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Hybrid modeling. Annemie Bogaerts Department of Chemistry, University of Antwerp (UA), Belgium Annemie.Bogaerts@ua.ac.be. College “Computational Plasma Physics” TU/e, 6/2/04. Contents. 1. Introduction Glow discharge + applications 2. Overview of models for GD plasma

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Hybrid modeling


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    1. Hybrid modeling Annemie Bogaerts Department of Chemistry, University of Antwerp (UA), Belgium Annemie.Bogaerts@ua.ac.be College “Computational Plasma Physics” TU/e, 6/2/04

    2. Contents 1. Introduction Glow discharge + applications 2. Overview of models for GD plasma Advantage of hybrid models 3. Hybrid modeling network for GD plasma Submodels + coupling 4. Particle-in-cell modeling for magnetron 5. Modeling network for laser ablation

    3. 1. Introduction Glow discharge: e.g.: cell dimensions: cm3 argon gas (p ~ 1 Torr) V ~ 1 kV, I ~ mA anode cathode negative glow anode zone cathode dark space

    4. 1. Introduction (continued) Processes in the glow discharge: second. electron emission anode sputtering cathode

    5. 1. Introduction (continued) • Applications of glow discharges: • Analytical spectrometry (source for MS, OES) • Semiconductor industry (deposition, etching) • Materials technology (deposition) • Lasers • Light sources • Flat plasma display panels • Environmental, biomedical applications • … Aim of our work: Better understanding to improve results  Modeling

    6. 2. Overview of models for GD plasma A. Analytical model Principle: Simple analytical formulas, valid for specific range of conditions Advantage:Simple, fast Disadvantage:Approximation, limited validity

    7. 2. Overview of models for GD plasma B. Fluid model Principle: Moment equations of Boltzmann equation: Conservation of mass, momentum, energy (or: drift-diffusion approximation) Advantage: Simple, fast Coupling to Poisson equation for self-consistent E-field Disadvantage: Approximation (therm. equilibrium)

    8. 2. Overview of models for GD plasma C. Collisional-radiative model Principle: ~ Fluid model Conservation equations for excited species (balance of production + loss processes) Advantage:Simple, fast Disadvantage: /

    9. 2. Overview of models for GD plasma D. Boltzmann model Principle: Full solution of Boltzmann transport equation (terms with E-gain, E-loss) Advantage: Accounts for non-equilibrium behavior Disadvantage: Mathematically complex

    10. 2. Overview of models for GD plasma E. Monte Carlo model • Principle: • Treats particles on lowest microscopic level • For every particle – during successive time-steps: • * trajectory by Newton’s laws • * collisions by random numbers Advantage: Accurate (non-equilibrium behavior) + simple Disadvantage: Long calculation time + not self-consistent

    11. 2. Overview of models for GD plasma F. Particle-in-cell / Monte Carlo model • Principle: • Similar to MC model (Newtons’ laws, random numbers) • But at every time-step: calculation of electric field • from positions of charged particles (Poisson equation) Advantage: Accurate + self-consistent Disadvantage: Even longer calculation time

    12. Which model should I use ? Every model has advantages + disadvantages… Solution: use combination of models !!

    13. 2. Overview of models for GD plasma G. Hybrid model • Principle: • Because some species = fluid-like, others = particle-like: •  Combination of above models, e.g.: • Monte Carlo for fast (non-equilibrium) species • Fluid model for slow (equilibrium) species + E-field • Advantage: • Combines the advantages + avoids the disadvantages • Accurate + self-consistent + reduced calculation time Disadvantage: /

    14. 3. Hybrid modeling network for GD plasma Combination of different models for various species Species: Model used: Ar0 atoms no model (assumed thermalized) or: heat conduction equation electrons MC for fast electrons fluid for slow electrons Ar+ ions fluid (with electrons + Poisson) MC in sheath Ar0f fast atoms MC in sheath Ar* excited atoms collisional-radiative model Cu0 atoms thermalization after sputtering: MC Cu*, Cu+(*), Cu++ collisional-radiative model Cu+ ions MC in sheath

