This book contains a comprehensive account of application of WKB method to pure Physics of nanostructures containing single or symmetric double barrier V(x) in their band model in presence of longitudinal magnetic field applied along x direction. It concentrates on effects on transmission coefficient of single and symmetric double barriers by three dimensional electron gas (3DEG). Analytical expressions for longitudinal magnetic field dependent transmission coefficient of single and symmetric double barrier of general shape are obtained first. These general expressions are then used to obtain analytical expressions of longitudinal magnetic field dependent transmission coefficient of single and symmetric double barriers of many different shapes we encounter in studying nanostructure Physics. This is followed by thorough numerical investigation to bring out effects of longitudinal magnetic field on transmission coefficient of all these barriers. Comparisons with standard results where available showed excellent agreements. Results of numerical investigation have been explained completely. The book makes well documented, with thorough calculation and discussion, pure Physics of semiconductor nanostructures.
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 Py op 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 Pz op 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
Objectives and Research Focus
This work aims to determine the effects of a longitudinal magnetic field on the transmission coefficient of various single and symmetric double barriers within nanostructure systems. By applying the WKB approximation to a 3D electron gas model, the research seeks to derive analytical expressions for transmission coefficients across different potential barrier shapes and validate these results through extensive numerical investigations.
- Application of the WKB method to determine transmission coefficients in the presence of longitudinal magnetic fields.
- Reduction of 3D quantum mechanical problems to 1D effective models using Landau gauge transformations.
- Analytical derivation of transmission probabilities for diverse barrier geometries including rectangular, parabolic, triangular, and Schottky profiles.
- Numerical analysis of how magnetic field strength and Landau level index influence tunneling characteristics.
- Comparative studies between WKB-derived analytical results and established standard transmission models.
Excerpt from the Book
1.1 WKB approximation
Wentzel-Kramers-Brillouin (WKB) approximation gives direct and approximate solution of Schroedinger equation if 1) potential V is such that 3D Schroedinger equation can be reduced to 1D differential equation, 2) potential V(x) is a slowly varying function of position x.
As such k(x) = sqrt(2m(E-V(x)))/h_bar = p(x)/h_bar is local wavenumber. Since k(x) is slowly varying function of x, |1/k * dk/dx| << k, i.e. fractional change of k is much smaller than k. Equation (1.2) is called condition of validity of WKB approximation.
If we use k(x) = 2*pi/lambda(x) where lambda(x) is local de Broglie wavelength, equation (1.2) becomes |lambda/2*pi * d/dx(2*pi/lambda)| << 2*pi/lambda, or, |lambda * 1/lambda^2 * d(lambda)/dx| << 2*pi/lambda, or, |d(lambda)/dx| << 2*pi, or, |d(lambda)/dx * lambda| << 2*pi*lambda, or, |d(lambda)/dx * lambda| << lambda, i.e. change of de Broglie wavelength over a wavelength is much smaller than de Broglie wavelength. Equation (1.4) is also called condition of validity of WKB approximation.
Summary of Chapters
Chapter I: Covers the derivation of the WKB solution for the Schroedinger equation and introduces the essential connection formulae used throughout the study.
Chapter II: Details the construction of the classical Hamiltonian function for a charged particle subjected to both electric and magnetic fields.
Chapter III: Focuses on the reduction of the 3D problem to a 1D eigenvalue equation using a specific Landau gauge, including the identification of the eigenvalue spectrum.
Chapter IV: Repeats the reduction process from Chapter III using an alternative Landau gauge to ensure consistency of the mathematical framework.
Chapter V: Derives general analytical expressions for the transmission coefficients of various barrier shapes, including rectangular, parabolic, and triangular potentials, under tunneling conditions.
Chapter VI: Presents numerical investigations and computational programs to visualize the impact of magnetic fields and Landau level indices on transmission coefficients across different barrier structures.
Keywords
WKB approximation, Longitudinal magnetic field, Transmission coefficient, Nanostructure Physics, 3DEG, Schroedinger equation, Landau gauge, Quantum tunneling, Tunnel barrier, Hamiltonian function, Potential profile, Landau levels, Fowler-Nordheim tunneling, Schottky barrier, Quantum Well.
Frequently Asked Questions
What is the fundamental objective of this research?
The work investigates how a longitudinal magnetic field affects the electron transmission coefficient through single and double barriers in nanostructure systems, specifically for a three-dimensional electron gas (3DEG).
What are the central thematic areas?
The core themes include quantum tunneling in nanostructures, the application of the WKB method, mathematical reduction of 3D eigenvalue problems, and numerical simulations of transmission properties.
What is the primary research methodology?
The study employs the WKB (Wentzel-Kramers-Brillouin) approximation to solve the Schroedinger equation and utilizes Landau gauge transformations to reduce dimensionality, followed by numerical verification using computational algorithms.
What types of barrier models are treated in the main text?
The main part of the book covers rectangular tunnel barriers, parabolic barriers, triangular tunnel barriers, and Schottky barriers, examining both single and symmetric double barrier configurations.
How does a longitudinal magnetic field impact electron behavior?
The magnetic field effectively modifies the energy spectrum through Landau levels, which in turn reduces the energy available for motion along the x-direction, thereby decreasing the transmission coefficient.
What are the key keywords characterizing this work?
The most important keywords are WKB approximation, longitudinal magnetic field, transmission coefficient, nanostructure physics, and Landau levels.
How do Landau levels affect the transmission coefficient?
An increase in the Landau level index 'n' or the magnetic field strength 'B' shifts the electron energy, reducing the effective energy component responsible for tunneling along the x-direction, which generally results in a lower transmission probability.
Are the results from the WKB method consistent with standard models?
The research concludes that the WKB analytical results agree with standard models within a factor of 2, with discrepancies being largely attributable to the steepness of potential variations near classical turning points.
- Quote paper
- Dr Sujaul Chowdhury (Author), Mahbub Alam (Author), Sudipta Saha (Author), 2011, Longitudinal Magnetic Field in WKB Method in Nanostructure Physics, Munich, GRIN Verlag, https://www.hausarbeiten.de/document/231376
