Computational Methods for Nanoscale Applications: Particles, Plasmons and Waves

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Springer Science & Business Media, Dec 24, 2007 - Technology & Engineering - 532 pages
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Computational Methods for Nanoscale Applications: Particles, Plasmons and Waves presents new perspectives on modern nanoscale problems where fundamental science meets technology and computer modeling. This book describes well-known computational techniques such as finite-difference schemes, finite element analysis and Ewald summation, as well as a new finite-difference calculus of Flexible Local Approximation MEthods (FLAME) that qualitatively improves the numerical accuracy in a variety of problems. Application areas in the book include long-range particle interactions in homogeneous and heterogeneous media, electrostatics of colloidal systems, wave propagation in photonic crystals, photonic band structure, plasmon field enhancement, and metamaterials with backward waves and negative refraction.

Computational Methods for Nanoscale Applications is accessible to specialists and graduate students in diverse areas of nanoscale science and technology, including physics, engineering, chemistry, and applied mathematics. In addition, several advanced topics will be of particular interest to the expert reader.

Key Features:

  • Utilizes a two-tiered style of exposition with intuitive explanation of key principles in the first tier and further technical details in the second
  • Bridges the gap between physics and engineering and computer science
  • Presents fundamentals and applications of computational methods, electromagnetic theory, colloidal systems and photonic structures
  • Covers "hot topics" in photonics, plasmonics, and metamaterials.
 

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Contents

Variational FLAME
231
472 The Model Problem
232
474 Summary of the VariationalDifference Setup
235
Coefficients of the 9Point TrefftzFLAME Scheme for the Wave Equation in Free Space
236
the Frechet Derivative
237
LongRange Interactions in Free Space
239
52 Real and Reciprocal Lattices
242
53 Introduction to Ewald Summation
243

