CHAPTER 2 NUMERICAL METHODS
2.8 Numerical Discretizations and Algorithms
2.8.4 Implementation of Parallel Computation
In the process of simulation, the resulting algebraic linear system of each equation is solved independently by the Krylov subspace method with or without parallel preconditioner provided by PETSc library [72] through domain decomposition technique implemented by using MPI [73]. Among the iterative linear solvers of Krylov subspace, we have employed the GMRES and BCGS to solve the linear systems obtained by discretization since it has been shown to be the most robust linear matrix solver for most of the cases. Besides, in the preconditioning we test the performance of two sub-domain solvers such as LU (direct) and incomplete LU (ILU;
iterative) factorizations. For preconditioners, we have selected the popular preconditioners such as Black Jacobi, successive over-relaxation (SOR), Additive Schwarz Preconditioners (ASM) and LU. The detail parallel performance will be presente using combination of Krylov subspace method and preconditioner later.
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CHAPTER 3 VALIDATION
AND
PARALLEL PERFORMANCE
In this chapter, we validate our developed code by comparing with previous simulation and experimental results of a typical cyclindrical ICP chamber. After validation, parallel performance of the developed fluid code for simulating an ICP source is presented in turn.
3.1 Validation
In this section, we presents validation of the developed fluid modeling code by comparing with previous simulation and experiment of Fukumoto et al. [26][74] in a typical cylindrical ICP chamber. Figure 8 shows the schematic diagram of the ICP reactor under consideration. The plasma reactor is 30 cm in diameter, and 9 cm in height. The dielectric window is at the top of the chamber with five turns of coil right above it, and the wafer is 20 cm in diameter at the bottom of the chamber. Table 10 summarized the corresponding test conditions. The major operating conditions consist of: a frequency of 13.56 MHz of power source, a gas pressure of 20 mTorr, an ICP input power of 250 W, and a gas flow rate of 200 sccm. In addition, because of the simple geometry of the ICP chamber, an exact solution of Maxwell’s equation based on the Biot-Savart’s law was used as the boundary condition at the plasma-dielectric window interface in [26].
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Figure 9 gives the prsent simulation results of chemical compositions of typical charged species averaged over the entire region of the ICP reactor along with the simulation results of Fukumoto et al. [26] for comparison purpose. In addition, Figure 10 gives their results of mass spectra measurements [74]. The results show that the trend and the quantity of the predicted concentrations of CFx+ (x=1~3) agree well with the experimental results [74], in which all show consistently the order of ion concentration as CFx+ > CFx+ > CFx+. Figure 10-15 show a series of the present simulation results of several important plasma properties along with previous simulation results of Fukumoto et al. [26]. These plasma properties include feedstock CF4 density, electron density, electron temperature, F- density, CF3+
density, and chemical compositions of ion species. Again, results showthat the present simulations agree reasonably well with those of Fukumoto et al. [26]. Thus, we conclude that the current fluid modeling code is valid at least under the similar conditions as simulated and measured by Fukumoto et al. [26][74].
3.2 Parallel Performance
Table 11 summarizes the computational time of the present parallel fluid modeling code for the CF4 ICP with a grid size of 122×179 with 32 species, which leads to about 700,000 unknowns in total. Two thousand time steps were run throughout the tests using the Generalized Minimal Residual Method (GMRES) as the linear equation solver in combination with various preconditioning techniques, including the Additive Swartz Method (ASM), block Jacobi, and Successive Over Relaxation (SOR), on an IBM-1350 clusters at NCHC (National Center of High-Performance Computing) of Taiwan. Note that the IBM-1350 consists of 2,048
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cores in total with 4 cores per node, in which Intel X5450 Processor 3.0 GHz Quad core of CPU with 16 GB of RAM is used per node.
Figure 16 shows the corresponding data of parallel performance tests as those summarized in Table 11. The results indicate that the combination of GMRES and block Jacobi (or ASM) using iLU as the sub-domain solver gives the best performance, and this combination is scalable for up to 26 processors for the problem size of 122×179 with 32 species. However, because there are 4 cores in a node of IBM-1350 with a restricting memory, the performance is not as good as the ideal value using 4 processors. In addition, we do not show the cases that combine subdomain solver LU for preconditioner with GMRES because this combination costs more than ILU due to its large grain size in the current test case with fewer numbers of processors. In brief summary, we can reduce greatly runtime at least 10-20 times with the developed fluid modeling code for studying very complex plasma physics and chemistry of in an ICP chamber.
