Commun. Comput. Phys., 6 (2009), pp. 699-729.


Spurious Solutions in the Multiband Effective Mass Theory Applied to Low Dimensional Nanostructures

B. Lassen 1, R. V. N. Melnik 2*, M. Willatzen 1

1 The Mads Clausen Institute, The University of Southern Denmark, Alsion 2, DK-6400, Sonderborg, Denmark.
2 M^2Net Lab and Department of Mathematics, Wilfrid Laurier University, 75 University Avenue West, Waterloo, Ontario N2L 3C5, Canada.

Received 21 September 2008; Accepted (in revised version) 5 January 2009
Available online 5 March 2009

Abstract

In this paper we analyze a long standing problem of the appearance of spurious, non-physical solutions arising in the application of the effective mass theory to low dimensional nanostructures. The theory results in a system of coupled eigenvalue PDEs that is usually supplemented by interface boundary conditions that can be derived from a variational formulation of the problem. We analyze such a system for the envelope functions and show that a failure to restrict their Fourier expansion coefficients to small k components would lead to the appearance of non-physical solutions. We survey the existing methodologies to eliminate this difficulty and propose a simple and effective solution. This solution is demonstrated on an example of a two-band model for both bulk materials and low-dimensional nanostructures. Finally, based on the above requirement of small k, we derive a model for nanostructures with cylindrical symmetry and apply the developed model to the analysis of quantum dots using an eight-band model.

AMS subject classifications: 35Q40, 81Q05, 65N25, 47N50, 33F05
PACS: 73.21.La, 73.22.Dj, 71.15.-m, 02.30.-f, 02.70.-c
Key words: Effective envelope theory, quantum confinement, abrupt interfaces, multiband models, k space, Fourier coefficients, highly oscillatory integrals, variational formulation, coupled systems of PDEs, multiple scales, continuum and atomistic models, eigenvalue problem, interface boundary conditions, band gap, spurious solutions, low dimensional semiconductor nanostructures.

*Corresponding author.
Email: benny@mci.sdu.dk (B. Lassen), rmelnik@wlu.ca (R. V. N. Melnik), willatzen@mci.sdu.dk (M. Willatzen)
 

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