# Ballistic electron transport in graphene nanodevices and billiards

by George Datseris

Date of Examination:2019-09-13

Date of issue:2019-10-17

Advisor:Prof. Dr. Theo Geisel

Referee:Prof. Dr. Theo Geisel

Referee:Prof. Dr. Stephan Herminghaus

Referee:Dr. Michael Wilczek

Referee:Prof. Dr. Jörg Enderlein

Referee:Prof. Dr. Ulrich Parlitz

Referee:Prof. Dr. Stefan Kehrein

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## Abstract

### English

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 nanodevice, 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 commensurability 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.\newline These analytic connections are further taken advantage of in dynamical billiards, systems typically 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. \newline 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 analytic 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 expectation 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 measurement. 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 undergo a special tunneling process called Klein tunneling. There, a particle incident on a potential 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 eigenfunction, 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. \newline 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 concepts 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 geometries. We also show how, through the Husimi function, one can recover the transmission probabilities of electronic wavefunctions being transmitted through a nanodevice, by interpreting 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. \newline 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.**Keywords:**ballistic; electron transport; graphene; billiards; chaos; dynamical systems; nanodevices