Numerical optimization of dielectric properties to achieve process uniformity in capacitively coupled plasma reactors

This paper presents the results of our numerical analysis to optimize the dielectric properties to achieve process uniformity in the thin film deposition process using capacitively coupled plasma. The difference in the plasma density distribution was analyzed by changing the wafer material from silicon to quartz (or Teflon). Similarly, aluminum was compared with aluminum nitride as the electrode material, and the sidewall material was varied from quartz to a perfect dielectric to study the effect on the plasma characteristics. A two-dimensional self-consistent fluid model was used to analyze the spatial distribution of the plasma parameters. In terms of the process conditions, the gas pressure was set to 400 Pa, the input power was fixed to 100 W, and a radio frequency of 13.56 MHz was used. SiH4/Ar was used as the gas mixture, and these conditions were used as input for numerical simulations of the deposition state of the hydrogenated amorphous silicon layer. The radial spatial distribution of plasma parameters was confirmed to be modified by dielectric elements with low dielectric constants regardless of the type of element. Despite the thin wafer thickness, the use of a wafer with low permittivity weakens the electric field near the electrode edge due to the stronger surface charging effect. Additionally, by changing the material of the sidewall to a perfect dielectric, a more uniform distribution of plasma could be obtained. This is achieved as the peak values of the plasma parameters are located away from the wafer edge. Interestingly, the case in which half of the sidewall was specified as comprising a perfect dielectric and the other half quartz had a more uniform distribution than the case in which the sidewalls consisted entirely of a perfect dielectric.


Introduction
Capacitively coupled plasmas (CCPs) are increasingly being used for a variety of critical thin-film processes with satisfactory uniformity and efficiency [1,2].Owing to its functional usefulness, CCP is often adopted in the plasma enhanced chemical vapor deposition (PECVD) process for the production of state-of-the-art microchips despite various technical difficulties in the design of CCP reactors [3][4][5].The recent requirement that the deposition unevenness of the film deposited on the wafer be less than 1%-3% has prompted growing interest in solving the technical problems surrounding the implementation of a uniform deposition process [6,7].Therefore, ensuring the uniformity of the process results makes it necessary to control the spatial distribution of plasma parameters such as the ionization rate, excitation rate, excitation transfer rate, and electron density [8,9].
The plasma uniformity is affected by various factors such as the type and physical properties of the source gas, the driving source, and the shape of the reactor.For example, the influence of standing waves generated by the radio frequency (RF) operating conditions such as the use of very high frequency can introduce non-uniformity in the spatial distribution of the plasma density near the center of the wafer.To address this, Yang and Kushner proposed differential conductivity electrodes using multiple layers of dielectric materials [10].Xia et al proposed a segmented non-uniform dielectric module design for uniformity control [11].As an alternative, Michna and Ellingboe proposed the use of a multi-tiled system where adjacent tiles are out of phase to ensure good uniformity [12].Additionally, according to Kim and Lee, a non-uniform temperature distribution of the source gas supplied from the showerhead electrode can result in a non-uniform spatial distribution of the plasma density near the center of the wafer [13].
More specifically, the non-uniformity of the spatial distribution of the plasma density observed near the edge of the wafer is often affected by the electrostatic effect resulting from the concentration of the electric field at the edge of the electrode [14,15].Due to the shape of a circular wafer, the number of chips produced for each radius increases as the edge of the wafer is approached.Many researchers have therefore attempted to develop a method for suppressing the edge effect to obtain uniform plasma to ultimately improve the productivity.Our previous studies showed that the spatial distribution of the plasma density around the wafer edge can be effectively controlled by changing the electrode spacing, the source gas supply method, and the temperature of the sidewall of the reactor [13,[16][17][18].Xu et al investigated the effect of controlling the phase shift of the RF bias with respect to the RF source and confirmed that it is possible to control the spatial distribution of the plasma density as well as that of the ion flux [19].
As mentioned above, the spatial distribution of the plasma density may be concentrated near the electrode edge because of electrostatic effects.The use of dielectric components is also effective for solving and alleviating this problem.The accumulation of ions and electrons on the dielectric surface is a consequence of the 'surface charging effect.'The relative permittivity and geometrical conditions (e.g. the effective dielectric thickness) of dielectric elements are responsible for the accumulation of charges, which affects the charge density.As a result, the potential of the dielectric surface is affected.In turn, the surface charging effect gives rise to the formation of an opposite electric field with a significant effect on the plasma behavior, especially the plasma uniformity.These effects are observed to become relatively more prominent, particularly near the electrode edge.
Various previous studies have focused on the role of dielectric elements in the spatial distribution of the plasma density near the electrode edge.Dalvie et al reported that the nonuniformity of the ionization rate near the reactor sidewall was mitigated by the use of a dielectric sidewall [20].Bi et al used a numerical method to assess the effect of enclosing the bottom electrode with a dielectric material and reported that this can suppress the edge effect [21].Kim and Manousiouthakis reported the effect of a dielectric material on the distribution of the ion flux in non-conductive wafers [22].By utilizing particle-in-cell/Monte Carlo collision (PIC/MCC) simulation, Liu et al analyzed the plasma non-uniformity in a symmetric CCP reactor with a dielectric sidewall [23].
Due to the physical importance, an in-depth investigation of the effect of the dielectric factors on the plasma parameters in an actual deposition reactor using a gas mixture is required.This motivated our study, in which we show that it is possible to change the spatial distribution of the plasma parameters by using CCP with a SiH 4 /Ar mixture and by varying the properties of dielectric components installed in the reactor.We discuss the results in terms of the change in the uniformity of the deposition rate profile measured on the wafer by changing the spatial distribution of this plasma parameter.The deposition rate of the amorphous silicon layer is used to determine the deposition rate profile.A numerical analysis was conducted on the basis of a 2D fluid model.
In fact, many studies concerned with various aspects of process optimization using SiH 4 /Ar plasma discharges have been reported.For example, some studies have attempted to establish the conditions that enable rapid deposition of an amorphous silicon layer [24][25][26].In addition, various studies have concentrated on the generation of silicon nanoparticles [27][28][29][30].However, it was difficult to find numerical research involving an in-depth analysis of the spatial distribution of the plasma parameters in SiH 4 /Ar CCPs.
First, we consider the change in the spatial distribution of the plasma parameters according to the electrical connection state of the grounded bottom electrode assembly and the material from which the wafer placed on top of the bottom electrode is constructed.The dependence of the plasma parameters on the combination of the dielectric parts constituting the sidewall that surrounds the space in which the plasma is formed inside the reactor along the circumferential direction is investigated.Although previous studies discussed the change in the spatial distribution of the plasma parameters by installing a dielectric part constructed from one material on the sidewall, the effect of combining dielectric parts consisting of different materials has not yet been reported.The results of our study are therefore of high importance.

