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.

Numerical coupling of thermal-electric network models and energy-transport equations including optoelectronic semiconductor devices

In this work the numerical coupling of thermal and electric network models with model equations for optoelectronic semiconductor devices is presented. Modified nodal analysis (MNA) is applied to model electric networks. Thermal effects are modeled by an accompanying thermal network. Semiconductor devices are modeled by the energy-transport model, that allows for thermal effects. The energy-transport model is expandend to a model for optoelectronic semiconductor devices. The temperature of the crystal lattice of the semiconductor devices is modeled by the heat flow eqaution. The corresponding heat source term is derived under thermodynamical and phenomenological considerations of energy fluxes. The energy-transport model is coupled directly into the network equations and the heat flow equation for the lattice temperature is coupled directly into the accompanying thermal network. The coupled thermal-electric network-device model results in a system of partial differential-algebraic equations (PDAE). Numerical examples are presented for the coupling of network- and one-dimensional semiconductor equations. Hybridized mixed finite elements are applied for the space discretization of the semiconductor equations. Backward difference formluas are applied for time discretization. Thus, positivity of charge carrier densities and continuity of the current density is guaranteed even for the coupled model.

Modeling surface-atmosphere exchange of trace gases and energy within and above the Amazon rain forest

Die vorliegende Dissertation untersucht die biogeochemischen Vorgänge in der Vegetationsschicht (Bestand) und die Rückkopplungen zwischen physiologischen und physikalischen Umweltprozessen, die das Klima und die Chemie der unteren Atmosphäre beeinflussen. Ein besondere Schwerpunkt ist die Verwendung theoretischer Ansätze zur Quantifizierung des vertikalen Austauschs von Energie und Spurengasen (Vertikalfluss) unter besonderer Berücksichtigung der Wechselwirkungen der beteiligten Prozesse. Es wird ein differenziertes Mehrschicht-Modell der Vegetation hergeleitet, implementiert, für den amazonischen Regenwald parametrisiert und auf einen Standort in Rondonia (Südwest Amazonien) angewendet, welches die gekoppelten Gleichungen zur Energiebilanz der Oberfläche und CO2-Assimilation auf der Blattskala mit einer Lagrange-Beschreibung des Vertikaltransports auf der Bestandesskala kombiniert. Die hergeleiteten Parametrisierungen beinhalten die vertikale Dichteverteilung der Blattfläche, ein normalisiertes Profil der horizontalen Windgeschwindigkeit, die Lichtakklimatisierung der Photosynthesekapazität und den Austausch von CO2 und Wärme an der Bodenoberfläche. Desweiteren werden die Berechnungen zur Photosynthese, stomatären Leitfähigkeit und der Strahlungsabschwächung im Bestand mithilfe von Feldmessungen