Studying the electronic band structure is necessary to understand the physics behind electron transport in devices made of the material under consideration. Graphene has been in the limelight of ground breaking research in the recent past and will continue to be in the future, as scientists around the world have been giving special attention to this remarkable material in pursuit of advancement in the semiconductor device industry by making use of many of its fascinating properties.

Although several different approaches have been proposed for the calculation of the band structure, the empirical methods have proven to be more convenient, since we can arrive at a reasonable result close to what has been found experimentally without a huge computational trade-off by varying some relevant parameters. Moreover, a method based on a plane wave basis has been found to be extremely compatible with advanced methods for device modeling and simulation such as a semi-classical Monte Carlo. The purpose of this study is to extract fitting parameters for the calculation of band structure of graphene using the empirical pseudopotential method, which can be used for further research, such as theoretical modeling and simulation of graphene-based devices. Since various methods have been proposed for the calculation of these pseudopotentials, we will also briefly study the effect of different pseudopotentials on the band structure of bilayer graphene.

In mesoscopic physics, conductance fluctuations are a quantum interference phenomenon that comes from the phase interference of electron wave functions scattered by the impurity disorder. During the past few decades, conductance fluctuations have been studied in various materials including metals, semiconductors and graphene. Since the patterns of conductance fluctuations is related to the distributions and configurations of the impurity scatterers, each sample has its unique pattern of fluctuations, which is considered as a sample fingerprint. Thus, research on conductance fluctuations attracts attention worldwide for its importance in both fundamental physics and potential technical applications. Since early experimental measurements of conductance fluctuations showed that the amplitudes of the fluctuations are on order of a universal value (e2/h), theorists proposed the hypothesis of ergodicity, e.g. the amplitudes of the conductance fluctuations by varying impurity configurations is the same as that from varying the Fermi energy or varying the magnetic field. They also proposed the principle of universality; e.g., that the observed fluctuations would appear the same in all materials. Recently, transport experiments in graphene reveal a deviation of fluctuation amplitudes from those expected from ergodicity.

Thus, in my thesis work, I have carried out numerical research on the conductance fluctuations in GaAs nanowires and graphene nanoribbons in order to examine whether or not the theoretical principles of universality and ergodicity hold. Finite difference methods are employed to study the conductance fluctuations in GaAs nanowires, but an atomic basis tight-binding model is used in calculations of graphene nanoribbons. Both short-range disorder and long-range disorder are considered in the simulations of graphene. A stabilized recursive scattering matrix technique is used to calculate the conductance. In particular, the dependence of the observed fluctuations on the amplitude of the disorder has been investigated. Finally, the root-mean-square values of the amplitude of conductance fluctuations are calculated as a basis with which to draw the appropriate conclusions. The results for Fermi energy sweeps and magnetic field sweeps are compared and effects of magnetic fields on the conductance fluctuations of Fermi energy sweeps are discussed for both GaAs nanowires and graphene nanoribbons.

What can classical chaos do to quantum systems is a fundamental issue highly relevant to a number of branches in physics. The field of quantum chaos has been active for three decades, where the focus was on non-relativistic quantumsystems described by the Schr¨odinger equation. By developing an efficient method to solve the Dirac equation in the setting where relativistic particles can tunnel between two symmetric cavities through a potential barrier, chaotic cavities are found to suppress the spread in the tunneling rate. Tunneling rate for any given energy assumes a wide range that increases with the energy for integrable classical dynamics. However, for chaotic underlying dynamics, the spread is greatly reduced. A remarkable feature, which is a consequence of Klein tunneling, arise only in relativistc quantum systems that substantial tunneling exists even for particle energy approaching zero. Similar results are found in graphene tunneling devices, implying high relevance of relativistic quantum chaos to the development of such devices. Wave propagation through random media occurs in many physical systems, where interesting phenomena such as branched, fracal-like wave patterns can arise. The generic origin of these wave structures is currently a matter of active debate. It is of fundamental interest to develop a minimal, paradigmaticmodel that can generate robust branched wave structures. In so doing, a general observation in all situations where branched structures emerge is non-Gaussian statistics of wave intensity with an algebraic tail in the probability density function. Thus, a universal algebraic wave-intensity distribution becomes the criterion for the validity of any minimal model of branched wave patterns. Coexistence of competing species in spatially extended ecosystems is key to biodiversity in nature. Understanding the dynamical mechanisms of coexistence is a fundamental problem of continuous interest not only in evolutionary biology but also in nonlinear science. A continuous model is proposed for cyclically competing species and the effect of the interplay between the interaction range and mobility on coexistence is investigated. A transition from coexistence to extinction is uncovered with a non-monotonic behavior in the coexistence probability and switches between spiral and plane-wave patterns arise. Strong mobility can either promote or hamper coexistence, while absent in lattice-based models, can be explained in terms of nonlinear partial differential equations.