    15. A. Heat transfer equation for Ar atoms Gas temperature = f (power input in Ar gas): • Power: calculated in MC models: • elastic collisions of Ar+ ions, Ar0f atoms, Cu atoms, • and electrons with Ar atoms • reflection of Ar+ ions and Ar0f atoms at cell walls  nAr = p / (k Tg)

    16. B. Monte Carlo model for fast electrons • During successive time-steps: Follow all electrons: • Trajectory by Newton’s laws: • Probability of collision: •  Compare with RN (0 - 1): If Prob < RN => no collision • If Prob > RN => collision

    17. B. Monte Carlo model for fast electrons (cont.) • Kind of collision: partial collision prob. + compare with RN • 0 1 • Pexcit Pioniz Pelast … • New energy + direction: scattering formulas + RN: • Repeat until electrons: (Ep + Ek) < Ethreshold • (typically in NG, where E-field weak) •  transfer to slow electron group

    18. C. Fluid model for slow electrons + Ar+ ions Continuity equations for ions + electrons: RAr+ and Re: from MC model Transport equations for ions + electrons (drift-diffusion): Poisson’s equation:

    19. D. MC model for Ar+ ions + Ar0f atoms • Only in CDS (where strong E-field) ! • During successive time-steps: Follow all ions + atoms: • Trajectory by Newton’s laws • Probability of collision, • Kind of collision, defined by RN’s • New energy + direction after coll. • Repeat this until: • Ar+ ions bombard cathode • Ar0f atoms: E < Ethermal

    20. E. Collisional-radiative model for Ar* atoms 65 Ar levels (individual or group of levels) For each level: continuity equation (balance equation): Production + loss processes for each level: * electron, Ar ion + Ar atom impact excitation, de-excitation, ionization * electron-ion three-body recombination, radiative recombination * radiative decay * Hornbeck-Molnar associative ionization (for Ar* levels with E > 14.7 eV) Additional processes for 4s levels: * Ar* - Ar* collisions -> (associative) ionization * Penning ionization of sputtered atoms * three-body collisions with Ar atoms * radiation trapping of resonant radiation

    21. Energy level scheme for Ar* CR model

    22. F. Cu0: sputtering, thermalization • Sputtering: • EDF of Ar+, Ar0, Cu+ (from MC models) • Sputtering yield Y(E): empirical formula • Result = Jsput • Thermalization: Monte Carlo model • cfr. electron Monte Carlo model • Result = FT • FT * Jsput: source term for Cu0 in next model

    23. G. Cu0, Cu*, Cu+, Cu+*: Collisional-radiative model • 8 Cu0, 7 Cu+ levels, 1 Cu++ level: • For each level: balance equation: • Production + loss processes: • Cu0: FT * Jsput • electron, atom impact excitation, de-excitation • electron, atom impact ionization, recombination • radiative decay • Penning ionization, asymmetric charge transfer • Transport: • diffusion for Cu0, diffusion + drift for Cu+ and Cu++

    24. Energy level scheme for Cu* CR model

    25. H. Cu+ ions in CDS: Monte Carlo model • cfr. Ar+ ion Monte Carlo model • Cu+ ions: important for sputtering !

    26. I. Coupling of the models

    27. Detailed coupling Start: Ar+ - e- fluid model: • Input: arbitrary creation rates: RAr+, Re • Output: • * Electric field: Eax, Erad • * Interface CDS – NG: dc (r) • * Ar+ ion flux entering CDS: jAr+,dc (r) • * Ar+ ion flux at cathode: jAr+,0 (r)

    28. This output = input in e-, Ar+, Ar0f MC models: • Electric field (Eax, Erad): to calculate trajectories of • electrons and ions (Newton’s laws) • Interface CDS – NG + Ar+ flux entering CDS: jAr+,dc (r): • to define Ar+ ions starting in Ar+ MC model • Ar+ ion flux at cathode: jAr+,0 (r): to define electron flux • starting at cathode (secondary electron emission): • je,0 (r) = -  jAr+,0(r)

