5. Two-dimensional Simulation of Silane/Hydrogen Gas Discharge in a Plasma En-
5.4 Multiscale Temporal Marching Scheme
To ease the computational cost caused by the large disparity of the time scales for electrons, ions and neutral species, we have designed a multiscale temporal marching scheme, which is sketched in Figure 5.2. In this special temporal marching scheme, we only implicitly solve the electron continuity equation, electron energy density equation and the Poisson’s equation together at each electron time step to obtain the instantaneous electric field and electron tem-perature. The electron time step usually decide by the minimum value of reciprocal plasma frequency or plasma relaxation time. Take the advantage of ion’s larger characteristics time than electron, we solve ions fluid equations after electrons march some number of steps, usu-ally is 10 in this study. In the small electron time step, the force induced by the variation of
electric field is generally small for ions since ions have larger inertia than electrons. Thus, we only record the data of the last electron time step and obtain the instantaneous transport coeffi-cients of ions using the latest electric field. At the same time, neutral species and ion densities are solved implicitly with much larger neutral time step size ( 10,000 electron time step size).
By using this strategy, a quasi-steady gas discharge with a steady neutral diffusion flow field can be obtained after several thousands of RF cycles.
5.5 Results and Discussion
Figure 5.3 shows the initial background density of H2and SiH4, gas temperature, and flow velocity. Results show that the temperature distribution across the electrode gap is almost lin-ear, which means that it is conduction dominated. This is reasonable since the chamber is under low-pressure condition (600 mtorr), which leads to low Reynolds number gas flows. In this case, the Reynolds number is only 0.035 with inflow velocity of silane/hydrogen and gap distance as the characteristic velocity and length, respectively. In addition, the density near the substrate surface decreases greatly because of the heated substrate at elevated temperature (250oC). Non-uniform background density is important in determining the ionization rate dur-ing the simulation.
Figure 5.4 shows the plasma potential at difference phase (a) φ = 0 (b) φ = 0.5π, φ = 1.5π and (d) φ = 2π. Results show that in most of the region between the electrodes the potential distribution is similar to a quasi-1D case. Electric field is found to be very strong near the edge of guard ring close to the ground electrode (e.g., x = 13 cm and y = 1 cm), although the corners of the guard rings were rounded.
Figure 5.5 show that the time-average potential distribution across the electrode gap at the center of the chamber, in which the plasma potential is 48.82 V. There is a 20 V averaged potential drop y = 0 0.7 cm due to the negative net charges accumulated on the glass surface, which provides protections from ion bombarding at the substrate surface.
Figure 5.6 and Figure 5.7 shows the electron and related electron temperature at difference phases respectively, which also shows high peak value close to the corner, where the strong elec-tric field is induced. Apart from the corner of guard ring close to power electrode where plasma
properties are strongly related to edge electric field, electron density and electron temperature show the homogeneity between the electrodes. It’s the benefit that a large scale PECVD can simplify the two or three dimensional effects into a pure one dimensional behavior. Thus, ho-mogeneous distribution of species densities above substrate glass are observed which help the uniformity of deposition. However, the non-uniformity of large scale PECVD with very high frequency (VHF) power source (i.e. microwave) are limited due to standing waves generated on the electrode surfaces [Schade et al., 2006], which needs further study in the future. Figure 5.8 shows the ion species densities include positive ions (a) H2+ (b) SiH2+(c) Si2H4+ and negative ion (d) SiH3−. The dominated positive ion is SiH2+, and negative charged species is SiH3−. Figure 5.9 show that the time-average charged densities distribution across the electrode gap at the center of the chamber. It shows that a quasi-neutral region is form in the center part of the gap with positive charge density in the sheath close to the electrodes. The bulk dominated by SiH2+and SiH3−at y = 0.85 cm, while electron density shows pick value of 5.2× 1014m−3 at y = 0.64 cm. Si2H4+species density is three order smaller than the density of SiH2+.
