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.

Analysis of optimal control problems for the incompressible MHD equations and implementation in a finite element multiphysics code

This thesis deals with the study of optimal control problems for the incompressible Magnetohydrodynamics (MHD) equations. Particular attention to these problems arises from several applications in science and engineering, such as fission nuclear reactors with liquid metal coolant and aluminum casting in metallurgy. In such applications it is of great interest to achieve the control on the fluid state variables through the action of the magnetic Lorentz force. In this thesis we investigate a class of boundary optimal control problems, in which the flow is controlled through the boundary conditions of the magnetic field. Due to their complexity, these problems present various challenges in the definition of an adequate solution approach, both from a theoretical and from a computational point of view. In this thesis we propose a new boundary control approach, based on lifting functions of the boundary conditions, which yields both theoretical and numerical advantages. With the introduction of lifting functions, boundary control problems can be formulated as extended distributed problems. We consider a systematic mathematical formulation of these problems in terms of the minimization of a cost functional constrained by the MHD equations. The existence of a solution to the flow equations and to the optimal control problem are shown. The Lagrange multiplier technique is used to derive an optimality system from which candidate solutions for the control problem can be obtained. In order to achieve the numerical solution of this system, a finite element approximation is considered for the discretization together with an appropriate gradient-type algorithm. A finite element object-oriented library has been developed to obtain a parallel and multigrid computational implementation of the optimality system based on a multiphysics approach. Numerical results of two- and three-dimensional computations show that a possible minimum for the control problem can be computed in a robust and accurate manner.

Functional equations of polylogarithms in motivic cohomology

For an infinite field F, we study the integral relationship between the Bloch group B_2(F) and the higher Chow group CH^2(F,3) by proving some relations corresponding to the functional equations of the dilogarithm. As a second result, the groups involved in Suslin’s exact sequence 0 → Tor^1(F^× ,F^×)∼ → CH^2(F,3) → B_2(F) → 0 are identified with homology groups of the cycle complex Z^2(F,•) computing Bloch’s higher Chow groups. Using these results, we give explicit cycles in motivic cohomology generating the integral motivic cohomology groups of some specific number fields and determine whether a given cycle in the Chow group already lives in one of the other groups of Suslin’s sequence. In principle, this enables us to find a presentation of the codimension two Chow group of an arbitrary number field. Finally, we also prove some relations in the higher Chow groups of codimension three modulo 2-torsion coming from relations in the higher Bloch group B_3(F) modulo 2-torsion. Further, we can prove a series of relations in CH^ 3(Q(zeta_p),5) for a primitive pth root of unity zeta_p.

Frobenius polynomials for Calabi-Yau equations

Sei $\pi:X\rightarrow S$ eine \"uber $\Z$ definierte Familie von Calabi-Yau Varietaten der Dimension drei. Es existiere ein unter dem Gauss-Manin Zusammenhang invarianter Untermodul $M\subset H^3_{DR}(X/S)$ von Rang vier, sodass der Picard-Fuchs Operator $P$ auf $M$ ein sogenannter {\em Calabi-Yau } Operator von Ordnung vier ist. Sei $k$ ein endlicher K\"orper der Charaktetristik $p$, und sei $\pi_0:X_0\rightarrow S_0$ die Reduktion von $\pi$ \uber $k$. F\ur die gew\ohnlichen (ordinary) Fasern $X_{t_0}$ der Familie leiten wir eine explizite Formel zur Berechnung des charakteristischen Polynoms des Frobeniusendomorphismus, des {\em Frobeniuspolynoms}, auf dem korrespondierenden Untermodul $M_{cris}\subset H^3_{cris}(X_{t_0})$ her. Sei nun $f_0(z)$ die Potenzreihenl\osung der Differentialgleichung $Pf=0$ in einer Umgebung der Null. Da eine reziproke Nullstelle des Frobeniuspolynoms in einem Teichm\uller-Punkt $t$ durch $f_0(z)/f_0(z^p)|_{z=t}$ gegeben ist, ist ein entscheidender Schritt in der Berechnung des Frobeniuspolynoms die Konstruktion einer $p-$adischen analytischen Fortsetzung des Quotienten $f_0(z)/f_0(z^p)$ auf den Rand des $p-$adischen Einheitskreises. Kann man die Koeffizienten von $f_0$ mithilfe der konstanten Terme in den Potenzen eines Laurent-Polynoms, dessen Newton-Polyeder den Ursprung als einzigen inneren Gitterpunkt enth\alt, ausdr\ucken,so beweisen wir gewisse Kongruenz-Eigenschaften unter den Koeffizienten von $f_0$. Diese sind entscheidend bei der Konstruktion der analytischen Fortsetzung. Enth\alt die Faser $X_{t_0}$ einen gew\ohnlichen Doppelpunkt, so erwarten wir im Grenz\ubergang, dass das Frobeniuspolynom in zwei Faktoren von Grad eins und einen Faktor von Grad zwei zerf\allt. Der Faktor von Grad zwei ist dabei durch einen Koeffizienten $a_p$ eindeutig bestimmt. Durchl\auft nun $p$ die Menge aller Primzahlen, so erwarten wir aufgrund des Modularit\atssatzes, dass es eine Modulform von Gewicht vier gibt, deren Koeffizienten durch die Koeffizienten $a_p$ gegeben sind. Diese Erwartung hat sich durch unsere umfangreichen Rechnungen best\atigt. Dar\uberhinaus leiten wir weitere Formeln zur Bestimmung des Frobeniuspolynoms her, in welchen auch die nicht-holomorphen L\osungen der Gleichung $Pf=0$ in einer Umgebung der Null eine Rolle spielen.

