Scattering in complex media : coherent transport at the crossover to Anderson localization

Abstract

Anderson-Lokalisierung ist ein Wellenphänomen, welches exponentielle Lokalisierung von elektronischen Wellenfunktionen oder klassischen Wellen in stark ungeordneten Medien vorhersagt. Dieses Fehlen von Diffusion begründet erstaunliche Eigenschaften lokalisierter Systeme, die Physiker seit der Entdeckung des Effekts durch P.W. Anderson im Jahr 1958 in Atem hält. Eine dieser Eigenschaften ist das Auftreten des sogenannten Ein-Kanal Transport-Regimes, welches wir numerisch und, in Zusammenarbeit mit der Universität von San Antonio (Texas), auch experimentell untersuchen. Das Erkennungsmerkmal des genannten Regimes ist die Konzentration des Transports auf nur einen einzigen der zur Verfügung stehenden Transmissionskanäle in einem ungeordneten Wellenleiter.Das Auftreten von Ein-Kanal-Transport wird durch die Analyse der Transport-Statistik sowie der Intensitätsverteilung innerhalb des ungeordneten Mediums gezeigt. Insbesondere verifizieren wir die Existenz sogenannter "necklace states" sowie deren Zusammenhang mit den internen Moden des offenen Systems.Während das Ein-Kanal-Regime eine direkte Folge von Anderson Lokalisierung darstellt, haben jüngste Forschungen gezeigt, dass bestimmte Systeme sehr robust gegen Lokalisierung sind. Solche "topologische Isolatoren" besitzen Randzustände mit genau dieser Eigenschaft. Darüberhinausgehend können diese Zustände nicht nur der Unordnung trotzen sondern sogar erst durch sie erzeugt werden. Dieses Phänomen, welches "topologischer Anderson-Isolator" (TAI) genannt wird, unter-suchen wir für den Fall eines räumlich korrelierten Potentials. Unsere numerischen Simulationen zeigen, dass solch eine Korrelation den Übergang in die TAI-Phase unterdrücken kann, was die Existenz des Phänomens in echten elektronischen Systemen in Frage stellt. Im Zuge dessen erweitern wir die bestehende analytische Theorie für den TAI auf den Fall räumlich korrelerierter Potentiale. Unsere zweite Studie zum Thema TAI befasst sich mit dem neuerlichen Ver-schwinden der TAI-Phase bei sehr starker Unordnung, welche zuvor mit einer Delokalisierung der Volumen-Zustände (also jener Zustände, die nicht entlang des Randes laufen) in Verbindung gebracht werden konnte. Wir zeigen, dass sowohl die Entstehung als auch das Abklingen der TAI-Phase in enger Verbindung mit sogenannten Perkolations-Zuständen steht, welche um die Hügel und Täler des Unordnungspotentials laufen. Weiters befindet sich in dieser Arbeit ein Vorschlag für die Realisierung eines Mach-Zehnder Interferometers in einem System topologischer Isolatoren. Ein solches könnte in Zukunft Anwendungen als Spin-Transistor finden. Neben Lokalisierung befassen wir uns auch mit einem zweiten fundamentalen Wellenphänomen genannt "branched flow". Dieser Effekt tritt für Wellen auf, welche durch ein langreichweitiges und schwaches Potential laufen. Dabei verteilt sich ihre Intensität in verästelter Weise anstatt sich gleichmäßig im System auszubreiten. In solchen Systeme, die in der Physik von der Nanoelektronik bis zu Monsterwellen im Ozean zu finden sind, führt der Effekt zur Bildung von Ästen konzentrierter Intensität ("branches") entlang welchen sich die Welle über große Distanzen ausbreiten kann. Wir befassen uns mit der Frage, inwieweit diese natürliche Eigenschaft zur Wellenkontrolle genutzt werden kann und Fokussieren durch ein ungeordnetes Medium ermöglicht. Unsere Methoden, die durch die Fortschritte in der Technologie mittlerweile auch schon im Experiment zugänglich sind, zeigen, dass durch gezielten Einschuss einzelne 'branches' selektiert werden können. Neben den physikalischen Anwendungen befasst sich diese Arbeit auch mit der erforderlichen Numerik, welche die Grundlage für die Erforschung der Transport-und Streueigenschaften ungeordneter Medien bildet. Die herkömmlichen Berechnungen verwenden die sogenannte 'modular-rekursive Greensfunktions-Methode',welche sehr effizient den numerischen Rechenaufwand minimiert. In dieser Arbeit konzentrieren wir uns hauptsächlich auf eine noch effizientere Parallelisierung der bekannten Algorithmen und erreichen eine signifikante Leistungssteigerung. Eine solche Verbesserung der Numerik erlaubt den Zugang zu größeren und stärker un-geordneten Systemen. Zusätzlich stellen wir einen Algorithmus namens 'parallel per-muation algorithm' (PPA) vor, welcher die Grundlage für die parallele Berechnungvon Wellenfunktionsbildern, lokaler Zustandsdichten und ungeordneten Superzell-Strukturen ist. Es werden für alle drei dieser Anwendungen Algorithmen implementiert, die hochparallelisierte Berechnungen ermöglichen und somit noch tieferen Einblick in die Streu- und Transporteigenschaften ungeordneter Systeme bieten - ein vielversprechender Ausgangspunkt für zukünftige Entwicklungen.

