@phdthesis{Zink2024,
  author    = {Zink, Johannes},
  title     = {Algorithms for Drawing Graphs and Polylines with Straight-Line Segments},
  doi       = {10.25972/OPUS-35475},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-354756},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2024},
  abstract  = {Graphs provide a key means to model relationships between entities. They consist of vertices representing the entities, and edges representing relationships between pairs of entities. To make people conceive the structure of a graph, it is almost inevitable to visualize the graph. We call such a visualization a graph drawing. Moreover, we have a straight-line graph drawing if each vertex is represented as a point (or a small geometric object, e.g., a rectangle) and each edge is represented as a line segment between its two vertices. A polyline is a very simple straight-line graph drawing, where the vertices form a sequence according to which the vertices are connected by edges. An example of a polyline in practice is a GPS trajectory. The underlying road network, in turn, can be modeled as a graph. This book addresses problems that arise when working with straight-line graph drawings and polylines. In particular, we study algorithms for recognizing certain graphs representable with line segments, for generating straight-line graph drawings, and for abstracting polylines. In the first part, we first examine, how and in which time we can decide whether a given graph is a stick graph, that is, whether its vertices can be represented as vertical and horizontal line segments on a diagonal line, which intersect if and only if there is an edge between them. We then consider the visual complexity of graphs. Specifically, we investigate, for certain classes of graphs, how many line segments are necessary for any straight-line graph drawing, and whether three (or more) different slopes of the line segments are sufficient to draw all edges. Last, we study the question, how to assign (ordered) colors to the vertices of a graph with both directed and undirected edges such that no neighboring vertices get the same color and colors are ascending along directed edges. Here, the special property of the considered graph is that the vertices can be represented as intervals that overlap if and only if there is an edge between them. The latter problem is motivated by an application in automated drawing of cable plans with vertical and horizontal line segments, which we cover in the second part. We describe an algorithm that gets the abstract description of a cable plan as input, and generates a drawing that takes into account the special properties of these cable plans, like plugs and groups of wires. We then experimentally evaluate the quality of the resulting drawings. In the third part, we study the problem of abstracting (or simplifying) a single polyline and a bundle of polylines. In this problem, the objective is to remove as many vertices as possible from the given polyline(s) while keeping each resulting polyline sufficiently similar to its original course (according to a given similarity measure).},
  subject      = {Graphenzeichnen},
  language  = {en}
}
@phdthesis{Schelter2012,
  author    = {Schelter, J{\"o}rg},
  title     = {The Aharonov-Bohm effect and resonant scattering in graphene},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-74662},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2012},
  abstract  = {In this thesis, the electronic transport properties of mesoscopic condensed matter systems based on graphene are investigated by means of numerical as well as analytical methods. In particular, it is analyzed how the concepts of quantum interference and disorder, which are essential to mesoscopic devices in general, are affected by the unique electronic and transport properties of the graphene material system. We consider the famous Aharonov-Bohm effect in ring-shaped transport geometries, and, besides providing an overview over the recent developments on the subject, we study the signatures of fundamental phenomena such as Klein tunneling and specular Andreev reflection, which are specific to graphene, in the magnetoconductance oscillations. To this end, we introduce and utilize a variant of the well-known recursive Green's function technique, which is an efficient numerical method for the calculation of transport observables in effectively non-interacting open quantum systems in the framework of a tight binding model. This technique is also applied to study the effects of a specific kind of disorder, namely short-range resonant scatterers, such as strongly bound adatoms or molecules, that can be modeled as vacancies in the graphene lattice. This numerical analysis of the conductance in the presence of resonant scatterers in graphene leads to a non-trivial classification of impurity sites in the graphene lattice and is further substantiated by an independent analytical treatment in the framework of the Dirac equation. The present thesis further contains a formal introduction to the topic of non-equilibrium quantum transport as appropriate for the development of the numerical technique mentioned above, a general introduction to the physics of graphene with a focus on the particular phenomena investigated in this work, and a conclusion where the obtained results are summarized and open questions as well as potential future developments are highlighted.},
