The Existence of Non-negatively Charged Dust Particles in Nonthermal Plasmas

Transcription

1 Plasma Chem Plasma Process (2017) 37: DOI /s ORIGINAL PAPER The Existence of Non-negatively Charged Dust Particles in Nonthermal Plasmas M. Mamunuru 1 R. Le Picard 2 Y. Sakiyama 1 S. L. Girshick 3 Received: 30 November 2016 / Accepted: 13 February 2017 / Published online: 22 February 2017 Springer Science+Business Media New York 2017 Abstract Particles in nonthermal dusty plasmas tend to charge negatively. However several effects can result in a significant fraction of the particles being neutral or positively charged, in which case they can deposit on surfaces that bound the plasma. Monte Carlo charging simulations were conducted to explore the effects of several parameters on the non-negative particle fraction of the stationary particle charge distribution. These simulations accounted for two effects not considered by the orbital motion limited theory of particle charging: single-particle charge limits, which were implemented by calculating electron tunneling currents from particles; and the increase in ion current to particles caused by charge-exchange collisions that occur within a particle s capture radius. The effects of several parameters were considered, including particle size, in the range 1 10 nm; pressure, ranging from 0.1 to 10 Torr; electron temperature, from 1 to 5 ev; positive ion temperature, from 300 to 700 K; plasma electronegativity, characterized in terms of n? /n e ranging from 1 to 1000; and particle material, either SiO 2 or Si. Within this parameter space, higher non-negative particle fractions are associated with smaller particle size, higher pressure, lower electron temperature, lower positive ion temperature, and higher electronegativity. Additionally, materials with lower electron affinities, such as SiO 2, have higher non-negative particle fractions than materials with lower electron affinities, such as Si. Keywords Dusty plasmas Particle charging Monte Carlo simulations Particle charge limits Electron tunneling Non-negative particles & S. L. Girshick slg@umn.edu Lam Research Corporation, SW Leveton Drive, Tualatin, OR 97062, USA Lam Research Corporation, 4650 Cushing Parkway, Fremont, CA 94538, USA Department of Mechanical Engineering, University of Minnesota, 111 Church St. S.E., Minneapolis, MN 55455, USA

2 702 Plasma Chem Plasma Process (2017) 37: Introduction In many nonthermal plasmas used for materials processing, charging of condensed-phase particles in the plasma is dominated by collisional attachment of electrons and ions. In that case, the much higher mobility of electrons compared to ions typically causes particles to charge negatively. For the same reason, walls that bound the plasma tend to charge to negative potential with respect to the plasma. The resulting electric potential profile then confines the negativelycharged particles in the plasma, preventing their diffusion to the walls. This phenomenon is beneficial in semiconductor processing, where it is typically desired to avoid deposition of dust particles onto the wafers being processed, as well as in nanoparticle synthesis, where particle losses to walls may limit process efficiency. Additionally, if all particles are charged negatively, then coagulation is considerably reduced by Coulomb repulsion, which may be beneficial for deliberate synthesis of monodisperse nanoparticles for applications. Conversely, phenomena that may cause a non-negligible fraction of particles in a plasma to be neutral or positively charged can have important consequences. As neutral particles are not electrostatically confined to the plasma they are free to diffuse out of the plasma, while positively-charged particles are actually accelerated by the sheath potential drop toward walls. The resulting deposition of these particles onto surfaces might result in unacceptable process contamination, particularly in the case of plasmas used for microelectronics fabrication. While the criterion for how small the non-negative charge fraction would have to be for it to be considered negligible is not well understood, it is reasonable to suppose that relatively small non-negative charge fractions might represent a potential contamination problem. Additionally, the existence of non-negative dust particles would promote coagulation, which broadens the particle size distribution and can lead to the formation of nonspherical agglomerates, effects that may be undesirable for controlled synthesis of nanoparticles [1], although the non-negative fraction would probably have to be higher, perhaps at least several percent, for this to become a significant concern. On the other hand, the existence of very small positively charged particles, with diameters around 2 nm or smaller, could potentially be exploited for controlled deposition of nanocrystalline films, as has been observed experimentally [2]. In this paper we examine several effects that may lead to the existence of a population of non-negative particles in processing plasmas, and conduct numerical simulations to obtain quantitative estimates of the non-negative particle fraction for various process conditions. As the existence of non-negative particle populations is likely to be most important for very small nanoparticles, we focus on particles with diameters in the range 1 10 nm. The regime considered involves nonthermal argon plasmas at pressures of Torr, and with electron temperatures of 1 5 ev. We consider the effects of several parameters, including pressure, electron temperature, ion temperature, plasma electronegativity, particle size and particle material. We do not here consider particle charging by UV-induced photodetachment or secondary electron emission. These phenomena, which can be important in plasmas with relatively high fluxes of UV and VUV photons, or with non-maxwellian electron energy distributions having overpopulated high-energy tails, respectively, can strongly shift the particle charge distribution toward positive charging [3 5], but are often relatively unimportant, compared to collisional attachment, under the conditions examined. Thus, insofar as our results show that the existence of non-negative particles can be important under some of the conditions considered, they indicate that this can be so even without these explicitly electron-emissive effects.

