Numerical Modeling of an RF Argon Silane Plasma with Dust Particle Nucleation and Growth

Transcription

1 Plasma Chem Plasma Process (2014) 34: DOI /s ORIGINAL PAPER Numerical Modeling of an RF Argon Silane Plasma with Dust Particle Nucleation and Growth Pulkit Agarwal Steven L. Girshick Received: 19 August 2013 / Accepted: 22 October 2013 / Published online: 5 December 2013 Ó Springer Science+Business Media New York 2013 Abstract A one-dimensional numerical model and simulation results are presented for a capacitively-coupled radio frequency parallel-plate argon silane dusty plasma. The model includes self-consistently coupled numerical modules, including a plasma fluid model, a sectional aerosol model, and a simple chemistry model to predict rates of particle nucleation and surface growth. Operating conditions considered include MHz frequency, 100 mtorr pressure, a 4-cm electrode gap, gas flow through the top electrode with a 30:1 ratio of argon to silane, and applied radio frequency voltage amplitude of either 100 or 250 V. In the higher voltage case two lobes of relatively large particles are formed by ion drag, while fresh nucleation occurs in the void between these lobes. It is shown that the reason that fresh nucleation occurs in the void involves an interplay among several coupled phenomena, including nanoparticle transport, the plasma potential profile, and trapping of silicon hydride anions that drive nucleation in this system. Keywords Dusty plasmas Silane Nanoparticles Introduction Nanodusty plasmas plasmas in which nanoparticles nucleate and grow remain the subject of considerable interest, both because unwanted dust formation in plasmas is a source of contamination in applications such as semiconductor processing and tokamak fusion reactors, and because deliberate synthesis of nanoparticles using plasmas is used or P. Agarwal S. L. Girshick (&) Department of Mechanical Engineering, University of Minnesota, 111 Church St. S.E., Minneapolis, MN 55455, USA slg@umn.edu Present Address: P. Agarwal Applied Materials, Santa Clara, CA, USA

2 490 Plasma Chem Plasma Process (2014) 34: being developed for applications such as solar cells, optoelectronics, catalysis, and biomedicine. Controlling or avoiding the formation of nanoparticles in plasmas can be aided by the development of predictive numerical models, which furthermore can afford fundamental insights into the behavior of these complex systems. The evolution of a nanoparticle cloud in a plasma involves nucleation, particle growth by chemical or physical vapor deposition (hereafter termed surface growth), coagulation, charging, and transport by effects that include neutral gas drag, ion drag, Brownian diffusion, electric force, thermophoresis and gravity. In addition to nucleation and surface growth being sinks for chemical species in the plasma, the fact that nanoparticles can acquire considerable charge causes the aerosol phase and the plasma to be strongly coupled. While numerical models of nanodusty plasmas have been developed for a number of different chemistries and types of plasmas, the most studied system has been the radio frequency (RF) capacitively-coupled plasma consisting of silane diluted in noble gas, with a parallel-plate configuration and at total pressure on the order of 100 mtorr, in which silicon nanoparticles nucleate and grow. Several numerical studies of this type of nanodusty plasma have been presented [1 7]. In each of these studies the particle size distribution was represented by means of a sectional model [7 9], that is, the particle size spectrum was divided into sections, or bins, with a population balance equation solved for each section. In cases where particle charge distributions are calculated as well, each size section is further divided into subsections to represent discrete charge states, and population balance equations are solved for each size-charge section. The particle size sections are typically spaced logarithmically by particle volume, with a small sectional width used to minimize numerical diffusion in systems with particle surface growth [9]. Further considering the bivariate particle size-charge distribution, the number of size-charge sections can be large e.g. several thousand, for cases where particles grow to larger than 100 nm and sectional models can thus be quite computation-intensive. For example, the 1-D simulations presented in Ref. [7], which divided a 4-cm RF electrode gap into 120 cells, and which covered 11 s of plasma time, with particles at some locations growing to over 300 nm in diameter at the tail of the size distribution, required approximately 500 h of cpu time, using 12 parallel processors on an Intel 6-core Xeon X GHz machine with 24 GB RAM. Given this computational expense, it is not surprising that each of the previous studies neglected or drastically simplified one or more important aspects of the plasma-chemistryaerosol system. Bhandarkar et al. [2, 3] developed a zero-dimensional (0-D; homogeneous plasma) model of this system, with a detailed chemical kinetic mechanism that considered clustering of silicon hydrides containing up to 10 Si atoms. The nucleation rate was assumed to equal the rate of formation of clusters containing 11 or more Si atoms. An aerosol sectional model was used to model particle nucleation, surface growth and charging, self-consistently coupled to a plasma fluid model. De Bleecker et al. [4] developed a one-dimensional (1-D; infinite parallel-plate) model of this system. They extended the silicon hydride clustering model of Bhandarkar et al. [2, 3] up to size 12 Si atoms, again coupled to a plasma fluid model and an aerosol sectional model. Their aerosol model did not consider particle surface growth, and the effect of charge on coagulation was neglected. They presented results for spatial profiles of nanoparticles of various sizes, but without any indication of time dependence, so it does not appear that the temporal evolution of this system was modeled. Warthesen and Girshick [5] developed a 1-D, transient model. Again an aerosol sectional model was coupled to a plasma fluid model, and the effect of nanoparticle charge on coagulation was included. However, chemistry was not

