A Sectional Model for Investigating Microcontamination in a Rotating Disk CVD Reactor

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1 Aerosol Science and Technology, 38: , 2004 Copyright c American Association for Aerosol Research ISSN: print / online DOI: / A Sectional Model for Investigating Microcontamination in a Rotating Disk CVD Reactor Z. Sun, 1 R. L. Axelbaum, 1 and R. W. Davis 2 1 Department of Mechanical Engineering, Washington University in St. Louis, St. Louis, Missouri, USA 2 Chemical Science and Technology Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland, USA A sectional model is employed to study the aerosol dynamics in a rotating disk CVD reactor. The aerosol model is implemented into a numerical code that employs detailed chemistry and transport and a one-dimensional similarity transform to simulate this system. The numerical results are compared to experimental data where conditions are such that particles are formed from gas-phase reactions. A thin particle layer is reproduced and the thickness is accurately determined. The location of the layer is predicted within 5%. The thin layer is a consequence of the particle stagnation region that develops due to convection towards the disk being opposed by a particle thermophoretic velocity away from the disk. The effect of sticking coefficient is examined, and it is shown that with a small sticking coefficient a smaller mean particle size and broader particle layer is obtained. It is also found that particle layers are driven farther away from the disk when the temperature of the disk is higher or the chamber pressure is lower, both of which result in higher thermophoretic velocities for the particles. The thickness of the particle layer is found to be insensitive to pressure. The sectional model also reveals that the particle size distributions are bimodal throughout most of the particle layer. This is a result of the broad regions for nucleation and surface growth as well as the opposing convective and thermophoretic velocities. INTRODUCTION With the rapid development of technology in the semiconductor industry, feature size decreases dramatically every year. Consequently, the allowable size of contaminant particle decreases as well, and submicron contaminant particles produced from the gas phase can cause failures. The mechanism behind the production and transport of microcontaminant particles is poorly understood. Thus, there is a need for a model to predict the Received 6 January 2004; accepted 4 October The authors gratefully acknowledge the National Institute of Standards and Technology for financial support. Present address of Z. Sun is LHP Software, Columbus, Indiana. Address correspondence to R. L. Axelbaum, Department of Mechanical Engineering, Washington University, St. Louis, MO 63130, USA. rla@me.wustl.edu amount and size of microcontaminant particles and determine how the operating parameters affect the production, transport, and growth of these particles. The rotating-disk chemical vapor deposition (CVD) reactor is commonly used in wafer processing. It contains a heated rotating disk and an axisymmetric flow that impinges on the disk. An important property of the rotating disk configuration is that under certain operating regimes the concentration and temperature gradients normal to the disk are equal everywhere on the disk. Thus, this configuration allows for highly uniform chemical vapor deposition (Hitchman et al. 1977; Pollard and Newman 1980). Due to its simple configuration and well-defined flow field, numerous numerical studies have been performed for both reacting and nonreacting flows (Coltrin et al. 1989; Breiland and Evans 1991; Jensen and Einset 1991; Ho et al. 1994). By employing a von Karman similarity transform, the equations describing the threedimensional spiral fluid motion can be reduced to a system of ordinary differential equations that only depend on the axial coordinate. This allows a more complex chemical mechanism to be studied without the need for expensive computational resources. Kremer et al. (2003) studied the behavior of microcontaminant particles in the rotating disk CVD reactor both numerically and experimentally. Elastic light-scattering measurements were performed to examine the particle size distribution along the center line of the CVD reactor. A moment method was employed to simulate particle dynamics, and a lognormal size distribution was assumed. Both the model and experiments showed a thin particle layer above the rotating disk. In the model, a sticking coefficient, α, was used to modify the rate at which surface growth occurs on particles, and the numerical results were found to be very sensitive to the sticking coefficient. The model was unable to predict the experimental results in terms of position and width of the thin particle layer without adjusting the sticking coefficient. A justification for adjusting α based on physical reasoning was not given. One possible reason for requiring adjustment of the sticking coefficient would be to correct for inaccuracies elsewhere in the 1161

