On simplifying approaches to the solution of the Boltzmann equation in spatially inhomogeneous plasmas

With the continuing progress of plasma modelling in recent years the desire arose to develop simplified approaches to the kinetic description of the electron component in weakly ionized plasmas. Methods which are based on the direct solution of the Boltzmann equation in some limiting situations, namely the `nonlocal' approximation in a weakly collisional plasma and the `local' approximation in a highly collisional plasma, may be much more efficient than conventional simulation techniques. In this paper the foundation of these approaches is reviewed. Their quantitative accuracy and their applicable range is examined on the basis of a comparison to the numerical solution of the complete, space-dependent Boltzmann equation for a positive column plasma.

[1]  J. Loureiro,et al.  Characteristics of high-frequency and direct-current argon discharges at low pressures: a comparative analysis , 1984 .

[2]  Vladimir Kolobov,et al.  Nonlocal electron kinetics in collisional gas discharge plasmas , 1995 .

[3]  M. Kushner.  Application of a particle simulation to modeling commutation in a linear thyratron , 1987 .

[4]  T. Holstein.  Energy Distribution of Electrons in High Frequency Gas Discharges , 1946 .

[5]  M. Capitelli,et al.  Electron energy distribution functions in radio-frequency collision dominated nitrogen discharges , 1988 .

[6]  V. Kolobov,et al.  Nonlocal electron kinetics in a low‐pressure inductively coupled radio‐frequency discharge , 1994 .

[7]  Yu. A. Lebedev,et al.  Computer simulation of microwave and DC plasmas: comparative characterisation of plasmas , 1992 .

[8]  H. Deutsch,et al.  Electron Kinetics of Weakly Ionized HF Plasmas. II. Decoupling in the Fourier Hierarchy and Simplified Kinetics at Higher Frequencies , 1984 .

[9]  J. Loureiro,et al.  Effects of electron-electron collisions on the characteristics of DC and microwave discharges in argon at low pressures , 1992 .

[10]  Charles K. Birdsall,et al.  Particle-in-cell charged-particle simulations, plus Monte Carlo collisions with neutral atoms, PIC-MCC , 1991 .

[11]  Boeuf,et al.  Transition between different regimes of rf glow discharges. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[12]  D. J. Rose,et al.  High-Frequency Gas Discharge Plasma in Hydrogen , 1955 .

[13]  J. Loureiro,et al.  Electron energy distributions and excitation rates in high-frequency argon discharges , 1983 .

[14]  J. Wilhelm,et al.  Multi-term solutions of Boltzmann’s equation for electrons in the real gases Ar, CH4 and CO2 , 1985 .

[15]  M. Capitelli,et al.  Electron Energy Distribution Functions in RF Molecular Plasmas: The Role of Second Kind Collisions , 1986 .

[16]  M. Lieberman,et al.  Capacitive RF discharges modelled by particle-in-cell Monte Carlo simulation. II. Comparisons with laboratory measurements of electron energy distribution functions , 1993 .

[17]  Henry Margenau,et al.  Theory of High Frequency Gas Discharges. II. Harmonic Components of the Distribution Function 1 , 1948 .

[18]  C. Ferreira,et al.  Self‐consistent modeling of surface wave produced discharges at low pressures , 1991 .

[19]  Sommerer,et al.  Self-consistent kinetic model of the cathode fall of a glow discharge. , 1989, Physical review. A, General physics.

[20]  Stephen D. Rockwood,et al.  Elastic and Inelastic Cross Sections for Electron-Hg Scattering from Hg Transport Data , 1973 .

[21]  M. Capitelli,et al.  Electron energy distribution function in He-CO radio-frequency plasmas: the role of vibrational and electronic superelastic collisions , 1987 .

[22]  I. Kaganovich,et al.  Low-pressure RF discharge in the free-flight regime , 1992 .

[23]  Charles K. Birdsall,et al.  Capacitive RF discharges modelled by particle-in-cell Monte Carlo simulation. I. Analysis of numerical techniques , 1993 .

[24]  U. Kortshagen.  On the influence of energy transfer efficiency on the electron energy distribution function in HF sustained rare gas plasmas: experimental and numerical study , 1993 .

[25]  Busch,et al.  Numerical solution of the spatially inhomogeneous Boltzmann equation and verification of the nonlocal approach for an argon plasma. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[26]  I. Bernstein,et al.  Electron Energy Distributions in Stationary Discharges , 1954 .

[27]  M. Moisan,et al.  The similarity laws for the maintenance field and the absorbed power per electron in low-pressure surface wave produced plasmas and their extension to HF plasmas in general , 1988 .

[28]  U. Kortshagen,et al.  Determination of electron energy distribution functions in surface wave produced plasmas. I. Modelling , 1991 .

[29]  K. Wiesemann.  Der Einfluß einer Blende auf die Verteilungsfunktion der Elektronen in einem Gasentladungsplasma. II Die Messung der Verteilungsfunktion der Elektronen in der Umgebung einer Blende , 1969 .

