t ω pe Debye cell was chosen as N d = 1000, totaling about computational particles used in the simulation. The calculations were carried out on a 16-processor Linux cluster in about 50 hours. D 69

4 a) b) c) d) e) f) Figure 4.2 Dynamics of the potential profile during one cycle; f =13.56 MHz, p = Torr, d = 4.25 cm, n = 10 cm, = 7 10 cm, n H2 e n CH = cm. The phase of cycle ϕ = ω t is a) ϕ = 2 b) ϕ = 43 c) ϕ = 92 d) ϕ = 182 e) ϕ = 223 f) ϕ =

5 Below we present the results of the simulation. In Fig 4.2 the potential dynamics in the system during the cycle is presented. The potential profiles calculated for 6 different times during the cycle are plotted. In Fig. 4.3 we present the potential profile averaged over the period. We can see that a steep potential drop, up to ϕmax 1100 V, takes place near the electrodes within oscillating positive space-charge layers of about L s 32λD 0 thick - the sheaths. The electric field in the bulk plasma is negligible in comparison with the field in the sheaths, where it is E 450V cm on average. This strong electric field in the sheath regions is directed toward the electrodes, preventing electrons from leaving the plasma for most of the cycle. The electrons are able to escape to electrodes only during a short time, when the sheath collapses (Figs. 4.2c, 4.2f). Figure 4.3 Potential, averaged through one cycle; f =13.56 MHz, p = Torr, d = 4.25 cm, n = 10 cm, n = 7 10 cm, n = cm. e CH 4 H2 71

6 a) b) c) d) Figure 4.4 Dynamics of the density profile of CH 4 + ion during one cycle; f =13.56 MHz, p = Torr, d = 4.25 cm, n = 10 cm, = 7 10 cm, n H2 e n CH = cm. The phase of cycle is a) ϕ = 2 b) ϕ = 43 c) ϕ = 92 d) ϕ = 133. We can follow the space charge dynamics in Figs. 4.4, 4.5, where the time evolution of CH 4 + ion and electron densities during the period of oscillation is presented. As we can expect, the ions due to high inertia (ion plasma frequency 72

7 being smaller than frequency ω pi 0 2 n = ie < ω ε m i ) are not able to react to the fast changing electric field. On their timescale ions respond only to the electric field averaged over cycle (Fig. 4.3), such that the ion density stays constant over the cycle. Thus, the flux of energetic ions, accelerated in the sheath electric field to energies of about average sheath potential drop, constantly flows to the electrodes. The electrons, being much more mobile, follow the changes of electric field during the cycle, oscillating between the electrodes on the static background of the positive space charge of the ions. In the bulk plasma the electron density during the cycle stays equal to the total ion density, maintaining the quasi-neutrality. In the sheath regions the positive space charge of the ions during most of the period remains uncompensated because the electrons reach the electrode only for a short time, during the collapse of the sheath potential (Figs. 4.5d, 4.5e), to balance the ion current on the wall. The change of the net space charge near the electrode during the cycle, resulting from the different response of ions and electrons to the applied voltage, is responsible for the dynamics of the sheath electric field. More information about particle behavior can be extracted from their velocity (energy) distribution dynamics. In Fig. 4.6 we plot the spectrum of the parallel CH 4 + ion energy depending on the longitudinal coordinate Y, calculated at 16 different times during the period. As we can see, in the bulk region the ions stay cold, their mean energy of chaotic motion ch CH E ev is close to the thermal energy 4 of the background gas. In the sheath regions low-energy ions from the bulk plasma are sharply accelerated in the strong electric field toward the electrodes up to the maximum energy of E max 480 ev, which is close to the average sheath potential drop U s 450 V (see Fig. 4.3). Thus, the ion energy distribution has the shape of a narrow ridge aligned along the 0-energy axis in the bulk region and bent from 0 to ±E max in the sheath regions. Due to modulation the maximum ion energy at the electrode position is oscillating through the cycle in the range of ev. In the sheath regions we can also distinguish additional branches in the ion energy distribution corresponding to lower ion energies. These secondary branches in our case are conditioned by electron impact ionization taking place in the sheath region, resulting in the appearance of the low-energy ions in the sheaths. In Figs 4.6 d-e we can see the ions with the energies close to zero appearing in the sheath ( Y < 32 λd ). 73

