Physik der Elektronen in Nanostrukturen und niedrigdimensionalen Systemen - Studie zweier Beispiele

Als Beispiele für die vielfältigen Phänomene der Physik der Elektronen in niedrigdimensionalen Systemen wurden in dieser Arbeit das Cu(110)(2x1)O-Adsorbatsystem und die violette Li0.9Mo6O17-Bronze untersucht. Das Adsorbatsystem bildet selbstorganisierte quasi-eindimensionale Nanostrukturen auf einer Kupferoberfläche. Die Li-Bronze ist ein Material, das aufgrund seiner Kristallstruktur quasi-eindimensionale elektronische Eigenschaften im Volumen aufweist. Auf der Cu(110)(2x1)O-Oberfläche kann durch Variation der Sauerstoffbedeckung die Größe der streifenartigen CuO-Domänen geändert werden und damit der Übergang von zwei Dimensionen auf eine Dimension untersucht werden. Der Einfluss der Dimensionalität wurde anhand eines unbesetzten elektronischen Oberflächenzustandes studiert. Dessen Energieposition (untere Bandkante) verschiebt mit zunehmender Einschränkung (schmalere CuO-Streifen) zu größeren Energien hin. Dies ist ein bekannter quantenmechanischer Effekt und relativ gut verstanden. Zusätzlich wurde die Lebensdauer des Zustandes auf der voll bedeckten Oberfläche (zwei Dimensionen) ermittelt und deren Veränderung mit der Breite der CuO-Streifen untersucht. Es zeigt sich, dass die Lebensdauer auf schmaleren CuO-Streifen drastisch abnimmt. Dieses Ergebnis ist neu. Es kann im Rahmen eines Fabry-Perot-Modells als Streuung in Zustände außerhalb der CuO-Streifen verstanden werden. Außer den gerade beschriebenen Effekten war es möglich die Ladungsdichte des diskutierten Zustandes orts- und energieabhängig auf den CuO-Streifen zu studieren. Die Li0.9Mo6O17-Bronze wurde im Hinblick auf das Verhalten der elektronischen Zustandsdichte an der Fermikante untersucht. Diese Fragestellung ist besonders wegen der Quasieindimensionalität des Materials interessant. Die Messungen von STS-Spektren in der Nähe der Fermienergie zeigen, dass die Elektronen in der Li0.9Mo6O17-Bronze eine sogenannte Luttingerflüssigkeit ausbilden, die anstatt einer Fermiflüssigkeit in eindimensionalen elektronischen Systemen erwartet wird. Bisher wurde Luttingerflüssigkeitsverhalten erst bei wenigen Materialien und Systemen experimentell nachgewiesen, obschon die theoretischen Voraussagen mehr als 30 Jahre zurückliegen. Ein Charakteristikum einer Luttingerflüssigkeit ist die Abnahme der Zustandsdichte an der Fermienergie mit einem Potenzgesetz. Dieses Verhalten wurde in STS-Spektren dieser Arbeit beobachtet und quantitativ im Rahmen eines Luttingerflüssigkeitsmodells beschrieben. Auch die Temperaturabhängigkeit des Phänomens im Bereich von 5K bis 55K ist konsistent mit der Beschreibung durch eine Luttingerflüssigkeit. Generell zeigen die Untersuchungen dieser Arbeit, dass die Dimensionalität, insbesondere deren Einschränkung, einen deutlichen Einfluss auf die elektronischen Eigenschaften von Systemen und Materialien haben kann.

A geometrical heuristic for drawing concept lattices

Concept lattices are used in formal concept analysis to represent data conceptually so that the original data are still recognizable. Their line diagrams should reflect the semantical relationships within the data. Up to now, no satisfactory automatic drawing programs for this task exist. The geometrical heuristic is the most successful tool for drawing concept lattices manually. It ueses a geometric representation as intermediate step between the list of upper covers and the line diagram of the lattice.

Exploration tools in formal concept analysis

The development of conceptual knowledge systems specifically requests knowledge acquisition tools within the framework of formal concept analysis. In this paper, the existing tools are presented, and furhter developments are discussed.

