Skip to main content
SpringerLink
Log in

Modeling of the effect of the presence of a free surface on transport structures and mixing during the dissolution process of silicon into germanium melt

  • Published:
Journal of Thermal Analysis and Calorimetry Aims and scope Submit manuscript

Abstract

In this article, the behavior of the dissolution process of silicon (Si) in molten germanium (Ge) was mathematically modeled and examined numerically. The transport phenomena during this process were modeled using the axisymmetric model (2D) and the equations of the model were solved numerically using the COMSOL multiphysics package. The numerical simulations were carried out exclusively to explain the experimental observations (carried out previously) on the effect of the presence of a free surface on the transport and the mixture of the solute and the shape of the interface of dissolution. The dissolution experimental work used a configuration in which the sample (source Si) was located at the bottom to mimic for instance the process in the melt replenishment Czochralski growth system. For the samples processed in the dissolution experiments, the dissolved heights of silicon were measured. This measurement gives the quantity of silicon dissolved in the experimental times and must be directly linked to the quantity of silicon transported in the melt. Measurement of the silicon composition profiles in the samples was carried out (in the experimental work previously carried out) using the energy dispersive X-ray spectrometer technique. The present numerical results confirm and complement the experimental observations and show that the effect indicates a tendency to more mixing and the presence of several complex convective melt flow regimes leading to rapid chaotic mixing with the presence of a free surface on the melt. In addition, the numerical and the experimental results reveal that it is necessary to take into account the geometry of the crystal growth system when the source Si material is located at the bottom. Indeed, the dissolution of silicon from the bottom of the melt in the presence of a free surface will occur much faster. This however may lead to instability and crystal growth with nonuniform composition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Abbreviations

\(C\) :

Concentration of solute (Si liquid) in the melt (mass%)

\(C^{*}\) :

Dimensionless concentration

\(c_{{\rm p}}\) :

Specific heat (J kg−1 K−1)

\(d\) :

Diameter (m)

\(D\) :

Mass diffusivity of the solute in the melt (m2 s−1)

\(D_{{\rm s}}\) :

Mass diffusivity of solute in the Si source “sample” (Si solid) (m2 s−1)

\(h\) :

Height (m)

\(g\) :

Gravitational acceleration (m s−2)

\(G_{{\rm C}}\) :

Radial gradient of concentration in the melt (mass% m−1)

\(G_{{\rm T}}\) :

Radial gradient of temperature in the melt (K m−1)

\(k\) :

Thermal conductivity (W m−1 K−1)

\({\mathbf{n}},{\mathbf{t}}\) :

Unit normal and tangent vectors

\(p\) :

Pressure (Pa)

\(p^{*}\) :

Dimensionless pressure

\(q,{\mathbf{q}}\) :

Heat flux and heat flux vector (W m−2)

\(R\) :

Diameter (length) of the free surface (m)

R D :

Dissolution rate (m s−1)

\(r,z\) :

Radial and axial coordinates (m)

\(r^{*} ,z^{*}\) :

Dimensionless radial and axial coordinates

\(S\) :

Surface (m2)

\(t\) :

Time (s)

\(t^{*}\) :

Dimensionless time

\(T\) :

Temperature (K)

\(T^{*}\) :

Dimensionless temperature

\(u,v\) :

Velocity components in r and z directions (m s−1)

\(u^{*} ,v^{*}\) :

Dimensionless velocity components in radial and axial directions

\({\mathbf{u}}\) :

Velocity vector (m s−1)

\({\mathbf{u}}^{*}\) :

Dimensionless velocity vector

\(V\) :

Volume (m3)

\(\alpha\) :

Thermal diffusivity (= k/ρcp) (m2 s−1)

\(\beta_{{\rm T}}\) :

Thermal expansion coefficient (K−1)

\(\beta_{{\rm C}}\) :

Solutal expansion coefficient ((mass% Si)−1)

\(\gamma\) :

Temperature coefficient (N m−1 K−1)

\(\mu\) :

Dynamic viscosity (Pa s)

\(\nu\) :

Kinematic viscosity (= μ/ρ ) (m2 s−1)

\(\rho\) :

Density (kg m−3)

\(\sigma\) :

Surface tension (N m−1)

Bd:

Dynamic Bond number

Ma:

Marangoni number

Gr:

Grashof number

GrC :

Solutal Grashof number

Pr:

Prandtl number

Ra:

Rayleigh number

RaC :

Solutal Rayleigh number

Re:

Thermocapillary Reynolds number (=Ma/Pr)

Sc:

Schmidt number

0:

Reference state, zero vector

1:

Symbol referring to the liquid material (liquid phase)

2:

