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Despite its precise agreement with the experiment, the validity of the standard model (SM) of elementary particle physics is ensured only up to a scale of several hundred GeV so far. Even more, the inclusion of gravity into an unifying theory poses a problem which cannot be solved by ordinary quantum field theory (QFT). String theory, which is the most popular ansatz for a unified theory, predicts QFT on noncommutative space-time as a low energy limit. Nevertheless, independently of the motivation given by string theory, the nonlocality inherent to noncommutative QFT opens up the possibility for the inclusion of gravity. There are no theoretical predictions for the energy scale Lambda_NC at which noncommutative effects arise and it can be assumed to lie in the TeV range, which is the energy range probed by the next generation of colliders. Within this work we study the phenomenological consequences of a possible realization of QFT on noncommutative space-time relying on this assumption. The motivation for this thesis was given by the gap in the range of phenomenological studies of noncommutative effects in collider experiments, due to the absence in the literature of Large Hadron Collider (LHC) studies regarding noncommutative QFTs. In the first part we thus performed a phenomenological analysis of the hadronic process pp -> Z gamma -> l^+l^- gamma at the LHC and of electron-positron pair annihilation into a Z boson and a photon at the International Linear Collider (ILC). The noncommutative extension of the SM considered within this work relies on two building blocks: the Moyal-Weyl star-product of functions on ordinary space-time and the Seiberg-Witten maps. The latter relate the ordinary fields and parameters to their noncommutative counterparts such that ordinary gauge transformations induce noncommutative gauge transformations. This requirement is expressed by a set of inhomogeneous differential equations (the gauge equivalence equations) which are solved by the Seiberg-Witten maps order by order in the noncommutative parameter Theta. Thus, by means of the Moyal-Weyl star-product and the Seiberg-Witten maps a noncommutative extension of the SM as an effective theory as expansion in powers of Theta can be achieved, providing the framework of our phenomenological studies. A consequence of the noncommutativity of space-time is the violation of rotational invariance with respect to the beam axis. This effect shows up in the azimuthal dependence of cross sections, which is absent in the SM as well as in other models beyond the SM. Thus, the azimuthal dependence of the cross section is a typical signature of noncommutativity and can be used in order to discriminate it against other new physics effects. We have found this dependence to be best suited for deriving the sensitivity bounds on the noncommutative scale Lambda_NC. By studying pp -> Z gamma -> l^+l^- gamma to first order in the noncommutative parameter Theta, we show in the first part of this work that measurements at the LHC are sensitive to noncommutative effects only in certain cases, giving bounds on the noncommutative scale of Lambda_NC > 1.2 TeV. Our result improved the bounds present in the literature coming from past and present collider experiments by one order of magnitude. In order to explore the whole parameter range of the noncommutativity, ILC studies are required. By means of e^+e^- -> Z gamma -> l^+l^- gamma to first order in Theta we have shown that ILC measurements are complementary to LHC measurements of the noncommutative parameters. In addition, the bounds on Lambda_NC derived from the ILC are significantly higher and reach Lambda_NC > 6 TeV. The second part of this work arose from the necessity to enlarge the range of validity of our model towards higher energies. Thus, we expand the neutral current sector of the noncommutative SM to second order in $\theta$. We found that, against the general expectation, the theory must be enlarged by additional parameters. The new parameters enter the theory as ambiguities of the Seiberg-Witten maps. The latter are not uniquely determined and differ by homogeneous solutions of the gauge equivalence equations. The expectation was that the ambiguities correspond to field redefinitions and therefore should vanish in scattering matrix elements. However, we proved that this is not the case, and the ambiguities do affect physical observables. Our conjecture is, that every order in Theta will introduce new parameters to the theory. However, only the experiment can decide to what extent efforts with still higher orders in Theta are reasonable and will also give directions for the development of theoretical models of noncommutative QFTs.