3 Plasma Chem Plasma Process (2017) 37: It should also be noted that the simulations reported here are not self-consistent, in that the plasma conditions are fixed and varied one at a time, regardless of the particle charge distribution. In reality the particle density and charge distribution would affect the plasma, parameters such as pressure would affect properties such as electron and ion temperatures, and so forth. Such self-consistent simulations of dusty plasmas have been reported (e.g. [5 7]) and can provide considerable insight. However such calculations can be quite computationally expensive, especially when performed for spatially non-uniform plasmas [7, 8], and it is also of interest to explore the independent effects of various fixed plasma parameters on particle charge distributions, based on computationally inexpensive numerical simulations, such as the Monte Carlo charging simulations reported here. While not self-consistent, this allows one to isolate the effects of each of the plasma parameters on the particle charge distribution, and thus can provide valuable insights as well as useful guidance to process designers. Numerical Model Overview A well-established theory known as the orbital motion limited (OML) theory is widely used to predict particle charging in plasmas by collisional attachment [9]. This theory assumes that dust particle radii are much smaller than the plasma Debye length, that the electrical sheath around each particle is collisionless, and that particles do not interact with each other. As particle charging is inherently a stochastic process, the charge of any particle fluctuates, and a population of particles exhibits a distribution of charge states. A steady state charge distribution exists when, on average, the positive and negative currents to a particle balance each other. OML theory allows one to quantitatively predict these stationary charge distributions [10, 11], which depend on parameters including the particle diameter d p, the electron and positive ion temperatures T e and T?, and the ratios of the number densities, n? /n e, and masses, m? /m e, of positive ions to electrons, negative ions being usually neglected in applications of the theory because of their much lower mobility compared to electrons. Although the ion and electron currents in OML theory explicitly depend on n? and n e, respectively, most estimates in the literature of particle charge based on OML theory assume that these two densities are equal (e.g. [10, 11]). However, in many cases the plasma may be electronegative [5], meaning that much of the negative charge is carried not by free electrons but by negative ions and/or by negatively-charged dust particles themselves [6, 12]. In such cases plasma quasi-neutrality requires the positive ion density to exceed the electron density, in some cases by a considerable factor. Therefore, as electrons have much higher mobility than ions, the average particle charge in electronegative plasmas can be expected to be less negative than in electropositive plasmas, potentially leading to an increase in the non-negative charge fraction. Additionally, two effects that may be important under many conditions are neglected by OML theory. First, depending on the pressure and the particle size, the assumption that the electrical sheath around each particle is collisionless may not be valid. Second, the theory neglects the fact that the number of electrons a dust particle can hold is limited [13]. For the solid particles and particle sizes considered here, the most important source of this charge limit is electron tunneling, which causes attached electrons to be emitted from the

