Chapter 5. Applications to Realistic Problems
5.3 Simulation on Magnetron Plasma
5.3.2 Three-Dimensional RF Magnetron Plasma
In this studying case, we use the same computational domain and permanent magnet system as shown in the previous subsection. The radio-frequency is 13.56MHz,V is rf 300 Volts. The electron timestep is
s f te
2000 10 1
695 .
3 × 11 =
=
∆ − and the number of
sub-cycling is 10. Initial ion temperature is 232 K and initial electron temperature is 0.5 eV. In this work, we still examine the effects of magnetization M and secondary electron emission coefficient γ on the discharge structure. Three cases are presented:
(1.) M=0.125T, γ=0., (2) M=0.125T, γ=0.06, and (3) M=0.25T, γ=0.06.
Fig. 5.35 Shows the potential distributions of these three cases, which illustrates that
the plasma potential is nearly constant and positive (≈0.5Vrf ) and axisymmetric sheaths are formed near the powered and grounded electrodes. As M orγincreases, the thickness of cathode sheath between N and S poles decreases. The electron and ion number densities of these three cases are shown in Fig. 5.36-5.38. Both show that the most of electrons and ions are confined between N and S poles, which also show the axisymmetrical electron and ion number densities distributions. As M orγ increases, both the electron and ion number densities increase. The electron and ion energies of these three cases are shown in Fig. 5.39-5.41. They show that the mean electron energy in the bulk region is 5~9eV. The electron energy is larger at the region between bulk and cathode sheath, where is located between N and S poles. This is
because electrons are strongly magnetized and exhibit E×B drift motion. The mean ion energy can be accelerated to 50eV by the strong electric field in the two sheath regions and ions are hardly magnetized due to their heavy mass.
5.4 Some Remarks
In this chapter, our developed parallel 3D PIC-FEM code has performed its superior capability in dealing with the 3D field emission display and 3D low-temperature plasma sources since their corresponding results agree with the previous experimental or numerical studies. Some important simulation results are summarized as follows:
1. In simulating the 3D FED without considering the space-charge effect, we use a triode-type CNT-based emitter without a focusing electrode as the studying case.
The primarily results are: The first is the spreading angle of electrons from the tip increases with the increasing gate voltage. The second is the emission current increases dramatically with the given CNT height. The third is a magnetic focusing design can optimally suppress the electron beam dispersion under a well-controlled magnetic field and the emission current to anode will not decrease by using this magnetic focusing method. The second studying case for considering the space-charge effect is simulation on the silicon field emission diode. The
primarily result shows that simulated I-V curve agrees with the experimental work, especially when simulation work function is set to 4.5eV.
2. In simulating 3D DC and RF gas discharge plasmas, the spatial distributions of plasma macroparameters are presented. The results show that sheath does play an important role in sustaining plasma, which providing the field to accelerate the particles. Due to the 3D geometric shape of electrodes, there is a very strong electric field existing around the edges of electrodes. In other words, this field may lead to unexpected ion bombardment.
3. In simulating 3D DC and RF magnetron plasmas, the spatial distributions of plasma macroparameters are presented. The concentric cylindrical magnets with different magnetization are solved in advance, and the results that magnetic field is proportional to magnetization. With these magnetic fields, electrons are confined between N and S poles and exhibit the E×B drift motion, which leads to high ionization rate in plasma under very low gas pressure. The results also show that as M or γincreases, the plasma density increases.
