On the modelling of semi-insulating GaAs including surface tension and bulk stresses

Necessary heat treatment of single crystal semi-insulating Gallium Arsenide (GaAs), which is deployed in micro- and opto- electronic devices, generate undesirable liquid precipitates in the solid phase. The appearance of precipitates is influenced by surface tension at the liquid/solid interface and deviatoric stresses in the solid. The central quantity for the description of the various aspects of phase transitions is the chemical potential, which can be additively decomposed into a chemical and a mechanical part. In particular the calculation of the mechanical part of the chemical potential is of crucial importance. We determine the chemical potential in the framework of the St. Venant--Kirchhoff law which gives an appropriate stress/strain relation for many solids in the small strain regime. We establish criteria, which allow the correct replacement of the St. Venant--Kirchhoff law by the simpler Hooke law. The main objectives of this study are: (i) We develop a thermo-mechanical model that describes diffusion and interface motion, which both are strongly influenced by surface tension effects and deviatoric stresses. (ii) We give an overview and outlook on problems that can be posed and solved within the framework of the model. (iii) We calculate non-standard phase diagrams, i.e. those that take into account surface tension and non-deviatoric stresses, for GaAs above 786 extdegreeC, and we compare the results with classical phase diagrams without these phenomena.</o:p

On the Becker/Döring theory of nucleation of liquid droplets in solids

Nucleation of liquid precipitates in semi-insulating GaAs is accompanied by deviatoric stresses resulting from the liquid/solid misfit. A competition of surface tension and stress deviators at the interface determines the nucleation barrier. The evolution of liquid precipitates in semi-insulating GaAs is due to diffusional processes in the vicinity of the droplet. The diffusion flux results from a competition of chemical and mechanical driving forces. The size distribution of the precipitates is determined by a Becker/Döring system. The study of its properties in the presence of deviatoric stresses is the subject of this study. The main tasks of this study are: (i) We propose a new Becker/Döring model that takes thermomechanical coupling into account. (ii) We compare the current model with already existing models from the literature. Irrespective of the incorporation of mechanical stresses, the various models differ by different environments where the evolution of precipitates takes place. (iii) We determine the structure of equilibrium solutions according to the Becker/Döring model, and we compare these solutions with those that result from equilibrium thermodynamics. </o:p

Towards thermodynamic modeling of nucleation and growth of droplets in crystals

Stress assisted diffusion in single crystal Gallium Arsenide (GaAs) leads to the formation and growth of unwanted liquid arsenic droplets in a solid matrix. This process happens during the heat treatment of single crystal GaAs, which is needed for its application in opto-electronic devices, and it is of crucial importance to pose and answer the question if the appearance of droplets can be avoided. To this end we start a thermodynamic simulation of this process. Special emphasis is given to the influence of mechanical effects on chemistry, diffusion and interface motion in GaAs

Diffusion in the vicinity of an evolving spherical arsenic droplet

We study the diffusion problem of liquid droplets in single crystal semi-insulating Gallium Arsenide (GaAs). This problem is posed by an industrial application, where the droplets, also called precipitates, appear during a necessary heat treatment of GaAs wafer. The subsequent dissolution of the droplets is mandatory, in order to use the wafer after the heat treatment as a substrate material for micro- and opto- electronic devices. In this study we consider a single droplet in a solid matrix, which is in contact with an arsenic gas, so that the arsenic can cross the solid/gas interface. The model equations have been derived by the authors. They consist of a nonlinear diffusion equation with diffusion controlled and kinetic boundary conditions, respectively, at the liquid/solid interface. Furthermore we study at the solid/gas interface alternatively zero flux and Dirichlet conditions. Surface tension at the liquid/solid interface and deviatoric stresses in the solid are taken into account. The latter appear due to different densities of liquid and solid GaAs. There is a large influence of these effects on diffusion, interface motion and phase diagrams, which are used to identify regions, where coexistence of liquid and solid phases is possible. In order to study the evolution of the droplet, and in particular possibilities to enforce its dissolution, we solve several initial and boundary value problems for the diffusion system.</o:p

