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Simplified numerical approach for estimation of effective segregation coefficient at the melt/crystal interface 
Anatoly I. Prostomolotov ^{1}, Nataliya A. Verezub ^{2}, Alexey E. Voloshin ^{3} 
1. Institute for Problems in Mechanics of RAS (IPMECH), Vernadskogo prospect 101, bl. 1, Moscow 119526, Russian Federation 
Abstract 
The effective segregation coefficient – K_{eff} is an important parameter for the analysis of a dopant inhomogeneity in crystals grown from a melt. Its value depends on a structure and intensity of a melt flow, which substantially affect on a dopant flux from a melt into a crystal. The estimations of K_{eff} value were done in many publications on the basis of two main approaches. The former of these approaches is quite simple, and therefore it is the most widely used in technological practice [1]. Its basis is the analytical expressions for calculation of K_{eff}, which is determined by the value of flow velocity near the melt/crystal interface, according to some analytical hydrodynamic formulas [23]. The second approach is based on the complete numerical simulation of crystallization process, requiring specialized program codes, large computational cost and high user qualification [46]. This work describes an intermediate approach for Bridgman GaSb(Te) crystal growth in microgravity [7]. For its implementation the numerical modeling is required, too. But it is much less costly than in second case. Its basis is a simplified model of convective heat and mass transfer in a melt on the assumption of a flat moving melt/crystal interface with taking into account of dopant flux from a melt into a crystal [8]. The crystallization process is considered for the constant velocity of the melt/crystal interface V_{S} in a flat melt layer of thickness D and length L at gravitational field g for following thermal conditions: T_{S} = 985 K (the melting point) and different values of T_{w} = 996 ¸ 1057 K (Fig. 1), which determine the variation of longitudinal temperature gradient. The equations of NavierStokesBoussinesq and heat and mass transfer in a melt may be written in the coordinate system associated with moving melt/crystal interface [8]. By solution of this system the velocity vector V, pressure P, temperature T, the dopant concentration C depending on the spatial coordinates and time are determined.
Fig. 1. Scheme of the simplified model: thermal boundary conditions, V_{S} – velocity of the melt/crystal interface, and the direction of melt flow (dashed line). On the boundaries of a melt: V = 0 and T is defined as shown in Fig. 1. The initial concentration of dopant in a melt: C = Co. The boundary conditions for dopant concentration are following: Dgrad(C) = (1 – K_{o})V_{S}C – at the melt/crystal interface, and C = 0 – for all other boundaries in the case of a continuous crystallization for a whole ingot length. For a GaSb (Te) melt the physical parameters were following: kinematic viscosity visc=0.0032 cm^{2}/s, thermal conductivity 1.02*10^{6} erg/cmKs, heat capacity 3.3*10^{6} erg/gK, thermal expansion coefficient b=9.6*10^{5} 1/K, tellurium diffusion coefficient D = 5*10^{5} cm^{2}/s, and equilibrium segregation coefficient K_{o} = 0.37. Crystallization rate was constant: V_{S} = 3*10^{4} cm/s, and microgravity was varied: g/g_{o} = 1.6*10^{5}  2.2*10^{3}; g_{o} = 980 cm/s^{2}. Desired value of the effective segregation coefficient is calculated by following formula: K_{eff} = K_{o}<C>/C_{o}, where <C> is an average dopant concentration at the melt/crystal interface. In this work the more simplified variant of the model was applied, which corresponds to the calculation of a discrete stage of crystallization (D = 1.5 cm, L = 4 cm), and the condition C = C_{o} at x = L (Fig. 1). The parametric calculations were carried out with the use of widespread program code AnsysFluent [9], which was supplemented by the author subroutines in C++ taking into account of crystallization model [8]. Numerical results have been compared with those of analytical models [2,3] and data of semianalytic model [8], in which the equation for dopant concentration is solved numerically at the analytical velocity field in a melt: V_{x}(x,y) = (Gr/6)[1/4 – (y – 1/2)^{2}](y – 1/2){1 – eαx[cos(βx) + (α/β)sin(βx)]}, V_{y}(x,y) = (Gr/24)[1/4 – (y – 1/2)^{2}]eαx(α2/β + β)sin(βx). Here: α=4.15, β=2.286, and Grashof’s number Gr=gb[(T_{W}T_{S})/L]D^{4}/visc^{2}.
Fig. 2. The dependences of effective segregation coefficient K_{eff} upon maximum velocity V_{max} in a melt: 1 – the simplified model, 2 and 3 – the estimations by [2] with different formulas for the diffusion layer thickness, 4, 5 – the estimations by [3] and [8] respectively. Fig. 2 illustrates the dependences of K_{eff} upon maximum velocity V_{max} in a melt as calculated by simplified model – 1, in accordance with [2] for the different formulas of the diffusion layer thickness – 2, 3, by formula [3] – 4, and with application of the analytical hydrodynamic model [8] – 5. May be noted the discrepancies of various analytical estimations of K_{eff} depending upon V_{max}. It is explained by an ambiguity at the choice of the formula for V_{max} in [2,3] and the approximate velocity field in [8], which corresponds to the thermal convection in a semiinfinite layer. By means of simplified approach (see 1 in Fig. 2) the best analytical model  2 of [2] has been selected for the analysis of measured tellurium concentration [1] in GaSb crystal, grown in microgravity conditions [7]. ACKNOWLEDGMENTS We thank the Russian Foundation for Basic Research for the support of projects: 110800966, 120201126. References [1] Voloshin A.E., Nishinaga T., Ge P. et al. Te distribution in space grown GaSb // J. Cryst. Growth. 2002. V. 234. P. 1224. [2] Burton J.A., Prim R.C., Slichter W.P. The distribution of solute in crystals grown from the melt. Part I. Theoretical // J. Chem. Phys. 1953. V. 21. N 11. P.19871991. [3] Ostrogorsky A.G., Muller G. A model of effective segregation coefficient, accounting for convection in the solute layer at the growth interface // J. Cryst. Growth. 1992. V. 121. P. 587598. [4] Lan C.W., Lee I.F., Yeh B.C. Threedimensional analysis of flow and segregation in vertical Bridgman crystal growth under axial and transversal magnetic fields // J. Cryst. Growth. 2003. V. 254. P. 503–515. [5] Verezub N.A., Marchenko M.P., Nutsubidze M.N., Prostomolotov A.I. Inluence of convective heat transfer on crystalmelt interface for Stockbarger method with step heater // Growth of Crystals. 1996. V. 20. New York: Consultants Bureau. P. 129138. [6] Strelov V.I., Zakharov B.G., Sidorov V.S. et al. Mathematical Modeling and Experimental Investigation of the Effect of Temperature Gradients on Crystallization Processes under Terrestrial and Space Conditions // Crystallography Reports. 2005. V. 50. N 3. P. 490498. [7]Ge P., Nishinaga T., Huo C. et al. Recrystallization of GaSb under microgravity during China returnable satellite No. 14 mission // Microgravity Q. 1993.V.3. N 24. P. 161165. [8] Polezhaev V.I., Bune A.V., Verezub N.A. et al. Mathematical Modeling of Convective Heat and Mass Transfer on the Basis of Navier–Stokes Equations. Moscow: Nauka. 1987. 272 p. [in Russian]. [9] ANSYS FLUENT Tutorial Guide: Release 14.0, ANSYS Inc. 2011. 
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