2 1162 Z. SUN ET AL. model. For example, the model assumes a lognormal distribution for particle size throughout the flow field. While this assumption is commonly made to simplify computations, it may not be appropriate for a rotating disk CVD reactor, and no measurements of particle size distribution are available to validate this assumption. Thus, in this study, we have employed a more sophisticated aerosol model to understand further the particle dynamics in the CVD reactor and to determine if the lognormal assumption is responsible for the variation in sticking coefficient reported in Kremer et al. (2003). In this study, a sectional model is employed so that no assumptions are needed vis-à-vis particle size distribution other than the size distribution within a section. The other aspects of the simulation were kept the same as those in Kremer et al. (2003). Briefly, the Sandia rotation-disk CVD code, SPIN, was employed to simulate the gas-phase processes in a vertical, rotating-disk CVD reactor and a National Institute of Standards and Technology (NIST) gas-phase chemical kinetic mechanism was employed for the thermal decomposition of silane. The organization of this article is as follows: First the implementation of the sectional method is discussed. Next, results from the model are compared with experiment and the effects of process parameters are examined. Finally the particle size distributions at different axial positions along the reactor are discussed. GAS-PHASE MODEL The configuration of the reactor is shown in Figure 1. The model assumes an infinite-radius rotating disk with the reactant Figure 1. Sketch of the rotating-disk CVD reactor and inlet boundary conditions. (SiH 4 ) injected at a height x = L with an axial velocity of u L. The reactant decomposes as it approaches the high-temperature disk. Silicon monomers are formed during decomposition and grow through coagulation and condensation. A one-dimensional model for the rotating disk reactor, based on a modified form of the von Karman similarity transformation, has been developed at Sandia National Laboratories (Coltrin et al. 1991). This code, the SPIN code, incorporates detailed chemistry and transport in the gas phase and computes velocity, temperature, species concentrations, and surface growth rates in the reactor. Gas-phase and surface chemical kinetics are handled by CHEMKIN and SURFACE CHEMKIN subroutine packages, respectively. The transport properties are determined from the TRANSPORT subroutine package. The silane reaction mechanism developed at NIST (Kremer et al. 2003) is used. This mechanism includes the decomposition of silane and subsequent chemistry, which eventually leads to the formation of silicon particles. The mechanism includes reactions involving formation of silicon clusters, Si 2 to Si 6, and their reactions with intermediates, SiH 2 and Si 2 H 2. Note that the clustering mechanism is only a simplified representation of a complex mechanism. The clustering reactions terminate at Si 6, and any reaction that produces a cluster with more than 6 silicon atoms is considered to form a silicon particle rather than a cluster. The formation rate of silicon particles is used as the nucleation source term for the smallest section in the sectional model. The silicon particles grow through surface reaction, which is proportional to the surface area and the partial pressure of the condensable species. Silicon hydride (Si 2 H 2 )isconsidered the surface-growth species because it is reactive and present in relatively high concentration. AEROSOL MODEL The sectional model employed was developed by Gelbard et al. (1980). The essential idea is to divide the aerosol size spectrum into n sections and assume a size distribution function within each section. An integral quantity Q for each section is solved. Q can be total number (n-model), total volume (v-model), the volume square (v 2 -model), or some other appropriate quantity in each section. In this work, the choice of integral quantity is dictated by the need to model accurately the surface-growth process. When simulating condensation and surface-growth processes the sectional method suffers from numerical diffusion (Warren and Seinfeld 1985). Numerical diffusion can be reduced by increasing the number of sections (decreasing the section size). Landgrebe and Pratsinis (1989) removed the geometric approximation employed in earlier works (Gelbard et al. 1980), which limited the number of sections. Their approach will be used in this study as it allows a finer resolution of sections, which will greatly reduce numerical diffusion. Wu and Biswas (1998) studied numerical diffusion in a discrete-sectional model. It was found that the v-model is more accurate than either the n-model or the v 2 -model in predicting the integral properties of the size

