Simple method to incorporate nonparabolicity effects in the Schrödinger equation of a quantum dot

2007 ◽  
Vol 101 (9) ◽  
pp. 093715 ◽  
Author(s):  
F. M. Gómez-Campos ◽  
S. Rodríguez-Bolívar ◽  
J. E. Carceller
Keyword(s):  
Quantum Dot ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Simple Method

Parabolic band approximation of the electron energy levels in a tetrahedral-shaped quantum dot

2008 ◽  
Vol 86 (11) ◽  
pp. 1327-1331
Author(s):  
T Pengpan ◽  
C Daengngam
Keyword(s):  
Quantum Dot ◽  
Schrödinger Equation ◽  
Effective Mass ◽  
Electron Energy ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Energy Levels ◽  
Nonlinear Schrodinger Equation ◽  
Gradient Term ◽  
Nonlinear Schrödinger

In more elaborate schemes, an electron’s effective mass in a heterostructure semiconductor quantum dot (QD) depends on both its position and its energy. However, the electron’s effective mass can be simply modeled by a parabolic band approximation — the electron’s effective mass inside the QD — which is assumed to be constant and differs from the one outside the QD, which is also assumed to be constant. The governing equation to be solved for the electron’s energy levels inside the QD is the nonlinear Schrödinger equation. With the approximation, the nonlinear Schrödinger equation for a tetrahedral-shaped QD is discretized by using the finite-volume method. The discretized nonlinear Schrödinger equation is solved for the electron energy levels by a computer program. It is noted that the resulting energy levels for the parabolic mass model are nondegenerate due to the mass-gradient term at the corners, edges, and surfaces of the tetrahedral-shaped QD.PACS Nos.: 02.60.Cb, 03.65.Ge, 81.07.Ta

Superstatistics of two electrons quantum dot

2019 ◽  
Vol 34 (03) ◽  
pp. 1950023 ◽  
Author(s):  
S. Sargolzaeipor ◽  
H. Hassanabadi ◽  
W. S. Chung
Keyword(s):  
Energy Spectrum ◽  
Thermodynamic Properties ◽  
Quantum Dot ◽  
Schrödinger Equation ◽  
Analytical Method ◽  
Schrodinger Equation ◽  
Boltzmann Factor ◽  
Dirac Delta ◽  
Uniform Distributions ◽  
Parabolic Quantum Dot

In this paper, we discussed the Schrödinger equation in the presence of the harmonic two electrons interaction for the parabolic quantum dot and the energy spectrum by an analytical method is obtained, then the effective Boltzmann factor in a deformed formalism for modified Dirac delta and uniform distributions are derived. We make use of the superstatistics for the two distributions in physics and the thermodynamic properties of the system are calculated. Ordinary results are recovered for the vanishing deformed parameter. Furthermore, the effect of all parameters in the problems are calculated and shown graphically.

scholarly journals SOLVING THE TIME-DEPENDENT SCHRÖDINGER EQUATION WITH ABSORBING BOUNDARY CONDITIONS AND SOURCE TERMS IN MATHEMATICA 6.0

2010 ◽  
Vol 21 (11) ◽  
pp. 1391-1406 ◽  
Author(s):  
F. L. DUBEIBE
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Evolution Operator ◽  
Matrix Inversion ◽  
Absorbing Boundary Conditions ◽  
Time Dependent ◽  
Source Terms ◽  
Absorbing Boundary ◽  
Simple Method ◽  
Numerical Computing

In recent decades a lot of research has been done on the numerical solution of the time-dependent Schrödinger equation. On the one hand, some of the proposed numerical methods do not need any kind of matrix inversion, but source terms cannot be easily implemented into these schemes; on the other, some methods involving matrix inversion can implement source terms in a natural way, but are not easy to implement into some computational software programs widely used by non-experts in programming (e.g. Mathematica). We present a simple method to solve the time-dependent Schrödinger equation by using a standard Crank–Nicholson method together with a Cayley's form for the finite-difference representation of evolution operator. Here, such standard numerical scheme has been simplified by inverting analytically the matrix of the evolution operator in position representation. The analytical inversion of the N × N matrix let us easily and fully implement the numerical method, with or without source terms, into Mathematica or even into any numerical computing language or computational software used for scientific computing.