    29. Coupling between e-, Ar+, Ar0f MC models: • Electron MC model (1) • Input: from fluid model • Output: electron imp. ionization rate = creation of Ar+ ions • Ar+ MC model (1): • Input: from fluid model + e- MC model (creation of Ar+) • Output: * Creation of Ar0f (elastic collisions) • * Creation of e- (ionization collisions) • Ar0f MC model (1): • Input: from Ar+ MC model • Output: ionization collisions: creation of e-, Ar+

    30. Coupling between e-, Ar+, Ar0f MC models (cont): • Ar+ MC model (2): • Extra input: from Ar0f MC model (creation of Ar+) • Same output: ( = more creation of Ar0f and e- ) • Ar0f MC model (2): • Extra input: from Ar+ MC model • Same output: (= more creation of e-, Ar+ ) • Ar+ MC model (3) • Ar0f MC model (3) • … Etc… until CVG reached (creation of e- constant)

    31. Coupling between e-, Ar+, Ar0f MC models (cont): • Electron MC model (2) • Input: from fluid + Ar+, Ar0f MC model (creation of e-) • Same output: (= more creation of Ar+ ions) • Again: Ar+ / Ar0f MC models • Again: e- MC model (3) • … Etc… until CVG reached: total creation of Ar+ ions and e- constant (typically: after 2-3 iterations)

    32. Coupling betw. e-, Ar+, Ar0f MC models (summary): e- MC model Creation of Ar+ Creation of Ar0f Ar+ MC model Ar0f MC model Creation of Ar+ Creation of Ar+ Creation of e- e- MC model

    33. Coupling between MC models + fluid model: • Global output of MC models: • Creation rates of Ar+, electrons: RAr+, Re • = Input in Ar+ - e- fluid model (2) • Output of Ar+ - e- fluid model (= same): • * Electric field: Eax, Erad • * dc (r) and jAr+, dc (r) • * Ar+ ion flux at cathode: jAr+,0 (r) • = Again input in MC models Etc… until CVG reached (E-field, jAr+,0 constant) (typically after 5-10 iterations)

    34. Coupling between MC + fluid models (Summary): Arbitrary creation of Ar+ , e- e- - Ar+ fluid model • E-field • dc(r), jAr+,dc(r) • jAr+,0 (r) e-, Ar+, Ar0f MC models Creation of Ar+, e- No CVG ? Yes

    35. Remark: This is “core” of hybrid model • Similar hybrid MC - fluid models for GD plasmas: • L.C. Pitchford and J.-P. Boeuf (Toulouse) • Z. Donko (Budapest)

    36. When Ar density ≠ constant, but calculated in heat transfer model  Additional model in loop: Arbitrary creation of Ar+ , e- + constant nAr Initial e- - Ar+ fluid model E-field, dc(r), jAr+,dc(r), jAr+,0 (r) e-, Ar+, Ar0f MC models • Creation of Ar+, e- ( Fluid model) • Power input in Ar gas ( Heat transf.model) No Ar heat transfer model New nAr (function of position) CVG ? Yes

    37. Coupling between MC + fluid models = strongest coupling • Hybrid MC – fluid model determines: • Structure of GD (CDS, NG) • Electric field distribution • Ar+ ion and electron densities • Electrical characteristics (I-V-p) • … Other models: more loosely coupled • Other models: • Do not affect electrical structure of GD • Are important for specific applications • (e.g., spectrochemistry)

    38. Results of the MC + fluid models used as input in the other models: • From e- MC model: • Electron impact ionization, excitation, de-excitation • rates of Ar ( used as populating + depopulating • terms in Ar CR model) • Electron impact ionization, excitation, de-excitation • rates of Cu ( used as populating + depopulating • terms in Cu CR model)

    39. Results of the MC + fluid models used as input in the other models (cont): • From Ar+ and Ar0f MC models: • Ar+ and Ar0f impact ionization, excitation, de-excitation • rates of Ar ( used as populating + depopulating • terms in Ar CR model) • Ar+ and Ar0f flux energy distributions at cathode • ( used to calculate flux of sputtered Cu atoms)

