Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T13:17:08.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Excitonic Eigenstates of Disordered Semiconductor Quantum Wires: Adaptive Wavelet Computation of Eigenvalues for the Electron-Hole Schrödinger Equation

Published online by Cambridge University Press:  03 June 2015

Christian Mollet ,
Angela Kunoth  and
Torsten Meier
Christian Mollet*
Affiliation:
Institut für Mathematik, Universität Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany
Angela Kunoth*
Affiliation:
Institut für Mathematik, Universität Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany
Torsten Meier*
Affiliation:
Department Physik, Universität Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
Get access

Abstract

A novel adaptive approach to compute the eigenenergies and eigenfunctions of the two-particle (electron-hole) Schrödinger equation including Coulomb attraction is presented. As an example, we analyze the energetically lowest exciton state of a thin one-dimensional semiconductor quantum wire in the presence of disorder which arises from the non-smooth interface between the wire and surrounding material. The eigenvalues of the corresponding Schrödinger equation, i.e., the one-dimensional exciton Wannier equation with disorder, correspond to the energies of excitons in the quantum wire. The wavefunctions, in turn, provide information on the optical properties of the wire.

We reformulate the problem of two interacting particles that both can move in one dimension as a stationary eigenvalue problem with two spacial dimensions in an appropriate weak form whose bilinear form is arranged to be symmetric, continuous, and coercive. The disorder of the wire is modelled by adding a potential in the Hamiltonian which is generated by normally distributed random numbers. The numerical solution of this problem is based on adaptive wavelets. Our scheme allows for a convergence proof of the resulting scheme together with complexity estimates. Numerical examples demonstrate the behavior of the smallest eigenvalue, the ground state energies of the exciton, together with the eigenstates depending on the strength and spatial correlation of disorder.

