# 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

Authors: Stefan Balint; Agneta M. Balint
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

A theoretical procedure for the creation of a stable drop-like static meniscus, appropriate for the growth of a single crystal tube, with a priori specified inner and outer radius, is presented. The method locates the controllable partp of the pressure difference across the free surface. It consists in a set of calculus, which leads to the determination of the melt column height (between the horizontal crucible melt level and the shaper top level) in function of the pressure of the gas flow (introduced in the furnace for release the heat) in order to obtain the desired meniscus. The procedure is presented in general and is numerically illustrated for InSb tubes. The novelty is the algorithm for the exact determination of p, which has to be used, the determination of the melt column height, and the evaluation of the effect of shaper radii. The setting of the thermal conditions, which assure that for the obtained static meniscus the solidification conditions are satisfied at the “right” places, is not considered here.


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## Body

## 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<αc<π/2; 0<αg<π/2; αc>π/2-αg.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 ri and constant outer radius re if the angle between the tangent lines to the free surface at the points (ri,zi(ri)),(re,ze(re)) (Figure 1 where the solidification conditions have to be assured) and the vertical is equal to the growth angle αg. 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 ri and outer radius re.The thermal problem concerning the setting of the thermal conditions, which assure that for the obtained static meniscus at the level zi(ri),ze(re) 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.Figure 1
Axisymmetric meniscus geometry in the tube growth by E.F.G. method.