evaluiert. Das Teilmodell zum Vertikaltransport wird im Detail unter Verwendung von 222-Radon-Messungen evaluiert. Die ``Vorwärtslösung'' und der ``inverse Ansatz'' des Lagrangeschen Dispersionsmodells werden durch den Vergleich von beobachteten und vorhergesagten Konzentrationsprofilen bzw. Bodenflüssen bewertet. Ein neuer Ansatz wird hergeleitet, um die Unsicherheiten des inversen Ansatzes aus denjenigen des Eingabekonzentrationsprofils zu quantifizieren. Für nächtliche Bedingungen wird eine modifizierte Parametrisierung der Turbulenz vorgeschlagen, welche die freie Konvektion während der Nacht im unteren Bestand berücksichtigt und im Vergleich zu früheren Abschätzungen zu deutlich kürzeren Aufenthaltszeiten im Bestand führt. Die vorhergesagte Stratifizierung des Bestandes am Tage und in der Nacht steht im Einklang mit Beobachtungen in dichter Vegetation. Die Tagesgänge der vorhergesagten Flüsse und skalaren Profile von Temperatur, H2O, CO2, Isopren und O3 während der späten Regen- und Trockenzeit am Rondonia-Standort stimmen gut mit Beobachtungen überein. Die Ergebnisse weisen auf saisonale physiologische Änderungen hin, die sich durch höhere stomatäre Leitfähigkeiten bzw. niedrigere Photosyntheseraten während der Regen- und Trockenzeit manifestieren. Die beobachteten Depositionsgeschwindigkeiten für Ozon während der Regenzeit überschreiten diejenigen der Trockenzeit um 150-250%. Dies kann nicht durch realistische physiologische Änderungen erklärt werden, jedoch durch einen zusätzlichen cuticulären Aufnahmemechanismus, möglicherweise an feuchten Oberflächen. Der Vergleich von beobachteten und vorhergesagten Isoprenkonzentrationen im Bestand weist auf eine reduzierte Isoprenemissionskapazität schattenadaptierter Blätter und zusätzlich auf eine Isoprenaufnahme des Bodens hin, wodurch sich die globale Schätzung für den tropischen Regenwald um 30% reduzieren würde. In einer detaillierten Sensitivitätsstudie wird die VOC Emission von amazonischen Baumarten unter Verwendung eines neuronalen Ansatzes in Beziehung zu physiologischen und abiotischen Faktoren gesetzt. Die Güte einzelner Parameterkombinationen bezüglich der Vorhersage der VOC Emission wird mit den Vorhersagen eines Modells verglichen, das quasi als Standardemissionsalgorithmus für Isopren dient und Licht sowie Temperatur als Eingabeparameter verwendet. Der Standardalgorithmus und das neuronale Netz unter Verwendung von Licht und Temperatur als Eingabeparameter schneiden sehr gut bei einzelnen Datensätzen ab, scheitern jedoch bei der Vorhersage beobachteter VOC Emissionen, wenn Datensätze von verschiedenen Perioden (Regen/Trockenzeit), Blattentwicklungsstadien, oder gar unterschiedlichen Spezies zusammengeführt werden. Wenn dem Netzwerk Informationen über die Temperatur-Historie hinzugefügt werden, reduziert sich die nicht erklärte Varianz teilweise. Eine noch bessere Leistung wird jedoch mit physiologischen Parameterkombinationen erzielt. Dies verdeutlicht die starke Kopplung zwischen VOC Emission und Blattphysiologie.

