Universität Stuttgart
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Item Open Access Fractional calculus for distributions(2024) Hilfer, Rudolf; Kleiner, TillmannFractional derivatives and integrals for measures and distributions are reviewed. The focus is on domains and co-domains for translation invariant fractional operators. Fractional derivatives and integrals interpreted as -convolution operators with power law kernels are found to have the largest domains of definition. As a result, extending domains from functions to distributions via convolution operators contributes to far reaching unifications of many previously existing definitions of fractional integrals and derivatives. Weyl fractional operators are thereby extended to distributions using the method of adjoints. In addition, discretized fractional calculus and fractional calculus of periodic distributions can both be formulated and understood in terms of -convolution.Item Open Access Physics-informed transformers for electronic quantum states(2025) Sobral, João Augusto; Perle, Michael; Scheurer, Mathias S.Neural-network-based variational quantum states, particularly autoregressive models, are powerful tools for describing complex many-body wave functions. However, their performance depends on the computational basis chosen and they often lack physical interpretability. We propose a modified variational Monte-Carlo framework which leverages prior physical information to construct a complete computational many-body basis containing a reference state that serves as a rough approximation to the true ground state. A Transformer is used to parametrize and autoregressively sample corrections to this reference state, giving rise to a more interpretable and computationally efficient representation of the ground state. We demonstrate this approach in a fermionic model featuring a metal-insulator transition by employing Hartree-Fock and a strong-coupling limit to define physics-informed bases. We also show that the Transformer’s hidden representation captures the natural energetic order of the different basis states. This work paves the way for more efficient and interpretable neural quantum-state representations.Item Open Access Ambient pressure oxidation of Ag(111) surfaces : an in-situ X-ray study(2008) Reicho, Alexander; Dosch, Helmut (Prof. Dr.)The oxidation of metals plays an outstanding role in everyday life. Typical phenomena are the formation of rust on steel or oxide scales on copper, showing up as a green patina. The formation of metal oxides is not always an unwanted process. The functionality of many materials is directly related to their controlled oxidation. The most prominent examples are passivating oxide layers on stainless steel. Relevant for this thesis are industrially applied heterogeneous catalytic reactions for the synthesis of many chemical products, where gaseous reactants are in contact with the solid surface of the catalyst. Oxidation reactions are very important in this context, leading to a big need of understanding of these processes in research and development. Thereby, the active oxygen species on the surface and selectivity and poisoning of the catalyst have to be studied on an atomic scale. The high temperature and high pressure oxidation of the 4d transition metals Ru, Rh, Pd and Ag is a matter of particular interest, because these metals are widely used as oxidation catalysts. On Ruthenium one observes the formation of RuO2(110) bulk oxide islands at elevated temperatures and oxygen pressure. In the case of the Pd(100) and Rh(111) surface oxidation can lead to the formation of so-called surface oxides. These oxides are structurally related to the bulk oxide of the respective element. Furthermore, surface oxides are ultra thin oxides containing one metallic layer surrounded by two oxygen layers, giving rise to an oxygen-metal-oxygen sequence perpendicular to the surface plane. A future vision is to get a direct microscopic control of the emerging surface structures and ultimately of the real-time oxidation/reduction dynamics allowing one to tailor such catalytic reactions to better performance. A necessary prerequisite to the microscopic control is the full atomistic understanding of the surface structures which form at high temperature and at high oxygen pressures. Silver plays a unique role in heterogeneous catalysis. Supported Ag catalysts are used for the selective oxidation ('epoxidation') of ethylene and for the partial oxidation of methanol to formaldehyde. Ethylene oxide and its derivates are basic chemicals for industry, used in a many technologies with a world-wide production of more than 10 million tons as in medicine for disinfection, sterilization, or fumigation, or in transport and energy technologies for engine antifreeze