3 SECTIONAL MODEL FOR CVD REACTOR 1163 distribution for systems with only condensation. For systems involving only coagulation, the v-model is more accurate for predicting total number concentration, total volume concentration, and geometric mean particle volume, and is reasonably accurate for predicting total volume square concentration. Thus, in this study, we choose the v-model and solve for total mass in each section. For the sectional analysis, it is assumed that the boundaries of the sections vary linearly on a log scale. Thus, if we have a total of n sections, with diameters ranging from d 0 to d n, respectively, the section boundaries are set by d l+1 = (d n /d 0 ) 1/n d l, [1] and section l will have boundaries m(l 1) and m(l)inparticle mass. The total mass of particles in section l, Q l (t), is defined as Q l (t) = ml m l 1 mn l (m, t)dm, l = 1,...,n, [2] where n l (m,t) represents the number density per unit mass of particles within section l. By assuming the distribution of n l, the total number density of particles in section l, N l, can be calculated from Q l.following the approach of Hall et al. (1997), we choose the form Thus, N l = ml n l (m, t) = m l 1 n l (m, t)dm = Q l m 2 ln(m / /m l 1 ). [3] ( Q l 1 1 ) = Q l, ln(m. /m l 1 ) m l 1 m l m l [4] 1 G l are intrasectional growth coefficients, 2 G l are intersectional growth coefficients, and S i is particle inception rate for the initial size class, and its value can be obtained from the formation rate of silicon particles. To obtain the Gs weassume a form for the growth rate due to surface deposition that is given by dm p dt = α Pρ pπd 2 p v m, [7] (2πmk B T ) 1/2 where m p is the particle mass, α is the sticking coefficient, P is the partial pressure of surface growth species, ρ p is the density of the particle, d p is the particle diameter, v m is the molecular volume of the deposited phase, m is the mass of the surface-growth species molecules (e.g., Si 2 H 2 ), k B is the Boltzmann constant, and T is the temperature (Friedlander 1977). The surface growth rate can be rewritten in the following form (Hall et al. 1997): dm p = G(t)m 2/3. [8] dt The intrasectional growth coefficient can be calculated analytically as 1 G l = 3 l G(t) ( 1 m 1/3 l 1 and the intersectional coefficients are ) 1 m 1/3, [9] l 2 G l = 1 G l m l+1 m l+1 m l. [10] where the average particle mass m l in section l is given by ml m m l = l 1 mn l (m, t)dm ml m l 1 n l (m, t)dm = ln(m l/m l 1 )(1/m l 1 1/m l ) 1. The dynamic balance equation for Q l can be represented as [5] dq l dt = 1 l 1 l 1 l β i, j,l Q i Q j Q l β i,l Q i, 2 i=1 j=1 i=1 1 M l 1 3 β l,l Ql Q l β i,l Q i + Q l β i,l Q i 2 i=l+1 i=1 + 1 G l Q l + 2 G l 1 Q l 1 2 G l Q l + δ 1 S i (t), (l = 1,...,n), [6] where β represents the sectional coagulation coefficient. Expressions for β can be found in Landgrebe and Pratsinis (1989). Figure 2. Temperature and species profiles for a typical run for a mixture of 0.25 molar fraction silane in helium, a rotational speed of 500 rpm, a flow rate of 19 slpm, a disk temperature of 1100 K, and a pressure of 200 torr.

4 1164 Z. SUN ET AL. TRANSPORT EQUATIONS FOR PARTICLES The governing equation for particles in section l is given by When the particle is in the transition or free-molecular regime the equation for v Tl is (Whitby et al. 1991) x (ρy l(v Tl + v Dl + v)) 1 r r (rρy lu) + Q l = 0, l = 1,...,n [11] where v Tl = F th B, [13] or d dx (ρy l(v Tl + v Dl )) ρu dy l dx + Q l = 0, l = 1,...,n, [12] where Y l is the sectional mass fraction and Q l = ρy l, ρ is the density of the gas, v is the convection velocity in the x direction and u in the radial direction, v Tl is the thermophoretic velocity and v Dl is the diffusion velocity of the particles in section l, and Q l was described in the section above. F th = Pλd 2 p T/ T, [14] P is the reactor pressure, and λ is the mean free path of the gas molecules. In Equation (13), B is the particle mobility and is defined as B = 8C C C D πρ g d 2 p (v p v ), [15] where C C is the Cunningham slip correction; C D is the drag coefficient for a sphere of diameter, d p ; and v p and v are particle Figure 3. Effect of number of sections on (a) mass concentration, (b) mean particle size, and (c) particle size distribution at 4.25 cm away from disk for the conditions of Figure 2.

