Modeling and simulation of sublimation growth of SiC bulk single crystals

Abstract

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.

Second-order sufficient optimality conditions for a semilinear optimal control problem with nonlocal radiation interface conditions

“…Due to the high temperatures and the low pressure, the powder sublimates and crystallizes at a single-crystalline seed located at the cooled top of the cavity, such that the desired single crystal grows into the reaction chamber. See [6] for more details.…”

Existence for the stationary MHD-equations coupled to heat transfer with nonlocal radiation effects

“…At the boundary of the transparent cavity Σ, the conductive heat flux has a discontinuity (14) equal to the difference between the heat radiation outgoing from the surface, denoted by R, and the heat radiation incoming at the surface, denoted by J. We follow the approach of [11], standard in crystal growth, to model the quantities R and J. The main assumption is that all the opaque materials involved are diffuse grey, so that their emission properties do not depend on the wavelength of the radiation.…”

“…We will address the problem of finding fields v, H, B, E, D, j and scalars p, θ that satisfy (2), (3), (12), (13), (5), (17), (23), (6), (7), (8), (9), (10), (11), (18), and (19), as Problem (P ).…”

“…The requirement (10) is a consequence of charge conservation in the absence of free charges. Finally, (11) are the constitutive relations characterizing linear media. The use of this type of model for the electromagnetic part of the problem is justified in the reference [1].…”

Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary p-summable right-hand side

“…Models of radiative heat transfer have recently been studied from the point of view of applied mathematics in different publications: e.g. [16], [5], [4] for modeling and numerics, [11] for control theory, [7] for analysis.…”