8 a) b) c) d) Figure 4.5 Dynamics of the electron density profile during one cycle; f = MHz, p = Torr, d = 4.25 cm, n = 10 cm, = 7 10 cm, n H2 e n CH = cm. The phase of cycle is a) ϕ = 2, b) ϕ = 22.5, c) ϕ = 43, d) ϕ = 71, e) ϕ = 92, f) ϕ = 112.5, g) ϕ = 133, h) ϕ = 161 (continued on p. 75). 74

9 e) f) g) h) Figure 4.5 (Continued). 75

10 a) b) c) d) Figure 4.6 Dynamics of the parallel CH4 + ion energy spectrum during one cycle; f =13.56 MHz, p = Torr, d = 4.25 cm, n = 10 cm, = 7 10 cm, n H2 e n CH = cm. The phase of cycle is: a) ϕ = 2, b) ϕ = 22.5, c) ϕ = 43, d) ϕ = 71, e) ϕ = 92, f) ϕ = 112.5, g) ϕ = 133, h) ϕ = 161, i) ϕ = 182, j) ϕ = 202.5, k) ϕ = 223, l) ϕ = 251, m) ϕ = 272, n) ϕ = 292.5, o) ϕ = 313, p) ϕ = 341. Continued on p.p

11 e) f) g) h) Figure 4.6 (Continued). 77

12 i) j) k) l) Figure 4.6 (Continued). 78

13 m) n) o) p) Figure 4.6 (Continued). 79

14 4 ion flux (a.u.) ion energy (ev) Figure 4.7 Ion energy distribution averaged over one cycle CH4 + calculated at the position of the grounded electrode; f =13.56 MHz, p = Torr, d = cm, n = 10 cm, n = 7 10 cm, n = cm. e CH 4 H2 This corresponds to the time interval when the sheath collapses and energetic electrons are able to penetrate deep in the sheath, ionizing neutrals and generating low-energy ions directly in the sheath region. During one cycle this group of low-energy ions is accelerated to the energy about 100 ev at the wall position, producing the next secondary branch in the ion energy distribution (see Fig. 4.6d). The time that the CH 4 + ion takes to traverse the sheath is about τ + CH 4 2Ls 2U m s CH ns, when the period is 2π τ = = ns. Thus it ω takes about 4 periods for low-energy ions generated in the sheath to escape to the electrode. During this time these low-energy ions are gradually accelerated up to energies of about E max, finally contributing to the primary branch of the ion distribution. As a result, a fan-like structure in the ion energy distribution forms in the sheath region, where the high-energy branch is contributed by ions being accelerated from the sheath edge, and secondary peaks correspond to ions generated inside the sheath. The number of the secondary branches should correspond to the number of periods that an ion takes to cross the sheath. In 80

15 total, 4 secondary branches in the ion energy distribution in the sheath region can be distinguished in Fig Figure 4.8 Ion energy distributions measured in a capacitively coupled discharge in argon. Figure from [Wild, 1991]. In Fig. 4.7 we plot the CH 4 + ion energy distribution averaged over the cycle calculated at the position of the grounded electrode. In the high-energy part of the distribution we can distinguish a saddle-like structure with two peaks at energies 430 ev and 530 ev, caused by modulation of the ions which experience the full sheath potential drop. The peaks at lower energies are contributed by low-energy ions produced due to collisions inside the sheath. Similar structures in the ion distribution were observed in experiments and obtained in models [Wild, 1990], [Snijkers, 1993; Kawamura, 1999]. In Fig. 4.8 we present ion energy distribution 81