## 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(2.1)ze′′=ρ·g·ze-peγ[1+(ze′)2]3/2-1r·[1+(ze′)2]·ze′;r∈[Rgi+Rge2,Rge],
and the inner free surface equation is
(2.2)zi′′=ρ·g·zi-piγ[1+(zi′)2]3/2-1r·[1+(zi′)2]·zi′;r∈[Rgi,Rgi+Rge2].
Here, γ is the surface tension of the melt; ρ is the melt density; g is the gravitational acceleration; ze,zi 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; Rge,Rgi are the outer and inner radius of the shaper, respectively; pe,pi are the  pressure difference across the outer and inner free surface, respectively:(2.3)pe=pm-pge-ρ·g·H,pi=pm-pgi-ρ·g·H.
In (2.3), pm denotes the hydrodynamic pressure in the meniscus melt due to the thermal and Marangoni convection; pge,pgi 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 solutionze=ze(r) of (2.1) has to satisfy the following conditions:(2.4a)z′e(re)=-tan(π2-αg),(2.4b)z′e(Rge)=-tanαc,(2.4c)ze(Rge)=0andze(r)isstrictlydecreasingon[re,Rge],
where re∈((Rgi+Rge)/2,Rge) is the tube outer radius; αg is the growth angle; αcis the contact angle between the outer free surface and the outer edge of the shaper top and 0<αc<π/2;0<αg<π/2;π/2-αg<αc (Figure 1).Condition (2.4a) expresses that at the point (re,ze(re)) (the left end of the outer free surface), where the solidification has to be realized, the angle between the tangent line to the free surface and the vertical is equal to the growth angle αg (i.e., the tangent to the tube outer wall is vertical).Condition (2.4b) expresses that at the point (Rge,0) (the right end of the outer free surface, where the free surface is attached to the shaper edge), the angle between the tangent line to the free surface and horizontal (i.e., the contact angle) is equal to αc.Condition (2.4c) expresses that at the point (Rge,0) the free surface is attached to the shaper edge.Moreover, the solutionze=ze(r) has to minimize the energy functional of the melt column:
(2.5)Ie(z)=∫reRge{γ·[1+(z′)2]1/2+12·ρ·g·z2-pe·z}·r·dr,z(re)=he>0,z(Rge)=0.
The solution zi=zi(r) of (2.2) has to satisfy the following conditions:
(2.6a)zi′(Rgi)=tanαc,(2.6b)zi′(ri)=tan(π2-αg),(2.6c)zi(Rgi)=0andzi(r)isstrictlyincreasingon[Rgi,ri],
where:  ri∈(Rgi,(Rgi+Rge)/2) is the tube inner radius; αg is the growth angle; αcis the contact angle between the inner free surface and the inner edge of the shaper top and 0<αc<π/2;0<αg<π/2;π/2-αg<αc (Figure 1).Condition (2.6a) expresses that at the point (Rgi,0) (the left end of the inner free surface, where the free surface is attached to the shaper edge), the angle between the tangent line to the free surface and horizontal (i.e., the contact angle) is equal to αc.Condition (2.6b) expresses that at the point (ri,zi(ri)) (the right end of the inner free surface), where the solidification has to be realized, the angle between the tangent line to the free surface and the vertical is equal to the growth angle αg (i.e., the tangent to the tube inner wall is vertical).Condition (2.6c) expresses that at the point (Rgi,0) the free surface is attached to the shaper edge.Moreover, the solutionzi=zi(r) has to minimize the energy functional of the melt column:
(2.7)Ii(z)=∫Rgiri{γ·[1+(z′)2]1/2+12·ρ·g·z2-pi·z}·r·dr,z(Rgi)=0,z(ri)=hi>0.Based on the mathematical theorems, rigorously proven in [25], the following statements, regarding the creation of an appropriate meniscus, can be formulated.Statement 1
If the solution of the initial value problem (IVP)(2.8)ze′′=ρ·g·ze-peγ[1+(ze′)2]3/2-1r·[1+(ze′)2]·ze′,ze(Rge)=0,ze′(Rge)=-tanαc
is convex, then it does not represent the outer free surface of an appropriate drop-like static meniscus.Comment 1.
The above statement shows that the solutions of IVP (2.8), which represent the outer free surface of an appropriate drop-like static meniscus, can be obtained only for those values of  pe for which the solution is not globally convex.Statement 2
Ifpe<(γ/Rge)·sinαc, then the solution of the IVP (2.8) is globally convex, and it does not represent the outer free surface of an appropriate drop-like static meniscus.Comment 2.
The above statement locates a set ofpe values which are not appropriate for the creation of a drop-like meniscus.Statement 3
If the solutionze=ze(r) of the nonlinear boundary value problem (NLBVP) (2.1), (2.4a), (2.4b), and (2.4c) represents the outer free surface of an appropriate concave static meniscus on the closed interval [Rge/n,Rge], with 1<n<(Rgi+Rge)/2, then the following inequalities hold:
(2.9)nn-1·γ·αc+αg-π/2Rge·cosαc+γRge·cosαg≤pe≤nn-1·γ·αc+αg-π/2Rge·sinαg+n-1n·ρ·g·Rge·tanαc+n·γRge·sinαc.Comment 3.
The above statement shows that in order to obtain the outer free surface of a concave static meniscus, appropriate for the growth of a tube of outer radiusre=Rge/n, pe has to be searched in the range defined by the inequalities (2.9).Consequence 1
Ifn=2·Rge/(Rgi+Rge) (re is the middle point of the interval [Rgi,Rge]), then re=Rge/n=(Rgi+Rge)/2 and inequalities (2.9) become
(2.10)2·γ·αc+αg-π/2Rge-Rgi·cosαc+γRge·cosαg≤pe≤2·γ·αc+αg-π/2Rge-Rgi·sinαg+ρ·g·(Rge-Rgi)2·tanαc+2·γRge+Rgi·sinαc.Comment 4.
The above consequence shows that in order to obtain the outer free surface of a concave static meniscus, appropriate for the growth of a tube of outer radiusre=(Rgi+Rge)/2, pe  has to be searched in the range defined by the inequalities (2.10).Consequence 2
Ifpe verifies the inequality
(2.11)pe<2·γ·αc+αg-π/2Rge-Rgi·cosαc+γRge·cosαg,
then there is no re in the closed interval [(Rgi+Rge)/2,Rge] for which the NLBVP (2.1), (2.4a), (2.4b), and (2.4c) possesses a concave solution.Comment 5.
The above consequence shows that ifpe is in the range defined by the inequality (2.11), then it is impossible to obtain a static meniscus having concave outer free surface which is appropriate for the growth of a tube of outer radius  re  situated in the range [(Rgi+Rge)/2,Rge].Consequence 3
Ifn→1, then  re=Rge/n→Rge and, according to (2.9), pe→+∞.Comment 6.
The above consequence shows that in order to obtain the outer free surface of a concave static meniscus, appropriate for the growth of a tube of outer radiusre≈Rge, pe has to be very high.Statement 4
Ifn and  pe verify the inequalities
(2.12)1<n<2·RgeRge+Rgi,pe>nn-1·γ·αc+αg-π/2Rge·sinαg+n-1n·ρ·g·Rge·tanαc+n·γRge·sinαc,
then there exist re∈[Rge/n,Rge]  and a concave solution of the NLBVP (2.1), (2.4a), (2.4b), and (2.4c) on the interval [re,Rge].Comment 7.
The above consequence shows that ifn  and   pe  verify (2.12), then a static meniscus having concave outer free surface, appropriate for the growth of a tube of outer radius re situated in the range [Rge/n,Rge], is obtained.Consequence 4
If for1<n′<n<2·Rge/(Rgi+Rge) and  pe the following inequalities hold:
(2.13)nn-1·γ·αc+αg-π/2Rge·sinαg+n-1n·ρ·g·Rge·tanαc+n·γRge·sinαc<pe<n′n′-1·γ·αc+αg-π/2Rge·cosαc+γRge·cosαg,
then there exist re in the interval [Rge/n,Rge/n′]  and a concave solution of the NLBVP (2.1), (2.4a), (2.4b), and (2.4c) on the interval [re,Rge].Comment 8.
