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Ballistic Electron Transport in Graphene Nanodevices and Billiards

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Ballistic Electron Transport in Graphene Nanodevices and Billiards Ballistic Electron Transport in Graphene Nanodevices and Billiards Dissertation (Cumulative Thesis) for the award of the degree Doctor rerum naturalium of the Georg-August-Universität Göttingen within the doctoral program Physics of Biological and Complex Systems of the Georg-August University School of Science submitted by George Datseris from Athens, Greece Göttingen 2019 Thesis advisory committee Prof. Dr. Theo Geisel Department of Nonlinear Dynamics Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Stephan Herminghaus Department of Dynamics of Complex Fluids Max Planck Institute for Dynamics and Self-Organization Dr. Michael Wilczek Turbulence, Complex Flows & Active Matter Max Planck Institute for Dynamics and Self-Organization Examination board Prof. Dr. Theo Geisel (Reviewer) Department of Nonlinear Dynamics Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Stephan Herminghaus (Second Reviewer) Department of Dynamics of Complex Fluids Max Planck Institute for Dynamics and Self-Organization Dr. Michael Wilczek Turbulence, Complex Flows & Active Matter Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Ulrich Parlitz Biomedical Physics Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Stefan Kehrein Condensed Matter Theory, Physics Department Georg-August-Universität Göttingen Prof. Dr. Jörg Enderlein Biophysics / Complex Systems, Physics Department Georg-August-Universität Göttingen Date of the oral examination: September 13th, 2019 Contents Abstract 3 1 Introduction 5 1.1 Thesis synopsis and outline.............................. 10 2 Fundamental Concepts 13 2.1 Nonlinear dynamics of antidot superlattices..................... 13 2.2 Lyapunov exponents in billiards............................ 20 2.3 Fundamental properties of graphene......................... 22 2.4 Why quantum?..................................... 31 2.5 Transport simulations and scattering wavefunctions................. 33 2.6 Quantum mechanics and the phase space....................... 34 3 1st paper: Robustness of ballistic transport in antidot superlattices 38 3.1 Hall resistance..................................... 48 4 Software to simulate dynamical billiards 51 4.1 Design.......................................... 51 4.2 Usage in my projects.................................. 53 4.3 Unique features..................................... 54 5 2nd paper: Estimating Lyapunov exponents in billiards 55 6 3rd paper: Phase space analysis of quantum transport in graphene 69 7 Husimi functions and Klein tunneling in magnetic fields 81 7.1 Husimi function in magnetic fields.......................... 81 7.2 Application to a graphene nanodevice........................ 84 7.3 Klein tunneling in a pn-junction with magnetic field................ 85 8 Summary & Outlook 89 8.1 Summary of results................................... 89 8.2 Outlook......................................... 90 Acknowledgments 92 Bibliography 92 Curriculum Vitae 101 2 Abstract Nanodevices are objects with broad relevance for today’s society, which bases its economy, operation, and communication onto digital resources. In ballistic nanodevices, disorder has either negligible or minor impact on electron transport. These ballistic devices are then typical components in computers and digital storage units. This cumulative thesis theoretically studies ballistic electron transport in graphene-based nanodevices and billiards. The thesis is divided into two main parts: the first part uses classical dynamical systems theory, while the second part applies semiclassical approaches to quantum transport. The first part of the thesis is motivated by recent graphene experiments, that show puzzling measurements of the resistance versus the magnetic field strength in an antidot superlattice nan- odevice, a two-dimensional periodic array of repellers patterned on top of a crystalline lattice (in this case, graphene). The resistance of an antidot superlattice shows pronounced peaks when the strength of an applied perpendicular magnetic field is varied. The peaks, called commensu- rability peaks, occur when the cyclotron radius becomes commensurable with the geometry of the superlattice. This effect was been studied before, and it was shown that classical theory and nonlinear dynamics are best suited to understand it. What is puzzling is that existing theories cannot account for the observed peaks, when the electron mean free time (average time between disorder-induced scatterings) in the device is as small as in the recent experiments. By finding the inherent timescales of the dynamical system that represents the nanodevice, we provide the necessary theoretical background to explain