Regularisierung und Renormierung in der Quantenfeldtheorie : Resultate aus dem Vergleich konsistenter und praktikabler Methoden

Die Arbeit beginnt mit dem Vergleich spezieller Regularisierungsmethoden in der Quantenfeldtheorie mit dem Verfahren zur störungstheoretischen Konstruktion der S-Matrix nach Epstein und Glaser. Da das Epstein-Glaser-Verfahren selbst als Regularisierungsverfahren verwandt werden kann und darüberhinaus ausschließlich auf physikalisch motivierten Postulaten basiert, liefert dieser Vergleich ein Kriterium für die Zulässigkeit anderer Regularisierungsmethoden. Zusätzlich zur Herausstellung dieser Zulässigkeit resultiert aus dieser Gegenüberstellung als weiteres wesentliches Resultat ein neues, in der Anwendung praktikables sowie konsistentes Regularisierungsverfahren, das modifizierte BPHZ-Verfahren. Dieses wird anhand von Ein-Schleifen-Diagrammen aus der QED (Elektronselbstenergie, Vakuumpolarisation und Vertexkorrektur) demonstriert. Im Gegensatz zur vielverwandten Dimensionalen Regularisierung ist dieses Verfahren uneingeschränkt auch für chirale Theorien anwendbar. Als Beispiel hierfür dient die Berechnung der im Rahmen einer axialen Erweiterung der QED-Lagrangedichte auftretenden U(1)-Anomalie. Auf der Stufe von Mehr-Schleifen-Diagrammen zeigt der Vergleich der Epstein-Glaser-Konstruktion mit dem bekannten BPHZ-Verfahren an mehreren Beispielen aus der Phi^4-Theorie, darunter das sog. Sunrise-Diagramm, daß zu deren Berechnung die nach der Waldformel des BPHZ-Verfahrens zur Regularisierung beitragenden Unterdiagramme auf eine kleinere Klasse eingeschränkt werden können. Dieses Resultat ist gleichfalls für die Praxis der Regularisierung bedeutsam, da es bereits auf der Stufe der zu berücksichtigenden Unterdiagramme zu einer Vereinfachung führt.

Numerical and analytical approaches to strongly correlated electron systems

In this thesis we consider three different models for strongly correlated electrons, namely a multi-band Hubbard model as well as the spinless Falicov-Kimball model, both with a semi-elliptical density of states in the limit of infinite dimensions d, and the attractive Hubbard model on a square lattice in d=2. In the first part, we study a two-band Hubbard model with unequal bandwidths and anisotropic Hund's rule coupling (J_z-model) in the limit of infinite dimensions within the dynamical mean-field theory (DMFT). Here, the DMFT impurity problem is solved with the use of quantum Monte Carlo (QMC) simulations. Our main result is that the J_z-model describes the occurrence of an orbital-selective Mott transition (OSMT), in contrast to earlier findings. We investigate the model with a high-precision DMFT algorithm, which was developed as part of this thesis and which supplements QMC with a high-frequency expansion of the self-energy. The main advantage of this scheme is the extraordinary accuracy of the numerical solutions, which can be obtained already with moderate computational effort, so that studies of multi-orbital systems within the DMFT+QMC are strongly improved. We also found that a suitably defined Falicov-Kimball (FK) model exhibits an OSMT, revealing the close connection of the Falicov-Kimball physics to the J_z-model in the OSM phase. In the second part of this thesis we study the attractive Hubbard model in two spatial dimensions within second-order self-consistent perturbation theory. This model is considered on a square lattice at finite doping and at low temperatures. Our main result is that the predictions of first-order perturbation theory (Hartree-Fock approximation) are renormalized by a factor of the order of unity even at arbitrarily weak interaction (U->0). The renormalization factor q can be evaluated as a function of the filling n for 00, the q-factor vanishes, signaling the divergence of self-consistent perturbation theory in this limit. Thus we present the first asymptotically exact results at weak-coupling for the negative-U Hubbard model in d=2 at finite doping.

