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APPENDIX A. NOTATION 129 ¯xi Spatial coordinate of ith electron ∆x Increment in space used in numerical approximation of PDE y Spatial variable y j jth time iterate for an M-dimensional ODE y(0) Initial condition for y(t) ∈ R M z Spatial variable zm mth Newton iterate z ∗ Root of nonlinear equation F : R M → R M Z(f) Part of fixed point map Z ′ Frechet derivative of Z(f) α Parameter in R Greek α m Coefficient in implicit Adams integration ¯α Constant used in fixed point theory β Small positive number β m Coefficient in BDF method ¯β Constant used in fixed point theory δ Small positive number ¯δ Constant used in fixed point theory ∆c(x) Potential barriers with RTD ɛ Dielectric permittivity of RTD
APPENDIX A. NOTATION 130 ɛc ɛk ɛs ηm Eigenvalue of Cayley transformation matrix Small positive number Eigenvalue of shift-invert transformation matrix Forcing term for inexact-Newton condition γ Small positive number γi Coordinate used to describe vector in Krylov space Kj ¯γ Constant that depends on integration method γ Vector in R j Γ Curve in C that contains all the eigenvalues of A λ Eigenvalue of matrix D ˜λ Ritz value µ0 µL Fermi energy at emitter (x =0) Fermi energy at collector (x = L) ¯µ Parameter used in spectral transformation ψ(x) Wavefunction of an electron (in position variable) φ(p) Wavefunction of an electron (in momentum variable) ρ(q, r) Density matrix ¯ρ Radius of a circle in the complex plane σ Line search parameter ¯σ Parameter used in spectral transformation θ Angular coordinate in polar coordinates Θ Quantum mechanical operator corresponding to physical observable ξ Sufficient decrease parameter
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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
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Chapter 2 Temporal Integration 2.1
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CHAPTER 2. TEMPORAL INTEGRATION 30
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Chapter 3 Bifurcation Analysis 3.1
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CHAPTER 3. BIFURCATION ANALYSIS 46
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Chapter 4 Theory 4.1 Steady-State T
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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
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: APPENDIX A. NOTATION 128 ∆t Incre
Page 145 and 146: APPENDIX B. TRILINOS 132 user-given
Page 147 and 148: APPENDIX B. TRILINOS 134 Trilinos,
Page 149 and 150: APPENDIX B. TRILINOS 136 Minimum Re
Page 151 and 152: APPENDIX B. TRILINOS 138 A very use
Page 153 and 154: Appendix C Guide to RTD Simulation
Page 155 and 156: APPENDIX C. GUIDE TO RTD SIMULATION
Page 163 and 164: Appendix D GMRES and Arnoldi Iterat
Page 165 and 166: APPENDIX D. GMRES AND ARNOLDI ITERA
Page 167 and 168: APPENDIX D. GMRES AND ARNOLDI ITERA
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