3 Plasma Chem Plasma Process (2014) 34: modeled; instead, the plasma was treated as pure argon, with nucleation of particles having the properties of silicon, and parameterized particle nucleation and surface growth rates were assumed, based on the predictions of Bhandarkar et al. s 0-D model [2] for similar conditions. The model of Warthesen and Girshick [5] was subsequently extended by Ravi and Girshick [6], who incorporated the effect of image potentials on coagulation [10]. This effect was found to be important, as coagulation in this system is dominated by interactions between very small (1 or 2 nm) neutral particles and larger, negatively charged particles, that are electrostatically trapped and thus can grow. In summary, numerical models of various degrees of complexity have been developed for RF argon silane plasmas in which nanoparticles nucleate and grow, but none of them have self-consistently modeled the plasma, the chemistry of particle nucleation and surface growth, and the aerosol dynamics (including particle size- and charge-dependent coagulation) to predict the spatiotemporal evolution of this system in a spatially nonuniform environment. In the present work, a chemistry model is added to the 1-D model of Girshick and coworkers [5 7] to predict rates of particle nucleation and surface growth in an argon silane plasma. The model for silicon hydride chemistry here is much simpler than in Refs. [2 4]: population balance equations are solved only for silicon hydrides (neutrals and ions) containing one Si atom, and the nucleation rate is equated to the sum of the rates of formation of the anions Si 2 H - 4 and Si 2 H - 5. The justification for this simplification is that a body of experimental work [11 16] and numerical modeling [2, 17 19] for this system indicates that nucleation proceeds primarily via two sequences of anion-molecule reactions, which grow successively larger silicon hydride anions, with stoichiometry Si n H 2n - and Si n H - 2n?1, respectively, for the two different sequences. As Si 2 H - 4 and Si 2 H - 5 lie at the roots of these two sequences, their rates of formation are closely correlated with the nucleation rate. Additionally, a simple model for particle growth by heterogeneous reactions on particle surfaces is included. While this model can in the future be improved by extension to more detailed silicon hydride clustering kinetics and surface chemistry, we believe that it already provides considerable insight into the spatiotemporal evolution of the coupled plasma-chemistryaerosol system. Numerical Model Overview The numerical model used here builds on previous work of Girshick and coworkers that is described in more detail in Refs. [5, 7, 20]. We use a time-slicing approach [21], in which separate modules are run, each with its own time step, according to the need to resolve the characteristic time scales for change in the phenomena treated in each of the modules. An overview of the model is shown in Fig. 1. A plasma module solves population balance equations for electrons and all ionic species, the electron energy equation under the assumption of a Maxwellian electron velocity distribution, and Poisson s equation for the electric field. The DC bias on the top electrode is determined by setting the net current at the top electrode to zero. An aerosol module solves the aerosol general dynamic equation, equivalent here to a population balance equation for nanoparticles in each size section and discrete charge state; and a chemistry module solves population balance equations for all neutral chemical species.