Chapter 6
Concluding Remarks
6.1 Summary
In this dissertation, a general parallel three-dimensional electrostatic particle-in-cell scheme with finite element method (PIC-FEM) using unstructured mesh is proposed and verified. A multi-level graph-partitioning technique is used to dynamically decompose the computational domain to improve the parallel performance during runtime. Completed parallelized PIC-FEM code is used to simulate several important physical problems, including field emission, DC/RF gas discharge and DC/RF magnetron plasmas. In brief summary, the major achievements in the present dissertation can be listed as follows:
1. A parallelized three-dimensional electrostatic Poisson’s equation solver using Galerkin finite element method with an unstructured mesh is developed and validated. Study of parallel performance of the parallelized PIC-FEM code is performed on the HP-IA64 clusters. With subdomain-by-subdomain scheme for parallel conjugate gradient method, parallel efficiency can reach 84% at 32 processors of HP PC clusters at NCHC. This code coupled with PAMR was used to accurately and efficiently simulate field emission from emitter with complicated geometry without considering space-charge effects, as demonstrated in Chapter 5.
2. A parallelized three-dimensional vector potential magnetostatic Poisson’s equation solver using Galerkin finite element method with an unstructured mesh is developed and validated. Study of parallel performance of the parallelized
PIC-FEM code is performed on the HP-IA64 clusters. With subdomain-by-subdomain scheme for parallel conjugate gradient method, parallel efficiency can reach 75% at 32 processors of HP PC clusters at NCHC.
This code was used to simulate the magnetic field around permanent magnets or coils for magnetron plasma simulation as demonstrated in Chapter 5.
3. A general parallelized three-dimensional PIC-FEM code is developed and validated. This PIC-FEM code integrates the parallelized Poisson’s equation solver with the PIC and Monte Carlo collision (MCC) schemes on an unstructured tetrahedral mesh. Charged particles can be traced either cell-by-cell on an unstructured mesh. This is achieved using leap-frog time-integration method and Boris rotational scheme when magnetic field is involved. Charge assignment and force (field) interpolation between charged particles and grid points is implemented using the same interpolation function originated from the FEM. In addition, dynamic domain decomposition (DDD) with weighting based on number of particles is used to balance the workload among processors during runtime. Study of parallel performance of the parallelized PIC-FEM code is performed on the HP-IA64 clusters. Results using DDD for a typical RF gas discharge show that parallel efficiency can reach 83% at 32 processors.
4. Completed parallelized PIC-FEM code was used to simulate several important problems to demonstrate its superior capability in handling practical problems.
These problems include field emission from a silicon tip under external electric field, two typical three-dimensional DC and RF gas-discharge plasmas, and two typical three-dimensional DC and RF magnetron plasmas with permanent magnets. Results are either compared well with experiments or demonstrate the correct physical pictures as expected.
6.2 Recommendation for Future Study
In the present dissertation, we have developed and tested a parallelized three-dimensional PIC-FEM code using an unstructured mesh on memory-distributed parallel machines. We have also applied this code to simulate several important physical problems. Based on the viewpoints of further improving this PIC-FEM code, several possible directions of research are recommended for the future study and are summarized as follows:
1. To implement a better preconditioner for parallel conjugate gradient method for solving the Poisson’s equation more efficiently to shorten the runtime and improve the speedup at higher number of processors.
2. To incorporate a Maxwell’s equation solver that uses edge-based finite element method into the present PIC-FEM code to further extend its applicability in plasma related simulation, such as ICP, ECR and microwave plasmas.
3. To incorporate a simulation module that can model realistic external circuits into the present PIC-FEM code, which are often coupled to a RF-type gas discharge.
4. To extend the database of collision data for other types of plasma, such as methane with hydrogen, which are very important in growing carbon nanotubes.
5. To incorporate a parallelized DSMC (direct simulation Monte Carlo) module into the PIC-FEM code to consider the neutral transport self-consistently that is
very important in some plasma flow, such as magnetron plasma.
Reference
[1] Abril, I., Gras-Marti, A., and Valles-Abarca, J. A., “Energy distributions of particles striking the cathode in a glow discharge,” Physical Review A, Vol. 28 pp. 3677-3678, 1983.
[2] Akarsu, E., Dincer, K., Haupt, T., and Fox, G. C., “Particle-in-Cell Simulation Codes in High Performance Fortran,” Proceeding of Super Computing Conference, Pittsburgh, PA, 1996.