Concept exploration - a tool for creating and exploring conceptual hierarchies

Concept exploration is a knowledge acquisition tool for interactively exploring the hierarchical structure of finitely generated lattices. Applications comprise the support of knowledge engineers by constructing a type lattice for conceptual graphs, and the exploration of large formal contexts in formal concept analysis.

Conceptual knowledge discovery with frequent concept lattices

Knowledge discovery support environments include beside classical data analysis tools also data mining tools. For supporting both kinds of tools, a unified knowledge representation is needed. We show that concept lattices which are used as knowledge representation in Conceptual Information Systems can also be used for structuring the results of mining association rules. Vice versa, we use ideas of association rules for reducing the complexity of the visualization of Conceptual Information Systems.

Conceptual structures represented by conceptual graphs and formal concept analysis

Conceptual Graphs and Formal Concept Analysis have in common basic concerns: the focus on conceptual structures, the use of diagrams for supporting communication, the orientation by Peirce's Pragmatism, and the aim of representing and processing knowledge. These concerns open rich possibilities of interplay and integration. We discuss the philosophical foundations of both disciplines, and analyze their specific qualities. Based on this analysis, we discuss some possible approaches of interplay and integration.

Fast computation of concept lattices using data mining techniques

We present a new algorithm called TITANIC for computing concept lattices. It is based on data mining techniques for computing frequent itemsets. The algorithm is experimentally evaluated and compared with B. Ganter's Next-Closure algorithm.

Intelligent structuring and reducing of association rules with formal concept analysis

Association rules are used to investigate large databases. The analyst is usually confronted with large lists of such rules and has to find the most relevant ones for his purpose. Based on results about knowledge representation within the theoretical framework of Formal Concept Analysis, we present relatively small bases for association rules from which all rules can be deduced. We also provide algorithms for their calculation.

Efficient data mining based on formal concept analysis

Formal Concept Analysis is an unsupervised learning technique for conceptual clustering. We introduce the notion of iceberg concept lattices and show their use in Knowledge Discovery in Databases (KDD). Iceberg lattices are designed for analyzing very large databases. In particular they serve as a condensed representation of frequent patterns as known from association rule mining. In order to show the interplay between Formal Concept Analysis and association rule mining, we discuss the algorithm TITANIC. We show that iceberg concept lattices are a starting point for computing condensed sets of association rules without loss of information, and are a visualization method for the resulting rules.

Conceptual knowledge processing with formal concept analysis and ontologies

Among many other knowledge representations formalisms, Ontologies and Formal Concept Analysis (FCA) aim at modeling ‘concepts’. We discuss how these two formalisms may complement another from an application point of view. In particular, we will see how FCA can be used to support Ontology Engineering, and how ontologies can be exploited in FCA applications. The interplay of FCA and ontologies is studied along the life cycle of an ontology: (i) FCA can support the building of the ontology as a learning technique. (ii) The established ontology can be analyzed and navigated by using techniques of FCA. (iii) Last but not least, the ontology may be used to improve an FCA application.

Ontology merging with formal concept analysis

Ontologies have been established for knowledge sharing and are widely used as a means for conceptually structuring domains of interest. With the growing usage of ontologies, the problem of overlapping knowledge in a common domain becomes critical. In this short paper, we address two methods for merging ontologies based on Formal Concept Analysis: FCA-Merge and ONTEX. --- FCA-Merge is a method for merging ontologies following a bottom-up approach which offers a structural description of the merging process. The method is guided by application-specific instances of the given source ontologies. We apply techniques from natural language processing and formal concept analysis to derive a lattice of concepts as a structural result of FCA-Merge. The generated result is then explored and transformed into the merged ontology with human interaction. --- ONTEX is a method for systematically structuring the top-down level of ontologies. It is based on an interactive, top-down- knowledge acquisition process, which assures that the knowledge engineer considers all possible cases while avoiding redundant acquisition. The method is suited especially for creating/merging the top part(s) of the ontologies, where high accuracy is required, and for supporting the merging of two (or more) ontologies on that level.