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List of Figures 1.1 Drawing of a Resonant Tunneling Diode and Its Electric Potential U 4 1.2 Boundary Conditions on Wigner Distribution f ............ 27 1.3 Grid Showing Modification of Composite Trapezoidal Rule . . . . . . 27 2.1 Flow Chart for Computing U ...................... 28 3.1 Steady-State Solution Branch of dw dt = h(w, α) =w2 − α ....... 47 3.2 Steady-State Solution Branch of dw dt = h(w, α) =w3 − αw ...... 48 3.3 Illustration of Pseudo-Arc Length Continuation . . . . . . . . . . . . 53 3.4 Current-VoltagePlot ........................... 62 3.5 Enlarged Portion of Curve Where Methods Were Compared . . . . . 63 3.6 I-V Curve Generated by Time-Accurate Simulation . . . . . . . . . . 65 3.7 Bias Window of Current Oscillation . . . . . . . . . . . . . . . . . . . 65 3.8 I-VCurveGeneratedbyLOCA ..................... 66 3.9 Eigenvalues Creating Oscillatory Solutions . . . . . . . . . . . . . . . 67 3.10 Comparison of Current-Voltage Plots for Varying Grids . . . . . . . . 68 x
3.11 I-V Curve with 30 Percent Reduced Correlation Length . . . . . . . . 69 3.12 I-V Curve with 25 Percent Reduced Correlation Length . . . . . . . . 70 3.13 I-V Curve with 20 Percent Reduced Correlation Length . . . . . . . . 70 3.14 I-V Curve with 10 Percent Increased Correlation Length . . . . . . . 71 3.15 I-V Curve with 20 Percent Increased Correlation Length . . . . . . . 71 xi
Page 1 and 2: Numerical Methods for the Wigner-Po
Page 3 and 4: NUMERICAL METHODS FOR THE WIGNER-PO
Page 5 and 6: Acknowledgments First and foremost,
Page 7 and 8: Table of Contents List of Tables ix
Page 9 and 10: 4.3 ApplicationofSchauder’sFixedP
Page 11: List of Tables 1.1 PhysicalConstant
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
Page 57 and 58: Chapter 3 Bifurcation Analysis 3.1
Page 59 and 60: CHAPTER 3. BIFURCATION ANALYSIS 46
Page 61 and 62: CHAPTER 3. BIFURCATION ANALYSIS 48
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CHAPTER 3. BIFURCATION ANALYSIS 50
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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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