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CHAPTER 4
RESULTS AND DISSCUSION
In this chapter, we employ our validated fluid model to study CF4 discharge in GECRC (section 4.1) and in dome-shaped reactor (section 4.2). Then, we employ developed fluid model to investigate the etching rate, coverage, and fluxes on the substrate in different geometries of ICP reactors: t typical reactor with planar (top) coils, typical reactor with cylindrical (side) coils, and dome-shaped reactors, which is described in the section 4.3.
4.1 CF
4Discharge in Gaseous Electronics Conference Reference Cell (GECRC)
Figure 18 shows the diagram of the GECRC, which is applied for etching SiO2. There are four turns of coil insulated from the plasma by a quartz window (1.2 cm in thickness and 8 cm in radius), which are driven by a current with a radio frequency (RF) of 13.56 MHz that induces electric field heating in the azimuthal direction. A wafer substrate is located at the central bottom of the reactor. A gas inlet ring for feedstock gas, CF4, is located at the outer top of the chamber, and the pumping port is located at the outer bottom of the chamber.
The test conditions for CF4 ICP discharge simulation in GECRC include a gas pressure of 30 mTorr, a CF4 flow rate of 150 sccm, and a deposited power of 150 W.
Several grid convergence tests show that a grid of 66 113 points suffices. The temperatures of the ion and the neutral species are assumed to be constant at 0.026 eV
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and 400 K, respectively. The ion bombardment energy is set at 100 eV at the substrate boundary and 20 eV on the quartz windows.
4.1.1 Spatial Profiles
In this section, we present the distributed profiles of species in CF4 GECRC discharge. The results are compared with the available experimental data. In the discussion below, the “core region” and the “edge region” refer to the central and the edge space, respectively, between the quartz window and the substrate. The “outer region” refers to the cylindrical space surrounding the parallel plates.
4.1.1.1 Induced Electric Field, Power Absorption and Electron Temperature
Figures 19(a) and 19(b) show the real and the imaginary parts of the induced electric field in the azimuthal direction, respectively. In addition, Figure 19(c) shows the power deposition through Ohmic heating of electrons. The real part of the electric field caused by plasma shows a maximum intensity of approximately 50 V/m in the core region, where the electron temperature is high. The imaginary part of the induced electric field caused by the coil current shows a maximum value of approximately 1,500 V/m near the coils and approximately 500 V/m in the gas phase, and it decays rapidly by an order of magnitude within several centimeters into the high density plasma, as expected. The total induced electric field is obtained from the magnitude of the complex induced electric field, which is given by Im2
2 Re
~ E E
E , which indicates that the total electric field induced by the coil current is dominant, but the electric filed induced by plasma can be ignored when considering the contribution of the region
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10~20% away from the core region. Therefore, absorption of inductive power is deposited mainly by the imaginary part of the induced electric field within a few centimeters of the dielectric roof. In this case, the plasma conductivity reaches its maximum value in the bulk plasma and decays toward the quartz, while the electric filed reaches its maximum value at the dielectric and decays towards the plasma.
However, the power deposition of the plasma is stronger than the square of the imaginary part of the electric field. Therefore, the peak power deposition is approximately 1.5 W at 1 cm below the quartz, and the power deposition becomes 0.2 W away from the quartz and toward the plasma bulk at a height of 2.5 cm. Figure 19(d) illustrates the contours of the electron temperature, which follow a toroidal shape from the quartz to the core region, corresponding to the power absorption of plasma. The electron temperature is highest near the coils in the core region, where the power deposition is the highest. Although the power peaks in the confined region near the dielectric, electron thermal conduction at the low operating pressure helps to heat the whole plasma. The electron is cooled in the outer region, due to either the lower electron thermal conductivity or the higher collision frequency.