  subject      = {Graphen},
  language  = {en}
}
@phdthesis{Pakkayil2017,
  author    = {Pakkayil, Shijin Babu},
  title     = {Towards ferromagnet/superconductor junctions on graphene},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-153863},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2017},
  abstract  = {This thesis reports a successful fabrication and characterisation of ferromagnetic/superconductor junction (F/S) on graphene. The thesis preposes a fabrication method to produce F/S junctions on graphene which make use of ALD grown Al2O3 as the tunnel barrier for the ferromagnetic contacts. Measurements done on F/G/S/G/F suggests that by injecting spin polarised current into the superconductor, a spin imbalance is created in the quasiparticle density of states of the superconductor which then diffuses through the graphene channel. The observed characteristic curves are similar to the ones which are already reported on metallic ferromagnet/superconductor junctions where the spin imbalance is created using Zeeman splitting. Further measurements also show that the curves loose their characteristic shapes when the temperature is increased above the critical temperature (Tc) or when the external magnetic field is higher then the critical field (Hc) of the superconducting contact. But to prove conclusively and doubtlessly the existence of spin imbalance in ferromagnet/superconductor junctions on graphene, more devices have to be made and characterised preferably in a dilution refrigerator.},
  subject      = {Graphen},
  language  = {en}
}
@phdthesis{Kiesel2012,
  author    = {Kiesel, Maximilian Ludwig},
  title     = {Unconventional Superconductivity in Cuprates, Cobaltates and Graphene: What is Universal and what is Material-Dependent in strongly versus weakly Correlated Materials?},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-76421},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2012},
  abstract  = {Eine allgemeing{\"u}ltige Theorie f{\"u}r alle unterschiedlichen Arten von unkonventionellen Supraleitern ist immer noch eine der ungel{\"o}sten Kernfragen der Festk{\"o}rperphysik. Momentan ist es nicht einmal bewiesen, dass es {\"u}berhaupt einen gemeinsamen grundlegenden Mechanismus gibt, sondern es m{\"u}ssen vielleicht mehrere verschiedene Ursachen f{\"u}r unkonventionelle Supraleitung ber{\"u}cksichtigt werden. Der Einfluss der Elektron-Phonon-Wechselwirkung ist dabei noch nicht abschließend gekl{\"a}rt. In dieser Dissertation wird ein rein elektronischer Paarungsmechanismus untersucht, in welchem die Paarung durch Spin-Fluktuationen vermittelt wird, was nach dem aktuellen Stand der Forschung auf dem Gebiet der unkonventionellen Supraleiter am wahrscheinlichsten ist. Der Schwerpunkt liegt dabei auf der Bestimmung von Material-unabh{\"a}ngigen Eigenschaften der supraleitenden Phase. Diese k{\"o}nnen durch eine Auswahl sehr unterschiedlicher Systeme herausgearbeitet werden. Eine Untersuchung der Phasendiagramme gibt außerdem Auskunft dar{\"u}ber, welche konkurrierenden Quantenfluktuationen den supraleitenden Zustand abschw{\"a}chen oder verst{\"a}rken. F{\"u}r diese Analyse von sehr unterschiedlichen supraleitenden Materialien ist der Einsatz einer einzelnen numerischen L{\"o}sungsmethode unzureichend. F{\"u}r diese Dissertation ist dies aber kein Nachteil, sondern vielmehr ein großer Vorteil, da der Einsatz verschiedener Techniken die Abh{\"a}ngigkeit der Ergebnisse von der verwendeten Numerik reduziert und dadurch der grundlegende Mechanismus besser untersucht werden kann. Im speziellen werden in dieser Dissertation die Kuprate mit der Variationellen Clustern{\"a}herung ausgewertet, weil die Elektronen hier eine starke Wechselwirkung untereinander besitzen. Besonders die Frage eines m{\"o}glichen Klebstoffs f{\"u}r die Cooper-Paare wird ausf{\"u}hrlich diskutiert, auch mit einer Unterscheidung in retardierte und nicht-retardierte Betr{\"a}ge. Den Kupraten werden das Kobaltat NaCoO sowie Graphen gegen{\"u}bergestellt. Diese Materialien sind jedoch schwach korrelierte Systeme, so dass hier die Funkionelle Renormierungsgruppe als numerisches Grundger{\"u}st dient. Die Ergebnisse sind reichhaltige Phasendiagramme mit vielen verschiedenen langreichweitigen Ordnungen, wie zum Beispiel d+id-wellenartige Supraleitung. Diese bricht die Zeitumkehr-Symmetrie und besitzt eine vollst{\"a}ndige Bandl{\"u}cke, welche im Falle von NaCoO jedoch eine stark Dotierungs-abh{\"a}ngige Anisotropie aufweist. Als letztes wird das Kagome-Gitter allgemein diskutiert, ohne ein konkretes Material zu beschreiben. Hier hat eine destruktive Interferenz zwischen den Elektronen auf verschiedenen Untergittern drastische Auswirkungen auf die Instabilit{\"a}ten der Fermi-Fl{\"a}che, so dass die {\"u}bliche Spin-Dichte-Welle und die damit verbundene d+id-wellenartige Supraleitung unterdr{\"u}ckt werden. Dadurch treten ungew{\"o}hnliche Spin- und Ladungsdichte-Ordnungen sowie eine nematische Pomeranchuck Instabilit{\"a}t hervor. Zusammengefasst bietet diese Dissertation einen Einblick in unterschiedliche Materialklassen von unkonventionellen Supraleitern. Dadurch wird es m{\"o}glich, die Material-spezifischen Eigenschaften von den universellen zu trennen.},
