A Mathematical Model for Single Crystal Cylindrical Tube Growth by the Edge-Defined Film-Fed Growth (EFG) Technique

Modern engineering does not only need crystals of arbitrary shapes but also plate, rod and tube-shaped crystals, i.e., crystals of shapes that allow their use as final products without additional machining. Therefore, the growth of crystals of specified sizes and shapes with controlled defect and impurity structures are required. In the case of crystals grown from the melt, this problem appears to be solved by profiled-container crystallization as in the case of casting. However, this solution is not always possible, for example growing very thin plate-shaped crystals from the melt (to say nothing of more complicated shapes), excludes container application completely.[ Tatarchenko, 1993]


Crystal growth from the melt by E.F.G. technique
Modern engineering does not only need crystals of arbitrary shapes but also plate, rod and tube-shaped crystals, i.e., crystals of shapes that allow their use as final products without additional machining. Therefore, the growth of crystals of specified sizes and shapes with controlled defect and impurity structures are required. In the case of crystals grown from the melt, this problem appears to be solved by profiled-container crystallization as in the case of casting. However, this solution is not always possible, for example growing very thin plate-shaped crystals from the melt (to say nothing of more complicated shapes), excludes container application completely. [ Tatarchenko, 1993] The techniques which allow the shaping of the lateral crystal surface without contact with the container walls are appropriate for the above purpose. In the case of these techniques the shapes and the dimensions of the grown crystals are controlled by the interface and meniscus-shaping capillary force and by the heat-and mass-exchange conditions in the crystal-melt system. The edge-defined film-fed growth (EFG) technique is of this type. Whenever the E.F.G. technique is employed, a shaping device is used (Fig. 1). In the device a capillary channel is manufactured ( Fig. 1) in which the melt raises and feeds the growth process. Frequently, a wettable solid body is used to raise the melt column above the shaper, where a thin film is formed. When a wettable body is in contact with the melt, an equilibrium liquid column embracing the surface of the body is formed. The column formation is caused by the capillary forces being present. Such liquid configuration is usually called a meniscus (Fig. 1) and in the E.F.G. technique, its lower boundary ( Fig. 1 point C) is attached to the sharp edge of the shaper.
Let be the temperature of the meniscus upper horizontal section ( Fig. 1 -AB ) the temperature of the liquid crystallization. So, above the plane of this section, the melt transforms in solid phase. Now set the liquid phase into upward motion with the constant rate, v, keeping the position of the phase-transition plane invariable by selection of the heat conditions. When the motion starts, the crystallized position of the meniscus will continuously form a solid upward or downward tapering body. In the particular case when the line tangent at the triple point B to the liquid meniscus surface makes a specific angle (angle of growth) with the vertical, the lateral wall of the crystal will be vertical. Thus, the initial body, called the seed, serves to form a meniscus which later on determines the form of the crystallized product, the phase transition position being fixed. Based on this description, a conclusion can be drawn that the dimensions and shapes of the specimens being pulled by the E.F.G. technique depend upon the following factors: (i) the shaper geometry; (ii) the pressure of feeding the melt to the shaper; (iii) the crystallization front position; (iv) the seed's shape. The seed's shape is only important for stationary pulling; in this case its cross-section should coincide with the desired product's crosssection. Frequently, especially when complicated profiles are grown, the pulling process is carried out under unstationary conditions by lowering the crystallization surface, which then enhances the dependence of the shapes on the crystal cross-section. With such an approach applied to the pulling process, the dimensions and the shape of the grown crystal are determined by the above-mentioned factors and by the pulling rate-to-crystallization front displacement ratio. [Tatarchenko, 1993].

