4 search hits
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Dynamics of vortices in the two-dimensional anisotropic Heisenberg model with magnetic fields.
(2003)
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Juan Pablo Zagorodny
- The subject of this work is the dynamics of a vortex in a classical 2-dimensional spin system with anisotropic exchange interaction under the combined action of magnetic fields and damping. Static as well as dynamic magnetic fields were employed (as dynamical field we used a homogeneous field which is rotating in the XY-plane). The most important goal of this work was to demonstrate that there is a coupling between the inner and translational freedom degrees of the vortex, coupling which is responsible for at least 2 phenomena that we study in detail in this Thesis: 1. the switching or flipping of the vortex polarization (for negative field frequency), and 2. the formation of stable orbits of the vortex center around the center of the system driven by the rotating field (for positive frequency). It was known to us that the polarization can change abruptly its sign under the action of a field rotating in the XY-plane, for p omega < 0 and appropriate field amplitudes. In the Chapter 4 we have investigated the possible underlying mechanisms for this phenomenon. Our main results can be summarized as follows: a) The flipping times do not depend essentially on the size of the system, provided that the lattice is large enough (radius L >~ 36 lattice constants). In other words, the switching of the vortex polarization is not much affected by the presence of boundaries. b) In our numerical simulations we observed a clear correlation between the core magnetization dynamics (the oscillations of the core spins in the out-of-plane direction) and the velocity of the vortex center in the plane of the lattice. c) A diagram of flipping events as a function of the field parameters, from extensive numerical simulations with an OP vortex in a rotating magnetic field, was presented. We found out that in the (omega, h) parameters space there is no well-defined curve which separates the regime where the flips do not occur from the regime where they do. We found intervals ("windows'') of intermittent flip and non-flip events. d) The switching of the vortex polarization can be achieved also by applying a static magnetic field with both in-plane (IP) and out-of-plane (OP) components. The IP component of the field sets the vortex into translational movement in the XY-plane, while the OP component breaks the vertical symmetry favoring one of the two possible orientations. e) The switching dynamics may be described in terms of a core model which takes into account a coupling between the vortex polarization dynamics and the motion of the vortex center. We showed that a reduced core model, which is valid near the threshold of the IP-OP vortex instability (lambda ~ lambda c), can be mapped to a generalized Thiele equation with an inertial term. f) It is plausible that the phenomenon of switching we described will not be essentially affected by the inclusion of a dipole-dipole interaction. The experimental works on nanodisks mentioned in the Introduction of this Thesis reported the observation of vortices in either of two polarization states, and the switching between them was forced by means of static fields perpendicular to the plane of the disks. Rotating magnetic fields might be used as well static fields with both IP and OP components to make this switching more favorable. In the Chapter 5 we turned to the study of the movement of the vortex in the XY plane, in the presence of the IP rotating field. Attention was directed to the existence of stable orbits, where the vortex stays inside the system in a stationary movement, forming circular limit cycles. We discussed then the failure of the conventional Thiele approach to describe this phenomenon, and this motivated us to formulate an extended collective coordinate Theory, which leads to a qualitative agreement with the results of the simulations. A diagram of the different types of trajectories, as a function of the field parameters, showed the presence of non-monotonous effects and "windows'', like in the case of the switching diagram. We are led to conclude that for some regions of the field parameters space, the system exhibits chaos -which is typical for many-body systems-, though no particular tool of the chaos theory was used to study our discrete and collective coordinate models, from this viewpoint. Our theoretical work qualitatively suggests that it would be interesting to apply in the experiments weak rotating fields like those used here, to control both the mean position of a vortex in larger magnetic dots (where the vortex center could show dynamics) and at the same time the sign of the out-of-plane core magnetization. Future directions of this work may include the use of inhomogeneous fields, particularly with a gaussian localization in a small region of the lattice or "spot'', as a model of the field of a laser beam.
