Transient numerical simulation of sublimation growth of SiC bulk single crystals. Modeling, finite volume method, results
- Philip, Peter
2010 Mathematics Subject Classification
- 35K60 47J35 65M99 78M25 80A17 80A20 80M25
2008 Physics and Astronomy Classification Scheme
- 02.60.Cb, 02.60.Lj, 44.05.+e, 44.10.+i, 44.40.+a, 81.10.Bk, 84.32.Hh
- numerical simulation, physical vapor transport, SiC single crystal, finite volume method, diffuse radiation, electromagneticheating, evolution equations, initial-boundary value problem, discrete solution
This work treats transient numerical simulation of growth of silicon carbide (SiC) bulk single crystals by physical vapor transport (also called the modified Lely method). A transient mathematical model of the growth process is presented. Subsequently, the finite volume method for the discretization of evolution equations, which constitutes the basis for the numerical simulations presented in this work, is studied mathematically, proving the existence of discrete solutions. All material data used for numerical simulations in this work are collected in the appendix. Starting with a description of the physical growth procedure, problems arising during the growth process are discussed as well as techniques that are used for process control. It is explained why numerical simulation is an important tool for control, and the advantages of a transient approach are considered. Within the presented transient model, continuous mixture theory is used to obtain balance equations for energy, mass, and momentum inside the gas phase. In particular, reaction-diffusion equations are deduced. Heat conduction is treated inside solid materials. Heat transport by radiation is modeled via the net radiation method for diffuse-gray radiation to allow for radiative heat transfer between the surfaces of cavities. The model includes the semi-transparency of the single crystal via a band approximation. Induction heating is modeled by an axisymmetric complex-valued magnetic scalar potential that is determined as the solution of an elliptic problem. The resulting heat source distribution is calculated from the magnetic potential. The heat sources are updated continuously during the solution of the transient problem for the temperature evolution to allow for changes in the electrical conductivity depending on temperature and for changes due to a moving induction coil. The finite volume method is treated in a rigorous mathematical framework. It allows the discretization of parabolic, hyperbolic, and elliptic partial differential equations, as they arise from the mathematical model of the growth process, including nonlocal contributions due to radiative heat transfer. The general abstract setting consists of a system of nonlinear evolution equations in arbitrary finite space dimension, each evolution equation living on a different polytope domain. In general, each evolution equation has diffusive and convective contributions as well as source and sink terms. Each contribution is permitted to depend on the solution. Discontinuities of the solution are allowed at domain interfaces. Interface conditions in terms of the solution and its flux are considered. Moreover, nonlocal interface conditions are considered. Outer boundary conditions include Dirichlet conditions, flux conditions, emission conditions, and nonlocal conditions. Time discretization is performed by an implicit Euler scheme, where an explicit discretization is allowed in certain dependencies such that the temperature-dependent emissivities can be taken from the previous time step. As usual, the space discretization is performed by integrating the evolution equations over control volumes and then using quadrature formulas. As an axisymmetric setting and cylindrical coordinates are used in the simulations, a treatment of change of variables is included in the abstract considerations. For the case that the evolution equations constitute nonlinear heat equations, still allowing nonlinear diffusion, convection, and source and sink terms, as well as nonlocal interface and boundary conditions as they arise from modeling radiative heat transfer, discrete L∞ - L1 a priori estimates are established for the system resulting from the finite volume discretization. A fixed point argument is then used to prove the existence and uniqueness of discrete solutions. The presented numerical simulations are conducted in an axisymmetric setting. They constitute transient investigations of control parameters affecting the temperature evolution during the heating of the growth apparatus. A cylindrically symmetric finite volume scheme provides the discretization for both the transient nonlinear heat problem and the stationary magnetic potential problem. For different heating powers and different vertical coil positions, the temperature evolution is monitored at the surface of the crystal and at the surface of the source powder as well as at the top and at the bottom of the growth apparatus. It is studied how the temperature difference between source and seed, which is highly relevant to the growth process, is related to the measurable temperature difference between bottom and top. Results concerning the time lack between the heating of the surface of the source powder and the heating of its interior are considered. Finally, the global evolution of temperature and heat sources is investigated.