The above consequence shows that ifpe  is in the range defined by the inequalities (2.13), then a static meniscus having concave outer free surface, appropriate for the growth of a tube of outer radius re  situated in the range [Rge/n,Rge/n′], is obtained.Statement 5
A concave solutionze=ze(r) of the NLBVP (2.1), (2.4a), (2.4b), and (2.4c) is a weak minimum of the energy functional of the melt column.Comment 9.
The above consequence shows that a static meniscus having a concave outer free surface, appropriate for the growth of a tube of outer radiusre  situated in the range [(Rgi+Rge)/2,Rge], is stable.Statement 6
Ifn and pe satisfy the inequalities
(2.14)1<n<2·RgeRge+Rgi,pe>n-1n·g·ρ·Rge·tanαc+n·γRge,
then the solution ze=ze(r) of the IVP (2.8) is concave on the interval  I∩[Rge/n,Rge]  where I is the maximal interval of the existence of   ze(r).Comment 10.
The above consequence shows that ifpe is in the range defined by the inequalities (2.14), then eventually it can be used for the creation of a drop-like static meniscus, appropriate for the growth of a tube of outer radius  re in the range [Rge/n,Rge].Statement 7
Ifpe>(γ/Rge)·sinαc and for a value re=Rge/n, which satisfies re∈((Rgi+Rge)/2,Rge), a static meniscus appropriate for the growth of a tube of outer radius    re   exists, then for pe the following inequalities hold:
(2.15)γRge·sinαc<pe<ρ·g·n-1n·Rge·tanαc+n·γRge·cosαc.Comment 11.
The above statement locates those values ofpe for which eventually nonglobally concave (drop-like) static meniscus, appropriate for the growth of a tube of outer radius re=Rge/n, exists.Statement 8
If the solution of the IVP(2.16)zi′′=ρ·g·zi-piγ[1+(zi′)2]3/2-1r·[1+(zi′)2]·zi′,zi(Rgi)=0,zi′(Rgi)=tanαc
is convex, then it does not represent the inner free surface of an appropriate drop-like static meniscus.Comment 12.
The above statement shows that the inner free surface of an appropriate drop-like static meniscus can be obtained only for those values ofpi for which the IVP (2.16) is not globally convex.Statement 9
Ifpi satisfies the inequality
(2.17)pi<-γRgi,
then the solution zi=zi(r) of the IVP (2.15) is convex on the interval  I∩[Rgi,(Rgi+Rge)/2], where I is the maximal interval of the existence of zi(r).Comment 13.
The above statement shows that ifpi is in the range defined by the inequality (2.17), then it is impossible to obtain a static meniscus, appropriate for the growth of a tube of inner radius ri situated in the range [Rgi,(Rgi+Rge)/2].Statement 10
If the solutionzi=zi(r)  of the NLBVP (2.2) and (2.5) represents the inner free surface of an appropriate concave static meniscus on the closed interval  [Rgi,m·Rgi]  with  1<m<(Rgi+Rge)/2·Rgi, then the following inequalities hold:
(2.18)1m-1γ·αc+αg-π/2Rgi·cosαc-γRgi·sinαc≤pi≤1(m-1)γ·αc+αg-π/2Rgi·sinαg+(m-1)·ρ·g·Rgi·tanαc-γm·Rgi·cosαg.Comment 14.
The above statement shows that in order to obtain the inner free surface of a concave static meniscus, appropriate for the growth of a tube of inner radiusri=m·Rgi, pi has to be searched in the range defined by the inequalities (2.18).Consequence 5
Ifm=(Rgi+Rge)/2·Rgi, then ri=(Rgi+Rge)/2 (i.e., ri  is the middle point of the interval ⌊Rgi,Rge⌋) and the inequalities (2.18) become:
(2.19)2·γ·αc+αg-π/2Rge-Rgi·cosαc-γRgi·sinαc≤pi≤2·γ·αc+αg-π/2Rge-Rgi·sinαg+ρ·g·[Rge-Rgi]2·tanαc-2·γRge+Rgi·cosαg.Comment 15.
The above consequence shows that in order to obtain the inner free surface of a concave static meniscus, appropriate for the growth of a tube of inner radiusri=(Rgi+Rge)/2, pi  has to be searched in the range defined by the inequalities (2.19).Consequence 6
Ifpi verifies:
(2.20)pi<2·γ·αc+αg-π/2Rge-Rgi·cosαc-γRgi·sinαc,
then there is no ri in the closed interval [Rgi,(Rgi+Rge)/2] for which the NLBVP (2.2) and (2.5) possesses a concave solution.Comment 16.
The above consequence shows that ifpi is in the range defined by the inequalities (2.19), then it is impossible to obtain a static meniscus, having concave inner free surface and appropriate for the growth of a tube of inner radius ri  situated in the range [Rgi,((Rgi+Rge)/2)].Consequence 7
Ifm→1, then ri=m·Rgi→Rgi and pi→+∞.Comment 17.
The above consequence shows that in order to obtain the inner free surface of a concave static meniscus appropriate for the growth of a tube of inner radiusri≈Rgi, pi has to be very high.Statement 11
Ifm  and  pi  verify the inequalities
(2.21)1<m<Rgi+Rge2·Rgi,pi>1m-1·γ·αc+αg-π/2Rgi·sinαg+(m-1)·ρ·g·Rgi·tanαc-γm·Rgi·cosαg,
then there exist  ri in the closed interval  [Rgi,m·Rgi]  and a concave solution of the NLBVP (2.2) and (2.5) on the interval  [Rgi,ri].Comment 18.
The above statement shows that ifm  and  pi  verify (2.21), then a static meniscus having concave inner free surface and appropriate for the growth of a tube of inner radius ri  in the range  (Rgi,m·Rgi),  is obtained.Consequence 8
If for1<m′<m<(Rgi+Rge)/2·Rgi and pi the following inequalities hold:
(2.22)1m-1·γ·αc+αg-π/2Rgi·sinαg+(m-1)·ρ·g·Rgi·tanαc-γm·Rgi·cosαg<pi<1m′-1·γ·αc+αg-π/2Rgi·cosαc+γRgi·sinαc,
then there exist  ri  in the interval [m′·Rgi,m·Rgi]  and a concave solution of the NLBVP (2.2) and (2.5) on the interval  [Rgi,ri].Comment 19.
The above consequence shows that ifpi is in the range defined by the inequalities (2.22), then a static meniscus having a concave inner free surface and appropriate for the growth of a tube of inner radius ri, situated in the range [m′·Rgi,m·Rgi], is obtained.Statement 12
A concave solutionzi=zi(r)   of the NLBVP (2.2) and (2.5) is a weak minimum of  the energy functional of the melt column.Comment 20.
The above statement shows that a static meniscus having a concave inner free surface appropriate for the growth of a tube of inner radiusri, situated in the range [Rgi,(Rgi+Rge)/2], is stable.Statement 13
Ifpi satisfies pi<(-γ/Rgi)·sinαc   and there exists   ri∈(Rgi,(Rgi+Rge)/2) such that on the interval  [Rgi,ri] an appropriate static meniscus exists, then pi verifies:
(2.23)-γRgi<pi<-γRgi·sinαc.Comment 21.
Inequality (2.23) locates the values of pi for which eventually nonglobally concave (convex-concave) static meniscus can be obtained.Statement 14
If(-γ/Rgi)·sinαc<pi and there exists ri=m·Rgi;m∈(1,(Rgi+Rge)/2·Rgi) such that on   [Rgi,ri]  an appropriate non-globally concave static meniscus exists, then   pi  satisfies the inequalities
(2.24)-γRgi·sinαc<pi<ρ·g·(m-1)·Rgi·tanαc+γRgi.Comment 22.
Inequality (2.24) locates the values of pi for which eventually concave-convex  static menisci can be obtained.Statement 15
Ifm and pi satisfy the inequalities
(2.25)1<m<Rgi+Rge2·Rgi,pi>γRgi+ρ·g·(m-1)·Rgi·tanαc,
then the solution zi(r) of the IVP (2.16) is concave on the interval I∩[Rgi,m·Rgi] where I is the maximal interval of the existence of zi(r).Comment 23.
Inequality (2.25) locates the values of pi for which we can obtain eventually appropriate static menisci.


