Longitudinal Magnetic Field in WKB Method in Nanostructure Physics

Inhaltsverzeichnis (Table of Contents)

• Chapter I: Derivation of WKB solution of Schroedinger equation and introduction to WKB connection formulae
  • 1.1 WKB approximation
  • 1.2 WKB solution of (1 dimensional) Schroedinger equation
  • 1.3 Classical turning point
  • 1.4 WKB connection formulae: case I
  • 1.5 WKB connection formulae: case II
• Chapter II: Construction of classical Hamiltonian function of a charged particle in an electric and a magnetic field
  • 2.1 Lagrange's equation, Lagrangian function and generalized potential
  • 2.2 Construction of Lagrangian function for a charged particle in an electric field and a magnetic field
  • 2.3 Construction of Hamiltonian function for a charged particle in an electric field and a magnetic field
• Chapter III: Reducing 3D problem to 1D problem: using one Landau gauge
  • 3.1 Description of the problem
  • 3.2 The general eigenvalue equation of energy
  • 3.3 Simplification of the general eigenvalue equation of energy
  • 3.4 Adapting the general eigenvalue equation of energy to our problem
  • 3.5 P_yop commutes with Hamiltonian operator of our problem
  • 3.6 Reducing 3D eigenvalue equation to 1D eigenvalue equation and obtaining eigenvalue spectrum and identification of different parts of eigenvalue spectrum
  • 3.7 Landau level index n is a constant of motion
• Chapter IV: Reducing 3D problem to 1D problem: using the other Landau gauge
  • 4.1 Description of the problem
  • 4.2 The general eigenvalue equation of energy
  • 4.3 Simplification of the general eigenvalue equation of energy
  • 4.4 Adapting the general eigenvalue equation of energy to our problem
  • 4.5 P_zop commutes with Hamiltonian operator of our problem
  • 4.6 Reducing 3D eigenvalue equation to 1D eigenvalue equation and obtaining eigenvalue spectrum and identification of different parts of eigenvalue spectrum
  • 4.7 Landau level index n is a constant of motion
• Chapter V: Obtaining analytical expressions for tunneling regime of longitudinal magnetic field dependent transmission coefficient of single and symmetric double barriers of general shape and of many different shapes we encounter in studying Nanostructure Physics using WKB method
  • 5.1 Results of 1D problem at zero magnetic field
  • 5.2 Single and symmetric double barriers of general shape
  • 5.3 Single rectangular tunnel barrier
  • 5.4 Symmetric rectangular double barrier
  • 5.5 Single rectangular barrier biased to Fowler Nordheim tunneling regime
  • 5.6 Moderately biased single rectangular tunnel barrier
  • 5.7 Single parabolic tunnel barrier
  • 5.8 Schottky barrier
  • 5.9 Single triangular tunnel barrier
  • 5.10 Two identical triangular tunnel barriers separated by a triangular Quantum Well
  • 5.11 Symmetric double barrier obtained by biasing asymmetric rectangular double barrier
• Chapter VI: Numerical investigation of longitudinal magnetic field dependent transmission coefficient of different types of single and symmetric double barriers encountered in Nanostructure Physics using WKB method
  • 6.1 Single rectangular tunnel barrier
  • 6.2 Symmetric rectangular double barrier
  • 6.3 Single rectangular barrier biased to Fowler Nordheim tunneling regime
  • 6.4 Moderately biased single rectangular tunnel barrier
  • 6.5 Single parabolic tunnel barrier
  • 6.6 Schottky barrier
  • 6.7 Single triangular tunnel barrier
  • 6.8 Two identical triangular tunnel barriers separated by a triangular Quantum well
  • 6.9 Symmetric double barrier obtained by biasing asymmetric rectangular double barrier

Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

The work aims to investigate the effects of a longitudinal magnetic field on the transmission coefficient of single and symmetric double barriers in nanostructure physics. This analysis is conducted using the Wentzel-Kramers-Brillouin (WKB) method. The study explores the transmission coefficient for various barrier shapes and configurations, including rectangular, parabolic, triangular, and Schottky barriers.

• The application of the WKB method in nanostructure physics, specifically for analyzing electron tunneling through potential barriers.
• The impact of a longitudinal magnetic field on the transmission coefficient of single and double barriers.
• The investigation of different barrier shapes and configurations, such as rectangular, parabolic, triangular, and Schottky barriers.
• The derivation of analytical expressions for the transmission coefficient in different tunneling regimes.
• The numerical analysis of the transmission coefficient for various barrier types.

Zusammenfassung der Kapitel (Chapter Summaries)

• Chapter I introduces the WKB approximation and its application to the one-dimensional Schrödinger equation. It defines the classical turning point and outlines the WKB connection formulae for different cases.
• Chapter II constructs the classical Hamiltonian function for a charged particle moving in both electric and magnetic fields. It utilizes Lagrange's equation and the concept of generalized potential to derive the Lagrangian and Hamiltonian functions.
• Chapter III focuses on reducing the three-dimensional problem to a one-dimensional problem using the Landau gauge. It discusses the general eigenvalue equation of energy and its simplification for the specific problem under consideration. The chapter further examines the commutation of the momentum operator with the Hamiltonian operator and derives the one-dimensional eigenvalue equation.
• Chapter IV continues the discussion of reducing the three-dimensional problem to a one-dimensional problem using a different Landau gauge. It follows a similar approach as Chapter III, analyzing the eigenvalue equation and the commutation properties of the relevant operators. The chapter also discusses the constant of motion associated with the Landau level index.
• Chapter V aims to obtain analytical expressions for the transmission coefficient of single and symmetric double barriers in different tunneling regimes. It explores the behavior of the transmission coefficient in the presence of a longitudinal magnetic field for various barrier shapes, including rectangular, parabolic, triangular, and Schottky barriers.

Schlüsselwörter (Keywords)

The main keywords and focus topics of this work are: WKB method, nanostructure physics, transmission coefficient, longitudinal magnetic field, single and symmetric double barriers, tunneling, rectangular barrier, parabolic barrier, triangular barrier, Schottky barrier, Fowler Nordheim tunneling, analytical expressions, numerical analysis.