4 492 Plasma Chem Plasma Process (2014) 34: Fig. 1 Overview of numerical model For the MHz RF plasma modeled here, a time step of s is used for the plasma module. A steady-state solution is typically achieved after RF cycles. The aerosol module, which runs with a time step of s, then calculates the response of the aerosol to the new plasma solution, with plasma properties time-averaged over the RF cycle. After each 20 time steps of the aerosol model, the chemistry module is run (hence, with a time step of 10-5 s) to recalculate the neutral species density profiles. The aerosol module runs until the nanoparticle charge concentration at any location has changed by at least 1 %, which typically occurs after 100 1,000 time steps. The plasma module is then re-run to determine the new plasma solution, and the process is repeated, with the solution marching forward in time. The aerosol model considers gas-to-nanoparticle conversion by nucleation and surface growth; size- and charge-dependent coagulation, including the effect of image potentials; particle charging by electron- and ion attachment, based on orbital motion limited theory [22], with the negative charge of single nanoparticles being limited according to their size [23]; and nanoparticle transport by neutral drag, ion drag, electric force, diffusion and gravity. We here focus on the models used for chemistry, nucleation and surface growth, which constitute the changes to the model relative to earlier work [5 7]. The reader is referred to Ref. [5] in particular for more details of the plasma model. Chemistry, Nucleation and Surface Growth Mass spectrometry measurements [11 14] have demonstrated that silicon hydride anion clusters can grow to large sizes (several 10 s of Si atoms) in RF silane-containing plasmas, while corresponding neutral and cation clusters are limited to much smaller sizes. Numerical simulations that used the 0-D model with detailed silicon hydride clustering chemistry of Bhandarkar et al. [2, 3] predict, for conditions similar to those assumed for the simulations reported here, that such large anion clusters are formed predominantly by two reaction sequences, involving silane addition to either a silylene anion or a silyl anion, growing the cluster by adding one Si atom while eliminating hydrogen:

5 Plasma Chem Plasma Process (2014) 34: Si n H 2n þ SiH 4 Si nþ1 H 2nþ2 þ H 2; ð1þ and Si n H 2nþ1 þ SiH 4 Si nþ1 H 2nþ3 þ H 2: Guided by this, we model the chemical kinetics of species containing one Si atom, and take the sum of the formation rates of Si 2 H 4 - and Si 2 H 5 - as a surrogate for the nucleation rate. Compared to a more detailed clustering model that extends to larger cluster sizes, the nucleation rate obtained by this simple model may be overpredicted, because of neglect of kinetic bottlenecks to produce silicon hydrides with more than two Si atoms. For example, in the 0-D simulations of Bhandarkar et al. [2, 3], extending the silicon hydride clustering mechanism to size Si 10 H m resulted in a reduction in the predicted nucleation rate by approximately 40 % compared to what would have been obtained if the approach of the present work had been used. However, it should be noted that there is considerable uncertainty in the values of rate constants for anion-neutral reactions that are primarily responsible for clustering in this system, and thus any correction factor one might apply for the predicted nucleation rate would itself be highly uncertain. The present approach should be qualitatively correct, and predict nucleation rates that are reasonable with regard to their spatial profiles and the effects of changes in operating conditions. By modeling the chemistry of Si 1 species, one obtains density profiles of Si-containing ions including SiH 2 -, SiH 3 - and SiH 3?. Aside from the importance of the anions for nucleation, these ions may be sufficiently abundant to significantly affect the plasma charge balance as well as nanoparticle charging, even with silane highly diluted in noble gas. Our gas-phase chemistry model thus considers reactions involving electrons, argon neutrals, metastables and singly-charged ions, and nine silicon hydride species (neutral, anion and cation) that contain one Si atom. The chemical reactions and rate parameters included in our model are given in Table 1. Individual nanoparticles can grow by heterogeneous reactions on their surfaces. We here use the simple surface reaction mechanism given in Table 2. Following the same approach as for coagulation between neutral and charged nanoparticles, we account for the induced dipole interaction in calculating the collision rate between neutral gas species and charged nanoparticles. The sticking of Si-species on the nanoparticle surface depends on the fractional coverage of hydrogen-terminated sites on the surface, denoted Si H(S) in Table 2, versus Si dangling bonds, denoted Si(S). In lieu of a more detailed model, which would have to track the particle surface coverage as particles coagulate, and lacking better information, we here assume that the fractional coverage of hydrogen-terminated sites is fixed at approximately 50 % [24]. As Si atoms are added to the nanoparticle, Si atoms at the surface are converted to Si atoms in the bulk of the particle, denoted Si(B). Boundary Conditions Pertinent boundary conditions include the following. At both electrodes, the number densities n(-) of negative ions and negative particles are set to zero, while the number densities n(?) of positive species (ions and particles) and all neutrals are all assigned zero concentration gradient. Thus, at both electrodes, nð Þ ¼ 0; ð3þ ð2þ

