The derivation of moment equations for the theoretical description of electrons is of interest for modelling of gas discharge plasmas and semiconductor devices. Usually, certain artificial closure assumptions are applied in order to derive a closed system of moment equations from the electron Boltzmann equation. Here, a novel four-moment model for the description of electrons in nonthermal plasmas is derived by an expansion of the electron velocity distribution function in Legendre polynomials. The proposed system of partial differential equations is consistently closed by definition of transport coefficients that are determined by solving the electron Boltzmann equation and are then used in the fluid calculations as function of the mean electron energy. It is shown that the four-moment model can be simplified to a new drift-diffusion approximation for electrons without loss of accuracy, if the characteristic frequency of the electric field alteration in the discharge is small in comparison with the momentum dissipation frequency of the electrons. Results obtained by the proposed fluid models are compared to those of a conventional drift-diffusion approximation as well as to kinetic results using the example of low pressure argon plasmas. It is shown that the results provided by the new approaches are in good agreement with kinetic results and strongly improve the accuracy of fluid descriptions of gas discharges.
KeywordsMoment EquationsPlasma ModellingElectron Transport
B. Eliasson, M. Hirth and U. Kogelschatz, “Ozone Synthesis from Oxygen in Dielectric Barrier Discharges,” Journal of Physics D: Applied Physics, Vol. 20, No. 11, 1987, pp. 1421-1437. doi:10.1088/0022-3727/20/11/010
U. Kogelschatz, “Dielectric-Barrier Discharges: Their History, Discharge Physics, and Industrial Applications,” Plasma Chemistry and Plasma Processing, Vol. 23, No. 1, 2003, pp. 1-46. doi:10.1023/A:1022470901385
G. Lister, J. Lawler, W. Lapatovich and V. Godyak, “The Physics of Discharge Lamps,” Reviews of Modern Physics, Vol. 76, No. 2, 2004, pp. 541-598. doi:10.1103/RevModPhys.76.541
J. Ehlbeck, U. Schnabel, M. Polak, J. Winter, T. von Woedtke, R. Brandenburg, T. von dem Hagen and K.-D. Weltmann, “Low Temperature Atmospheric Pressure Plasma Sources for Microbial Decontamination,” Journal of Physics D: Applied Physics, Vol. 44, No. 1, 2011, Article ID: 13002. doi:10.1088/0022-3727/44/1/013002
M. J. Kushner, “Monte-Carlo Simulation of Electron Properties in rf Parallel Plate Capacitively Coupled Discharges,” Journal of Applied Physics, Vol. 54, No. 9, 1983, pp. 4958-4965. doi:10.1063/1.332763
D. B. Graves, “Fluid Model Simulations of a 13.56-MHz rf Discharge: Time and Space Dependence of Rates of Electron Impact Excitation,” Journal of Applied Physics, Vol. 62, No. 1, 1987, pp. 88-94. doi:10.1063/1.339111
J.-P. Boeuf, “A Two-Dimensional Model of dc Glow Discharges,” Journal of Applied Physics, Vol. 63, No. 5, 1988, pp. 1342-1349. doi:10.1063/1.339961
G. J. M. Hagelaar, G. M. W. Kroesen, U. van Slooten and H. Schreuders, “Modeling of the Microdischarges in Plasma Addressed Liquid Crystal Displays,” Journal of Applied Physics, Vol. 88, No. 5, 2000, pp. 2252-2262. doi:10.1063/1.1287529
R. Bussiahn, S. Gorchakov, H. Lange, D. Loffhagen and D. Uhrlandt, “Ac Operation of Low-Pressure He Xe Lamp Discharges,” Journal of Physics D: Applied Physics, Vol. 40, 2007, Article ID: 3882. doi:10.1088/0022-3727/40/13/S07
M. Wendt, S. Peters, D. Loffhagen, A. Kloss and M. Kettlitz, “Breakdown Characteristics of High Pressure Xenon lamps,” Journal of Physics D: Applied Physics, Vol. 42, No. 18, 2009, Article ID: 185208. doi:10.1088/0022-3727/42/18/185208
S. Chapman and T. G. Cowling, “The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases,” 3rd Edition, Cambridge University Press, Cambridge, 1998.
M. Wilcoxson and V. J. Manousiouthakis, “Simulation of a Three-Moment Fluid Model of a Two-Dimensional Radio Frequency Discharge,” Chemical Engineering Science, Vol. 51, No. 7, 1996, pp. 1089-1106.