Figure 5.10 shows a series of time-average spatial distribution of several important radi-cal species related to the a-Si deposition in the PECVD chamber. These include: (a) H (b) SiH2 and (c) SiH3. Note their concentrations are almost stationary and do not move with the oscillating field because they are neutral species. It is clear that SiH3 is the most dominant radical species include silicon atom, which is important in film deposition, as found in earlier simulations and experiments [Bleecker et al., 2004a]. The predicted uniform radical density distribution above the substrate glass can lead to a uniform a-Si deposition rate over the glass, which will be shown later. Figure 5.9 show that the time-average radical densities distribution across the electrode gap at the center of the chamber. H and SiH3 radical have peak value at y = 0.85 cm of 2.9×1018m−3and 2.02×1018m−3respectively, while SiH2+has 8.7×1015m−3 at y = 0.64 cm.
Figure 5.12 present the comparison of deposition rate from numerical simulation and ex-perimental data as well as SiH3 radical species flux to the glass. The numerical deposition rate is estimated by combining the simulated SiH3density and the deposition rate coefficient [Lin,
2010],
SiH3(g)+ db→ SiH3(a) (5.1)
with its reaction rate coefficient
kgas = 6.39× 10−7Tg−1.2exp(−159.62
Tg ) (5.2)
where the gas temperature is assumed 500 K. The surface rate coefficient can be expressed as ksurf ace = kgas
As (5.3)
where Asis the area of the gas on the site which is assumed as 1015cm2, and θ is the fraction of available surface sites. The numerical deposition rate can obtain by
d[X]surf ace
dt = ksurf ace[X]g (5.4)
While the fraction θ is assumed as 0.015 in order to coincide the experimental result. In conclu-sion, both numerical and experimental results qualitatively and quantitatively show the uniform deposition rate on the glass with the choosing of relevant number fraction θ.
Figure 5.12 shows that, with the present simulation data, the calculated deposition rate agrees very well with the experimental data under the same test conditions [provided by Prof.
Tsai at NCTU], if the fraction θ is assumed as 0.015. In addition, the deposition rates of both the simulation and experiment at the edge increase slightly probably due to the enhanced electric field as shown in Figure 5.4.
5.6 Brief Summary of This Chapter
In this chapter, we have demonstrated a 2D-axisymmetric chamber-scale simulation of a realistic silane/hydrogen PECVD chamber using the developed parallel fluid modeling based on a set of fairly complicated plasma chemistry. Major finding of this chapter can be summarized as follows:
1. In most of the region between the electrodes, the plasma can be approximated by a quasi-1D discharge, in which a quasi-neutral region in the center is formed with sheaths near the electrodes.
2. SiH3 is the most dominant radical species include silicon atom predicted in the simula-tion, which coincides with previous findings using simulations and experiments.
3. Calculated deposition rate on the substrate agree very well with the measured value if the fraction of available reaction site is set to 0.015.
4. Slightly increased deposition rate at the edge of the substrate was found due to the en-hanced electric field in this region, which also have been observed in the experiments.
Chapter 6
Conclusion and Recommendations for Future Study
6.1 Summaries of the Thesis
In this thesis, development of parallelized 1D/1D-axisymmetric and 2D/2D-axisymmetric fluid modeling codes using fully implicit finite-difference method with hybrid analytical-numerical Jacobian evaluation for low-temperature, non-equilibrium plasma simulation has been reported.
