# Inequalities for Single Crystal Tube Growth by Edge-Defined Film-Fed Growth Technique

- Stefan Balint
^{1}and - Agneta M. Balint
^{2}Email author

**2009**:732106

https://doi.org/10.1155/2009/732106

© S. Balint and A. M. Balint. 2009

**Received: **3 January 2009

**Accepted: **29 March 2009

**Published: **7 May 2009

## Abstract

The axi-symmetric Young-Laplace differential equation is analyzed. Solutions of this equation can describe the outer or inner free surface of a static meniscus (the static liquid bridge free surface between the shaper and the crystal surface) occurring in single crystal tube growth. The analysis concerns the dependence of solutions of the equation on a parameter which represents the controllable part of the pressure difference across the free surface. Inequalities are established for which are necessary or sufficient conditions for the existence of solutions which represent a stable and convex outer or inner free surfaces of a static meniscus. The analysis is numerically illustrated for the static menisci occurring in silicon tube growth by edge-defined film-fed growth (EFGs) technique. This kind of inequalities permits the adequate choice of the process parameter . With this aim this study was undertaken.

## 1. Introduction

The growth process was scaled up by Kaljes et al. in [2] to grow 15 cm diameter silicon tubes. It has been realized that theoretical investigations are necessary for the improvement of the technology. Since the growth system consists of a small die type (1 mm width) and a thin tube (order of m wall thickness), the width of the melt/solid interface and the meniscus are accordingly very small. Therefore, it is essential to obtain accurate solution for the free surface of the meniscus, the temperature, and the liquid-crystal interface position in this tinny region.

Here is the melt surface tension, denotes the melt density, is the gravity acceleration, denote the main normal curvatures of the free surface at a point of the free surface, is the coordinate of with respect to the axis, directed vertically upwards, and is the pressure difference across the free surface. For the outer free surface, and for the inner free surface, .

Here denotes the hydrodynamic pressure in the meniscus melt, denote the pressure of the gas flow introduced in the furnace in order to release the heat from the outer and inner walls of the tube, respectively, and denotes the melt column height between the horizontal crucible melt level and the shaper top level. When the shaper top level is above the crucible melt level, then , and when the crucible melt level is above the shaper top level, then (see Figure 1).

To calculate the outer and inner free surface shapes of the static meniscus, it is convenient to employ the Young-Laplace (1.1) in its differential form. This form of the (1.1) can be obtained as a necessary condition for the minimum of the free energy of the melt column [3].

which is the Euler equation for the energy functional

The state of the arts at the time 1993-1994, concerning the dependence of the shape of the meniscus free surface on the pressure difference for small and large bond numbers, in the case of the growth of single crystal rods by E.F.G. technique, are summarized in [4]. According to [4], for the general differential equation (1.2), (1.4) describing the free surface of a liquid meniscus, there are no complete analysis and solution. For the general equation only numerical integrations were carried out for a number of process parameter values that were of practical interest at the moment.

Later, in 2001, Rossolenko shows in [5] that the hydrodynamic factor is too small to be considered in the automated single crystal tube growth. Finally, in [6] the authors present theoretical and numerical study of meniscus dynamics under axi-symmetric and asymmetric configurations. In [6] the meniscus free surface is approximated by an arc of constant curvature, and 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 among tube wall thickness, effective melt height, pull-rate, die-top temperature, and crystal environmental temperature.

In the present paper the shape of the inner and outer free surfaces of the static meniscus is analyzed as function of , the controllable part of the pressure difference across the free surface, and the static stability of the free surfaces is investigated. The novelty with respect to the considerations presented in literature consists in the fact that the free surface is not approximated as in [1, 6], by an arc with constant curvature, and the pressure of the gas flow introduced in the furnace for releasing the heat from the tube wall is taken in consideration. The setting of the thermal conditions is not considered in this paper.

## 2. Meniscus Outer Free Surface Shape Analysis in the Case of Tube Growth

Definition 2.1.

- (a)
- (b)
- (c)

Theorem 2.2.

Proof.

Equality (2.5) and inequalities (2.6) imply inequalities (2.2).

Corollary 2.3.

Corollary 2.4.

Theorem 2.5.

on the interval describes the convex outer free surface of a static meniscus.

Proof.

Consider the solution of the initial value problem (2.10). Denote by the maximal interval on which the function exists and by the function defined on . Remark that for the equality (2.3) holds.

It is clear that and for any inequalities (2.12) hold.

for some . Hence This last inequality is impossible, since according to the inequality (2.14), we have . Therefore, , defined by (2.14), satisfies .

what is impossible.

In this way we obtain that the equality holds.

For the solution of the initial value problem (2.8) on the interval describes a convex outer free surface of a static meniscus.

Corollary 2.6.

then there exists in the interval such that the solution of the initial value problem (2.10) on the interval describes a convex outer free surface of a static meniscus.

Corollary 2.7.

then there exists in the interval such that the solution of the initial value problem (2.10) on the interval describes a convex outer free surface of a static meniscus. The existence of and the inequality follows from Theorem 2.5. The inequality follows from Corollary 2.3.

Remark 2.8.

Moreover, if , the solution of the initial value problem (2.8) is convex everywhere (i.e., for ). That is because the change of convexity implies the existence of such that and , what is impossible.

Theorem 2.9.

Proof.

Since (2.1) is the Euler equation for (2.23), it is sufficient to prove that the Legendre and Jacobi conditions are satisfied in this case.

