WIAS Preprints

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Modeling and simulation of strained quantum wells in semiconductorlasers
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2000) Bandelow, Uwe; Kaiser, Hans-Christoph; Koprucki, Thomas; Rehberg, Joachim
A model allowing for efficiently obtaining band structure information on semiconductor Quantum Well structures will be demonstrated which is based on matrix-valued kp-Schrödinger operators. Effects such as confinement, band mixing, spin-orbit interaction and strain can be treated consistently. The impact of prominent Coulomb effects can be calculated by including the Hartree interaction via the Poisson equation and the bandgap renormalization via exchange-correlation potentials, resulting in generalized (matrix-valued) Schrödinger-Poisson systems. Band structure information enters via densities and the optical response function into comprehensive simulations of Multi Quantum Well lasers. These device simulations yield valuable information on device characteristics, including effects of carrier transport, waveguiding and heating and can be used for optimization.
Transient numerical study of termperature gradients during sublimation growth of SiC: Dependence on apparatus design
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2005) Geiser, Jürgen; Klein, Olaf; Philip, Peter
Using a transient mathematical heat transfer model including heat conduction, radiation, and radio frequency (RF) induction heating, we numerically investigate the time evolution of temperature gradients in axisymmetric growth apparatus during sublimation growth of silicon carbide (SiC) bulk single crystals by physical vapor transport (PVT) (modified Lely method). Temperature gradients on the growing crystal's surface can cause defects. Here, the evolution of these gradients is studied numerically during the heating process, varying the apparatus design, namely the amount of the source powder charge as well as the size of the upper blind hole used for cooling of the seed. Our results show that a smaller upper blind hole can reduce the temperature gradients on the surface of the seed crystal without reducing the surface temperature itself.
Stochastic spectral and Fourier-wavelet methods for vector Gaussian random fields
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2005) Kurbanmuradov, Orazgeldi; Sabelfeld, Karl
Randomized Spectral Models (RSM) and Randomized Fourier-Wavelet Models (FWM) for simulation of homogeneous Gaussian random fields based on spectral representations and plane wave decomposition of random fields are developed. Extensions of FWM to vector random processes are constructed. Convergence of the constructed Fourier-Wavelet models (in the sense of finite-dimensional distributions) under some general conditions on the spectral tensor is given. A comparative analysis of RSM and FWM is made by calculating Eulerian and Lagrangian statistical characteristics of a 3D isotropic incompressible random field through an ensemble and space averaging
Metastability : a potential theoretic approach
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2005) Bovier, Anton
Metastability is an ubiquitous phenomenon of the dynamical behaviour of complex systems. In this talk, I describe recent attempts towards a model-independent approach to metastability in the context of reversible Markov processes. I will present an outline of a general theory, based on careful use of potential theoretic ideas and indicate a number of concrete examples where this theory was used very successfully. I will also indicate some challenges for future work.
Stationary solutions to an energy model for semiconductor devices where the equations are defined on different domains
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Glitzky, Annegret; Hünlich, Rolf
We discuss a stationary energy model from semiconductor modelling. We accept the more realistic assumption that the continuity equations for electrons and holes have to be considered only in a subdomain $Omega_0$ of the domain of definition $Omega$ of the energy balance equation and of the Poisson equation. Here $Omega_0$ corresponds to the region of semiconducting material, $OmegasetminusOmega_0$ represents passive layers. Metals serving as contacts are modelled by Dirichlet boundary conditions. We prove a local existence and uniqueness result for the two-dimensional stationary energy model. For this purpose we derive a $W^1,p$-regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem.
Modeling of drift-diffusion systems
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Stephan, Holger
We derive drift-diffusion systems describing transport processes starting from free energy and equilibrium solutions by a unique method. We include several statistics, heterostructures and cross diffusion. The resulting systems of nonlinear partial differential equations conserve mass and positivity, and have a Lyapunov function (free energy). Using the inverse Hessian as mobility, non-degenerate diffusivity matrices turn out to be diagonal, or
A shape calculus analysis for tracking type formulations for tracking type formulations in electrical impedance tomography
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Eppler, Karsten
In the paper [17], the authors investigated the identification of an obstacle or void of perfectly conducting material in a two-dimensional domain by measurements of voltage and currents at the boundary. In particular, the reformulation of the given nonlinear identification problem was considered as a shape optimization problem using the Kohn and Vogelius criterion. The compactness of the complete shape Hessian at the optimal inclusion was proven, verifying strictly the ill-posedness of the identification problem. The aim of the paper is to present a similar analysis for the related least square tracking formulations. It turns out that the two-norm-discrepancy is of the same principal nature as for the Kohn and Vogelius objective. As a byproduct, the necessary first order optimality condition are shown to be satisfied if and only if the data are perfectly matching. Finally, we comment on possible consequences of the two-norm-discrepancy for the regularization issue.