[3] Amestoy, P. R., Duff, I. S., and L'Excellent, J. Y., “Multifrontal parallel distributed symmetric and unsymmetric solvers,” Computer Methods in Applied Mechanical Engineering, Vol. 184, pp. 501-520, 2000.
[4] Barnard, S. T. and Simon, H. D., “A Fast Multilevel Implementation of Recursive Spectral Bisection for Partitioning Unstructured Problems,”
Concurrence. Practice Experience, Vol. 6, pp.101-117, 1994.
[5] Barrett, R. and Berry, M., “Templates for the solution of linear systems: Building blocks for iterative methods,” SIAM, Philadelphia, 1994.
[6] Beltzer, A. I., “Variational and finite element methods: a symbolic computation approach,＂ Springer-Verlag, Berlin, 1990.
[7] Birdsall, C. K. and Langdon, A. B., “Plasma Physics via Computer Simulation,”
Institute of Physics Publishing, Bristol, 1991.
[8] Birdsall, C. K., “Particle-in-cell charged-particle simulations, plus Monte Carlo collisions with neutral atoms, PIC-MCC,” IEEE Transaction on Plasma Science, Vol. 19, pp. 65-85, 1991.
[9] Bogaerts, A., Neyts, E., Gijbels, R., and Mullen J. V., “Gas discharge plasmas and their applications,” Spectrochimica Acta Part B, Vol. 57, pp. 609-658, 2002.
[10] Brackbill, J. U. and Cohen, B. I., “Multiple Time Scales,” Academic Press, 1985.
[11] Bukowski, J. D. and Graves, D. B., “Two-dimensional fluid model of an inductively coupled plasma with comparison to experimental spatial profiles,”
Journal of Applied Physics, Vol. 80, pp. 2614-2623, 1996.
[12] Burnett, D. S., “Finite element analysis: from concepts to applications,＂
Addison-Wesley, Reading Mass., 1987.
[13] Cheng, D. K., “Field and wave electromagnetics,” Addison Wesley, 2nd edition, 1989.
[14] Choi, W. B., Chung, D. S., and Kang, J. H., “Fully sealed, high-brightness carbon-nanotube field-emission display” Applied Physics Letter, Vol. 75, pp.
3129-3131, 1999.
[15] Decyk, V. K., “Skeleton PIC Codes for parallel computers”, Computer Physics Communications, Vol.87, pp. 87-94, 1995.
[16] Decyk, V. K. and Norton, C. D., “UCLA Parallel PIC Framework,” Computer
Physics Communications, Vol. 164, pp. 80-85, 2004.
[17] de Heer, W. A., Chatelain, A., and Ugarte, D., “A Carbon Nanotube Field-Emission Electron Source,” Science Vol. 270, pp.1179-1180, 1995.
[18] Deshmukh, S. C. and Economou, D. J., “Factors affecting the Cl atom density in a chlorine discharge,” Journal of Applied Physics, Vol. 72, pp. 4597-4607, 1992.
[19] Economou, D. J., “Modeling and simulation of plasma etching reactors for microelectronics,” Thin solid film, Vol. 365, pp.348-367, 2000.
[20] Farhat, C., Maman, N., and Brown, G., “Mesh Partitioning for Implicit Computations via Domain Decomposition,” International Journal of Numerical Methods in Engineering, Vol. 38, pp.989-1000, 1995.
[21] Fenner, R. T., “Finite element methods for engineers,＂ Imperial College Press, London, 1996.
[22] Font, G. I., Boyd, I. D., and Bakakriahnan,J., “Effects of wall recombination on the etch rate and plasma composition of an etch reactor,” Journal of Vacuum Science and Technology B, Vol. 16, pp. 2057-2064, 1998.
[23] Fowler, R. H. and Nordheim, L., “Electron Emission in Intense Electric Fields,”
Proc. Royal Society A, Vol. 119, pp.173-181, 1928.