## 3. Creation of an Appropriate Drop-Like Static Meniscus for the Growth of a Tube of Inner Radiusriand Outer Radius re

In this section it will be shown in which kind the explicit formulas presented in the above section can be used for the creation of an appropriate drop-like meniscus whenαc,αg,ρ,γ,Rgi,Rge,riandre are given a priori. In the same time the melt column “height” H, which has to be used, is found in function of the pressure of the gas flow introduced in the furnace for release the heat.


### 3.1. Creation of the Outer Free Surface

For the creation of the appropriate outer free surface, the following limits were considered:(3.1)L1e(n)=nn-1·γ·αc+αg-π/2Rge·cosαc+γRge·cosαg,L2e(n)=nn-1·γ·αc+αg-π/2Rge·sinαg+n-1n·ρ·g·Rge·tanαc+n·γRge·sinαc,L3e(n)=n-1n·ρ·g·Rge·tanαc+n·γRge·cosαc,L4e(n)=n-1n·ρ·g·Rge·tanαc+n·γRge,l1e=2·γ·αc+αg-π/2Rge-Rgi·cosαc+γRge·cosαg,l2e=2·γ·αc+αg-π/2Rge-Rgi·sinαg+ρ·g·(Rge-Rgi)2·tanαc+2·γRge-Rgi·sinαc,l3e=γRge·sinαc
for n>1.These limits are represented forn∈(1;1.1) in Figure 2. The computations were performed in MathCAD V.13 using the following numerical data
(3.2)Rge=4.8·10-3[m];Rgi=4.2·10-3[m];αc=63.80=1.1135[rad];αg=28.90=0.5044[rad];ρ=6582[kg/m3];γ=4.2·10-1[N/m];g=9.81[m/s2].
When n1=1.03226, that is, re1=4.65·10-3[m], we have the following.Figure 2
The possiblepe ranges which have to be investigated for n∈(1;1.1).(i)
If there exists a concave outer free surface, appropriate for the growth of a tube of outer radiusre1=4.65·10-3[m], then according to Statement 3 this can be obtained for a value of  pe which is in the range (L1e(n),L2e(n))=(134.85;164.49)[Pa].(ii)
Taking into account the above fact, in order to create a concave outer free surface, appropriate for the growth of a tube of which outer radius is equal tore1=4.65·10-3[m], we have solved the IVP (2.8) for different values of pein the range (134.85;164.49)[Pa].More precisely, we have integrated the following system:(3.3)dzedr=-tanαedαedr=-1cosαe·[g·ρ·ze-peγ+1r·sinαe]
for ze(Rge)=0, ze′(Rge)=-tanαc and different pe. The obtained outer radii re versus pe are represented in Figure 3, which shows that the desired outer radius re1=4.65·10-3[m] is obtained for pe′=149.7[Pa].Figure 3
Outer radiire versus pe in the range (134.85;164.49)[Pa].Actually, as it can be seen on the same figure, forpe′=149.7[Pa], we can obtain also a second outer radius re2=3.8·10-3[m], which is not anymore in the desired range ((Rgi+Rge)/2,Rge). Moreover, the outer free surface of this meniscus is not globally concave; it is a convex-concave meniscus (Figure 4).Nonglobally concave outer free surface obtained forpe′=149.7[Pa].
(a)(b)Taking into accountpm≈0 [7, 23, 24], the melt column height in this case is He′=-[1/ρ·g]·[pe′+pge], where pge≥0 is the pressure of the gas flow (introduced in the furnace for release the heat from the outer side of the tube wall). When pge=0  [7, 24], then He′ is negative, He′=-2.31·10-3[m]; that is, the crucible melt level has to be with -He′=2.31·10-3[m] above the shaper top level. When pge=800[Pa], then  He′=-15·10-3[m]; that is, the crucible melt level has to be with -He′=15·10-3[m] above the shaper top level.Additional remarks are as follows.(i)
Forpe in the range (-∞,l3e)=(-∞,78.51)[Pa], according to Statement 2, the outer free surface of the meniscus is globally convex, and, according to Statement 1, such a meniscus is not appropriate for the growth of a tube of outer radius re1=4.65·10-3[m].(ii)
Forpe in the range (l3e,l1e)=(78.51;105.73)[Pa], according to Consequence 2, it is impossible to obtain concave outer free surface, which is appropriate for the growth of a tube of outer radius re1=4.65·10-3[m].(iii)
Forpe in the range (l1e,L1e(n))=(105.73;134.85)[Pa], according to the Statement 3, it is impossible to obtain concave outer free surface, which is appropriate for the growth of a tube of outer radius re1=4.65·10-3[m].(iv)
Forpe in the range (L2e(n);+∞)=(164.49;+∞)[Pa], according to the Statement 4, the outer free surface is concave and is appropriate for the growth of a tube which outer radius re is higher than re1=4.65·10-3[m].