Background history of tube growth from the melt by E.F.G. method
The technology of growing tubes can have a significant impact for example on the solar cell technology. The growth of silicon tubes by E.F.G. process was first reported by Erris et al. [Erris et al.,1980] et al.,1980] a theory of tube growth by the E.F.G. process is developed to show the dependence of the tube wall thickness on the growth variables. The theory concerns the calculations of the shape of the liquid-vapor interface (or meniscus) and of the heat flow in the system. The inner and outer meniscus shapes, (Fig.1), are both calculated from Laplace's capillary equation, in which the pressure difference Δp across a point on meniscus is considered to be Δp = ρ ּ◌g ּ◌H eff = constant, where H eff represents the effective height of the growth interface above the horizontal liquid level in the crucible (Fig.1). According to [Surek et al.,1977], [Swartz et al., 1975], it includes the effects of the viscous flow of the melt in the shaper capillary and in the meniscus film, as well as that of the hydrostatic head. The above approximation for Δp is valid for silicon ribbon growth [Surek et al.,1977], [Kalejs et al., 1990], when H eff >> h, where h is the height of the growth interface above the shaper top (i.e. the meniscus height). Another approximation used in [Erris et al.,1980], concerning the meniscus, is that the inner and outer meniscus shapes are approximated by circular segments. With these relatively tight tolerances concerning the menisci in conjunction with the heat flow calculation in the system, the predictive model developed in [Erris et al.,1980] has been shown to be a useful tool in understanding the feasible limits of wall thickness control. A more precise predictive model would require an increase of the acceptable tolerance range introduced by approximation.
Later, this process was scaled up by Kaljes et al. [Kalejs et al., 1990] to grow   2 15 10 [m] diameter silicon tubes, and the stress behavior in the grown tube was investigated. It has been realized that numerical investigations are necessary for the improvement of the technology. Since the growth system consists of a small die tip (   3 1 10 m width) and a thin tube (order of   6 200 10 [m] wall thickness) the width of the melt/solid interface and meniscus are accordingly very small. Therefore, it is essential to obtain an accurate solution for the temperature and interface position in this tiny region.
In [Rajendran et al., 1993] an axisymmetric finite element model of magnetic and thermal field was presented for an inductively heated furnace. Later the same model was used to determine the critical parameters controlling silicon carbide precipitation on the die wall [Rajendran et al., 1994]. Rajendran et al. also developed a three dimensional magnetic induction model for an octagonal E.F. G, system. Recently, in [Roy et al., 2000a], [Roy et al., 2000b], a generic numerical model for an inductively heated large diameter Si tube growth system was reported. In [Sun et al., 2004] a numerical model based on multi-block method and multi-grid technique is developed for induction heating and thermal transport in an E.F.G. system. The model is applied to investigate the growth of large octagon silicon tubes of up to   2 50 10 m diameter. A 3D dynamic stress model for the growth of hollow silicon polygons is reported in [Behnken et al., 2005]. In [Mackintosh et al., 2006]  250 10 300 10 m is described. In [Kasjanow et al., 2010] the authors present a 3D coupled electromagnetic and thermal modeling of E.F.G. silicon tube growth, successfully validated by experimental tests with industrial installations.
The state of the art at 1993-1994 concerning the calculation of the meniscus shape in general in the case of the growth by E.F.G. method is summarized in [Tatarchenko, 1993]. According to [Tatarchenko, 1993], for the general equation describing the surface of a liquid meniscus possessing axial symmetry, there is no complete analysis and solution. For the general equation only numerical integration was carried out for a number of process parameter values that are of practical interest at the moment. The authors of papers [Borodin&Borodin&Sidorov&Petkov, 1999], [Borodin&Borodin&Zhdanov, 1999] consider automated crystal growth processes based on weight sensors and computers. They give an expression for the weight of the meniscus, contacted with a crystal and shaper of arbitrary shape, in which there are two terms related to the hydrodynamic factor. In [Rosolenko et al., 2001] it is shown that the hydrodynamic factor is too small to be considered in the automated crystal growth and it is not clear what equation (of non Laplace type) was considered for the meniscus surface. Finally, in [Yang et al., 2006] the authors present theoretical and numerical study of meniscus dynamics under symmetric and asymmetric configurations. A meniscus dynamics model is developed to consider meniscus shape and its dynamics, heat and mass transfer around the die-top and meniscus. Analysis reveals the correlations between tube thickness, effective melt height, pull-rate, die-top temperature and crystal environmental temperature.
The purpose of this chapter is the mathematical description of the growth process of a single crystal cylindrical tube grown by the edge-defined film-fed growth (EFG) technique. The mathematical model defined by a set of three differential equations governing the evolution of the outer radius and the inner radius of the tube and of the crystallization front level is the one considered in [Tatarchenko, 1993]. This system contains two functions which represent the angle made by the tangent line to the outer (inner) meniscus surface at the three-phase point with the horizontal. The meniscus surface is described mathematically by the solution of the axi-symmetric Young-Laplace differential equation. The analysis of the dependence of solutions of the Young-Laplace differential equation on the pressure difference across the free surface, reveals necessary or sufficient conditions for the existence of solutions which represent convex or concave outer or inner free surfaces of a meniscus. These conditions are expressed in terms of inequalities which are used for the choice of the pressure difference, in order to obtain a single-crystal cylindrical tube with specified sizes.
A numerical procedure for determining the functions appearing in the system of differential equations governing the evolution is presented.
Finally, a procedure is presented for setting the pulling rate, capillary and thermal conditions to grow a cylindrical tube with prior established inner and outer radius. The right hand terms of the system of differential equations serve as tools for setting the above parameters. At the end a numerical simulation of the growth process is presented.
The results presented in this chapter were obtained by the authors and have never been included in a book concerning this topic.
Since the calculus and simulation in this model can be made by a P.C., the information obtained in this way is less expressive than an experiment and can be useful for experiment planing.