and heat transfer. Because of its ability to kill most bacteria, formaldehyde is extensively used as disinfectant and as preservative in vaccinations. Therefore, the optimisation of these two Ag-supported catalytic reactions is of paramount importance. Current strategies employed in the industrial process to enhance selectivity include the empirical use of inhibitors (Cl) and promoters (Cs), however, on the way to a knowledge-based control of these reactions one has first to understand the surface structure of oxidized silver under relevant conditions in full detail. The formation of extended Ag(111) facets is observed on polycrystalline silver during the above industrial catalytic oxidation reactions, in turn fundamental research (experiment and theory) has been devoted to the detailed understanding of oxidation of this surface. The formation of an oxygen induced p(4x4) reconstruction on the Ag(111) surface is known since the early 70s. A surface oxide trilayer model, based on a three-layer slab of Ag2O(111), was proposed. Accordingly, the Ag(111) surface seemed to show a similar behaviour like Pd and Rh, being neighbours in the periodic table. Further theoretical calculations predicted the stability of this reconstruction under industrially relevant conditions. Nevertheless, several questions remained unsolved: the stability of the p(4x4) reconstruction under industrially relevant conditions was not checked experimentally, the structural model of the p(4x4) structure was not proven by a crystallographic method and previously unknown structures might play an important role for the catalytic activity of Ag(111) facets. Our experimental approach is based on the nowadays routinely available highly brilliant x-ray radiation produced by third generation synchrotron light sources. This radiation is used by us in three surface sensitive x-ray techniques. In-situ surface x-ray diffraction (SXRD) allows the identification and determination of structural models of surface reconstructions under industrially relevant conditions. This technique is combined with high resolution core level spectroscopy (HRCLS) and normal incidence x-ray standing wave absorption (NIXSW), giving insight into the local binding geometry of the oxygen and silver atoms.Item Open Access Charge regulation and swelling of weak polyelectrolyte nanogels in divalent salt solutions(2026) Beyer, David; Holm, ChristianWe use computer simulations to investigate the behavior of a weak polyelectrolyte nanogel in a solution containing divalent salt. In our simulations, we systematically vary the bulk pH value and the bulk concentration of divalent salt, allowing us to study the influence of charge regulation and divalent ions on the ionization behavior, ion partitioning, and nanogel swelling. With regard to the ionization behavior, we observe that, with an increasing concentration of divalent salt, the suppression of ionization becomes weaker. Moreover, we find that the strongly non‐uniform ionization profile observed in the absence of divalent counterions becomes increasingly uniform as the concentration of divalent salt is increased. We also study the partitioning of monovalent and divalent counterions between the nanogel and the bulk solution; our analysis shows that the uptake of divalent ions may be enhanced by up to tenfold as compared to the mean‐field prediction. Finally, we consider the influence of divalent ions on the pH‐dependent swelling behavior of the nanogel. Here, we observe a two‐stage swelling driven by charge regulation and ion partitioning. Overall, our results highlight the complex interplay of ionization equilibria, valency effects, and ion partitioning in weak polyelectrolyte systems.Item Open Access Resonant Raman scattering from superconducting single crystals of (BEDT-TTF)2I3(1993) Pokhodnia, Konstantin I.; Graja, Andrzej; Weger, Meir; Schweitzer, DieterA study of low energetic resonant Raman scattering of (BEDT-TTF)2I3 superconducors was performed. The vanishing of phonon bands accompanied by a decrease of the electronic background was observed below Tc. We propose a theoretical explanation for this novel effect in terms of the Balseiro-Falicov model of phonon-superconducting amplitude mode interaction.Item Open Access Colloidal monolayers on quasiperiodic laser fields(2010) Mikhael, Jules; Bechinger, Clemens (Prof. Dr.)Quasicrystals are somewhat paradoxical structures which exhibit many amazing properties distinguishing them from ordinary crystals. Although the atoms are not localized at periodic positions, quasicrystals posses perfect long-range order. Until the early 1980s it was unanimously established that ordered matter is always periodic. Accordingly, the rotational symmetry in real space was thought to be limited to n=2,3,4 and 6. However