#
Numerical Modelling of the Czochralski Growth of β-Ga_{2}O_{3}

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## Abstract

**:**

## 1. Introduction

## 2. Material Properties

#### 2.1. Properties of β-Ga${}_{2}$O${}_{3}$ Crystal

Exp. [21] | Exp. [22] | Calc. [23] | |

a | 12.23 | 12.23 | 12.14 |

b | 3.04 | 3.04 | 3.14 |

c | 5.80 | 5.81 | 5.85 |

β | 103.7 | 103.85 | 103.7 |

Property | Symbol | Value | Reference |

Density | ρ | $5950\phantom{\rule{3.33333pt}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$ | [24] |

Specific heat | ${c}_{p}\phantom{\rule{3.33333pt}{0ex}}(@\phantom{\rule{3.33333pt}{0ex}}300\phantom{\rule{3.33333pt}{0ex}}\mathrm{K})$ | $560\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}\xb7{\mathrm{kg}}^{-1}\xb7{\mathrm{K}}^{-1}$ | [6] |

${c}_{p}\phantom{\rule{3.33333pt}{0ex}}(@\phantom{\rule{3.33333pt}{0ex}}1500\phantom{\rule{3.33333pt}{0ex}}\mathrm{K})$ | $720\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}\xb7{\mathrm{kg}}^{-1}\xb7{\mathrm{K}}^{-1}$ | [6] | |

Melting point temperature | ${T}_{m}$ | $(2093\pm 20)\phantom{\rule{3.33333pt}{0ex}}\mathrm{K}$ | ${}^{1}$ [7] |

${T}_{m}$ | $2066\phantom{\rule{3.33333pt}{0ex}}\mathrm{K}$ | ${}^{2}$ [25] | |

Latent heat | $\Delta H$ | $4.6\times {10}^{5}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}/\mathrm{kg}$ | as YAG [13] |

Refraction index | $\mathcal{R}$ | 1.9 | [26] |

Absorption coefficient | α | $400\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-1}$ | ${}^{3}$ [6] |

#### 2.1.1. Thermal Conductivity

[6] | [28] | [29] | ||

λ (W·m^{−1}·K^{−1}) | λ (W·m^{−1}·K^{−1}) | λ (W·m^{−1}·K^{−1}) | ||

a | [100] | - | $11\pm 1$ | $11\pm 1$ |

b | [010] | 21 | $27\pm 2$ | $29\pm 2$ |

c | [001] | - | $15\pm 2$ | $21\pm 2$ |

$\left[\overline{2}01\right]$ | - | $13\pm 1$ | - |

#### 2.1.2. Thermal Expansion

#### 2.1.3. Elastic Stiffness Coefficients

#### 2.2. Properties of Ga${}_{2}$O${}_{3}$ Melt

density | ρ | $6000\phantom{\rule{3.33333pt}{0ex}}\mathrm{kg}\xb7{\mathrm{m}}^{-3}$ |

specific heat | ${c}_{p}$ | $800\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}\xb7{\mathrm{kg}}^{-1}\xb7{\mathrm{K}}^{-1}$ |

thermal expansion | β | $1.8\times {10}^{-5}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{K}}^{-1}$ |

Prandtl number | Pr | 4 |

dynamic viscosity | μ | $0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{Pa}\xb7\mathrm{s}$ |

thermal conductivity | λ | 20 $\mathrm{W}/\mathrm{m}\mathrm{K}$ |

## 3. Computation

- CrysMAS and CGsim: different values for radial and vertical directions;
- Ansys-cfx: different values for x-, y-, and z-directions;
- comsol: full matrix.

- CrysMAS: Unstructured triangles for the entire geometry—in regions with convection, a structured rectangular grid is used for solving the Navier–Stokes equation (not applied in this paper).
- CGsim: Every domain can have an unstructured triangle or structured rectangular grid. At interfaces of domains, grids do not need to match, but values are interpolated. Only at the crystal melt must interface grids match. For the view factor calculation, an independent grid definition is used.
- Ansys-cfx: We used a fully structured grid. In all computations presented in this paper, we had a total amount of 1,330,756 nodes. In the crystal, the number of grid elements was 140,676. In order to save computational time, for the Monte Carlo ray tracing, the number of grid points can be reduced. Treating the crystal as transparent, we reduced the number of elements for ray tracing to 19,810 in the crystal.