  subject      = {Supraleitung},
  language  = {en}
}
@phdthesis{Herrmann2016,
  author    = {Herrmann, Oliver},
  title     = {Graphene-based single-electron and hybrid devices, their lithography, and their transport properties},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-146924},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2016},
  abstract  = {This work explores three different aspects of graphene, a single-layer of carbon atoms arranged in a hexagonal lattice, with regards to its usage in future electronic devices; for instance in the context of quantum information processing. For a long time graphene was believed to be thermodynamically unstable. The discovery of this strictly two-dimensional material completed the family of carbon based structures, which had already been subject of intensive research with focus on zero-dimensional fullerenes and one-dimensional carbon nanotubes. Within only a few years of its discovery, the field of graphene related research has grown into one of today's most diverse and prolific areas in condensed matter physics, highlighted by the award of the 2010 Nobel Prize in Physics to A.K. Geim and K. Noveselov for "their groundbreaking experiments regarding the two-dimensional material graphene". From the point of view of an experimental physicist interested in the electronic properties of a material system, the most intriguing characteristic of graphene is found in the Dirac-like nature of its charge carriers, a peculiar fact that distinguishes graphene from all other known standard semiconductors. The dynamics of charge carriers close to zero energy are described by a linear energy dispersion relation, as opposed to a parabolic one, which can be understood as a result of the underlying lattice symmetry causing them to behave like massless relativistic particles. This fundamentally different behavior can be expected to lead to the observation of completely new phenomena or the occurrence of deviations in well-known effects. Following a brief introduction of the material system in chapter 2, we present our work studying the effect of induced superconductivity in mesoscopic graphene Josephson junctions by proximity to superconducting contacts in chapter 3. We explore the use of Nb as the superconducting material driven by the lack of high critical temperature and high critical magnetic field superconductor technology in graphene devices at that time. Characterization of sputter-deposited Nb films yield a critical transition temperature of \(T_{C}\sim 8{\rm \,mK}\). A prerequisite for successful device operation is a high interface quality between graphene and the superconductor. In this context we identify the use of an Ti as interfacial layer and incorporate its use by default in our lithography process. Overall we are able to increase the interface transparency to values as high as \(85\\%\). With the prospect of interesting effects in the ballistic regime we try to enhance the electronic quality of our Josephson junction devices by substrate engineering, yet with limited success. We achieve moderate charge carrier mobilities of up to \(7000{\rm \,cm^2/Vs}\) on a graphene/Boron-nitride heterostructure (fabrication details are covered in chapter 5) putting the junction in the diffusive regime (\(L_{device}<L_{\rm{mfp}}\)). We speculate that either inhomogeneities in the graphene channel or lithography residues are responsible for this observation. Furthermore we study the Josephson effect and Andreev reflection related physics in this device by low-temperature transport measurements. The junction carries a bipolar supercurrent which remains finite at the charge neutrality point. The genuine Josephson character is confirmed by the modulation of the supercurrent as a function of an out-of-plane magnetic field resembling that of a Fraunhofer-like pattern. This is further supported by the response of the junction to microwave radiation in the form of Shaprio steps. Surprisingly we find a strongly reduced superconducting energy gap of approximately \(\Delta = 400{\rm \,\mu eV}\) by quantitatively analyzing data of multiple Andreev reflections. We show this result to be consistent by careful analysis of the device parameters and comparison of these to a theoretical model. More experiments will be needed to determine the origin of this reduction and if the presence of the Ti interfacial layer plays an important role in that. With regards to possible usability of superconducting contacts in more complex hybrid structures we can conclude that our work establishes the necessary preconditions while still leaving room for improvements; especially in terms of device quality. In the second part of this work we are primarily interested in electrical transport properties