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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:kunoth@math.uni-paderborn.de
Email:mollet@math.uni-paderborn.de
Email:torsten.meier@uni-paderborn.de
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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

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