Anwendung der von Kármán'schen Plattentheorie und der Hertz'schen Pressung für die Spannungsanalyse zur Biegung von GaAs-Wafern im modifizierten Doppelringtest

In der vorliegenden Arbeit werden verschiedene Aspekte der Modellierung eines speziellen Biegetests von Gallium-Arsenid-Wafern diskutiert. Wafer sind dünne Kreisscheiben aus einkristallinem Material, aus denen in der Mikro- und Optoelektronikindustrie eingesetzt wird. Bei den hierzu notwendigen Herstellungsprozessen treten mechanische Beanspruchungen auf, die zum Bruch führen können. Der Biegetest dient dem Wafer-Hersteller zur Ermittlung der Bruchfestigkeit und zur Homogenitätsanalyse. In der Arbeit werden unter anderem experimentell gegebene Last-Durchbiegungskurven unter Verwendung der von Kármán'schen Plattentheorie numerisch mit hoher Genauigkeit verifiziert. Das Kontaktproblem, welches bei der Lastaufbringung mit einer Druckkugel entsteht, wird durch Hertz'sche Pressung modelliert. In diesem Zusammenhang werden insbesondere anisotrope Effekte untersucht. Es werden numerisch berechnete isotrope Ersatzkonstanten zur Lösung des elastischen Biegeproblems angegeben, die sich nicht aus einer klassischen Mittelungstheorie, z.B. nach Voigt oder Reuss, herleiten lassen. Die von Kármán'sche Theorie wird zuerst auf klassischem Wege durch Formulierung bestimmter Kleinheitsannahmen aus dem 3D-Problem hergeleitet. Diese Annahmen werden anschließend durch formal asymptotische Entwicklung der Verschiebungen gerechtfertigt. Die numerische Umsetzung des Biegeproblems wird mit Hilfe von reduzierten Hermite-Dreieckselementen realisiert. Das resultierende nichtlineare Finite-Element-Schema wird vollständig beschrieben. Dabei wird bei der Formulierung der Randbedingungen auf dem frei drehbaren Stützrand dem Babu ka-Plattenparadox Rechnung getragen, so dass hier trotz Verwendung geradliniger Plattenelemente zur Approximation eines krummlinigen Stützrandes Konvergenz erreicht wird. Die numerischen Untersuchungen zeigen, dass in der Umgebung der Kontaktfläche, die unter der Druckkugel entsteht, die Voraussetzungen zur Gültigkeit der Plattentheorie verletzt sind. Deshalb werden am Schluss der Arbeit analytische Lösungen für den isotropen linearen Fall nach der 2D-Theorie gemäß Kirchhoff und der 3D-Theorie für dicke Platten unter Verwendung der Ansätze von Papkowitsch und Neuber verglichen. Insbesondere wird die Größe des Fehlers der berechneten Maximalspannungen in Abhängigkeit vom Kontaktradius bei Verwendung einer 2D-Theorie untersucht

A mathematical model for impulse resistance welding

We present a mathematical model of impulse resistance welding. It accounts for electrical, thermal and mechanical effects, which are nonlinearly coupled by the balance laws, constitutive equations and boundary conditions. The electrical effects of the weld machine are incorporated by a discrete oscillator circuit which is coupled to the field equations by a boundary condition. We prove the existence of weak solutions for a slightly simplified model which however still covers most of its essential features, e.g. the quadratic Joule heat term and a quadratic term due to non-elastic energy dissipation. We discuss the numerical implementation in a 2D setting, present some numerical results and conclude with some remarks on future research