16 measured in argon discharge for various pressures from [Wild, 1990]. At lower pressures we can see both the saddle structure in high-energy part of the spectrum and the secondary peaks at lower energies. Going to higher pressures the collisional effects in the sheath start to dominate, populating the low energy part of the ion energy spectrum. In Fig. 4.9 we present profiles of the electron parallel velocity component distribution function along the system, calculated at 8 different times during the half-period. The velocity is scaled by the electron thermal velocity v kt e0 te 0 =, me calculated for T e0 = 20 ev. We can clearly distinguish two groups of electrons on these plots: the time independent formation of cold electrons in the middle of system, and the tail of high-energy electrons oscillating between the electrodes. The electrons from the low energy group (with average energy about 0.6 ev) are not energetic enough to overcome the ambipolar potential barrier and penetrate the sheath region, where they could be accelerated by the strong electric field. Thus they are locked in the middle of the system. The energy of these electrons is far below the energy threshold for the majority of inelastic collision processes, thus they are not participating in collisions with neutrals, except for elastic scattering. Eventually, due to elastic collisions, these electrons diffuse to the sheath regions, where they are accelerated in the strong sheath electric field. This group of electrons is most likely populated by the low-energy secondary electrons, produced in electron-neutral ionization collisions. Unlike the electrons from the low-temperature group, the electrons from the high-energy tail can easily overcome the ambipolar potential barrier and penetrate into the region of strong electric field in the sheath. As in our case the mean free path for electron-neutral elastic collisions of magnitude as the system length d = λ en cm is the same order σ n n en 4.25 cm, these electrons can oscillate between the sheaths, getting reflected from them by the strong retarding electric field. Although during single reflection from the sheath, an electron can both gain and loose energy, depending on the phase of the field, but on average, electrons can be accelerated due to stochastization of their motion, following the Fermi acceleration mechanism [Lieberman, 1998]. 82

17 a) b) c) d) Figure 4.9 Dynamics of the distribution of the parallel component of electron velocity during one cycle; f =13.56 MHz, p = Torr, d = 4.25 cm, n e = 10 cm, n CH4 = 7 10 cm, n H2 = cm. The phase of cycle is: a) ϕ = 2, b) ϕ = 22.5, c) ϕ = 43, d) ϕ = 71, e) ϕ = 92, f) ϕ = 112.5, g) ϕ = 133, h) ϕ = 161, i) ϕ = 182, j) ϕ = 202.5, k) ϕ = 223, l) ϕ = 251, m) ϕ = 272, n) ϕ = 292.5, o) ϕ = 313, p) ϕ = 341. Continued on p.p

18 e) f) g) h) Figure 4.9 (Continued). 84

19 i) j) k) l) Figure 4.9 (Continued). 85

20 m) n) o) p) Figure 4.9 (Continued). Originally, Fermi proposed the idea of stochastic heating to explain the origin of cosmic rays [Fermi, 1949] the flux of charged particles with super high energies E ~ ev. In 1949 Fermi suggested, that cosmic rays are originated and accelerated primarily in the interstellar space of the galaxy by collisions against moving magnetic fields. A cosmic ray particle can gain energy from such collisions if the magnetic cloud is moving toward the particle. In the opposite case, when the 86

21 4 Modeling of capacitive discharge region of high magnetic field is moving away from particle, it will lose energy due to the collision. The net result will be average gain, primarily for the reason that headon collisions are more frequent than overtaking collisions because the relative velocity is larger in the former case. Ulam suggested [Ulam, 1961] a simple model problem to illustrate the mechanism of stochastic acceleration: a ball bouncing between one fixed and one regularly oscillating horizontal wall (also known as the Ulam-Fermi problem). Although during one reflection from the oscillating wall the ball can both loose and gain energy, depending on the phase of the oscillating wall, due to dynamic randomization of the collision phase the motion of the ball can become stochastic and after a series of reflections it can be accelerated as it was shown in [Zaslavsky, 1965; Lieberman, 1998]. Godyak applied the Fermi acceleration to electron heating in discharges [Godyak, 1971], proposing that stochastic heating of electrons oscillating between sheaths becomes the dominant heating mechanism in low-pressure capacitive discharges. Randomization of electron motion in discharges can also arise due to electron-neutral collisions. Collisions with neutrals can become the dominant randomization mechanism, when condition for dynamic stochasticity is not satisfied [Lieberman, 1998]. Such collisional randomization can be responsible for stochastic heating of electrons by sheaths even in case of low collisionality: λen d [Kaganovich, 1996; Lieberman, 1998]. In our system λ en < d, thus collisions with neutral gas should play an important role in the randomization of electron motion and hence the stochastic heating. In Fig 4.10 we present the time-averaged electron energy probability function (EEPF), calculated in the middle of the system. EEPF is defined as: 3 2 2E Fe ( E ) = 4π 2 me fe, (4.1) m e with normalization: F E EdE n ( ) e = e. (4.2) 0 87