4 704 Plasma Chem Plasma Process (2017) 37: particle, and is related to the particle s material-dependent electron affinity [14 16]. In this work we find that both of these effects the effect of pressure on ion currents, and the existence of charge limits can under some circumstances strongly increase the fraction of particles that are not charged negatively. Effect of Pressure Regarding the effect of pressure, the assumption in OML theory that particles are surrounded by a collisionless sheath may be valid at sufficiently low pressure but breaks down as pressure is increased. At high pressure one is in a fully collisional, hydrodynamic regime. At intermediate pressures, one is in a collision-enhanced regime where positive ions may experience a charge-exchange collision with a neutral that occurs within the particle s capture radius. While the original ion may have sufficient energy to escape the attractive potential well of the negative particle, the newly created ion may have less energy, increasing the probability that it will be collected by the particle. Hence the positive ion current to the particle increases, causing the particle to be less negatively charged than it would be otherwise. Khrapak et al. [17] proposed that the pressure-dependent transition between the collisionless and collision-enhanced regimes can be characterized in terms of a particle Knudsen number Kn R0 based on a capture radius R 0, defined such that the potential distribution around the particle has a minimum at Kn R0 1. Gatti and Kortshagen [18] extended this concept over a wide range of pressure and collisionality, encompassing the collisionless regime (OML), the collisional-enhanced regime (CE), and the hydrodynamic regime (HY). They expressed the positive ion current to a negativelycharged particle as a weighted function of the currents in these three regimes [18]: I þ ¼ P 0 I OML þ þ P 1 I CE þ þ P [ 1I HY þ ; where Iþ OML, ICE þ and IHY þ denote the positive ion currents (s-1 ) to a particle in each of the three regimes, and the terms P 0, P 1 and P [ 1 represent the corresponding probabilities that a positive ion experiences zero, one, or more than one collision within a particle s capture radius. The probabilities are given by and ð1þ P 0 ¼ exp 1 ; Kn R0 ð2þ P 1 ¼ 1 exp 1 Kn R0 Kn R0 ð3þ P [ 1 ¼ 1 P 0 P 1 : ð4þ The ion currents in Eq. (1) are given by Iþ OML ¼ pr 2 n þ v þ;th 1 e/ p ; ð5þ kt þ and Iþ CE ¼ par2 0 nþ v þ;th ; ð6þ

5 Plasma Chem Plasma Process (2017) 37: I HY þ ¼ 4pRn þl þ / p ; ð7þ where R is the ordinary particle radius, v?,th is the ion thermal velocity, e is the elementary charge, / p is the particle potential, k is the Boltzmann constant, a is a constant equal to 1.22 for a Maxwellian ion velocity distribution, and l? is the ion mobility. For Ar? ions in an argon plasma, l? = m 2 V -1 s -1 [19]. The capture radius depends on the ion mean free path, which itself depends on the ionneutral collision cross section. The total Ar? cross section in an argon plasma equals *10-14 cm 2 over a wide range of ion energies, up to *400 ev, which encompasses the conditions considered in our simulations. Then for purposes of estimating the capture radius, assuming a heavy species temperature of 300 K, the mean free path of Ar? ions in an argon plasma can be approximated by k þ ¼ 1 330p ; ð8þ where k? is in units of cm and the pressure p is in units of Torr [19]. Equations (1) (4) guarantee that the ion currents for the two limiting cases of the collisionless regime and the hydrodynamic regime are correctly expressed. Equation (6) gives the ion current for the intermediate regime where the incoming ion experiences exactly one charge-exchange collision inside the particle s capture radius, creating a new ion. In this case it is assumed that all such newly-created ions are eventually collected by the particle. As the probability increases that an incoming ion will experience multiple collisions, the ion current for the hydrodynamic regime is gradually phased in, as given by Eq. (1). In simulations reported here the electron current to particles (s -1 ) is always assumed to lie in the OML regime, and is therefore expressed as Ie OML ¼ pr 2 n e v e;th 1 e/ p : ð9þ kt e Anion currents to particles are neglected, because of the much higher mobility of electrons compared to ions. We checked this assumption by including anion currents in some of the simulations, assuming that the total negative charge carried by nanoparticles is negligible so that the anion densities have their highest possible value for given values of n? /n e. The contribution of the anion current to charging was found to be quite small even at n? /n e = 1000, and would be even smaller if one accounted for the fact that much of the negative charge in a dusty plasma would be carried by nanoparticles. Particle Charge Limits The maximum number of electrons that can coexist on a single particle is limited. Various expressions for charge limits are reviewed in [13]. In simulations of particle charging reported in that work, the charge limit was either taken as an arbitrary parameter or was based on an expression for the effective electron affinity of a particle [20]. As recently pointed out, for the small solid nanoparticles of interest here the main source of such charge limits is the tunneling of electrons attached to the particle through the potential barrier posed by the particle s negative charge [15]. The resulting emission of electrons from the particle, or tunneling current, depends on the particle s charge and on its electron affinity, which in turn depends on the electron affinity of the bulk (flat) material and on the