The system of differential equations which governs the evolution of the tube's inner radius r i , outer radius r e and the level of the crystallization front h
According to [Tatarchenko, 1993] the system of differential equations which governs the evolution of the tube's inner radius i r , the outer radius e r and the level of the crystallization front h is:   Fig.1 b),  g is the growth angle ( Fig.   1), e p ( i p ) is the controllable part of the pressure difference across the free surface given by: where m p is the hydrodynamic pressure in the melt under the free surface, which can be neglected in general, with respect to the hydrostatic pressure    In the equation (1) 3 :  is the latent melting heat;  1 ,  2 are the thermal conductivity coefficients in the melt and the crystal respectively; 1 j G , 2 j G are the temperature gradients at the interface in the melt (i=1) and in the crystal (i=2) respectively, given by the formulas: en T -the environment temperature at = 0 z , k -the vertical temperature gradient in the furnace, e rthe outer radius of the tube equal to the upper radius of the outer meniscus, i r -the inner radius of the tube equal to the upper radius of the inner meniscus, L -the tube length and and COSH are the hyperbolic sine and hyperbolic cosine functions.
Due to the supercooling in this gradients it is assumed that In the following sections we will show in which way  ( , , )

The choice of the pressure of the gas flow and the melt level in silicon tube growth
In a single crystal tube growth by edge-defined film-fed growth (E.F.G.) technique, in hydrostatic approximation, the free surface of a static meniscus is described by the Young-Laplace capillary equation [Finn, 1986]: Here γ is the melt surface tension, ρ denotes the melt density, g is the gravity acceleration, 1 2 1 / , 1 / R R denote the mean normal curvatures of the free surface at a point M of the free surface, z is the coordinate of M with respect to the Oz axis, directed vertically upwards, p is the pressure difference across the free surface. To calculate the outer and inner free surface shape of the static meniscus it is convenient to employ the Young-Laplace eq.(5) in its differential form. This form of the eq.(5) can be obtained as a necessary condition for the minimum of the free energy of the melt column [Finn, 1986].For a tube of outer radius , the axi-symmetric differential equation of the outer free surface is given by: which is the Euler equation for the energy functional The axi-symmetric differential equation of the inner free surface is given by: which is the Euler equation for the energy functional: In papers [Balint & Balint, 2009b], [Balint&Balint&Tanasie, 2008], [Balint & Tanasie, 2008] , Balint, Tanasie, 2011] some mathematical theorems and corollaries have been rigorously proven regarding the existence of an appropriate meniscus. These results are presented in Appendixes. In the following we will shown in which way the inequalities can be used for creation of the appropriate meniscus.