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Maximum Entropy Method in Superspace Crystallography
(2003)
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Lukas Palatinus
- This thesis discusses several aspects of the combination of the Maximum Entropy Method (MEM) for the reconstructions of the electron density with the superspace approach to the description of structures of aperiodic crystals. It is shown that the MEM in superspace provides a parameter-free reconstruction of the modulation functions with sufficient accuracy. The MEM in superspace has been applied to diffraction data of several compounds. The computer program BayMEM was developed for this purpose. BayMEM allows electron densities of the ordinary 3D structures and the superspace electron densities of the aperiodic structures to be reconstructed using the same general principles. The program has been extended by adding features improving its versatility and accuracy of the results. The improvements include attaching of the set of subroutines MemSys5 to BayMEM, implementation of the method of the Generalized F-constraints and the static weighting, implementation of the G-constraints, of the Prior-derived F-constraints and of the two-channel entropy. The second major computer program EDMA is a software tool for analysis of the electron densities in arbitrary dimension. The program analyzes the MEM electron density and extracts quantitative information about the atoms according to Bader's formalism “Atoms in molecules“. Two new variants of the constraints in the MEM have been developed in order to solve the problems with artifacts in the MEM reconstructions. The two methods are the Generalized F-constraints and the Prior-derived F-constraints. The concept of the Generalized F-constraints is based in the observation, that the standard F-constraint is not sufficiently strong to constrain the histogram of the normalized residuals of the structure factors to the expected Gaussian shape. Higher moments of the distribution of the normalized residuals were therefore used as the constraint in the MEM calculations. With these constraints significantly improved histograms were obtained. The source of some artifacts in the MEM electron densities was identified to be the tendency of the MEM to estimate incorrectly those structure factors, that are not included in the experimental dataset. It is shown that the missing structure factors can successfully be replaced by the structure factors derived from the procrystal electron density. If the structure factors derived from the procrystal prior electron density (the Prior-derived F-constraints) are used as additional constraints in the MEM calculation, the result is free of sharp artifacts and the quality of the reconstruction of the electron density is comparable with the results of multipole refinements. To test the accuracy of the MEM in superspace, the method was applied to the dataset of the misfit-layer composite structure of (LaS)1.14NbS2. It has been shown, that the MEM on the model structure factors reproduces the model modulation functions with accuracy better that 10% of the pixel size of the grid, on which the electron density was sampled. The structure of the high-pressure phase III of Bi provided a prominent example illustrating the advantages of the MEM in superspace over the standard structure refinements. The MEM in superspace was applied to the diffraction data of Bi-III to extract more information about the modulation than obtained from the standard structure refinement. The modulation functions extracted from the MEM electron density revealed a block-wave-like shape of the modulation function of the Bi atom of the host structure that indicates shifts of the atom between two stable environments rather than smooth harmonic variation of the position indicated by the modulation function from the standard refinement. Secondly, the MEM modulation function of the Bi atoms in channels allowed to better understand the nature of the most prominent feature of the modulated structure - the occurrence of the pairs of Bi atoms along the channels. The incommensurately modulated structure of ammonium tetrafluoroberyllate (NH4)2BeF4, stable between 175K and 182K, was solved and refined in superspace. The known two-fold low-temperature superstructure of (NH4)2BeF4, that is stable below 175K has been described in superspace as a commensurately modulated structure. With aid of this description the close relationship between the two structures has been found. The MEM was applied to the incommensurate structure to test the appropriateness of the refined harmonic structure model. The MEM has shown that the harmonic model is very accurate. The MEM in superspace was established in this thesis as a reliable tool for the structure solutions of the modulated structure. The individual chapters together form a framework that allows to use the MEM in superspace to extract novel information from the diffraction data of both the periodic and aperiodic structures, that cannot be obtained from the structure refinements.