Implementation and validations against earlier simulations and experimental data are described in detail. Applications with wide range of pressures and frequencies (radio frequency in mega Hertz and alternating current in kilo Hertz) are demonstrated, compared with experimental data wherever possible, and related plasma physics and chemistry are discussed therein. Validated codes are the applied to simulate one-dimensional helium dielectric barrier discharge driven by realistic distorted sinusoidal voltages and two-dimensional silane/hydrogen gas discharge in an PECVD chamber. Major findings of the thesis can be summarized as follows:
1. Parallelized 1D and 2D fluid modeling code using finite-difference method was developed and validated against experiments and previous simulations. (Chapter 3)
2. Parallel speedup was demonstrated to be super-linear for the number of processors up to 144 using a GEC chamber-scale simulation with 120,048 degrees of freedom with a suitable combination of preconditioning and matrix solvers. (Chapter 3)
3. Major Towsend-like behavior along with complicated mode transitions were found for a helium dielectric barrier discharge driven by a 20 KHz distorted sinusoidal power source under atmospheric-pressure condition. (Chapter 4)
4. In the silane/hydrogen gas discharge in a low-pressure PECVD chamber driven by a RF power source (27.12 MHz), SiH3 was found to be the most dominant radical species with silicon atom, which leads to the deposition of silicon film on the substrate. Relatively uniform film deposition rate was found across the substrate, except slightly increased value near the edge. (Chapter 5)
6.2 Recommendations for Future Work
Base on the viewpoint for improving the fluid modeling code, there are several possible directions of research are recommended for further studies and are summarized as follows:
1. To apply the fluid modeling code for simulating several challenging gas discharge prob-lems in the frequency range of RF and AC.
2. To further reduce the computational time in larges-scale low-pressure (or diffusion domi-nated) fluid modeling problem that may involve large number of ion and neutral species, we may adopt the following multiscale temporal marching scheme:
(a) Solve the electron continuity equation, electron energy density equation and Pois-son’s equation together with fully implicit scheme with an electron time step and repeat until an ion time step size (∼10-50 electron time steps) is reached.
(b) Solve the (linearized) continuity equations for all ion species implicitly one by one using the most updated (or mean) electric field for evaluating the transport coeffi-cients of ions.
(c) Repeat Step a and Step b until a neutral time step size (∼10,000-100,000 electron time steps) is reached.
(d) Solve the (linearized) continuity for all neutral species implicitly one by one using the most updated (or mean) electric field for evaluating the transport coefficients of ions.
(e) Repeat Steps a through d until the preset steady-state condition is reached.
In Step b, the steady-state continuity equation for each neutral species may be solved at each time step to further reduce the computational cost to reach steady-state flow condi-tion.
3. For treating gas discharges at higher pressure, in which the convection by the neutral flow is important, the Navier-Stokes equation needs to be solved at each neutral time step in the above after Step d.
4. For treating gas discharged with complex geometry, the fluid modeling equations need to be solved in the curvilinear coordinate frame.
5. To develop a Maxwell equation solver for solving discharge involving EM waves such as high-frequency and large-area PECVD and inductively coupled plasma (ICP) problems.
6. To incorporate an automatic DC-bias adjustment function into the fluid modeling code.
7. To couple an external circuit module with the fluid modeling code.
8. To extend the fluid modeling code into a three-dimensional version and couple with a time-dependent Maxwell equation solver (e.g., time-dependent finite-difference, TDFD) for several realistic PECVD cases, such as solar cell film deposition.
Figure 3.1: Comparison of simulated and experimental voltages and currents for atmospheric-pressure discharge with 1 mm gap spacing using sinusoidal 13.56 M Hz power source.
Figure 3.2: Current-voltage characteristic of numerical results and experimental data, using helium gas dielectric barrier discharge at 760 torr, applied wave frequency 60 KHz.
(a)
(b)
(c)
Figure 3.3: Comparison of simulated and measured discharged currents along with photo
im-Figure 3.4: Spatial-average temporal discharge properties of nitrogen DBD (60 kHz, d = 0.7 mm).
(a) (b)
(c) (d)
(e) (f)
Figure 3.5: Simulated cycle averaged plasma properties of helium GEC including (a) electron, (b) He+, (c) He+2, (d) He∗2, (e) He∗, and (f) Hemeta.
Applied Voltage (V) ElectronDensity(1015 cm-3 )
0 50 100 150 200 250
100 101 102 103
Simulation Present
Exp. M. E. Riley et. al. (1993) Theory M. E. Riley et. al. (1993) Symbol Method Source
Figure 3.6: A comparison of the simulated peak electron densities with the theoretical prediction and the experimental data [Riley et al., 1994] for various applied voltages
Number of Processors
Figure 3.7: The parallel performance including (a) speedup analysis and (b) runtime per time step as a function of the number of processors
(a)
(b)
Figure 4.1: Schematic diagram of (a) simple and (b) complicated helium plasma chemistry based on the magnitude of energy level.