more than a hundred complex metal alloys, for instance the discretely diffracting icosahedral AlPdMn or decagonal AlNiCo, have defied these crystallographic rules and self-organized into quasicrystals. Although the majority of the identified quasicrystals are complex metal alloys synthesized in the laboratory, recent experimental results proved that quasiperiodic order is not limited to metals. Matter also organizes itself aperiodically at larger length scales where thermal fluctuations play an important role. Recent experiments have shown that quasiperiodic order is also oberved in soft matter systems, such as micellars, polymers, and binary nanoparticles. Quasicrystals show many interesting properties which are quite different from that of periodic crystals. Accordingly, they are considered as materials with high technological potential e.g. as surface coatings, thermal barriers, catalysts or photonic materials. Quasicrystalline structures have been theoretically predicted also in systems with a single type of particles. Nevertheless, experimentally their spontaneous formation has been only observed in binary, ternary or even more complex alloys. Accordingly, their surfaces exhibit a high degree of structural and chemical complexity and show intriguing properties. In order to understand the origin of those characteristics it would be helpful to disentangle structural and chemical aspects which can be achieved by growing single-element monolayers to quasicrystalline surfaces. Apart from understanding how quasicrystalline properties can be transferred to such monolayers, this approach might allow fabrication of materials with novel properties. First heteroepitatic growth experiments on decagonal and icosahedral surfaces indeed demonstrate the formation of Pb, Bi and Sb monolayers with a high degree of quasicrystalline order as determined by low-energy electron diffraction and elastic helium atom scattering experiments. Compared to reciprocal space studies, only recently atomically resolved scanning tunneling microscopy investigations of the adsorbate morphology became possible. Even then, however, it is difficult to relate the structure of the adsorbate to that of the underlying substrate. In that respect, the study of the phase behaviour of colloidal particles interacting with quasiperiodic laser fields can throw new light on fundamental problems of broad interest in the physics of quasicrystals and in condensed matter physics. In fact colloidal systems are meanwhile established as excellent model for atomic systems and colloidal physics have demonstrated that such systems can give answers to many basic physics questions. Depending on the pair-interaction and the concentration, colloidal systems show analogues of all the states of atomic systems: gas, liquid and solid states. The mesoscopic size (nm-µm), the time scales (ms-s) and the tunability of the pair interaction in colloidal systems make them a convenient model system for experimental and theoretical studies. As a consequence, real space analysis by means of video microscopy allows tracking the trajectories of the individual particles and makes the time evolution of the system accessible in detail. Such information is inaccessible in systems investigated by diffraction experiments, as the scattering information is available only averaged over the scattering area. Because in a colloidal system there is direct access to real space information, the strength and nature of the different interactions, the origins of the complex phase behavior could be in different examples identified. In conclusion, the study of the rich phase behavior of colloidal suspensions provides ideal conditions for experimental and theoretical studies. In this Thesis, we report on a real-space investigation of the phase behaviour of charged colloidal monolayers interacting with quasicrystalline decagonal or tetradecagonal substrates created by interfering five or seven laser beams. Different starting configurations, such as dense fluid and triangular crystals with different densities, are prepared. At low intensities and high particle densities, the electrostatic colloidal repulsion dominates over the colloid-substrate interaction and the crystalline structure remains mainly intact. As expected, at very high intensities the colloid-substrate interaction dominates and a quasiperiodic ordering is observed. Interestingly, at intermediate intensities we observe the alignment of crystalline domains along the 5 directions of the quasicrystalline substrate. This is in agreement with observations of Xenon atoms adsorbed on the ten-fold decagonal Al-Ni-Co surface and numerical simulations of weakly adsorbed atomic systems. Intermediate phases are observed for colloid-substrate interactions strong enough to produce