EXCITON STATES IN A QUANTUM DOT WITH PARABOLIC CONFINEMENT

2011 ◽  
Vol 25 (32) ◽  
pp. 4489-4497 ◽  
Author(s):  
Ü. DOĞAN ◽  
S. SAKİROĞLU ◽  
A. YILDIZ ◽  
K. AKGÜNGÖR ◽  
H. EPİK ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Quantum Dot ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Center Of Mass ◽  
High Accuracy ◽  
Light Hole ◽  
Parabolic Confinement ◽  
Radial Variable ◽  
Parabolic Quantum Dot

In this study the electronic eigenstructure of an exciton in a parabolic quantum dot (QD) has been calculated with a high accuracy by using Finite element method (FEM). We have converted the coordinates of electron–light-hole system to relative and center of mass coordinate, then placed the Spherical Harmonics into Schrödinger equation analytically and obtained the Schrödinger equation which depends only on the radial variable. Finally we used FEM with only radial variable in order to get the accurate numerical results. We also showed first 21 energy level spectra of exciton depending on confinement and Coulomb interaction parameters.

scholarly journals Quantum dot–ring nanostructure — A comparison of different approaches

2016 ◽  
Vol 30 (13) ◽  
pp. 1642013 ◽  
Author(s):  
Iwona Janus-Zygmunt ◽  
Barbara Kȩdzierska ◽  
Anna Gorczyca-Goraj ◽  
Elżbieta Zipper ◽  
Maciej M. Maśka
Keyword(s):  
Quantum Dot ◽  
Schrödinger Equation ◽  
Transport Properties ◽  
Schrodinger Equation ◽  
Cylindrical Symmetry ◽  
Two Dimensions ◽  
Quantum Ring ◽  
Tight Binding Approximation ◽  
The Third ◽  
Confining Potential

It has been shown recently that a nanostructure composed of a quantum dot (QD) surrounded by a quantum ring (QR) possesses a set of very unique characteristics that make it a good candidate for future nanoelectronic devices. Its main advantage is the ability to easily tune transport properties on demand by so-called “wavefunction engineering”. In practice, the distribution of the electron wavefunction in the nanostructure can be controlled by, e.g., electrical gating. In order to predict some particular properties of the system, one has to know the exact wavefunctions for different shapes of the confining potential that defines the structure. In this paper, we compare three different methods that can be used to determine the energy spectrum, electron wavefunctions and transport properties of the system under investigation. In the first approach, we utilize the cylindrical symmetry of the confining potential and solve only the radial part of the Schrödinger equation; in the second approach, we discretize the Schrödinger equation in two dimensions and find the eigenstates with the help of the Lanczös method; in the third approach, we use package Kwant to solve a tight-binding approximation of the original system. To study the transport properties in all these approaches, we calculate microscopically the strength of the coupling between the nanosystem and leads. In the first two approaches, we use the Bardeen method, in the third one calculations are performed with the help of package Kwant.

scholarly journals Schrodinger Equation for the Hydrogen Atom - A Simplified Treatment

2008 ◽  
Vol 5 (3) ◽  
pp. 659-662 ◽  
Author(s):  
L. R. Ganesan ◽  
M. Balaji
Keyword(s):  
Hydrogen Atom ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Polar Coordinates ◽  
Mechanical Problem ◽  
Mathematical Concepts ◽  
Simple Method ◽  
Cartesian Coordinates ◽  
Alternative Method ◽  
Normal Method

A simple method is presented here for solving the wave mechanical problem of the hydrogen atom. The normal method of converting the Cartesian coordinates into polar coordinates is tedious and also requires an understanding of the Legendre and Lagurre polynomials. In this paper we are using an alternative method, which requires only minimal familiarity with mathematical concepts and techniques.

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