Figure 4.2: Schematic diagram of (a) simple and (b) complicated helium plasma chemistry based on the magnitude of energy level.
Figure 4.3: Comparison of simulated and measured discharge currents in a quasi-pulse AC cycle (20 kHz).
(a)
(b)
Figure 4.4: (a) Comparison between experimental current and simulation using the simple plasma chemistry. (b) Power absorption by various mechanisms.
(a)
(b)
Figure 4.5: Snapshots of distribution of (a) plasma properties and (b) rate of generation of species in several reaction channels in region A (Long Townsend like) of a helium DBD driven by a quasi-pulse power source (20 kHz).
(a)
(b)
Figure 4.6: Snapshots of distribution of (a) plasma properties and (b) rate of generation of species in several reaction channels in region B (Dark current like) of helium DBD driven by a quasi-pulse power source (20 kHz)
(a)
(b)
Figure 4.7: Snapshots of distribution of (a) plasma properties and (b) rate of generation of species in several reaction channels in region C (Primary short Townsend like) of a helium DBD driven by a quasi-pulse power source (20 kHz).
(a)
(b)
Figure 4.8: Snapshots of distribution of (a) plasma properties and (b) rate of generation of species in several reaction channels in region D (Secondary short Townsend like discharge) of a helium DBD driven by a quasi-pulse power source (20 kHz)
Figure 4.9: Time-average spatial power absorption by various mechanisms.
Figure 4.10: Spatial profiles of cycle-averaged discharge parameters
Figure 4.11: Spatial-average temporal power absorption by various mechanisms.
Figure 4.13: Phase diagram of electron number density distribution.
Figure 4.14: Phase diagram of He+2 number density distribution.
Figure 4.15: Phase diagram of electron temperature distribution.
Figure 5.1: Sketch of the PECVD chamber
Figure5.2:Thesketchofspecialtemporalmarchingscheme
(a)
(b)
(c)
(d)
Figure 5.3: Fluid modeling initial conditions which are obtained form Navier-Stock equations solver include: (a) Gas temperature (b) Background gas flow velocity (c) H2 density distribution and (d) SiH4 density distribution.
(a)
(b)
(c)
(d)
Figure 5.4: Plasma potential at difference phase of a RF cycle, where (a) φ = 0 (b) φ = 0.5π, φ = 1.5π and (d) φ = 2π.
Y (cm)
AveragedPotential(V)
0 0.5 1 1.5
-20 -10 0 10 20 30 40 50
Figure 5.5: Cycle averaged potential profile across the electrode gap at the center of the cham-ber.
(a)
(b)
(c)
(d)
Figure 5.6: Electron density at difference phase of a RF cycle, where (a) φ = 0 (b) φ = 0.5π, φ = 1.5π and (d) φ = 2π.
(a)
(b)
(c)
(d)
Figure 5.7: Electron temperature at difference phase of a RF cycle, where (a) φ = 0 (b) φ = 0.5π, φ = 1.5π and (d) φ = 2π.
(a)
(b)
(c)
(d)
Figure 5.8: Ion species distributions include positive ions (a) H2+ (b) SiH2+ (c) Si2H4+ and negative ion (d) SiH3−.
Y (cm) ChargeDensities(1015 m-3 )
0 0.5 1 1.5
0 0.5 1 1.5 2 2.5 3 3.5
Electron H2+ SiH2+ Si2H4+ SiH3
-Figure 5.9: Cycle averaged charged densities profile across the electrode gap at the center of the chamber.
(a)
(b)
(c)
Figure 5.10: Important radical species relate to s-Si deposition, include(a) H(b) SiH2 and (c) SiH3.
