On the creation of a stable drop-like static meniscus, appropriate for the growth of a single crystal tube with prior specified inner and outer radii.(Research Article)(Report)

1. Introduction

The conventional melt growth techniques, as Bridgman growth [1-3] or Czochralski pulling [4-6] of single crystals, typically produce ingots of circular or square cross-sections which need to be cut in hundreds of slices to produce wafers. Using these processes, it is difficult to produce thin wafers from an ingot without wasting 40%-50% of material as kerfs during the cutting process. For this reason the E.F.G. technology can be more appropriate to produce single crystals with prescribed shapes and sizes which can be used without additional machining.

The growth of silicon tubes by E.F.G. process was first reported by Erris et al. [7] .In [7] a theory of tube growth by E.F.G. process is developed to show the dependence of tube wall thickness on the growth variables. The theory uses approximation reported in [8,9], and it has been shown to be a useful tool understanding the feasible limits of the wall thickness control. A more accurate predictive model would require an increase of the acceptable tolerance range introduced by approximation.

Later, the heat flow in a tube growth system was analyzed in [10-19].

The state of the arts at the time 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 [20]. According to [20], for the general differential equation describing the free surface of a liquid meniscus, possessing axial symmetry, 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. The authors of [21, 22] consider automated crystal growth processes based on weight sensors and computers. They give an expression for the weight of the meniscus, contacted with crystal and shaper of arbitrary shape, in which there are two terms related to the hydrodynamic factor.

In [23] it is shown that the hydrodynamic factor is too small to be considered in the automated crystal growth. In [24] a theoretical and numerical study of meniscus dynamics, under symmetric and asymmetric configurations, is presented. 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 thickness, effective melt height, pull rate, die top temperature, and crystal environmental temperature.

In [25] the effect of the controllable part of the pressure difference on the free surface shape of the static meniscus is analyzed for the tube growth by E.F.G. method for materials for which 0 < [[alpha].sub.c] < [pi]/2; 0 < [a.sub.g] <[pi]/2; [[alpha].sub.c] >[pi]/2 - [a.sub.g].

The present paper concerns also the shape and the stability of the free surface of a static meniscus (pulling rate equal to zero). More precisely, it is shown in which kind the explicit formulas reported in [25] can be combined in order to create a stable static meniscus having a free surface with prescribed size and shape, which is appropriate for the growth of a single crystal tube having a priori specified inner and outer radii. The free surface of a static meniscus is appropriate for the growth of a single crystal tube of constant inner radius [r.sub.i] and constant outer radius [r.sub.e] if the angle between the tangent lines to the free surface at the points ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]) (Figure 1 where the solidification conditions have to be assured) and the vertical is equal to the growth angle [a.sub.g]. Moreover, the function describing the free surface has to minimize the energy functional of the melt column (i.e., the meniscus has to be stable). In this paper we give a procedure for the choice of the melt column height, between the horizontal crucible melt level and shaper top level and of the pressure of the gas flow introduced in the furnace (for release the heat), in order to create a static meniscus of which free surface is appropriate for the growth of a single crystal tube of constant inner radius [r.sub.i] and outer radius [r.sub.e]. The thermal problem concerning the setting of the thermal conditions, which assure that for the obtained static meniscus at the level [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] the solidification conditions are satisfied is not considered in this paper. The novelty consists in the fact that the free surface is not approximated by an arc with constant curvature, the computation takes into account the pressure of the gas flow, and the stability of the free surface is assured.

2. The Free Surfaces Equations and the Pressure Difference Limits

For a single crystal tube growth by E.F.G. technique, in hydrostatic approximation, the outer free surface equation of the static meniscus is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (2.1)

and the inner free surface equation is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (2.2)

Here, [gamma] is the surface tension of the melt; p is the melt density; g is the gravitational acceleration; [z.sub.e],[z.sub.i] are the coordinates with respect to the Oz axis, directed vertically upwards; r is the radial coordinate with respect to the Or axis, oriented horizontal; [R.sub.ge].[R.sub.gi] are the outer and inner radius of the shaper, respectively; pe.pr are the pressure difference across the outer and inner free surface, respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (2.3)

In (2.3), [[rho].sub.m] denotes the hydrodynamic pressure in the meniscus melt due to the thermal and Marangoni convection; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] are the pressure of the gas flow introduced in the exterior and in the interior of the tube, respectively, for releasing the heat from the inner and outer side of the tube wall; H denotes the melt column "height" between the horizontal crucible melt level, and the shaper top level (Figure 1). H is positive when the crucible melt level is under the shaper top level and it is negative when the shaper top level is under the crucible melt level.

The solution [z.sub.e] = [z.sub.e](r) of (2.1) has to satisfy the following conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (2.4a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (2.4b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (2.4c)

where [r.sub.e] e (([R.sub.gi] + [R.sub.ge])/2,[R.sub.ge]) is the tube outer radius; ag is the growth angle; acis the contact angle between the outer free surface and the outer edge of the shaper …