Numerical model
CCP with a SiH 4 /Ar mixture was simulated based on a 2D fluid model.The details of the fluid model used in this study were described in our previous papers [13,[16][17][18]31].For comparison with the experimental values, non-isothermal conditions were used in the analysis of the gas temperature and density, and the transport phenomena of various neutral and ionic species were precisely considered.In particular, since the analysis of the density distribution of Ar * and the density distribution of Si 2 H 5 and Si 3 H 7 radicals is important for a gas pressure of 100 Pa or more, convective transfer and the diffusion of multiple gas species were considered.For brevity of notation, the corresponding governing equations are briefly described here.
The conservation of ion density and conservation of ion momentum are considered as follows: where n i is the ion density, t is the time, v i is the ion velocity, S i is the ion source term, m i is the ion mass, p i is the ion pressure, e is the elementary charge, E is the electric field, and M i is the momentum transfer term.
The conservation of the electron density is considered by the following equations: where n e , t, J e , S e , D e , µ e , and ϕ represent electron density, the time, the electron flux, the source of the electrons, the electron diffusion coefficient, the electron mobility, and the electric potential, respectively.The conservation of the electron energy is considered by the following equation: where ε, v e , and S ε represent the electron energy, the electron velocity, and the electron energy source term, respectively.
To obtain the electric potential ϕ, the Poisson equation is considered as follows: where e, ε 0 , n p , and n q represent the elementary charge, the permittivity of free space, the positive ion density, and negative ion density, respectively.
The overall mass continuity and the conservation of momentum equations are considered as follows: where ρ, v, p g , τ, f, µ, and I are the overall density, the overall velocity, the gas pressure, the viscous stress tensor, the net force per unit volume, the molecular viscosity, and the unit matrix, respectively.The density distributions of neutral species are considered as follows: where Y i , j i C , j i T , G i , L i , and M i represent the mass fraction of species i, the flux of species i resulting from the concentration gradients, the thermal diffusion flux of species i resulting from the temperature gradients, the rate at which species i is generated, and the rate at which species i is depleted, and the molar mass of species i, respectively.
The dielectric surface is considered as follows: where σ s is the surface charge, q i is the ion species charge unit, J i,n is the flux of the ion species i, J e,n is the flux of the electron, and ε d is the permittivity of the insulator, respectively, note that subscript d represents the insulator.

Numerical methods
The governing equations related to the conservation of the density of neutral species were discretized using the finite volume method.The semi-implicit method for pressure-linked equations (SIMPLE) algorithm was used to consider the conservation of the momentum of gases.
Using the Scharfetter-Gummel exponential scheme, the governing equations of electron density conservation and electron energy conservation were discretized.We calculated the electron impact reaction rates and electron transport parameters by solving the 0D Boltzmann equation for the electron energy distribution function.The solution of the 0D Boltzmann equation was first performed as a preprocessing step, and a lookup table was created as a result of the calculation.This generated lookup table was used during the 2D CCP deposition simulation to reduce the computation time.In this study, we considered various neutral and charged species that can be observed in CCP discharges of a SiH 4 /Ar mixture.The species we considered include radical species, positive and negative ionic species, and also excited molecules generated from SiH 4 and Ar.Surface phenomena were analyzed with the aid of a sticking model.Thus, the deposition rate can be calculated by using the equations below: where subscript i is the species index, Φ i is the deposition rate of species i, α i is the stoichiometry of species i, s i is the sticking coefficient of species i, γ i is the recombination probability of species i, n i,s is the density of species i at the surface, k B is the Boltzmann constant, T g is the gas temperature, M i is the mass of species i, n s is the surface density of the film, N A is the Avogadro number, M is the molar mass of the film, and ρ is the mass density of the film, respectively.Note that the loss probability β is defined as β = s + γ.The β, s, and γ values of the considered radicals had to be carefully selected from among those reported in the literature, as listed in table 1 [8,32,33].
To ensure the completeness of the simulation results, the chemical reaction model of the SiH 4 /Ar mixture was constructed as detailed as possible: a total of more than 200 gas-phase chemical reactions were considered.Since we used Ar as the dilution gas, we used the detailed Ar chemistry from our previous work on torr-regime Ar discharges [34].A detailed set of chemical reactions of SiH 4 was constructed from a subset of our SiH 4 /He detailed chemistry.Electron impact reactions that were considered include ionization, dissociation, and dissociation attachments.For brevity of notation, table A1 in the appendix lists only selected gaseous reactions that are considered to play an important role in the formation of deposition precursors under conditions where the gas pressure exceeds 100 Pa.

