Numerical Simulation of Species Segregation and 2D Distribution in the Floating Zone Silicon Crystals

The distribution of dopants and impurities in silicon grown with the floating zone method determines the electrical resistivity and other important properties of the crystals. A crucial process that defines the transport of these species is the segregation at the crystallization interface. To investigate the influence of the melt flow on the effective segregation coefficient as well as on the global species transport and the resulting distribution in the grown crystal, we developed a new coupled numerical model. Our simulation results include the shape of phase boundaries, melt flow velocity and temperature, species distribution in the melt and, finally, the radial and axial distributions in the grown crystal. We concluded that the effective segregation coefficient is not constant during the growth process but rather increases for larger melt diameters due to less intensive melt mixing

Mathematical modeling of Czochralski type growth processes for semiconductor bulk single crystals

This paper deals with the mathematical modeling and simulation of crystal growth processes by the so-called Czochralski method and related methods, which are important industrial processes to grow large bulk single crystals of semiconductor materials such as, e.,g., gallium arsenide (GaAs) or silicon (Si) from the melt. In particular, we investigate a recently developed technology in which traveling magnetic fields are applied in order to control the behavior of the turbulent melt flow. Since numerous different physical effects like electromagnetic fields, turbulent melt flows, high temperatures, heat transfer via radiation, etc., play an important role in the process, the corresponding mathematical model leads to an extremely difficult system of initial-boundary value problems for nonlinearly coupled partial differential equations ..

Mathematical modeling of Czochralski type growth processes for semiconductor bulk single crystals

This paper deals with the mathematical modeling and simulation of crystal growth processes by the so-called Czochralski method and related methods, which are important industrial processes to grow large bulk single crystals of semiconductor materials such as, e.\,g., gallium arsenide (GaAs) or silicon (Si) from the melt. In particular, we investigate a recently developed technology in which traveling magnetic fields are applied in order to control the behavior of the turbulent melt flow. Since numerous different physical effects like electromagnetic fields, turbulent melt flows, high temperatures, heat transfer via radiation, etc., play an important role in the process, the corresponding mathematical model leads to an extremely difficult system of initial-boundary value problems for nonlinearly coupled partial differential equations. In this paper, we describe a mathematical model that is under use for the simulation of real-life growth scenarios, and we give an overview of mathematical results and numerical simulations that have been obtained for it in recent years

Two- and three-dimensional transient melt-flow simulation in vapour-pressure-controlled Czochralski crystal growth

Flow and thermal properties associated with semiconductor melt flow in an axisymmetric crucible container are studied numerically. Axisymmetric and three-dimensional computational solutions are obtained using a standard-Galerkin, finite-element solver. The crucible and crystal are optionally rotated, and the influence of gravity (through buoyancy) is accounted for via a Boussinesq approximation in the controlling Navier-Stokes equations. The results indicate a strong dependence of the flow on both rotation and buoyancy. Results for axisymmetric flows, computed in both flat and curved geometries, are presented first, and strongly suggest that rotation of crystal and crucible in the same direction (iso-rotation) is most favourable for producing a desired convexity for the crystal/melt interface. Three-dimensional results are then presented for higher Reynolds numbers, and, in particular, reveal that for iso-rotation under moderate buoyancy, the flow undergoes a switch from a steady, 2D state to an unsteady 3D state, and that the temperature becomes non-trivially advected by the flow beneath the crystal. Further evidence reveals however, that on a time scale more appropriate to the crystal growth process, the (time-averaged) flow has a weaker three-dimensionality, in relation to its axisymmetric mode, and there is only slight distortion to the temperature field beneath the crystal

Numerical study of surface tension driven convection in thermal magnetic fluids

Microgravity conditions pose unique challenges for fluid handling and heat transfer applications. By controlling (curtailing or augmenting) the buoyant and thermocapillary convection, the latter being the dominant convective flow in a microgravity environment, significant advantages can be achieved in space based processing. The control of this surface tension gradient driven flow is sought using a magnetic field, and the effects of these are studied computationally. A two-fluid layer system, with the lower fluid being a non-conducting ferrofluid, is considered under the influence of a horizontal temperature gradient. To capture the deformable interface, a numerical method to solve the Navier???Stokes equations, heat equations, and Maxwell???s equations was developed using a hybrid level set/ volume-of-fluid technique. The convective velocities and heat fluxes were studied under various regimes of the thermal Marangoni number Ma, the external field represented by the magnetic Bond number Bom, and various gravity levels, Fr. Regimes where the convection were either curtailed or augmented were identified. It was found that the surface force due to the step change in the magnetic permeability at the interface could be suitably utilized to control the instability at the interface.published or submitted for publicationis peer reviewe