Complex dynamical systems consisting interacting dynamical units are ubiquitous in nature and society. Predicting and reconstructing nonlinear dynamics of units and the complex interacting networks among them serves the base for the understanding of a variety of collective dynamical phenomena. I present a general method to address the two outstanding problems as a whole based solely on time-series measurements. The method is implemented by incorporating compressive sensing approach that enables an accurate reconstruction of complex dynamical systems in terms of both nodal equations that determines the self-dynamics of units and detailed coupling patterns among units. The representative advantages of the approach are (i) the sparse data requirement which allows for a successful reconstruction from limited measurements, and (ii) general applicability to identical and nonidentical nodal dynamics, and to networks with arbitrary interacting structure, strength and sizes. Another two challenging problem of significant interest in nonlinear dynamics: (i) predicting catastrophes in nonlinear dynamical systems in advance of their occurrences and (ii) predicting the future state for time-varying nonlinear dynamical systems, can be formulated and solved in the framework of compressive sensing using only limited measurements. Once the network structure can be inferred, the dynamics behavior on them can be investigated, for example optimize information spreading dynamics, suppress cascading dynamics and traffic congestion, enhance synchronization, game dynamics, etc. The results can yield insights to control strategies design in the real-world social and natural systems. Since 2004, there has been a tremendous amount of interest in graphene. The most amazing feature of graphene is that there exists linear energy-momentum relationship when energy is low. The quasi-particles inside the system can be treated as chiral, massless Dirac fermions obeying relativistic quantum mechanics. Therefore, the graphene provides one perfect test bed to investigate relativistic quantum phenomena, such as relativistic quantum chaotic scattering and abnormal electron paths induced by klein tunneling. This phenomenon has profound implications to the development of graphene based devices that require stable electronic properties.

ABSTRACT An Ensemble Monte Carlo (EMC) computer code has been developed to simulate, semi-classically, spin-dependent electron transport in quasi two-dimensional (2D) III-V semiconductors. The code accounts for both three-dimensional (3D) and quasi-2D transport, utilizing either 3D or 2D scattering mechanisms, as appropriate. Phonon, alloy, interface roughness, and impurity scattering mechanisms are included, accounting for the Pauli Exclusion Principle via a rejection algorithm. The 2D carrier states are calculated via a self-consistent 1D Schrödinger-3D-Poisson solution in which the charge distribution of the 2D carriers in the quantization direction is taken as the spatial distribution of the squared envelope functions within the Hartree approximation. The wavefunctions, subband energies, and 2D scattering rates are updated periodically by solving a series of 1D Schrödinger wave equations (SWE) over the real-space domain of the device at fixed time intervals. The electrostatic potential is updated by periodically solving the 3D Poisson equation. Spin-polarized transport is modeled via a spin density-matrix formalism that accounts for D'yakanov-Perel (DP) scattering. Also, the code allows for the easy inclusion of additional scattering mechanisms and structural modifications to devices. As an application of the simulator, the current voltage characteristics of an InGaAs/InAlAs HEMT are simulated, corresponding to nanoscale III-V HEMTs currently being fabricated by Intel Corporation. The comparative effects of various scattering parameters, material properties and structural attributes are investigated and compared with experiments where reasonable agreement is obtained. The spatial evolution of spin-polarized carriers in prototypical Spin Field Effect Transistor (SpinFET) devices is then simulated. Studies of the spin coherence times in quasi-2D structures is first investigated and compared to experimental results. It is found that the simulated spin coherence times for GaAs structures are in reasonable agreement with experiment. The SpinFET structure studied is a scaled-down version of the InGaAs/InAlAs HEMT discussed in this work, in which spin-polarized carriers are injected at the source, and the coherence length is studied as a function of gate voltage via the Rashba effect.