Band structures of ultrathin films of heavy elements, £\-Sn and Sb, were investigated using first-principle calculations with the inclusion of spin-orbit coupling. The band structures were gradually varied as the physical parameters were adjusted. The band inversion was obtained at the high symmetry point in Brillouin zone, making a topological phase transition. In this study, the band inversion at £F point of the Brillouin zone was predicted in single bi-layer of Sb(111) and single bi-layer and two bi-layers of £\-Sn(111). The topological phase transition is from trivial insulator to topological insulator for single bi-layer of Sb(111). Finally, the topological phase transition is from trivial semi-metal to topological semi-metal for single bi-layer of £\-Sn(111), whereas as it is from topological semi-metal to trivial metal for two bi-layers of £\-Sn(111).

We discuss the appearance of quantum orders in the Hubbard model for interacting electrons, at half-filling. Such phases do not have local order parameters and need to be characterized by the quantum mechanical properties of their ground state. On one hand, we study the Mott transition from a metal to a spin liquid insulator in two dimensions, of potential relevance to some layered organic compounds. The correlation-driven transition occurs at fixed filling and involves fractionalization of the electron: upon entering the insulator, a Fermi surface of neutral spinons coupled to an internal gauge field emerges. We focus on the transport properties near the quantum critical point and find that the emergent gauge fluctuations play a key role in determining the universal scaling. Second, motivated by a class of three-dimensional transition metal oxides, the pyrochlore iridates, we study the interplay of non-trivial band topology and correlations. Building on the strong spin orbit coupling in these compounds, we construct a general microscopic Hubbard model and determine its mean-field phase diagram, which contains topological insulators, Weyl semimetals, axion insulators and various antiferromagnets. We also discuss the effects many-body correlations on theses phases. We close by examining a fractionalized topological insulator that combines the two main themes of the thesis: fractionalization and non-trivial band topology. Specifically, we study how the two-dimensional protected surface states of a topological Mott insulator interact with a three-dimensional emergent gauge field. Various correlation effects on observables are identified.

We discuss the appearance of quantum orders in the Hubbard model for interacting electrons, at half-filling. Such phases do not have local order parameters and need to be characterized by the quantum mechanical properties of their ground state. On one hand, we study the Mott transition from a metal to a spin liquid insulator in two dimensions, of potential relevance to some layered organic compounds. The correlation-driven transition occurs at fixed filling and involves fractionalization of the electron: upon entering the insulator, a Fermi surface of neutral spinons coupled to an internal gauge field emerges. We focus on the transport properties near the quantum critical point and find that the emergent gauge fluctuations play a key role in determining the universal scaling. Second, motivated by a class of three-dimensional transition metal oxides, the pyrochlore iridates, we study the interplay of non-trivial band topology and correlations. Building on the strong spin orbit coupling in these compounds, we construct a general microscopic Hubbard model and determine its mean-field phase diagram, which contains topological insulators, Weyl semimetals, axion insulators and various antiferromagnets. We also discuss the effects many-body correlations on theses phases. We close by examining a fractionalized topological insulator that combines the two main themes of the thesis: fractionalization and non-trivial band topology. Specifically, we study how the two-dimensional protected surface states of a topological Mott insulator interact with a three-dimensional emergent gauge field. Various correlation effects on observables are identified.