252 Symplectic Schemes for Hamiltonian Systems
37
26 Schemes for OneDimensional Boundary Value Problems
39
262 Using Constraints to Derive Difference Schemes
40
263 FluxBalance Schemes
42
264 Implementation of 1D Schemes for Boundary Value Problems
46
27 Schemes for TwoDimensional Boundary Value Problems
47
272 FluxBalance Schemes
48
273 Implementation of 2D Schemes
50
274 The Collatz Mehrstellen Schemes in 2D
51
28 Schemes for ThreeDimensional Problems
55
283 FluxBalance Schemes in 3D
56
284 Implementation of 3D Schemes
57
285 The Collatz Mehrstellen Schemes in 3D
58
29 Consistency and Convergence of Difference Schemes
59
210 Summary and Further Reading
64
The Finite Element Method
68
32 The Weak Formulation and the Galerkin Method
75
33 Variational Methods and Minimization
81
332 The Galerkin Solution and the Energy Functional
82
34 Essential and Natural Boundary Conditions
83
Convergence LaxMilgram and Ceas Theorems
86
36 Local Approximation in the Finite Element Method
89
37 The Finite Element Method in One Dimension
91
372 HigherOrder Elements
102
38 The Finite Element Method in Two Dimensions
105
382 HigherOrder Triangular Elements
120
39 The Finite Element Method in Three Dimensions
122
310 Approximation Accuracy in FEM
123
311 An Overview of System Solvers
129
312 Electromagnetic Problems and Edge Elements
139
3122 The Definition and Properties of WhitneyNedelec Elements
142
3123 Implementation Issues
145
3124 Historical Notes on Edge Elements
146
Several Common Families of Tetrahedral Edge Elements
147
313 Adaptive Mesh Refinement and Multigrid Methods
148
3132 Hierarchical Bases and Local Refinement
149
3133 A Posteriori Error Estimates
151
3134 Multigrid Algorithms
154
Element Shape and Approximation Accuracy
158
Eigenvalue and Singular Value Conditions
160
3143 Geometric Implications of the Singular Value Condition
171
3144 Condition Number and Approximation
179
3145 Discussion of Algebraic and Geometric a priori Estimates
180
Generalized FEM
181
3152 Tradeoffs
183
316 Summary and Further Reading
184
Generalized Curl and Divergence
186
Flexible Local Approximation MEthods FLAME
189
42 Perspectives on Generalized FD Schemes
191
Approximating the Solution Not the Equation
192
Multivalued Approximation
193
425 Why Flexible Approximation?
195
the 1D Laplace Equation
197
43 Trefftz Schemes with Flexible Local Approximation
198
432 Construction of the Schemes
200
433 The Treatment of Boundary Conditions
202
434 TrefftzFLAME Schemes for Inhomogeneous and Nonlinear Equations
203
435 Consistency and Convergence of the Schemes
205
Case Studies
206
442 The 1D Heat Equation with Variable Material Parameter
207
443 The 2D and 3D Laplace Equation
208
444 The Fourth Order 9point Mehrstellen Scheme for the Laplace Equation in 2D
209
445 The Fourth Order 19point Mehrstellen Scheme for the Laplace Equation in 3D
210
447 Superhighorder FLAME Schemes for the 1D Schrodinger Equation
212
448 A Singular Equation
213
449 A Polarized Elliptic Particle
215
4410 A Line Charge Near a Slanted Boundary
216
4411 Scattering from a Dielectric Cylinder
217
45 Existing Methods Featuring Flexible or Nonstandard Approximation
219
451 The Treatment of Singularities in Standard FEM
221
453 Homogenization Schemes Based on Variational Principles
222
455 Homogenization Schemes in FDTD
223
456 Meshless Methods
224
457 Special Finite Element Methods
225
458 Domain Decomposition
226
4510 Special FD Schemes
227
46 Discussion
228
531 A Boundary Value Problem for Charge Interactions
246
532 A Reformulation with Clouds of Charge
248
533 The Potential of a Gaussian Cloud of Charge
249
534 The Field of a Periodic System of Clouds
251
535 The Ewald Formulas
252
536 The Role of Parameters
254
54 Gridbased Ewald Methods with FFT
256
542 On Numerical Differentiation
262
543 ParticleMesh Ewald
264
544 Smooth ParticleMesh Ewald Methods
267
545 ParticleParticle ParticleMesh Ewald Methods
269
546 The YorkYang Method
271
547 Methods Without Fourier Transforms
272
55 Summary and Further Reading
274
The Fourier Transform of Periodized Functions
277
An Infinite Sum of Complex Exponentials
278
LongRange Interactions in Heterogeneous Systems
280
62 FLAME Schemes for Static Fields of Polarized Particles in 2D
285
621 Computation of Fields and Forces for Cylindrical Particles
289
WellSeparated Particles
291
Small Separations
294
63 Static Fields of Spherical Particles in a Homogeneous Dielectric
303
Spherical Particle in Uniform Field
306
64 Introduction to the PoissonBoltzmann Model
309
65 Limitations of the PBE Model
313
66 Numerical Methods for 3D Electrostatic Fields of Colloidal Particles
314
67 3D FLAME Schemes for Particles in Solvent
315
68 The Numerical Treatment of Nonlinearity
319
69 The DLVO Expression for Electrostatic Energy and Forces
321
610 Notes on Other Types of Force
324
611 Thermodynamic Potential Free Energy and Forces
328
612 Comparison of FLAME and DLVO Results
332
613 Summary and Further Reading
337
Thermodynamic Potential for Electrostatics in Solvents
338
Generalized Functions Distributions
343
Applications in NanoPhotonics
349
73 OneDimensional Problems of Wave Propagation
353
732 Signal Velocity and Group Velocity
355
733 Group Velocity and Energy Velocity
358
74 Analysis of Periodic Structures in 1D
360
75 Band Structure by Fourier Analysis Plane Wave Expansion in 1D
375
76 Characteristics of Bloch Waves
379
762 Fourier Harmonics and the Poynting Vector
380
764 Energy Velocity for Bloch Waves
382
77 TwoDimensional Problems of Wave Propagation
384
78 Photonic Bandgap in Two Dimensions
386
PWE FEM and FLAME
389
792 The Role of Polarization
390
793 Accuracy of the Fourier Expansion
391
794 FEM for Photonic Bandgap Problems in 2D
393
Band Structure Using FEM
397
796 Flexible Local Approximation Schemes for Waves in Photonic Crystals
401
797 Band Structure Computation Using FLAME
405
Comparison with the 2D Case
411
7102 FEM for Photonic Bandgap Problems in 3D
415
7103 Historical Notes on the Photonic Bandgap Problem
416
711 Negative Permittivity and Plasmonic Effects
417
7111 Electrostatic Resonances for Spherical Particles
419
Electrostatic Approximation
421
7113 Wave Analysis of Plasmonic Systems
423
7115 TrefftzFLAME Simulation of Plasmonic Particles
426
7116 Finite Element Simulation of Plasmonic Particles
429
712 Plasmonic Enhancement in Scanning NearField Optical Microscopy
433
7121 Breaking the Diffraction Limit
434
7122 Apertureless and DarkField Microscopy
439
7123 Simulation Examples for Apertureless SNOM
441
7132 Negative Permittivity and the Perfect Lens Problem
451
7133 Forward and Backward Plane Waves in a Homogeneous Isotropic Medium
456
7134 Backward Waves in Mandelshtams Chain of Oscillators
459
7135 Backward Waves and Negative Refraction in Photonic Crystals
465
7136 Are There Two Species of Negative Refraction?
471
The Bloch Transform
477
Eigenvalue Solvers
478
Conclusion Plenty of Room at the Bottom for Computational Methods
487
References
489
Index
523
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