Chaotic bound state of localized structures in the complex Ginzburg-Landau equation
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Turaev, Dmitry; Vladimirov, Andrei; Zelik, Sergey
A new type of stable dynamic bound state of dissipative localized structures is found. It is characterized by chaotic oscillations of distance between the localized structures, their phase difference, and the center of mass velocity.
Multi-pulse evolution and space-time chaos in dissipative systems
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Zelik, Sergey; Mielke, Alexander
We study semilinear parabolic systems on the full space Rn that admit a family of exponentially decaying pulse-like steady states obtained via translations. The multi-pulse solutions under consideration look like the sum of infinitely many such pulses which are well separated. We prove a global center-manifold reduction theorem for the temporal evolution of such multi-pulse solutions and show that the dynamics of these solutions can be described by an infinite systems of ODEs for the positions of the pulses. As an application of the developed theory, we verify the existence of Sinai-Bunimovich space-time chaos in 1D space-time periodically forced Swift-Hohenberg equation.
The von Mises model vor one-dimensional elastoplastic beams and Prandtl-Ishlinskii hysteresis operators
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Krejčí, Pavel; Sprekels, Jürgen
In this paper, the one-dimensional equation for the transversal vibrations of an elastoplastic beam is derived from a general three-dimensional system. The plastic behavior is modeled using the classical three-dimensional von Mises plasticity model. It turns out that this single-yield model without hardening leads after a dimensional reduction to a multi-yield one-dimensional hysteresis model with kinematic hardening, given by a hysteresis operator of Prandtl-Ishlinskii type whose density function can be determined explicitly. This result indicates that the use of Prandtl-Ishlinskii hysteresis operators in the modeling of elastoplasticity is not just a questionable phenomenological approach, but in fact quite natural. In addition to the derivation of the model, it is shown that the resulting partial differential equation with hysteresis can be transformed into an equivalent system for which the existence and uniqueness of a strong solution is proved. The proof employs techniques from the mathematical theory of hysteresis operators.
Continuum descriptions for the dynamics in discrete lattices : derivation and justification
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Giannoulis, Johannes; Herrmann, Michael; Mielke, Alexander
The passage from microscopic systems to macroscopic ones is studied by starting from spatially discrete lattice systems and deriving several continuum limits. The lattice system is an infinite-dimensional Hamiltonian system displaying a variety of different dynamical behavior. Depending on the initial conditions one sees quite different behavior like macroscopic elastic deformations associated with acoustic waves or like propagation of optical pulses. We show how on a formal level different macroscopic systems can be derived such as the Korteweg-de Vries equation, the nonlinear Schroedinger equation, Whitham's modulation equation, the three-wave interaction model, or the energy transport equation using the Wigner measure. We also address the question how the microscopic Hamiltonian and the Lagrangian structures transfer to similar structures on the macroscopic level. Finally we discuss rigorous analytical convergence results of the microscopic system to the macroscopic one by either weak-convergence methods or by quantitative error bounds.
A thin film model for corotational Jeffreys fluids under strong slip
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Münch, Andreas; Wagner, B.; Rauscher, M.; Blossey, R.
We derive a thin film model for viscoelastic liquids under strong slip which obey the stress tensor dynamics of corotational Jeffreys fluids.
Multiple disorder problems for Wiener and compound Poisson processes with exponential jumps
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Gapeev, Pavel V.
The multiple disorder problem consists of finding a sequence of stopping times which are as close as possible to the (unknown) times of 'disorder' when the distribution of an observed process changes its probability characteristics. We present a formulation and solution of the multiple disorder problem for a Wiener and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial optimal switching problems to the corresponding coupled optimal stopping problems and solving the equivalent coupled free-boundary problems by means of the smooth- and continuous-fit conditions.
Attractors for the semiflow associated with a class of doubly nonlinear parabolic equations
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Schimperna, Giulio; Segatti, Antonio
A doubly nonlinear parabolic equation of the form [alpha](ut)-[delta]u+W'(u)=f, complemented with initial and either Dirichlet or Neumann homogeneous boundary conditions, is addressed. The two nonlinearities are given by the maximal monotone function [alpha] and by the derivative W' of a smooth but possibly nonconvex potential W; f is a given known source. After defining a proper notion of solution and recalling a related existence result, we show that from any initial datum emanates at least one solution which gains further regularity for t>0. Such regularizing solutions contitute a semiflow S for which unqueness is satisfied for strictly positive times and we can study long time behaviour properties,. In particular, we can prove existence of both global and exponential attractors and investigate the structure of [omega]-limits of single trajectories.