6 494 Plasma Chem Plasma Process (2014) 34: Table 1 Gas-phase reaction mechanism Reaction A (cm 3 s -1 mol -1 ) b E A (cal mol -1 ) Refs. 1. Ar? e? Ar?? 2e ,860 [29] 2. Ar? e? Ar*? e ,662 [29] 3. Ar*? e? Ar?? 2e ,191 [29] 4. Ar*? Ar*? Ar? Ar?? e [30] 5. Ar*? e? Ar? e [31] 6. SiH 4? e? SiH 3? H? e ,430 [29] 7. SiH 4? e? SiH 2? 2H? e ,430 [29] 8. SiH 4? e? SiH - 3? H ,540 [2] 9. SiH 4? e? SiH - 2? 2H ,540 [2] 10. SiH 4? e? SiH? 3? H? 2e ,910 [24] 11. SiH 3? e? SiH - 2? H [2] 12. SiH 3? e? SiH? 3? 2e ,403 [24] 13. SiH - 3? e? SiH 3? 2e ,070 [32] 14. SiH - 2? e? SiH 2? 2e ,300 [32] SiH 2? e? SiH [24] 16. H 2? e? 2H? e ,356 [29] 17. H 2? e? H? 2? 2e ,590 [29] 18. SiH 4? Ar*? SiH 3? H? Ar [33] 19. SiH 4? Ar*? SiH 2? 2H? Ar [33] 20. SiH 3? Ar*? SiH 2? H? Ar [33] 21. SiH 2? Ar*? SiH? H? Ar [33] 22. H 2? Ar*? 2H? Ar [33] 23. SiH 3? SiH 3? SiH 2? SiH [29] 24. SiH 4? SiH 3? Si 2 H 5? H ,400 [29] 25. SiH 2? H 2? SiH [4] 26. SiH 2? Si? H ,241 [34] 27. SiH 4? H? H 2? SiH ,190 [29] 28. SiH 2? SiH 2? Si 2 H 2? H [29] 29. SiH 2? H? SiH? H [29] 30. SiH 3? SiH? H ,770 [29] 31. SiH 3? H? SiH 2? H ,500 [29] 32. SiH - 2? H? 2? SiH 2? H [24] 33. SiH - 3? H? 2? SiH 3? H [24] 34. SiH - 3? SiH 3? Si 2 H - 4? H [2] 35. SiH - 3? SiH? 3? Si 2 H [2] 36. SiH - 2? SiH 4? Si 2 H - 4? H [2] 37. SiH - - 2? SiH 3? SiH 2? SiH [2] 38. SiH - 3? SiH 2? Si 2 H - 3? H [2] 39. SiH - 3? SiH 4? Si 2 H - 5? H [2] 40. SiH - 2? SiH? 3? Si 2 H [2] 41. SiH - 2? Ar?? SiH 2? Ar [32]

