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CHAPTER 3. BIFURCATION ANALYSIS 67 Imaginary Part 0.02 0.015 0.01 0.005 0 −0.005 −0.01 −0.015 0.240 0.242 0.244 0.246 0.248 0.250 0.252 0.254 0.256 Bifurcating Eigenvalues on nx=86,nk=72 Grid −5 −4 −3 −2 −1 0 1 2 x 10 −3 −0.02 Real Part Figure 3.9: Eigenvalues Creating Oscillatory Solutions current-voltage plots. Figure 3.10 compares the previous grid results for Nx = 86, Nk = 72 (about 6,000 unknowns) to the results for the grids Nx = 512, Nk = 2048 (about 1 million unknowns) and Nx = 1024, Nk = 2048 (about 2 million unknowns). The current-voltage plots of the 1 million unknown and 2 million unknown grids lie on top of one another. These grids present new physics in the current voltage-plot as previously shown from the coarser grid. The plot in general is shifted down, having a smaller global maximum than the coarser grid plot. There are more local minima and maxima in the refined grid plots as well.
CHAPTER 3. BIFURCATION ANALYSIS 68 Current Density (A/cm 2 ) x Grid 105 7 6 5 4 3 2 1 0 Refinement nx=512, nk=2048 nx=1024, nk=2048 nx=86, nk=72 −1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Applied Voltage (V) Figure 3.10: Comparison of Current-Voltage Plots for Varying Grids 3.10 Correlation Length Study 3.10.1 Current Voltage Relationship The current-voltage (I-V) curve generated by the fine mesh simulation looked un- physical when evaluated by the physicists in our research group. As shown in Figure 3.10, the fine mesh I-V curve has several peaks, and the physicists do not expect these. In the original coarse grid simulation (Nx = 86, Nk = 72), the correlation length was varied until the simulation produced a physical I-V curve. Since we have refined the mesh, the correlation length needs to be changed again so that the I-V curve agrees with what physicists believe to be correct. Figures 3.11-3.15 show sev-
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Numerical Methods for the Wigner-Po
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NUMERICAL METHODS FOR THE WIGNER-PO
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Acknowledgments First and foremost,
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Table of Contents List of Tables ix
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4.3 ApplicationofSchauder’sFixedP
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List of Tables 1.1 PhysicalConstant
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3.11 I-V Curve with 30 Percent Redu
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CHAPTER 1. OVERVIEW 2 being heavily
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CHAPTER 1. OVERVIEW 4 Doping Spacer
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CHAPTER 1. OVERVIEW 6 and identfyin
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CHAPTER 1. OVERVIEW 8 for predictin
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CHAPTER 1. OVERVIEW 10 by Schrödin
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CHAPTER 1. OVERVIEW 12 ranging term
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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 79: CHAPTER 3. BIFURCATION ANALYSIS 66
Page 87 and 88: Chapter 4 Theory 4.1 Steady-State T
Page 89 and 90: CHAPTER 4. THEORY 76 If we write ou
Page 91 and 92: CHAPTER 4. THEORY 78 So if we write
Page 93 and 94: CHAPTER 4. THEORY 80 Combining Equa
Page 95 and 96: CHAPTER 4. THEORY 82 We now estimat
Page 97 and 98: CHAPTER 4. THEORY 84 pendent of x,
Page 99 and 100: CHAPTER 4. THEORY 86 interval we ha
Page 101 and 102: CHAPTER 4. THEORY 88 So an estimate
Page 103 and 104: CHAPTER 4. THEORY 90 We want to cho
Page 105 and 106: CHAPTER 4. THEORY 92 The exact same
Page 107 and 108: CHAPTER 4. THEORY 94 Since un → u
Page 109 and 110: CHAPTER 4. THEORY 96 thereexistsaco
Page 111 and 112: CHAPTER 4. THEORY 98 For 0
Page 113 and 114: CHAPTER 4. THEORY 100 denote this d
Page 115 and 116: CHAPTER 4. THEORY 102 Now, we can t
Page 117 and 118: CHAPTER 4. THEORY 104 operator. 4.2
Page 119 and 120: CHAPTER 4. THEORY 106 we have at th
Page 121 and 122: CHAPTER 4. THEORY 108 we have at th
Page 123 and 124: CHAPTER 4. THEORY 110 4.3 Applicati
Page 125 and 126: CHAPTER 4. THEORY 112 We need to fi
Page 127 and 128: Chapter 5 Conclusion In this work,
Page 129 and 130: 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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