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Full Description
The central focus of this textbook is the elucidation of the interplay between the principle of stationary action and Schrödinger's equation, and its solution using the finite element method (FEM), a method of solving differential equations, in physical systems whose dimensions are on the order of nanometers. The treatment of the dynamics of electrons in such systems deserves a quantum mechanical description and typical applications at the nanoscale also require the modeling of electrodynamic fields. For instance, nanoscale semiconductor laser design requires the interplay between electrons and photons to be modeled simultaneously.
Aimed at graduate students and researchers in nanoscale systems, materials growth, optoelectronics, engineering, physics, and chemistry, as well as electrical engineers, mechanical engineers, computational scientists, and quantum computer developers, this book explores the development of variational methods and their implementation for several physical examples in the framework of the FEM and addresses issues that are very common in modeling nanoscale systems.
Contents
Part I - The Action Integral in Quantum Mechanics
1: Schrödinger's equation and the action
2: Action, FEM and BCs
3: Element geometries for 2D and 3D
4: Boundary conditions at material interfaces
5: Accidental degeneracy in cubic semiconductor quantum dots
Part II - Quantum Scattering
6: Quantum scattering in 1D revisited
7: 2D quantum waveguides
8: Quantum scattering in 2D waveguides
9: Open domain quantum scattering with sources and absorbers
Part III - Wavefunction Engineering
10: Wavefunction engineering of semiconductor nanostructures
11: Schrödinger-Poisson self-consistency in layered semiconductor nanostructures
Part IV - Steady-state current distributions
12: The Extraordinary Magneto-Resistance effect in metal- semiconductor structures
13: Read-head design based on the EMR effect
Part V - Electrodynamics
14: Fields in electromagnetic waveguides
15: Modeling photonic crystals with Hermite FEM
16: Cavity Electrodynamics and symmetries
17: Dimensional continuation of EM singularities in structures with re-entrant geometry
18: The gauge degree of freedom in Electrodynamics
Part VI - Further applications of FEM
A: Derivation of shape functions using group theory
B: Shape functions for 1D, 2D, and 3D finite elements
C: Hermite Least Squares Data Fitting
