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Sandia. I was able to get a better understanding of the type of work mathematicians do at a government lab and would not have been as successful at incorporating LOCA into the RTD simulation as well as parallelizing the RTD simulation without their help. Eric Sills and Gary Howell deserve many thanks for answering my numerous com- puter questions and installing Trilinos (several times) on a cluster of computers at N.C. State University. I also want to thank Denise Seabrooks, Brenda Currin, Rory Loycano, Lesa Denning, and Mary Byrd for helping me through all the red tape and administrative processes involved with the graduate school. I would also like to thank my fellow graduate students Quinton Anderson, Todd Coffey, Dan Finkel, Katie Ka- vanagh Fowler, Nathan George, Rachel Levy, and Jill Reese for helping me through classes and showing me the ropes with using Matlab and LaTex. I want to thank my mom, dad, and brother Michael and my close friends Bill Bryan, Danielle Parker, and Jeremy Slatton for all their support through graduate school. Finally, my girlfriend Myra Durham’s daily care and devotion helped me keep my sanity when school was daunting and taxing, and I will forever appreciate her for that. iv
Table of Contents List of Tables ix List of Figures x 1 Overview 1 1.1 Introduction................................ 1 1.2 Simulating Nanoscale Semiconductors . . . . . . . . . . . . . . . . . . 2 1.3 QuantumMechanics ........................... 7 1.4 QuantumStatisticalMechanics ..................... 10 1.5 TheWigner-PoissonEquations ..................... 17 1.6 Numerical Solution: Finite Difference Method . . . . . . . . . . . . . 21 2 Temporal Integration 28 2.1 IntegratorSelection............................ 28 2.2 Newton-KrylovMethods ......................... 32 2.3 GMRES:Matrix-FreeKrylovMethod.................. 34 v
Page 1 and 2: Numerical Methods for the Wigner-Po
Page 3 and 4: NUMERICAL METHODS FOR THE WIGNER-PO
Page 5: Acknowledgments First and foremost,
Page 9 and 10: 4.3 ApplicationofSchauder’sFixedP
Page 11 and 12: List of Tables 1.1 PhysicalConstant
Page 13 and 14: 3.11 I-V Curve with 30 Percent Redu
Page 15 and 16: CHAPTER 1. OVERVIEW 2 being heavily
Page 17 and 18: CHAPTER 1. OVERVIEW 4 Doping Spacer
Page 19 and 20: CHAPTER 1. OVERVIEW 6 and identfyin
Page 21 and 22: CHAPTER 1. OVERVIEW 8 for predictin
Page 23 and 24: CHAPTER 1. OVERVIEW 10 by Schrödin
Page 25 and 26: CHAPTER 1. OVERVIEW 12 ranging term
Page 27 and 28: CHAPTER 1. OVERVIEW 14 But because
Page 29 and 30: CHAPTER 1. OVERVIEW 16 = h 2π ,we
Page 31 and 32: CHAPTER 1. OVERVIEW 18 Table 1.1: P
Page 33 and 34: CHAPTER 1. OVERVIEW 20 where q is t
Page 35 and 36: CHAPTER 1. OVERVIEW 22 get K(f)
Page 37 and 38: CHAPTER 1. OVERVIEW 24 rule. The pr
Page 39 and 40: CHAPTER 1. OVERVIEW 26 The weights
Page 41 and 42: Chapter 2 Temporal Integration 2.1
Page 43 and 44: CHAPTER 2. TEMPORAL INTEGRATION 30
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Chapter 3 Bifurcation Analysis 3.1
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CHAPTER 3. BIFURCATION ANALYSIS 46
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Chapter 4 Theory 4.1 Steady-State T
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CHAPTER 4. THEORY 76 If we write ou
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CHAPTER 4. THEORY 78 So if we write
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CHAPTER 4. THEORY 80 Combining Equa
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CHAPTER 4. THEORY 82 We now estimat
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CHAPTER 4. THEORY 84 pendent of x,
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CHAPTER 4. THEORY 86 interval we ha
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CHAPTER 4. THEORY 88 So an estimate
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CHAPTER 4. THEORY 90 We want to cho
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CHAPTER 4. THEORY 92 The exact same
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CHAPTER 4. THEORY 94 Since un → u
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CHAPTER 4. THEORY 96 thereexistsaco
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CHAPTER 4. THEORY 98 For 0
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CHAPTER 4. THEORY 100 denote this d
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CHAPTER 4. THEORY 102 Now, we can t
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CHAPTER 4. THEORY 104 operator. 4.2
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CHAPTER 4. THEORY 106 we have at th
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CHAPTER 4. THEORY 108 we have at th
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CHAPTER 4. THEORY 110 4.3 Applicati
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CHAPTER 4. THEORY 112 We need to fi
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Chapter 5 Conclusion In this work,
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CHAPTER 5. CONCLUSION 116 • Findi
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REFERENCES 118 ODE solver. Technica
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REFERENCES 120 [21] C. T. Kelley an
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REFERENCES 122 [37] Homer F. Walker
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APPENDIX A. NOTATION 124 ˜B Genera
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APPENDIX A. NOTATION 126 k Wave num
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APPENDIX A. NOTATION 128 ∆t Incre
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APPENDIX A. NOTATION 130 ɛc ɛk ɛ
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APPENDIX B. TRILINOS 132 user-given
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APPENDIX B. TRILINOS 134 Trilinos,
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APPENDIX B. TRILINOS 136 Minimum Re
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APPENDIX B. TRILINOS 138 A very use
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Appendix C Guide to RTD Simulation
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APPENDIX C. GUIDE TO RTD SIMULATION
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Appendix D GMRES and Arnoldi Iterat
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APPENDIX D. GMRES AND ARNOLDI ITERA
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APPENDIX D. GMRES AND ARNOLDI ITERA
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