5 SECTIONAL MODEL FOR CVD REACTOR 1165 velocity and gas velocity, respectively. An expression for C C that is valid for all size regimes (Friedlander 1977) is C C = 1 + Kn[ exp( 0.87/Kn)]. [16] For the particles in the Stokes regime (Re p < 0.1), C D = 24/ Re p, [17] where Re p = ρ g d p v p v /µ. The diffusion velocity is given by (Whitby et al. 1991) v dl = 1 D d(ρy l), [18] ρy l dz where D = k BT ( Kn ). [19] 3πµd p Particle growth by surface deposition will deplete the local concentration of the growth species k,sothere is a sink term, S g,in the gas-phase species equation: ρ Y k t where = (ρy kv k ) x S g = ρu Y k x + M k ω + S g = 0, [20] n ( ) dql. [21] dt cond l=1 Figure 4. Normalized scattering intensity profiles as a function of distance from disk for a rotational speed of (a) 500 rpm, (b) 750 rpm, and (c) 1000 rpm. The experimental results of Kremer et al. (2003) are shown with the solid lines and the numerical results are shown with dashed lines.

6 1166 Z. SUN ET AL. The species equations are solved simultaneously with the sectional equations. RESULTS AND DISCUSSION Numerical simulations were first performed for experimental conditions corresponding to injection of 19 slpm of a mixture of 0.25 mole percent silane in helium, a pressure of 26.7 kpa (200 torr) and a disk temperature of 1100 K (Kremer et al. 2003). Figure 2 shows the temperature profile and species profiles for a rotational speed of 500 rpm. As seen in Figure 2, the temperature is highest at the disk surface and decreases nearly linearly with distance from the disk. The concentration of Si 2 H 2 is relatively high and peaks at about 0.25 cm. For comparison, results are also shown without surface growth to reveal the level of depletion of Si 2 H 2 resulting from heterogeneous reaction. The depletion is significant for this case, and since growth rate is proportional to concentration of Si 2 H 2, the coupling between gas-phase chemistry and particle dynamics is strong. The Si 2 H 2 concentration has a broad distribution and surface growth occurs over a wide region. Mass concentration and mean particle size distributions are shown in Figure 3 for the operating condition of Figure 2. As seen in Figure 3a, most of the particle mass is concentrated in a narrow region to form a thin particle layer. Particles that are formed are transported by convection, thermophoresis, and to a lesser extent diffusion. The thermophoretic velocity, which varies with T/T, slowly decreases with distance from the disk, while the convective velocity rapidly decreases. Convection carries the particles towards the disk, while thermophoresis carries particles away from the disk. The convective velocity is large far away from the disk, and thermophoretic velocity is large in the region near the disk surface. Thus, the particles are transported to a particle stagnation plane where drag-induced convective velocity and thermophoretic velocity are of equal magnitude but opposite sign. The particles grow much larger in this region due to smaller net velocity and longer residence time, thus forming a particle layer. The effect of the number of sections is also examined in Figure 3. The number of sections per decade is varied from 6 to 24. The curves for spatial distribution of mass concentration are very similar (Figure 3a), even with only 6 sections per decade, indicating that the sectional method is highly effective at describing total mass distribution for this system. However, to obtain accurate results for mean particle size (Figure 3b), at least 12 sections per decade are needed. Figure 3c shows the particle size distribution at 4.25 cm away from the disk with 6, 12, and 24 sections per decade. Numerical diffusion is evident in the figure, as the size distribution with 6 sections per decade is broader than that with 24 sections per decade. Numerical diffusion is intrinsic in the sectional method and cannot be eliminated, but it can be reduced by using more sections. Note that the total mass concentration at this position is very close for all three cases, as seen in Figure 3a. To compare the measured and predicted position and thickness of the particle layer, the normalized sixth moment is plotted. Sixth moment is proportional to scattering intensity, and since the only known particle data are laser scattering measurements, the normalized sixth moment is directly compared with normalized experimental data. The results for different rotational speeds (500, 750, and 1000 rpm, respectively) are compared with the experiment results of Kremer et al. (2003) in Figure 4. For all cases, a sticking coefficient of 1.0 is used. As seen in Figure 4, the simulation reproduces the thickness of the particle layer observed in the experiments and predicts the location of the layer within 5%. The agreement is good and within the uncertainties in the model and experiment. Uncertainties arise from modeling the flow with the similarity solution, modeling thermophoresis, the kinetic model and experimental uncertainties. The profiles of convective and thermophoretic velocities at 500 and 1000 rpm are plotted in Figure 5. Since the convective velocities are negative, the absolute values are plotted. The intersection of the curves of convective and thermophoretic velocity (0.56 cm and 0.40 cm from the disk for 500 and 1000 rpm, respectively) corresponds to the position in Figure 4 where the particle layer resides. With an increase in rotational speed from 500 to 1000 rpm, the gradient in convective velocity near the disk is almost doubled. At the same time, the thermophoretic velocity increases by 33% due to the increase in temperature gradient. Since the convective velocity increases faster than the thermophoretic velocity, the particle layer is closer to the disk at higher rpm. Figure 5. Convective and thermophoretic velocity profile as a function of distance from disk for rotational speeds of 500 and 1000 rpm.