Renormalization of fermion mixing

Precision measurements of phenomena related to fermion mixing require the inclusion of higher order corrections in the calculation of corresponding theoretical predictions. For this, a complete renormalization scheme for models that allow for fermion mixing is highly required. The correct treatment of unstable particles makes this task difficult and yet, no satisfactory and general solution can be found in the literature. In the present work, we study the renormalization of the fermion Lagrange density with Dirac and Majorana particles in models that involve mixing. The first part of the thesis provides a general renormalization prescription for the Lagrangian, while the second one is an application to specific models. In a general framework, using the on-shell renormalization scheme, we identify the physical mass and the decay width of a fermion from its full propagator. The so-called wave function renormalization constants are determined such that the subtracted propagator is diagonal on-shell. As a consequence of absorptive parts in the self-energy, the constants that are supposed to renormalize the incoming fermion and the outgoing antifermion are different from the ones that should renormalize the outgoing fermion and the incoming antifermion and not related by hermiticity, as desired. Instead of defining field renormalization constants identical to the wave function renormalization ones, we differentiate the two by a set of finite constants. Using the additional freedom offered by this finite difference, we investigate the possibility of defining field renormalization constants related by hermiticity. We show that for Dirac fermions, unless the model has very special features, the hermiticity condition leads to ill-defined matrix elements due to self-energy corrections of external legs. In the case of Majorana fermions, the constraints for the model are less restrictive. Here one might have a better chance to define field renormalization constants related by hermiticity. After analysing the complete renormalized Lagrangian in a general theory including vector and scalar bosons with arbitrary renormalizable interactions, we consider two specific models: quark mixing in the electroweak Standard Model and mixing of Majorana neutrinos in the seesaw mechanism. A counter term for fermion mixing matrices can not be fixed by only taking into account self-energy corrections or fermion field renormalization constants. The presence of unstable particles in the theory can lead to a non-unitary renormalized mixing matrix or to a gauge parameter dependence in its counter term. Therefore, we propose to determine the mixing matrix counter term by fixing the complete correction terms for a physical process to experimental measurements. As an example, we calculate the decay rate of a top quark and of a heavy neutrino. We provide in each of the chosen models sample calculations that can be easily extended to other theories.

Exact quantum Monte Carlo methods for correlated electron systems

Thema dieser Arbeit ist die Entwicklung und Kombination verschiedener numerischer Methoden, sowie deren Anwendung auf Probleme stark korrelierter Elektronensysteme. Solche Materialien zeigen viele interessante physikalische Eigenschaften, wie z.B. Supraleitung und magnetische Ordnung und spielen eine bedeutende Rolle in technischen Anwendungen. Es werden zwei verschiedene Modelle behandelt: das Hubbard-Modell und das Kondo-Gitter-Modell (KLM). In den letzten Jahrzehnten konnten bereits viele Erkenntnisse durch die numerische Lösung dieser Modelle gewonnen werden. Dennoch bleibt der physikalische Ursprung vieler Effekte verborgen. Grund dafür ist die Beschränkung aktueller Methoden auf bestimmte Parameterbereiche. Eine der stärksten Einschränkungen ist das Fehlen effizienter Algorithmen für tiefe Temperaturen.rnrnBasierend auf dem Blankenbecler-Scalapino-Sugar Quanten-Monte-Carlo (BSS-QMC) Algorithmus präsentieren wir eine numerisch exakte Methode, die das Hubbard-Modell und das KLM effizient bei sehr tiefen Temperaturen löst. Diese Methode wird auf den Mott-Übergang im zweidimensionalen Hubbard-Modell angewendet. Im Gegensatz zu früheren Studien können wir einen Mott-Übergang bei endlichen Temperaturen und endlichen Wechselwirkungen klar ausschließen.rnrnAuf der Basis dieses exakten BSS-QMC Algorithmus, haben wir einen Störstellenlöser für die dynamische Molekularfeld Theorie (DMFT) sowie ihre Cluster Erweiterungen (CDMFT) entwickelt. Die DMFT ist die vorherrschende Theorie stark korrelierter Systeme, bei denen übliche Bandstrukturrechnungen versagen. Eine Hauptlimitation ist dabei die Verfügbarkeit effizienter Störstellenlöser für das intrinsische Quantenproblem. Der in dieser Arbeit entwickelte Algorithmus hat das gleiche überlegene Skalierungsverhalten mit der inversen Temperatur wie BSS-QMC. Wir untersuchen den Mott-Übergang im Rahmen der DMFT und analysieren den Einfluss von systematischen Fehlern auf diesen Übergang.rnrnEin weiteres prominentes Thema ist die Vernachlässigung von nicht-lokalen Wechselwirkungen in der DMFT. Hierzu kombinieren wir direkte BSS-QMC Gitterrechnungen mit CDMFT für das halb gefüllte zweidimensionale anisotrope Hubbard Modell, das dotierte Hubbard Modell und das KLM. Die Ergebnisse für die verschiedenen Modelle unterscheiden sich stark: während nicht-lokale Korrelationen eine wichtige Rolle im zweidimensionalen (anisotropen) Modell spielen, ist in der paramagnetischen Phase die Impulsabhängigkeit der Selbstenergie für stark dotierte Systeme und für das KLM deutlich schwächer. Eine bemerkenswerte Erkenntnis ist, dass die Selbstenergie sich durch die nicht-wechselwirkende Dispersion parametrisieren lässt. Die spezielle Struktur der Selbstenergie im Impulsraum kann sehr nützlich für die Klassifizierung von elektronischen Korrelationseffekten sein und öffnet den Weg für die Entwicklung neuer Schemata über die Grenzen der DMFT hinaus.