[24] Godyak, V. A., Piejak, R. B., and Alexandrovich, B. M. “Measurements of electron energy distribution in low-pressure RF discharges,” Plasma Sources
Science and Technology, Vol.1, pp. 36-38, 1992.
[25] Gogolides, E. and Sawin, H. H., “Continuum Modeling of Radio-Frequency Glow Discharges I: Theory and Results for Electropositive and Electronegative Gases,” Journal of Applied Physics, Vol. 72, pp. 3971-3987, 1992.
[26] Golub, G. H. and Van Loan, C. F., “Matrix computations,＂ Johns Hopkins University Press, Baltimore, 1996.
[27] Gullerud, A. and Dodds, R. J., “MPI-based implementation of a PCG solver using an EBE architecture and preconditioner for implicit, 3-D finite element analysis,” Computers and Structures, Vol. 79, pp. 553-575, 2001.
[28] Halbach, K., “Design of Permanent Multipole Magnets with. oriented Rare Earth Cobalt Material,”Nuclear Instruments and Methods, Vol. 169, pp.1-10, 1980.
[29] Hockney, R. W. and Eastwood, J. W., “Computer simulation using particles,＂
Adam Hilger, Bristol & New York, 1988.
[30] Hodgson, D. and Jimack, P., “Efficient Mesh Partitationing for Parallel P.D.D.
Solvers on Distributed Memory Machines,” Proceedings of the sixth SIAM conference on Parallel Processing for Scientific Computing, 1993.
[31] Hong, D., Aslam, M., Feldmann, M., and Olinger, M. “Simulations of fabricated field emitter structures,” Journal of Vacuum Science and Technology B, Vol. 12, pp.764-769, 1994.
[32] Hu, Y. and Huang. C. H., “Computer simulation of the field emission properties of multiwalled carbon nanotubes for flat panel displays,” Journal of Vacuum Science and Technology B, Vol. 21, pp. 1648-1654 , 2003.
[33] Itoh, S., Tanaka, M., and Tonegawa, T., “Development of field emission displays,” Journal of Vacuum Science and Technology B, Vol., 22, pp.1362-1366, 2004.
[34] Jimack, P. and Touheed, N., “Developing Parallel Finite Element software Using MPI,” Department of Scientific computing, University of Leeds, UK, 1997.
[35] Jin, J., “The finite element method in electromagnetics,＂ John Wiley & Sons, New York, 2002.
[36] Karypis, G.. and Kumar, V., “Metis: Unstructured Graph Partitioning and Sparse Matrix Ordering, Version 2.0 User Manual,” Minneapolis MN55455, Department of Computer Science, University of Minnesota, U.S.A, 1995.
[37] Karypis, G., Schloegel, K., and Kumar, V., “ParMetis: Parallel version of METIS,” Department of Computer Science, University of Minnesota, September 1998.
[38] Khan, A. and Topping, B., “Parallel finite element analysis using Jacobi-conditioned conjugate gradient algorithm,” Advance in Engineering Soft ware, Vol. 25, pp. 309-319, 1996.
[39] Kawamura, E., Birdsall, C. K., and Vahedi, V., “Methods in speeding up PIC-MCC codes applied to RF plasma discharges,” ERL report, University of California at Berkeley, 2000.
[40] Kline, L. E., Partlow, W. D., and Bies, W. E., “Electron and chemical kinetics in methane rf glow-discharge deposition plasmas,” Journal of Applied Physics, Vol.
65, pp. 70-78, 1989.
[41] Kushner, M. J., “Modeling of Microdischarge Devices: Plasma and Gas Dynamics “, Journal of Physics D, Vol. 38, pp. 1633-1643, 2005.
[42] Kushner, M. J., Collison, W .Z., Grapperhaus, M. J., Holl, J. P., and Barnes, M.
S., “A 3-dimensional Model for Inductively Coupled Plasma Etching Reactors:
Azimuthal Symmetry and Coil Properties”, Journal Applied Physics, Vol. 80, pp.
