Abstract

In some recent papers Li, Voskoboynikov, Lee, Sze and Tretyak suggested an iterative scheme for computing the electronic states of quantum dots and quantum rings taking into account an electron effective mass which depends on the position and electron energy level. In this paper we prove that this method converges globally and linearly in an alternating way, i.e. yielding lower and upper bounds of a predetermined energy level in turn. Moreover, taking advantage of the Rayleigh functional of the governing nonlinear eigenproblem, we propose a variant which converges even quadratically thereby reducing the computational cost substantially. Two examples of finite element models of quantum dots of different shapes demonstrate the efficiency of the method.

Keywords: Quantum dot; Electronic structure; Electron states; Computer simulation; Nonlinear eigenproblem; Schrödinger equation; Rayleigh functional

PACS: 73.20.At; 73.61.Ey

Article Outline

1. Introduction

2. The governing Schrödinger equation

3. Convergence results

4. Numerical results

4.1. Conical quantum dot

4.2. Pyramidal quantum dot

5. Conclusions

Appendix A. Proof of Theorem 3.1

References