A new double laser pulse pumping scheme for transient collisionally excited plasma soft X-ray lasers

Within this thesis a new double laser pulse pumping scheme for plasma-based, transient collisionally excited soft x-ray lasers (SXRL) was developed, characterized and utilized for applications. SXRL operations from ~50 up to ~200 electron volt were demonstrated applying this concept. As a central technical tool, a special Mach-Zehnder interferometer in the chirped pulse amplification (CPA) laser front-end was developed for the generation of fully controllable double-pulses to optimally pump SXRLs.rnThis Mach-Zehnder device is fully controllable and enables the creation of two CPA pulses of different pulse duration and variable energy balance with an adjustable time delay. Besides the SXRL pumping, the double-pulse configuration was applied to determine the B-integral in the CPA laser system by amplifying short pulse replica in the system, followed by an analysis in the time domain. The measurement of B-integral values in the 0.1 to 1.5 radian range, only limited by the reachable laser parameters, proved to be a promising tool to characterize nonlinear effects in the CPA laser systems.rnContributing to the issue of SXRL pumping, the double-pulse was configured to optimally produce the gain medium of the SXRL amplification. The focusing geometry of the two collinear pulses under the same grazing incidence angle on the target, significantly improved the generation of the active plasma medium. On one hand the effect was induced by the intrinsically guaranteed exact overlap of the two pulses on the target, and on the other hand by the grazing incidence pre-pulse plasma generation, which allows for a SXRL operation at higher electron densities, enabling higher gain in longer wavelength SXRLs and higher efficiency at shorter wavelength SXRLs. The observation of gain enhancement was confirmed by plasma hydrodynamic simulations.rnThe first introduction of double short-pulse single-beam grazing incidence pumping for SXRL pumping below 20 nanometer at the laser facility PHELIX in Darmstadt (Germany), resulted in a reliable operation of a nickel-like palladium SXRL at 14.7 nanometer with a pump energy threshold strongly reduced to less than 500 millijoule. With the adaptation of the concept, namely double-pulse single-beam grazing incidence pumping (DGRIP) and the transfer of this technology to the laser facility LASERIX in Palaiseau (France), improved efficiency and stability of table-top high-repetition soft x-ray lasers in the wavelength region below 20 nanometer was demonstrated. With a total pump laser energy below 1 joule the target, 2 mircojoule of nickel-like molybdenum soft x-ray laser emission at 18.9 nanometer was obtained at 10 hertz repetition rate, proving the attractiveness for high average power operation. An easy and rapid alignment procedure fulfilled the requirements for a sophisticated installation, and the highly stable output satisfied the need for a reliable strong SXRL source. The qualities of the DGRIP scheme were confirmed in an irradiation operation on user samples with over 50.000 shots corresponding to a deposited energy of ~ 50 millijoule.rnThe generation of double-pulses with high energies up to ~120 joule enabled the transfer to shorter wavelength SXRL operation at the laser facility PHELIX. The application of DGRIP proved to be a simple and efficient method for the generation of soft x-ray lasers below 10 nanometer. Nickel-like samarium soft x-ray lasing at 7.3 nanometer was achieved at a low total pump energy threshold of 36 joule, which confirmed the suitability of the applied pumping scheme. A reliable and stable SXRL operation was demonstrated, due to the single-beam pumping geometry despite the large optical apertures. The soft x-ray lasing of nickel-like samarium was an important milestone for the feasibility of applying the pumping scheme also for higher pumping pulse energies, which are necessary to obtain soft x-ray laser wavelengths in the water window. The reduction of the total pump energy below 40 joule for 7.3 nanometer short wavelength lasing now fulfilled the requirement for the installation at the high-repetition rate operation laser facility LASERIX.rn

Airborne in situ measurements of ice particles in the tropical tropopause layer

Ice clouds have a strong effect on the Earth-atmosphere radiative energy balance, on the distribution of condensable gases in the atmosphere, as well as on the chemical composition of the air. The ice particles in these clouds can take on a variety of shapes which makes the description of the cloud microphysical properties more difficult. In the tropical upper troposphere/lower stratosphere (UTLS), a region where ice cloud abundance is relatively high, different types of ice clouds can be observed. However, in situ measurements are rare due to the high altitude of these clouds and the few available research aircraft, only three worldwide, that can fly at such altitudes.rnThis work focuses on in situ measurements of the tropical UTLS clouds performedrnwith a Cloud Imaging Probe (CIP) and a Forward Scattering Spectrometer Probern(FSSP-100), whereof the CIP is the key instrument of this thesis. The CIP is anrnairborne in situ instrument that obtains two-dimensional shadow images of cloud particles. Several cloud microphysical parameters can be derived from these measurements, e.g. number concentrations and size distributions. In order to obtain a high quality data set, a careful image analysis and several corrections need to be applied to the CIP observations. These methods are described in detail.rnMeasurements within the tropical UTLS have been performed during two campaigns:rnSCOUT-O3, 2005 in Northern Australia and SCOUT-AMMA, 2006 inWest Africa. Thernobtained data set includes first observations of subvisible cirrus clouds over a continental area and observations of the anvils of deep convective clouds. The latter can be further divided into clouds in mesoscale convective system outflows of different ages and clouds in overshooting cloud turrets that even penetrated the stratosphere. The microphysical properties of these three cloud types are discussed in detail. Furthermore, the vertical structure of the ice clouds in the UTLS is investigated. The values of the microphysical parameters were found to decrease with increasing altitude in the upper troposphere. Particle numbers and maximum sizes were also decreasing with increasing age of the outflow clouds. Further differences between the deep convective clouds and subvisible cirrus were found in the particle morphology as well as in the ratio of the observed aerosol particles to cloud particles which indicates that the different freezing processes (deposition, contact, immersion freezing) play different roles in the formation of the respective clouds. For the achievementrnof a better microphysical characterisation and description numerical fits have been adjusted onto the cloud particle size distributions of the subvisible cirrus as well as on the size distributions of the clouds at different altitudes in the UTLS.