108 Sn studied with intermediate-energy Coulomb excitation

In dieser Arbeit wurde der instabile, Neutronenarme Kern 108Sn mit Hilfe der Coulomb-Anregung bei intermediaeren Energien in inverser Kinematik studiert. Diese Methode wurde bisher zur Untersuchung der ersten angeregten 2+ Zustaende und deren E2 Zerfallsraten in Kernen mit Kernladungszahl Z< 30 angewendet. 108Sn ist somit der Kern mit der groeßten Kernladungszahl, bei dem diese Studien bisher stattfanden. Das Ziel dieses Experiments war die Messung der unbekannten reduzierten Uebergangswahrscheinlichkeit B(E2,0+ -> 2+). Der B(E2)-Wert von 0.230(57) e2b2 wurde relativ zu dem bekannten Wert des Isotops 112Sn bestimmt. Das Experiment wurde an der GSI Darmstadt mit Hilfe des RISING Detektors und des Fragmentseperators (FRS) durchgefuehrt. Sekundaere Strahlen (108Sn, 112Sn) mit einer Energie von ca. 150 MeV pro Nukleon wurden auf ein 386 mg/cm2 dickes 197Au Target geschossen. Die Projektilfragmente wurden mit Hilfe des Fragmentseparators selektiert und identifiziert. Zur Selektion des Reaktionskanals und zur Bestimmung des Winkels der gestreuten Fragmente wurde das Teilchenteleskop CATE, das sich hinter dem Target befand, verwendet. Gammastrahlung, die in Koinzidenz mit den Projektilrestkernen emittiert wurde, wurde in den Germanium-Cluster Detektoren des RISING Detektors nachgewiesen. Der gemessene B(E2,0+ -> 2+)-Wert von 108Sn ist in Uebereinstimmung mit neueren Schalenmodellrechnungen, die auf realistischen effektiven Wechselwirkungen basieren und im Rahmen eines verallgemeinerten Seniorit¨ats-Schemas erklaert werden.

Finite-element discretization of 3D energy-transport equations for semiconductors

In this thesis a mathematical model was derived that describes the charge and energy transport in semiconductor devices like transistors. Moreover, numerical simulations of these physical processes are performed. In order to accomplish this, methods of theoretical physics, functional analysis, numerical mathematics and computer programming are applied. After an introduction to the status quo of semiconductor device simulation methods and a brief review of historical facts up to now, the attention is shifted to the construction of a model, which serves as the basis of the subsequent derivations in the thesis. Thereby the starting point is an important equation of the theory of dilute gases. From this equation the model equations are derived and specified by means of a series expansion method. This is done in a multi-stage derivation process, which is mainly taken from a scientific paper and which does not constitute the focus of this thesis. In the following phase we specify the mathematical setting and make precise the model assumptions. Thereby we make use of methods of functional analysis. Since the equations we deal with are coupled, we are concerned with a nonstandard problem. In contrary, the theory of scalar elliptic equations is established meanwhile. Subsequently, we are preoccupied with the numerical discretization of the equations. A special finite-element method is used for the discretization. This special approach has to be done in order to make the numerical results appropriate for practical application. By a series of transformations from the discrete model we derive a system of algebraic equations that are eligible for numerical evaluation. Using self-made computer programs we solve the equations to get approximate solutions. These programs are based on new and specialized iteration procedures that are developed and thoroughly tested within the frame of this research work. Due to their importance and their novel status, they are explained and demonstrated in detail. We compare these new iterations with a standard method that is complemented by a feature to fit in the current context. A further innovation is the computation of solutions in three-dimensional domains, which are still rare. Special attention is paid to applicability of the 3D simulation tools. The programs are designed to have justifiable working complexity. The simulation results of some models of contemporary semiconductor devices are shown and detailed comments on the results are given. Eventually, we make a prospect on future development and enhancements of the models and of the algorithms that we used.

