2007 年計算數學研討會 中山大學

1. 2007年計算數學研討會中山大學 Ren-Chuen Chen陳仁純 高雄師範大學 O. Voskoboynikov霍斯科 交通大學 Modeling and Simulation ofClassical and Quantum Computer Devices Jen-Hao Chen陳人豪交通大學 Jinn-Liang Liu 劉晉良高雄大學

2. Part 1. Classical Computer Microprocessor Microchips MOSFET

3. MOSFET (Metal Oxide Semiconductor Field Effect Transistor)

4. Semiconductor A semiconductor is a material that can behave as a conductor or an insulator depending on what is done to it. We can control the amount of current that can pass through a semiconductor. Kingfisher Science Encyclopedia

7. Czochralski Crystal Growth

8. Sand Ingot Wafer Doping IC 12吋矽晶圓 Gold Ingots Silicon Ingot

9. Silicon Crystal

10. Si Si Si Extra Si As Si Si Si Si - Doping Impurities (n-Type) Conducting band, Ec Ed ~ 0.05 eV Electron Eg = 1.1 eV Valence band, Ev

11. Conducting band, Ec Si Si Si Hole Eg = 1.1 eV Si B Si Ea ~ 0.05 eV Si Si Si - Valence band, Ev Electron Doping Impurities (p-Type)

12. S. Roy and A. Asenov, Science 2005 MOSFET (Metal Oxide Semiconductor Field Effect Transistor) 2003 L = 4 nm Research 2005 L = 45 nm Production 2018 L = 7 nm Production 3D, 30nm x 30nm

13. Gate Length: 90 nm (2005 In Production) (Device Size) 65 nm (2006 In Production) 34 nm (This Talk)

14. Device Sizes Vs. Models

15. Quantum Corrected Energy Transport Model R.-C. Chen and J.-L. Liu, JCP 2005 L=IJ=34nm

16. Doping Concentration

17. Physical Models Drift diffusion model (3 PDEs) Energy transport model (5 PDEs) Hydrodynamic Model (7 PDEs)

18. Energy Transport Model • the electrostatic potential • n the electron concentration • p the hole concentration • J the current density • S the energy flux • E the electric field • R the generation- recombination rate

19. Auxiliary Relationships

20. Density Gradient Theory (Bohm Quantum Potentials)

21. Self-Adjoint Formulation New Variables

24. Self-Adjoint DGET Model

25. Adaptive Algorithm Finite Element Method Monotone Iteration Exponential Fitting

26. The Final Adaptive Mesh

27. Electron Concentration

28. Electron Temperature

29. Hole Quantum Potential

30. Electron Current Density

31. Drain Current for MOSFET

32. Conclusion • New Model (DGET, Self-Adjointness) • Better Approximation • Global Convergence (Monotone Iterative Method) • Efficiency • Easy Implementation