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Analysis And Efficient Solution Of Stationary Schrödinger Equation Governing Electronic States Of Quantum Dots And Rings In Magnetic Field

Published online by Cambridge University Press:  20 August 2015

Marta M. Betcke  and
Heinrich Voss
Marta M. Betcke*
Affiliation:
Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK
Heinrich Voss*
Affiliation:
Institute of Numerical Simulation, Hamburg University of Technology, D-21071 Hamburg, Germany
*
Corresponding author.Email:rn.betcke@ucl.ac.uk
Email:voss@tu-harburg.de
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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•i. 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.

Keywords

65F1565F50Quantum dotquantum ringnonlinear eigenvalue problemminmax characterizationiterative projection methodelectronic statespin orbit interactionmagnetic field
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 5 , May 2012 , pp. 1591 - 1617
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Arioli, G. and Szulkin, A.A semilinear Schrödinger equation in the presence of a magnetic field. Arch. Rational Mech. Anal., 170:277–295, 2003.Google Scholar
[2]Babuska, I. and Osborn, J.Eigenvalue problems. In Ciarlet, P.G. and Lions, J.L., editors, Handbook of Numerical Analysis, volume II, pages 641–923. North Holland, Amsterdam, 1991.Google Scholar
[3]Bastard, G.Wave Mechanics Applied to Semiconductor Heterostructures. Les editions de physique, Les Ulis Cedex, 1988.Google Scholar
[4]Betcke, M.Iterative Projection Methods for Nonlinear Eigenvalue Problems with Applications. Dessertation.de, 2007. PhD Thesis, Hamburg University of Technology.Google Scholar
[5]Betcke, M. M. and Voss, H.Stationary Schrödinger equations governing electronic states of quantum dots in the presence of spin-orbit splitting. Applications of Mathematics, 52(3):267–284, 2007.CrossRefGoogle Scholar
[6]Betcke, M. M. and Voss, H.Efficient solution of stationary Schrödinger equation governing electronic states of quantum dots and rings in magnetic field. Techn. Report 143, Inst. Numer. Simulation, TU Hamburg-Harburg, 2010.Google Scholar
[7]Betcke, M. M. and Voss, H.Numerical simulation of electronic properties of coupled quantum dots on wetting layers. Nanotechnology, 19(16):165204 (10pp), 2008.Google Scholar
[8]Betcke, T. and Voss, H.A Jacobi-Davidson-type projection method for nonlinear eigenvalue problems. Future Generation Computer Systems, 20(3):363–372, 2004.Google Scholar
[9]Chuang, S. L.Physics of Optoelectronic Devices. John Wiley & Sons, New York, 1995.Google Scholar
[10]Estaban, M. J. and Lions, P.-L.Partial Differential Equations and the Calculus of Variations, volume 1, chapter Stationary solutions of nonlinear Schrödinger equations with an external magnetic field. Birkhäuser, Berlin, 1989.Google Scholar
[11] FEMLAB, Version 3.1. COMSOL, Inc., Burlington, MA, USA, 2004.Google Scholar
[12]Filikhin, I., Deyneka, E., Melikian, G., and Vlahovic, B.Electron states of semiconductor quantum ring with geometry and size variations. Molecular Simulation, 31:779–785, 2005.Google Scholar
[13]Filikhin, I., Deyneka, E., and Vlahovic, B.Energy dependent effective mass model of InAs/GaAs quantum ring. Model. Simul. Mater. Sci. Eng., 12:1121–1130, 2004.Google Scholar
[14]Hermann, C. and Weisbuch, C.k.p perturbation theory in III-V compounds and alloys: a reexamination. Phys. Rev. B, 15(2):823–833, 1977.Google Scholar
[15]Hwang, T. M., Lin, W. W., Wang, W. C., and Wang, W.Numerical simulation of three dimensional quantum dot. J. Comput. Phys., 196:208–232, 2004.Google Scholar
[16]Kane, E. O.Chapter 3 in Semiconductors and Semimetals, Vol. 1, edited by Willardson, R. K. and Beer, A. C., The k.p Method. Academic Press, New York, 1966.Google Scholar
[17]Kressner, D.A block Newton method for nonlinear eigenvalue problems. Numer. Math., 114:355–372, 2009.Google Scholar