Validation
As mentioned above, because Ar was used as the dilution gas, the plasma chemistry of Ar needed to be considered in detail.This was accomplished by verifying the plasma chemistry of Ar by conducting a quantitative comparison with the measurement results obtained with the Langmuir probe in the previous study.In addition, the validity of the SiH 4 /Ar reaction model was confirmed as follows.The deposition rate profile of the amorphous silicon layer deposited using CCP simulation was compared with the experimental values.This comparison is presented in figure 12 of section 3.4.

Results
The shape and structural characteristics of the cylindrical CCP reactor used in this study are depicted in figure 1(a).The sidewall of the reactor is composed of a ring-shaped dielectric, of which the inner surface serves as the sidewall.Because it is a CCP reactor, it is equipped with a showerhead electrode and a bottom electrode.As the voltage of 13.56 MHz RF is applied to the top electrode, this electrode serves as the powered electrode.On the other hand, the bottom electrode is grounded, and on top of this grounded bottom electrode, a silicon (Si, lossy dielectric; dielectric constant, k = 11.7)wafer, Teflon (non-conductive dielectric; dielectric constant, k = 2) wafer, or quartz (SiO 2 , non-conductive dielectric; dielectric constant, k = 3.9) wafer is placed.
In the semiconductor industry, wafers are often heated to adjust the properties of thin films, and the temperature of the walls of the reactor is varied to lower the density of contaminants generated inside the reactor or to control the density distribution of the source gas.For this reason, the wall of the reactor is specified as a non-isothermal boundary.When the lower electrode is heated to increase the temperature of the wafer, the heated lower electrode is referred to as a heater.To achieve process uniformity, the effect of electrical discontinuity that is inevitably formed near the wafer edge must be minimized, and to minimize this, the wafer is surrounded by a lossy dielectric or dielectric ring (a so-called edge ring).
Since the shape of the reactor is axisymmetric, it is considered in the r-z plane, where r is the radial coordinate and z is the axial coordinate.The left boundary of the computational domain is the axis of symmetry, and the right boundary is the sidewall.It is assumed that the SiH 4 /Ar mixture (50 sccm SiH 4 and 5000 sccm Ar) is uniformly supplied through multiple showerhead holes applied to the showerhead electrode.The gas pressure is 400 Pa.The amplitude of the potential (V rf ) is applied to the showerhead until the supplied power (i.e. the power delivered to the charged species) equals the value specified by the preset input parameter of 100 W (or 200 W).The sidewall temperature (T sw ), heater temperature (T h ), and showerhead temperature (T sh ) are set to 423, 673, and 473 K, respectively.The input conditions of the 18 cases considered in this study are listed in table 2.
Figures 1(b) and (c) show the time-averaged Ar * density profile and SiH 3 density profile for Case 1.Because Ar is used as the diluent gas, the generation of SiH 3 by Ar * is as important as the generation of SiH 3 by electron impact.The reaction coefficient for SiH 3 generation is considered as: where k r is the reaction rate coefficient [35].

The base case
Figure 2 shows the time-averaged results for Case 1 (base case with the aluminum bottom electrode, see table 2).In figures 2(a)-(c), spatial profiles of the time-averaged horizontal electric field, vertical ion (Ar + ) flux, and plasma potential are depicted for Case 1, respectively.Figures 2(d) and (e) show the excitation rate (Ar * production: Ar + e − → Ar * + e − ) and excitation transfer rate (Si 2 H 4 production: The figure shows that the distribution is characterized by the concentration of large values near the edge of the bottom electrode due to the electrostatic effect.This effect refers to the effect whereby electrons are heated as a result of the spatial gradient of the plasma potential that exists at the corner at the edge of the electrode, as noted in equation (5).Therefore, the radial spatial distributions are non-uniform near the bottom electrode edge.These reaction rate distributions have their maxima near the corner where the axial sheath along the right side of the bottom electrode and the radial sheath along the upper side of the bottom electrode overlap.Furthermore, the reaction rates are non-uniformly distributed near the top and bottom of the sidewall of the reactor.In particular, the distributions of reaction rates are concentrated near the upper and lower edges.The reason for focusing on these two specific areas is that the higher power deposition in the sheaths increased the reaction efficiency.Furthermore, as mentioned above, similar to the way in which the electrostatic effect increases the reaction efficiency at the edge of the bottom electrode, it also effectuates a highly efficient reaction near the edge of the top electrode.
The time-averaged spatial distribution of the electron density N e (m −3 ) is shown in figure 2(f), which confirms the offaxis maximum (N eo ) of the electron density to have reached 1.5 × 10 16 (m −3 ).On the other hand, the electron density distribution near the reactor wall has a small value of approximately zero compared to N eo .The spatial variation of the Si 4 H 9 density (m −3 ) in Case 1 is confirmed by the results in figure 2(g).In fact, in the previous study involving the SiH 4 /He mixture, understanding the spatial distribution of SiH 3 was the most important factor explaining the formation of the hydrogenated amorphous silicon layer.However, the importance of Si 4 H 9 with respect to determining the uniformity also increased for the process conditions being considered in this study (i.e. the use of a SiH 4 /Ar mixture and gas pressure in excess of 100 Pa).The common point found in figures 2(f) and (g) is that their peak values are observed near the edge of the lower electrode.