The calcite(10-14) surface: a versatile substrate for molecular self-assembly

This thesis presents a detailed and successful study of molecular self-assembly on the calcite CaCO3(10-14) surface. One reason for the superior applicability of this particular surface is given by reflecting the well-known growth modes. Layer-by-layer growth, which is a necessity for the formation of templated two-dimensional (2D) molecular structures, is particularly favoured on substrates with a high surface energy. The CaCO3(10-14) surface is among those substrates and, thus, most promising. rnrnAll experiments in this thesis were performed using the non-contact atomic force microscope (NC-AFM) under ultra-high vacuum conditions. The acquisition of drift-free data became in this thesis possible owing to the herein newly developed atom-tracking system. This system features a lateral tip-positioning precision of at least 50pm. Furthermore, a newly developed scan protocol was implemented in this system, which allows for the acquisition of dense three-dimensional (3D) data under room-temperature conditions. An entire 3D data set from a CaCO3(10-14) surface consisting of 85x85x500 pixel is discussed. rnrnThe row-pairing and (2x1) reconstructions of the CaCO3(10-14) surface constitute most interesting research subjects. For both reconstructions, the NC-AFM imaging was classified to a total of 12 contrast modes. Eight of these modes were observed within this thesis, some of them for the first time. Together with literature findings, a total of 10 modes has been observed experimentally to this day. Some contrast modes presented themselves as highly distance-dependent and at least for one contrast mode, a severe tip-termination influence was found. rnrnMost interestingly, the row-pairing reconstruction was found to break a symmetry element of the CaCO3(10-14) surface. With the presence of this reconstruction, the calcite (10-14) surface becomes chiral. From high-resolution NC-AFM data, the identification of the enantiomers is here possible and is presented for one enantiomer in this thesis. rnrnFive studies of self-assembled molecular structures on calcite (10-14) surfaces are presented. Only for one system, namely HBC/CaCO3(10-14), the formation of a molecular bulk structure was observed. This well-known occurence of weak molecule-insulator interaction hinders the investigation of two-dimensional molecular self-assembly. It was, however, possible to force the formation of an island phase for this system upon following a variable-temperature preparation. rnFor the C60/CaCO3(10-14) system it is most notably that no branched island morphologies were found. Instead, the first C60 layer appeared to wet the calcite surface. rnrnIn all studies, the molecules arranged themselves in ordered superstructures. A templating effect due to the underlying calcite substrate was evident for all systems. This templating strikingly led either to the formation of large commensurate superstructures, such as (2x15) with a 14 molecule basis for the C60/CaCO3(10-14) system, or prevented the vast growth of incommensurate molecular motifs, such as the chicken-wire structure in the trimesic acid (TMA)/CaCO3(10-14) system. rnrnThe molecule-molecule and the molecule-substrate interaction was increased upon choosing molecules with carboxylic acid moieties in the third, fourth and fifth study, using terephthalic acid, TMA and helicene molecules. In all these experiments, hydrogen-bonded assemblies were created. rnrnDirected hydrogen bond formation combined with intermolecular pi-pi interaction is employed in the fifth study, where the formation of uni-directional molecular "wires" from single helicene molecules succeeded. Each "wire" is composed of heterochiral helicene pairs, well-aligned along the [01-10] substrate direction and stabilised by pi-pi interaction.