6 706 Plasma Chem Plasma Process (2017) 37: particle s size. Here, instead of imposing charge limits we calculate the tunneling current, which effectively imposes charge limits. In Ref. [15] it is assumed that attached electrons bounce around a particle with an average velocity that is based on their being in thermal equilibrium at the particle temperature. The electron tunneling current from a particle is then estimated as rffiffiffiffiffiffiffiffiffi 2kT p 1 I e;tunnel ¼ jqj T; m e 2R ð10þ where q is the particle charge, T p is particle temperature, and T is the tunneling probability, given by T exp 4p Z rt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m e ½/ ðþ / r ðr t Þ dr : ð11þ h R Here h is the Planck constant, / ðþis r the electric potential of an electron at distance r from the center of the particle, and r t is the location where the particle s electron affinity equals zero, which is the location to which the electron must tunnel to escape the particle. At this location the electron s potential energy is given by / ðr t Þ ¼ / ðþ A r n ; where A n, the electron affinity of the neutral particle, is given by ð12þ A n ¼ A pe 0 R : ð13þ Here A? is the electron affinity of the bulk (flat) material, and e 0 is the permittivity of free space. The exponential form of Eq. (11) results in a tunneling current that increases by many orders of magnitude for very small increases in the magnitude of the negative particle charge. Since charge is an integer quantity, this effectively imposes a sharp charge limit. From Eqs. (10) (13), one finds that this limit is related to the bulk electron affinity of which the particle is composed, with lower values of bulk electron affinity implying more severe charge limits. Thus in the present work we compare two related materials with quite different bulk electron affinities: Si, with A? = 4.05 ev, and SiO 2, with A? = 1.0 ev. Numerical Method The Monte Carlo charging model used in this work is based on Ref. [21], a model originally developed to calculate particle charge distributions due to electron and ion attachment in low-pressure plasmas. We extend that work by considering electron tunneling as well as the effect of gas pressure on the ion current to a particle. The model calculates the transient evolution of particle charge, starting from a neutral state at t = 0s. The time for the next collision charging event between the particle and a positive ion or electron is chosen randomly based on the currents given by Eqs. (1) and (9). A random number r 1 that lies between 0 and 1 is generated, and is compared to the ratio r 2 ¼ I þ : ð14þ I þ þ I e If r 1 [ r 2, a positive ion attaches to the particle, and the particle charge is incremented by one at the time t ¼ t þ Iþ 1. Otherwise an electron attaches to the particle and the particle e 2