Effects of the dielectric wafer and dielectric bottom electrode
In Case 1, the bottom electrode was set as aluminum, and both the top and lateral surfaces of the bottom electrode were set to be electrically grounded.In addition, the effect of the wafer (i.e. the effect of a thin dielectric layer placed on the bottom electrode) was not considered.However, in the CCP reactor for PECVD used in the semiconductor industry, a wafer is placed on the lower electrode, which often consists of aluminum or a ceramic material.Typically, this electrode is fabricated from the ceramic materials aluminum oxide (Al 2 O 3 ) or aluminum nitride (AlN).The reason for the selection of these materials is the strong deformation resistance to heat and low strain rates of ceramic electrodes in situations in which wafers must be exposed to high temperatures depending on the process conditions.In addition, the top and lateral surfaces of the bottom electrode, which are made of ceramic material, serve as dielectric surfaces that promote the uniformity of plasma distribution.Therefore, the AlN (dielectric constant, k = 8.9) bottom electrode is used in Cases 2, 3, 4, and 5 to understand the advantage of a dielectric surface.In Case 2 and Case 3, the effects of the wafer material are investigated, and in Case 4 and Case 5, the effects of the structure of the AlN lower electrode are studied.As mentioned above, to investigate the effects observed on the wafer surface according to the change in the electrical properties of the bottom electrode, the wafer was placed on top of the bottom electrode and in Cases 2, 3, 4, and 5, 'the bottom electrode assembly (the wafer and the bottom electrode)' was changed as follows.In Case 2, a 'metal wafer' was placed on top of the AlN bottom electrode, whereas in Case 3, the AlN bottom electrode was retained and the metal wafer was replaced with a Si (lossy dielectric, k ≈ 12) wafer.This is to check whether the difference in electrical properties between the metal and Si affects the plasma density distribution despite the thin thickness of the Si wafer.In Case 4 and Case 5, a Teflon wafer (k = 2) was used.Importantly, in Cases 2, 3, and 4, the bottom of the wafer was grounded, whereas in Case 5, the bottom of the AlN electrode was grounded.The different settings selected for Case 4 and Case 5 were expected to confirm the differences in the spatial distribution of their plasma parameters.This difference would arise because of the difference in the 'effective dielectric thickness' between these two cases.As confirmed in table 2, the remaining input conditions of Case 5 were the same as those of Case 2, Case 3, and Case 4.   As a dielectric wafer is used, for a wafer of which the impedance exceeds that of Case 2, the effective sizes of the two electrodes are more similar to each other.Therefore, the degree of asymmetry of the spatial distributions of the excitation rate and excitation transfer rate in the inter-electrode region is reduced.In particular, in Case 5, the spatial distributions of the excitation rate and excitation transfer rate in the bulk plasma region do not differ significantly from those of the previous cases.However, the values of both of these rates decrease near the lower sheath (indicated in figure 5(d)) formed near the lower electrode.In addition, as the 'effective dielectric thickness' increased, the 'effective dielectric constant' significantly decreased.As a result, the locations of the maximum values of the excitation rate and excitation transfer rate are observed to move away from the wafer edge.According to equation ( 12), the dielectric thickness and dielectric constant simultaneously affect the surface charging effect, thus an increase in the 'effective dielectric thickness' corresponds to a decrease in the 'effective dielectric constant.'In the area marked with a red circle in figure 5(d), the excitation transfer rate near the bottom electrode edge of Case 5 was observed to be approximately 12% lower than that of Case 2. Additionally, the values were observed to be distributed in a relatively radially outward direction compared to Case 3. As illustrated in figure 4, these combined effects produce a uniform ion flux in the radial direction near the electrode edge without significantly affecting the distribution of the bulk plasma.
Figure 6 shows the spatial distribution of the time-averaged electron density (N e ) for Cases 2, 3, 4 and 5.As shown in figure 6, the spatial distribution of the electron density is also uniform at r < 140 mm in the radial direction.As described above, in Case 2, the top surface of the bottom electrode is grounded and the lateral surface is dielectric.On the other hand, in Case 5, both the top and side surfaces of the bottom electrode were set to dielectric and have the thickest effective dielectric thickness.The difference in the spatial distributions of the plasma parameters between Case 2 and Case 5 is therefore noticeably large.This difference is attributed to the mitigating tendency of the electron density profile to reach a maximum due to the effect of the accumulation of charged species in the region near the electrode edge.This improves the uniformity of the radical density and flux profiles in the radial direction, similar to what has already been shown in the previous figures.
The distribution of the excitation rates in Case 5 tends to be concentrated near the upper and lower sidewalls; thus, the vertical symmetry of this distribution is more obvious than the corresponding distributions in the other cases.Moreover, because the surface charging effect on the dielectric surface is more pronounced at the wafer edge in Case 5 (dielectric wafer   and thick AlN electrode), the dielectric induced electric field attenuates the axial electric field in the plasma more effectively.Note that even though the same amount of RF power (100 W) was applied to Cases 2, 3, 4, and 5, the peak value of the electron density was the smallest in Case 5.In Case 5, the ion flux near the edge of the bottom electrode decreased because this electrode was the thickest, as depicted in figure 4.
However, compared to Cases 2, 3, and 4, the density of the bulk plasma in the r < 0.14 m region increases in Case 5. Therefore, in Case 5, the electron density is the most uniformly distributed in the radial direction.Additionally, this change results in a relative decrease in the axial radical flux near the wafer edge.
Figure 7 shows the spatial distribution of the time-averaged Si 4 H 9 density for Cases 2, 3, 4 and 5.As shown in figure 7, the spatial distribution of the Si 4 H 9 density is also uniform at r < 140 mm in the radial direction.Similar to what is observed in figure 6, the tendency of the maximum value of the Si 4 H 9 density profile of Case 5 to be formed due to the effect of the accumulation of charged species in the electrode edge region is alleviated.Therefore, the radial density and flux profiles in Case 5 are more uniform than those of the other three cases (Cases 2-4).