Keywords

81Q0565N2565N5065Zxx
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 1 , July 2013 , pp. 21 - 47
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adams, R.A. and Fournier, J.J.F.Sobolev spaces. Academic press, second edition, 2003.Google Scholar
[2]Baranovskii, S.D. and Efros, A.L.Sov. Phys. Semicond., 12:1328, 1978.Google Scholar
[3]Chuang, S.L.Physics of Optoelectronic Devices. John Wiley & Sons, Inc., New York, 1995.Google Scholar
[4]Cohen, A.Numerical Analysis of Wavelet Methods. Studies in Mathematics and its Applications 32, Elsevier, 2003.Google Scholar
[5]Cohen, A., Dahmen, W., and DeVore, R.Adaptive wavelet methods for elliptic operator equations: Convergence rates. Math. Comp., 70(233):27–75, 2001.Google Scholar
[6]Cohen, A., Dahmen, W., and DeVore, R.Adaptive wavelet schemes for nonlinear variational problems. SIAM J. Numer. Anal., 41(5):1785–1823, 2003.Google Scholar
[7]Cohen, A., Dahmen, W., and DeVore, R.Sparse evaluation of compositions of functions using multiscale expansions. SIAM J. Math. Anal., 35(2):279–303, 2003.Google Scholar
[8]Dahmen, W.Wavelet and multiscale methods for operator equations. Acta Numerica, 6:55–228, 1997.CrossRefGoogle Scholar
[9]Dahmen, W., Kunoth, A., and Urban, K.Biorthogonal spline-wavelets on the interval - Stability and moment conditions. Applied and Comp. Harmonic Analysis, 6(2):132–196, 1999.Google Scholar
[10]Dahmen, W., Rohwedder, T., Schneider, R., and Zeiser, A.Adaptive eigenvalue computation: Complexity estimates. Numer. Math., 110(3):277–312, 2008.Google Scholar
[11]Dautray, R. and Lions, J.-L.Mathematical Analysis and Numerical Methods for Science and Technology, volume 2: Functional and Variational Methods. Springer, second edition, 1988.Google Scholar
[12]Dautray, R. and Lions, J.-L.Evolution Problems I, volume 5 of Mathematical Analysis and Numerical Methods for Science and Technology. Springer, 1992.Google Scholar
[13]Davies, E. B.Spectral Theory and Differential Operators. Cambridge studies in advanced mathematics, 1995.Google Scholar
[14]DeVore, R. and Kunoth, A.Multiscale, Nonlinear and Adaptive Approximation. Springer Berlin Heidelberg, 2009.Google Scholar
[15]Fox, M.Quantum Optics: An Introduction. Oxford University Press, 2006.Google Scholar
[16] T.Gantumur, Harbrecht, H., and Stevenson, R.An optimal adaptive wavelet method without coarsening of the iterands. Math. Comp., 76(258):615–629, 2006.Google Scholar
[17]Goögh, N., Thomas, P., Kuznetsova, I., Meier, T., and Varga, I.Annalen der Physik, 18:905, 2009.Google Scholar
[18]Hackbusch, W.Elliptic Differential Equations: Theory and Numerical Treatment. Springer, 2003.Google Scholar
[19]Harbrecht, H. and Li, J.A fast deterministic method for stochastic elliptic interface problems based on low-rank approximation. Research Report No. 2011-24, Seminar für Angewandte Mathematik, ETH Zürich, Switzerland, 2011.Google Scholar
[20]Haug, H. and Koch, S.W.Quantum Theory of the Optical and Electronic Poperties of Semiconductors. World Scientific Publishing, Singapore, fourth edition, 2004.CrossRefGoogle Scholar
[21]Knyazev, A.V., Argentati, M.E., Lashuk, I., and Ovtchinnikov, E.E.Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in Hypre and PETSc. SIAM J. Sci. Comput., 29(5):2224–2239, 2007.Google Scholar
[22]Knyazev, A.V. and Neymeyr, K.Gradient flow approach to geometric convergence analysis of preconditioned eigensolvers. SIAM J. Matrix Anal. Appl., 31(2):621–628, 2009.Google Scholar
[23]Knyazev, V.A. and Neymeyr, K.A geometric theory for preconditioned inverse iteration III: A short and sharp convergence estimate for generalized eigenvalue problems. Linear Algebra Appl., 358(1-3):95–114, 2003.Google Scholar
[24]Kunoth, A. and Mollet, Chr.Adaptive computation of eigenvalues for the electronic Schrödinger equation based on wavelets. Manuscript in preparation, 2012.Google Scholar
[25]Kuznetsova, I., Gögh, N., Förstner, J., Meier, T., Cundiff, S.T., Varga, I., and Thomas, P.Phys. Rev. B, 81:075307, 2010.Google Scholar
[26]Mehrmann, V. and Miedlar, A.Adaptive computation of smallest eigenvalues of self-adjoint elliptic partial differential equations. Numerical Linear Algebra, 18(3):387–409, 2011.Google Scholar
[27]Meier, T., Thomas, P., and Koch, S.W.Coherent Semiconductor Optics. Springer, 2007.CrossRefGoogle Scholar
[28]Mollet, C.A Space-Time Adaptive Wavelet Method for the Electron-Hole Schrödinger Equation with Random Disorder. PhD thesis, Institut für Mathematik, Universität Paderborn, Germany, in preparation.Google Scholar
[29]Mollet, Chr.Excitonic Eigenstates in Disordered Semiconductor Quantum Wires: Adaptive Computation of Eigenvalues for the Electronic Schrödinger Equation Based on Wavelets. Shaker-Verlag, DOI: 10.2370/OND000000000098, 2011.Google Scholar
[30]Mollet, Chr., Meier, T., and Kunoth, A.Wavelet-based adaptive computations of the excitonic eigenstates of disordered semiconductor quantum wires. AIP Conference Proceedings Volume 1398, 2011, pp. 156–158.Google Scholar
[31]Mollet, Chr. and Pabel, R.Efficient application of nonlinear stationary operators in adaptive wavelet methods - The isotropic case. Numerical Algorithms, 2012, to appear, DOI: 10.1007/s11075-012-9645-z.Google Scholar
[32]Pabel, R.Adaptive Wavelet Methods for PDE Constrained Nonlinear Elliptic Control Problems (working title). PhD thesis, Institut für Mathematik, Universität Paderborn, Germany, In preparation.Google Scholar
[33]Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P.Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, third edition, 2007.Google Scholar
[34]Rohwedder, T., Schneider, R., and Zeiser, A.Perturbed preconditioned inverse iteration for operator eigenvalue problems with application to adaptive wavelet discretization. Adv. Comput. Math., 34(1):43–66, 2011.Google Scholar
[35]Schwab, C. and Stevenson, R.Space-time adaptive wavelet methods for parabolic evolution problems. Math. Comp., 78(267):1293–1318, 2009.CrossRefGoogle Scholar
[36]Vorloeper, J.Adaptive Wavelet Methoden für Operator Gleichungen - Quantitative Analyse und Softwarekonzepte (in German). VDI-Verlag, 2010.Google Scholar
[37]Yserentant, H.On the electronic Schrödinger equation. Lecture notes, Universitaät Tübingen, Germany, 2003.Google Scholar
[38]Zhu, H., Y., Chen., S., Song., and Hu, H.Erratum to: Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations. Applied Numerical Mathematics, 61:974–976, 2011.CrossRefGoogle Scholar
[39]Zimmermann, R., Große, F., and Runge, E.Pure & Appl. Chem., 69:1179, 1997.Google Scholar