Y (cm) RadicalDensities(m-3 )
0 0.5 1 1.5
0 0.5 1 1.5 2 2.5 3
H ( x 10
19) SiH
2( x 10
16) SiH
3( x 10
18)
Figure 5.11: Cycle averaged radical densities profile across the electrode gap at the center of the chamber.
Figure 5.12: Comparison of deposition rate from numerical simulation and experiment data as well as SiH3flux to the subtract glass.
Table 3.1: Nitrogen plasma chemistry reaction channels.
No. Reaction Channel Threshold Energy (eV) Rate Coefficient
1 e + N2→ e + N2 0.00 cross section
Table4.1:Summaryofsimpleandcomplicatedheliumplasmachemistry. NoReactionTypeReactionChannelsComplicatedSimpleThresholdEnergy(eV) 00Momentumtransfere−+He→e−+HeBOLSIG+BOLSIG+0 01e-impactexcitation(2S)e−+He→e−+He∗ meta(3S1)BOLSIG+2.308×10−16T0.31 eexp(−2.297×105 Te)19.82 02e-impactexcitation(2S)e−+He→e−+He∗ meta(1S1)BOLSIG+20.61 03e-impactexcitation(23P)e−+He→e−+He∗∗(23P)BOLSIG+0 04e-impactexcitation(21P)e−+He→e−+He∗∗(21P)BOLSIG+0 05e-impactexcitation(3SPD)e−+He→e−+He∗∗(3SPD)BOLSIG+0 06e-impactexcitation(4SPD)e−+He→e−+He∗∗(4SPD)BOLSIG+0 07e-impactexcitation(5SPD)e−+He→e−+He∗∗(5SPD)BOLSIG+0 08e-impactionizatione−+He→2e−+He+BOLSIG+2.584×10−18T0.68 eexp(−2.854×105 Te)24.58 09e-impactionizatione−+He∗ meta→2e−+He+BOLSIG+4.611×10−16T0.6 eexp(−5.546×104 Te)4.78 10e-impactde-excitatione−+He∗ meta→e−+He2.9×10−151.099×10−17T0.31 e-19.8 11e-impactdissociatione−+He∗ 2→e−+2He3.8×10−151.268×10−18T0.71 eexp(−3.945×104 Te)-17.9 12e-ionrecombination2e−+He+→e−+He∗ meta6×10−32-4.78 13e-iondissociativerecombination2e−+He+ 2→e−+He∗ meta+He2.8×10−325.386×10−13T−0.5 e0 14e-iondissociativerecombinatione−+He+ 2+He→He∗ meta+2He3.5×10−390 15e-ionrecombination2e−+He+ 2→e−+He∗ 21.2×10−330 16e-ionrecombinatione−+He+ 2+He→He∗ 2+He1.5×10−390 17Hombeck-MolnarassociativeionizationHe∗∗+He→e−+He+ 21.5×10−170 18Metastable-metastableassociativeionizationHe∗ meta+He∗ meta→e−+He+ 22.03×10−15-18.2 19Metastable-metastableionizationHe∗ meta+He∗ meta→e−+He++He8.7×10−162.7×10−16-15.8 20ionconversionHe++2He→He+ 2+He6.5×10−442.584×10−18T0.68 eexp(1×1045 Te)0 21Metastable-inducedassociationHe∗ meta+2He→He∗ 2+He1.9×10−461.3×10−450 22Metastable-induceddissociativeionizationHe∗ meta+He∗ 2→e−+He++2He5×10−16-13.5 23Metastable-inducedionizationHe∗ meta+He∗ 2→e−+He+ 2+He2×10−15-15.9 24Dimer-induceddissociativeionizationHe∗ 2+He∗ 2→e−+He++3He3×10−16-11.3 25Dimer-inducedionizationHe∗ 2+He∗ 2→e−+He+ 2+2He1.2×10−15-13.7 26He-atominduceddissociationHe∗ 2+He→3He4.9×10−220
Table 5.1: Silane/Hydrogen plasma chemistry reaction channels.
No. Reaction Rate Coefficient Threshold Energy(eV)
01 SiH4+ e−→ SiH+2 + 2H + 2e− cross section 11.9
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