Convex free surface creation
In this section, it will be shown in which way the inequalities presented in Appendix 1  Theorem 5 (Appendix 1) shows that a static meniscus having a convex outer free surface, appropriate for the growth of a tube of outer radius e r situated in the range Inequalities (A.1.6) establish the range where the pressure difference i p has to be chosen in order to obtain a static meniscus with convex inner free surface appropriate for the growth of a tube of inner radius equal to  gi m R .
If the pressure difference i p satisfies (A.1.7) then a static meniscus with convex inner free surface is obtained which is appropriate for the growth of a tube of inner radius If the pressure difference i p satisfies the inequality (A.1.9) and the value of i p is close to the value of the right hand term of the inequality (A.1.9) then a static meniscus with convex inner free surface is obtained which is appropriate for the growth of a tube of inner radius If the pressure difference i p satisfies inequality (A.1.10) then a static meniscus with convex inner free surface is obtained which is appropriate for the growth of a tube of inner radius Theorem 10 (Appendix 1) shows that a static meniscus having a covex inner free surface appropriate for the growth of a tube of inner radius i r situated in the range    (10) and (11)  It follows that the pressure of the gas flow, introduced in the furnace for releasing the heat from the inner wall of the tube has to be higher than the pressure of the gas flow, introduced in the furnace for releasing the heat from the outer wall of the tube and we have to take:

Concave free surface creation
In this section, it will be shown in which way the inequalities presented in Appendix 2 can be used for the creation of an appropriate static concave meniscus by the choice of e p and i p [Balint&Balint, 2009a].
Moreover, the outer free surface of this meniscus is not globally concave; it is a convexconcave meniscus (Fig.6).
Taking into account  0 m p [Eriss, 1980], [Rossolenko, 2001], [Yang, 2006] [Eriss et al., 1980], [Rossolenko et al., 2001], [Yang et al. 2006  The deviation of the tangent to the crystal outer (inner) free surface at the triple point from the vertical is the difference Due to the nonlinearity, the above described procedure can't be realized analytically. This is the reason why for the construction of the function   ( ; , ) e e c e r p in [Balint&Tanasie, 2010] the following numerical procedure was conceived:

The angles
Step 1 s i n t a n 2 1 cos 2 Step 3. For e p the range     , e e p p defined by: is considered.
Step 4. In the range   Step 5. In the range     , e e p p a set of m different values of e p is chosen.
Step 6. In a given range   Step 8 Step 10 Step s i n 1 t a n 2 1 sin 2 ( , ) cos cos 1 Step 2.
Step 3. For i p the range     , i i p p defined by: is considered.
Step 4. In the range   Step 5. In the range     , i i p p a set of n different values of i p are chosen.
Step 6. In a given range   Step 7. For a given k i p ,  1, k n and  iq c ,  1, q l the solution of the system (11) which satisfies the conditions: is found numerically obtaining the functions (profiles curves Ref. [Tatarchenko, 1993] For the case of a silicon tube and the outer free surface the function:  Step 1-Step 3. present the some differences. For the outer free surface we have to consider: Step 1 is considered.
For the inner free surface we have to make: Step Step 2.

Setting the pulling rate, the thermal and capillary conditions
For the growth of a silicon tube with convex profile curves the following numerical data will be used:   [Tatarchenko, 1993] i.e.:  a a a  a a a a a a  a a  a  a a a a a a a a  a a a a a  The values of the numbers ij a in the considered cases are given in Table 2. It is easy to verify that in all cases the Hurwitz condition are satisfied.
In Fig. 10 simulations of the silicon tube growth is presented when the seed length is

Conclusions
Knowing the material constants (density, heat conductivity, etc), the size of the single crystal tube which will be grown from that material, the size of the shaper which will be used and the cooling gas temperature at the entrance, it is possible to predict values of pulling rate, temperature at the meniscus basis, cooling gas temperature at the exit, vertical temperature gradient in the furnace, inner and outer walls cooling gas pressure differences, melt column height differences, crystallization front level, which can be used for a stable growth.
According to the model the predicted values are not unique i.e. there are several possibility to obtain a tube with prior given size from a given material using the same shaper. So, even if in our computation the material and the size of the shaper and tube is the same as in [Eriss] experiment, the computed data given in Table 1 can be different from that used in the real experiment. For this reason our purpose is not rely to compare the computed results with the experimental data. Moreover we want to reveal that a tube of prior given size can be obtained by different settings and the model permit to compute such settings. The choice of a specific setting is the practical crystal grower decision. The model provide possible settings and can be helpful in a new experiment planning.
Concerning the limits of the model it is clear that it is limited in applicability, as all models. The main limits are those introduced by approximations made in equations defining the model.

Appendix 1. Inequalities for single crystal tube growth by E.F.G. technique -Convex outer and inner free surface
Consider the differential equation (6)