Commun. Comput. Phys., 11 (2012), pp. 1591-1617.

Analysis and Efficient Solution of Stationary Schrodinger Equation Governing Electronic States of Quantum Dots and Rings in Magnetic Field

Marta M. Betcke 1*, Heinrich Voss 2

1 Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK.
2 Institute of Numerical Simulation, Hamburg University of Technology, D-21071 Hamburg, Germany.

Received 11 September 2010; Accepted (in revised version) 25 May 2011
Available online 6 January 2012


In this work the one-band effective Hamiltonian governing the electronic states of a quantum dot/ring in a homogenous magnetic field is used to derive a pair/quadruple of nonlinear eigenvalue problems corresponding to different spin orientations and in case of rotational symmetry additionally to quantum number ± l. We show, that each of those pair/quadruple of nonlinear problems allows for the minmax characterization of its eigenvalues under certain conditions, which are satisfied for our examples and the common InAs/GaAs heterojunction. Exploiting the minmax property we devise efficient iterative projection methods simultaneously handling the pair/quadruple of nonlinear problems and thereby saving up to 40% of the computational time as compared to the nonlinear Arnoldi method applied to each of the problems separately.

AMS subject classifications: 65F15, 65F50

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Key words: Quantum dot, quantum ring, nonlinear eigenvalue problem, minmax characterization, iterative projection method, electronic state, spin orbit interaction, magnetic field.

*Corresponding author.
Email: (M. M. Betcke), (H. Voss)

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