H. C. Kim and V. I. Manousiouthakis, “Dually Driven Radio Frequency Plasma Simulation with a Three Moment Model,” Journal of Vacuum Science & Technology A, Vol. 16, No. 4, 1998, p. 2162. doi:10.1116/1.581324
S. Elaissi, M. Yousfi, H. Helali, S. Kazziz, K. Charrada and M. Sassi, “Radio-Frequency Electronegative Gas Discharge Behaviour in a Parallel-Plate Reactor for Material Processing,” Plasma Devices and Operations, Vol. 14, No. 1, 2006, pp. 27-45. doi:10.1080/10519990500493874
M. M. Becker, D. Loffhagen and W. Schmidt, “A Stabilized Finite Element Method for Modeling of Gas Discharges,” Computer Physics Communications, Vol. 180, No. 8, 2009, pp. 1230-1241. doi:10.1016/j.cpc.2009.02.001
G. K. Grubert, M. M. Becker and D. Loffhagen, “Why the Local-Mean-Energy Approximation Should Be Used in Hydrodynamic Plasma Descriptions Instead of the Local-Field Approximation,” Physical Review E, Vol. 80, No. 3, 2009, Article ID: 036405. doi:10.1103/PhysRevE.80.036405
A. Derzsi, P. Hartmann, I. Korolov, J. Karacsony, G. Bano, and Z. Donko, “On the accuracy and limitations of fluid models of the cathode region of dc glow discharges,” Journal of Physics D: Applied Physics, Vol. 42, No. 22, 2009, Article ID: 225204. doi:10.1088/0022-3727/42/22/225204
I. Rafatov, E. A. Bogdanov and A. A. Kudryavtsev, “On the Accuracy and Reliability of Different Fluid Models of the Direct Current Glow Discharge,” Physics of Plasmas, Vol. 19, No. 3, 2012, Article ID: 033502.
A. Bogaerts, R. Gijbels and W. J. Goedheer, “Hybrid Monte Carlo-Fluid Model of a Direct Current Glow Discharge,” Journal of Applied Physics, Vol. 78, No. 4, 1995, pp. 2233-2241. doi:10.1063/1.360139
Z. Donko, “Hybrid Model of a Rectangular Hollow Cathode Discharge,” Physical Review E, Vol. 57, No. 6, 1998, pp. 7126-7137. doi:10.1103/PhysRevE.57.7126
D. Loffhagen and F. Sigeneger, “Advances in Boltzmann Equation Based Modelling of Discharge Plasmas,” Plasma Sources Science and Technology, Vol. 18, No. 3, 2009, Article ID: 034006. doi:10.1088/0963-0252/18/3/034006
R. E. Robson, R. D. White and Z. L. Petrovic, “Colloquium: Physically Based Fluid Modeling of Collisionally Dominated Low-Temperature Plasmas,” Reviews of Modern Physics, Vol. 77, No. 4, 2005, pp. 1303-1320. doi:10.1103/RevModPhys.77.1303
P. Nicoletopoulos and R. E. Robson, “Periodic Electron Structures in Gases: A Fluid Model of the ‘Window’ Phenomenon,” Physical Review Letters, Vol. 100, No. 12, 2008, Article ID: 124502. doi:10.1103/PhysRevLett.100.124502
L. L. Alves, “Fluid Modelling of the Positive Column of Direct-Current Glow Discharges,” Plasma Sources Science and Technology, Vol. 16, No. 3, 2007, p. 557. doi:10.1088/0963-0252/16/3/015
M. Gnybida, D. Loffhagen and D. Uhrlandt, “Fluid Modeling and Analysis of the Constriction of the DC Positive Column in Argon,” IEEE Transactions on Plasma Science, Vol. 37, No. 7, 2009, pp. 1208-1218. doi:10.1109/TPS.2009.2021419
R. E. Robson and K. F. Ness, “Velocity Distribution Function and Transport Coefficients of Electron Swarms in Gases: Spherical-Harmonics Decomposition of Boltzmann’s Equation,” Physical Review A, Vol. 33, No. 3, 1986, pp. 2068-2077. doi:10.1103/PhysRevA.33.2068
R. E. Robson, R. Winkler and F. Sigeneger, “Multiterm Spherical Tensor Representation of Boltzmann’s Equation for a Nonhydrodynamic Weakly Ionized Plasma,” Physical Review E, Vol. 65, No. 5, 2002, Article ID: 056410. doi:10.1103/PhysRevE.65.056410
R. Winkler, J. Wilhelm and V. Schüller, “Legendre Polynomial Expansion and General Spherical Harmonics Expansion in the Boltzmann Equation of the Lorentz Plasma,” Contributions to Plasma Physics, Vol. 10, No. 1, 1970, pp. 51-77. doi:10.1002/ctpp.19700100105
S. Arndt, D. Uhrlandt and R. Winkler, “Space-Dependent Kinetics of Electrons in the Anode region of a Glow Discharge,” Journal of Physics D: Applied Physics, Vol. 34, No. 13, 2001, pp. 1982-1992. doi:10.1088/0022-3727/34/13/309
D. Loffhagen and R. Winkler, “Spatiotemporal Relaxation of Electrons in Non-Isothermal Plasmas,” Journal of Physics D: Applied Physics, Vol. 34, No. 9, 2001, pp. 1355-1366. doi:10.1088/0022-3727/34/9/312