### 3.2. Creation of an Appropriate Inner Free Surface

For the creation of the appropriate inner free surface, the following limits were considered(3.4)L1i(m)=1m-1·γ·αc+αg-π/2Rgi·cosαc-γRgi·sinαc,L2i(m)=1m-1·γ·αc+αg-π/2Rgi·sinαg+(m-1)·ρ·g·Rgi·tanαc-1m·γRgi·cosαg,L3i(m)=ρ·g·(m-1)·Rgi·tanαc+γRgi,l1i=2·γ·αc+αg-π/2Rge-Rgi·cosαc-γRgi·sinαc,l2i=2·γ·αc+αg-π/2Rge-Rgi·sinαg+ρ·g·(Rge-Rgi)2·tanαc-2·γRge+Rgi·cosαg,l3i=-γRgi·sinαc,l4i=-γRgi,
for m>1.These limits are represented form∈(1;1.1) in Figure 5. The computations were performed in MathCAD V.13 using the same numerical data as for the creation of the appropriate outer free surface.Figure 5
The possiblepi ranges which have to be investigated for m∈(1;1.1).Whenm1=1.03571, that is, ri′=0.00435[m], we obtain the following.(i)
If there exists a concave inner free surface for the growth of a tube of inner radiusri′=0.00435[m], then, according to the Statement 10, this can be obtained for a value of pi  which is in the range (L1i(m1),L2i(m1))=(-31.46,-1.07)[Pa].Taking into account the above fact, in order to create a concave inner free surface, appropriate for the growth of a tube which inner radius is equal tori′=0.00435[m], we have solved the IVP (2.16) for different values of pi in the range (L1i(m1),L2i(m1))=(-31.46,-1.07)[Pa]. More precisely, we have integrated the following system:
(3.5)dzidr=tanαi,dαidr=1cosαi·[g·ρ·zi-piγ-1r·sinαi]
for zi(Rgi)=0, zi′(Rgi)=tanαc and different pi. The obtained inner radii ri versus pi are represented in Figure 6, which shows that the desired inner radius ri1=4.35·10-3[m] is obtained for pi′=-16.2[Pa].Figure 6
Inner radiiri versus pi in the range (L1i(m1),L2i(m1))=(-31.46,-1.07)[Pa].Takingpm≈0 [7, 23, 24], the melt column height in this case is Hi′=-(1/ρ·g)·[pi′+pgi], where pgi≥0 is the pressure of the gas flow (introduced in the furnace for release the heat from the inner side of the tube wall). When pgi=0 [7, 24], then Hi′ is positive, Hi′=0.25·10-3[m]; that is, the crucible melt level has to be with Hi′=0.25·10-3[m] under the shaper top level. When pgi=800[Pa], then  Hi′=-12.1·10-3[m]; that is, the crucible melt level has to be with -He′=12.1·10-3[m] above the shaper top level.Additional remarks are as follows.(i)
Forpi in the range (-∞,l4i)=(-∞,-100)[Pa], according to Statement 9, the meniscus inner free surface is convex on the interval I∩[Rgi,((Rgi+Rge)/2)], and according to Statement 8, such a meniscus is not appropriate for the growth of a tube of inner radius ri′=0.00435[m].(ii)
Forpi in the range (l4i,l3i)=(-100,89.72)[Pa] if an appropriate static meniscus exists, then the inner free surface of the meniscus is not globally concave.(iii)
Ifpi is in the range (l3i,+∞)=(89.72,+∞)[Pa] and an appropriate nonglobally concave inner free surface exists, then pi is less than L3i(m1)=119.68[Pa].(iv)
Forpi in the range (L2i(m),+∞)=(-1.07,+∞)[Pa] the obtained inner free surface is appropriate for the growth of a tube of inner radius ri which is less than the desired radius ri′=0.00435[m].