Topological phases of matter represent a new phase which cannot be understood in terms of Landau’s theory of symmetry breaking and are characterized by non-local topological properties emerging from purely local (microscopic) degrees of freedom. It is the non-trivial topology of the bulk band structure that gives rise to topological phases in condensed matter systems. Quantum Hall systems are prominent examples of such topological phases. Different quantum Hall states cannot be distinguished by a local order parameter. Instead, non-local measurements are required, such as the Hall conductance, to differentiate between various quantum Hall states. A signature of a topological phase is the existence of robust properties that do not depend on microscopic details and are insensitive to local perturbations which respect appropriate symmetries. Examples of such properties are the presence of protected gapless edge states at the boundary of the system for topological insulators and the remarkably precise quantization of the Hall conductance for quantum Hall states. The robustness of these properties can be under-stood through the existence of a topological invariant, such as the Chern number for quantum Hall states which is quantized to integer values and can only be changed by closing the bulk gap. Two other examples of topological phases of matter are topological superconductors and Weyl semimetals. The study of transport in various kinds of junctions of these topological materials is highly interesting for their applications in modern electronics and quantum computing. Another intriguing area to study is how to generate new kind of gapless edge modes in topological systems. In this thesis I have studied various aspects of topological phases of matter, such as electronic transport in junctions of topological insulators and topological superconductors, the generation of new kinds of boundary modes in the presence of granularity, and the effects of periodic driving in topological systems. We have studied the following topics. 1. transport across a line junction of two three-dimensional topological insulators, 2. transport across a junction of topological insulators and a superconductor, 3. surface and edge states of a topological insulator starting from a lattice model, 4. effects of granularity in topological insulators, 5. Majorana modes and conductance in systems with junctions of topological superconducting wires and normal metals, and 6. generation of new surface states in a Weyl semimetal in the presence of periodic driving by the application of electromagnetic radiation. A detailed description of each chapter is given below. • In the first chapter we introduce a number of concepts which are used in the rest of the thesis. We will discuss the ideas of topological phases of matter (for example, topological insulators, topological superconductors and Majorana modes, and Weyl semimetals), the renormalization group theory for weak interactions, and Floquet theory for periodically driven systems. • In the second chapter we study transport across a line junction which separates the surfaces of two three-dimensional topological insulators. The velocities of the Dirac electrons on the two surfaces may be unequal and may even have opposite signs. For a time-reversal invariant system, we show that the line junction is characterized by an arbitrary real parameter α; this determines the scattering amplitudes (reflection and transmission) from the junction. The physical origin of α is a potential barrier that may be present at the junction. If the surface velocities have the same sign, edge states exist that propagate along the line junction with a velocity and orientation of the spin which depend on α and the ratio of the velocities. Next, we study what happens if the two surfaces are at an angle φ with respect to each other. We study the scattering and differential conductance across the line junction as functions of φ and α. We also show that there are edge states which propagate along the line junction with a velocity and spin orientation which depend on φ. Finally, if the surface velocities have opposite signs, we find that the electrons must necessarily transmit into the two-dimensional interface separating the two topological insulators. • In the third chapter we discuss transport across a line junction lying between two orthogonal topological insulator surfaces and a superconductor which can have either s-wave (spin-singlet) or p-wave (spin-triplet) pairing symmetry. This junction is more complicated than the line junction discussed in the previous chapter because of the presence of the superconductor. In a topological insulator spin-up and spin-down electrons get coupled while in a superconductor electrons and holes get coupled. Hence we have to use a four-component spinor formalism to describe both spin and particle-hole degrees of freedom. The junction can have three time-reversal invariant barriers on the three sides. We compute the subgap charge conductance across such a junction and study their behaviors as a function of the bias voltage applied across the junction and the three parameters which characterize the barriers. We find that the presence of topological insulators and a superconductor leads to both Dirac and Schrodinger-like features in the charge conductances. We discuss the effects of bound states on the superconducting side on the conductance; in particular, we show that for triplet p-wave superconductors such a junction may be used to determine the spin state of its Cooper pairs. • In the fourth chapter we derive the surface Hamiltonians of a three-dimensional topological insulator starting from a microscopic model. (This description was not discussed in the previous chapters where we directly started from the surface Hamiltonians without deriving them form a bulk Hamiltonian). Here we begin from the bulk Hamiltonian of a three-dimensional topological insulator Bi2Se3. Using this we derive the surface Hamiltonians on various surfaces of the topological insulator, and we find the states which appear on the different surfaces and along the edge between pairs of surfaces. The surface Hamiltonians depend on the orientation of the surfaces and are therefore quite different from the previous chapters. We use both analytical methods based on the surface Hamiltonians (which are derived from the bulk Hamiltonian) and numerical methods based directly on a lattice discretization of the bulk Hamiltonian in order to find surface and edge states. We find that the application of a potential barrier along an edge can give rise to states localized at that edge. These states have an unusual energy-momentum dispersion which can be controlled by applying a potential along the edge; in particular, the velocity of these states can be tuned to zero. The scattering and conductance across the edge are