7 Plasma Chem Plasma Process (2017) 37: charge is decreased by one at the time t ¼ t þ Ie 1. The next time for an electron to be emitted by tunneling from the particle is based on the tunneling current given by Eq. (10). Only the ratio n? /n e not the value of n? itself affects the stationary particle charge distribution. However, in all of the simulations presented here we assume that n? = 10 9 cm -3. The reason for this is that the ion and electron densities affect the characteristic particle charging times and hence the time required to reach steady state. Based on our choice of n?, the typical collision/emission time is on the order of a microsecond. The time step of the simulation was set to 10-9 s, much smaller than the time associated with the greatest charging frequency. In the simulations presented, the number of discrete charging events per simulation ranged from *2 to5910 6, providing statistically meaningful data for the stationary charge distribution. Similarly, under the assumption that particles are electrically noninteracting, the stationary particle charge distribution is not directly affected by the particle number density. The charge distribution can, however, be affected indirectly, because the charge accumulated on dust particles affects the ratio n? /n e. Results and Discussion We first consider a base case that involves SiO 2 particles with given values of particle diameter d p, pressure p, cation density n?, cation-to-electron density ratio n? /n e, electron temperature T e and cation temperature T?. We then examine the effects of varying each of these parameters, with the exception of n?, as well as the effect of particle material, where we compare results for SiO 2 and Si. Note that we characterize the plasma electronegativity, usually defined as n /n e, n being the anion density, in terms of n? /n e. This is because, as noted above, anions make only a small contribution to particle charging under the conditions considered, and are neglected in the simulations. Base Case As shown in Table 1, our base case involves SiO 2 particles with the following conditions: d p = 2 nm, p = 1 Torr, n? = 10 9 cm -3, n? /n e = 10, T e = 2 ev, T? = 300 K, T p = 300 K. Figure 1 shows the calculated stationary particle charge distribution with and without considering the effect of charge limits, i.e. with and without including the electron tunneling current, which effectively establishes the limit. Without accounting for tunneling, charge fractions exceeding 10-4 exist out to a charge of -5. However, accounting for tunneling, the particle charge limit in this case is seen to equal -2. As a result, the neutral and positive particle fractions in this case both increase by about 70% compared to the Table 1 Base case conditions Particle material SiO 2 Particle diameter 2 nm Pressure 1 Torr n? /n e 10 Electron temperature 2 ev Ion temperature 300 K Ion density 10 9 cm -3

8 708 Plasma Chem Plasma Process (2017) 37: Fig. 1 Effect of particle charge limits (i.e., electron tunneling), on particle charge distribution for base case conditions (Table 1) simulation without tunneling. We find that accounting for tunneling makes a much larger difference under some of the other conditions examined, consistent with the results presented in [13]. As tunneling is a real phenomenon that limits particle charge at the small sizes considered here, it is included in all of the other simulations discussed below. Effect of Particle Size With all other conditions the same as in the base case, we ran simulations for particle diameters ranging from 1 to 10 nm. The results for the fractions of neutral and positive particles are shown in Fig. 2. As can be seen, the fractions of both neutral and positive particles are quite strong functions of particle size. Approximately 50% of 1-nm particles are predicted to be neutral, while at 10 nm the neutral fraction equals only a few ppm. As neutral nanoparticles are not trapped in the plasma and can diffuse to walls, this implies Fig. 2 Effect of particle size on fraction of particles that are neutral or positively-charged. Refer to Table 1 for other conditions

9 Plasma Chem Plasma Process (2017) 37: that particle deposition will be dominated by the smallest nanoparticles, assuming that a range of particle sizes exists in the plasma. Moreover this behavior will be amplified by the strong size-dependence of the particle diffusion coefficient D, which, from kinetic theory for particles in the free molecule regime, scales with particle diameter d p approximately as D / dp 2 [22]. Hence this analysis suggests that quite significant fluxes to walls of very small nanoparticles can be expected in plasmas in which nanoparticles nucleate. Indeed this has been reported in experimental studies of growth by plasma-enhanced chemical vapor deposition of polymorphous silicon films, which consist of very small nanoparticles embedded in an amorphous silicon matrix [23]. Once nanoparticles that nucleate in a plasma grow beyond a certain size in the base case, about 6 or 7 nm, but it depends on the plasma conditions and the particle material the neutral fraction becomes negligibly small, but charge fluctuations for the smallest particles, combined with their high diffusivity, can be expected to result in significant fluxes of these particles to walls and film substrates. The positive charge fraction for all sizes seen in Fig. 2 lies about two orders of magnitude below the neutral fraction. However this does not necessarily imply that the positive charge fraction is unimportant, as positive particles are accelerated to walls by the wall sheath potential, reaching a drift velocity that is governed by ambipolar diffusion in the sheath electric field, thereby increasing their flux to walls compared to neutral particles with the same number density [24]. Effect of Pressure With all other conditions the same as in the base case, we conducted simulations with pressures ranging from 0.1 to 10 Torr, and for particle diameters of 2, 5 and 10 nm. Figure 3 shows the results in terms of the non-negative particle fraction, i.e. the sum of the neutral and positive particle fractions. As noted above, the non-negative fraction is very close to the neutral fraction. For given particle size, increasing pressure over the range Torr is seen to strongly increase the non-negative particle fraction. Since the ion and electron densities here are fixed, and do not scale with pressure, this behavior can be attributed to the fact that Fig. 3 Effect of pressure on non-negative particle fraction, for various particle diameters. Refer to Table 1 for other conditions

