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CHAPTER 4. THEORY 79 given by Kmax −Kmax |u(x, k ′ )|dk ′ = ≤ Kmax −Kmax Kmax −Kmax (1)|u(x, k ′ )|dk ′ ≤ 2KmaxuX. 1 2 dk ′ 1 2 Kmax −Kmax |u(x, k ′ )| 2 dk ′ 1 2 We will now get an estimate for T (x, k − k ′ ) for u ∈ L 2 ([0,L] × [−Kmax,Kmax]). By Equation (1.16), we know that U(x) is the sum of the electrostatic potential up(x)and the potential barriers ∆c(x). The dependence of T (x, k −k ′ )onu is through Poisson’s equation (Equation (1.13)). We can solve Poisson’s equation explicitly through the use of a Green’s function [13]. The Green’s function for Poisson’s equation in one dimension for the domain [0,L]isgivenby Therefore, the exact solution up(x) isgivenby L up(x) = 0 ⎧ ⎪⎨ z( J(x, z) = ⎪⎩ x − 1), if 0 ≤ z ≤ x L x( z . (4.4) − 1), if x ≤ z ≤ L L 2 q J(x, z) Nd(z) − ɛ 1 Kmax u(z,k 2π −Kmax ′ )dk ′ dz.
CHAPTER 4. THEORY 80 Combining Equation 1.16 for U(x) and Equation 1.9 for T (x, z), we get that the explicit dependence of T (x, k − k ′ )onu is given by T (x, k − k ′ )= q2 Lc 2 2πɛ 0 L Kmax 0 [J(x − y, z) − J(x + y, z)] sin(2y(k − k −Kmax ′ ))u(z,k ′′ )dk ′′ dzdy Lc L 2 + q2 ɛ 0 + 0 Lc 2 0 [J(x + y, z) − J(x − y, z)] sin(2y(k − k ′ ))Nd(z)dzdy dy sin(2y(k − k ′ )) ∆c(x + y) − ∆c(x − y) − 2Vy . L We note that T (x, k − k ′ ) is a continuous function. To get an upper bound on ||T ||∞, we will look at each term of T (x, k − k ′ ) individually. Define T1(x, k − k ′ )as T1(x, k−k ′ )= q2 Lc 2 2πɛ 0 So, |T1(x, k − k)| = q2 2πɛ ≤ q2 2πɛ L Kmax Lc 2 0 Lc 2 0 0 ≤ q2 2πɛ −Kmax [J(x−y, z)−J(x+y, z)] sin(2y(k−k ′ ))u(z,k ′′ )dk ′′ dzdy. L Kmax 0 −Kmax [J(x − y, z) − J(x + y, z)] sin(2y(k − k′ ))u(z,k ′′ )dk ′′ dzdy L Kmax 0 −Kmax |[J(x − y, z) − J(x + y, z)] sin(2y(k − k′ ))u(z,k ′′ )|dk ′′ dzdy Lc 2 0 ≤ q2 2πɛ Lc 2 0 L Kmax 0 −Kmax 2||J||∞|u(z,k ′′ )|dk ′′ dzdy L Kmax 0 −Kmax 4||J||2∞dk ′′ dz 1 2 L 0 = q2Lc 4πɛ 2J∞ √ 2KmaxLu2. Kmax −Kmax u2 (z,k ′′ )dk ′′ 1 2 dz
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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
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CHAPTER 1. OVERVIEW 16 = h 2π ,we
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CHAPTER 1. OVERVIEW 18 Table 1.1: P
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CHAPTER 1. OVERVIEW 20 where q is t
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CHAPTER 1. OVERVIEW 22 get K(f)
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CHAPTER 1. OVERVIEW 24 rule. The pr
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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 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: CHAPTER 4. THEORY 78 So if we write
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
Page 131 and 132: REFERENCES 118 ODE solver. Technica
Page 133 and 134: REFERENCES 120 [21] C. T. Kelley an
Page 135 and 136: REFERENCES 122 [37] Homer F. Walker
Page 137 and 138: APPENDIX A. NOTATION 124 ˜B Genera
Page 139 and 140: APPENDIX A. NOTATION 126 k Wave num
Page 141 and 142: 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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