1337-1344, 1996.
[43] Kushner, M. J., “Consequences of Asymmetric Pumping in Low Pressure Plasma Processing Reactors: A 3-dimensional Modeling Study”, Journal Applied.
Physics, Vol. 82, pp. 5312-5320, 1997.
[44] Kythe, P. K., “An introduction to boundary element methods,＂ CRC Press, Boca Raton, 1995
[45] Lan, Y. C., Lai, J. T., Chen, S. H., Wang, W. C., Tsai, C. H., Tsai, K. L., and Sheu, C. Y., “Simulation of focusing field emission devices,” Journal of Vacuum
Science and Technology B, Vol. 18, pp. 911-913, 2000.
[46] Lan, Y. C., Lee, C. T., Hu, Y., Chen, S. H., Lee, C. C., Tsui, B. Y., and Lin, T. L.,
“Simulation study of carbon nanotube field emission display with under-gate and planar-gate structures,” Journal of Vacuum Science and Technology B, Vol. 22, pp. 1244-1249 ,2004.
[47] Leupold, H. A., Tilak, A. S., and PotenzianiII, E., “Adjustable multi-tesla perma-.
nent magnet field sources,” IEEE Transaction on Magnetics., Vol. 29, pp.
2902-2904, 1993.
[48] Leupold, H. A., Tilak, A. S., and PotenzianiII, E., “Permanent magnet spheres:
Design, construction, and application,” Journal of Applied Physics, Vol. 87, pp.
4730-4734, 2000.
[49] Lee, S. Y. and Azari, N. G., “Hybrid task decomposition for Particle-In-Cell methods on message passing systems,” Proceedings of International Conference on Parallel Processing, St. Charles, IL, Vol. 3, pp. 141-144,1992.
[50] Lei, W., Wang, B., and Yin, H., “Simulation study on performance of field emitter array,” Journal of Vacuum Science and Technology B, Vol. 16, pp.
2881-2886 ,1998.
[51] Lian, Y. Y., Hsu, K. H., Shao, Y. L., Lee, Y. M., Jeng, Y. W., and Wu, J. S.,
“Parallel Adaptive Mesh-Refining Scheme on Three-dimensional Unstructured
Tetrahedral Mesh and Its Applications," Computer Physics Communications, 2006 (accepted).
[52] Lieberman, M. A. and Lichtenberg, A. J. “Principles of Plasma Discharges and Materials Processing,” Wiley, New York, 1994.
[53] Liewer, P. C. and Decyk, V. K., “A general Concurrent Algorithm for Plasma Particle-In-Cell Simulation Codes,” Journal of Computational Physics, Vol. 85, pp. 302-322,1989.
[54] Lymberopoulos, D. P. and Economou, D. J., “Two-dimensional simulation of polysilicon etching with chlorine in a high density plasma reactor,” IEEE Transaction on Plasma Science, Vol.23, pp. 573-580, 1995.
[55] Mahony, C. M. O., McFarland, J., Steen, P. G., and Graham, W. G., “Structure observed in measured electron energy distribution functions in capacitively coupled radio frequency hydrogen plasmas,” Applied Physics Letters, Vol. 75 , pp. 331-333, 1999.
[56] Meeks, E., Larson, R. S., Vosen, S. R., and Shon, J. W., “Modeling Chemical Downstream Etch Systems for NF3/O2 Mixtures,” Journal of the Electrochemical
Society, Vol.144, pp. 357-366, 1997.
[57] Meeks, E. and Shon, J. W., “Modeling of plasma-etch processes using well stirred reactor approximations and including complex gas-phase and surface
reactions,” IEEE Transaction on Plasma Science, Vol. 23, pp. 539-549, 1995.
[58] Meyyappan, M., “Computational Modeling in Semiconductor Processing”
Artech House, Boston, 1994.