10 710 Plasma Chem Plasma Process (2017) 37: increasing pressure shifts the positive ion current from the collisionless toward the collision-enhanced regime discussed above. For 2-nm-diameter particles the non-negative particle fraction ranges from about 0.4% at 0.1 Torr to over 20% at 10 Torr. For 5-nm particles the non-negative fraction, compared to 2-nm particles, ranges from about three orders of magnitude lower at 0.1 Torr to one order of magnitude lower at 10 Torr. Effect of Electron Temperature Higher electron temperatures lead to higher electron currents to particles and higher average negative particle charge, and hence to lower fractions of non-negative particles. This is shown in Fig. 4, for electron temperatures ranging from 1 to 5 ev, and for pressures ranging from 0.1 to 10 Torr. Reducing T e from 2 to 1 ev approximately doubles the nonnegative charge fraction over the pressure range considered. Effect of Positive Ion Temperature As with increasing T e, increasing T? causes the non-negative charge fraction to decrease, as seen in Fig. 5. This perhaps counterintuitive behavior was previously noted by Matsoukas and Russell [10]. As T? increases, the ion has enough energy to escape the attractive field of a negatively charged particle, causing the cross section for ion capture to decrease, resulting in a higher average negative charge and thus a lower non-negative charge fraction. In Fig. 5, an increase in ion temperature from 300 to 500 K causes the non-negative particle fraction to decrease by a factor ranging from *5 at 0.1 Torr to *2at 10 Torr. Note that in all these simulations we assumed a fixed particle temperature of 300 K. However, insofar as ion temperatures are often close to the neutral gas temperature, one might expect that particle temperatures would tend to be close to the ion temperature. From Eq. (10), higher particle temperatures would result in higher tunneling currents, and hence more severe charge limits, potentially leading to higher non-negative charge fractions. Moreover, especially for the smallest nanoparticles considered here, a recent body of work Fig. 4 Effect of electron temperature on non-negative particle fraction, over a range of pressure. Refer to Table 1 for other conditions

11 Plasma Chem Plasma Process (2017) 37: Fig. 5 Effect of positive ion temperature on non-negative particle fraction, over a range of pressure. Refer to Table 1 for other conditions indicates that particle temperatures may exceed the gas temperature by up to several hundred K [7, 25, 26]. Effect of Plasma Electronegativity Because of the accumulation of negative charge on dust particles, dusty plasmas are inherently electronegative, with studies finding positive ion densities exceeding the electron density by factors ranging from several [5, 12] to several hundred [8]. Additionally, aside from the depletion of electrons due to attachment on nanoparticles, in electronegative gases anions can be relatively abundant carriers of negative charge, which also causes the positive ion density to exceed the electron density. Figure 6 shows the effect of electronegativity, as characterized here by n? /n e, on the fractions of particles that are either neutral or positive, with all other conditions the same as in the base case. As expected, increasing electronegativity strongly increases the neutral and positive charge fractions, with the neutral charge fraction exceeding 10% for values of n? /n e greater than about 30. The positive charge fraction rises even more steeply with increasing electronegativity. Thus, as n? /n e increases, the ratio of positive to neutral particles increases, ranging from a few tenths of a percent at n? /n e = 1to*25% at n? / n e = The effect of electronegativity on the non-negative particle fraction depends on the pressure, as seen in Fig. 7. At very high values of n? /n e, *10 3, the electronegativity dominates, as the non-negative particle fraction is quite high regardless of the pressure, exceeding 50% for pressures greater than 1 Torr, and falling only slightly at pressures below 1 Torr. At more modest values of n? /n e, in the 1 10 range, both higher electronegativity and higher pressure are positively correlated with increasing values of the non-negative particle fraction. Effect of Particle Material The material of which the particle is composed affects the particle charge limit, via the electron tunneling current, which depends on the material s bulk electron affinity. As noted