defects in the crystal. These defects adapt the form of rows of quadratic tiles. Surprisingly, for specific particle densities (at which the colloid-substrate interaction is minimized) we identify a novel pseudomorphic ordering. This intermediate phase which exhibits likewise crystalline and quasicrystalline structural properties can be described by an Archimedean-like tiling consisting of alternating rows of quadratic and triangular tiles. The calculated diffraction pattern of this phase is in agreement with recent observations of copper adsorbed on icosahedral AlPdMn surfaces. Interestingly, we also observe the formation of the same phase on tetradecagonal substrates also at densities for which the potential energy of the colloidal system is minimized. Although the structure can also be described by rows of triangles and rows of squares, a closer analysis reveals substantial differences. Here, large domains with almost periodic ordering are found. We show that this behavior is closely related to the low density of highly symmetric local motifs in the substrate potential. In the second part of this Thesis the conditions under which quasicrystals form are investigated. Currently, it is not clear why most quasicrystals hold 5- or 10-fold symmetry but no single example with 7 or 9-fold symmetry has ever been observed. Since the properties of quasicrystals are strongly connected to their atomic structure, a better understanding of their growth mechanisms is of great importance. In contrast to crystals which are periodic in all three dimensions, quasiperiodicity is always (except for icosahedral quasicrystals) restricted to two dimensions. Accordingly, three-dimensional quasicrystals are comprised of a periodic stacking of quasiperiodic layers and any hurdle in the formation of quasiperiodic order within a single layer will eventually prohibit their growth along the periodic direction. In this Thesis, we also report on geometrical constraints which impede the formation of quasicrystals with certain symmetries in a colloidal model system. This is achieved by subjecting a colloidal monolayer to N=5- and 7-beam quasiperiodic potential landscapes. Our results clearly demonstrate that quasicrystalline order is much easier established for N = 5 compared to N = 7. With increasing laser intensity we observe that the colloids first adopt quasiperiodic order at local areas which then laterally grow until an extended quasicrystalline layer forms. As nucleation sites where quasiperiodicity originates, we identify highly symmetric motifs in the laser pattern. We find that their density strongly varies with n and surprisingly is smallest exactly for those quasicrystalline symmetries which have never been observed in atomic systems. Since such high symmetry motifs also exist in atomic quasicrystals where they act as preferential adsorption sites, this suggests that it is indeed the deficiency of such motifs which accounts for the absence of e.g. materials with 7-fold symmetry. In addition to the fundamental aspects, we report in this Thesis on the fabrication of large colloidal quasiperiodic layers incorporated in a polymer hydrogel matrix. Because quasicrystals have higher point group symmetry than ordinary crystals, micrometer-scale quasicrystalline materials are expected to exhibit large and isotropic photonic bandgaps in the visible range. In our case, the quasiperiodic symmetries are induced using extended light fields. The reported gelled colloidal quasicrystals are unique in that they have large sizes as well as good optical uniformity. With laser diffraction the in situ variable length scale of such materials is demonstrated. In conclusion, we have studied the phase behavior of charged colloidal particles interacting with quasiperiodic laser fields. We showed that novel pseudomorphic growth can lead to the formation of a phase which exhibits likewise crystalline and quasicrystalline structural properties. We also performed unconventional measurements in order to understand why the formation of quasicrystals is limited to specific rotational symmetries. We have found that geometrical hurdles play a crucial role in the proliferation of quasiperiodicity and that such hurdles can hindered or even prohibited the formation of e.g. 7- or 9-fold symmetry. And finally, we have shown that the combination of extended light fields and hydrogel matrices leads to the formation of large quasiperiodically ordered colloidal materials.Item Open Access Towards spin injection into silicon(2007) Dash, Saroj Prasad; Carstanjen, Heinz Dieter (Prof. Dr.)The efficient spin injection into semiconductors could pave the way to a new generation of electronics devices such as spin memories, spin transistors, and spin quantum computers. The most important semiconductor