### 3.3. Creation of a Concave Meniscus, Appropriate for the Growth of a Tube of Inner Radiusri and Outer Radius re

For creating a concave meniscus, appropriate for the growth of a tube with outer radiusre′=0.00465 [m] (n1=Rge/re=1.03226) and inner radius  ri′=0.00435 [m] (m1=Rgi/ri=1.03571) the melt column heights (with respect to the crucible melt level) have to be
(3.6)Hi1=-1ρ·g·[pi1+pgi]=-1ρ·g·[-16.2+pgi],He1=-1ρ·g·[pe1+pge]=-1ρ·g·[149.7+pge].In the following we will present three different procedures to create such a meniscus.Procedure A [He1=Hi1; pgi≠pge]
When the shaper outer top is at the same level as the shaper inner top, with respect to the crucible melt level, then the relationHe1=Hi1 holds and hence: (1/ρ·g)·[-16.2+pgi]=(1/ρ·g)·[149.7+pge]. It follows that the pressure of the gas flow, introduced in the furnace for releasing the heat from the inner wall of the tube, pgi, has to be higher than the pressure of the gas flow introduced in the furnace for release the heat from the outer wall of the tube, pge; pgi-pge=149.7+16.2=165.9[Pa].The results of the integration of the system (3.3) and (3.5) for pe=pe1=149.7[Pa] and pi=pi1=-16.2[Pa] when the shaper outer top is at the same level as the shaper inner top, with respect to the crucible melt level, are represented in Figure 7.Figure 7 shows also that the inner free surface is higher than the outer free surface: zi(ri1)=hia=2.865·10-4[m];ze(re1)=hea=2.860·10-4[m];hia-hea=0.005·10-4[m].Figure 7
zi(r) and ze(r) for pe=149.7[Pa];pi=-16.2[Pa] and He1=Hi1.Procedure B [pge=pgi; He1≠Hi1]
When the pressure of the gas flow introduced in the furnace for releasing the heat from the outer wall of the tube,pge, is equal to the pressure of the gas flow introduced in the furnace for release the heat from the inner wall of the tube pgi, then the equalities pge=-pe1-ρ·g·He1=-149.7-ρ·g·He1 and pgi=-pi1-ρ·g·Hi1=16.2-ρ·g·Hi1 imply the equality
(3.7)Hi1-He1=(1/ρ·g)·165.9=2.569·10-3[m].
.In other words, whenpge=pgi, then the level difference between the shaper inner top and outer top has to be Hi1-He1=2.569·10-3[m]. In this condition the inner surface top level hib with respect to the shaper outer top level is hib=Hi1-He1+hia=(1/ρ·g)·(-pe1+pi1)+hia=2.855·10-3[m] (see Figure 8).Figure 8
zi(r) and ze(r) for pi=-16.2[Pa] and pe=149.7[Pa] and pge=pgi.Hence the level difference between the inner free surface tophib and outer free surface  top heb=hea in this case is hib-heb=(1/ρ·g)·(-pe1+pi1)+(hia-hea)=2.5695·10-3[m].Due to the difficulties which can appear in this case in the creation of an appropriate thermal field, which assures the solidification in the right places, it is more appropriate to useHe1=Hi1 and gas flows having the property  (Procedure A) pgi=pge+(pi1-pe1)=pge+165.9[Pa]. For this choice the difference between the inner free surface height and outer free surface height with respect to the shaper top level is only hi-he=0.0005·10-3[m], and the creation of the thermal field, which assures the solidification in the right places, can be easier.Procedure C [pge≠pgi; He1≠Hi1]
Finally, to create an appropriate concave meniscus for which the inner meniscus is equal to the outer meniscus height, we will proceed by changing both the pressurespgi, pge of the gas flows and the inner and  outer top levels Hi,He of the shaper.
For example, if we takepge=100[Pa] for creating an appropriate outer free surface, then the crucible melt level has to be with -He1=(1/ρ·g)·[149.7+100][m]=38.67·10-4[m] above the shaper outer top level.
Now we takeHi1=He1-(hia-hea)=-38.67·10-4-0.005·10-4=-38.675·10-4[m].  After that, in order to satisfy pi=-pgi-ρ·g·Hi1=-16.2[Pa], we take the pressure pgi equal to pgi=16.2-ρ·g·(-38.675·10-4)=265.92[Pa]. For this choice, we will have an appropriate drop-like meniscus for which hic=hec, as it is shown in Figure 9, obtained by integration of systems (3.3) and (3.5) for the given numerical data.Figure 9
zi(r) and ze(r) for pi=-16.2[Pa]  and pe=149.7[Pa]; pge=100[Pa];pgi=265.92[Pa];He1=-38.67·10-4[m];Hi1=-38.675·10-4[m].


## 3.1. Creation of the Outer Free Surface

For the creation of the appropriate outer free surface, the following limits were considered:(3.1)L1e(n)=nn-1·γ·αc+αg-π/2Rge·cosαc+γRge·cosαg,L2e(n)=nn-1·γ·αc+αg-π/2Rge·sinαg+n-1n·ρ·g·Rge·tanαc+n·γRge·sinαc,L3e(n)=n-1n·ρ·g·Rge·tanαc+n·γRge·cosαc,L4e(n)=n-1n·ρ·g·Rge·tanαc+n·γRge,l1e=2·γ·αc+αg-π/2Rge-Rgi·cosαc+γRge·cosαg,l2e=2·γ·αc+αg-π/2Rge-Rgi·sinαg+ρ·g·(Rge-Rgi)2·tanαc+2·γRge-Rgi·sinαc,l3e=γRge·sinαc
for n>1.These limits are represented forn∈(1;1.1) in Figure 2. The computations were performed in MathCAD V.13 using the following numerical data
(3.2)Rge=4.8·10-3[m];Rgi=4.2·10-3[m];αc=63.80=1.1135[rad];αg=28.90=0.5044[rad];ρ=6582[kg/m3];γ=4.2·10-1[N/m];g=9.81[m/s2].
When n1=1.03226, that is, re1=4.65·10-3[m], we have the following.Figure 2
The possiblepe ranges which have to be investigated for n∈(1;1.1).(i)
If there exists a concave outer free surface, appropriate for the growth of a tube of outer radiusre1=4.65·10-3[m], then according to Statement 3 this can be obtained for a value of  pe which is in the range (L1e(n),L2e(n))=(134.85;164.49)[Pa].(ii)
Taking into account the above fact, in order to create a concave outer free surface, appropriate for the growth of a tube of which outer radius is equal tore1=4.65·10-3[m], we have solved the IVP (2.8) for different values of pein the range (134.85;164.49)[Pa].More precisely, we have integrated the following system:(3.3)dzedr=-tanαedαedr=-1cosαe·[g·ρ·ze-peγ+1r·sinαe]
for ze(Rge)=0, ze′(Rge)=-tanαc and different pe. The obtained outer radii re versus pe are represented in Figure 3, which shows that the desired outer radius re1=4.65·10-3[m] is obtained for pe′=149.7[Pa].Figure 3
Outer radiire versus pe in the range (134.85;164.49)[Pa].Actually, as it can be seen on the same figure, forpe′=149.7[Pa], we can obtain also a second outer radius re2=3.8·10-3[m], which is not anymore in the desired range ((Rgi+Rge)/2,Rge). Moreover, the outer free surface of this meniscus is not globally concave; it is a convex-concave meniscus (Figure 4).Nonglobally concave outer free surface obtained forpe′=149.7[Pa].
(a)(b)Taking into accountpm≈0 [7, 23, 24], the melt column height in this case is He′=-[1/ρ·g]·[pe′+pge], where pge≥0 is the pressure of the gas flow (introduced in the furnace for release the heat from the outer side of the tube wall). When pge=0  [7, 24], then He′ is negative, He′=-2.31·10-3[m]; that is, the crucible melt level has to be with -He′=2.31·10-3[m] above the shaper top level. When pge=800[Pa], then  He′=-15·10-3[m]; that is, the crucible melt level has to be with -He′=15·10-3[m] above the shaper top level.Additional remarks are as follows.(i)
Forpe in the range (-∞,l3e)=(-∞,78.51)[Pa], according to Statement 2, the outer free surface of the meniscus is globally convex, and, according to Statement 1, such a meniscus is not appropriate for the growth of a tube of outer radius re1=4.65·10-3[m].(ii)
Forpe in the range (l3e,l1e)=(78.51;105.73)[Pa], according to Consequence 2, it is impossible to obtain concave outer free surface, which is appropriate for the growth of a tube of outer radius re1=4.65·10-3[m].(iii)
Forpe in the range (l1e,L1e(n))=(105.73;134.85)[Pa], according to the Statement 3, it is impossible to obtain concave outer free surface, which is appropriate for the growth of a tube of outer radius re1=4.65·10-3[m].(iv)
Forpe in the range (L2e(n);+∞)=(164.49;+∞)[Pa], according to the Statement 4, the outer free surface is concave and is appropriate for the growth of a tube which outer radius re is higher than re1=4.65·10-3[m].


