On eigenvalue problems arising from the modeling of semiconductor nanostructures
Thomas Koprucki
Weierstrass-Institut für Angewandte Analysis und Stochastik
Mohrenstr. 39, D-10117 Berlin
Abstract: The design of modern optoelectronic devices like semiconductor lasers is based on semiconductor nanostructures. The key to the understanding and to the prediction of the physical properties of such devices is the computation of their electronic states.
A widely used tool for the quantum mechanical modeling of the electronic states in semiconductor nanostructures is the kp-method in combination with the envelope function approximation. The wave function within the nanostructure is approximated in terms of envelope functions, which are eigenfunctions of matrix-valued kp Schroedinger operators with discontinous coefficients.
We give an overview on the kp Schroedinger operators for one-dimensional nanostructures like Quantum Wells and on their particular spectral properties [1]. For the numerical solution of the corresponding eigenvalue problems of kp Schroedinger operators, we developed the open tool box KPLIB, which is based upon the tool box PDELIB [2,3]. To demonstrate properties like eigenvalue distribution and multiplicity of eigenvalues, we present results on the electronic states of some III-V single and multi quantum well structures obtained with KPLIB.
So far we use for the numerical solution of the resulting algebraic eigenvalue problems standard packages like LAPACK or ARPACK. To work out KPLIB to an highly efficient tool available for physicists and engineers, we plan to improve the performance by using iterative eigenvalue algorithms customized to these problems. On the background of our theoretical knowledge and practical experience, we discuss the requirements for such algorithms concerning efficiency and accuracy. These considerations will play a crucial role for numerical treatment of kp Schroedinger operators for two- and three-dimensional nanostructures like Quantum Wires (2d) and Quantum Dots (3d).
References:
[1] U. Bandelow, H.-Chr. Kaiser, Th. Koprucki, J. Rehberg: Spectral properties of kp Schroedinger operators in one space dimension, WIAS-Preprint No. 494, Berlin, to appear in Numer. Func. Anal. Optim.
[2] J. Fuhrmann, Th. Koprucki, H. Langmach: PDELIB: An Open Modular Tool Box for the Numerical Solution of Partial Differential Equations. Design Patterns, to appear in: Procceedings of the 14th GAMM Seminar on Concepts of Numerical Software (1998), Notes of Numerical Fluid Mechanics, Vieweg Verlag Braunschweig.
[3] http://www.wias-berlin.de/products/pdelib