Commun. Comput. Phys., 11 (2012), pp. 797-830.

Computational Methods for Electromechanical Fields in Self-Assembled Quantum Dots

D. Barettin 1, S. Madsen 1, B. Lassen 1*, M. Willatzen 1

1 Alsion 2, Mads Clausen Institute, University of Southern Denmark, DK-6400 Sonderborg, Denmark.

Received 11 November 2010; Accepted (in revised version) 11 April 2011
Available online 28 October 2011


A detailed comparison of continuum and valence force field strain calculations in quantum-dot structures is presented with particular emphasis to boundary conditions, their implementation in the finite-element method, and associated implications for electronic states. The first part of this work provides the equation framework for the elastic continuum model including piezoelectric effects in crystal structures as well as detailing the Keating model equations used in the atomistic valence force field calculations. Given the variety of possible structure shapes, a choice of pyramidal, spherical and cubic-dot shapes is made having in mind their pronounced shape differences and practical relevance. In this part boundary conditions are also considered; in particular the relevance of imposing different types of boundary conditions is highlighted and discussed. In the final part, quantum dots with inhomogeneous indium concentration profiles are studied in order to highlight the importance of taking into account the exact In concentration profile for real quantum dots. The influence of strain, electric-field distributions, and material inhomogeneity of spherical quantum dots on electronic wavefunctions is briefly discussed.

AMS subject classifications: 82-08
PACS: 62.20.-x, 68.65.Hb, 77.65.Ly
Key words: Quantum dots, electromechanical fields, continuum model, valence force field, boundary conditions.

*Corresponding author.
Email: (D. Barettin), (S. Madsen), (B. Lassen), (M. Willatzen)

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