4.1.1.2 Production Rate
Figure 20(a) illustrates the profile of the feeding gas CF4, which shows the gradient profile from the gas inlet to the gas outlet, and the feeding gas is mostly consumed by electron collisions and pumping outlet in the reactor. While the electron temperature and the feeding background gas CF4 are determined, the reaction rate for production and destruction of each species are specified to influence the species distributions during computation. Therefore, the absolute reaction rates are studied to obtain information on the absolute importance of a reaction, which tells both the
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important process and its importance level. In the following, we present the absolute reaction concentration rates (i.e., concentration per second) of the most important reactions in steady state, which are the product of the reaction rate and the reactant concentrations in s-1m-3. Figure 20(b) presents the production rate of the momentum transfer reaction, where the electrons collide with the background feedstock gas CF4. Obviously, the highest frequencies of electron collision occur in two regions: the core region, where electron has a high concentration, and the gas inlet, where CF4 has a relatively high concentration. Figure 21(a), Figure 21(b) and Figure 21(c) show the relative contributions of dissociative ionization, which are significant production of electron and charge ions. In Figure 21(a), it can be observed that the dissociative ionization that dissociates CF4 contributes for 90% in generating electron and CF3+ because its threshold energy (14.8 eV) is lower than those in the reaction channels to produce CF2+ (20.8 eV) and CF+ (23.9 eV). Although dissociative ionizations to produce CF2+ and CF+ are not dominant reaction processes in CF4 discharge, they have a significant influence on the production of F in the core region, as shown in Figures 21(b) and 21(c). The production rate of CFx (x=1~3) from dissociating CF4 is shown in Figures 21(d), 21(e) and 21(f). The production of CFx (x=1~3) decreases with decreasing x because their dissociative thresholds increase with decreasing x. In Figure 21(d), CF3 is completely governed by the electron impact dissociation from CF4 because it has the lowest threshold energy among the CFx, which has threshold energies of 5.67 eV (CF3), 9.32 eV ( CF2) and 14.7 eV (CF). The dissociations of CF4 that generate CF and CF2 provide relatively lower productions than the dissociation of CF4 that generates CF3. The productions of F- are shown in Figures 22(a), 22(b), 22(c) and 22(d), which are the dissociative attachment of CF4, CF3, CF2 and F2, respectively. We see that dissociative attachment of CF4 clearly dominates the production of F- in the
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discharge. This type of reaction boosts up the production of F- and the destruction of electron, and F- shows high concentration in the outer region near gas inlet due to the threshold energy of attachment is relatively low.
4.1.1.3 Electron and Negative Charge Ions
The concentrations of negatively charged species like electron and F- were illustrated in Figures 23(a) and 23(b) respectively. The electron density peaks at 2.5x1017 1/m3 at the center of the core region, where electrons are produced in high quantity by the relative ionized reactions with high electron temperature (F01~F10).
Most electrons are lost by diffusion into the chamber walls, and a small amount of electron is lost by dissociative attachment to CF3 and CF4 to form negative ions F -(F01~F03). The static electric fields push the electrons toward the center, causing the electron density to decrease. The predicted high concentration of F- is 4 x 1016 m-3 in the simulation. The profile and quantity of electron and F- are reasonably consistent with the results measured by Rao et al.[48]. F- ions are mainly generated from two reaction paths. One path is dissociative attachment to the feedstock gas CF4, and the other is dissociative attachment to CF3, which were dissociated from CF4 near the gas inlet because it has a lower threshold energy than the other reaction paths. Because F- is lost significantly via ion recombination, its density is lower in the core region, where CF3+ has maximum value, than at the gas inlet. For the other negative ion O-, the main source of O- is dissociative electron attachment to O2 (Ox18 and Ox19), and it presents the largest concentration in the core region. However, this ion is a minor negative charge species, due to its low concentration of 1013 m-3, and we do not show its contour here.
All negative ions are related to attachment and recombination of negative and positive charges.
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4.1.1.4 Positive Ions
Figure 24 shows the distributing profiles of (a) F+, (b) CF+, (c) CF2+
, and (d) CF3+
. The concentrations of CFx+ (x=0~3) peak at the core region due to electron relating ionization from the feedstock gas CF4 (F01~F10). The comparison between CFx+
gives CF3+(~2x1017 m-3) > CF2+(~4x1016 m-3) > CF+(~1x1016 m-3) > F+(~1x1014 m-3), which was observed by some experiments [42][48]. The phenomenon is consistent with the threshold energy of electron-impact dissociative ionization reacting to the feedstock gas CF4: 23.9 (CF+) > 20.8 (CF2+
) > 14.8 (CF+). Moreover, the CF3+
concentration peaks not only at the core region but also at the outer region near the gas inlet because of its low threshold energy of dissociative ionization. Figure 25 shows the oxygen-containing charged species: (a) O+, (b) O2+ and (c) CO+, and Figure 26 shows the silicon-contain charge species: (a) SiF+, (b) SiF2+
and (c) SiF3+
, which are produced from volatile oxygen-containing and silicon-contain neutral species liberated in the etching process. O+ and O2+ are generated via ionization of their mother gases O and O2 (Ox01, Ox04, Ox19, Ox20 and Ox21). CO+ is the product of ionized CO, which is produced from the gas phase of COF and COF2 (Ox28). SiFx+ (x=1~3) are products of dissociative ionization from their mother gases SiFx (SF01~SF06), and they were restricted in the central region by the electric field of ambipolar diffusion. The concentration of SiF3+ is higher than those of SiF+ and SiF2+ because the threshold energy of SiF3+ is lower than those of SiF+ and SiF2+.