Large-Scale Numerical Modeling of Melt and Solution Crystal Growth

We present an overview of mathematical models and their large-scale numerical solution for simulating different phenomena and scales in melt and solution crystal growth. Samples of both classical analyses and state-of-the-art computations are presented. It is argued that the fundamental multi-scale nature of crystal growth precludes any one approach for modeling, rather successful crystal growth modeling relies on an artful blend of rigor and practicality

Mathematical modelling of the Czochralski crystal growth process

Includes bibliographical references (leaves 142-149).In this document a mathematical model for the Czochralski crystal growth process is developed. The trend in current research involves developing cumbersome numerical simulations that provide little or no understanding of the underlying physics. We attempt to review previous research methods, mainly devoted to silicon, and develop a novel analytical tool for indium antimonide (lnSb) crystal growth. This process can be subdivided into two categories: solidification and fluid mechanics. Thus far, crystal solidification of the Czochralski process has been described in the literature mainly qualitatively. There has been little work in calculating actual solidification dynamics. Czochralski crystal growth is a very sensitive process, particularly for lnSb, so it is crucial to describe the system as accurately as possible. A novel ID quasi-steady method is proposed for the shape and temperature field of an lnSb crystal, incorporating the effects of the melt. The fluid mechanics of the Czochralski melt have been modelled by numerous researchers,with calculations performed using commercial software. However, a descriptionof the buoyancy and rotation interaction in the melt has not been adequatelyperformed. Many authors have presented flow patterns but none have indicated either: melt conditions preferential for crystal growth or at least a description of a typical melt structure. In this work, a scale analysis is performed that implies an idealized flow structure. An asymptotic model is then derived based on this order of magnitude analysis, resulting in a fast and efficient fluid flow calculation. The asymptotic model is validated against a numerical solution to ensure that the macroscopic features of the flow structure are present. The asymptotic model does not show exact agreement, but does provide an estimate of the melt heat flux that is necessary for the solidification calculation. The asymptotic model is also used to predict macroscopic changes in the melt due to rotation