Symbol referring to solid materials (solid phases)

3D:

Three dimensional model

max:

Maximal value

min:

Minimal value

mean:

Average value

melt:

Melt

eq:

Equilibrium

i:

Symbol referring to refer to material phases (solids and liquid)

ini:

Initial value

int:

Interface melt-vacuum (free surface)

s:

Symbol that refers to the solid taken into account among solid phases

sat:

Saturation

Si:

Source sample (Si seed) “Si solid”

\(\Delta T\) :

Temperature variation (or temperature scale) (K)

\(\Delta C\) :

Concentration variation (or concentration scale) (mass%)

\({\mathbf{0}}\) :

Zero vector

\({\mathbf{\nabla }}\) :

Del “nabla” operator (gradient)

References

  1. Adam AY, Oumer AN, Najafi G, Ishak M, Firdaus M, Aklilu TB. State of the art on flow and heat transfer performance of compact fin-and-tube heat exchangers. J Therm Anal Calorim. 2020;139:2739–68. https://doi.org/10.1007/s10973-019-08971-6.

    Article  CAS  Google Scholar 

  2. Rajendran DR, Ganapathy Sundaram E, Jawahar P, Sivakumar V, Mahian O, Bellos E. Review on influencing parameters in the performance of concentrated solar power collector based on materials, heat transfer fluids and design. J Therm Anal Calorim. 2020;140:33–51. https://doi.org/10.1007/s10973-019-08759-8.

    Article  CAS  Google Scholar 

  3. Jegadheeswaran S, Sundaramahalingam A, Pohekar SD. High-conductivity nanomaterials for enhancing thermal performance of latent heat thermal energy storage systems. J Therm Anal Calorim. 2019;138:1137–66. https://doi.org/10.1007/s10973-019-08297-3.

    Article  CAS  Google Scholar 

  4. Tahmasbi M, Siavashi M, Abbasi Akhlaghi M. Mixed convection enhancement by using optimized porous media and nanofluid in a cavity with two rotating cylinders. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09604-z.

    Article  Google Scholar 

  5. Giwa SO, Sharifpur M, Ahmadi MH, Meyer JP. A review of magnetic field influence on natural convection heat transfer performance of nanofluids in square cavities. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09832-3.

    Article  Google Scholar 

  6. Menni Y, Azzi A, Chamkha A. Enhancement of convective heat transfer in smooth air channels with wall-mounted obstacles in the flow path. J Therm Anal Calorim. 2019;135:1951–76. https://doi.org/10.1007/s10973-018-7268-x.

    Article  CAS  Google Scholar 

  7. Deymi-Dashtebayaz M, Akhoundi M, Ebrahimi-Moghadam A, Arabkoohsar A, Moghadam AJ, Farzaneh-Gord M. Thermo-hydraulic analysis and optimization of CuO/water nanofluid inside helically dimpled heat exchangers. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09398-0.

    Article  Google Scholar 

  8. Noorbakhsh M, Zaboli M, Mousavi Ajarostaghi SS. Numerical evaluation of the effect of using twisted tapes as turbulator with various geometries in both sides of a double-pipe heat exchanger. J Therm Anal Calorim. 2020;140:1341–53. https://doi.org/10.1007/s10973-019-08509-w.

    Article  CAS  Google Scholar 

  9. Raj CR, Suresh S, Bhavsar RR, Singh VK. Recent developments in thermo-physical property enhancement and applications of solid solid phase change materials. J Therm Anal Calorim. 2020;139:3023–49. https://doi.org/10.1007/s10973-019-08703-w.

    Article  CAS  Google Scholar 

  10. Turkyilmazoglu M. Cooling of particulate solids and fluid in a moving bed heat exchanger. J Heat Trans ASME. 2019;141(11):114501–5. https://doi.org/10.1115/1.4044590.

    Article  CAS  Google Scholar 

  11. Turkyilmazoglu M. Suspension of dust particles over a stretchable rotating disk and two-phase heat transfer. Int J Multiph Flow. 2020;127:103260. https://doi.org/10.1016/j.ijmultiphaseflow.2020.103260.

    Article  CAS  Google Scholar 

  12. Bhatti MM, Khalique CM, Bég TA, Bég OA, Kadir A. Numerical study of slip and radiative effects on magnetic Fe3O4-water-based nanofluid flow from a nonlinear stretching sheet in porous media with Soret and Dufour diffusion. Mod Phys Lett B. 2019;34(2):2050026. https://doi.org/10.1142/S0217984920500268.