## 3.2. Creation of an Appropriate Inner Free Surface

For the creation of the appropriate inner free surface, the following limits were considered(3.4)L1i(m)=1m-1·γ·αc+αg-π/2Rgi·cosαc-γRgi·sinαc,L2i(m)=1m-1·γ·αc+αg-π/2Rgi·sinαg+(m-1)·ρ·g·Rgi·tanαc-1m·γRgi·cosαg,L3i(m)=ρ·g·(m-1)·Rgi·tanαc+γRgi,l1i=2·γ·αc+αg-π/2Rge-Rgi·cosαc-γRgi·sinαc,l2i=2·γ·αc+αg-π/2Rge-Rgi·sinαg+ρ·g·(Rge-Rgi)2·tanαc-2·γRge+Rgi·cosαg,l3i=-γRgi·sinαc,l4i=-γRgi,
for m>1.These limits are represented form∈(1;1.1) in Figure 5. The computations were performed in MathCAD V.13 using the same numerical data as for the creation of the appropriate outer free surface.Figure 5
The possiblepi ranges which have to be investigated for m∈(1;1.1).Whenm1=1.03571, that is, ri′=0.00435[m], we obtain the following.(i)
If there exists a concave inner free surface for the growth of a tube of inner radiusri′=0.00435[m], then, according to the Statement 10, this can be obtained for a value of pi  which is in the range (L1i(m1),L2i(m1))=(-31.46,-1.07)[Pa].Taking into account the above fact, in order to create a concave inner free surface, appropriate for the growth of a tube which inner radius is equal tori′=0.00435[m], we have solved the IVP (2.16) for different values of pi in the range (L1i(m1),L2i(m1))=(-31.46,-1.07)[Pa]. More precisely, we have integrated the following system:
(3.5)dzidr=tanαi,dαidr=1cosαi·[g·ρ·zi-piγ-1r·sinαi]
for zi(Rgi)=0, zi′(Rgi)=tanαc and different pi. The obtained inner radii ri versus pi are represented in Figure 6, which shows that the desired inner radius ri1=4.35·10-3[m] is obtained for pi′=-16.2[Pa].Figure 6
Inner radiiri versus pi in the range (L1i(m1),L2i(m1))=(-31.46,-1.07)[Pa].Takingpm≈0 [7, 23, 24], the melt column height in this case is Hi′=-(1/ρ·g)·[pi′+pgi], where pgi≥0 is the pressure of the gas flow (introduced in the furnace for release the heat from the inner side of the tube wall). When pgi=0 [7, 24], then Hi′ is positive, Hi′=0.25·10-3[m]; that is, the crucible melt level has to be with Hi′=0.25·10-3[m] under the shaper top level. When pgi=800[Pa], then  Hi′=-12.1·10-3[m]; that is, the crucible melt level has to be with -He′=12.1·10-3[m] above the shaper top level.Additional remarks are as follows.(i)
Forpi in the range (-∞,l4i)=(-∞,-100)[Pa], according to Statement 9, the meniscus inner free surface is convex on the interval I∩[Rgi,((Rgi+Rge)/2)], and according to Statement 8, such a meniscus is not appropriate for the growth of a tube of inner radius ri′=0.00435[m].(ii)
Forpi in the range (l4i,l3i)=(-100,89.72)[Pa] if an appropriate static meniscus exists, then the inner free surface of the meniscus is not globally concave.(iii)
Ifpi is in the range (l3i,+∞)=(89.72,+∞)[Pa] and an appropriate nonglobally concave inner free surface exists, then pi is less than L3i(m1)=119.68[Pa].(iv)
Forpi in the range (L2i(m),+∞)=(-1.07,+∞)[Pa] the obtained inner free surface is appropriate for the growth of a tube of inner radius ri which is less than the desired radius ri′=0.00435[m].