22 Here fe ( v ) is the electron velocity distribution function, E and n e electron energy and local density respectively. Representation of the energy distribution in this form is convenient as it shows Maxwellian distribution as a straight line. As we can see in Fig the electron distribution can be quite well represented as a sum of two Maxwellian distributions with temperatures T 1 = 0.39 ev, T 2 = 3 ev and densities n 1 = cm -3, n 2 = 10 9 cm -3. The low temperature part corresponds to the static group of cold electrons in the bulk region, whereas the high-temperature component is contributed by the energetic electrons, oscillating between sheaths. Similar bi-maxwellian electron distributions were experimentally found in lowpressure capacitive discharges [Godyak, 1990; Turner, 1993; Mahony, 1999]. In Fig we present EEPF measured in the middle of the capacitive discharge in argon at p = 0.1 Torr with electrode spacing d = 1.2 cm [Godyak, 1990]. This figure clearly indicates two groups of electrons in the discharge: low energy bulk and highenergy tail, resulting from stochastic heating. The electrons from the high-energy tail bouncing between sheaths have enough energy to participate in relevant inelastic collisions with neutrals: ionization, dissociation and excitation. (The ionization energy for methane is E ich 4 = 12.6 ev and dissociation energy - E dch 4 = 10 ev). As the mean free path for such collisions is in our case longer than the system length λi 6 cm > d = 4.25 cm, the inelastic processes are distributed throughout the whole bulk region. In Fig we present a profile of the calculated ionization rate along the system axis: ( ) ( ) n = n vσ E F E EdE, (4.3) ei n i e 0 where n n is neutral density, ( ) σ i E is the electron impact ionization cross-section, and v is the absolute electron velocity. As we can see in Fig. 4.12, ionization is spread rather uniformly in the bulk region, but decreases fast in the sheaths because electrons penetrate the sheath only during short time when the sheath collapses. 88

23 T 1 = 0.39 ev, n 1 = cm T 2 = 3 ev, n 2 = 10 9 cm eepf (ev- 3/2 cm -3 ) T 2 T electron energy (ev) Figure 4.10 The time-spatial averaged electron energy probability function calculated for a capacitive discharge; f =13.56 MHz, p = Torr, d = cm, n = 10 cm, = 7 10 cm, = cm. e n CH4 n H2 ` Figure 4.11 The electron energy probability function measured in a capacitive discharge in argon: f =13.56 MHz, p = 0.1 Torr, d = 2 cm. Figure from [Godyak, 1990]. 89

24 n' ei, cm -3 c -1 8x x x x Y, λ D Figure 4.12 Ionization rate along the system; f =13.56 MHz, p = Torr, d = 4.25 cm, n = 10 cm, n = 7 10 cm, n = cm. e CH 4 H2 In order to study the influence of the electron-neutral collisionality on discharge behavior, we performed a simulation for background gas pressure, increased by a factor of 10 ( p = 0.85 Torr), with neutral densities cm 7 10 cm 15-3 n CH = and n H =. The results of this simulation are presented in Figs In Fig we present the dynamics of the potential during the cycle, showing the potential profile calculated at 6 different phases of cycle. Comparing the potential dynamics with the case of lower gas pressure (Fig. 4.2), we can see that, in the high pressure case, the collapse of the sheath leads to the reversal of electric field in the sheath region (Fig 4.13 b, e). During the short interval of the cycle, the electric field in the sheath region changes direction and accelerates electrons toward the electrode. Because electrons reach the electrode only during the short time when the sheath collapses (see dynamics of electron density in Fig. 4.16), in the case of high working gas pressure, when electron mobility is reduced due to electron-neutral collisions, the accelerating electric field is necessary at this time to provide electron current sufficient to keep balance with generally constant ion current. 90