studied as a function of the edge potential. We show that a magnetic field applied in a particular direction can also give rise to zero energy states on certain edges. We point out possible experimental ways of looking for the various edge states. • In the fifth chapter we study a system made of topological insulator (TI) nanocrystals which are coupled to each other. Our theoretical studies are motivated by the following experimental observations. Electrical transport measurements were carried out on thin films of nanocrystals of Bi2Se3 which is a TI. The measurements reveal that the entire system behaves like a single TI with two topological surface states at the two ends of the system. The two surface states are found to be coupled if the film thickness is small and decoupled above a certain film thickness. The surface state penetration depth is found to be unusually large and it decreases with increasing temperature. To explain all these experimental results we propose a theoretical model for this granular system. This consists of multiple grains of Bi2Se3 stacked next to each other in a regular array along the z-direction (the c-axis of Bi2Se3 nanocrystals). We assume translational invariance along the x and y directions. Each grain has top and bottom surfaces on which the electrons are described by Hamiltonians of the Dirac form which can be derived from the bulk Hamiltonian known for this material. We introduce intra-grain tunneling couplings t1 between the opposite surfaces of a single grain and inter-grain couplings t2 between nearby surfaces of two neighboring grains. We show that when t1 < t2 the entire system behaves like a single topological insulator whose outermost surfaces have gapless spectra described by Dirac Hamiltonians. We find a relation between t1, t2 and the surface state penetration depth λ which explains the properties of λ that are seen experimentally. We also present an expression for the surface state Berry phase as a function of the hybridization between the surface states and a Zeeman magnetic field that may be present in the system. At the end we theoretically studied the surface states on one of the side surfaces of the granular system and showed that many pairs of surface states can exist on the side surfaces depending on the length of the unit cell of the granular system. • In the sixth chapter we present our work on junctions of p-wave superconductors (SC) and normal metals (NM) in one dimension. We first study transport in a system where a SC wire is sandwiched between two NM wires. For such a system it is known that there is a Majorana mode at the junction between the SC and each NM lead. If the p-wave pairing changes sign at some point inside the SC, two additional Majorana modes appear near that point. We study the effect of all these modes on the subgap conductance between the leads and the SC. We derive an analytical expression as a function of and the length L of the SC for the energy shifts of the Majorana modes at the junctions due to hybridization between them; the energies oscillate and decay exponentially as L is increased. The energies exactly match the locations of the peaks in the conductance. We find that the subgap conductances do not change noticeably with the sign of . So there is no effect of the extra Majorana modes which appear inside the SC (due to changes in the signs of Δ) on the subgap conductance. Next we study junctions of three p-wave SC wires which are connected to the NM leads. Such a junction is of interest as it is the simplest system where braiding of Majorana modes is possible. Another motivation for studying this system is to see if the subgap transport is affected by changes in the signs of . For sufficiently long SCs, there are zero energy Majorana modes at the junctions between the SCs and the leads. In addition, depending on the signs of the Δ’s in the three SCs, there can also be one or three Majorana modes at the junction of the three SCs. We show that the various subgap conductances have peaks occurring at the energies of all these modes; we therefore get a rich pattern of conductance peaks. Next we study the effects of interactions between electrons (in the NM leads) on the transport. We use a renormalization group approach to study the effect of interactions on the conductance at energies far from the SC gap. Hence the earlier part of this chapter where we studied the transport at an energy E inside the SC gap (so that − < E < Δ) differs from this part where we discuss conductance at an energy E where |E| ≫ . For the latter part we assume the region of three SC wires to be a single region whose only role is to give rise to a scattering matrix for the NM wires; this scattering matrix has both normal and Andreev elements (namely, an electron can be reflected or transmitted as either an electron or a hole). We derive a renormalization group equation for the elements of the scattering matrix by assuming the interaction to be sufficiently weak. The fixed points of the renormalization group flow and their stabilities are studied; we find that the scattering matrix at the stable fixed point is highly symmetric even when the microscopic scattering matrix and the interaction strengths are not symmetric. Using the stability analysis we discuss the dependence of the conductances on the various length scales of the problem. Finally we propose an experimental realization of this system which can produce different signs of the p-wave pairings in the different SCs. • In the seventh chapter we show that the application of circularly polarized electro-magnetic radiation on the surface of a Weyl semimetal can generate states at that surface. The surface states can be characterized by their momenta due to translation invariance. The Floquet eigenvalues of these states come in complex conjugate pairs rather than being equal to ±1. If the amplitude of the radiation is small, we find some unusual bulk-boundary relations: the Floquet eigenvalues of the surface states lie at the extrema of the Floquet eigenvalues of the bulk system when the latter are plotted as a function of the momentum perpendicular to the surface, and the peaks of the Fourier transforms of the surface state wave functions lie at the momenta where the bulk Floquet eigenvalues have extrema. For the case of zero surface momentum, we can analytically derive interesting scaling relations between the decay lengths of the surface states and the amplitude and penetration depth of the radiation. For topological insulators, we again find that circularly polarized radiation can generate states on the surfaces; these states have much larger decay lengths (which can be tuned by the radiation amplitude) than the topological surface states which are present even in the absence of radiation. Finally, we show that radiation can generate surface states even for trivial insulators.

