Calibration of a Melt Flow Model for Silicon Crystal Growth with the Floating Zone Method
The numerical modelling of the melt flow in Si crystal growth plays an important role for improving the resistivity distribution of crystals grown in industrial processes. However, recent series of experiments have shown that the existing numerical model—a finite volume solver with incompressible la...
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MDPI AG
2025-07-01
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| Series: | Crystals |
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| Online Access: | https://www.mdpi.com/2073-4352/15/7/667 |
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| author | Kirils Surovovs Stanislavs Luka Strozevs Maksims Surovovs Robert Menzel Gundars Ratnieks Janis Virbulis |
| author_facet | Kirils Surovovs Stanislavs Luka Strozevs Maksims Surovovs Robert Menzel Gundars Ratnieks Janis Virbulis |
| author_sort | Kirils Surovovs |
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| description | The numerical modelling of the melt flow in Si crystal growth plays an important role for improving the resistivity distribution of crystals grown in industrial processes. However, recent series of experiments have shown that the existing numerical model—a finite volume solver with incompressible laminar approximation of the melt flow—is not always accurate enough to describe the experimental results for 4″ crystals. To improve the simulation results, material properties have been revised. For some of them, such as the Marangoni or thermal expansion coefficients, the literature suggests different values varying by more than a factor of two. Therefore, simulations using different combinations of parameters were run to perform parameter calibration. The study demonstrated that the description of induced heat on the open melting front needs to be modified to obtain the shape of phase boundaries that provides the best agreement to the experiment. It was concluded that new values should be assigned to several material properties in the model, most importantly the Marangoni coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>=</mo><mrow><mo>−</mo><mn>1.2</mn><mspace width="0.166667em"></mspace><mi>·</mi><mspace width="0.166667em"></mspace><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow><mspace width="0.166667em"></mspace><mstyle scriptlevel="0" displaystyle="true"><mfrac><mi mathvariant="normal">N</mi><mrow><mi mathvariant="normal">m</mi><mspace width="0.166667em"></mspace><mi>·</mi><mspace width="0.166667em"></mspace><mi mathvariant="normal">K</mi></mrow></mfrac></mstyle></mrow></semantics></math></inline-formula>, and that an appropriate turbulence model may help to describe the dopant transport more precisely. |
| format | Article |
| id | doaj-art-636d7389666e450e990a337e59e5df3d |
| institution | DOAJ |
| issn | 2073-4352 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Crystals |
| spelling | doaj-art-636d7389666e450e990a337e59e5df3d2025-08-20T03:07:57ZengMDPI AGCrystals2073-43522025-07-0115766710.3390/cryst15070667Calibration of a Melt Flow Model for Silicon Crystal Growth with the Floating Zone MethodKirils Surovovs0Stanislavs Luka Strozevs1Maksims Surovovs2Robert Menzel3Gundars Ratnieks4Janis Virbulis5Institute of Numerical Modelling, University of Latvia, Jelgavas 3, LV-1004 Riga, LatviaInstitute of Numerical Modelling, University of Latvia, Jelgavas 3, LV-1004 Riga, LatviaInstitute of Numerical Modelling, University of Latvia, Jelgavas 3, LV-1004 Riga, LatviaLeibniz-Institut für Kristallzüchtung, Max-Born-Straße 2, 12489 Berlin, GermanySiltronic AG, Einsteinstraße 172, 81677 München, GermanyInstitute of Numerical Modelling, University of Latvia, Jelgavas 3, LV-1004 Riga, LatviaThe numerical modelling of the melt flow in Si crystal growth plays an important role for improving the resistivity distribution of crystals grown in industrial processes. However, recent series of experiments have shown that the existing numerical model—a finite volume solver with incompressible laminar approximation of the melt flow—is not always accurate enough to describe the experimental results for 4″ crystals. To improve the simulation results, material properties have been revised. For some of them, such as the Marangoni or thermal expansion coefficients, the literature suggests different values varying by more than a factor of two. Therefore, simulations using different combinations of parameters were run to perform parameter calibration. The study demonstrated that the description of induced heat on the open melting front needs to be modified to obtain the shape of phase boundaries that provides the best agreement to the experiment. It was concluded that new values should be assigned to several material properties in the model, most importantly the Marangoni coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>=</mo><mrow><mo>−</mo><mn>1.2</mn><mspace width="0.166667em"></mspace><mi>·</mi><mspace width="0.166667em"></mspace><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow><mspace width="0.166667em"></mspace><mstyle scriptlevel="0" displaystyle="true"><mfrac><mi mathvariant="normal">N</mi><mrow><mi mathvariant="normal">m</mi><mspace width="0.166667em"></mspace><mi>·</mi><mspace width="0.166667em"></mspace><mi mathvariant="normal">K</mi></mrow></mfrac></mstyle></mrow></semantics></math></inline-formula>, and that an appropriate turbulence model may help to describe the dopant transport more precisely.https://www.mdpi.com/2073-4352/15/7/667silicon single crystalsfloating zonenumerical modellinghydrodynamics |
| spellingShingle | Kirils Surovovs Stanislavs Luka Strozevs Maksims Surovovs Robert Menzel Gundars Ratnieks Janis Virbulis Calibration of a Melt Flow Model for Silicon Crystal Growth with the Floating Zone Method Crystals silicon single crystals floating zone numerical modelling hydrodynamics |
| title | Calibration of a Melt Flow Model for Silicon Crystal Growth with the Floating Zone Method |
| title_full | Calibration of a Melt Flow Model for Silicon Crystal Growth with the Floating Zone Method |
| title_fullStr | Calibration of a Melt Flow Model for Silicon Crystal Growth with the Floating Zone Method |
| title_full_unstemmed | Calibration of a Melt Flow Model for Silicon Crystal Growth with the Floating Zone Method |
| title_short | Calibration of a Melt Flow Model for Silicon Crystal Growth with the Floating Zone Method |
| title_sort | calibration of a melt flow model for silicon crystal growth with the floating zone method |
| topic | silicon single crystals floating zone numerical modelling hydrodynamics |
| url | https://www.mdpi.com/2073-4352/15/7/667 |
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