Effects of optimization of the dielectric sidewall
In section changes in the electrical properties of 'the bottom electrode assembly' were shown to induce changes in the spatial distributions of the plasma parameters.Despite the effectiveness of changes in the electrical properties of the bottom electrode and changes in the wafer material, as observed for the results of Case 5, a noticeable change was possible only when the effective dielectric thickness of the bottom electrode was sufficient.Therefore, in this section, we analyze the effect of the sidewall to determine whether the dielectric components could be utilized more efficiently.This sidewall refers to the component that surrounds the space in which the plasma is formed inside the reactor in the circumferential direction (i.e. in a tangential direction along the circumference).The dependence of the plasma parameters on the combination of dielectric components that make up this sidewall is discussed.
The changes in the spatial distributions of the plasma parameters by using a combination of dielectric components with different dielectric constants to fabricate the component that plays the role of an electrode, such as the showerhead, were previously studied [10][11][12].However, studies of the effect of using a combination of dielectric parts on the sidewall do not seem to have been reported.In addition, considering that a very large number of small holes must be machined in the showerhead, manufacturing the part for experimental implementation would be expected to be difficult.In contrast, the spatial change of the dielectric constant would be expected to be relatively easy to implement in the case of the sidewall.
In the cases considered above, a thickness of 45 mm was consistently used for the quartz sidewall, but in Case 6 and Case 7, respectively, our intention was to analyze the effects of the thickness and material of the sidewall.In Case 6, a quartz sidewall with a thickness of 30 mm was used and, in Case 7, the sidewall material was replaced with a perfect dielectric to investigate the effect thereof.A SiO 2 wafer was used in both Case 6 and Case 7. Additionally, AlN was used as the bottom electrode in both of these two cases, and the bottom surface of the wafer was grounded in both cases.The other process conditions in Case 6 and Case 7 were the same as those applied in Cases 1-5.
The time-averaged results for Case 6 (30-mm-thick quartz sidewall) and Case 7 (perfect dielectric sidewall) are provided in figures 8(a) and (b).As shown in figure 8(a), the excitation rate of Case 6 is enhanced near the edge of both the top and bottom electrodes, and thus is distributed non-uniformly in the radial direction.The use of a perfect dielectric as the material of the sidewall in the CCP reactor increases the impedance compared to that of the sidewall in the previous cases.This means that the effective sizes of the upper and lower electrodes become similar to each other.Therefore, as observed in figure 8(b) (Case 7), the axial asymmetry of the excitation rate is effectively reduced, and the radial uniformity of the excitation rate is also improved efficiently.The spatial distributions of the excitation rates in the bulk plasma region do not change significantly compared to those of the previous cases, but a significant decrease in the value is observed near the surface of the bottom electrode.The maxima of the excitation rates are detected at locations slightly radially outward from the reactor corner where the axial and radial sheaths overlap.Therefore, the spatial distribution of the excitation rate in Case 7 is more localized near the upper and lower sidewalls, and has a more vertically symmetrical distribution than that of Case 6.The excitation rates plotted along the vertical direction at r = 0.155 m are compared in figure 9.
By reducing the maximum values of the spatial distributions of the excitation rate and excitation transfer rate in the inter-electrode region, the distributions of the corresponding values at the electrode edge were homogenized.This results in more uniform radial densities and flux profiles.
In Case 7, the perfect dielectric sidewall effectively improves the plasma parameter distribution.Based on this fact, we introduced ways to further improve the uniformity of this distribution.As described above, although the change in the spatial distribution of plasma parameters by installing dielectric parts on the sidewall was previously studied, the effect of combining dielectric elements under different conditions does not seem to have been reported.Therefore, in this study, the sidewall was divided into upper and lower parts.In Case 8, the material of the upper part of the sidewall was set to perfect dielectric, whereas that of the lower part was set to SiO 2 .Conversely, in Case 9, the material of the upper part of the sidewall was set to SiO 2 and that of the lower part was set to perfect dielectric.As observed in figure 8(c), interestingly, compared to the result of Case 7 where the perfect dielectric was applied to the entire sidewall, the distribution of plasma parameters in Case 8 (perfect dielectric applied only to the upper sidewall) was more uniform.However, as observed in figure 8(d), in Case 9, the distributions were not as uniform as in Case 8.Although these results are not shown here, the effects of changes in the sidewall conditions on the distribution of the excitation rate were similar to those observed for the excitation transfer rate.
Figures 10(a) and (b) show the spatial distributions of the electron density for Cases 6 and 7, respectively.In particular, a more uniform gradient of the contour is observed near the bottom electrode in Case 7. As observed in figure 10(c), the maximum value observed at the electrode edge in Case 8 is  Figure 11 shows the spatial distributions of the timeaveraged Si 4 H 9 density (m −3 ) for Cases 6, 7, 8, and 9. Noticeable differences between the distributions shown by the contours are not observed in the bulk region.Considering that the spatial distribution of the plasma parameters is already sufficiently uniform, these results suggested that the density distributions of Si 4 H 9 do not differ significantly.However, a visible difference exists in the distribution near the wafer, with a relatively larger difference observed near the electrode edge.The thickness of the boundary layer in Case 6 observed on the surface of the electrode edge is thinner than the thickness of the boundary layer observed in Case 8.The boundary layer of the Si 4 H 9 density contour refers to the line at which the density has a value of 1.0 × 10 18 m −3 .These lines are drawn on the contours in figures 11(a) and (c) for Case 6 and Case 8, respectively.The surface flux of radicals is proportional to the density of radicals and inversely proportional to the distance between the electrode surface and the area with higher density.Therefore, a greater distance from the boundary layer means a lower deposition rate.Based on these results, the deposition rate profile of Case 8 was expected to be the most uniform.Furthermore, an important feature evident from the density distribution of Si 4 H 9 in Case 8 is that the position of the maximum value is relatively far from the wafer edge.