This dissertation works out the vibrational properties of epitaxial silicene, which was discovered by Vogt et al. in 2012 by the epitaxial synthesis on the silver substrate. Its two-dimensional (2D) character is modified in comparison to the free-standing silicene due to its epitaxial nature, since the underlying substrate alters the physical properties of silicene as a result of the strong hybridization of the electronic levels of the substrate and adlayer. The growth of silicene layers is complicated by the sensitivity of the Si structures to the experimental conditions, mainly temperature, resulting in the formation of several seemingly different surface reconstructions. Another Si structure appears on the Ag surface at a supramonolayer coverage. The Raman spectroscopy was utilized to understand the relation between different Si structures and reveal their origin as well as to investigate the phonon-related physical properties of two-dimensional Si sheets. The central core of this work is the growth and characterization of these 2D silicene monolayers on the Ag (111) surface as well as the formation of silicene multilayer structures. The characterization of these materials was performed using in situ surface-sensitive measurement methods such as Raman spectroscopy and low-energy electron diffraction under ultra-high vacuum conditions due to high chemical reactivity of epitaxial silicene. Additional characterization was done ex situ by means of scanning force microscopy. The experimentally determined spectral signature of the prototypical epitaxial (3x3)/(4x4) silicene structure was confirmed by ab initio calculations, in collaboration with theory groups. The Raman signatures of the other 2D and 3D Si phases on Ag (111) were determined which allowed us to provide a clear picture of their formation depending on the preparation conditions. The monitoring of the silicene multi-layer growth yielded the vibrational signature of the top layer, reconstructed in a (√3x√3) fashion. It was compared to the inverse, (√3x√3)-Ag/Si(111), system showing the vast amount of similarities, which suggest that the (√3x√3) reconstruction belong to the silver layer. The chemical and physical properties of this surface structure additionally strengthen this equivalence. The possibility of functionalization of epitaxial silicene was demonstrated via exposure to the atomic hydrogen under UHV conditions. The adsorbed hydrogen covalently bonds to the silicene lattice modifying it and reducing its symmetry. As shown by Raman spectroscopy, such modification can be reversed by thermal desorption of hydrogen. The excitation-dependent Raman measurements also suggest the change of the electronic properties of epitaxial silicene upon hydrogenation suggesting that its originally semi-metallic character is modified into a semiconducting one. / Die experimentellen Forschungsarbeiten zum Thema Silicen basieren auf den 2012 von Vogt et al. durchgeführten Untersuchungen zu dessen Synthese auf Silbersubstraten. Diese Untersuchungen lieferten die Grundlage, auf der zweidimensionales (2D) epitaktisches Silicen sowie weitere 2D Materialien untersucht werden konnten. In den anfänglichen Arbeiten konnte dabei gezeigt werden, dass sich die Eigenschaften von epitaktischem Silicen gegenüber den theoretischen Vorhersagen von frei-stehendem Silicen unterscheiden. Darüber hinaus verkomplizieren sich die experimentellen Untersuchungen dieses 2D Materials, da auf dem Ag(111) Wachstumssubstrat sechs verschiedene 2D Si Polytypen existieren. Eine detaillierte Darstellung dieser Untersuchungen findet sich in dem einführenden Kapitel der vorliegen Promotionsschrift. Der zentrale Kern dieser Arbeit beschäftigt sich mit dem Wachstum und der Charakterisierung dieser 2D Silicen Monolagen auf Ag(111) Oberflächen sowie der Bildung von Silicen- Multilagen Strukturen. Die Charakterisierung dieser Materialien wurde in situ mit oberflächenempfindlichen Messmethoden wie der Raman Spektroskopie und der niederenergetischen Elektronenbeugung unter Ultrahochvakuum-Bedingungen durchgeführt. Eine