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Instabilities in layered liquids induced by external fields
(2003)
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Günter Auernhammer
- In this thesis, we have shown that the inclusion of a nematic degree of freedom in the macroscopic hydrodynamic description of smectic-A-like liquids leads to a number of interesting results. While the director and the layer normal are coupled such that they are parallel in equilibrium, in non-equilibrium situations, the director needs not be parallel to the smectic layer normal. This is in contrast to standard smectic-A hydrodynamics. Using irreversible thermodynamics and symmetry arguments, we derived a complete set of macroscopic hydrodynamic equations for the director variables, the layer displacement, the velocity field, and the moduli of the nematic and smectic order parameters. Recent experiments find that the parallel orientation of smectic-A- like liquids is destabilized by an applied shear. After destabilization, two typical scenarios are observed in a steady state situation: i) The layers are oriented perpendicular to the vorticity direction of the flow, i.e., they lie in the plane spanned by the velocity and the gradient direction (`perpendicular' orientation). ii) Closed multi-lamellar vesicles (`onions') form. A number of experiments indicate that the onset of this reorientation is controlled by the applied shear rate. In contrast to standard smectic-A hydrodynamics where shear in the parallel orientation has no effect on the layers, this destabilizing effect comes out naturally from our extended smectic-A hydrodynamics. The argumentation goes along the following lines. The shear field exerts a torque on the director that must be balanced by the coupling to the layer normal. In the limit of small angles, balancing these torques leads, in the steady state, to a shear-induced director tilt proportional to the shear rate. The preferred thickness of a smectic layer is directly connected to the projection of the averaged molecular axes on the layer normal, or, in terms of our model, the thickness is proportional to the projection of the director on the layer normal. If the director is tilted, this projection is shorter. This decrease of the projection is equivalent to an effective dilation, because the actual layer thickness is larger than the preferred layer thickness. Similar to the case of low molecular weight smectic-A liquid crystals under a dilative strain, this effective dilation leads to an undulation instability. To investigate the stability of the parallel alignment, we performed a linear and weakly non-linear analysis of the governing equations. The initial state is the above described spatially homogeneous director tilt with the smectic layers in the parallel orientation. The linear stability analysis showed an undulation instability which sets in above a critical tilt angle (or equivalently, a critical shear rate). This critical tilt angle turned out to depend strongly on the material parameters. For a typical low molecular weight thermotropic liquid crystal, we estimated the critical tilt angle to be on the order of a few degrees. The linear stability analysis also revealed that the nematic and smectic order is modulated close to the boundaries. Since the probability for the formation of defects is larger in regions with a decreased modulus of the order parameter, these variations in the modulus of the order parameter open the way for a destabilization of the layered structure. We note that a detailed investigation of this point is beyond the scope of the present work. Finally, we could exclude an oscillatory instability for all physically reasonable regions in parameter space. The weakly non-linear analysis shows that the bifurcation is supercritical for most physically relevant regions in the parameter space. A detailed comparison to an independent approach was undertaken in a collaboration with simulation physicists from the Max-Planck- Institute for Polymer Research in Mainz. In a molecular dynamics simulation, a model layered liquid consisting of chains of four particles (AABB) was considered. The interaction potential of particles not connected by springs is attractive for like particles and repulsive for particles of a different nature. The simulation demonstrated the two main predictions of our analytic theory: The director tilts in the flow direction and, above a critical shear rate, the layers show stationary undulations with a wave vector in the vorticity direction. Besides this good qualitative agreement, a reasonable quantitative agreement for the critical shear rate was found.
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Der Einfluss zusätzlicher Fallenzustände auf die Transporteigenschaften niedermolekularer Lochleiter und Hybrid-Solarzellen
(2003)