7 SECTIONAL MODEL FOR CVD REACTOR 1167 Figure 6. Effect of sticking coefficient. Run conditions are the same as those for Figure 2. The sticking coefficient, α, is not known and can be any value between 0 and 1. To study the effect of α, additional computations using α = 0.1 were made, and the results are presented in Figure 6. Other conditions are the same as those of Figure 2. As seen in Figure 6, the smaller sticking coefficient yields a broader distribution, because with a small sticking coefficient the growth rate due to condensation is smaller, and thus the final particle size is smaller. Smaller particles have larger diffusivities, which results in a broader spatial distribution of mass. To understand better the aerosol dynamics in this system, contours of particle number density are shown on a logarithmic scale in Figure 7. The contours show particle number density as a function of particle size and distance. Also shown in Figure 7 is the particle production rate, which indicates that particles are formed in a broad region and the production rate reaches a maximum on the disk side of the particle stagnation (P S) plane. Particles on both sides of the P S plane grow due to coagulation and surface growth while being transported towards the plane, which is located at 0.53 cm. In the vicinity of the P S plan and on both sides of it, the particles have a bimodal size distribution. This is readily evident at positions 0.25 and 0.75 cm. The smaller size mode consists mostly of monomers; the other mode consists of much larger particles ( nm) and is formed due to surface growth and particle coagulation. Despite a large particle production rate at 0.4 cm, there are few monomers in this region. This is due to the very high surfacegrowth rates, and to a lesser extent the coagulation rates, in this region. The number density of the monomers is determined by competition between production and consumption, and in this region the two are balanced, leading to a small number of monomers because the monomers are consumed as quickly as they are formed. On the disk side of the P S plane, the particles are, on average, much larger than those on the ambient side. This leads to the interesting result that most of the mass found Figure 7. Production rate and contour of particle number density (10 8 #/cm 3 )asafunction of particle size and distance for the run conditions of Figure 2.

8 1168 Z. SUN ET AL. Figure 8. Particle size distributions at different axial positions for the run conditions of Figure 2. in the particle layer actually originates from the disk side, not the ambient side of the P S plane. The mass distribution of particles at six different axial positions is shown in Figure 8. The solid curve corresponds to the position where the sixth moment is a maximum. The distances shown are relative to the location of the particle stagnation plane (i.e., x = x x S P, where x is the distance from the disk and x S P is the location of the P S plane) A negative sign indicates that the location is on the disk side. It can be seen that at x = and mm the mass distributions are very close to lognormal. The mass concentration and average particle size are larger for the locations on the disk side of the layer because the particles are transported from regions with higher Si 2 H 2 concentration and higher particle production rate. Closer to the P S plane, the distributions are clearly not lognormal. The distributions are skewed and the development of a bimodal size distribution can be seen at x = mm and mm. The reason for this is that particles from both sides are being transported to the particle stagnation plane, and each side has a different size distribution. The mixing of these particles produces a bimodal mass distribution. Since there is more mass coming from the disk side than the ambient side, the mass distribution on the ambient size is influenced more by this mixing and thus shows a distinctly bimodal distribution, while the distribution on the disk side is just skewed (see x = mm). Figures 7 and 8 indicate that throughout the rotating disk CVD reactor the size and mass distributions are clearly not lognormal and in many regions they are bimodal. This indicates that an assumption of lognormal distribution is inappropriate for modeling this process. The size distribution function in this region cannot be determined a prior, and thus models such as the sectional method are needed to understand and predict the behavior of this aerosol. In fact, by using the sectional model we have found that, unlike the case where the lognormal assumption was used, it is not necessary to force the sticking coefficient to vary in order to obtain agreement with experiment. Recall that when Kremer et al. applied the moment method to this system with identical chemistry, it was necessary to vary the sticking coefficient to get agreement with experiment. It is possible that the Figure 9. The effect of disk temperature: (a) normalized sixth moments for disk temperatures of 1100 K, 1200 K, and 1300 K, and (b) convective and thermophoretic velocities for disk temperatures of 1100 and 1300 K.