Deposition rate profiles
This section describes the effect of dielectric elements and different combinations of these elements on the deposition rate profile.Figure 12(a) shows the deposition rate profile for Case 1 and figure 12(b) presents the corresponding profiles for Case 2, Case 3, Case 4, and Case 5. Figure 12(c) shows the deposition rate profiles for Case 6, Case 7, Case 8, and Case 9.In addition, the experimental data of Case 3 and Case 10 are overlapped with the profiles of Case 3 and Case 10 in figure 12(d).The deposition rate profiles are normalized to the corresponding mean values to conveniently compare the characteristics of the spatial distribution.
What we need to pay attention to is that in figure 12(d), in Case 10, the simulation results and the experimental results match well, while in Case 3, there is a difference between the simulation results and the experimental results in the r < 0.1 m area.The experimental results for Case 3 showed that the deposition rate decreases by about 0.4% in the outward direction in the region from r = 0 to r = 0.1 m.However, in Case 10, where the RF power was 200 W, the deposition rate was observed to increase outward in the same area.This slight decrease is considered to occur because the surface heating due to the ion flux in Case 3 is lower than that in Case 10.Because of the less efficient heat transfer, the region from r = 0 m to r = 0.1 m may have lower film density and a lower sticking coefficient.And we assumed that the sticking coefficient and film density were uniformly distributed in the radial direction.
As shown in figures 12(a) and (b), the deposition rate profiles of Cases 1, 2, 3, 4 and 5 have similar shapes with a faster increase near the edge of the electrode.However, as shown in figure 12(c), in the deposition rate profiles of Cases 7, 8, and 9, the uniformity is improved by integrating dielectric elements such as the SiO 2 wafer, a bottom electrode made of AlN, and a perfect dielectric sidewall.Note that Case 8 has the most uniform deposition rate profile (less than 2.0% nonuniformity).Non-uniformity is calculated by dividing the difference between the maximum and minimum values of the profile by the average value and multiplying the divided value by 100.