    40. Results of the MC + fluid models used as input in the other models (cont): • From Ar+ - e- fluid model: • densities of Ar+ ions and electrons • ( used in some populating + depopulating terms • in Ar CR model and Cu CR model, i.e., recombination) • density of Ar+ ions ( used in Cu CR model, • for the rate of asymmetric charge transfer ionization: • Ar+ + Cu0  Ar0 + Cu+) • E-field distribution ( used in Cu CR model, • to calculate transport of Cu+ ions by migration)

    41. Using all this input: the Ar CR model + 3 Cu models (i.e., Cu MC thermalization model, Cu CR model and Cu+ MC model) are solved in coupled way • (1) Ar CR model: • Input: cfr. above, + constant Cu atom density • Output (among others): Ar metastable atom density • ( used in Cu CR model, for the rate of • Penning ionization: Ar*m + Cu0  Ar0 + Cu+ + e-)

    42. Coupling of the 3 Cu models (i.e., Cu MC thermalization model, Cu CR model and Cu+ MC model) • (1) MC model for Cu thermalization after sputtering: • Input: * Ar+ and Ar0f flux EDFs (from MC models) • * Ar density (constant or from heat transf.model) • Output: Thermalization profile • ( used in Cu CR model, for initial distribution of • Cu atoms (product: FT * Jsput))

    43. Coupling of the 3 Cu models (cont) • (2) Cu CR model: • Input: * from MC + fluid models (cfr. above) • * FT*Jsput from Cu thermalization MC model) • Output (among others): • * Flux of Cu+ ions entering CDS from NG • * Creation rate of Cu+ ions in CDS • ( both used in Cu+ MC model in CDS)

    44. Coupling of the 3 Cu models (cont) • (3) Cu+ MC model in CDS: • Input: from Cu CR model (cfr. above) • Output (among others): • Cu+ ion flux EDF at cathode • ( used to calculate flux of sputtered Cu atoms)

    45. Coupling of the 3 Cu models (cont) • With this extra output of Cu+ MC model (i.e., Cu+ EDF): • Again calculation of : • Sputtering flux (empirical formula) • MC model for Cu thermalization • Cu CR model • Cu+ MC model in CDS Etc… until cvg reached (i.e., when sputter flux = constant) (typically after 2 – 3 iterations)

    46. Coupling to the Ar CR model Output of the 3 Cu models (among others): Cu0 density ( used in Ar CR model, to calculate rate of Penning ionization: Cu0 + Ar*m Cu+ + Ar0 + e-) Output of Ar CR model (Ar metastable atom density): again used as input in the 3 Cu models Etc… until cvg reached (i.e., when Cu0 and Cu+ density constant) (typically after 2 iterations)

    47. Summary: Coupling of Ar CR model and the 3 Cu models Output from Ar+ / e- MC + fluid models Ar*m density Cu sputtering flux (emp.formula) MC model for Cu thermalization Ar CR model FT * Jsput Cu CR model • Cu+flux entering CDS • Cu+ creation rate in CDS Cu0 density (coupling back until CVG) Cu+ MC model in CDS Cu+ EDF at cathode

    48. Results of the Ar CR model + 3 Cu models used as input in the Ar/e- MC + fluid models: • From Ar CR model: • Populations of Ar metastable + other excited levels • ( used to recalculate the e-, Ar+ and Ar0f impact • ionization, excitation, de-excitation rates of Ar • in the MC models) • Some production + loss terms of Ar excited levels • ( can influence densities of Ar+ and e- in fluid model)

    49. Results of the Ar CR model + 3 Cu models used as input in the Ar/e- MC + fluid models: • From the 3 Cu models: • Cu atom density • ( used to recalculate the e- impact ionization, • excitation, de-excitation rate of Cu in MC model) • Cu+ ion density • ( can influence E-field distribution in fluid model) • Ionization rates of Cu (EI, PI, aCT) ( can influence • densities of Ar+ and e- in fluid model)