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Claus Jäger
- Im Rahmen dieser Arbeit sollte untersucht werden, inwieweit sich die Beweglichkeit von niedermolekularen Lochleitern auf der Basis von Triphenyldiaminen durch Zumischen von zusätzlichen Fallenzuständen manipulieren und mit der Modellvorstellung des Hopping-Transports beschreiben lässt. Die Materialmischungen sollten dann in Hybrid-Solarzellen eingebaut werden und so der Einfluss der Ladungsträgerbeweglichkeit auf die photovoltaischen Parameter bestimmt werden. Die Time-of-Flight-Messungen konnten an einer Vielzahl von Lochleitermischungen durchgeführt werden, wobei die Fallenkonzentration und -tiefe variiert wurden. Es zeigte sich beim reinen Matrixmaterial DMe-TPD, dass ausschließlich das Hopping-Modell die Daten gut beschreiben kann, da die Steigungen im Nachtransitbereich so groß sind, dass das Steigungskriterium von -2 für das CTRW- und Multiple-Trapping-Modell nicht erfüllt ist. Bei Dotierung mit tiefen Fallen (NTDATA) wurde mit zunehmender Konzentration ein Übergang von dispersivem zu nicht-dispersivem Ladungsträgertransport beobachtet, wobei sich auch das Verhalten der Steigungswerte im Vor- und Nachtransitbereich der Transienten ändert. Bei dispersivem Transport ist die Feldabhängigkeit gering, während bei hohen Fallenkonzentrationen c eine deutliche Feldabhängigkeit nachzuweisen ist. Die Beweglichkeit lässt sich durch Dotierung mit NTDATA im Vergleich zum reinem Material um bis zu vier Größenordnungen erniedrigen, was eine Anwendung dieser Lochleitersysteme in Hybrid-Solarzellen möglich macht. Über temperaturabhängige Messungen konnten die charakteristischen Parameter im Hopping-Modell bestimmt werden. Die energetische Breite sigma weist eine logarithmische Abhängigkeit von der Fallenkonzentration auf. Im Rahmen des Hoesterey-Letson-Formalismus konnten die Messdaten gut beschrieben werden und es zeigt sich, dass sich die gesamte energetische Breite der Zustandsdichte aus Wirt- und Gastmolekülen weitestgehend durch eine effektive Breite sigma-eff charakterisieren lässt. Ebenso wurden die Grenzen dieses Modells dargestellt, da bei flachen Fallen die angewandte Näherung ungültig wird und die Voraussage der c-1-Abhängigkeit der Beweglichkeit im Experiment nicht mehr beobachtet wird. Am System mit tiefen Fallen (NTDATA) wurde in Übereinstimmung mit der Literatur eine überlineare Konzentrationsabhängigkeit der Beweglichkeit nachgewiesen. Als Grund wurde hierfür der Übergang von der dispersiven zur nicht-dispersiven Transientenform identifiziert. Eine sehr gute Übereinstimmung mit den experimentellen Daten ergab sich mit der neueren EMA-Theorie. Der Verlauf der konzentrationsabhängigen Beweglichkeiten für unterschiedliche Fallentiefen konnte bis auf die Abweichungen durch die dispersiven Transienten bei NTDATA hervorragend wiedergegeben werden. Im fallendominierten Bereich ergibt sich als Grenzfall die c-1-Abhängigkeit des HLF. Die temperaturabhängigen Nullfeldbeweglichkeiten für NTDATA konnten im Rahmen der Messgenauigkeit gut simuliert werden. Dabei zeigte sich sowohl im Experiment als auch in der Theorie unterhalb der kritischen Temperatur Tcr ein verändertes Steigungsverhalten. Der einzig frei wählbare Parameter ist die energetische Unordnung der Fallenzustände sigma1, die mit höherer Fallenkonzentration zunimmt. Die in diesem Rahmen erweiterte Hoesterey-Letson-Gleichung konnte ebenfalls gut auf die Messdaten angewandt werden und beschreibt diese in einem weiten Konzentrationsbereich. Insbesondere für höheren Fallendichten von NTDATA nimmt in der Theorie die effektive Breite sigma-eff im Vergleich zu den gemessenen Werten schneller ab, da der Beitrag des Interfallentransports in der Simulation überschätzt wird bzw. die Bestimmung von sigma-eff im Experiment durch lineare Regression nicht eindeutig ist. Die Anwendung der molekular-dotierten Photoleiter in Hybrid-Solarzellen zeigte einen zunächst unerwartet geringen Zusammenhang zwischen Beweglichkeit und den charakteristischen Kenngrößen der Solarzelle. Der Grund hierfür liegt darin, dass im Vergleich zu den Time-of-Flight-Messungen bei der photovoltaischen Charakterisierung eine deutlich höhere Ladungskonzentration im Lochleiter vorliegt, die eine Besetzung der zusätzlichen Fallenzustände verursacht. Da diese so den Transport nicht verlangsamen können, ist die effektive Beweglichkeit deutlich erhöht. Intensitätsabhängige Messungen an den Solarzellen zeigen, dass ausschließlich monomolekulare Rekombination während des Transports stattfindet und die Fallenzustände deshalb nicht als Rekombinationszentren wirken. Damit konnte in dieser Arbeit erstmals eindeutig nachgewiesen werden, dass eine durch Fallen reduzierte Beweglichkeit nicht der limitierende Faktor bei der Solarzelleneffizienz ist, solange keine zusätzlichen bimolekularen Rekombinationsprozesse im Material stattfinden.