In this PhD thesis, we develop models for the numerical simulation of epitaxial crystal growth, as realized, e.g., in molecular beam epitaxy (MBE). The basic idea is to use a discrete lattice gas representation of the crystal structure, and to apply kinetic Monte Carlo (KMC) simulations for the description of the growth dynamics. The main advantage of the KMC approach is the possibility to account for atomistic details and at the same time cover MBE relevant time scales in the simulation. In chapter 1, we describe the principles of MBE, pointing out relevant physical processes and the influence of experimental control parameters. We discuss various methods used in the theoretical description of epitaxial growth. Subsequently, the underlying concepts of the KMC method and the lattice gas approach are presented. Important aspects concerning the design of a lattice gas model are considered, e.g. the solid-on-solid approximation or the choice of an appropriate lattice topology. A key element of any KMC simulation is the selection of allowed events and the evaluation of Arrhenius rates for thermally activated processes. We discuss simplifying schemes that are used to approximate the corresponding energy barriers if detailed knowledge about the barriers is not available. Finally, the efficient implementation of the MC kinetics using a rejection-free algorithm is described. In chapter 2, we present a solid-on-solid lattice gas model which aims at the description of II-VI(001) semiconductor surfaces like CdTe(001). The model accounts for the zincblende structure and the relevant surface reconstructions of Cd- and Te-terminated surfaces. Particles at the surface interact via anisotropic nearest and next nearest neighbor interactions, whereas interactions in the bulk are isotropic. The anisotropic surface interactions reflect known properties of CdTe(001) like the small energy difference between the c(2x2) and (2x1) vacancy structures of Cd-terminated surfaces. A key element of the model is the presence of additional Te atoms in a weakly bound Te* state, which is motivated by experimental observations of Te coverages exceeding one monolayer at low temperatures and high Te fluxes. The true mechanism of binding excess Te to the surface is still unclear. Here, we use a mean-field approach assuming a Te* reservoir with limited occupation. In chapter 3, we perform KMC simulations of atomic layer epitaxy (ALE) of CdTe(001). We study the self-regulation of the ALE growth rate and demonstrate how the interplay of the Te* reservoir occupation with the surface kinetics results in two different regimes: at high temperatures the growth rate is limited to one half layer of CdTe per ALE cycle, whereas at low enough temperatures each cycle adds a complete layer. The temperature where the transition between the two regimes occurs depends mainly on the particle fluxes. The temperature dependence of the growth rate and the flux dependence of the transition temperature are in good qualitative agreement with experimental results. Comparing the macroscopic activation energy for Te* desorption in our model with experimental values we find semiquantitative agreement. In chapter 4, we study the formation of nanostructures with alternating stripes during submonolayer heteroepitaxy of two different adsorbate species on a given substrate. We evaluate the influence of two mechanisms: kinetic segregation due to chemically induced diffusion barriers, and strain relaxation by alternating arrangement of the adsorbate species. KMC simulations of a simple cubic lattice gas with weak inter-species binding energy show that kinetic effects are sufficient to account for stripe formation during growth. The dependence of the stripe width on control parameters is investigated. We find an Arrhenius temperature dependence, in agreement with experimental investigations of phase separation in binary or ternary material systems. Canonical MC simulations show that the observed stripes are not stable under equilibrium conditions: the adsorbate species separate into very large domains. Off-lattice simulations which account for the lattice misfit of the involved particle species show that, under equilibrium conditions, the competition between binding and strain energy results in regular stripe patterns with a well-defined width depending on both misfit and binding energies. In KMC simulations, the stripe-formation and the experimentally reported ramification of adsorbate islands are reproduced. To clarify the origin of the island ramification, we investigate an enhanced lattice gas model whose parameters are fitted to match characteristic off-lattice diffusion barriers. The simulation results show that a satisfactory explanation of experimental observations within the lattice gas framework requires a detailed incorporation of long-range elastic interactions. In the appendix we discuss supplementary topics related to the lattice gas simulations in chapter 4.