Modelling, Numerical Analysis and Simulation of kp-Schrödinger equations


Universität Münster Leerraum DFG Forschungszentrum Berlin Leerraum Technische Universität Berlin Leerraum WIAS Berlin
Project Members: A. Arnold, Numerische Mathematik, Universität Münster
M. Ehrhardt, Institut für Mathematik, Technische Universität Berlin
Th. Koprucki, Weierstrass Institute for Applied Analysis and Stochastics, Berlin
A. Zisowsky, Institut für Mathematik, Technische Universität Berlin

Description

We are a group of scientists working in mathematics and physics, joint together in a work on systems of Schrödinger-type equations in one space dimension. These occur e.g. in the physics of layered semiconductor devices ([C],[K]) as the so called k·p-Schrödinger equations, which are a well established tool for band structure calculations [CH91]. The k·p method in combination with an envelope function approximation ([BA],[S],[CH95],[C]) is frequently used to calculate the near band edge electronic band structure of semiconductor heterostructures ([S],[WZ],[CC]), such as quantum wells. In this project we are mainly interested in the so-called Double-Barrier Stepped-Quantum-Well (DBSQW). In the notation we slightely deviate by some abbreviations from Bandelow, Kaiser, Koprucki and Rehberg, who performed in [BKKR] a rigorous analysis of spectral properties for the spatially one-dimensional k·p-Schrödinger operators. Thus, the Schrödinger equation we are concerned with reads

gleichung Schrödinger-System

This PDE is discetized by a theta-scheme and a discrete TBC was derived by Zisowsky [ZI] based on the ideas of Arnold and Ehrhardt [A], [EA]. To reduce the numerical effort of the TBCs we use the sum-of-exponential ansatze from [AES] to construct an effective approximative TBC. Here we introduce this ansatz to the numerical solution of special systems of Schrödinger equations [BKKR] (Kohn-Luttinger model of the valence band)

Simulation Double-Barrier Stepped-Quantum-Well

References

[A] A. Arnold,
Numerically Absorbing Boundary Conditions for Quantum Evolution Equations, VLSI Design 6 (1998), 313-319.
[AES] A. Arnold, M. Ehrhardt and I. Sofronov,
Discrete Transparent Boundary Conditions for the Schrödinger Equation: Fast calculation, approximation, and stability, Comm. Math. Sci. 1 (2003), 501-556.
[BA] G. Bastard,
Wave Mechanics Applied to Semiconductor Heterostructures, Hasted Press, 1988.
[BKKR] U. Bandelow, H.-C. Kaiser, Th. Koprucki and J. Rehberg,
Spectral properties of k-p Schrödinger operators in one space dimension, Numer. Funct. Anal. Optimization 21 (2000), 379-409.
[C] M. Cardona,
Fundamentals of Semiconductors, Springer-Verlag, Berlin, 1996.
[CC] C. Y.-P. Chao and S. L. Chuang,
Spin-orbit-coupling e ects on the valence-band structure of strained semiconductor quantum wells, Phys. Rev. B, 46(7):4110-4122, 1992.
[CH91] S. L. Chuang,
Efficient band-structure calculations of strained quantum wells, Phys. Rev. B, 43(12):9649-9661, 1991.
[CH95] S. L. Chuang,
Physics of optoelectronic Devices, Wiley & Sons, New York, 1995.
[DDVL] V. Duez, O. Dupois, O. Vanbésien and D. Lippens
Band Gap Engineering for High Frequency Quantum Devices, Proceedings of the 11th III-V Semiconductor Device Simulation Workshop, Lille, 1994.
[E] M. Ehrhardt,
Discrete Artificial Boundary Conditions, Ph.D. Thesis, Technische Universität Berlin, 2001.
[EA] M. Ehrhardt and A. Arnold,
Discrete transparent boundary conditions for the Schrödinger equation, Rivista di Matematica della Universit a di Parma, 6:57-108, 2001.
[K] E. O. Kane,
Energy Band Theory,In W. Paul, editor, Handbook on Semiconductors, volume 1, chapter 4a, 193-217. North-Holland, Amsterdam, New York, Oxford, 1982.
[Ke] J. Kefi,
Analyse mathématique et numérique de modèles quantiques pour les semiconducteurs, Ph.D. Thesis, Université Toulouse III - Paul Sabatier, 2003.
[S] J. Singh,
Physics of semiconductors and their heterostructures. McGraw-Hill, New York, 1993.
[WZ] D. M. Wood and A. Zunger,
Successes and failures of the k·method: A direct assessment for GaAs=AlAs quantum structures, Phys. Rev. B, 53(12):7949-7963, 1996.
[ZG] J.-F. Zhang and B.Y. Gu,
Temporal characteristics of electron tunneling in double-barrier stepped-quantum-well structures, Phys. Rev. B 43 (1991), 5028-5034.
[ZI] A. Zisowsky,
Discrete Transparent Boundary Conditions for Systems of Evolution Equations, Ph.D. Thesis, Technische Universität Berlin, 2003.

Publications in this Project

[1] A. Zisowsky, A. Arnold, M. Ehrhardt and Th. Koprucki,
Discrete Transparent Boundary Conditions for Time-Dependent Systems of Schrödinger Equations, ( PDF-Format), Matheon-Preprint No. 104, March 2004.
[2] A. Zisowsky, A. Arnold, M. Ehrhardt and Th. Koprucki,
Discrete Transparent Boundary Conditions for transient kp-Schrödinger Equations with Application to Quantum-Heterostructures, ( PDF-Format), Journal of Applied Mathematics and Mechanics (ZAMM), Vol. 85, No. 11, (2005), 793-805.


mailbox ehrhardt@math.tu-berlin.de