IMEX finite volume methods for the shallow water equations

Die Flachwassergleichungen (SWE) sind ein hyperbolisches System von Bilanzgleichungen, die adäquate Approximationen an groß-skalige Strömungen der Ozeane, Flüsse und der Atmosphäre liefern. Dabei werden Masse und Impuls erhalten. Wir unterscheiden zwei charakteristische Geschwindigkeiten: die Advektionsgeschwindigkeit, d.h. die Geschwindigkeit des Massentransports, und die Geschwindigkeit von Schwerewellen, d.h. die Geschwindigkeit der Oberflächenwellen, die Energie und Impuls tragen. Die Froude-Zahl ist eine Kennzahl und ist durch das Verhältnis der Referenzadvektionsgeschwindigkeit zu der Referenzgeschwindigkeit der Schwerewellen gegeben. Für die oben genannten Anwendungen ist sie typischerweise sehr klein, z.B. 0.01. Zeit-explizite Finite-Volume-Verfahren werden am öftersten zur numerischen Berechnung hyperbolischer Bilanzgleichungen benutzt. Daher muss die CFL-Stabilitätsbedingung eingehalten werden und das Zeitinkrement ist ungefähr proportional zu der Froude-Zahl. Deswegen entsteht bei kleinen Froude-Zahlen, etwa kleiner als 0.2, ein hoher Rechenaufwand. Ferner sind die numerischen Lösungen dissipativ. Es ist allgemein bekannt, dass die Lösungen der SWE gegen die Lösungen der Seegleichungen/ Froude-Zahl Null SWE für Froude-Zahl gegen Null konvergieren, falls adäquate Bedingungen erfüllt sind. In diesem Grenzwertprozess ändern die Gleichungen ihren Typ von hyperbolisch zu hyperbolisch.-elliptisch. Ferner kann bei kleinen Froude-Zahlen die Konvergenzordnung sinken oder das numerische Verfahren zusammenbrechen. Insbesondere wurde bei zeit-expliziten Verfahren falsches asymptotisches Verhalten (bzgl. der Froude-Zahl) beobachtet, das diese Effekte verursachen könnte.Ozeanographische und atmosphärische Strömungen sind typischerweise kleine Störungen eines unterliegenden Equilibriumzustandes. Wir möchten, dass numerische Verfahren für Bilanzgleichungen gewisse Equilibriumzustände exakt erhalten, sonst können künstliche Strömungen vom Verfahren erzeugt werden. Daher ist die Quelltermapproximation essentiell. Numerische Verfahren die Equilibriumzustände erhalten heißen ausbalanciert.rnrnIn der vorliegenden Arbeit spalten wir die SWE in einen steifen, linearen und einen nicht-steifen Teil, um die starke Einschränkung der Zeitschritte durch die CFL-Bedingung zu umgehen. Der steife Teil wird implizit und der nicht-steife explizit approximiert. Dazu verwenden wir IMEX (implicit-explicit) Runge-Kutta und IMEX Mehrschritt-Zeitdiskretisierungen. Die Raumdiskretisierung erfolgt mittels der Finite-Volumen-Methode. Der steife Teil wird mit Hilfe von finiter Differenzen oder au eine acht mehrdimensional Art und Weise approximniert. Zur mehrdimensionalen Approximation verwenden wir approximative Evolutionsoperatoren, die alle unendlich viele Informationsausbreitungsrichtungen berücksichtigen. Die expliziten Terme werden mit gewöhnlichen numerischen Flüssen approximiert. Daher erhalten wir eine Stabilitätsbedingung analog zu einer rein advektiven Strömung, d.h. das Zeitinkrement vergrößert um den Faktor Kehrwert der Froude-Zahl. Die in dieser Arbeit hergeleiteten Verfahren sind asymptotisch erhaltend und ausbalanciert. Die asymptotischer Erhaltung stellt sicher, dass numerische Lösung das "korrekte" asymptotische Verhalten bezüglich kleiner Froude-Zahlen besitzt. Wir präsentieren Verfahren erster und zweiter Ordnung. Numerische Resultate bestätigen die Konvergenzordnung, so wie Stabilität, Ausbalanciertheit und die asymptotische Erhaltung. Insbesondere beobachten wir bei machen Verfahren, dass die Konvergenzordnung fast unabhängig von der Froude-Zahl ist.