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4.1.1.5 Neutral and Radical Species
The contours of F and CFx (1~3) were shown in Figure 27. These contours are dominant neutral species to etch SiO2, and they have relatively high concentrations, which are only less than the feedstock gas concentration. Additionally, the threshold energy and the rate coefficients of the dissociated reactions strongly influence the distributions of these radical species, and the threshold energy of dissociating feedstock gas CF4 by electron is 14.7 eV(CF) > 9.32 eV(CF2) > 5.67 eV(CF3). Neutral F atoms are produced principally through electron-impact dissociation reactions of the feedstock CF4 (F11~F13) in the core region, where the electron density and electron temperature are relatively high. Because a higher threshold energy of reaction is necessary to dissociate the feedstock CF4 to form CF or CF2, the distribution peaks in the region of high electron temperature, i.e., in the core region. Because the necessary threshold energy to dissociate CF4 to generate fragment CF3 is lower than those to produce CF and CF2, the concentration of CF3 is high in both the core region and the outer region near the gas inlet.
When the reactive radicals CFx (0~3) and the reactive ions approach the SiO2
layers, including the substrate and the dielectric windows, the etching products are computed to produce through the surface model. Such etching products as O, O2, COF, COF2 and CO are liberated into the plasma as the reactive radicals bombard. Once these
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etching products merge into the gas, the properties of the plasma will change, due to the impact of electrons on these etching products. The following discussion will include O, O2, COF, COF2 and CO, which spill from the etching processes of SiO2, and their gas-phase products, such as O(1D), O2(a) and CO2, which are generated from electron-impact and gas-phase reactions. First, Figure 28(b) gives the distribution of O2, which has a strong downhill-like gradient from the core region toward the edge region on the dielectric and the substrate surface. O2 is mainly produced through the surface mechanisms of ion-enhance chemical etching by F (S3) and ion-enhanced chemical sputtering (S4), leading to a high F surface coverage, which is discussed in next section.
O2 is lost via electron-impact reactions (Ox10, Ox16 and Ox17), which produce considerable quantities of O, O(1D) and O2(a). The profile of O is given in Figure 28(a).
The maximum value of O concentration in the central core region occurs due to a high dissociation of O2. Although O is liberated from physical sputtering (S1), it also recombines to form O2 on the substrate surface (W15). Figures 28(c) and 28(d) show the profiles of O(1D) and O2(a), which are excited species of O and O2 (Ox02, Ox10), and they follow the distributions of O and O2 struck by electron to form excitation reactions depending on the electron. Meanwhile, the distributions of carbonous oxide products from the etching substrate, such as COF, COF2, CO and CO2, are given in Figure 29. COF is mainly produced in the surface reactions with fluorocarbon radicals,
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but it is lost via collision with such radicals as CFx. While a portion of COF2 is produced on the substrate via etching, most of it is produced in the reaction between CF3 and O near the gas inlet region, where the concentration of CF3 is high. CO and CO2 are mainly distributed in the core region, where the gas reaction of COFx and CFx
(Ox31, Ox32, Ox34, Ox36, Ox38 and Ox39) and the surface reactions (S7) occur.
Figures 30(a)-(e) show the distributions of SiFx (x=0~4), which have a strong downhill-like gradient from the central core region to the substrate edge region because these species escape through the etching processes, depending on the fluxes of radicals and ions transported to the SiO2 layer. SiFx (x=0~4) are also produced on the SiO2
dielectric windows and on the substrate surface as x increases. SiF4 is the dominant etching product while etching SiO2 in CF4 discharge, which was also observed in many experiments [37][45][45]. Although the volatile SiFx (x=0~3) strongly depend on the surface coverage of CFx (θCFx), the volatile SiF4 strongly depends on the surface coverage of F (θF). When θF is the dominant coverage on either the substrate or the dielectric windows, θCFx only covers the substrate because the low ion energy is presented at the dielectric windows. The distribution of F2 in Figure 30(f) shows a large concentration near the gas inlet because F2 molecules mainly come from the detached recombination of F- and F (FN11), which are rich near the gas inlet, and it is assumed that F2 would not interact with wall, due to its full chemical bond.
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4.1.2 Chemical Ingredients
The chemical ingredients of the charged and the neutral species averaged over the core region of the reactor were shown in Figure 31 and Figure 32 respectively. The positive ion CF3+ is the obvious dominant charge with a concentration of ~1017 1/m3. Regarding the significant charges CFx+
, the trend of CF3+
> CF2+
> CF+ is observed in the simulation because of their threshold energies to fragment CF4, which is consistent with the experiments [48][49][68]. The concentration of the negative ion F- in simulation of CF4 discharge is about 1016 1/m3, and it is comparable to the concentration of electron in our simulation, which was also measured by Rao et al. [48].
Moreover, Because CF4 is fed from the gas inlet, it is the most numerous species as the
Moreover, Because CF4 is fed from the gas inlet, it is the most numerous species as the