7 Plasma Chem Plasma Process (2014) 34: Table 1 continued Reaction A (cm 3 s -1 mol -1 ) b E A (cal mol -1 ) Refs. 42. SiH - 3? Ar?? SiH 3? Ar [32] Rate constants k for each reaction follow the Arrhenius form, k ¼ AT b expð E A =RTÞ, where R is the molar gas constant. Ar* denotes electronically excited (metastable) argon Table 2 Heterogeneous reaction mechanism at particle surfaces Surface reaction mechanism Sticking coefficient Refs. 1. SiH 4? 2Si(S)? 2SiH(S)? Si(B)? H [35] 2. SiH 2? Si(B)? H [32] 3. SiH 3? Si(S)? SiH(S)? Si(B)? H [32] 4. SiH? 3? Si(S) - Z P e? SiH(S)? Si(B)? H 2 - (Z P - 1)e 1.0 Est. 5. Ar*? Particle? Ar? Particle 1.0 Est. and dnðþþ ¼ 0: ð4þ dx The fluxes at the electrodes of electrons, J e, and of electron energy, q e, are given by the corresponding thermal fluxes: and J e ¼ n ev th 4 ; ð5þ 5 2 q e ¼ n ev th T e ; ð6þ 4 where n e and T e are electron density and temperature, respectively, and v th is the electron thermal velocity, i.e. the mean velocity of Maxwellian electrons at temperature T e. The lower electrode is grounded and the upper electrode is RF powered according to the applied voltage given below. The net currents at each of the electrodes is set to zero to account for capacitive coupling. Operating Conditions The simulation results shown below are for a capacitively-coupled RF plasma operating at MHz frequency with an applied voltage of either 100 or 250 V (amplitude) on the top (showerhead) electrode. The bottom electrode is grounded. Gas consisting of a 30:1 argon silane mixture is introduced through the top electrode with a velocity of m/s, corresponding to a flow rate of 31 sccm through a 12-cm-diameter electrode. A stagnation flow profile is assumed with respect to the bottom electrode. The total pressure equals 100 mtorr (13.33 Pa).

8 496 Plasma Chem Plasma Process (2014) 34: Results and Discussion We present a comparison of simulation results for the two different RF voltage amplitudes, 100 and 250 V, but then focus on the 250-V case for more detailed results. In previous simulations without chemistry, the spatiotemporal evolution of the nanodusty plasma was discussed for a case with identical conditions except for a lower voltage [7]. Here we focus mainly on results at a time t = 2.0 s following initiation of the plasma and introduction of the gas flow. While major qualitative features of the temporal evolution in these simulations are similar to those discussed in [7], the use of chemistry to self-consistently calculate rates of nucleation and surface growth should, in principle, improve the simulations with regard to quantitative results, while also affording insights that are not possible without the addition of chemistry. The reason for focusing on the results at t = 2.0 s is that by this time the major features of the predicted distributions of plasma and aerosol properties have developed. Particle nucleation and growth continue beyond this time, but the profiles of major plasma and aerosol properties change only slowly. At much longer times, when particles have grown to diameters around 100 nm or larger, interesting new features were found to emerge in the simulations without chemistry [7], but following the present simulations to such long times is beyond the scope of the present work. Figure 2 shows the spatial distribution across the electrode gap of the particle size distribution function, dn/d log(d p ), where N is nanoparticle concentration and d p is particle diameter, for the two different applied voltages. White contour lines show the average particle charge. As previously discussed [7], it can be seen that two distinct populations of nanoparticles develop: very small (1 or 2 nm) particles, many of which are neutral or even positively charged, that diffuse toward both electrodes, and negatively charged particles that are electrostatically trapped in the plasma, and thus can grow to larger sizes, in this case several tens of nm in diameter. In both cases shown, the cloud of larger, negatively charged nanoparticles is pushed by the gas flow toward the lower electrode. The cloud pushes right up to the lower sheath edge, to a location where the repulsive sheath potential balances the neutral drag and other forces pushing the particles toward the electrode. Fig. 2 Particle size distribution function, dn/dlog(d p ), with N representing particles per cm 3 and d p particle diameter in nm, and average particle charge (white lines) at 2.0 s following plasma initiation. Applied RF voltage amplitude equals a 100 V, and b 250 V