On phase change of a vapor bubble in liquid water

We consider a bubble of vapor and inert gas surrounded by the corresponding liquid phase. We study the behavior of the bubble due to phase change, i.e. condensation and evaporation, at the interface. Special attention is given to the effects of surface tension and heat production on the bubble dynamics as well as the propagation of acoustic elastic waves by including slight compressibility of the liquid phase. Separately we study the influence of the three phenomena heat conduction, elastic waves, and phase transition on the evolution of the bubble. The objective is to derive relations including the mass, momentum, and energy transfer between the phases. We find ordinary differential equations, in the cases of heat transfer and the emission of acoustic waves partial differential equations, that describe the bubble dynamics. From numerical evidence we deduce that the effect of phase transition and heat transfer on the behavior of the radius of the bubble is negligible. It turns out that the elastic waves in the liquid are of greatest importance to the dynamics of the bubble radius. The phase transition has a strong influence on the evolution of the temperature, in particular at the interface. Furthermore the phase transition leads to a drastic change of the water content in the bubble, so that a rebounding bubble is only possible, if it contains in addition an inert gas. In a forthcoming paper the equations derived are sought in order to close equations for multi-phase mixture balance laws for dispersed bubbles in liquids involving phase change. Also the model is used to make comparisons with experimental data on the oscillation of a laser induced bubble. For this case it was necessary to include the effect of an inert gas in the thermodynamic modeling of the phase transition

On phase change of a vapor bubble in liquid water

We consider a bubble of vapor and inert gas surrounded by the corresponding liquid phase. We study the behavior of the bubble due to phase change, i.e. condensation and evaporation, at the interface. Special attention is given to the effects of surface tension and heat production on the bubble dynamics as well as the propagation of acoustic elastic waves by including slight compressibility of the liquid phase. Separately we study the influence of the three phenomena heat conduction, elastic waves, and phase transition on the evolution of the bubble. The objective is to derive relations including the mass, momentum, and energy transfer between the phases. We find ordinary differential equations, in the cases of heat transfer and the emission of acoustic waves partial differential equations, that describe the bubble dynamics. From numerical evidence we deduce that the effect of phase transition and heat transfer on the behavior of the radius of the bubble is negligible. It turns out that the elastic waves in the liquid are of greatest importance to the dynamics of the bubble radius. The phase transition has a strong influence on the evolution of the temperature, in particular at the interface. Furthermore the phase transition leads to a drastic change of the water content in the bubble, so that a rebounding bubble is only possible, if it contains in addition an inert gas. In a forthcoming paper the equations derived are sought in order to close equations for multi-phase mixture balance laws for dispersed bubbles in liquids involving phase change. Also the model is used to make comparisons with experimental data on the oscillation of a laser induced bubble. For this case it was necessary to include the effect of an inert gas in the thermodynamic modeling of the phase transitio

On unwanted nucleation phenomena at the wall of a VGF chamber

This is preliminary study on a phenomenon that happens during crystal growth of GaAs in a vertical gradient freeze (VGF) device. Here unwanted polycrystals nucleate at the chamber wall and move into the interior of the crystal. This happens within an undercooled region in the vicinity of the triple point, where the liquid-solid interface meets the chamber wall. The size and shape of that region is modelled by the Gibbs-Thomson law, which will be rederived in this paper. Hereafter we identify the crucial parameter, whose proper adjustment may minimize the undercooled region. Finally we give a simple estimate to calculate and evaluate the energy barrier for homogeneous and heterogeneous nucleation of a solid nucleus in the undercooled mel

Stress analysis and bending tests for GaAs wafer

Wafer made from single crystal Gallium Arsenide (GaAs) are used as substrate materials in micro- and opto- electronic devices. During the various processes of manufacturing, the wafer are subjected to mechanical loads which may lead to fracture. The characterization of the fracture toughness of the wafer needs bending tests and a theoretical calculation of various stress distributions within the wafer. In this study we show that the nonlinear von K'arm'an theory may serve as an appropriate tool to calculate the stress distributions as functions of the external load, while the Kirchhoff theory has turned out to be completely inappropriate. Our main focus is devoted to (i) calculation of the contact area between the load sphere and the wafer, (ii) study of the influence of the anisotropic character of the material, (iii) study of the important geometric nonlinearity. Finally we compare the calculated and theoretical load-flexure relations in order to demonstrate the high accuracy of the von K'arm'an theory and its Finite Element implementation