Spurious Solutions in the Multiband Effective Mass Theory Applied to Low Dimensional Nanostructures
B. Lassen 1, R. V. N. Melnik 2*, M. Willatzen 11 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
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.
Email: email@example.com (B. Lassen), firstname.lastname@example.org (R. V. N. Melnik), email@example.com (M. Willatzen)