1 / 45

The Gallium Arsenide Wafer Problem

The Gallium Arsenide Wafer Problem. Industrial Needs versus Mathematical Capabilities. Margarita Naldzhieva joint work with Wolfgang Dreyer, Barbara Niethammer. DFG Research Center M ATHEON Mathematics for key technologies.

lula
Télécharger la présentation

The Gallium Arsenide Wafer Problem

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Gallium Arsenide Wafer Problem Industrial Needs versus Mathematical Capabilities Margarita Naldzhieva joint work with Wolfgang Dreyer, Barbara Niethammer DFG Research Center MATHEONMathematics for key technologies BMS Days Berlin 18 / 02 / 2008

  2. Industrial problem The Becker-Döring model From Becker-Döring to Fokker-Planck Quasi-stationaryand longtime behaviour of solutions Outline BMS Days Berlin 18 / 02 / 2008

  3. Arsen concentration = 0.500082 The Gallium Arsenide Wafer Problem Single crystal gallium arsenide BMS Days Berlin 18 / 02 / 2008

  4. The Gallium Arsenide Wafer Problem Single crystal gallium arsenide Distribution of liquid droplets BMS Days Berlin 18 / 02 / 2008

  5. The Gallium Arsenide Wafer Problem Single crystal gallium arsenide Distribution of liquid droplets Modeling of liquid droplets BMS Days Berlin 18 / 02 / 2008

  6. The Gallium Arsenide Wafer Problem Single crystal gallium arsenide Distribution of liquid droplets Modeling of liquid droplets BMS Days Berlin 18 / 02 / 2008

  7. The Gallium Arsenide Wafer Problem Single crystal gallium arsenide Distribution of liquid droplets Modeling of liquid droplets BMS Days Berlin 18 / 02 / 2008

  8. + 1 n n + 1 + n 1 n-1 The Becker – Döring Process BMS Days Berlin 18 / 02 / 2008

  9. + 1 n n + 1 + n 1 n-1 Variables density ofn- cluster The Becker – Döring Process BMS Days Berlin 18 / 02 / 2008

  10. + 1 n n + 1 + n 1 n-1 Variables Becker-Döring system density ofn- cluster with The Becker – Döring Process BMS Days Berlin 18 / 02 / 2008

  11. + 1 n n + 1 + n 1 n-1 Variables Becker-Döring system density ofn- cluster with The Becker – Döring Process BMS Days Berlin 18 / 02 / 2008

  12. Transition rates Lyapunov Function Becker-Döring system with Different Strategies, I: Ball Carr Penrose Equilibria BMS Days Berlin 18 / 02 / 2008

  13. Lyapunov Function Becker-Döring system with Different Strategies, II: Dreyer Duderstadt BMS Days Berlin 18 / 02 / 2008

  14. Lyapunov Function n n Different Strategies, II: Dreyer Duderstadt BMS Days Berlin 18 / 02 / 2008

  15. Transition rates Lyapunov Function Becker-Döring system with Different Strategies, II: Dreyer Duderstadt BMS Days Berlin 18 / 02 / 2008

  16. Transition rates Lyapunov Function 2nd law of thermodynamics Different Strategies, II: Dreyer Duderstadt BMS Days Berlin 18 / 02 / 2008

  17. Condensation rates 2nd law of thermodynamics Evaporation rates Comparison of Transition Rates BMS Days Berlin 18 / 02 / 2008

  18. Lyapunov Functions Lyapunov Functions with BMS Days Berlin 18 / 02 / 2008

  19. Water droplets in vapour An p0 T0 n NNNN

  20. Quasi stationary Flux An p0 T0 n J25(t) p0 nmax = 70 nmax = 50 nmax = 40 t NNNN

  21. Quasi stationary Flux monomer density constant Model system for calculation Skim of droplets with nmax + 1 atoms: zn + 1= 0 max J25(t) J25(t) p0 p0 nmax = 70 nmax = 50 nmax = 40 T0 NNNN

  22. Quasi stationary Flux monomer density constant Model system for calculation Skim of droplets with nmax + 1 atoms: zn + 1= 0 max J25(t) p0 p0 T0 t t J25(t) nmax = 70 Model system nmax = 50 nmax = 40 NNNN

  23. Equilibrium solutions : Jn=0 Conservation of mass number n-clusters total volume number n-clusters total number Becker-Döring system with Variables Fluxes BMS Days Berlin 18 / 02 / 2008

  24. Equilibrium solutions : Jn=0 Constraints Equilibrium solutions BMS Days Berlin 18 / 02 / 2008

  25. Equilibrium solutions : Jn=0 Convergence radii RBCP and RDD Constraints Equilibrium solutions BMS Days Berlin 18 / 02 / 2008

  26. Equilibrium solutions : Jn=0 Convergence radii RBCP and RDD Equilibrium solutions Constraints Equilibrium conditions BMS Days Berlin 18 / 02 / 2008

  27. Longtime behaviour of solutions I: Penrose et al. Equilibrium condition Equilibrium solutions BMS Days Berlin 18 / 02 / 2008

  28. Longtime behaviour of solutions I: Penrose et al. Equilibrium condition Equilibrium solutions Asymptotical behaviour BMS Days Berlin 18 / 02 / 2008

  29. Longtime behaviour of solutions I: Penrose et al. Asymptotical behaviour • existence of metastable states (Penrose 1989) • excess density described by the LSW equations (Penrose 1997, Niethammer 2003) • convergence rate to equilibrium e-ct1/3 (Niethammer, Jabin 2003) BMS Days Berlin 18 / 02 / 2008

  30. Simplified Dreyer/Duderstadt Model Current state of the art Mathematical results for a modified DD model! Modified Flux Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  31. Simplified Dreyer/Duderstadt Model New time scale Modified Becker-Döring system with Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  32. Longtime behaviour of solutions II: mod. Dreyer Equilibrium condition Equilibrium solutions BMS Days Berlin 18 / 02 / 2008

  33. Longtime behaviour of solutions II: mod. Dreyer Equilibrium condition Equilibrium solutions depending on the Availability of the System: Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  34. Longtime behaviour of solutions: GaAs wafer Assumption n Distribution of liquid droplets BMS Days Berlin 18 / 02 / 2008

  35. Longtime behaviour of solutions: GaAs wafer Assumption Equilibrium solutions relevant? Distribution of liquid droplets BMS Days Berlin 18 / 02 / 2008

  36. Longtime behaviour of solutions: GaAs wafer Assumption J25(t) nmax = 70 nmax = 50 nmax = 40 t relevant? Distribution of liquid droplets J25(t) BMS Days Berlin 18 / 02 / 2008

  37. From Becker-Döring to Fokker-Planck Becker-Döring system with Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  38. From Becker-Döring to Fokker-Planck Becker-Döring system with Continuous System (Duncan) Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  39. From Becker-Döring to Fokker-Planck Continuous System (Duncan) Dreyer/Duderstadt Continuous thermodynamics with Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  40. From Becker-Döring to Fokker-Planck Continuous System (Duncan) Dreyer/Duderstadt Continuous thermodynamics Lyapunov Funktion, minimal at equilibria! Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  41. Mixed System Mixed System Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  42. Mixed System Mixed System Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  43. Mixed System Thermodynamics and mass conservation Mixed System Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  44. Mixed System Boundary values Mixed System Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008

  45. Becker-Döring model for homogeneous nucleation thermodynamically consistent nucleation rates equilibrium solutions and asymptotical behaviour existence of a metastable phase befor equilibrium? Duncan`s PDE approximation of the discrete System? Summary and outlook BMS Days Berlin 18 / 02 / 2008

More Related