9 Plasma Chem Plasma Process (2014) 34: However, the 250-V case is seen to be notably different than the 100-V case in one important respect. In the 250-V case two spatially separate lobes of the nanoparticle cloud are observed, with the larger particles pushed not only toward the lower electrode, in the direction of gas flow, but also toward the upper electrode, against the gas flow. The reason for this is that ion drag is stronger in the 250-V case. Ion drag pushes negatively charged particles in both directions, from the center of the plasma (the location where the electric field equals zero) toward the electrodes. The strength of ion drag depends on the electric field, which is higher in the 250-V case, and on particle size and charge. Thus, in the 250-V case, particles that are sufficiently large and carry sufficient negative charge experience strong enough ion drag to be pushed toward the upper electrode, against the direction of gas flow. As these two lobes of large nanoparticles develop, a relative void opens up between the lobes, at a location x & 2.35 cm above the lower electrode. In this void region, a high concentration of very small particles is predicted. (Thus the term void here can be interpreted as a region of much lower density of particle surface area or mass rather than number density.) The reason for this is that fresh nucleation is concentrated in the void region. Indeed, experimental studies have observed the formation of voids (based on scattered light intensity) in dusty plasmas and the occurrence of fresh nucleation in these voids [25]. Whereas the relatively large particles that constitute each lobe are almost all negatively charged, many of the freshly nucleated particles, being only 1 or 2 nm in diameter, are neutral. This is seen in Fig. 3, which shows the density profiles of negative, neutral and positive nanoparticles. The predicted density of negatively-charged nanoparticles equals *10 10 cm -3 in the region of fresh nucleation and *10 9 cm -3 in the two lobes containing mostly larger nanoparticles. Meanwhile, the density of neutral nanoparticles is predicted to equal *10 9 cm -3 in the nucleation region but drops to much lower values outside this region. The density of positively-charged nanoparticles equals * cm -3 over the whole of the bulk plasma. These latter values may be underpredicted, as processes such as secondary electron emission and UV photoemission from nanoparticles are not included in the model. Fig. 3 Density profiles of negatively-charged, neutral and positively-charged nanoparticles, in the 250-V case at t = 2.0 s

10 498 Plasma Chem Plasma Process (2014) 34: Fig. 4 Profiles of particle surface area density and nucleation rate in the 250-V case at t = 2.0 s Fig. 5 Density profiles of selected neutral species in the 250-V case at t = 2.0 s Figure 4 shows profiles of particle surface area density and nucleation rate, again for the 250-V case at t = 2.0 s. Here the void is clearly defined, and a strong negative correlation is seen between surface area density and nucleation rate. In neutral aerosols, it is typically the case that high particle surface area density quenches nucleation, because nucleation and surface growth compete for the same chemical species [26]. For example in thermal (non-plasma) decomposition of silane, SiH 2 plays a key role in both nucleation and surface growth [27]. As can be seen in Fig. 5, which shows density profiles of neutral species at t = 2.0 s, SiH 2 and SiH 3 (an abundant radical in plasmas but not thermal decomposition) are both depleted in the regions of high particle surface area density. However, dusty plasmas are different than neutral aerosols. In the following, we show that the reason nucleation is concentrated in the void is not because the high particle