9 SECTIONAL MODEL FOR CVD REACTOR 1169 sticking coefficient was ultimately a fitting parameter to correct for inaccuracies in the model that were induced by the lognormal assumption. To assess the effect of disk temperature, additional cases with disk temperatures of 1200 K and 1300 K were simulated, and the results are shown in Figure 9 along with those of 1100 K. All other conditions are the same as those of Figure 2. As seen in Figure 9a, the particle layers are farther away from the disk at higher disk temperature, and the thickness of the particle layers is not strongly dependent on temperature. Convective and thermophoretic velocities at 1100 K and 1300 K are plotted in Figure 9b. This figure shows that the profile of convective velocity in the region close to disk is not sensitive to temperature, but the thermophoretic velocity increases with higher disk temperature due to higher temperature gradients. Thus, the particle layer is driven away from the disk at higher temperature. The effect of chamber pressure is also examined. The results with a pressure of 150, 200, and 320 torr are shown in Figure 10. Figure 10a reveals that the particle layer moves away from the disk as pressure is decreased. As seen in Figure 10b, the convective velocities at distances greater than 1.0 cm from the disk are affected by pressure, but in the region close to the disk they are Figure 10. The effect of pressure: (a) normalized sixth moments for pressures of 150, 200, and 320 torr, (b) convective and thermophoretic velocities at 150 and 320 torr, and (c) Si 2 H 2 partial pressures and temperatures at 150 and 320 torr.

10 1170 Z. SUN ET AL. nearly independent of pressure. The thermophoretic force is not sensitive to pressure, but the mobility of the particle is higher at lower pressure. Thus, the thermophoretic velocity increases with decreasing pressure, and the particle layer is found farther away from the disk. The thickness of the particle layers is not sensitive to pressure. As seen in Figure 10c, although the maximum value of the partial pressure of Si 2 H 2 is increased by a factor of 2 when total pressure is increased from 150 to 320 torr, at the positions where the particle layers reside and particles experience the greatest amount of growth the partial pressure of Si 2 H 2 at 320 torr is only slightly higher than at 150 torr. This is because at lower pressure the diffusivities are higher and thus the distributions are broader. The temperatures at the two positions are also very close. The calculated average particle size in the particle layer increases from about 40 nm at 150 torr to 50 nm at 320 torr, and thus changes in the particle are small. In Figure 10b, we see that at the particle layer the convective and thermophoretic velocities are higher at 150 torr than those at 320 torr. Thus, the particle layer is more compressed by convection and thermophoresis under lower pressure, but it can also expand due to the slightly smaller particle size and higher diffusivity. The two effects appear to cancel and thus we did not observe a significant change in thickness of the particle layer with pressure. CONCLUSIONS In this study, a sectional method was implemented into a CVD code in order to model and understand the aerosol dynamics of the rotating-disk CVD reactor. The model was compared to previous numerical results employing a moment method and to existing experimental data. The results are consistent with the experimental results and correctly predict the thickness and location of the thin particle layer. The results also indicate weaknesses of employing a moment method to study this system. The particle layer was found to be due to opposing directions of the convective and thermophoretic velocities, and the thickness of the layer was accurately predicted by the model. The location of the layer was predicted within 5% for three different rotational speeds. The effect of sticking coefficient was examined and it was shown that with a small