A jump-diffusion Libor model and tits robust calibration
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Belomestrny, Denis; Schoenmakers, John G.M.
In this paper we propose a jump-diffusion Libor model with jumps in a high-dimensional space and test a stable non-parametric calibration algorithm which takes into account a given local covariance structure. The algorithm returns smooth and simply structured Lévy densities, and penalizes the deviation from the Libor market model. In practice, the procedure is FFT based, thus fast, easy to implement, and yields good results, particularly in view of the ill-posedness of the underlying inverse problem.
Mathematical modelling of indirect measurements in periodic diffractive optics and scatterometry
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Gross, Hermann; Model, Regine; Bär, Markus; Wurm, Matthias; Bodermann, Bernd; Rathsfeld, Andreas
In this work, we illustrate the benefits and problems of mathematical modelling and effective numerical algorithms to determine the diffraction of light by periodic grating structures. Such models are required for reconstruction of the grating structure from the light diffraction patterns. With decreasing structure dimensions on lithography masks, increasing demands on suitable metrology techniques arise. Methods like scatterometry as a non-imaging indirect optical method offer access to the geometrical parameters of periodic structures including pitch, side-wall angles, line heights, top and bottom widths. The mathematical model for scatterometry is based on the Helmholtz equation derived as a time-harmonic solution of Maxwell's equations. It determines the incident and scattered electric and magnetic fields, which fully specify the light propagation in a periodic two-dimensional grating structure. For numerical simulations of the diffraction patterns, a standard finite element method (FEM) or a generalized finite element method (GFEM) is used for solving the elliptic Helmholtz equation. In a first step, we performed systematic forward calculations for different varying structure parameters to evaluate the applicability and sensitivity of different scatterometric measurement methods ...
External cavity modes in Lang-Kobayashi and traveling wave models
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Radziunas, Mindaugas; Wünsche, Hans-Jürgen; Krauskopf, Bernd; Wolfrum, Matthias
We investigate a semiconductor laser with delayed optical feedback due to an external cavity formed by a regular mirror. We discuss similarities and differences of the well-known Lang--Kobayashi delay differential equation model and the traveling wave partial differential equation model. For comparison we locate the continuous wave states in both models and analyze their stability.
Energy estimates for electro-reaction-diffusion systems with partly fast kinetics
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Glitzky, Annegret
We start from a basic model for the transport of charged species in heterostructures containing the mechanisms diffusion, drift and reactions in the domain and at its boundary. Considering limit cases of partly fast kinetics we derive reduced models. This reduction can be interpreted as some kind of projection scheme for the weak formulation of the basic electro--reaction--diffusion system. We verify assertions concerning invariants and steady states and prove the monotone and exponential decay of the free energy along solutions to the reduced problem and to its fully implicit discrete-time version by means of the results of the basic problem. Moreover we make a comparison of prolongated quantities with the solutions to the basic model.
Maximal convergence theorems for functions of squared modulus holomorphic type and various applications
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Kraus, Christiane
In this paper we extend the theory of maximal convergence introduced by Walsh to functions of squared modulus holomorphic type. We introduce in accordance to the well-known complex maximal convergence number for holomorphic functions a real maximal convergence number for functions of squared modulus holomorphic type and prove several maximal convergence theorems. We achieve that the real maximal convergence number for F is always greater or equal than the complex maximal convergence number for g and equality occurs if L is a closed disk in R^2. Among other various applications of the resulting approximation estimates we show that for functions F of squared holomorphic type which have no zeros in a closed disk B_r the relation limsupntoinftysqrt[n]En(Br,F)=limsupntoinftysqrt[n]En(partialBr,F) is valid, where E_n is the polynomial approximation error.
Slip-controlled thin film dynamics
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Fetzer, Renate; Rauscher, M.; Münch, A.; Wagner, B.; Jacobs, K.
In this study, we present a novel method to assess the slip length and the viscosity of thin films of highly viscous Newtonian liquids. We quantitatively analyse dewetting fronts of low molecular weight polystyrene melts on Octadecyl- (OTS) and Dodecyltrichlorosilane (DTS) polymer brushes. Using a thin film (lubrication) model derived in the limit of large slip lengths, we can extract slip length and viscosity. We study polymer films with thicknesses between 50 nm and 230 nm and various temperatures above the glass transition. We find slip lengths from 100 nm up to 1 $mu$m on OTS and between 300 nm and 10 $mu$m on DTS covered silicon wafers. The slip length decreases with temperature. The obtained values for the viscosity are consistent with independent measurements.

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