Olaf Klein

Radiation- and convection-driven transient heat transfer during sublimation growth of silicon carbide single crystals

This article presents transient numerical simulations of heat transfer during sublimation growth of SiC single crystals via physical vapor transport (also called the modified Lely method), investigating the respective influence of radiative and convective contributions and of the semi-transparency of the growing crystal. For radiative heat transfer, we use the net radiation model. Semi-transparency is included via the band approximation model. We briefly describe the corresponding numerical methods. A complete documentation of the material data used is included.

Transient numerical investigation of induction heating during sublimation growth of silicon carbide single crystals

This article presents transient numerical simulations of the temperature evolution during sublimation growth of SiC single crystals via physical vapor transport (also called the modified Lely method) including diffusion and radiation, investigating the influence of induction heating. Using the imposed voltage as input data, the heat sources are computed via an axisymmetric complex-valued magnetic scalar potential that is determined as the solution of an elliptic PDE. The presented results include stationary simulations of magnetic potential distributions and resulting heat sources as well as transient simulations of the temperature evolution during the heating process. We examine the effects of imposed voltage (i.e. heating power), of different coil positions, and of a moving induction coil on the evolution of the global temperature field and on the temperature at the source, at the seed, and at the blind holes. All material data used are either included or referenced.

Optimal Control of Three-Dimensional State-Constrained Induction Heating Problems with Nonlocal Radiation Effects

The paper is concerned with a class of optimal heating problems in semiconductor single crystal growth processes. To model the heating process, time-harmonic Maxwell equations are considered in the system of the state. Due to the high temperatures characterizing crystal growth, it is necessary to include nonlocal radiation boundary conditions and a temperature-dependent heat conductivity in the description of the heat transfer process. The first goal of this paper is to prove existence and uniqueness of the state. The regularity analysis associated with the time-harmonic Maxwell equations is also studied. In the second part of the paper, existence and uniqueness of the solution of the corresponding linearized equation are shown. With this result at hand, the differentiability of the control-to-state operator is derived. Finally, based on the theoretical results, first order necessary optimality conditions for an associated optimal control problem are established.

Modeling and simulation of sublimation growth of SiC bulk single crystals

We present a transient mathematical model for the sublimation growth of silicon carbide (SiC) single crystals by the physical vapor transport (PVT) method. The model of the gas phase consists of balance equations for mass, momentum, and energy, as well as reaction-diffusion equations. Due to physical and chemical reactions, the gas phase is encompassed by free boundaries. Nonlinear heat transport equations are considered in the various solid components of the growth system. Discontinuous and nonlocal interface conditions are formulated to account for temperature steps between gas and solid as well as for diffuse-gray radiative heat transfer between cavity surfaces. An axisymmetric induction heating model is devised using a magnetic scalar potential. For a nonlinear evolution problem arising from the model, a finite volume scheme is stated, followed by a discrete existence and uniqueness result. We conclude by presenting and analyzing results of transient numerical experiments relevant to the physical growth process.

A transient model for the sublimation growth of silicon carbide single crystals

We present a transient model for the Modified Lely Method for the sublimation growth of SiC single crystals which consists of all conservation laws including reaction–diffusion equations. The model is based on a mixture theory for the gas phase. First numerical results illustrate the influence of the geometrical set-up inside the reactor on the evolution of the temperature distribution.

TRANSIENT CONDUCTIVE–RADIATIVE HEAT TRANSFER: DISCRETE EXISTENCE AND UNIQUENESS FOR A FINITE VOLUME SCHEME

This article presents a finite volume scheme for transient nonlinear heat transport equations coupled by nonlocal interface conditions modeling diffuse-gray radiation between the surfaces of (both open and closed) cavities. The model is considered in three space dimensions; modifications for the axisymmetric case are indicated. Proving a maximum principle as well as existence and uniqueness for roots to a class of discrete nonlinear operators that can be decomposed into a scalar-dependent sufficiently increasing part and a benign rest, we establish a discrete maximum principle for the finite volume scheme, yielding discrete L∞-L∞ a priori bounds as well as a unique discrete solution to the finite volume scheme. We present results of numerical experiments to illustrate the effectiveness of the considered scheme.

Correct voltage distribution for axisymmetric sinusoidal modeling of induction heating with prescribed current, voltage, or power

We consider the problem of determining the voltage in coil rings, which arise as an axisymmetric approximation of a single connected induction coil during modeling of induction heating. Assuming axisymmetric electromagnetic fields with sinusoidal time dependence, we compute the voltages from the condition that the total current must be equal in each ring. Depending on whether total current, total voltage, or total power is prescribed, different linear systems of complex equations represent the ring voltages. In two sets of numerical simulations, varying the number of coil rings, we compare results using the correct voltage distribution to the corresponding results using the simple homogeneous voltage distribution.

Existence and approximation of solutions to an anisotropic phase field system for the kinetics of phase transitions

This paper is concerned with a phase field system of Penrose–Fife type for a non-conserved order parameter with a kinetic relaxation coefficient depending on the gradient of the order parameter. This system can be used to model the anisotropic solidification of liquids. A time-discrete scheme for an initial-boundary value problem to this system is presented. By proving the convergence of this scheme, the existence of a solution to the problem is shown.

Optimal Control of Sublimation Growth of SiC Crystals

The project aims at providing numerical tools to control and optimize sublimation growth of SiC bulk single crystals via the Modified Lely Method. It is in cooperation with the experimental group of Dr. Dietmar Siche at the Institute of Crystal Growth in Berlin. In the course of the project the Modified Lely Method is mathematically modeled and numerically simulated. We present a transient model which for the gas phase consists of balance equations for mass, momentum and energy, and reaction-diffusion equations. The model for the solid components takes into account heat transfer via conduction inside the solid materials and via radiation between solid surfaces of cavities. Results of transient numerical simulations of the temperature evolution inside the growth apparatus are depicted, illustrating the paramount influence of radiation at growth temperature.

Transient Numerical Simulation of Sublimation Growth of SiC Single Crystals

This article presents a brief description of a transient model for the Modified Lely Method of sublimation growth of silicon carbide single crystals. For the gas phase the model consists of balance equations for mass, momentum and energy, including reaction-diffusion equations. The model for the solid components takes into account heat transfer via conduction inside the solid materials and via radiation between solid surfaces of cavities. Results of transient numerical simulations of the temperature evolution inside the growth apparatus are depicted, illustrating the paramount influence of radiation at growth temperature.