25 a) b) c) d) e) f) Figure 4.13 Dynamics of the potential profile during one cycle; f =13.56 MHz, p = 0.85 Torr, d = 4.25 cm, n = 10 cm, n = 7 10 cm, cm 15-3 n H =. The phase of cycle is: a) 2 2 ϕ = 182 e) ϕ = 223 f) ϕ = 272. e CH 4 ϕ = b) ϕ = 43 c) ϕ = 92 d) 91

26 Figure 4.14 Potential, averaged during one cycle; Torr, d = 4.25 cm, 10 cm 10-3 n e =, 7 10 cm 15-3 n CH =, 4 f =13.56 MHz, p = n = cm. H2 The time evolution of the CH 4 + ion density profile is plotted in Fig The ion density shows the same static behavior as in the case of lower neutral gas pressure (see Fig. 4.3), but now the ion density in the sheath region has grown about an order in magnitude. Because the averaged sheath potential drop U s 380 V (see averaged through the cycle potential profile in Fig. 4.14) did not change much in comparison with the case of lower gas pressure, the strong increase of the ion density in the sheath shows the presence of intensive ionization in this region. 92

27 a) b) c) d) Figure 4.15 Dynamics of the density profile of CH 4 + ion during one cycle; f =13.56 MHz, p = 0.85 Torr, d = 4.25 cm, n = 10 cm, n = 7 10 cm, 15-3 n = cm. The phase of cycle is: a) ϕ = 2 b) ϕ = 43 c) ϕ = 92 d) H2 ϕ = 133. e CH 4 93

28 a) b) c) d) Figure 4.16 Dynamics of the electron density profile during one cycle; f =13.56 MHz, p = 0.85 Torr, d = 4.25 cm, n = 10 cm, n = 7 10 cm, 15-3 n = cm. The phase of cycle is: a) ϕ = 2, b) ϕ = 22.5, c) ϕ = 43, H2 d) ϕ = 71, e) ϕ = 92, f) ϕ = 112.5, g) ϕ = 133, h) ϕ = 161. Continued on p. 95. e CH 4 94

29 e) f) g) h) Figure 4.16 (Continued). 95

30 a) b) c) d) Figure 4.17 Dynamics of the parallel CH 4 + ion energy spectrum during one cycle; f =13.56 MHz, p = 0.85 Torr, d = 4.25 cm, 10 cm 10-3 n e =, n = 7 10 cm, n = cm. The phase of cycle is: a) ϕ = 2, b) CH 4 H2 ϕ = 22.5, c) ϕ = 43, d) ϕ = 71, e) ϕ = 92, f) ϕ = 112.5, g) ϕ = 133, h) ϕ = 161, i) ϕ = 182, j) ϕ = 202.5, k) ϕ = 223, l) ϕ = 251, m) ϕ = 272, n) ϕ = 292.5, o) ϕ = 313, p) ϕ = 341. Continued on p.p

31 e) f) g) h) Figure 4.17 (Continued). 97

32 i) j) k) l) Figure 4.17 (Continued). 98

33 m) n) o) p) Figure 4.17 (Continued). 99

34 In Fig the dynamics of the parallel energy of CH 4 + ions during the period is presented. We can see that the ion energy distribution shows a pattern similar to the lower pressure case (see Fig. 4.6) with cold bulk ions and fan-like structures in the sheaths. But now the low-energy branches corresponding to ions produced within the sheaths are considerably higher due to higher ionization rate in the sheath regions. As the ion transit time through a sheath is τ ns, i.e. CH 4 about two periods, only two secondary branches in the ion distribution can be seen in addition to a primary branch contributed by ions which are experiencing acceleration from the sheath edge. In Fig 4.18 we present the dynamics of the profile of the electron parallel velocity component distribution as a function of longitudinal coordinate Y. As we can see, the behavior of electrons has changed considerably in comparison with the low-pressure case (see Fig. 4.9). Now all heating of electrons is taking place in the sheath regions near the electrodes. The mean free path of electron-neutral elastic collisions is now λ 0.05 cm, which is much smaller than the system length en d = 4.25 cm and considerably smaller then the sheath width L 16λ cm. Thus, the Ohmic heating in the sheath region, when the electrons are accelerated in the strong electric field between successive elastic collisions with neutrals, becomes the dominating mechanism of electron heating. As we can see, electrons are heated in both half-periods of the cycle: during sheath expansion (Fig 4.18 a-d at Y = Y max ) and reversal of the sheath electric field (Fig 4.18 b-d at Y = 0 ). In Fig we present the electron parallel velocity component distribution averaged during the cycle. Here we can clearly distinguish two zones in the sheath where the electrons are heated. The broadening of the velocity distribution at the distance ~16λ D from the electrode corresponds to electrons heated by the retarding field at the edge of the expanding sheath. The electrons, accelerated in the reversed electric field during the sheath collapse contribute to the hot group directly in front of the electrode. The mean free path for electron-neutral inelastic collisions λ 1 n σ 0.5 cm s D i n i is close to the sheath width, so electrons should be quickly cooled down by inelastic collisions after they leave the sheath region. In Fig we can see that electrons in the bulk region are considerably colder than in the sheaths, and the cooling down takes place on a characteristic length of about the sheath width. 100