of graphene nanodevices and their application in graphene-superconductor hybrid structures. The fact that graphene is mechanically stable down to a few tens of nanometers in width while exhibiting a finite conductance makes it an appealing choice as host for single-electron devices, also known as quantum dots. Our work on this topic is covered in chapter 4 where we first develop a high-resolution lithography process for the fabrication of single electron devices with critical feature sizes of roughly \(50{\rm \,nm}\). To this end we use a resist etch mask in combination with a reactive-ion etch process for device patterning. Carrier confinement in graphene is known to be hindered by the Klein tunneling phenomenon, a challenge that can be overcome by using all-graphene nano-constrictions to decouple the source and drain contacts from the central island. The traditionally used constriction design is comprised of long and narrow connections. We argue that a design with very short and narrow constrictions could be beneficial for the quantum dot performance as the length merely affects the overall conductance and requires extended side-gates to control their transmission. We confirm the functionality of two different devices in low-temperature measurements, which differ in the size of their central island with \(d=250{\rm \,nm}\) for device no. 1 and \(d=400{\rm \,nm}\) for device no. 2. Coulomb blockade measurements conducted at \(20{\rm \,mK}\) on both devices reveal clear sequences of Coulomb peaks with amplitudes of up to \(0.8\rm{\,e}^2/\rm{h}\), a value significantly larger than what is commonly reported for similar devices. We interpret this as an indication of rather homogeneous constrictions, resulting from the modified design. Coulomb diamond measurements display the behavior expected for a lithographically designed single quantum dot revealing no features related to the presence of an additional dot. Using the stability diagram we determine the addition energies of the two dots and find them to be in good agreement with values reported in the literature for devices of similar size. Using the normalized Coulomb peak spacing as a figure of merit for the device quality we find that device no. 1 quantitatively compares well with a similar device fabricated on a superior hexagonal boron-nitride substrate. This result underlines the importance of non-substrate related extrinsic disorder sources and emphasizes the cleanliness of our lithography process. Superconductor-graphene quantum dot hybrid structures employing Nb and Al electrodes were successfully fabricated from a lithography point of view, yet no evidence of any superconducting related effect was found in transport measurements. We assign the missing observation to interface issues that require careful analysis and likely a revision of the fabrication process. A property equally important in graphene Josephson Junctions and quantum dots is the electronic quality of the device, as has been addressed in the previous paragraphs. It turns out that the \(\rm{SiO}_{2}\;\) substrate and lithography residues constitute the two major sources of disorder in graphene. In chapter 5 we present an approach based on the original work of Dean et al. who utilize hexagonal-Boron nitride as a replacement substrate for \(\rm{SiO}_{2}\). This idea was then extended by Wang et al. who also used this material as a shield to protect the graphene surface from contaminations during the lithography process. These structures are commonly referred to as van der Waals heterostructures and are assembled by stacking individual crystals on top of each other. For this purpose we build a mechanical transfer system based on an optical microscope equipped with an additional micro-manipulator stage allowing precise alignment of two micrometer sized crystals with high precision. We demonstrate the functionality of this setup on the basis of successfully fabricated heterostructures. Furthermore a variation on the traditional method for single graphene/boron nitride structures is presented. Based on a reversed stacking order this method yields large areas of homogeneous graphene, however it comes with the drawback of limited yields. A common type of problem accompanying the fabrication of encapsulated graphene structures is the formation of contamination spots (also referred to as bubbles in the literature) at the interfaces between BN and graphene. We experience similar issues which we are unable to prevent and thus pose a limit to the maximum available device size. In the next step we develop a full lithography paradigm including high-resolution device patterning by electron beam lithography combined with reactive ion etching and two different ways to establish electrical contact to the encapsulated graphene flake. In this context we explore the use of three different types of etch masks and find a double layer of PMMA/HSQ best suited for our purposes. Our low power plasma etch process utilizes a combination of \(\rm{O}_{2}\;\) and \(\rm{CHF}_{3}\;\) and is optimized to show reproducible etch results. A widely used method for electrical contacts relies on