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The numerous technical applications in electronic and optoelectronic devices, such as lasers, diodes, and sensors, demand high-quality silicon carbide (SiC) bulk single crystal for industrial applications. We consider an SiC crystal growth process by physical vapor transport (PVT), called modified Lely method. We deal with a model for the micro- and macroscales of the sublimation processes within the growth apparatus. The macroscopic model is based on the heat equation with heat sources due to induction heating and nonlocal interface conditions, representing the heat transfer by radiation. The microscopic model is based on the quantum interatomic potential and is computed with molecular dynamics. We study the temperature evolution in the apparatus and reflect the growth behavior of the microscopic model. We present results of some numerical simulations of the micro- and macromodels of our growth apparatus.

The motivation for this study comes from the technical
demand to simulate a crystal growth apparatus for SiC single crystals. The
single crystals are used as a high-valued and expensive material for
optoelectronics and electronics (cf. [

The silicon carbide (SiC) bulk single crystals are
produced by a growth process through physical vapor transport (PVT), called
modified Lely method. The modeling for the thermal processes within the growth
apparatus is done in [

In the following, we discuss the macroscopic model, which is based on continuum equations for the heat-flux.

The underlying equations of the model are given as follows.

(a) In this work, we assume that the temperature
evolution inside the gas region

(b) The temperature evolution inside the region of
solid materials

The equations hold in the domains of the respective
materials and are coupled by interface conditions, for example, requiring the
continuity for the temperature and for the normal components of the heat flux
on the interfaces between opaque solid materials. On the boundary of the gas
domain, that is, on the interface between the solid material and the gas
domain, we consider the interface condition

The density of the heat source induced by the
induction heating is determined by solving Maxwell’s equations. We deal with
these equations under the simplifying assumption of an axisymmetric geometry,
axisymmetric electromagnetic fields, and a sinusoidal time dependence of the
involved electromagnetic quantities, following [

In this paper, we focus on the discretization and
material properties, which are important for realistic simulations. Our
underlying software tool WIAS-HiTNIHS (cf. [

In the next section, we describe the used discretization.

For the discretization of the heat equation (diffusion
equation), we apply the implicit Euler method in time and the finite volume
method for the space discretization (cf.
[

More details of the discretization and of dealing with
the interface conditions are presented in [

In the next section, the properties of the materials in the crystal growth apparatus are described.

For the technical realization of the apparatus, we
implement the axisymmetric geometry given in [

The growth apparatus’ dimensions:

Within the following specific material functions and
parameters for the processes, the thermal conductivity

(i) For the gas phase (argon), we have

(ii) For graphite felt insulation, we
have

(iii) For the graphite, we have

(iv) For the SiC crystal, we have

(v) For the SiC powder, we have

The functions are programmed in our flexible software package WIAS-HiTNIHS.

In the next section, we discuss the microscopic model.

Often simple coupling via the parameters (e.g., target-temperature and growth velocity of the bulk) is enough for the problem.

Here we propose a new idea of coupling the model equations together, on the one hand the heat equations and on the other hand the kinetic equations for molecules.

For a first idea, we deal with abstract operators, which include the heat- and the kinetics equations.

Using our two standard codes of the macro- and
micromodels, we could implement a coupled model, by a so-called iterative
operator-splitting method. Such a proposed method couples the two physical
processes of the thermal situation in the growth apparatus and their important
geometrical differences at the deposition layer with the kinetic molecular
model. The benefits are a numerical algorithm, that exchanged the underlying
operators of the thermal situation and the kinetic molecular situation, which
are computed by each software code independently and coupled via an iterative
solver step; see a detailed coupling analysis in [

In the following algorithm, an iteration method is
proposed, with fixed splitting discretization step-size

Due to the underlying multiscale problem of kinetics
and heat processes, we have to solve fine time scales of kinetic equations and
coarse time scales for heat equations. On a time interval

In the following, we give an overview to the accuracy of the method, which is given in the convergence and the rate of the convergence.

Let us consider the abstract
Cauchy problem in a Banach space

The proof is given in [

In the next subsection, we present the methods for the microscopic model.