for industrial application, Si has been studied for the purpose of spin injection extensively in this thesis. Three different concepts for spin injection into Si have been addressed: (1) spin injection through a ferromagnet-Si Schottky contact, (2) spin injection using MgO tunnel barriers in between the ferromagnet and Si, and (3) spin injection from Mn-doped Si (DMS) as spin aligner. (1) FM-Si Schottky contact for spin injection: In a heterostructure of a ferromagnetic thin film on a Si substrate, any structural disorder at the interface would drastically reduce the spin polarization at the interface and, hence, the spin injection efficiency. To be able to improve the interface qualities one needs to understand the atomic processes involved in the formation of such silicide phases. In order to obtain more detailed insight into the formation of such silicide phases the initial stages of growth of Co and Fe were studied in situ by HRBS with monolayer depth resolution. As understood, it was important to prohibit the in-diffusion of Co into interstitial sites at the initial stages of growth and the out-diffusion of Si atoms in the latter stages. So in order to control and improve the interface, equilibrium growth conditions were followed (i) by lowering the growth temperature and (ii) by surfactant-mediated growth. Low temperature growth of Co on Si (100): Already at very low coverage Co diffusion into the bulk Si has been observed. The amount of in-diffused Co is, however, less than at room temperature. In contradiction to room temperature growth, Co atoms form layers of pure Co on top of the Si surface already at very low coverage. Every second Si layer, starting with the first Si layer, is Co depleted. This leads to an oscillatory Co distribution in the Si lattice which is preserved up to higher coverages (1.3 ML). Surfactant-mediated growth of Co on Si (100) : The lower surface free energy of Sb in comparison to Co and Si, makes it a potential candidate for surfactant mediated growth. By the use of one monolayer of Sb adsorbed on a Si (100) surface, Co-Si intermixing at the interface is strongly reduced in comparison to the interface without Sb as surfactant. The improved interface quality with Sb-mediated growth is also reflected in magnetic measurements. Co with Sb-mediated growth shows a higher magnetic moment. It was shown that simple solutions can reduce the FM-Si inter diffusion at the interface and improve the interface quality. However these non-equilibrium growth conditions could not stop the silicide formation completely. (2) MgO tunnel barrier for spin injection into Si: On the other hand, using an ultra-thin tunnel barrier between FM and Si will have three advantages: (i) form a chemical barrier between the FM and Si, (ii) circumvent the conductivity mismatch problem, and (iii) in addition act as a spin filter. The fabrication and characterization of ultra-thin crystalline MgO tunnel barriers on Si (100) was presented. Some of the important properties required for tunnel barriers on Si have been addressed. Ultra-thin stoichiometric MgO tunnel barriers with sharp interface with Si (100), very homogeneous, without pin-holes, and crystalline in structure could be fabricated by reactive molecular beam epitaxy. Co and Fe on an ultra thin MgO tunnel barrier were found to have island-like growth with a rough surface. Ultra-thin Co and Fe films are found to be thermally quite stable up to 450 °C. (3) Mn doped Si for spin injection: For spin injection purpose, instead of contacting the Si with a ferromagnetic metal, the contact could be made with another semiconductor, one with ferromagnetic properties. This solves the conductivity mismatch problem by ensuring that the resistivities of the materials on both side of the interface are comparable in magnitude. Si-based diluted magnetic semiconductor samples were prepared by doping Si with Mn by two different methods i) by Mn ion implantation and ii) by in-diffusion of Mn atoms (solid state growth). In the case of implanted samples, Mn atoms do not substitute Si sites. The implanted samples show room temperature ferromagnetism as measured by a SQUID magnetometer. The magnetic moment per Mn atom is found to decrease with increasing implantation dose. It has been observed that the implanted samples show carrier mediated ferromagnetism and, more importantly, mediated by both holes and electrons in contrast to statements in the literature. Solid state growth of Mn doped Si : For evaporation of Mn on Si (100), Mn atoms diffuse deep into the Si bulk already at room temperature, even for very low coverage (0.25 ML) with an oscillatory concentration depth profile as observed by HRBS with monolayer depth resolution. This results in natural MnxSi1-x/Si digital layers on the