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CHAPTER 3. BIFURCATION ANALYSIS 63 Current Density (10 5 A/cm 2 ) 4.6 4.5 4.4 4.3 4.2 4.1 4 3.9 3.8 3.7 3.6 0.23 0.232 0.234 0.236 Voltage Difference (V) 0.238 0.24 0.242 Figure 3.5: Enlarged Portion of Curve Where Methods Were Compared number of nonlinear (Newton) iterations, and total time to take 7 pseudo arc-length continuation steps. Table 3.3: Comparison of Bordering and Householder Methods Method No. of GMRES Its. No. of Newton Its. Solution Time (sec) Bordering 8826 21 5905 Householder 3026 15 2016 We previously expected a factor of 2 savings from using Householder continuation instead of the bordering algorithm. The run times show a factor of 3 savings. An explanation of this is that, as previously stated, the Householder continuation linear solves were better conditioned than those of the bordering method. This is why
CHAPTER 3. BIFURCATION ANALYSIS 64 GMRES was able to more efficiently solve the linear systems in the Householder continuation than in the bordering method. 3.9 Results Once the simulator was connected with LOCA, the first thing we wanted to do was analyze the original grid Nx = 86, Nk = 72 that our project’s physicists and engi- neers had used in reporting results to the scientific community. These results [8], [40] reported hystersis in the current-voltage (I-V) curves and current oscillation for some values of applied bias. Figure 3.6 shows a forward sweep in bias from V =0to V =0.480, with increments of 0.08 taken in bias. As you can see, the I-V curve drops suddenly at 0.318 volts to a lower curve and continues on. In the reverse bias case, wherewestartatV =0.480 and reduce by 0.08 volts to V = 0, the I-V curve follows out this lower branch, but does not jump up at 0.318 volts. Instead, it continues along this lower curve, until it jumps back up to the upper curve at 0.256 volts. Figure 3.7 shows the window of voltage that creates current oscillation within the device. We can see the oscillation onsets between 0.240 volts and 0.248 volts and turns off again between 0.256 volts and 0.264 volts. Using LOCA’s pseudo arc-length continuation algorithm and eigensolver, we were able to explain why these phenomena occur. Figure 3.8 shows the I-V curve generated by LOCA. LOCA has found two turning points: one at 0.256 volts and one at 0.318 volts. The branch connecting the two is an unstable steady-state solution branch,
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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
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 75: CHAPTER 3. BIFURCATION ANALYSIS 62
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
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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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