Partial Discharge Phenomena in Converter and Traction Transformers: Identification and Reliability

After the development of power electronics converters, the number of transformers subjected to non-sinusoidal stresses (including DC) has increased in applications such as HVDC links and traction (electric train power cars). The effects of non-sinusoidal voltages on transformer insulation have been investigated by many researchers, but still now, there are some issues that must be understood. Some of those issues are tackled in this Thesis, studying PD phenomena behavior in Kraft paper, pressboard and mineral oil at different voltage conditions like AC, DC, AC+DC, notched AC and square waveforms. From the point of view of converter transformers, it was found that the combined effect of AC and DC voltages produces higher stresses in the pressboard that those that are present under pure DC voltages. The electrical conductivity of the dielectric systems in DC and AC+DC conditions has demonstrated to be a critical parameter, so, its measurement and analysis was also taken into account during all the experiments. Regarding notched voltages, the RMS reduction caused by notches (depending on firing and overlap angles) seems to increase the PDIV. However, the experimental results show that once PD activity has incepted, the notches increase PD repetition rate and magnitude, producing a higher degradation rate of paper. On the other hand, the reduction of mineral oil stocks, their relatively low flash point as well as environmental issues, are factors that are pushing towards the use of esters as transformer insulating fluids. This PhD Thesis also covers the study of two different esters with the scope to validate their use in traction transformers. Mineral oil was used as benchmark. The complete set of dielectric tests performed in the three fluids, show that esters behave better than mineral oil in practically all the investigated conditions, so, their application in traction transformers is possible and encouraged.

Arbitrary Waveform Multilevel Generator for High Voltage High Frequency Plasma Actuators

This dissertation presents the theory and the conducted activity that lead to the construction of a high voltage high frequency arbitrary waveform voltage generator. The generator has been specifically designed to supply power to a wide range of plasma actuators. The system has been completely designed, manufactured and tested at the Department of Electrical, Electronic and Information Engineering of the University of Bologna. The generator structure is based on the single phase cascaded H-bridge multilevel topology and is comprised of 24 elementary units that are series connected in order to form the typical staircase output voltage waveform of a multilevel converter. The total number of voltage levels that can be produced by the generator is 49. Each level is 600 V making the output peak-to-peak voltage equal to 28.8 kV. The large number of levels provides high resolution with respect to the output voltage having thus the possibility to generate arbitrary waveforms. Maximum frequency of operation is 20 kHz. A study of the relevant literature shows that this is the first time that a cascaded multilevel converter of such dimensions has been constructed. Isolation and control challenges had to be solved for the realization of the system. The biggest problem of the current technology in power supplies for plasma actuators is load matching. Resonant converters are the most used power supplies and are seriously affected by this problem. The manufactured generator completely solves this issue providing consistent voltage output independently of the connected load. This fact is very important when executing tests and during the comparison of the results because all measures should be comparable and not dependent from matching issues. The use of the multilevel converter for power supplying a plasma actuator is a real technological breakthrough that has provided and will continue to provide very significant experimental results.

Design and Fabrication of Bond Wire Micro-Magnetics

This thesis presents a new approach for the design and fabrication of bond wire magnetics for power converter applications by using standard IC gold bonding wires and micro-machined magnetic cores. It shows a systematic design and characterization study for bond wire transformers with toroidal and race-track cores for both PCB and silicon substrates. Measurement results show that the use of ferrite cores increases the secondary self-inductance up to 315 µH with a Q-factor up to 24.5 at 100 kHz. Measurement results on LTCC core report an enhancement of the secondary self-inductance up to 23 µH with a Q-factor up to 10.5 at 1.4 MHz. A resonant DC-DC converter is designed in 0.32 µm BCD6s technology at STMicroelectronics with a depletion nmosfet and a bond wire micro-transformer for EH applications. Measures report that the circuit begins to oscillate from a TEG voltage of 280 mV while starts to convert from an input down to 330 mV to a rectified output of 0.8 V at an input of 400 mV. Bond wire magnetics is a cost-effective approach that enables a flexible design of inductors and transformers with high inductance and high turns ratio. Additionally, it supports the development of magnetics on top of the IC active circuitry for package and wafer level integrations, thus enabling the design of high density power components. This makes possible the evolution of PwrSiP and PwrSoC with reliable highly efficient magnetics.