35 a) b) c) d) Figure 4.18 Dynamics of the distribution of parallel component of electron velocity 10-3 during one cycle; f =13.56 MHz, p = 0.85 Torr, d = 4.25 cm, n e = 10 cm, n = 7 10 cm, n = cm. The phase of cycle is: a) ϕ = 2, b) CH 4 H2 ϕ = 22.5, c) ϕ = 43, d) ϕ = 71, e) ϕ = 92, f) ϕ = 112.5, g) ϕ = 133, h) ϕ = 161, i) ϕ = 182, j) ϕ = 202.5, k) ϕ = 223, l) ϕ = 251, m) ϕ = 272, n) ϕ = 292.5, o) ϕ = 313, p) ϕ = 341. Continued on p.p

36 e) f) g) h) Figure 4.18 (Continued). 102

37 i) j) k) l) Figure 4.18 (Continued). 103

38 m) n) o) p) Figure 4.18 (Continued). 104

39 Figure 4.19 Distribution of parallel component of electron velocity averaged over 10-3 one cycle; f =13.56 MHz, p = 0.85 Torr, d = 4.25 cm, n = 10 cm, n = 7 10 cm, n = cm. CH 4 H2 e In Fig we present a profile of the ionization rate averaged over the cycle. We can see that in contrast to the previous case (see Fig. 4.11) the ionization is taking place only in the sheath regions. We can also distinguish at each sheath two maxima, contributed by electrons accelerated at different phases of the sheath electric field. We can see this effect in more detail in Fig. 4.21, were the timeresolved ionization rate close to the electrode during one cycle is plotted. Here we can clearly see that the ionization peak near the electrode takes place at time when the electric field near the wall is reversed, and the ionization peak at the sheath edge appears when the repulsive sheath builds up. 105

40 n' ei, cm -3 c x x x x x x Y, λ D Figure 4.20 Ionization rate profile along the system averaged over one cycle; f =13.56 MHz, p = 0.85 Torr, d = 4.25 cm, n = 10 cm, n = 7 10 cm, 15-3 n = cm. H2 e CH 4 Y, mm n' ei, cm -3 c -1 7E16 6E16 4.8E16 3.6E16 2.4E16 1.2E time, ns Figure 4.21 Spatiotemporal distribution of ionization rate in the vicinity of the powered electrode of the capacitive discharge; f =13.56 MHz, p = 0.85 Torr, d = 4.25 cm, n = 10 cm, n = 7 10 cm, n = cm. e CH 4 H2 106

41 Figure 4.22 (a) Spatially and temporally resolved Balmer-alpha (653.3 nm) emission from a hydrogen plasma operating at 133 Pa (1 Torr) with a power of 15 W. This corresponds to a voltage amplitude of 116 V. D is the distance from the driven electrode. (b) Spatially and temporally resolved nm excitation rate derived from (a). (c) The temporal dependence of the electrode potential relative to the plasma for four different powers. Figures from [Mahony, 1997]. 107

42 eepf (ev -3/2 cm -3 ) neutral gas pressure (Torr) electron energy (ev) Figure 4.23 The electron energy probability functions calculated for various pressures of CH 4 - H 2 mix, f =13.56 MHz, n e = cm -3, d = 3 cm. 108