12 712 Plasma Chem Plasma Process (2017) 37: Fig. 6 Effect of plasma electronegativity, characterized by n? /n e, on fractions of neutral and positive particles. Refer to Table 1 for other conditions Fig. 7 Effect of n? /n e on nonnegative particle fraction, over a range of pressure. Refer to Table 1 for other conditions above in the section on particle charge limits, Si has a bulk electron affinity approximately four times higher than that of SiO 2. Hence Si nanoparticles can be expected to have less severe charge limits than SiO 2, potentially leading to smaller non-negative charge fractions for Si than for SiO 2. Figure 8 shows a comparison of the stationary particle charge distribution calculated for 4-nm-diameter particles composed of either Si or SiO 2. All other conditions are the same as in the base case. Because of the difference in electron tunneling, the effective charge limit for the Si particles equals 6, while it equals 2 for the SiO 2 particles. As a result, the neutral charge fraction for the SiO 2 particles is seen to exceed that for the Si particles by more than one order of magnitude. The non-negative particle fraction comparing Si with SiO 2 for a range of particle sizes is shown in Fig. 9. For 1- and 2-nm particles, the non-negative particle fraction is about twice as high for SiO 2 as for Si. For particles that are around 4 nm and larger, the nonnegative SiO 2 particle fraction is more than an order of magnitude larger than for Si. This

13 Plasma Chem Plasma Process (2017) 37: Fig. 8 Effect of particle material on charge distributions of 4-nmdiameter particles. Refer to Table 1 for other conditions Fig. 9 Effect of particle material on non-negative particle fraction over a range of particle sizes. Refer to Table 1 for other conditions behavior is a direct consequence of the quite different charge limits for each material at each size. Based on inspection of the calculated particle charge distributions, for SiO 2 the charge limits for particles of 1, 2, 4, and 6 nm diameter are given by -1, -2, -2, and -3, respectively; for Si the corresponding charge limits are given by -2, -4, -6, and -9. The fact that the charge limit for SiO 2 particles is the same, -2, for both 2- and 4-nm particles is the reason that the curve in Fig. 9 appears non-smooth in that region. Essentially, this behavior is related to the integer nature of charge. Summary and Conclusions In this work we conducted Monte Carlo charging simulations to calculate stationary particle distributions for a variety of conditions in dusty plasmas. We focused specifically on the fraction of particles that are not charged negatively, as these particles are not electrostatically

14 714 Plasma Chem Plasma Process (2017) 37: confined in the plasma and can freely diffuse (in the case of neutral particles) or be electrostatically attracted (in the case of positive particles) to surfaces bounding the plasma. The simulations accounted for two deviations from orbital motion limited theory the existence of single-particle charge limits due to electron tunneling, and the effect of pressure on the positive ion current to particles of given size due to charge-exchange collisions that occur with the particle s capture radius. The effect of several parameters on the non-negative particle fraction was considered, including particle size, in the range 1 10 nm; pressure, ranging from 0.1 to 10 Torr; electron temperature, from 1 to 5 ev; positive ion temperature, from 300 to 700 K; plasma electronegativity, characterized in terms of n? /n e ranging from 1 to 1000; and particle material, either SiO 2 or Si. Within the parameter space examined, higher non-negative particle fractions are associated with smaller particle size, higher pressure, lower electron temperature, lower positive ion temperature, and higher electronegativity. Additionally, materials with lower electron affinities, such as SiO 2, have higher non-negative particle fractions than materials with lower electron affinities, such as Si. This is caused by the more severe charge limits that result from lower electron affinities. In conclusion, we find that under many conditions that are pertinent to microelectronics fabrication and other processing plasmas the fraction of nanoparticles that are not negatively charged is high enough to imply significant fluxes of these particles to walls and material substrates, which may be either a serious contamination concern or a feature that could deliberately be exploited for synthesis of nanostructured materials. It should also be noted that these simulations assumed that particle charging is dominated by collisional attachment of electrons and ions, and did not consider explicitly electron-emissive effects such as UV and VUV photodetachment and secondary electron emission. These latter effects, if significant, would increase the non-negative particle fraction even further. Acknowledgements This work was partially supported by the Lam Research Foundation, the U.S. National Science Foundation (CHE ), and the U.S. Dept. of Energy Office of Fusion Energy Science (DE- SC ). References 1. Kortshagen UR, Sankaran RM, Pereira RN, Girshick SL, Wu JJ, Aydil ES (2016) Nonthermal plasma synthesis of nanocrystals: fundamental principles, materials, and applications. Chem Rev 116: Chaabane N, Suendo V, Vach H, Roca i Cabarrocas P (2006) Soft landing of silicon nanocrystals in plasma enhanced chemical vapor deposition. Appl Phys Lett 88: Rosenberg M, Mendis DA (1996) Use of UV to reduce particle trapping in process plasmas. IEEE Trans Plasma Sci 24: Kortshagen U, Bhandarkar U (1999) Modeling of particulate coagulation in low pressure plasmas. Phys Rev E 60: Denysenko IB, Ostrikov K, Xu S, Yu MY, Diong CH (2003) Nanopowder management and control of plasma parameters in electronegative SiH 4 plasmas. J Appl Phys 94: Agarwal P, Girshick SL (2014) Numerical modeling of the spatiotemporal behavior of an rf argonsilane plasma with dust particle nucleation and growth. Plasma Chem Plasma Process 34: Le Picard R, Markosyan AH, Porter DH, Girshick SL, Kushner MJ (2016) Synthesis of silicon nanoparticles in nonthermal capacitively-coupled flowing plasmas: processes and transport. Plasma Chem Plasma Process 36: Agarwal P, Girshick SL (2012) Sectional modeling of nanoparticle size and charge distributions in dusty plasmas. Plasma Sources Sci Technol 21:055023