In this work, we investigated effects that are closely related to Anderson localization or the robustness against the latter. In a collaborate project with the University of San Antonio, Texas, we were concerned with, what we called, the single-channel regime of transport. Our numerical and experimental results validate a close relation between the single transmission eigenchannel typical of this regime and the internal modes of the system. We showed that in this regime a single transmission eigenchannel is either formed by a single such mode or by a combination of few modes resulting in a necklace state. By statistical means we could show that a single-channel sample can be described as a one-dimensional system with a renormalized localization length, thereby providing analytical means for the description of such, in general complicated, systems. In addition, we showed that the crossover to the single-channel regime is unique and can be charted in terms of a wellsuited quantity that shows the desired behavior in two-dimensional numerics and our three-dimensional microwave experiments. This cross-over could even be shown in the time-domain when the samples are excited by Gaussian pulses. For pulses with suitably chosen width, the single-channel transport through necklace states could be identified at short time-delays, whereas at large time delays single-channel transport through single long-lived localized modes could be observed. The results are fundamental for the understanding of waves in random media and provide opportunities for enhancing energy transfer through strongly-scattering systems. Apart from Anderson localization, we investigated another wave-effect that has concerned physicists for the past 15 years: Branched flow occurs through random media in which the disorder potential is weak but smooth. Such systems develop branches along which the flow is travelling through the medium. This time, we did not focus on principal knowledge about these systems but asked the question how to control and utilize the effect for future applications. In an exemplary optical system we developed a method that allows for the injection of a wave in such a way that almost all the flow travels just along a single such branch. We showed that this finding is not trivial and fairly stable over a large frequency range. Although our methods rely on the scattering matrix, we found that already small parts of it are sufficient for a fair amount of control of branched flow. The amount of information that needs to be measured should thus be within experimental reach and our results might provide an important step forward in wavefront shaping and wave control. In the second part of this thesis, we were concerned with a new class of materials that promise exciting properties in terms of transport. In such two-dimensional topological insulators, edge states that are robust against Anderson localization are present. Moreover, this edge states can even occur as a result of disorder which leads to enhanced transport properties. Our calculations show that a finite spatial correlation of the disorder potential enhances finite-size effects and may entirely suppress the aforementioned regime of quantized conductance known as the topological Anderson insulator (TAI) phase. We thereby extend previous studies in which only uncorrelated disorder potentials were considered. To describe the observed boundaries of quantized conductance theoretically, we perform a scaling analysis and adapt an existing effective medium theory to the case of spatially correlated potentials which yields quantitative agreement with our numerics. Our results suggest that for observing the topological Anderson insulator phase experimentally, it will be necessary to work with comparatively large samples (to suppress finite size effects) and with very short ranged disorder potentials as any long-range correlations may strongly suppress this topologically nontrivial phase. We speculate that spatial correlations might also be an important impediment to eliminate the bulk conductance in three-dimensional topological insulators. This would certainly constitute an interesting topic for further investigations. In a follow-up work, we linked the phenomenon with a quantum percolation transition that we found to occur in the limit of correlated strong disorder . While the reason for the emergence of the TAI had already been understood, its breakdown could so far only be vaguely connected to a delocalization of bulk states. Here we showed that in a spatially correlated potential this delocalization is caused primarily by bulk states, that are localized when circumnavigating the hills of the disorder potential, but that become connected with each other when passing a percolation threshold. These connections and thus also the delocalization transition are consolidated by local edge states that can internally form in the disordered sample. By showing how the localized bulk states derive from flat bands in the valence band structure of the clean sample without disorder, we clarified that the same physics is at work also in the well-studied case of an uncorrelated disorder potential. Additionally, we investigated the effects of magnetic fields on two-dimensional topological insulators. In accordance with literature we found that a perpendicular field has strong impact onto transport. In contrast, an in-plane field has hardly changes the topological insulator's electronic properties. We proposed a device that could be used for basic interference experiments using the edge states of a twodimensional topological insulator. The setup relies on building a quantum well structure in a cylindrical shape or possibly by using topological insulator thin films. By the application of two quantum point contacts (QPCs) the geometry resembles a classical Mach-Zehnder interferometer which lets us observe oscillations of the transmission (conductance) as a function of Fermi energy. When applying a very weak and in-plane magnetic field through the core of the cylinder, we find that the device can in addition use the Aharonov-Bohm effect in order to serve as a spinfilterand spin-transistor. Again the numerical results agree very nicely with thesimple model and the 4 possible settings of our spin-transistor were demonstrated numerically. The realization of such a device would open up countless opportunities for measurements of topological insulator properties and technological applications in spin-tronics and low-power information processing. The last chapter of this work is dedicated to the improvements that we did in the numerics. As the numerical solution of scattering problems comes down to the calculation of surface Green's functions (GFs) of the disordered region, we were concerned with making this task as efficient as possible. While the calculation effort is already minimal in the commonly used modular recursive Green's function method, we concentrated on efficient parallelization of the required tasks. Since the fastest way of calculating the required surface GFs (called parallel Dyson algorithm - PDA) still leaves some of the calculation power in a parallel computation unused, we developed an algorithm (parallel permutation algorithm - PPA) that fills these gaps. While suffering some overhead due to additional communication effort, this extratime proves well invested when, in addition to the scattering problem, scattering wave functions need to be computed. Moreover, we developed an algorithm for the efficient calculation of the local density of states. For this application, the PPA paves the way to parallelization that would be otherwise not possible. All the results are fortified by extensive benchmarks that proved our methods to allow access to larger and better-resolved systems than before. As another numerical task we addressed the evaluation of so-called super-cell structures. Such a problem considers the task of periodically stringing together a disordered region (super-cell) to itself which results in the existence of a band structure. We found an efficient way to calculate such band structures by the use of surface Green's functions. In addition, we also showed how one can calculate the associated Bloch functions (eigenstates) in a parallel and efficient manner. For this problem we found that the PPA is once more useful since our parallel algorithm for the calculation of scattering wave functions can, with some adjustments, also be employed for this task. Both the aforementioned applications are promising candidates for a future better understanding of disordered system. Our contributions provide access to large systems that could not be investigated previously.