[59] Midha, V. and Economou, D. J., “Spatio-temporal evolution of a pulsed chlorine discharge,” Plasma Sources Science and Technology, Vol. 9, pp. 256-269, 2000.
[60] Nanbu, K., “Probability theory of electron-molecule, ion-molecule, molecule-molecule, and Coulomb collisions for particle modeling of materials processing plasmas and Gases,” IEEE Transaction on Plasma Science, Vol. 28 , pp. 971- 990, 2000.
[61] Nanbu, K., “Simple method to determine collisional event in Monte Carlo simulation of electron-molecule collision,” Japanese Journal of Applied Physics, Vol. 33, pp. 4752-4753, 1994.
[62] Nedelea, T. and Urbassek, H. M., “Parametric study of ion acceleration in a one-dimensional plasma expansion using the particle-in-cell simulation,”
Physical Review E, Vol. 69, pp. 056408-1~6, 2004.
[63] Nilsson, L., Groening, O., Emmenegger, C., Kuettel, O., Schaller, E., Schlapbach L., Kind, H., Bonard, J.M. ,and Kern, K., “Scanning field emission from patterned carbon nanotube films,” Applied Physics Letter, Vol. 76, pp.
2071-2073, 2000.
[64] Nishimura, K., Shen, Z., Fujikawa, M., Hosono, A., Hashimoto, N., Kawamoto, S., Watanabe, S., and Nakata, S., Journal of Vacuum Science and Technology B, Vol. 22, pp. 1377-1381, 2004.
[65] Onate, E., Periauxand, J., and Samuelsson, A., “The finite element method in the 1990's,＂ Springer-Verlag, Berlin, 1991.
[66] Pirio, G., Legagneux, P., Pribat, D., Teo, K. B. K., Chhowalla, M., Amaratunga, G.
A. J., and Milne, W. I., “Fabrication and electrical characteristics of carbon nanotube field emission microcathodes with an integrated gate electrode,”
Nanotechnology, Vol. 13, pp.1-4, 2002.
[67] Raizer, Y. P., Shneider, M. N., and Yatsenko, N. A., “Radio-Frequency Capacitive Discharges,” CRC Press, Boca Raton, London, Tokyo, 1995.
[68] Rao, A.M., Jacques, D., Haddon, R.C., Zhu, W., Bower, C., and Jin, S., “In situ-grown carbon nanotube array with excellent field emission characteristics,”
Applied Physics Letter, Vol. 76, pp. 3813-3815, 2000.
[69] Saad, Y., “Iterative methods for sparse linear systems,” Society for Industrial and Applied Mathematics, Philadelphia, 2003.
[70] Santi, M., Cheng S., Celik, M., Martinez-Sanchez, M., and Peraire J., “further development and preliminary results of the aquila hall thruster plume model,”
39th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit
Huntsville, AL, pp.20-23, July 2003.
[71] Seidel, D. B., Plimpton, S. J., Pasik, M. F., Coats, R. S., and Montry, G. R.,
“Dynamic load-balancing for a parallel electromagnetic particle-in-cell code,”
Pulsed Power Plasma Science, Vol. 2 pp. 1000-1003, 2002.
[72] Serikov, V. V. and Nanbu, K., “The analysis of background gas heating in ditect current sputtering discharge via particle simulation,” Journal of Applied Physics, Vol. 82, pp. 5948-5957, 1997.
[73] Shao, Y. L., “Development and Verification of a 3-D Parallelized Poisson-Boltzmann Equation Solver Using Finite Element Method with Adaptive Mesh Refinement, ＂ PhD dissertation, Department of Mechanical Engineering, National Chiao-Tung University, Taiwan, 2006.
[74] Shon, C H. and Lee, J. K., “Modeling of Magnetron Sputter Plasmas,” Applied Surface Science, Vol.192, pp. 258-270, 2002
[75] Shon, C. H., Park, J. S., and Lee, J. K., “Kinetic and Steady-State Properties of Magnetron Sputter with Three-Dimensional Field,” Japanese Journal Applied Physics Part I, Vol. 38, pp. 4440-4446, 1999.