## 4. Results

Vertical | Horizontal | |||

${\mathit{\lambda}}_{\mathit{z}}$ | ${\mathit{\lambda}}_{\mathit{x}}$ | ${\mathit{\lambda}}_{\mathit{y}}$ | ${\mathit{\lambda}}_{\mathit{r}}$ | |

Case 1 | 2.11 | 1.07 | 1.79 | 1.43 |

Case 2 | 1.23 | 2.61 | 1.63 | 1.87 |

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Unit cell of β-Ga${}_{2}$O${}_{3}$ (

**left**) and standard coordinate system (

**right**). Ga atoms in green (large) and oxygen ones in red (small). The figure is drawn with VESTA [27].

**Figure 4.**Temperature profile for Case 1 along the axis for runs with three different software tools. The curve in magenta is from the Ansys-cfx run (3D calculation). The temperature at the top is higher because we used the boundary conditions from a CrysMAS run without melt convection. The red and blue curves are the results of the runs with CGsim and CrysMAS, respectively.

**Figure 5.**Temperature fields in melt, crystal, and seed for the two cases of different growth directions, as observed by 3D calculations with Ansys-cfx. The temperature profile at the center of the crystal is plotted on the right-hand side. Please note that the temperature scales for melt and crystal are given seperately to ensure a more detailed view. Because the temperature varies slightly along the crystal–melt interface, the minimum temperature in the melt and the maximum one in crystal are not identical but overlap.

**Figure 6.**Temperature profiles in the crystal for different runs (isolines every 10 K): (

**A**) CGsim computation with $\alpha =1\times {10}^{-5}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-1}$ for Case 1; (

**B**) CGsim computation with opaque crystal for Case 1; (

**C**) CGsim computation with $\alpha =400\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-1}$ for Case 1; (

**D**) CGsim computation with $\alpha =400\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-1}$ for Case 2; (

**E**) CFX computation with $\alpha =400\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-1}$ for Case 2; and (

**F**) CFX computation with transparent crystal for Case 2. Please note that thermal conductivity is unique in the horizontal direction for the CGsim calculations, whereas, in the 3D calculation, it depends on the direction.

**Figure 7.**Comparison of interface shapes. The thick line with colours indicates the preset interface for the 3D calculations, and the colours exhibit the temperatures along this line from the computation for Case 1. The dotted line is the isoline of $T=2081\phantom{\rule{3.33333pt}{0ex}}\mathrm{K}$. We used axisymmetric calculations with CGsim to compute the crystal–melt interface. The black line is the interface for the same case as the 3D one. The dotted line in magenta is the interface for the case without melt convection.

**Figure 8.**Von Mises stress for Case 1 (

**left**) and Case 2 (

**right**). The thermal expansion coefficients of [22] have been used. The thin black lines indicate the orginal size of the crystal. The displacements due to the different temperatures are visualized by the surface with the colours of the von Mises stress (deformation magnification factor, (

**left**): 100; (

**right**): 20).

**Figure 9.**Von Mises stress for Case 1 (

**left**) and Case 2 (

**right**), the deformation of the geometry is claryfied by a magnification factor (

**left**): 100; (

**right**): 20. Thermal expansion coefficients used as measured by us. The thin black lines of the shorter undeformed crystal geometry are painted over by the color graphics: the displacements due to the different temperatures are visualized by the surface with the colours of the von Mises stress.

© 2017 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Miller, W.; Böttcher, K.; Galazka, Z.; Schreuer, J. Numerical Modelling of the Czochralski Growth of *β*-Ga_{2}O_{3}. *Crystals* **2017**, *7*, 26.
https://doi.org/10.3390/cryst7010026

**AMA Style**

Miller W, Böttcher K, Galazka Z, Schreuer J. Numerical Modelling of the Czochralski Growth of *β*-Ga_{2}O_{3}. *Crystals*. 2017; 7(1):26.
https://doi.org/10.3390/cryst7010026

**Chicago/Turabian Style**

Miller, Wolfram, Klaus Böttcher, Zbigniew Galazka, and Jürgen Schreuer. 2017. "Numerical Modelling of the Czochralski Growth of *β*-Ga_{2}O_{3}" *Crystals* 7, no. 1: 26.
https://doi.org/10.3390/cryst7010026