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Volume 11, Issue 5
Analysis and Efficient Solution of Stationary Schrödinger Equation Governing Electronic States of Quantum Dots and Rings in Magnetic Field

Marta M. Betcke & Heinrich Voss

DOI: 10.4208/cicp.110910.250511a

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

Published online: 2012-11

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  • Abstract

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 min-max 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. 

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@Article{CiCP-11-1591, author = {Marta M. Betcke and Heinrich Voss}, title = {Analysis and Efficient Solution of Stationary Schrödinger Equation Governing Electronic States of Quantum Dots and Rings in Magnetic Field}, journal = {Communications in Computational Physics}, year = {2012}, volume = {11}, number = {5}, pages = {1591--1617}, abstract = {

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 min-max 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. 

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.110910.250511a}, url = {http://global-sci.org/intro/article_detail/cicp/7426.html} }
TY - JOUR T1 - Analysis and Efficient Solution of Stationary Schrödinger Equation Governing Electronic States of Quantum Dots and Rings in Magnetic Field AU - Marta M. Betcke & Heinrich Voss JO - Communications in Computational Physics VL - 5 SP - 1591 EP - 1617 PY - 2012 DA - 2012/11 SN - 11 DO - http://doi.org/10.4208/cicp.110910.250511a UR - https://global-sci.org/intro/article_detail/cicp/7426.html KW - AB -

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 min-max 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. 

Marta M. Betcke and Heinrich Voss. (2012). Analysis and Efficient Solution of Stationary Schrödinger Equation Governing Electronic States of Quantum Dots and Rings in Magnetic Field. Communications in Computational Physics. 11 (5). 1591-1617. doi:10.4208/cicp.110910.250511a
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