Numerical schemes for three-dimensional irregular shape quantum dots over curvilinear coordinate systems

Tsung-Min Hwang, Wei Cheng Wang, Weichung Wang

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

In this article, we present efficient and stable numerical schemes to simulate three-dimensional quantum dot with irregular shape, so that we can compute all the bound state energies and associated wave functions. A curvilinear coordinate system that fits the target quantum dot shape is first determined. Three finite difference discretizations of the Schrödinger equation are then developed on the original and the skewed curvilinear coordinate system. The resulting large-scale generalized eigenvalue systems are solved by a modified Jacobi-Davidson method. Intensive numerical experiments show that the scheme using both grid points on the original and skewed curvilinear coordinate system can converge to the eigenpairs quickly and stably with second-order accuracy.

Original languageEnglish
Pages (from-to)754-773
Number of pages20
JournalJournal of Computational Physics
Volume226
Issue number1
DOIs
Publication statusPublished - 2007 Sep 10

Fingerprint

spherical coordinates
Semiconductor quantum dots
quantum dots
Wave functions
Electron energy levels
eigenvalues
grids
wave functions
Experiments
energy

Keywords

  • Bound state energies and wave functions
  • Curvilinear coordinate system
  • Finite difference
  • Large-scale generalized eigenvalue problem
  • The Schrödinger equation
  • Three-dimensional irregular shape quantum dot

ASJC Scopus subject areas

  • Numerical Analysis
  • Modelling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

Cite this

Numerical schemes for three-dimensional irregular shape quantum dots over curvilinear coordinate systems. / Hwang, Tsung-Min; Wang, Wei Cheng; Wang, Weichung.

In: Journal of Computational Physics, Vol. 226, No. 1, 10.09.2007, p. 754-773.

Research output: Contribution to journalArticle

@article{6842f47b514d49a5bfb5713d848e3f75,
title = "Numerical schemes for three-dimensional irregular shape quantum dots over curvilinear coordinate systems",
abstract = "In this article, we present efficient and stable numerical schemes to simulate three-dimensional quantum dot with irregular shape, so that we can compute all the bound state energies and associated wave functions. A curvilinear coordinate system that fits the target quantum dot shape is first determined. Three finite difference discretizations of the Schr{\"o}dinger equation are then developed on the original and the skewed curvilinear coordinate system. The resulting large-scale generalized eigenvalue systems are solved by a modified Jacobi-Davidson method. Intensive numerical experiments show that the scheme using both grid points on the original and skewed curvilinear coordinate system can converge to the eigenpairs quickly and stably with second-order accuracy.",
keywords = "Bound state energies and wave functions, Curvilinear coordinate system, Finite difference, Large-scale generalized eigenvalue problem, The Schr{\"o}dinger equation, Three-dimensional irregular shape quantum dot",
author = "Tsung-Min Hwang and Wang, {Wei Cheng} and Weichung Wang",
year = "2007",
month = "9",
day = "10",
doi = "10.1016/j.jcp.2007.04.022",
language = "English",
volume = "226",
pages = "754--773",
journal = "Journal of Computational Physics",
issn = "0021-9991",
publisher = "Academic Press Inc.",
number = "1",

}

TY - JOUR

T1 - Numerical schemes for three-dimensional irregular shape quantum dots over curvilinear coordinate systems

AU - Hwang, Tsung-Min

AU - Wang, Wei Cheng

AU - Wang, Weichung

PY - 2007/9/10

Y1 - 2007/9/10

N2 - In this article, we present efficient and stable numerical schemes to simulate three-dimensional quantum dot with irregular shape, so that we can compute all the bound state energies and associated wave functions. A curvilinear coordinate system that fits the target quantum dot shape is first determined. Three finite difference discretizations of the Schrödinger equation are then developed on the original and the skewed curvilinear coordinate system. The resulting large-scale generalized eigenvalue systems are solved by a modified Jacobi-Davidson method. Intensive numerical experiments show that the scheme using both grid points on the original and skewed curvilinear coordinate system can converge to the eigenpairs quickly and stably with second-order accuracy.

AB - In this article, we present efficient and stable numerical schemes to simulate three-dimensional quantum dot with irregular shape, so that we can compute all the bound state energies and associated wave functions. A curvilinear coordinate system that fits the target quantum dot shape is first determined. Three finite difference discretizations of the Schrödinger equation are then developed on the original and the skewed curvilinear coordinate system. The resulting large-scale generalized eigenvalue systems are solved by a modified Jacobi-Davidson method. Intensive numerical experiments show that the scheme using both grid points on the original and skewed curvilinear coordinate system can converge to the eigenpairs quickly and stably with second-order accuracy.

KW - Bound state energies and wave functions

KW - Curvilinear coordinate system

KW - Finite difference

KW - Large-scale generalized eigenvalue problem

KW - The Schrödinger equation

KW - Three-dimensional irregular shape quantum dot

UR - http://www.scopus.com/inward/record.url?scp=34548424467&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34548424467&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2007.04.022

DO - 10.1016/j.jcp.2007.04.022

M3 - Article

AN - SCOPUS:34548424467

VL - 226

SP - 754

EP - 773

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 1

ER -