Discussion
We found the introduction of dielectric elements to be useful for optimizing the spatial distribution and peak value positions of the excitation rates and excitation transfer rates in processes using Ar as the diluent gas.The generation of the maximum excitation rate is governed by the presence of the largest electric field at the edge of the electrode as a result of electrostatic effects.Based on the electron heating rate defined by −e J e •E (where J e is the electron flux), an increase in the field strength increases both the electron density and the electron temperature.Therefore, increasing both the electron density and the electron temperature enhances the excitation rate.
In this study, since Ar * contributes greatly to the dissociation of source gases such as SiH 4 or Si 2 H 6 , it is important to control the location of both the maximum electron density (N eo ) and the maximum excitation rate.The formation of the maximum electron density (N eo ) in the CCP reactor considered in this study is dominated by the effect of the RF breakdown between the bottom electrode and the lower sidewall, the reason why the excitation rate has the highest value in this region.The movement of electrons generated by step ionization is strongly influenced by the enhanced electric field observed near the electrode edge, resulting in the radial drift of the electron density accumulated near the electrode edge.Importantly, despite the use of a perfect dielectric on part of the sidewall, this local accumulation of electron density is most clearly observed in Case 9.
As stated previously, our process was based on the use of highly reactive radicals to form an amorphous silicon layer.The experimentally observed adsorption probabilities of radicals we applied were obtained from the literature [8,32,33].The sticking coefficient was varied to determine the influence thereof on the deposition rate profile.Although it varied over a wide range, the uniformity of the deposition rate profile did not change significantly, as shown in figure 13.The reason for the insensitivity to these changes is the high density of radicals generated under the given pressure conditions, and at such high densities, depletion reactions due to gas phase reactions actively occur.
Secondary electron emission was taken into account by setting the secondary electron emission coefficient (SEEC) to 0.1 in this study [36].Even increasing the SEEC of the sidewall composed of dielectric material to 0.8 did not significantly affect the deposition rate profiles, as depicted in figure 14.
The floating bottom (axial) sheath induced by the nonconductive wafer serves to partially reject the RF current without significantly contributing to the overall electron heating.Therefore, because it has the thickest effective dielectric thickness, Case 5 has the lowest excitation rate among Cases 1, 2, 3, 4, and 5 near the wafer edge.Among Cases 1, 2, 3, 4, and 5, Case 5, with the bottom electrode assembly with the thickest effective dielectric thickness, excessive current passes through a narrow area adjacent to the radial sheath more efficiently.In fact, directly varying the thickness of the electrode would be a more appropriate way to observe the effect of the thickness of the AlN electrode rather than changing the connection state inside the electrode.However, our reactor has a diffusion space between the edge of the bottom electrode and the sidewall, as noted in figure 1(a).This diffusion space was created for the purpose of controlling the process uniformity.Owing to the characteristics of this structure, adjusting the thickness of the bottom electrode would affect the diffusion area.
Therefore, directly changing the thickness of the bottom electrode was not considered to be appropriate to observe the effect on the plasma distribution.Since the dielectric thickness and dielectric constant simultaneously affect the surface charging effect according to equation (12), the material of the bottom electrode was changed to quartz (Case 17) and Teflon (Case 18) to observe the effect on the plasma distribution, as depicted in figure 15.As shown in the figure, the lower the dielectric constant of the material used for the bottom electrode, the more uniform the distribution of plasma parameters such as the excitation rate and electron density become.
In Case 7, where the sidewall material was changed to perfect dielectric, the excitation rate distribution was observed to be more concentrated in a narrow area between the edge of the bottom electrode and the bottom sidewall than in the previous  cases.This indicates greater (vertical) penetration of the electron density as well as the ion current, both of which lead to a decrease in the axial (vertical) asymmetry of the plasma parameter distribution.This led us to contemplate ways to further enhance the effect of improving the distribution of the plasma parameters of the perfect dielectric sidewall, which was confirmed in Case 7. Consequently, in Case 8, the material of the upper and lower parts of the sidewall was set to perfect dielectric and SiO 2 , respectively.In Case 9, the placement of these two materials was reversed, with SiO 2 and the perfect dielectric specified as the upper and lower parts of the sidewall, respectively.
Interestingly, compared to the results of Cases 7, where the perfect dielectric was applied to the entire sidewall, the distribution of the plasma parameters in Case 8, where the perfect dielectric was applied only to the upper part, was more uniform.The uniformity improved by setting the upper part of the sidewall in Case 8 to a perfect dielectric and the lower part to quartz.This can be understood by comparing the ion (Ar + ) flux distributions of Case 7 and Case 8, as depicted in figure 16.In Case 8, the use of quartz for the lower part of the sidewall resulted in the relative concentration of the horizontal ion (Ar + ) flux near the lower part of the sidewall.As a result, the vertical ion (Ar + ) flux formed near the edge of the bottom electrode is relatively reduced in Case 8. Therefore, the uniformity is improved.
In summary, the improvement in the uniformity of the plasma density distribution owing to the variation of the bottom electrode is attributable to the decrease in N eo rather than the change in the position of N eo .On the other hand, the improvement in the uniformity of the plasma density distribution by changing the properties of the sidewall is ascribed to the position of N eo away from the wafer edge rather than the reduction of this parameter.

Conclusions
In this paper, we used a two-dimensional self-consistent fluid model to discuss the effects of the spatial distributions of the plasma parameters of dielectric elements that can be installed in a CCP reactor.The dielectric elements we considered are conductive (metal, Si)/non-conductive (SiO 2 , Teflon) wafers, non-conductive sidewalls, and ceramic (AlN) electrodes.
In Case 1, because the aluminum lower electrode was used without a non-conductive wafer, the spatial distributions of the plasma parameters were severely localized.As a result, the formation of the maximum values of these parameters was concentrated near the edge of the lower electrode (or near the wafer edge).AlN was therefore used as the bottom electrode in subsequent cases.In Cases 2 and 3, metal and Si were selected as the wafer material.In Cases 4, 5, 6, 7, 8, and 9, dielectric wafers were used, and in Case 10, a Si wafer was used.Despite the thin wafer thickness, the electric field near the edge of the dielectric wafer is weakened due to the stronger surface charging effect.
In Case 6, the plasma density distribution was non-uniform because the 30-mm-thick quartz was applied to the sidewall.However, in Case 7, the plasma density distribution was uniform because the perfect dielectric was applied to the entire sidewall.In Cases 8 and 9, different dielectric properties were specified for the upper and lower parts of the sidewall, respectively.Interestingly, compared to the results of Case 7, where the perfect dielectric was applied to the entire sidewall, the distribution of plasma parameters in Case 8, in which the perfect dielectric was applied only to the upper part, became more uniform.
In conclusion, the optimal combination of dielectric elements can achieve plasma uniformity and thus deposition uniformity.

Figure 2 .
Figure 2. Contour plots for the spatial profiles of the time-averaged (a) horizontal electric field (V m −1 ), (b) vertical ion (Ar + ) flux (mA cm −2 ), (c) plasma potential (V), (d) excitation rate (kmol m −3 s −1 ), (e) excitation transfer rate (kmol m −3 s −1 ), (f) electron density (m −3 ), and (g) Si 4 H 9 density (m −3 ) for Case 1.In (d), the axial and radial sheaths are indicated with the black dotted line.Due to the local concentration of the electric field, lines were added to facilitate observation of the electric field distribution.