The swelling behaviour of polyelectrolyte networks

Being basic ingredients of numerous daily-life products with significant industrial importance as well as basic building blocks for biomaterials, charged hydrogels continue to pose a series of unanswered challenges for scientists even after decades of practical applications and intensive research efforts. Despite a rather simple internal structure it is mainly the unique combination of short- and long-range forces which render scientific investigations of their characteristic properties to be quite difficult. Hence early on computer simulations were used to link analytical theory and empirical experiments, bridging the gap between the simplifying assumptions of the models and the complexity of real world measurements. Due to the immense numerical effort, even for high performance supercomputers, system sizes and time scales were rather restricted until recently, whereas it only now has become possible to also simulate a network of charged macromolecules. This is the topic of the presented thesis which investigates one of the fundamental and at the same time highly fascinating phenomenon of polymer research: The swelling behaviour of polyelectrolyte networks. For this an extensible simulation package for the research on soft matter systems, ESPResSo for short, was created which puts a particular emphasis on mesoscopic bead-spring-models of complex systems. Highly efficient algorithms and a consistent parallelization reduced the necessary computation time for solving equations of motion even in case of long-ranged electrostatics and large number of particles, allowing to tackle even expensive calculations and applications. Nevertheless, the program has a modular and simple structure, enabling a continuous process of adding new potentials, interactions, degrees of freedom, ensembles, and integrators, while staying easily accessible for newcomers due to a Tcl-script steering level controlling the C-implemented simulation core. Numerous analysis routines provide means to investigate system properties and observables on-the-fly. Even though analytical theories agreed on the modeling of networks in the past years, our numerical MD-simulations show that even in case of simple model systems fundamental theoretical assumptions no longer apply except for a small parameter regime, prohibiting correct predictions of observables. Applying a "microscopic" analysis of the isolated contributions of individual system components, one of the particular strengths of computer simulations, it was then possible to describe the behaviour of charged polymer networks at swelling equilibrium in good solvent and close to the Theta-point by introducing appropriate model modifications. This became possible by enhancing known simple scaling arguments with components deemed crucial in our detailed study, through which a generalized model could be constructed. Herewith an agreement of the final system volume of swollen polyelectrolyte gels with results of computer simulations could be shown successfully over the entire investigated range of parameters, for different network sizes, charge fractions, and interaction strengths. In addition, the "cell under tension" was presented as a self-regulating approach for predicting the amount of swelling based on the used system parameters only. Without the need for measured observables as input, minimizing the free energy alone already allows to determine the the equilibrium behaviour. In poor solvent the shape of the network chains changes considerably, as now their hydrophobicity counteracts the repulsion of like-wise charged monomers and pursues collapsing the polyelectrolytes. Depending on the chosen parameters a fragile balance emerges, giving rise to fascinating geometrical structures such as the so-called pear-necklaces. This behaviour, known from single chain polyelectrolytes under similar environmental conditions and also theoretically predicted, could be detected for the first time for networks as well. An analysis of the total structure factors confirmed first evidences for the existence of such structures found in experimental results.