43 Figure 4.24 The electron energy probability function evolution with pressure measured in a capacitively coupled argon plasma. Figure from [Godyak, 1990]. A similar effect of double hot electron layers was observed in experiments with capacitive discharges as double emissive layers near the electrodes. In Fig we plot results of a time resolved measurement of Balmer-alpha (653.3 nm) emission from a discharge in hydrogen at 1 Torr from [Mahony, 1997]. Fig 4.22a shows the measured intensity of the radiation during 2 cycles in the vicinity of the electrode. In Fig. 4.22b the electron-impact excitation rate, deduced from these measurements is presented. We can again see two zones of high excitation rate, corresponding to regions of hot electrons, which are correlated with the sheath 109

44 potential dynamics shown in Fig. 4.22c. The region of high excitation near the electrode appears at moments when the plasma potential becomes lower than the electrode potential (reversed electric field in the sheath). The second region of high excitation rate further from the electrode is caused by electrons heated at the edge of the growing sheath. In order to investigate the transition of the electron heating mechanism with the increase of background gas pressure, we performed simulations for the same H 2 - CH 4 1.3:1 mix, changing the gas pressure. In Fig we summarize these simulations, presenting EEPF s averaged over the cycle in the bulk plasma for gas pressures Torr, Torr, Torr, 0.17 Torr, 0.34 Torr and 0.85 Torr. As we can see in Fig. 4.23, at low pressures the EEPFs are essentially bi- Maxwellian, revealing the stochastic electron heating mechanism, leading to the formation of cold bulk and oscillating hot tail electrons. With increase of the neutral gas pressure between 0.34 Torr and 0.85 Torr EEPF transforms to a convex, Druyvesteyn type distribution with a high-energy part depleted by inelastic collisions, corresponding to a regime when the Ohmic heating in the sheath regions is the dominant mechanism of the electron heating. Such changes of electron energy distribution were observed experimentally in capacitive discharges [Godyak, 1990; Godyak, 1992; Turner, 1993]. In Fig we present the EEPFs measured by [Godyak, 1990] in an argon capacitive plasma for different pressures. We can see that the bi-maxwellian distribution measured at low neutral gas pressures changes to a Druyvesteyn type distribution at pressure of Torr. This pressure is somewhat lower than values obtained in our simulations, which can be explained by larger cross-sections for electronneutral elastic collisions for argon. 4.2 Principal results A capacitively coupled radio-frequency () discharge was studied in collaboration with an experimental group at the University of Bochum. This discharge type has special importance for plasma technology (etching, deposition). We were able to follow the time and spatially resolved dynamics of the plasma particles. For the low pressure of the neutral gas the electron distribution is a sum of two Maxwellian distributions. The low temperature part corresponds to the static 110

45 group of cold electrons in the bulk region, whereas the high-temperature component is contributed by the stochastically heated electrons, oscillating between sheaths. Similar bi-maxwellian electron distributions were experimentally found in low-pressure capacitive discharges. In the high-pressure case the Ohmic heating in the sheath region becomes the dominating mechanism of electron heating. Due to electric field reversal in the sheath region, electrons are heated in both half-periods of the cycle, which cause the two peaks in the intensity of electron-induced inelastic processes (ionization, excitation) near the electrodes. A similar effect of double hot electron layers was observed in experiments with capacitive discharges as double emissive layers near the electrodes. As the neutral gas pressure increases, electron distribution transforms from bi- Maxwellian (resulting from the stochastic heating) to a Druyvesteyn type distribution (corresponding to Ohmic heating). Such changes of electron energy distribution were observed experimentally in capacitive discharges. The electron energy distribution function (EEDF) determines the chemical kinetic processes in the reactive discharges and hence is of vital importance for reactive plasmas (film deposition). The ability to calculate the EEDFs provides improved possibilities for predictive modeling of such systems [Möller, 2003]. The ion energy distribution (IED) at the wall position calculated within our model indicates a multi-peak structure due to ion modulation in the sheath. This is confirmed by the IED observed in experiments. The knowledge of IED is of special interest for plasma etching processes. 111