Figures 3
Figures 3 and 4 show the time-averaged results for Cases 2, 3, 4, and 5. Figure 3 compares the spatial distribution of the time-averaged excitation (Ar * production) rate under four different bottom electrode conditions.The spatial distributions of the time-averaged excitation rates for Cases 2 and 3 are shown in figures 3(a) and (b), respectively.Similar to figure 2(d), the radial distributions of the excitation rates in these two contours are uniform for r < 140 mm.As a result of electrostatic effects, a common feature is observed: the excitation rates are concentrated near the edges of the top and bottom electrodes.The spatial distributions of the time-averaged excitation rates for Cases 4 and 5 are shown in figures 3(c) and (d).Excitation rates similar to those observed in figures 3(a) and (b) are found near the edges of the top and bottom electrodes.Importantly, in Case 2, the highest excitation rate was observed near the edge of the top electrode, but in Case 5, the highest excitation rate was observed near the edge of the bottom electrode.As stated above, in figure 4, spatial profiles of the timeaveraged plasma parameters such as the horizontal electric field, vertical ion flux, and plasma potential are depicted for Cases 2, 3, 4, and 5.As depicted in figure 4, the horizontal electric field was relatively less enhanced on the side of the lower electrode in Cases 2 and 3 compared to Case 1.These reductions are attributed to the dielectric surface of the sidewall of the AlN lower electrode.As a result, the excitation rate on the side of the lower electrode decreased in Case 2 and Case 3.A comparison of the spatial distributions of the excitation rates in Case 1 and Cases 2 & 3 therefore reveals no significant difference in the inter-electrode region (r ⩽ 150 mm); instead, a decrease in the maximum excitation rate near the edge of the bottom electrode is observed.Note that, even though the lateral surface of the bottom electrode was specified as a dielectric surface, the excitation rates of Cases 2 & 3 show a more uneven distribution than the excitation rate of Case 1 on the top surface of the bottom electrode.This non-uniformity arises because the distribution of the ion flux is concentrated on the top surface of the bottom electrode as a result of the large difference in the electrical conditions between the top surface and

Figure 3 .
Figure 3. Contour plots for the spatial profiles of the time-averaged excitation rate for (a) Case 2, (b) Case 3, (c) Case 4, and (d) Case 5 for r ⩾ 135 mm.The grounded surfaces are indicated with the red line, while the dielectric surfaces are indicated with the green line.

Figure 5 .
Figure 5. Contour plots for the spatial profiles of the time-averaged excitation transfer rate for (a) Case 2, (b) Case 3, (c) Case 4, and (d) Case 5 for r ⩾ 135 mm.In the area marked with a red circle in (d), the excitation transfer rate near the bottom electrode edge of Case 5 was observed to be approximately 12% lower than that of Case 2.

Figure 6 .
Figure 6.Contour plots for the spatial profiles of the time-averaged electron density (Ne; m −3 ) for (a) Case 2, (b) Case 3, (c) Case 4, and (d) Case 5 for r ⩾ 135 mm.In the area marked with a red in (d), the electron density near the bottom electrode edge of Case 5 was observed to be approximately 5% lower than that of Case 2.

Figure 7 .
Figure 7. Contour plots for the spatial profiles of the time-averaged Si 4 H 9 density (m −3 ) for (a) Case 2, (b) Case 3, (c) Case 4, and (d) Case 5 for r ⩾ 135 mm.In the area marked with a red circle in (d), the Si 4 H 9 density near the bottom electrode edge of Case 5 was observed to be approximately 3% lower than that of Case 2.

Figure 8 .
Figure 8. Contour plots for the spatial profiles of the time-averaged excitation rate for (a) Case 6, Case 7, (c) Case 8, and (d) Case 9 for r ⩾ 135 mm.

Figure 9 .
Figure 9. Excitation rates for Cases 6 and 7, plotted along the vertical direction at r = 0.155 m.

Figure 11 .
Figure 11.Contour plots for the spatial profiles of the time-averaged Si 4 H 9 density (m −3 ) for (a) Case 6, (b) Case 7, (c) Case 8, and (d) Case 9 for r ⩾ 135 mm.The boundary layer of the Si 4 H 9 density contour refers to the line along which the density has a value of 1.0 × 10 18 m −3 .These lines are drawn on the contours in (a) and (c) for Case 6 and Case 8,

Figure 12 .
Figure 12.Deposition rate profiles of amorphous silicon layer (a-Si:H) for (a) Case 1; (b) Case 2, Case 3, Case 4 and Case 5; (c) Case 6, Case 7, Case 8, and Case 9. (d) Experimental data of Case 3 and Case 10 superimposed on the profiles of Case 3 and Case 10.

Figure 13 .
Figure 13.The series of results shows that loss probability (β) does not affect the uniformity of the deposition rate profile.(a) Deposition rate profiles are plotted for Case 3, Case 11, Case 12, and Case 13.Spatial profiles of the time-averaged SiH 3 density are depicted for (b) Case 11 and (c) Case 13.

Figure 14 .
Figure 14.The series of results shows that the secondary electron emission coefficient (SEEC) set on the sidewall does not affect the uniformity of the deposition rate profile.(a) Deposition rate profiles are plotted for Case 3, Case 14, Case 15, and Case 16.Spatial profiles of the time-averaged electron density are depicted for (b) Case 14 and (c) Case 16.

Figure 15 .
Figure 15.Contour plots for the spatial profiles of the time-averaged (a) excitation rate and (b) electron density for Case 17, and (c) excitation rate and (d) electron density for Case 18 for r ⩾ 135 mm.

Table 1 .
In the sticking model, the following probability coefficients were used.Coefficients were determined by other researchers using experimental studies.

Table 2 .
List of cases considered in this study.