*3.16. Simulation of PVT Bulk Growth*

To determine the growth conditions present inside the growth cell during the sublimation growth of 3C-SiC, numerical modeling of the temperature field and mass transportrelated phenomena were performed. The basic aim was to first identify the growth conditions existing in the 50 mm apparatus and then to use such data to ensure stable growth conditions to enlarge wafer sizes of 100 mm and greater. The study examined the effect of the appropriate SiC characteristics and the various carbon materials serving as process values in the computational modeling for both the temperature profile and the associated mass transport. In order to implement the thermal field and mass transport effects, computer simulation was performed through COMSOL Multiphysics. The appropriate selection of the physical parameters related to graphite-based components, as well as for the carbon isolation characteristics over 2000 ◦C, appeared to not be an easy task. The main issue concerns the ambiguity, if not a complete absence of reliable data, on the temperature behavior of electrical and thermal conductivity at the growth thermal conditions.

Nonetheless, numerical modeling allows for not only the calculation of the small growth cell but also the simulation of the whole growth reactor. Besides the calculation of temperature fields, mass transport, and supersaturation, simulation with COMSOL Multiphysics provided insight into the behavior of magnetic fields as well as into the formation of hot zones.

Worthy to note, despite the fact that there is no unambiguous data on the behavior of thermal conductivity at higher temperatures for graphite crucibles and insulating components, it is possible to assert using calculations and experimental calibrations that the thermal gradient in the gas phase has no effect, unless for the second order of approximation. As a consequence, it is expected that calculating, for example, supersaturation in front of the growth interface would offer accurate findings useful for the design and optimization of growth cells.

The supersaturation of the SiC<sup>2</sup> gas species plays an important role, influencing the growth-limiting parameter at the seed-growth front. This supersaturation can be calculated using the partial pressure of the gas species at the seed and source. Different approaches for the calculation of the supersaturation can be found in literature, for example, by Lilov [78] or Avrov [79]. Each experiment is similar in that for the calculations, the knowledge of the actual temperatures during the growth process is required. A comparison between the simulation data and the measured temperatures for the 50 mm-CS-PVT growth setup at different heating powers, showing a good agreement, is depicted in Figure 23a. A similar trend could be observed for the 100 mm-growth cell. Additionally, an example of the temperature field present in the growth setup can be seen in Figure 23b.

**Figure 23.** (**a**) Comparisons between simulations using COMSOL Multiphysics and the measured temperatures at the crucible top during growth runs for different heating powers in the 50 mm-CS-PVT setup. (**b**) Typical temperature field for CS-PVT. Some isotherms are indicated with the corresponding temperatures.

Using the temperatures obtained at the seed and source from the simulations, the supersaturations present during the sublimation growth can be calculated using the equation mentioned by Rankl et al. [80]. They found that for the heteroepitaxial growth of 3C-SiC on (0001)-oriented 6H—SiC, a supersaturation as high as s = 0.4 is necessary to achieve a high yield. Based on the development regarding the carbon materials databases, this value was revised to s = 0.24 [32]. In the case of homoepitaxial sublimation growth on seeds already containing the cubic polytype, a supersaturation higher than s = 0.1 was found to be suitable to ensure stable growth. It was also found that for this purpose, a source to seed a distance of 1 mm or smaller is necessary depending on the growth temperature as the supersaturation will decrease with the increasing spacing [31].

A global model for the evaluation of the processes' results in terms of the material growth rate can be obtained from the estimates of the mass transfer rate from the seed to the substrate once the temperature field is evaluated by the chamber simulation, as discussed in the previous section. Assuming that ballistic transport conditions occur for the Si-C molecules, which sublimate at different rates at the two interfaces, approximate estimates of the growth rate for the 3C-SiC in the different positions of the growing substrate (in a fully symmetric configuration) can be obtained from the balance between the atomic species' effective deposition fluxes (jdep), derived from the sources and ruled by the source temperature, and the evaporation flux (jev), ruled by the substrate local temperature. Due to the composition of the SiC vapor pressure, Si-rich conditions are usually assumed (see Avrov et al. [78]) and the growth rate can be estimated by:

$$Gr = \frac{\rho\_{\text{SiC}}}{M\_{\text{SiC}}} \left( j\_{d\text{exp}} - j\_{\text{ev}} \right) Gr \tag{1}$$

using the atomic carbon effective flux only. In Equation (1), *ρSiC* is the SiC density and *MSiC* is the SiC molar mass. The expression for *jdep* and *jev* are given by

$$\frac{j\_{dep} = j\_{dep}(\mathbf{C}) = j\_{dep}(\text{Si}\_2\mathbf{C}) + 2j\_{dep}(\text{Si}\mathbf{C}\_2)}{\sqrt{\frac{2\pi R T\_{\text{Sourc}}}{M\_{\text{Si}\_2\text{C}}} \exp\left(\frac{A\_{\text{Si}\_2\text{C}}}{T\_{\text{Source}}} + B\_{\text{Si}\_2\text{C}}\right)} + 2\sqrt{\frac{2\pi R T\_{\text{Sourc}}}{M\_{\text{Si}\text{C}\_2}} \exp\left(\frac{A\_{\text{Si}\text{C}\_2}}{T\_{\text{Source}}} + B\_{\text{Si}\text{C}\_2}\right)}\tag{2}$$

$$\begin{split} \mathbf{j}\_{ev} &= \mathbf{j}\_{ev}(\mathbf{C}) = j\_{ev}(\mathrm{Si}\_{2}\mathbf{C}) + 2\mathbf{j}\_{ev}(\mathrm{Si}\mathbf{C}\_{2}) \\ &= \sqrt{\frac{2\pi R \mathbf{T}\_{\mathrm{Sub}}}{M\_{\mathrm{Si}\_{2}\mathbb{C}}}} \exp\left(\frac{A\_{\mathrm{Si}\_{2}\mathbb{C}}}{T\_{\mathrm{Sub}}} + B\_{\mathrm{Si}\_{2}\mathbb{C}}\right) + 2\sqrt{\frac{2\pi R \mathbf{T}\_{\mathrm{Sub}}}{M\_{\mathrm{Si}\mathbb{C}\_{2}}}} \exp\left(\frac{A\_{\mathrm{Si}\mathbb{C}\_{2}}}{T\_{\mathrm{Sub}}} + B\_{\mathrm{Si}\mathbb{C}\_{2}}\right) \end{split} \tag{3}$$

where *TSource* and *TSub* are the source and the substrate temperatures; *MSi*2*<sup>C</sup>* and *MSiC*<sup>2</sup> are the molar masses of the *Si2C* and *SiC<sup>2</sup>* molecules in the vapor phase; and *A<sup>x</sup>* and *B<sup>x</sup>* are the experimental parameters that rule the partial pressures of the *X* species in the vapor mixtures at the thermodynamic equilibrium with the solid counterpart. Different calibrations for the partial pressures-related parameters can be found in the literature and by using the one in Reference [78], a quantitative estimation of the growth rate in accordance with the experimental results can be obtained.