[76] Shon, C. H., Lee, J. K., Lee, H. J., Shin, Y. K., Yang, Y., and Chung, T. H., IEEE Transaction on Plasma Science, Vol. 26, pp.1635-1644, 1998.
[77] Shon, C. H., Lee, H. J., and Lee, J. K., “Method to increase the simulation speed
of particle-in-cell (PIC) code”, Computer Physics Communications, Vol. 141, pp.323-329, 2001.
[78] Silvester, P. P. and Pelosi, G., “Finite elements methods and techniques for wave electromagnetics,＂ IEEE, New York, 1994.
[79] Simon, H., “Partitioning of Unstructured Problems for Parallel Processing,”
Computing Systems in Engineering, Vol. 2, pp. 135-148, 1991.
[80] Spindt, C. A., “A Thin-Film Field-Emission Cathode,” Journal of Applied Physics Vol. 39, pp.3504-3505, 1968.
[81] Spirkin, A. and Gatsonis, N., “Unstructured 3D PIC simulation of the flow in a retarding potential analyzer,” Computer Physics. Communication, Vol. 164, pp.
383-389, 2004.
[82] Sullivan, D. M., “Electromagnetic Simulation Using FDTD Method,” IEEE Press, New York, 2000.
[83] Surendra, M., Graves, D. B., and Morey, I. J., “Electron heating in low-pressure rf glow discharges,” Applied Physics Letters, Vol. 56, pp.1022-1024, 1990.
[84] Thiagarajan, G. and Aravamuthan, V., “Parallelization Strategies for Element-by-Element Preconditioned Conjugate Gradient Solver Using High-Performance Fortran for Unstructured Finite-Element Application On Linux Clusters,” Journal of Computing in Civil Engineering, Vol. 16, pp.1-4,
2002.
[85] Tseng, K. C., “Development of a General-Purpose Parallel Three-Dimensional DSMC code (PDSC) Using Unstructured Tetrahedral Mesh,” PhD dissertation, Department of Mechanical Engineering, National Chiao-Tung University, Taiwan, 2005.
[86] Turner, M. M., Doyle, R. A., and Hopkins, M. B., “Measured and simulated electron energy distribution functions in a low-pressure radio frequency discharge in argon,,” Applied Physics Letters, Vol. 62, pp. 3247-3249, 1993.
[87] Vahedi, V., DiPeso, G., Birdsall, C. K., Lieberman, M. A., and Ronglien, T. D.
“Capacitive RF discharges modeled by particle-in-cell Monte Carlo simulation. 1.
Analysis of numerical techniques,” Plasma Sources Science and Technology, Vol. 2, pp. 261-268, 1993.
[88] Umashankar, K., “Computational electromagnetics,＂ Artech House, Boston 1993.
[89] Ventzek, P. L. G., Sommerer, T. J., Hoekstra, R. J., and Kushner, M. J., “A 2-dimensional Hybrid Model of Inductively Coupled Plasma Sources for Etching”, Applied Physics Letter, Vol. 62, pp.605-607, 1993.
[90] Ventzek, P. L. G., Hoekstra, R. J., and Kushner, M. J., “2-Dimensional Modeling of High Plasma Density Inductively Coupled Sources for Materials Processing,”
Journal of Vacuum Science and Technology B, Vol. 12, pp. 461-477, 1994.
[91] Vahedi, V. and Surendra, M., “A monte carlo collision model for the particle-in-cell method: applications to argon and oxygen discharges,” Computer Physics Communications, Vol. 87, pp. 179-198, 1995.
[92] Vahedi, V., DiPeso, G., Birdsall, C.K., Lieberman, M.A., and Roglien, T.D.,
“Capacitive RF discharges modeled by particle-in-cell Monte Carlo simulation.
“Capacitive RF discharges modeled by particle-in-cell Monte Carlo simulation.