Dirk Klindworth


Discrete Transparent Boundary Conditions for Multiband Effective Mass Approximations


Diplomarbeit Technomathematik, TU Berlin


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Abstract

This diploma thesis is concerned with the derivation and numerical testing of discrete transparent boundary conditions (DTBCs) for multiband effective mass approximations (MEMAs). MEMAs are used to model electronic states in semiconductor nanostructures. This thesis is focused on the stationary case and comprises MEMAs such as the scalar Schrödinger equation, as a representative of single-band effective mass approximations, the two-band Kane-model and systems of kp-Schrödinger equations.

An analysis of the continuous problem is given and transparent boundary conditions (TBCs) are introduced. The discretization of the differential equations is done with the help of finite difference schemes. A fully discrete approach is used in order to develop DTBCs that are completely reflection-free. The analytical and discrete dispersion relations are analyzed in depth and the limitations of the numerical computations are shown.

The results of earlier works on DTBCs for the scalar Schrödinger equation are extended by alternative finite difference schemes. Existence and uniqueness of the numerical solutions are shown. The introduced schemes and their corresponding DTBCs are tested on numerical examples such as the single barrier potential and the tunneling effect at the double barrier potential.

The two-band Kane-model and the two-band kp-model with inter-band coupling are introduced as particular examples of MEMAs. DTBCs for the general d-band kp-model are derived and the numerical results are tested on a quantum well nanostructure.


Zusammenfassung

Die vorliegende Diplomarbeit beschäftigt sich mit der Herleitung von diskreten transparenten Randbedingungen für Effektive-Massen-Approximationen von Multiband-Systemen. Effektive-Massen-Approximationen finden Anwendung in der Berechnung der elektrischen Zustände in Halbleitern. Der Fokus dieser Arbeit liegt dabei auf dem stationären Fall. Es werden Modelle wie die skalare Schrödinger Gleichung, das Zwei-Band Kane-Modell sowie Multiband-kp-Modelle behandelt.

Zunächst werden die Modelle auf kontinuierlicher Ebene untersucht und anschließend transparente Randbedingungen hergeleitet. Die Diskretisierung der Modelle erfolgt mit Hilfe des Finite-Differenzen-Verfahrens. Auf Grundlage der diskreten Lösungen im Außenraum werden diskrete transparente Randbedingungen hergeleitet. Dieser sogenannte diskrete Ansatz verspricht reflektionsfreie Randbindungen, während eine ad-hoc Diskretisierung der transparenten Randbedingungen zu fehlerhaften Reflektionen an den künstlichen Rändern führen kann. Die diskreten Dispersionsrelationen der verschiedenen Modelle werden mit der analytischen Dispersionrelation verglichen und es werden daran die Grenzen der numerischen Schemata aufgezeigt.

Die Ergebnisse früherer Arbeiten zu diskreten transparenten Randbedingungen für die skalare Schrödinger Gleichung werden durch alternative Finite-Differenzen-Verfahren ergänzt. Für jedes dieser Verfahren wird die Existenz und Eindeutigkeit der numerischen Lösung gezeigt. Die eingeführten skalaren Schemata werden an einer einfachen Potentialbarriere sowie am Tunneleffekt der Doppelbarriere numerisch getestet.

Das Zwei-Band Kane-Modell sowie das Zwei-Band kp-Modell mit Interband-Kopplung werden als Beispiele von Multiband-Systemen eingeführt und auf kontinuierlichem sowie diskretem Niveau untersucht. Abschließend wird das allgemeine kp-Modell mit d Bändern analysiert und die zugehörigen diskreten transparenten Randbedingungen hergeleitet. Für das Acht-Band kp-Modell wird ein physikalisch realistisches Beispiel einer Quantum-Well-Struktur behandelt und die numerischen Ergebnisse mit den analytischen verglichen.


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mailbox ehrhardt@math.uni-wuppertal.de