Stabilized finite element approximation of the incompressible MHD equations

No es frecuente encontrar un campo donde dos ramas principales de la Física estén involucradas. La Magnetohidrodinámica es uno de tales campos debido a que involucra a la Mecánica de Fluidos y al Electromagnetismo. Aun cuando puede parecer que esas dos ramas de la Física tienen poco en común, comparten similitudes en las ecuaciones que gobiernan los fenómenos involucrados en ellas. Las ecuaciones de Navier-Stokes y las ecuaciones de Maxwell, ambas en la raíz de la Magnetohidrodinámica, tienen una condición de divergencia nula y es esta condición de divergencia nula sobre la velocidad del fluido y el campo magnético lo que origina algunos de los problemas numéricos que surgen en la modelación de los fenómenos donde el flujo de fluidos y los campos magnéticos están acoplados.El principal objetivo de este trabajo es desarrollar un algoritmo eficiente para la resolución mediante elementos finitos de las ecuaciones de la Magnetohidrodinámica de fluidos incompresibles.Para lograr esta meta, los conceptos básicos y las características de la Magnetohidrodinámica se presentan en una breve introducción informal.A continuación, se da una revisión completa de las ecuaciones de gobierno de la Magnetohidrodinámica, comenzando con las ecuaciones de Navier-Stokes y las ecuaciones de Maxwell. Se discute la aproximación que da origen a las ecuaciones de la Magnetohidrodinámica y finalmente se presentan las ecuaciones de la Magnetohidrodinámica.Una vez que las ecuaciones de gobierno de la Magnetohidrodinámica han sido definidas, se presentan los esquemas numéricos desarrollados, empezando con la linealización de las ecuaciones originales, la formulación estabilizada y finalmente el esquema numérico propuesto. En esta etapa se presenta una prueba de convergencia.Finalmente, se presentan los ejemplos numéricos desarrollados durante este trabajo.Estos ejemplos pueden dividirse en dos grupos: ejemplos numéricos de comparación y ejemplos de internes tecnológico. Dentro del primer grupo están incluidas simulaciones del flujo de Hartmann y del flujo sobre un escalón. El segundo grupo incluye simulaciones del flujo en una tobera de inyección de colada continua y el proceso Czochralski de crecimiento de cristales.It is not frequent to find a field where two major branches of Physics are involved. Magnetohydrodynamics is one of such fields because it involves Fluid Mechanics and Electromagnetism. Although those two branches of Physics can seem to have little in common, they share similarities in the equations that govern the phenomena involved. The Navier-Stokes equations and the Maxwell equations, both at the root of Magnetohydrodynamics, have a divergence free condition and it is this divergence free condition over the velocity of the fluid and the magnetic field what gives origin to some of the numerical problems that appear when approximating the equations that model the phenomena where fluids flow and magnetic fields are coupled.The main objective of this work is to develop an efficient finite element algorithm for the incompressible Magnetohydrodynamics equations.In order to achieve this goal the basic concepts and characteristics of Magnetohydrodynamics are presented in a brief and informal introduction.Next, a full review of the governing equations of Magnetohydrodynamics is given, staring from the Navier-Stokes equations and the Maxwell equations. The MHD approximation is discussed at this stage and the proper Magnetohydrodynamics equations for incompressible fluid are reviewed.Once the governing equations have been defined, the numerical schemes developed are presented, starting with the linearization of the original equations, the stabilization formulations and finally the numerical scheme proposed. A convergence test is shown at this stage.Finally, the numerical examples performed while this work was developed are presented. These examples can be divided in two groups: numerical benchmarks and numerical examples of technological interest. In the first group, the numerical simulations for the Hartmann flow and the flow over a step are included. The second group includes the simulation of the clogging in a continuous casting nozzle and Czochralski crystal growth process.Postprint (published version

Index to 1984 NASA Tech Briefs, volume 9, numbers 1-4

Short announcements of new technology derived from the R&D activities of NASA are presented. These briefs emphasize information considered likely to be transferrable across industrial, regional, or disciplinary lines and are issued to encourage commercial application. This index for 1984 Tech B Briefs contains abstracts and four indexes: subject, personal author, originating center, and Tech Brief Number. The following areas are covered: electronic components and circuits, electronic systems, physical sciences, materials, life sciences, mechanics, machinery, fabrication technology, and mathematics and information sciences

Microdefects in Czochralski Silicon

Disertační práce se zabývá studiem defektů v monokrystalech Czochralskiho křemíku legovaných bórem. Práce studuje vznik kruhových obrazců vrstevných chyb pozorovaných na povrchu křemíkových desek po oxidaci. Hlavním cílem práce je objasnit mechanismy vzniku pozorovaného rozložení vrstevných chyb na studovaných deskách a vyvinout metody pro řízení tohoto jevu. Na základě experimentálních analýz a rozborů obecných mechanismů vzniku defektů jsou objasňovány vazby mezi vznikem defektů různého typu. Tyto jsou pak diskutovány v souvislosti s parametry krystalu i procesu jeho růstu. Takto sestavený model je využit pro vývoj procesu růstu krystalů, kterým je potlačen nadměrný vznik defektů ve studovaných deskách. Za účelem studia defektů jsou zaváděny a vyvíjeny nové analytické metody. Disertační práce byla vytvořena za podpory ON Semiconductor Czech Republic, Rožnov pod Radhoštěm.The doctoral thesis deals with analyses of defects in single crystals of Czochralski silicon doped with boron. Mechanisms of formation of circular patterns of oxidation induced stacking faults are studied. The main goal of the work is to explain the mechanisms of formation of the observed defect patterns and to develop methods for control of this phenomenon. Mechanisms of defect formation in silicon are analyzed and the material is experimentally studied in order to explain relations between formation of defects of various kinds and to link these processes to parameters of the crystal and its growth. A qualitative model capturing all these relations is built and utilized to develop an optimized crystal growth process for suppression of excessive formation of the oxidation induced stacking faults. Novel methods are developed and implemented to support effective analyses of crystal defects. This doctoral thesis was written with the support of ON Semiconductor Czech Republic, Rožnov pod Radhoštěm.