11 Plasma Chem Plasma Process (2014) 34: surface area density outside the void quenches nucleation, as had been assumed in previous modeling that neglected chemistry [5 7]. Rather, it has to do with the two-way coupling between the electric field and transport of charged species and nanoparticles. As discussed above, both experimental studies and detailed 0-D numerical modeling indicate that cluster growth in silane plasmas involves growth of silicon hydride anions. In our kinetic mechanism, where the nucleation rate is equated to the sum of the rates of formation of Si 2 H - 4 and Si 2 H - 5, the dominant reactions that produce these species are ionmolecule reactions involving silane addition with hydrogen elimination: SiH 2 þ SiH 4! Si 2 H 4 þ H 2; ð7þ and SiH 3 þ SiH 4! Si 2 H 5 +H 2: ð8þ Now, SiH - 2 and SiH - 3 can be produced by electron attachment to SiH 2 and SiH 3, but under the conditions of these simulations the rates of these reactions are several orders of magnitude lower than the production rates of SiH - 2 and SiH - 3 by dissociative attachment to silane: SiH 4 þ e! SiH 2 þ 2H, ð9þ and SiH 4 þ e! SiH 3 þ H: ð10þ Therefore, neutral SiH 2 and SiH 3, important species in our mechanism for particle surface growth, are relatively unimportant here for nucleation. Conversely, SiH - 2 and SiH - 3 are unimportant for surface growth, because they are repelled by the mostly negatively-charged nanoparticles, but are predicted to be crucial for nucleation. Why, then, does nucleation occur in the void? First, consider Fig. 6, which shows density profiles of all charge carriers, for the 250-V case at t = 2.0 s. It can be seen that all of these profiles peak in unison around x & 2.35 cm. The most abundant of these charge carriers is the net negative charge carried by nanoparticles, which has a local maximum at the location where fresh nucleation is occurring, because of the large total particle density at that location. The densities of Fig. 6 Density profiles of electrons, ions and net negative charge on nanoparticles, in the 250-V case at t = 2.0 s. Curve labeled negative ions sums the densities of SiH 2 - and SiH 3 -

12 500 Plasma Chem Plasma Process (2014) 34: Fig. 7 Profile of plasma potential, in the 250-V case at t = 2.0 s. Inset shows same profile with voltage scale magnified. Asymmetry in sheath potentials is caused by DC self-bias at upper electrode positive ions also peak at the same location, maintaining charge quasi-neutrality. Meanwhile, the free electron density is over two orders of magnitude lower than the net negative charge carried by nanoparticles. The negative ion density (the sum of SiH 2 - and SiH 3 - )is especially sharply peaked at the same location, where it is almost as high as the electron density. The fact that all of these density profiles peak at the same location can be explained by the strongly coupled behavior of nanoparticle transport, the electric field, and chemistry. To further understand this, consider Fig. 7, which shows the plasma potential profile across the electrode gap for the same case and time. The peak in potential (see inset of Fig. 7, which magnifies the voltage scale), equivalent to the location of zero electric field, occurs at x = 2.35 cm, the same as the peaks in all of the charge density profiles. The location of the peak in the density profile of net negative charge on nanoparticles is determined mainly by a combination of neutral drag and ion drag. As nanoparticles are by far the dominant carrier of negative charge in these simulations, their charge density profile powerfully influences the location of zero electric field, via Poisson s equation. In turn, the location of zero electric field determines the dividing line regarding the direction of ion drag, and hence the location of the void. Additionally the zero electric field location corresponds to a negative potential well that traps negative ions. Finally, these trapped anions drive nucleation at this location, completing the circle. As discussed above, nucleation in our mechanism is strongly related to the production rate of SiH 2 - and SiH 3 -, for which the dominant reaction is dissociative attachment to silane. Figure 8 shows the profile of the rate of this reaction, in this case for production of SiH 2 -, again for the 250-V case at 2.0 s. In fact the peak in this profile does not occur at x = 2.35 cm, but slightly higher, at x = 2.57 cm. The rate of this reaction depends on the product of the densities of SiH 4 and electrons, and on electron temperature. While the peak in electron density does occur at x = 2.35 cm, the silane density monotonically decreases from the upper electrode (showerhead inlet) to the lower electrode. Hence the reaction rate