zusätzliche Charakterisierung erfolgte ex situ mittels Raster-KraftMikroskopie. Die experimentell bestimmte spektrale Raman-Signatur der prototypischen epitaktischen (3x3)/(4x4) Silicene Struktur wurde durch ab initio Rechnungen, in Zusammenarbeit mit Theoriegruppen, bestätigt. Durch diesen Vergleich wir die zweidimensionale Natur der epitaktischen Silicen-Schichten vollständig bestätigt, wodurch andere mögliche Interpretationen ausgeschlossen werden können. Darüber hinaus wurden die Ramans-Signaturen der weiteren 2D und 3D Siliziumphasen auf Ag(111) bestimmt, wodurch sich ein klares Bild der Bildung dieser Strukturen in Abhängigkeit von den Präparationsbedingungen ergibt. Um die Möglichkeit der Funktionalisierung von Silicen und der weiteren 2D Si Strukturen zu testen, wurden diese unter UHV Bedingungen atomarem Wasserstoff ausgesetzt. Durch die Bindung zu den Wasserstoffamen wird die kristalline Struktur der Silicen-Schichten modifiziert und die Symmetrie reduziert, was sich deutlich in der spektralen Raman-Signatur zeigt. Wie mittels Raman Spektroskopie gezeigt werden konnte, kann diese Modifikation durch thermische Desorption des Wasserstoffs rückgängig gemacht werden, ist also reversibel. Raman Messungen mit verschiedenen Anregungswellenlängen deuten darüber hinaus auf die Änderung der elektronischen Eigenschaften der Silicen-Schichten durch die Hydrierung hin. Der ursprüngliche halbmetallische Charakter der epitaktischen Silicen-Schicht geht möglicherweise in einen halbleitenden Zustand über. Das Wachstum von Silicen Multilagen wurde ebenfalls mit in situ Ramanspektroskopie verfolgt. Die sich dabei ergebene Raman-Signatur wurde mit der Raman-Signatur von Ag terminiertem Si(111) verglichen. Hier zeigen sich große Ähnlichkeiten, die auf eine ähnliche atomare Struktur hindeuten und zeigen, dass Ag Atome für die Ausbildung der Oberflächenstruktur während des Wachstums der Si-Lagen verantwortlich sind. Die chemischen und physikalischen Eigenschaften dieser Struktur bestärken zusätzlich diese Äquivalenz.

This dissertation works out the vibrational properties of epitaxial silicene, which was discovered by Vogt et al. in 2012 by the epitaxial synthesis on the silver substrate. Its two-dimensional (2D) character is modified in comparison to the free-standing silicene due to its epitaxial nature, since the underlying substrate alters the physical properties of silicene as a result of the strong hybridization of the electronic levels of the substrate and adlayer. The growth of silicene layers is complicated by the sensitivity of the Si structures to the experimental conditions, mainly temperature, resulting in the formation of several seemingly different surface reconstructions. Another Si structure appears on the Ag surface at a supramonolayer coverage. The Raman spectroscopy was utilized to understand the relation between different Si structures and reveal their origin as well as to investigate the phonon-related physical properties of two-dimensional Si sheets. The central core of this work is the growth and characterization of these 2D silicene monolayers on the Ag (111) surface as well as the formation of silicene multilayer structures. The characterization of these materials was performed using in situ surface-sensitive measurement methods such as Raman spectroscopy and low-energy electron diffraction under ultra-high vacuum conditions due to high chemical reactivity of epitaxial silicene. Additional characterization was done ex situ by means of scanning force microscopy. The experimentally determined spectral signature of the prototypical epitaxial (3x3)/(4x4) silicene structure was confirmed by ab initio calculations, in collaboration with theory groups. The Raman signatures of the other 2D and 3D Si phases on Ag (111) were determined which allowed us to provide a clear picture of their formation depending on the preparation conditions. The monitoring of the silicene multi-layer growth yielded the vibrational signature of the top layer, reconstructed in a (√3x√3) fashion. It was compared to the