15 Plasma Chem Plasma Process (2017) 37: Allen JE (1992) Probe theory the orbital motion approach. Phys Scr 45: Matsoukas T, Russell M (1995) Particle charging in low-pressure plasmas. J Appl Phys 77: Matsoukas T, Russell M (1997) Fokker-Planck description of particle charging in ionized gases. Phys Rev E 55: Bilik N, Anthony R, Merritt BA, Aydil ES, Kortshagen UR (2015) Langmuir probe measurements of electron energy probability functions in dusty plasmas. J Phys D 48: Le Picard R, Girshick SL (2016) The effect of single-particle charge limits on charge distributions in dusty plasmas. J Phys D 49: Gallagher A (2000) A model of particle growth in silane discharges. Phys Rev E 62: Heijmans LCJ, Wetering FMJH, Nijdam S (2016) Comment on The effect of single-particle charge limits on charge distributions in dusty plasmas. J Phys D 49: Le Picard R, Girshick SL (2016) Reply to Comment on The effect of single-particle charge limits on charge distributions in dusty plasmas. J Phys D 49: Khrapak SA, Ratynskaia SV, Zobnin AV, Usachev AD, Yaroshenko VV, Thoma MH, Kretschmer M, Hofner H, Morfill GE, Petrov OF, Fortov VE (2005) Particle charge in the bulk of gas discharges. Phys Rev E 72: Gatti M, Kortshagen U (2008) Analytical model of particle charging in plasmas over a wide range of collisionality. Phys Rev E 78: Lieberman M, Lichtenberg A (2005) Principles of plasma discharges and materials processing, 2nd edn. Wiley, New York 20. Bouchoule A (ed) (1999) Dusty plasmas: physics, chemistry and technological impacts in plasma processing. Wiley, New York 21. Goree J (1992) Ion trapping by a charged dust grain in a plasma. Phys Rev Lett 69: Epstein PS (1924) Phys Rev 23: Roca i Cabarrocas P, Gay P, Hadjadj A (1996) Experimental evidence for nanoparticle deposition in continuous argon-silane plasmas: effects of silicon nanoparticles on film properties. J Vac Sci Technol A 14: Larriba-Andaluz C, Girshick SL (2017) Controlled fluxes of silicon nanoparticles to a substrate in pulsed radio-frequency argon silane plasmas. Plasma Chem Plasma Process 37: Mangolini L, Kortshagen U (2009) Selective nanoparticle heating: another form of nonequilibrium in dusty plasmas. Phys Rev E 79: Kramer NJ, Anthony RJ, Mamunuru M, Aydil ES, Kortshagen UR (2014) Plasma-induced crystallization of silicon nanoparticles. J Phys D 47:075202