    Article  CAS  Google Scholar 

  13. Waqas H, Khan SU, Bhatti MM, Imran M. Significance of bioconvection in chemical reactive flow of magnetized Carreau-Yasuda nanofluid with thermal radiation and second-order slip. J Therm Anal Calorim. 2020;140:1293–306. https://doi.org/10.1007/s10973-020-09462-9.

    Article  CAS  Google Scholar 

  14. Shafee A, Sheikholeslami M, Jafaryar M, Selimefendigil F, Bhatti MM, Babazadeh H. Numerical modeling of turbulent behavior of nanomaterial exergy loss and flow through a circular channel. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09568-0.

    Article  Google Scholar 

  15. Turkyilmazoglu M. Single phase nanofluids in fluid mechanics and their hydrodynamic linear stability analysis. Comput Methods Prog Biomed. 2020;187:105171. https://doi.org/10.1016/j.cmpb.2019.105171.

    Article  Google Scholar 

  16. Kasper E, Paul DJ. Silicon quantum integrated circuits: silicon-germanium heterostructure devices: basics and realisations. 1st ed. Berlin: Springer; 2005.

    Google Scholar 

  17. Singh R, Oprysko MM, Harame D. Silicon Germanium: Technology, Modeling, and Design. 1st ed. Piscataway (NJ): Wiley-IEEE Press; 2004.

    Google Scholar 

  18. Dost S, Lent B. Single crystal growth of semiconductors from metallic solutions. 1st ed. Amsterdam: Elsevier; 2007.

    Google Scholar 

  19. Armour N, Yildiz M, Yildiz E, Dost S. Liquid phase diffusion growth of SiGe single crystals under magnetic fields. ECS Trans. 2008;16(10):135–46. https://doi.org/10.1149/1.2986760.

    Article  CAS  Google Scholar 

  20. Dhanaraj G, Byrappa K, Prasad V, Dudley M, editors. Springer handbook of crystal growth. 1st ed. Berlin: Springer; 2010.

    Google Scholar 

  21. Suzuki T, Nakajima K, Kusunoki T, Katoh T. Multicomponent zone melting growth of ternary InGaAs bulk crystal. J Electron Mater. 1996;25(3):357–61. https://doi.org/10.1007/BF02666602.

    Article  CAS  Google Scholar 

  22. Kodama S, Furumura Y, Kinoshita K, Kato H, Yoda S. Single crystalline bulk growth of In0.3Ga0.7As on GaAs seed using the multicomponent zone melting method. J Cryst Growth. 2000;208:165–70. https://doi.org/10.1016/S0022-0248(99)00413-3.

    Article  CAS  Google Scholar 

  23. Nishijima Y, Nakajima K, Otsubo K, Ishikawa H. InGaAs single crystal with a uniform composition in the growth direction grown on an InGaAs seed using the multicomponent zone growth method. J Cryst Growth. 2000;208:171–8. https://doi.org/10.1016/S0022-0248(99)00407-8.

    Article  CAS  Google Scholar 

  24. Yildiz MA. Combined experimental and modeling study for the growth of SixGe1-x single crystals by liquid phase diffusion (LPD). Doctoral dissertation, University of Victoria, Canada; 2005.

  25. Leverton WF. Floating crucible technique for growing uniformly doped crystals. J Appl Phys. 1958;29:1241–4. https://doi.org/10.1063/1.1723411.

    Article  CAS  Google Scholar 

  26. Nishijima Y, Nakajima K, Otsubo K, Ishikawa H. InGaAs single crystal with a uniform composition in the growth direction grown on an InGaAs seed using the multicomponent zone growth method. J Cryst Growth. 2000;208(4):171–8. https://doi.org/10.1016/S0022-0248(99)00407-8.

    Article  CAS  Google Scholar 

  27. Nakajima K, Kusunoki T. Constant temperature LEC growth of InGaAs ternary bulk crystals using the double crucible method. J Cryst Growth. 1996;169(2):217–22. https://doi.org/10.1016/S0022-0248(96)00379-X.

    Article  CAS  Google Scholar 

  28. Kusunoki T, Takenaka C, Nakajima K. Growth of ternary In0.14Ga0.86As bulk crystal with uniform composition at constant temperature through GaAs supply. J Cryst Growth. 1991;115:723–7. https://doi.org/10.1016/0022-0248(91)90834-R.

    Article  CAS  Google Scholar 

  29. Tanaka A, Watanabe A, Kimura M, Sukegawa T. The solute-feeding Czochralski method for homogeneous GaInSb bulk alloy pulling. J Cryst Growth. 1994;135:269–72. https://doi.org/10.1016/0022-0248(94)90750-1.

    Article  CAS  Google Scholar 

  30. Prasad V, Zhang H, Anselmo AP. Transport phenomena in Czochralski crystal growth processes. Adv Heat Transf. 1997;30:313–86. https://doi.org/10.1016/S0065-2717(08)70254-5.