Atomic-scale characterization of diamond surfaces and fullerene self-assembly

In this thesis, elemental research towards the implantation of a diamond-based molecular quantum computer is presented. The approach followed requires linear alignment of endohedral fullerenes on the diamond C(100) surface in the vicinity of subsurface NV-centers. From this, four fundamental experimental challenges arise: 1) The well-controlled deposition of endohedral fullerenes on a diamond surface. 2) The creation of NV-centers in diamond close to the surface. 3) Preparation and characterization of atomically-flat diamondsurfaces. 4) Assembly of linear chains of endohedral fullerenes. First steps to overcome all these challenges were taken in the framework of this thesis. Therefore, a so-called “pulse injection” technique was implemented and tested in a UHV chamber that was custom-designed for this and further tasks. Pulse injection in principle allows for the deposition of molecules from solution onto a substrate and can therefore be used to deposit molecular species that are not stable to sublimation under UHV conditions, such as the endohedral fullerenes needed for a quantum register. Regarding the targeted creation of NV-centers, FIB experiments were carried out in cooperation with the group of Prof. Schmidt-Kaler (AG Quantum, Physics Department, Johannes Gutenberg-Universität Mainz). As an entry into this challenging task, argon cations were implanted into (111) surface-oriented CaF2 crystals. The resulting implantation spots on the surface were imaged and characterized using AFM. In this context, general relations between the impact of the ions on the surface and their valency or kinetic energy, respectively, could be established. The main part of this thesis, however, is constituted by NCAFM studies on both, bare and hydrogen-terminated diamond C(100) surfaces. In cooperation with the group of Prof. Dujardin (Molecular Nanoscience Group, ISMO, Université de Paris XI), clean and atomically-flat diamond surfaces were prepared by exposure of the substrate to a microwave hydrogen plasma. Subsequently, both surface modifications were imaged in high resolution with NC-AFM. In the process, both hydrogen atoms in the unit cell of the hydrogenated surface were resolved individually, which was not achieved in previous STM studies of this surface. The NC-AFM images also reveal, for the first time, atomic-resolution contrast on the clean, insulating diamond surface and provide real-space experimental evidence for a (2×1) surface reconstruction. With regard to the quantum computing concept, high-resolution NC-AFM imaging was also used to study the adsorption and self-assembly potential of two different kinds of fullerenes (C60 and C60F48) on aforementioned diamond surfaces. In case of the hydrogenated surface, particular attention was paid to the influence of charge transfer doping on the fullerene-substrate interaction and the morphology emerging from self-assembly. Finally, self-assembled C60 islands on the hydrogen-terminated diamond surface were subject to active manipulation by an NC-AFM tip. Two different kinds of tip-induced island growth modes have been induced and were presented. In conclusion, the results obtained provide fundamental informations mandatory for the realization of a molecular quantum computer. In the process it was shown that NC-AFM is, under proper circumstances, a very capable tool for imaging diamond surfaces with highest resolution, surpassing even what has been achieved with STM up to now. Particular attention was paid to the influence of transfer doping on the morphology of fullerenes on the hydrogenated diamond surface, revealing new possibilities for tailoring the self-assembly of molecules that have a high electron affinity.

Microemulsion: prediction of the phase diagram with a modified Helfrich free energy

Microemulsions are thermodynamically stable, macroscopically homogeneous but microscopically heterogeneous, mixtures of water and oil stabilised by surfactant molecules. They have unique properties like ultralow interfacial tension, large interfacial area and the ability to solubilise other immiscible liquids. Depending on the temperature and concentration, non-ionic surfactants self assemble to micelles, flat lamellar, hexagonal and sponge like bicontinuous morphologies. Microemulsions have three different macroscopic phases (a) 1phase- microemulsion (isotropic), (b) 2phase-microemulsion coexisting with either expelled water or oil and (c) 3phase- microemulsion coexisting with expelled water and oil.rnrnOne of the most important fundamental questions in this field is the relation between the properties of the surfactant monolayer at water-oil interface and those of microemulsion. This monolayer forms an extended interface whose local curvature determines the structure of the microemulsion. The main part of my thesis deals with the quantitative measurements of the temperature induced phase transitions of water-oil-nonionic microemulsions and their interpretation using the temperature dependent spontaneous curvature [c0(T)] of the surfactant monolayer. In a 1phase- region, conservation of the components determines the droplet (domain) size (R) whereas in 2phase-region, it is determined by the temperature dependence of c0(T). The Helfrich bending free energy density includes the dependence of the droplet size on c0(T) as

Feasibility studies for accessing nucleon structure observables with the PANDA experiment at the future FAIR facility