the experiments and understand how it is possible to observe the effect. In the process of doing so, we also resolve a long-standing controversy on the origin of the effect and its connection with the various parts of the electron dynamics in an antidot superlattice. In addition, analytic connections are made between characteristic timescales in dynamical systems and the volume of sets in the phase space, and the impact of volume conservation of Hamiltonian mechanics is demonstrated. These analytic connections are further taken advantage of in dynamical billiards, systems typi- cally used to model real nanodevices. There, we show that generically in billiards the Lyapunov exponent, a number quantifying the chaotic dynamics of a system, has a leading contribution inversely proportional to the chaotic phase space volume. This finding has theoretical value in itself, but it also allows one to readily connect the Lyapunov exponent with the parameters of the billiard, which has implications for using billiards to model real physical systems. The results of the first part of the thesis have implications in understanding transport in many models following classical dynamics, because of the very simple principles used in deriving ana- lytic relations between transport timescales and phase space volumes. Also, our results suggest that if a relevant dynamic timescale is larger than the mean free time, then a reasonable expec- tation is that the dynamic feature will not be observable at all in experiments. Thus, knowing exactly which is the important timescale is crucial even before starting an experimental mea- surement. The second part of this thesis is motivated by quantum processes specific to graphene, as well as questions regarding quantum-classical correspondence. Specifically, electrons in graphene 3 undergo a special tunneling process called Klein tunneling. There, a particle incident on a po- tential barrier can penetrate the barrier with perfect transmission (100% probability), provided that the incident particle reaches the barrier with zero angle of incidence (i.e. normal incidence). Oddly, this is true irrespectively of the characteristics of the potential. From its definition, Klein tunneling is a process that depends on the angle of incidence of the particle, a concept that has classical intuition. Furthermore, even though possible to define in a trivial plane wave eigen- function, defining the angle of incidence in a complicated quantum transport setting is not at all trivial. To be able to define concepts such as the angle of incidence, and obtain classical intuition about them in a quantum transport numerical experiment, in this thesis we use the Husimi function, in both the absence and presence of magnetic fields. The Husimi function is a tool that can transform a quantum wavefunction into a probability distribution over the phase space of position and momentum, which is useful for defining con- cepts such as the angle of incidence. The Husimi function also helps to quantitatively define more complicated concepts, like intervalley scattering, and even in devices with non-trivial ge- ometries. We also show how, through the Husimi function, one can recover the transmission probabilities of electronic wavefunctions being transmitted through a nanodevice, by interpret- ing the marginal distributions of the Husimi function as weighting probabilities of a transmission formula. We then quantitatively measure the effect that pn-junctions and geometric scattering have on intervalley scattering. Surprisingly, we show that pn-junctions can intervalley-scatter only one of the two valleys of graphene. In the literature, it has been qualitatively suggested that the armchair termination of graphene is the strongest geometric intervalley scatterer. Here, we prove this to be true using quantitative measurements. Lastly, for the first time in the literature, we extend the Husimi function for electrons moving in magnetic fields while respecting fundamental energy considerations. We further use this new tool to study Klein tunneling in magnetic fields. Our results demonstrate that Klein tunneling in magnetic fields is not well understood yet since our numeric study does not align with the existing theory on the effect. The entire second part of this thesis showcases that the Husimi function is very likely to help unravel transport phenomena in nanodevices. Thus, it could be of general interest for condensed matter theory, currently a field that does not utilize the Husimi function at all. 4 1| Introduction The 21st century has been dubbed by many the Information Age or Digital Age [1], due to the rapid shift from the previous industry-based economy to a society that bases its economy, operation and communication onto digital resources. It is not an exaggeration to say that in the present day it would
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