surface. Surprisingly, the samples prepared by this solid state diffusion process show room-temperature ferromagnetism having a magnetic moment of 1.8 µB per Mn atom, which is much higher than that of the ion-implanted samples. In contrast to ion-implanted samples the ferromagnetism in these samples does not show any carrier mediation.Item Open Access Nanoscale mapping of magnetic auto-oscillations with a single spin sensor(2025) Hache, Toni; Anshu, Anshu; Shalomayeva, Tetyana; Richter, Gunther; Stöhr, Rainer; Kern, Klaus; Wrachtrup, Jörg; Singha, AparajitaSpin Hall nano-oscillators convert DC to magnetic auto-oscillations in the microwave regime. Current research on these devices is dedicated to creating next-generation energy-efficient hardware for communication technologies. Despite intensive research on magnetic auto-oscillations within the past decade, the nanoscale mapping of those dynamics remained a challenge. We image the distribution of free-running magnetic auto-oscillations by driving the electron spin resonance transition of a single spin quantum sensor, enabling fast acquisition (100 ms/pixel). With quantitative magnetometry, we experimentally demonstrate for the first time that the auto-oscillation spots are localized at magnetic field minima acting as local potential wells for confining spin-waves. By comparing the magnitudes of the magnetic stray field at these spots, we decipher the different frequencies of the auto-oscillation modes. The insights gained regarding the interaction between auto-oscillation modes and spin-wave potential wells enable advanced engineering of real devices.Item Open Access The role of dimensionality and geometry in quench-induced nonequilibrium forces(2021) Nejad, Mehrana Raeisian; Khalilian, Hamidreza; Rohwer, Christian M.; Moghaddam, Ali GhorbanzadehWe present an analytical formalism, supported by numerical simulations, for studying forces that act on curved walls following temperature quenches of the surrounding ideal Brownian fluid. We show that, for curved surfaces, the post-quench forces initially evolve rapidly to an extremal value, whereafter they approach their steady state value algebraically in time. In contrast to the previously-studied case of flat boundaries (lines or planes), the algebraic decay for curved geometries depends on the dimension of the system. Specifically, steady-state values of the force are approached in time as t-d/2 in d-dimensional spherical (curved) geometries. For systems consisting of concentric circles or spheres, the exponent does not change for the force on the outer circle or sphere. However, the force exerted on the inner circles or sphere experiences an overshoot and, as a result, does not evolve to the steady state in a simple algebraic manner. The extremal value of the force also depends on the dimension of the system, and originates from curved boundaries and the fact that particles inside a sphere or circle are locally more confined, and diffuse less freely than particles outside the circle or sphere.Item Open Access Direct electron beam patterning of electro-optically active PEDOT:PSS(2024) Doshi, Siddharth; Ludescher, Dominik; Karst, Julian; Floess, Moritz; Carlström, Johan; Li, Bohan; Mintz Hemed, Nofar; Duh, Yi-Shiou; Melosh, Nicholas A.; Hentschel, Mario; Brongersma, Mark; Giessen, HaraldThe optical and electronic tunability of the conductive polymer poly(3,4-ethylenedioxythiophene):poly(styrene sulfonate) (PEDOT:PSS) has enabled emerging applications as diverse as bioelectronics, flexible electronics, and micro- and nano-photonics. High-resolution spatial patterning of PEDOT:PSS opens up opportunities for novel active devices in a range of fields. However, typical lithographic processes require tedious indirect patterning and dry etch processes, while solution-processing methods such as ink-jet printing have limited spatial resolution. Here, we report a method for direct write nano-patterning of commercially available PEDOT:PSS through electron-beam induced solubility modulation. The written structures are water stable and maintain the conductivity as well as electrochemical and optical properties of PEDOT:PSS, highlighting the broad utility of our method. We demonstrate the potential of our strategy by preparing prototypical nano-wire structures with feature sizes down to 250 nm, an order of magnitude finer than previously reported direct write methods, opening the possibility of writing chip-scale microelectronic and optical devices. We finally use the high-resolution writing capabilities to fabricate electrically-switchable optical diffraction gratings. We show active switching in this archetypal system with >95 % contrast at CMOS-compatible voltages of +2 V and -3 V, offering a route towards highly-miniaturized dynamic optoelectronic devices.