The availability of a high-intensity antiproton beam with momentum up to 15,GeV/c at the future FAIR will open a unique opportunity to investigate wide areas of nuclear physics with the $overline{P}$ANDA (anti{$overline{P}$}roton ANnihilations at DArmstadt) detector. Part of these investigations concern the Electromagnetic Form Factors of the proton in the time-like region and the study of the Transition Distribution Amplitudes, for which feasibility studies have been performed in this Thesis. rnMoreover, simulations to study the efficiency and the energy resolution of the backward endcap of the electromagnetic calorimeter of $overline{P}$ANDA are presented. This detector is crucial especially for the reconstruction of processes like $bar pprightarrow e^+ e^- pi^0$, investigated in this work. Different arrangements of dead material were studied. The results show that both, the efficiency and the energy resolution of the backward endcap of the electromagnetic calorimeter fullfill the requirements for the detection of backward particles, and that this detector is necessary for the reconstruction of the channels of interest. rnrnThe study of the annihilation channel $bar pprightarrow e^+ e^-$ will improve the knowledge of the Electromagnetic Form Factors in the time-like region, and will help to understand their connection with the Electromagnetic Form Factors in the space-like region. In this Thesis the feasibility of a measurement of the $bar pprightarrow e^+ e^-$ cross section with $overline{P}$ANDA is studied using Monte-Carlo simulations. The major background channel $bar pprightarrow pi^+ pi^-$ is taken into account. The results show a $10^9$ background suppression factor, which assure a sufficiently clean signal with less than 0.1% background contamination. The signal can be measured with an efficiency greater than 30% up to $s=14$,(GeV/c)$^2$. The Electromagnetic Form Factors are extracted from the reconstructed signal and corrected angular distribution. Above this $s$ limit, the low cross section will not allow the direct extraction of the Electromagnetic Form Factors. However, the total cross section can still be measured and an extraction of the Electromagnetic Form Factors is possible considering certain assumptions on the ratio between the electric and magnetic contributions.rnrnThe Transition Distribution Amplitudes are new non-perturbative objects describing the transition between a baryon and a meson. They are accessible in hard exclusive processes like $bar pprightarrow e^+ e^- pi^0$. The study of this process with $overline{P}$ANDA will test the Transition Distribution Amplitudes approach. This work includes a feasibility study for measuring this channel with $overline{P}$ANDA. The main background reaction is here $bar pprightarrow pi^+ pi^- pi^0$. A background suppression factor of $10^8$ has been achieved while keeping a signal efficiency above 20%.rnrnrnPart of this work has been published in the European Physics Journal A 44, 373-384 (2010).rn

Systematic functionalization of molecules for molecular self-assembly

This thesis describes the investigation of systematically varied organic molecules for use in molecular self-assembly processes. All experiments were performed using high-resolution non-contact atomic force microscopy under UHV conditions and at room temperature. Using this technique, three different approaches for influencing intermolecular and molecule-surface interaction on the insulating calcite(10.4) surface were investigated by imaging the structure formation at the molecular scale. I first demonstrated the functionalization of shape-persistent oligo(p-benzamide)s that was engineered by introducing different functional groups and investigating their effect on the structural formation on the sample surface. The molecular core was designed to provide significant electrostatic anchoring towards the surface, while at the same time maintaining the flexibility to fine-tune the resulting structure by adjusting the intermolecular cohesion energy. The success of this strategy is based on a clear separation of the molecule-substrate interaction from the molecule-molecule interaction. My results show that sufficient molecule-surface anchoring can be achieved without restricting the structural flexibility that is needed for the design of complex molecular systems. Three derivatives of terephthalic acid (TPA) were investigated in chapter 7. Here, the focus was on changing the adhesion to the calcite surface by introducing different anchor functionalities to the TPA backbone. For all observed molecules, the strong substrate templating effect results in molecular structures that are strictly oriented along the calcite main crystal directions. This templating is especially pronounced in the case of 2-ATPA where chain formation on the calcite surface is observed in contrast to the formation of molecular layers in the bulk. At the same time, the amino group of 2-ATPA proved an efficient anchor functionality, successfully stabilizing the molecular chains on the sample surface. These findings emphasizes, once again, the importance of balancing and fine-tuning molecule-molecule and molecule-surface interactions in order to achieve stable, yet structurally flexible molecular arrangements on the sample surface. In the last chapter, I showed how the intrinsic property of molecular chirality decisively influences the structure formation in molecular self-assembly. This effect is especially pronounced in the case of the chiral heptahelicene-2-carboxylic acid. Deposition of the enantiopure molecules results in the formation of homochiral islands on the sample surface which is in sharp contrast to the formation of uni-directional double rows upon deposition of the racemate onto the same surface. While it remained uncertain from these previous experiments whether the double rows are composed of hetero- or homochiral molecules, I could clearly answer that question here and demonstrate that the rows are of heterochiral origin. Chirality, thus, proves to be another important parameter to steer the intermolecular interaction on surfaces. Altogether, the results of this thesis demonstrate that, in order to successfully control the structure formation in molecular self-assembly, the correct combination of molecule and surface properties is crucial. This is of special importance when working on substrates that exhibit a strong influence on the structure formation, such as the calcite(10.4) surface. Through the systematic variation of functional groups several important parameters that influence the balance between molecule-surface and molecule-molecule interaction were identified here, and the results of this thesis can, thus, act as a guideline for the rational design of molecules for use in molecular self-assembly.