sticking coefficient and thus lower condensation rate, a smaller mean particle size is obtained and a broader particle layer is produced, due to the higher diffusivity of the smaller particles. It was also found that when the temperature of the disk is higher or the pressure is lower, the thermophoretic velocity is higher and the particle layer is driven farther away from the disk, and results in a particle stagnation plane. The sectional model also revealed that the particle size distribution is not lognormal throughout most of the particle layer. In regions far from the particle stagnation plane, the distributions on both sides of the plane are bimodal due to rapid particle formation and rapid surface growth, leading to a fine mode and a coarse mode. Close to the particle stagnation plane the distribution is also bimodal, but here it is due to the mixing of particles from both sides. Since particles deposited on the surface of the disk can lead to failures of electronic devices, and failure is dependent on particle size, it is critical to know the exact size distribution of the particles in the gas phase. Thus, it is strongly advised that the sectional model be used for numerical studies of rotating-disk CVD reactors. REFERENCES Breiland, W. G., and Evans, G. H. (1991). Design and Verification of Nearly Ideal Flow and Heat Transfer in a Rotating Disk Chemical Vapor Deposition Reactor, J. Electrochem. Soc. 138: Coltrin, M. E., Kee, R. J., and Evans, G. H. (1989). Mathematical Model of the Fluid Mechanics and Gas-Phase Chemistry in a Rotating Disk Chemical Vapor Deposition Reactor, J. Electrochem. Soc. 136: Coltrin, M. E., Kee, R. J., Evans, G. H., Meeks, E., Rupley, F. M., and Grcar, J. F. (1991). SPIN (Version 3.83): A FORTRAN Program for Modeling One- Dimensional Rotating-Disk/Stagnation-Flow Chemical Vapor Depostion Reactors, Sandia National Laboratories, Report SAND Friedlander, S. K. (1977). Smoke, Dust and Haze, John Wiley & Sons, New York. Gelbard, F., Tambour, Y., and Seinfeld, J. H. (1980). Sectional Representations for Simulating Aerosol Dynamics, J. Colloid Interface Sci. 76: Hall, R. J., Smooke, M. D., and Colket, M. B. (1997). Predictions of Soot Dynamics in Opposed Jet Diffusion Flames, in Physical and Chemical Aspects of Combustion: A Tribute to Irvin Glassman, edited by R. F. Sawyer and F. L. Dryer, Combustion Science and Technology Book Series. Gordon and Breach, Amsterdam, Hitchman, M. L., Curtis, B. J., Brunner, H. R., and Eichenberger, V. (1977). The Study of Chemical Vapor Deposition Processes with Rotating Disks, in Physicochemical Hydrodynamics, Vol. 2, edited by D. B. Spalding. Advanced Publications, London, Ho, P., Coltrin, M. E., and Breiland, W. G. (1994). Laser-Induced Fluorescence Measurements and Kinetic Analysis of Si Atom Formation in a Rotating Disk Chemical Vapor Deposition Reactor. J. Phys. Chem. 98: Jensen, K. F., and Einset, E. O. (1991). Ann Rev. Fluid Mech. 23: Kremer, D. R., Davis, R. W., Moore, E. F., Maslar, J. E., Burgess, D. M., and Ehrman, S. H. (2003). An Investigation of Particle Dynamics in a Rotating Disk Chemical Vapor Deposition Reactor, J. Electrochem. Soc. 150(2):G127 G139. Landgrebe, J., and Pratsinis, S. (1989). Gas-Phase Manufacture of Particulates: Interplay of Chemical Reaction and Aerosol Coagulation in the Free- Molecular Regime, Ind. Eng. Chem. Res. 28: Pollard, R., and Newman, J. (1980). Silican Deposition on a Rotating Disk, J. Electrochem. Soc. 127: Warren, D. R., and Seinfeld, J. H. (1985). Simulation of Aerosol Size Distribution Evolution in Systems with Simultaneous Nucleation, Condensation, and Coagulation, Aerosol Sci. Technol. 4: Whitby, E. R., McMurry, P. H., Shankar, U., and Binkowski, F. S. (1991). Modal Aerosol Dynamics Modeling, EPA Report Wu, C. Y., and Biswas, P. (1998). Study of Numerical Diffusion in a Discrete- Sectional Model and Its Application to Aerosol Dynamics Simulation, Aerosol Sci. Technol. 29:

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