[30]  M. Surendra,et al.  Particle simulations of radio-frequency glow discharges , 1991 .

[31]  Y. Lebedev,et al.  The influence of the electron transport cross sectional shape on electron energy distribution functions in DC and microwave plasmas , 1988 .

[32]  G. J. Parker,et al.  Kinetic modeling of the α to γ transition in radio frequency discharges , 1993 .

[33]  I. Kaganovich,et al.  The space-time-averaging procedure and modeling of the RF discharge II. Model of collisional low-pressure RF discharge , 1992 .

[34]  Kolobov,et al.  Analytic model of the cathode region of a short glow discharge in light gases. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[35]  H. Deutsch,et al.  Electron kinetics of weakly ionized HF plasmas. I - Direct treatment and Fourier expansion , 1984 .

[36]  U. Kortshagen,et al.  FAST TWO-DIMENSIONAL SELF-CONSISTENT KINETIC MODELING OF LOW-PRESSURE INDUCTIVELY COUPLED RF DISCHARGES , 1994 .

[37]  Sommerer,et al.  Electron heating mechanisms in helium rf glow discharges: A self-consistent kinetic calculation. , 1989, Physical review letters.

[38]  Daniel J. Koch,et al.  An efficient scheme for convection-dominated transport , 1989 .

[39]  V. Godyak,et al.  Paradoxical spatial distribution of the electron temperature in a low pressure rf discharge , 1993 .

[40]  M. Bachynski,et al.  The Particle Kinetics of Plasmas , 1966 .

[41]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[42]  L. Schlie.  Electron velocity distributions, fractional energy transfer, and collision rates for potential Na, NaHe, and NaXe electrically excited laser discharges , 1976 .

[43]  S. Brown,et al.  High-frequency gas-discharge breakdown in helium , 1949 .

[44]  M. Capitelli,et al.  Electron kinetics of collision dominated r.f. bulk plasma in CO: The role of second-kind collisions , 1988 .

[45]  L. Pitchford,et al.  Effect of electrons produced by ionization on calculated electron-energy distributions , 1983 .

[46]  V. A. Feoktistov,et al.  RF discharge modeling considering time dependence and spatial nonlocality of the electron energy spectrum , 1991 .

[47]  H. Dreicer.  Electron Velocity Distributions in a Partially Ionized Gas , 1960 .

[48]  L. Tsendin.  Electron kinetics in non-uniform glow discharge plasmas , 1995 .

[49]  Robert J. Hoekstra,et al.  Two‐dimensional hybrid model of inductively coupled plasma sources for etching , 1993 .

[50]  William M. MacDonald,et al.  Fokker-Planck Equation for an Inverse-Square Force , 1957 .

[51]  J. Rumble,et al.  Extended Boltzmann analysis of electron swarm experiments , 1981 .

[52]  M. Kushner,et al.  Radially dependent solutions of Boltzmann’s equation in low‐temperature plasmas using a modified two‐term expansion , 1993 .

[53]  High frequency gas discharge breakdown in hydrogen , 1949 .

[54]  Kolobov,et al.  Electron distribution function in a low-pressure inductively coupled plasma. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[55]  A. D. Macdonald,et al.  Microwave Breakdown in Air, Oxygen, and Nitrogen , 1963 .

[56]  David B. Graves,et al.  Capacitively coupled glow discharges at frequencies above 13.56 MHz , 1991 .

[57]  Kortshagen.  Experimental evidence on the nonlocality of the electron distribution function. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[58]  Tsendin,et al.  Experimental investigation and fast two-dimensional self-consistent kinetic modeling of a low-pressure inductively coupled rf discharge. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[59]  M. Capitelli,et al.  Attachment and ionization in a self-consistent treatment of the electron kinetics of a collision-dominated rf plasma , 1987 .

[60]  Robert J. Hoekstra,et al.  Two‐dimensional modeling of high plasma density inductively coupled sources for materials processing , 1994 .

[61]  D. Graves,et al.  Two‐dimensional fluid model of high density inductively coupled plasma sources , 1994 .

[62]  H. Margenau.  Theory of High Frequency Gas Discharges. I. Methods for Calculating Electron Distribution Functions , 1948 .

[63]  A. V. Phelps,et al.  Comparative calculations of electron-swarm properties in N2 at moderate E/N values , 1982 .

[64]  M. Capitelli,et al.  Time-dependent solution of Boltzmann equation in rf plasmas: A comparison with the effective field approximation , 1987 .

[65]  James E. Lawler,et al.  Physical and numerical verification of discharge calculations , 1993 .

[66]  P. Morse,et al.  Velocity Distributions for Elastically Colliding Electrons , 1935 .

[67]  Mark J. Kushner,et al.  Monte‐Carlo simulation of electron properties in rf parallel plate capacitively coupled discharges , 1983 .