[18]Lee, B. C., Voskoboynikov, O., and Lee, C. P.III-IV simiconductor nano-rings. Physica E, 24:87–91, 2004.Google Scholar
[19]Li, Y.Numerical calculation of electronic structure for three-dimensional nanoscale semiconductor quantum dots and rings. J. Comput. Electronics, 2:49–57, 2003.Google Scholar
[20]Li, Y., Lu, H.-M., Voskoboynikov, O., Lee, C.P., and Sze, S.M.Dependence of energy gap on magnetic fields in semiconductor nano-scale quantum rings. Surface Science, 532-535:811–815, 2003.CrossRefGoogle Scholar
[21]Li, Y., Voskoboynikov, O., Lee, C. P., Sze, S. M., and Tretyak, O.Effect of shape and size on electron transition energiesof InAs semiconductor quantum dots. Jpn.J.Appl.Phys., 41:2698–2700, 2002.Google Scholar
[22]Lieb, E. H. and Loss, M.Analysis. Americal Mathematical Society, 2001.Google Scholar
[23]Lew Yan Voon, L. C. and Willatzen, M.The k. p Method: Electronic Properties of Semiconductors. Springer-Verlag Berlin Heidelberg, 2009.Google Scholar
[24]Markiewicz, M. and Voss, H.Electronic states in three dimensional quantum dot/wetting layer structures. In Gavrilova, M.et al. (eds.), editor, Proceedings of ICCSA 2006, volume 3980 of Lecture Notes on Computer Science, pages 684–693, Berlin, 2006. Springer Verlag.Google Scholar
[25]Meerbergen, K.Locking and restarting quadratic eigenvalue solvers. SIAM J. Sci. Comput., 22(5):1814–1839, 2000.Google Scholar
[26]Meerbergen, K.The quadratic Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM Journal on Matrix Analysis and Applications, 30(4):1463–1482, 2008.Google Scholar
[27]Neumaier, A.Residual inverse iteration for the nonlinear eigenvalue problem. SIAM J. Numer. Anal., 22:914–923, 1985.Google Scholar
[28]Niendorf, V. and Voss, H.Detecting hyperbolic and definite matrix polynomials. Linear Algebra Appl., 432:1017–1035, 2010.Google Scholar
[29]Opic, B. and Kufner, A.Hardy-type inequalities. Longman Scientific & Technical, Essex, England, 1990.Google Scholar
[30]Sirenko, A. A., Ruf, T., Ledentsov, N. N., Egorov, A. Yu.Kop’ev, P. S., Ustinov, V. M., and Zhukov, A. E.Resonant spin-flip Raman scattering and localized exciton luminescence in sub-monolayer InAs-GaAs structures. Solid State Comm., 97, 1996.Google Scholar
[31]Sleijpen, G. L., Booten, G. L., Fokkema, D. R., and van der Vorst, H. A.Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT, 36:595–633, 1996.Google Scholar
[32]Su, Y. and Bai, Z.Solving rational eigenvalue problems via linearization. Techn. Report UC Davis, 2008.Google Scholar
[33]Van de Walle, C. G.Band lineups and deformation potentials in the model-solid theory. Phys. Rev. B 39(3):1871–1883, 1989.CrossRefGoogle Scholar
[34]Voskoboynikov, O., Li, Y., Lu, H.-M., Shih, C.-F., and Lee, C. P.Energy states and magnetization in nanoscale quantum rings. Phys. Rev. B, 66:155306–1 -155306-6, 2002.CrossRefGoogle Scholar
[35]Voskoboynikov, O., Wijers, C. M. J., Liu, J. L., and Lee, C. P.Magneto-optical response of layers of semiconductor quantum dots and rings. Phys. Rev. B, 71:245332–1 -245332-12, 2005.CrossRefGoogle Scholar
[36]Voss, H.An Arnoldi method for nonlinear symmetric eigenvalue problems. In Online Proceedings of the SIAM Conference on Applied Linear Algebra, Williamsburg., http://www.siam.org/meetings/laa03/, 2003.Google Scholar
[37]Voss, H.An Arnoldi method for nonlinear eigenvalue problems. BIT Numerical Mathematics, 44:387–401, 2004.Google Scholar
[38]Voss, H.A rational eigenvalue problem governing relevant energy states of a quantum dots. J. Comput. Phys., 217:824–833, 2006.Google Scholar
[39]Voss, H.A minmax principle for nonlinear eigenproblems depending continuously on the eigenparameter. Numer. Lin. Algebra Appl., 15:899–913, 2006.Google Scholar
[40]Voss, H. and Werner, B.A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems. Math. Meth. Appl. Sci., 4:415–424, 1982.Google Scholar
[41]Voss, H. and Werner, B.Solving sparse nonlinear eigenvalue problems. Technical Report Inst. f. Angew. Mathematik, Universität Hamburg, 82/4, 1982.Google Scholar