13 Plasma Chem Plasma Process (2014) 34: Fig. 8 Profile of reaction rate of silane dissociative attachment to form SiH 2 -, in the 250-V case at t = 2.0 s Fig. 9 Profiles of electron density and electron temperature in the 250-V case, at t = 0.1 and 2.0 s peaks at a location higher than the peak in electron density. However, the anions produced by this reaction are pushed by the electric field toward the bottom of the negative potential well, causing the anion density to be sharply peaked at that location, and hence nucleation as well. Anions, having much lower mobility than electrons, are more effectively trapped in the negative potential well. Finally, Fig. 9 shows the predicted profiles of electron density and electron temperature for the 250-V case at two times, t = 0.1 s and 2.0 s. The predicted electron temperatures in the sheaths should be considered approximate, as our model considers neither secondary

14 502 Plasma Chem Plasma Process (2014) 34: electron emission nor stochastic heating, although it should be noted that electron heating in dusty plasmas is expected to be more resistive than stochastic [28]. In the bulk of the plasma, however, where the plasma fluid model should at least be qualitatively reasonable, one sees that as the electron density drops the electron temperature increases, going from *2.4 ev at t = 0.1 s to *3.8 ev at t = 2.0 s near the lower sheath edge, where the electron density is a minimum due to the nanoparticle charge density being a maximum (cf. Fig. 6). The outward spread of the electron density between 0.1 and 2.0 s is caused by the corresponding outward spread of the nanoparticle cloud, pushed toward both electrodes by ion drag. This causes the sheath thickness to decrease with time. At longer times the outward spread of the nanoparticle cloud becomes progressively slower, as ion drag on the nanoparticles is balanced by the increasingly strong potential gradient at the sheath edge. Summary and Conclusions In this work we have modeled a capacitively-coupled RF argon silane parallel-plate plasma, using a 1-D model with coupled numerical modules for the plasma, the aerosol and chemistry. Simulation results are compared for two different values of applied RF voltage. At the higher voltage, increased ion drag causes the development of two spatially separate lobes, each of which contains predominantly negatively-charged nanoparticles that are trapped by the electric field and growing. Between these lobes a region that constitutes a relative void a region with a much lower density of nanoparticle mass or surface area develops, and in this void fresh nucleation occurs. Detailed examination of the simulation results shows that the reason that fresh nucleation occurs inside the void involves a rich interplay among several coupled phenomena, including nanoparticle transport, the profile of electric field, and trapping of silicon hydride anions that drive nucleation in this system. Acknowledgments This work was partially supported by the US National Science Foundation (Grant CHE ), US Department of Energy Office of Fusion Energy Science (Grant DE-SC ), and the Minnesota Supercomputing Institute (MSI). Helpful discussions with Dr. David Porter of MSI are gratefully acknowledged. References 1. Kortshagen U, Bhandarkar U (1999) Phys Rev E 60: Bhandarkar U, Swihart MT, Girshick SL, Kortshagen U (2000) J Phys D 33: Bhandarkar U, Kortshagen U, Girshick SL (2003) J Phys D 36: De Bleecker K, Bogaerts A, Goedheer W (2006) New J Phys 8: Warthesen SJ, Girshick SL (2007) Plasma Chem Plasma Process 27: Ravi L, Girshick SL (2009) Phys Rev E 79: Agarwal P, Girshick SL (2012) Plasma Sources Sci Technol 21: Gelbard F, Tambour Y, Seinfeld JH (1980) J Coll Interface Sci 76: Warren DR, Seinfeld JH (1985) Aerosol Sci Technol 4: Huang DD, Seinfeld JH, Okuyama K (1991) J Coll Interface Sci 141: Howling AA, Dorier J-L, Hollenstein C (1993) Appl Phys Lett 62: Howling AA, Sansonnens L, Dorier J-L, Hollenstein C (1993) J Phys D 26: Howling AA, Sansonnens L, Dorier J-L, Hollenstein C (1994) J Appl Phys 75: Howling AA, Courteille C, Dorier J-L, Sansonnens L, Hollenstein C (1996) Pure Appl Chem 68: Watanabe Y, Shiratani M, Fukuzawa T, Kawasaki H, Ueda Y, Singh S, Ohkura H (1996) J Vac Sci Technol, A 14:

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