Simulation of the in luence of gas low on melt convection and phase boundaries in FZ silicon single crystal growth

Transcription

1 Simulation of the in luence of gas low on melt convection and phase boundaries in FZ silicon single crystal growth Andrejs Sabanskis, Jānis Virbulis University of Latvia, Riga, Latvia E-MRS 2014 Spring Mee ng, Lille, France, The present work is carried out at the University of Latvia and has been supported by the European Regional Development Fund, project contract No 2013/0051/2DP/21110/13/APIA/VIAA/009

2 Contents A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) Influence of gas flow on melt convec on in FZ Si crystal growth 1/16 1 Introduc on 2 2D axisymmetric mathema cal model for turbulent argon flow in FZ puller using OpenFOAM 3 Calcula on example of ICG (Berlin, Dr H Riemann) floa ng zone system with crystal diameter 4 inches 4 Conclusions

3 Si single crystal growth with FZ technique A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) Influence of gas flow on melt convec on in FZ Si crystal growth 2/16 Courtesy of Dr H Riemann (ICG, Berlin) W Zulehner Mater Sci Eng B, 73(1):7-15, 2000

4 Axisymmetric quasi-stationary FZ model A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) Influence of gas flow on melt convec on in FZ Si crystal growth 3/16 Inductor Polycrystal Melt Monocrystal High-frequency electromagne c field (BEM), 2D or 3D Thermal radia on, including reflector (view factor model) Heat transfer in crystals, melt and reflector (FEM) Free melt surface shape (itera ve approach) Solid-liquid interface posi ons Open mel ng front model Model implemented in the program FZone (Ratnieks, 2003) Further development of the model is s ll ongoing

5 Actual mathematical model A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) Influence of gas flow on melt convec on in FZ Si crystal growth 4/16 R F, R C, V pull, zone height 3D inductor shape FZone 2D axis-sym calcula on 2D phase boundaries FV mesh HF current 3D BEM mesh q m 3D FV mesh OpenFOAM gas flow calcula on Temperature 3D HF EM field calcula on 3D temperature in melt (BEM) OpenFOAM 3D HD U, T, C calcula on U, T qgas q HF f HF p HF 3D free surface calcula on U, T C, ρ

6 Gas low: Introduction A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) Influence of gas flow on melt convec on in FZ Si crystal growth 5/16 Density: Ideal gas law Large temperature and density varia ons ρ p M R T ρ = p M RT density pressure molar mass universal gas constant absolute temperature Viscosity: Sutherland s law µ(t ) = A S T 1 + T S /T Parameter values Argon Helium M g mol 1 c p J kg 1 K 1 A S µpa s K 1/2 T S K Pr Thermal conduc vity and kinema c viscosity of helium is 8 mes higher than of argon Turbulence SST k-omega turbulence model

7 Gas low: Basic equations (steady-state) A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) Influence of gas flow on melt convec on in FZ Si crystal growth 6/16 Con nuity equa on (ρu) = 0 Navier-Stokes equa ons Pressure p rgh = p ρ g x ( (ρu U) = p rgh + 2 ) 3 µ eff U g x ρ+ + [ µ eff ( U + U T )], µ eff = µ + µ t Enthalpy equa on ρu (h + U 2 ) = (α eff h), α eff = α + α t 2

8 FZone and OpenFOAM iteration algorithm A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) Influence of gas flow on melt convec on in FZ Si crystal growth 7/16 1 Calcula on of the quasi-sta onary shape of phase boundaries and temperature field in silicon using FZone 2 Crea on of the geometry for argon flow calcula ons and automa c mesh genera on 3 Interpola on of the temperature field on the silicon surfaces of the generated finite volume mesh 4 Calcula on of the axisymmetric steady state argon flow using OpenFOAM standard solver buoyantsimplefoam 5 Calcula on of the quasi-sta onary shape of phase boundaries with argon cooling taken into account using FZone

9 Melt low A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) Influence of gas flow on melt convec on in FZ Si crystal growth 8/16 Melt Current and temperature maximum Buoyancy force Feed rod Marangoni force EM force Crystal Inductor u = 0 u t + (u )u = p ρ 0 + ν 2 u + gβ(t T 0 ) T t + (u )T = f Ma = γ T λ ρc p 2 T Free surface: EM and Marangoni surface forces f EM = µ 0δ (i 2 ) 4 τ T τ = M T τ

10 In luence of Marangoni coef icient Introduc on Mathema cal Model Calcula on example Conclusions Marangoni coefficient M = γ, γ surface tension T M is not exactly known, therefore several coefficients were used No HD T, K1723 M = N/(m K) 5 cm/s M = N/(m K) 5 cm/s M = N/(m K) 5 cm/s A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) Influence of gas flow on melt convec on in FZ Si crystal growth 9/16

11 4 FZ process: Global view A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) Influence of gas flow on melt convec on in FZ Si crystal growth 10/16 Mesh Argon flow Helium flow 13 m 03 m, ca cells

12 T and U ields near the inductor: Argon A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) Influence of gas flow on melt convec on in FZ Si crystal growth 11/16 1 m/s T, K T ind = 400 K 1200 T max = 1710 K T 0 = 1687 K

13 T and U ields near the inductor: Helium A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) Influence of gas flow on melt convec on in FZ Si crystal growth 12/16 1 m/s T, K T ind = 400 K 1200 T max = 1710 K T 0 = 1687 K

14 Cooling by argon and helium A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) Influence of gas flow on melt convec on in FZ Si crystal growth 13/ Feed 2 Feed 0 0 ITP z, cm -2 Melt z, cm -2 Melt ETP -4-4 Crystal r, cm -6-8 Crystal Net radia on Argon cooling Helium cooling q, kw/m 2

15 In luence of gas cooling Introduc on Mathema cal Model Calcula on example Conclusions Witout HD With HD 5 cm/s No cooling 5 cm/s Argon 5 cm/s Helium A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) T, K 1723 Influence of gas flow on melt convec on in FZ Si crystal growth 14/16

16 In luence of gas cooling A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) Influence of gas flow on melt convec on in FZ Si crystal growth 15/16 Phase boundaries No cooling, H C 14 mm Argon, H C 13 mm Helium, H C 13 mm EM force r, cm fem, N/m 2 Marangoni force r, cm fma, N/m 2

17 Conclusions A Sabanskis, J Virbulis ( University of Latvia, Riga, Latvia ) Influence of gas flow on melt convec on in FZ Si crystal growth 16/16 1 Axisymmetric calcula ons of argon and helium flow in FZ system have been carried out using OpenFOAM 2 Argon cooling power density can be as large as 10% of net radia on heat flux density, for helium it can reach about 50% 3 Argon flow only slightly changes f EM and f Ma, while strong helium cooling requires significantly larger inductor current, no ceably increasing EM force 4 Shear stress of gas flow is by at least an order of magnitude smaller than f EM and f Ma (002 N/m 2 for argon and 0008 N/m 2 for helium) and therefore has a weak influence