one-dimensional edge contacts whose functionality crucially depends on the use of Cr as the interface layer. For compatibility reasons with superconducting materials, e.g. Nb, we develop a self-aligned contact process that instead of only Cr is also compatible with Ti. We achieve this by modifying the plasma etch parameters such that the etch process exhibits extremely low graphene etch rates while keeping a high etch rate for h-BN. This allows clearing of a narrow stripe of graphene at the edge of the structure by using a thick PMMA layer as etch mask as replacement of the PMMA/HSQ combination. The purpose of this PMMA mask is two-fold since it also serves as lift-off mask during metalization. The quality of the edge contacts fabricated with either method is excellent as determined from transport measurements at room and cryogenic temperatures. With typical contact resistances of a few hundred \({\rm \,}\Omega\mu{\rm m}\) and a record low of \(100{\rm \,}\Omega\mu{\rm m}\) the contacts can be considered to be state-of-the-art. The positive effect of encapsulation on the electronic quality is confirmed on a device exhibiting charge carrier mobilities exceeding \(10^5{\rm \,cm^2/Vs}\), one magnitude larger than what is commonly achieved on \(\rm{SiO}_{2}\). The investigation of induced superconductivity in graphene Josephson Junctions, quantum dots, and high mobility heterostructures underlines the versatility of this material system, while covering only a tiny fraction of its prospects. Combination of the acquired knowledge regarding the physical effects and the developed lithography processes lay the foundation towards the fabrication and study of novel graphene hybrid devices.},
  subject      = {Graphen},
  language  = {en}
}
@phdthesis{Fuchs2016,
  author    = {Fuchs, Moritz Jakob},
  title     = {Spin dynamics in the central spin model: Application to graphene quantum dots},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-136079},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2016},
  abstract  = {Due to their potential application for quantum computation, quantum dots have attracted a lot of interest in recent years. In these devices single electrons can be captured, whose spin can be used to define a quantum bit (qubit). However, the information stored in these quantum bits is fragile due to the interaction of the electron spin with its environment. While many of the resulting problems have already been solved, even on the experimental side, the hyperfine interaction between the nuclear spins of the host material and the electron spin in their center remains as one of the major obstacles. As a consequence, the reduction of the number of nuclear spins is a promising way to minimize this effect. However, most quantum dots have a fixed number of nuclear spins due to the presence of group III and V elements of the periodic table in the host material. In contrast, group IV elements such as carbon allow for a variable size of the nuclear spin environment through isotopic purification. Motivated by this possibility, we theoretically investigate the physics of the central spin model in carbon based quantum dots. In particular, we focus on the consequences of a variable number of nuclear spins on the decoherence of the electron spin in graphene quantum dots. Since our models are, in many aspects, based upon actual experimental setups, we provide an overview of the most important achievements of spin qubits in quantum dots in the first part of this Thesis. To this end, we discuss the spin interactions in semiconductors on a rather general ground. Subsequently, we elaborate on their effect in GaAs and graphene, which can be considered as prototype materials. Moreover, we also explain how the central spin model can be described in terms of open and closed quantum systems and which theoretical tools are suited to analyze such models. Based on these prerequisites, we then investigate the physics of the electron spin using analytical and numerical methods. We find an intriguing thermal flip of the electron spin using standard statistical physics. Subsequently, we analyze the dynamics of the electron spin under influence of a variable number of nuclear spins. The limit of a large nuclear spin environment is investigated using the Nakajima-Zwanzig quantum master equation, which reveals a decoherence of the electron spin with a power-law decay on short timescales. Interestingly, we find a dependence of the details of this decay on the orientation of an external magnetic field with respect to the graphene plane. By restricting to a small number of nuclear spins, we are able to analyze the dynamics of the electron spin by exact diagonalization, which provides us with more insight into the microscopic details of the decoherence. In particular, we find a fast initial decay of the electron spin, which asymptotically reaches a regime governed by small fluctuations around a finite long-time average value. Finally, we analytically predict upper bounds on the size of these fluctuations in the framework of quantum thermodynamics.},
  subject      = {Elektronenspin},
  language  = {en}
}