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CHAPTER 1. OVERVIEW 27 Figure 1.2: Boundary Conditions on Wigner Distribution f f (k) 1 k x k < 0 (Adapted from Frensley, Phys. Rev. B, Vol. 36, 1987) x=0 x=L I II I II I Doped Region Doped Region Figure 1.3: Grid Showing Modification of Composite Trapezoidal Rule x v h h 1 2 x y k > 0 f (k) 2 x v+1
Chapter 2 Temporal Integration 2.1 Integrator Selection To advance in time, we give the system of ODEs coming from the discretization to a numerical integrator. Since we want to reduce computation time, the choice of integrator is important. We tested the problem with two integrators: ROCK4 and VODEPK. In Figure 2.1, a flow chart is given to show how the nonlinearity in the problem is handled (that is, how the potential U is computed from f). Integrators are computer software that numerically solve an ODE. The typically problem they solve come in the form dy dt = G(y, t) with given initial condition y(0) = y0 where y ∈ R M . These integrators are given a time t ∗ , the initial condition y 0 ∈ R M , Current f-value Compute n(x) Poisson solve Add band-offset for u(x) Figure 2.1: Flow Chart for Computing U 28 New U(x)
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 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: CHAPTER 1. OVERVIEW 26 The weights
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 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
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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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Page 161 and 162:
Page 163 and 164:
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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