    Article  CAS  Google Scholar 

  31. Armour N, Dost S. Seed production and melt replenishment for the Czochralski growth of silicon germanium. Acta Phys Pol, A. 2013;124:198–212.

    Article  CAS  Google Scholar 

  32. Yildiz M, Dost S, Lent B. Growth of bulk SiGe single crystals by liquid phase diffusion. J Cryst Growth. 2005;280:151–60. https://doi.org/10.1016/j.jcrysgro.2005.03.030.

    Article  CAS  Google Scholar 

  33. Mechighel F, Armour N, Dost S, Kadja M. Axisymmetric and 3-D numerical simulations of the effects of a static magnetic field on dissolution of silicon into germanium. CMES Comput Model Eng. 2014;97(1):53–80. https://doi.org/10.3970/cmes.2014.097.053.

    Article  Google Scholar 

  34. Sheremet MA, Pop I. Marangoni natural convection in a cubical cavity filled with a nanofluid. J Therm Anal Calorim. 2019;135:357–69. https://doi.org/10.1007/s10973-018-7069-2.

    Article  CAS  Google Scholar 

  35. Armour N, Dost S, Lent B. Effect of free surface and gravity on silicon dissolution in germanium melt. J Cryst Growth. 2007;299:227–33. https://doi.org/10.1016/j.jcrysgro.2006.11.240.

    Article  CAS  Google Scholar 

  36. Armour N. Transport phenomena in liquid phase diffusion growth of silicon germanium. Doctoral dissertation, University of Victoria, Canada; 2012.

  37. Armour N, Dost S. Effect of an applied static magnetic field on silicon dissolution into a germanium melt. J Cryst Growth. 2009;311:780–2. https://doi.org/10.1016/j.jcrysgro.2008.09.156.

    Article  CAS  Google Scholar 

  38. Yildiz M, Dost S. A continuum model for the liquid phase diffusion growth of bulk SiGe single crystals. Int J Eng Sci. 2005;43:1059–80. https://doi.org/10.1016/j.ijengsci.2005.06.001.

    Article  CAS  Google Scholar 

  39. Mechighel F, Armour N, Dost S. Modeling of combined natural/thermocapillary convection flows in germanium melt at high temperatures. TWMS J Apl Eng Math. 2018;8(1a):193–207.

    Google Scholar 

  40. Mechighel F, Armour N, Dost S, Kadja M. Mathematical modeling of the dissolution process of silicon into germanium melt. TWMS J Appl Eng Math. 2011;1(2):200–22.

    Google Scholar 

  41. COMSOL Multiphysics user’s guide. Comsol. 2008. https://www.comsol.com.

  42. Ercenk E. The effect of clay on foaming and mechanical properties of glass foam insulating material. J Therm Anal Calorim. 2017;127:137–46. https://doi.org/10.1007/s10973-016-5582-8.

    Article  CAS  Google Scholar 

  43. Qader IN, Kok M, Cirak ZD. The effects of substituting Sn for Ni on the thermal and some other characteristics of NiTiSn shape memory alloys. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09758-w.

    Article  Google Scholar 

  44. Kuhlmann HC. Thermocapillary convection in models of crystal Growth. 1st ed. Berlin: Springer; 1999.

    Google Scholar 

  45. Kanchana C, Siddheshwar PG, Zhao Y. Regulation of heat transfer in Rayleigh-Bénard convection in Newtonian liquids and Newtonian nanoliquids using gravity, boundary temperature and rotational modulations. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09325-3.

    Article  Google Scholar 

Download references

Acknowledgements

The financial support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canada Research Chairs (CRC) Program is gratefully acknowledged.

Author information

Authors and Affiliations

  1. LR3MI Laboratory, Department of Mechanical Engineering, Faculty of Engineering Sciences, Annaba University, BP. 12, 23000, Annaba, Algeria

    Farid Mechighel

  2. Crystal Growth Laboratory (CGL), University of Victoria, Victoria, BC, V8W 3P6, Canada

    Neil Armour & Sadik Dost

  3. SPCTS Laboratory, University of Limoges, Limoges, France

    Farid Mechighel

Authors
  1. Farid Mechighel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Neil Armour
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sadik Dost
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Farid Mechighel or Sadik Dost.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mechighel, F., Armour, N. & Dost, S. Modeling of the effect of the presence of a free surface on transport structures and mixing during the dissolution process of silicon into germanium melt. J Therm Anal Calorim 146, 61–91 (2021). https://doi.org/10.1007/s10973-020-09957-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10973-020-09957-5

Keywords

Search

Navigation