V. I. Kolobov and R. R. Arslanbekov, “Simulation of Electron Kinetics in Gas Discharges,” IEEE Transactions on Plasma Science, Vol. 34, No. 3, 2006, pp. 895-909. doi:10.1109/TPS.2006.875850
D. Loffhagen and R. Winkler, “Multi-Term Treatment of the Temporal Electron Relaxation in He, Xe and N2 Plasmas,” Plasma Sources Science and Technology, Vol. 5, No. 4, 1996, pp. 710-719. doi:10.1088/0963-0252/5/4/013
D. Loffhagen, F. Sigeneger and R. Winkler, “Study of the Electron Kinetics in the Anode Region of a Glow Discharge by a Multiterm Approach and Monte Carlo Simulations,” Journal of Physics D: Applied Physics, Vol. 35, No. 14, 2002, pp. 1768-1776. doi:10.1088/0022-3727/35/14/319
G. K. Grubert and D. Loffhagen, “Nonequilibrium Properties of Electrons in Oxygen Plasmas,” IEEE Transactions on Plasma Science, Vol. 35, No. 5, 2007, pp. 1215-1222. doi:10.1109/TPS.2007.905115
M. M. Becker, G. K. Grubert and D. Loffhagen, “Boundary Conditions for the Electron Kinetic Equation Using Expansion Techniques,” The European Physical Journal Applied Physics, Vol. 51, No. 1, 2010, p. 11001-11007. doi:10.1051/epjap/2010073
G. M. Petrov and J. L. Giuliani, “Inhomogeneous Model of an Ar-Hg Direct Current Column Discharge,” Journal of Applied Physics, Vol. 94, No. 1, 2003, pp. 62-75. doi:10.1063/1.1576895
D. Uhrlandt, M. Schmidt, J. F. Behnke and T. Bindemann, “Self-Consistent Description of the dc Column Plasma Including Wall Interaction,” Journal of Physics D: Applied Physics, Vol. 33, No. 19, 2000, pp. 2475-2482. doi:10.1088/0022-3727/33/19/318
D. Loffhagen, S. Arndt, F. Sigeneger, D. Uhrlandt and R. Winkler, “Electron Kinetics and Self-Consistent Description of Inhomogeneous and Nonstationary Plasmas,” Contributions to Plasma Physics, Vol. 45, No. 5-6, 2005, pp. 309-318. doi:10.1002/ctpp.200510036
G. J. M. Hagelaar and L. C. Pitchford, “Solving the Boltzmann Equation to Obtain Electron Transport Coefficients and Rate Coefficients for Fluid Models,” Plasma Sources Science and Technology, Vol. 14, No. 4, 2005, p. 722. doi:10.1088/0963-0252/14/4/011
L. L. Alves, G. Gousset and S. Vallee, “Nonequilibrium Positive Column Revisited,” IEEE Transactions on Plasma Science, Vol. 31, No. 4, 2003, pp. 572-586. doi:10.1109/TPS.2003.815484
F. Sigeneger and R. Winkler, “Nonlocal Transport and Dissipation Properties of Electrons in Inhomogeneous Plasmas,” IEEE Transactions on Plasma Science, Vol. 27, No. 5, 1999, pp. 1254-1261. doi:10.1109/27.799801
J. H. Ingold, “Moment Method Applied to Gaseous Electronics,” Physical Review A, Vol. 40, No. 7, 1989, pp. 3855-3863. doi:10.1103/PhysRevA.40.3855
M. Surendra and M. Dalvie, “Moment Analysis of rf Parallel-Plate-Discharge Simulations Using the Particle-in-Cell with Monte Carlo Collisions Technique,” Physical Review E, Vol. 48, No. 5, 1993, pp. 3914-3924. doi:10.1103/PhysRevE.48.3914
V. E. Golant, A. P. Zilinskij, I. E. Sacharov and S. C. Brown, “Fundamentals of Plasma Physics,” Wiley, New York, 1980.
R. E. Robson, “Physics of Reacting Particle Swarms in Gases,” Journal of Chemical Physics, Vol. 85, No. 8, 1986, pp. 4486-4501. doi:10.1063/1.451769
R. E. Robson, P. Nicoletopoulos, B. Li and R. D. White, “Kinetic Theoretical and Fluid Modelling of Plasmas and Swarms: The Big Picture,” Plasma Sources Science and Technology, Vol. 17, No. 2, 2008, Article ID: 024020. doi:10.1088/0963-0252/17/2/024020
K. Blotekjaer, “Transport Equations for Electrons in Two-Valley Semiconductors,” IEEE Transactions on Electron Devices, Vol. 17, No. 1, 1970, pp. 38-47. doi:10.1109/T-ED.1970.16921
S. F. Liotta and H. Struchtrup, “Moment Equations for Electrons in Semiconductors: Comparison of Spherical Harmonics and Full Moments,” Solid-State Electronics, Vol. 44, No. 1, 2000, pp. 95-103. doi:10.1016/S0038-1101(99)00215-4
A. Bringer and G. Sch?n, “Extended Moment Equations for Electron Transport in Semiconducting Submicron Structures,” Journal of Applied Physics, Vol. 64, No. 5, 1988, pp. 2447-2455. doi:10.1063/1.341680