The density-functional tight-binding (DFTB) method is
employed as the quantum interatomic potential in our molecular dynamics (MD)
simulations, using atomic and diatomic parameters obtained from density
functional theory; see [

For time propagation we employed a velocity Verlet
integrator with a time step of 1.209 fs (50 atomic units) and used a
Nose-Hoover chain thermostat to generate a canonical ensemble for target
temperature

Regarding the atomistic structure of the employed
surface model systems, we have chosen the C-face of the same square SiC(000-1)
slab unit cell as in our previous study, [

Optimized geometry of the C-face of the (000-1) SiC surface as initial starting point for QM/MD simulations. Blue spheres correspond to silicon atoms, purple spheres correspond to carbon atoms, and white spheres correspond to hydrogen atoms terminating the slab model in bulk direction. The model is the unit cell used in periodic boundary calculations with infinite surface extension.

During MD simulations, the movements of hydrogen
terminating atoms were frozen. Using such an approach, we have effectively
introduced a steep temperature gradient from the deepest bulk-side SiC layer to
the atoms lying above on the surface. The slab model was then annealed at

In the microscopic model, we can derive the growth
rate

In the next section, we present results of our numerical experiments.

We present in the following our macro- and microscopic simulations, where the microscopic simulations take into account the target temperature of the macroscopic model.

For the numerical results, we apply the parameter
functions in Section

Transient
results for the temperature differences

The further computations are based on the stationary
case, dealing with (

Computations on
different grids for the errors analysis with absolute differences (cf. (

Grid | Grid point | Grid point | |||
---|---|---|---|---|---|

Level | Number of nodes | Solution | Absolute difference | Solution | Absolute difference |

0 | 1532 | 2408.11 | 2813.29 | ||

1 | 23017 | 2409.78 | 1.67 | 2812.78 | 1.01 |

2 | 91290 | 2410.35 | 0.57 | 2811.79 | 0.49 |

3 | 364225 | 2410.46 | 0.11 | 2811.60 | 0.19 |

The result of the refinement indicates the reduction of the absolute difference as it is demanded for the convergence of the discretization method. The method is stabilized in the presented refinement by reducing the differences.

In Figure

Temperature field for the apparatus simulated for the stationary case with

The total time of the three condensation simulations
was 24.02 picoseconds. This is admittedly a time
too short for the study of crystal growth, which would ideally require
annealing simulations on the order of several 100 nanoseconds, but this study
is focusing on the initial stages of SiC aggregation and tries to identify key
features in the condensation process. As such, this is at present rather a
preliminary study exploring the applicability of QM/MD simulations for SiC
crystal growth. We have first concentrated on the polar C-surface of SiC (0001)
since it has a maximum of dangling bonds with highest reactivity. The Si-face
and other nonpolar surfaces are much less reactive; see [

Simulation of
the addition of 10 SiC atoms on the C-face of the (000-1) SiC surface from
Figure

Once attached, the Si atoms on the surface prove to be
highly mobile, as their bond radius is larger than the case of carbon, and the
binding energies are lower [

We have presented a model for the heat transport inside a technical apparatus for crystal growth of SiC single crystals. We introduce the heat equation and the radiation of the apparatus and the coupled situation of the different materials. The equations are discretized by the finite volume method and the complex material functions are embedded in this method. Transient and stationary results are presented leading to some information about the processes within the technical apparatus. We present numerical results for the stationary case to support the accuracy of our solutions. We also presented atomistic quantum chemical molecular dynamics (QM/MD) simulations based on the density-functional tight-binding (DFTB) method for initial reactions of gaseous SiC on the polar C-face of SiC(000-1). In our future work, we concentrate on further implementations and numerical methods for a crystal growth model and use kinetic data obtained from more accurate microscopic model simulations in the simulation of the heat transport. Once longer and a larger number of trajectories are obtained in our microsimulations, it will be possible to deduct an accurate QM/MD-based estimate for the bulk growth, in dependence on the temperature to our macrosimulations. This data will then enter the iterative solution of the heat and kinetics equations of the coupled macroscopic and microscopic models.