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

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 = {}, 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 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 & Heinrich Voss. (2020). 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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