inverse, (√3x√3)-Ag/Si(111), system showing the vast amount of similarities, which suggest that the (√3x√3) reconstruction belong to the silver layer. The chemical and physical properties of this surface structure additionally strengthen this equivalence. The possibility of functionalization of epitaxial silicene was demonstrated via exposure to the atomic hydrogen under UHV conditions. The adsorbed hydrogen covalently bonds to the silicene lattice modifying it and reducing its symmetry. As shown by Raman spectroscopy, such modification can be reversed by thermal desorption of hydrogen. The excitation-dependent Raman measurements also suggest the change of the electronic properties of epitaxial silicene upon hydrogenation suggesting that its originally semi-metallic character is modified into a semiconducting one. / Die experimentellen Forschungsarbeiten zum Thema Silicen basieren auf den 2012 von Vogt et al. durchgeführten Untersuchungen zu dessen Synthese auf Silbersubstraten. Diese Untersuchungen lieferten die Grundlage, auf der zweidimensionales (2D) epitaktisches Silicen sowie weitere 2D Materialien untersucht werden konnten. In den anfänglichen Arbeiten konnte dabei gezeigt werden, dass sich die Eigenschaften von epitaktischem Silicen gegenüber den theoretischen Vorhersagen von frei-stehendem Silicen unterscheiden. Darüber hinaus verkomplizieren sich die experimentellen Untersuchungen dieses 2D Materials, da auf dem Ag(111) Wachstumssubstrat sechs verschiedene 2D Si Polytypen existieren. Eine detaillierte Darstellung dieser Untersuchungen findet sich in dem einführenden Kapitel der vorliegen Promotionsschrift. Der zentrale Kern dieser Arbeit beschäftigt sich mit dem Wachstum und der Charakterisierung dieser 2D Silicen Monolagen auf Ag(111) Oberflächen sowie der Bildung von Silicen- Multilagen Strukturen. Die Charakterisierung dieser Materialien wurde in situ mit oberflächenempfindlichen Messmethoden wie der Raman Spektroskopie und der niederenergetischen Elektronenbeugung unter Ultrahochvakuum-Bedingungen durchgeführt. Eine zusätzliche Charakterisierung erfolgte ex situ mittels Raster-KraftMikroskopie. Die experimentell bestimmte spektrale Raman-Signatur der prototypischen epitaktischen (3x3)/(4x4) Silicene Struktur wurde durch ab initio Rechnungen, in Zusammenarbeit mit Theoriegruppen, bestätigt. Durch diesen Vergleich wir die zweidimensionale Natur der epitaktischen Silicen-Schichten vollständig bestätigt, wodurch andere mögliche Interpretationen ausgeschlossen werden können. Darüber hinaus wurden die Ramans-Signaturen der weiteren 2D und 3D Siliziumphasen auf Ag(111) bestimmt, wodurch sich ein klares Bild der Bildung dieser Strukturen in Abhängigkeit von den Präparationsbedingungen ergibt. Um die Möglichkeit der Funktionalisierung von Silicen und der weiteren 2D Si Strukturen zu testen, wurden diese unter UHV Bedingungen atomarem Wasserstoff ausgesetzt. Durch die Bindung zu den Wasserstoffamen wird die kristalline Struktur der Silicen-Schichten modifiziert und die Symmetrie reduziert, was sich deutlich in der spektralen Raman-Signatur zeigt. Wie mittels Raman Spektroskopie gezeigt werden konnte, kann diese Modifikation durch thermische Desorption des Wasserstoffs rückgängig gemacht werden, ist also reversibel. Raman Messungen mit verschiedenen Anregungswellenlängen deuten darüber hinaus auf die Änderung der elektronischen Eigenschaften der Silicen-Schichten durch die Hydrierung hin. Der ursprüngliche halbmetallische Charakter der epitaktischen Silicen-Schicht geht möglicherweise in einen halbleitenden Zustand über. Das Wachstum von Silicen Multilagen wurde ebenfalls mit in situ Ramanspektroskopie verfolgt. Die sich dabei ergebene Raman-Signatur wurde mit der Raman-Signatur von Ag terminiertem Si(111) verglichen. Hier zeigen sich große Ähnlichkeiten, die auf eine ähnliche atomare Struktur hindeuten und zeigen, dass Ag Atome für die Ausbildung der Oberflächenstruktur während des Wachstums der Si-Lagen verantwortlich sind. Die chemischen und physikalischen Eigenschaften dieser Struktur bestärken zusätzlich diese Äquivalenz.