Hence, the Legendre condition is satisfied.

is a Sturm type upper bound for (2.27) [7].

has only one zero on the interval [7]. Hence the Jacobi condition is satisfied.

Definition 2.10.

A solution of (2.1) which describes the outer free surface of a static meniscus is said to be stable if it is a weak minimum of the energy functional of the melt column.

Remark 2.11.

Theorem 2.9 shows that if describes a convex outer free surface of a static meniscus on the interval , then it is stable.

Theorem 2.12.

If the solution of the initial value problem (2.10) is concave (i.e., ) on the interval , then it does not describe the outer free surface of a static meniscus on .

Proof.

Theorem 2.13.

Proof.

Denote by the solution of the initial value problem (2.10) which is assumed to represent the outer free surface of a static meniscus on the closed interval . Let be defined as in Theorem 2.2. for . Since , we have . Hence and therefore for , close to . Taking into account the fact that it follows that there exists such that .

Therefore . Since and , the following inequality holds: . On the other hand . Using the above evaluations we obtain inequalities (2.31).

Remark 2.14.

Theorem 2.15.

then the solution of the initial value problem (2.10) is concave on the interval where is the maximal interval of the existence of .

Proof.

## 3. Meniscus Inner Free Surface Shape Analysis in the Case of Tube Growth

Definition 3.1.

- (a)
- (b)
- (c)

Theorem 3.2.

Proof.

Equality (3.5) and inequalities (3.6) imply inequalities (3.2).

Corollary 3.3.

Corollary 3.4.

Theorem 3.5.

on the interval describes the convex inner free surface of a static meniscus.

Proof.

for some . Hence and it follows that there exists such that and . This last inequality is impossible according to the definition of .

Therefore, defined by (3.14) satisfies .

Since for , it follows that the equality is impossible.

what is impossible.

In this way we obtain that the equality holds.

For the solution of the initial value problem (3.10) on the interval describes a convex inner free surface of a static meniscus.

Corollary 3.6.

then there exists in the interval such that the solution of the initial value problem (3.10) on the interval describes a convex inner free surface of a static meniscus.

Corollary 3.7.

then there exists in the interval such that the solution of the initial value problem (3.10) on the interval describes a convex inner free surface of a static meniscus.

The existence of and the inequality follows from Theorem 3.5. The inequality follows from the Corollary 3.3.

Theorem 3.8.

Proof.

It is similar to the proof of Theorem 2.9.

Definition 3.9.

A solution of (3.1) which describes the inner free surface of a static meniscus is said to be stable if it is a weak minimum of the energy functional of the melt column.

Remark 3.10.

Theorem 3.8 shows that if describes a convex inner free surface of a static meniscus on the interval , then it is stable.

Remark 3.11.

Theorem 3.12.

- (a)
- (b)

Proof.

Remark 3.13.

If the solution of the initial value problem (3.10) is concave (i.e., ) on the interval , then it does not describe the inner free surface of a static meniscus on .

Theorem 3.14.

then the solution of the initial value problem (3.10) is concave on the interval where is the maximal interval of the existence of .

Proof.

## 4. Numerical Illustration

The objective was to verify if the necessary conditions are also sufficient, or if the sufficient conditions are also necessary. Moreover, the above data were used in experiments and the computed results can be tested against the experiments in order to evaluate the accuracy of the theoretical predictions. This test is not the subject of this paper.

For the same numerical data inequality (2.9) becomes . We have already obtained that for there exists in the interval such that the solution of (2.10) describes a convex outer free surface of a static meniscus on the interval . Hence, inequality (2.9) is not a necessary condition.

For the same numerical data inequality (2.33) becomes . We have already obtained that for (Figure 10) the solution of (2.10) is not anymore the outer free surface of a static meniscus. Hence, inequality (2.33) is not a sufficient condition.

For the considered numerical data (3.9) becomes . We have already obtained that for there exists such that the solution of (3.10) describes a convex inner free surface of a static meniscus on . Hence, inequality (3.9) is not a necessary condition.

For the considered numerical data inequality (3.24) becomes . Integration shows that for there exists such that the solution of (3.10) is a nonglobally convex inner free surface of a static meniscus on , but for it is not anymore the inner free surface of a nonglobally static meniscus (Figures 20 and 21). Hence, (3.24) is not a sufficient condition.

## 5. Conclusions

(1) Inequalities (2.2) and (2.7) localize regions on the pressure axis where the outer pressure has to be taken in order to obtain convex outer free surface.

Inequalities (2.9), (2.19), and (2.20) localize regions on the pressure axis, having the property that if the outer pressure is taken in this region, then a convex outer free surface is obtained.

Inequalities (2.31) and (2.32) localize region on the pressure axis, where the outer pressure has to be taken in order to obtain a convex-concave outer free surface.

(2) Inequalities (3.2) and (3.7) localize regions on the pressure axis where the inner pressure has to be taken in order to obtain convex inner free surface.

Inequalities (3.5), (3.20), and (3.21) localize regions on the pressure axis, having the property that if the inner pressure is taken in this region, then a convex inner free surface is obtained.

Inequality (3.24) localizes region on the pressure axis, where the inner pressure has to be taken in order to obtain a convex-concave inner free surface.

(3) By computation these values are found in a real case, and the "accuracy" (sufficiency or necessity) of the reported inequalities is discussed.

## Declarations

### Acknowledgments

The authors thank the anonymous referees for their valuable comments and suggestions, which led to an improvement of the manuscript. They are grateful to the Romanian National Authority for Research supporting this research under the Grant ID 354 no. 7/2007.

## Authors’ Affiliations

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