## 3.3. Creation of a Concave Meniscus, Appropriate for the Growth of a Tube of Inner Radiusri and Outer Radius re

For creating a concave meniscus, appropriate for the growth of a tube with outer radiusre′=0.00465 [m] (n1=Rge/re=1.03226) and inner radius  ri′=0.00435 [m] (m1=Rgi/ri=1.03571) the melt column heights (with respect to the crucible melt level) have to be
(3.6)Hi1=-1ρ·g·[pi1+pgi]=-1ρ·g·[-16.2+pgi],He1=-1ρ·g·[pe1+pge]=-1ρ·g·[149.7+pge].In the following we will present three different procedures to create such a meniscus.Procedure A [He1=Hi1; pgi≠pge]
When the shaper outer top is at the same level as the shaper inner top, with respect to the crucible melt level, then the relationHe1=Hi1 holds and hence: (1/ρ·g)·[-16.2+pgi]=(1/ρ·g)·[149.7+pge]. It follows that the pressure of the gas flow, introduced in the furnace for releasing the heat from the inner wall of the tube, pgi, has to be higher than the pressure of the gas flow introduced in the furnace for release the heat from the outer wall of the tube, pge; pgi-pge=149.7+16.2=165.9[Pa].The results of the integration of the system (3.3) and (3.5) for pe=pe1=149.7[Pa] and pi=pi1=-16.2[Pa] when the shaper outer top is at the same level as the shaper inner top, with respect to the crucible melt level, are represented in Figure 7.Figure 7 shows also that the inner free surface is higher than the outer free surface: zi(ri1)=hia=2.865·10-4[m];ze(re1)=hea=2.860·10-4[m];hia-hea=0.005·10-4[m].Figure 7
zi(r) and ze(r) for pe=149.7[Pa];pi=-16.2[Pa] and He1=Hi1.Procedure B [pge=pgi; He1≠Hi1]
When the pressure of the gas flow introduced in the furnace for releasing the heat from the outer wall of the tube,pge, is equal to the pressure of the gas flow introduced in the furnace for release the heat from the inner wall of the tube pgi, then the equalities pge=-pe1-ρ·g·He1=-149.7-ρ·g·He1 and pgi=-pi1-ρ·g·Hi1=16.2-ρ·g·Hi1 imply the equality
(3.7)Hi1-He1=(1/ρ·g)·165.9=2.569·10-3[m].
.In other words, whenpge=pgi, then the level difference between the shaper inner top and outer top has to be Hi1-He1=2.569·10-3[m]. In this condition the inner surface top level hib with respect to the shaper outer top level is hib=Hi1-He1+hia=(1/ρ·g)·(-pe1+pi1)+hia=2.855·10-3[m] (see Figure 8).Figure 8
zi(r) and ze(r) for pi=-16.2[Pa] and pe=149.7[Pa] and pge=pgi.Hence the level difference between the inner free surface tophib and outer free surface  top heb=hea in this case is hib-heb=(1/ρ·g)·(-pe1+pi1)+(hia-hea)=2.5695·10-3[m].Due to the difficulties which can appear in this case in the creation of an appropriate thermal field, which assures the solidification in the right places, it is more appropriate to useHe1=Hi1 and gas flows having the property  (Procedure A) pgi=pge+(pi1-pe1)=pge+165.9[Pa]. For this choice the difference between the inner free surface height and outer free surface height with respect to the shaper top level is only hi-he=0.0005·10-3[m], and the creation of the thermal field, which assures the solidification in the right places, can be easier.Procedure C [pge≠pgi; He1≠Hi1]
Finally, to create an appropriate concave meniscus for which the inner meniscus is equal to the outer meniscus height, we will proceed by changing both the pressurespgi, pge of the gas flows and the inner and  outer top levels Hi,He of the shaper.
For example, if we takepge=100[Pa] for creating an appropriate outer free surface, then the crucible melt level has to be with -He1=(1/ρ·g)·[149.7+100][m]=38.67·10-4[m] above the shaper outer top level.
Now we takeHi1=He1-(hia-hea)=-38.67·10-4-0.005·10-4=-38.675·10-4[m].  After that, in order to satisfy pi=-pgi-ρ·g·Hi1=-16.2[Pa], we take the pressure pgi equal to pgi=16.2-ρ·g·(-38.675·10-4)=265.92[Pa]. For this choice, we will have an appropriate drop-like meniscus for which hic=hec, as it is shown in Figure 9, obtained by integration of systems (3.3) and (3.5) for the given numerical data.Figure 9
zi(r) and ze(r) for pi=-16.2[Pa]  and pe=149.7[Pa]; pge=100[Pa];pgi=265.92[Pa];He1=-38.67·10-4[m];Hi1=-38.675·10-4[m].


## 4. The Effect of the Choice of the Shaper Radii

In this sequence we intend to evaluate numerically the effect of the choice of the shaper radii for the growth of a tube which outer radius is equal tore, and inner radius is equal to ri.For this purpose, the procedure described in sequence 3 will be applied for the growth of a tube having the same outer radiusre′=0.00465 [m] and inner radius ri′=0.00435 [m], but the shaper radii changed: the inner radius of the shaper now is Rgi=4·10-3[m] (instead of 4.2·10-3[m]), and the outer radius of the shaper now is Rge=5·10-3[m]  (instead of 4.8·10-3[m]).Applying the procedure in this case we find that to create a static meniscus which outer free surface is appropriate for the growth,pe has to be pe=124.38[Pa],  and to create a static meniscus which inner free surface is appropriate for the growth, pi has to be pi=-41.35[Pa]. The so-obtained outer and inner free surfaces are stable. The inner free surface height is hi=6.57·10-4[m], and the outer free surface height is he=6.603·10-4[m].When we take the shaper inner top at the same level as the shaper outer top, then the pressure difference between the pressure of the inner gas flowpgi and the outer gas flow pge has to be pgi-pge=pe-pi=165.73[Pa], and the level difference between the inner and outer free surface top is equal to hi-he=-0.033·10-4[m].When we take the pressure of the inner gas flowpgi equal to the pressure of the outer gas flow pge, then the level difference between the shaper inner top Hi and the shaper outer top He has to be Hi-He=(1/ρ·g)·[pi-pe]=2.567·10-3[m], and the level difference between the inner and outer free surface top is equal to the difference Hi+hi-(He+he)=(1/ρ·g)·[pi-pg]+hi-he=2.5637·10-3[m].


## 5. Conclusions

(i)
Stable and drop-like static meniscus, appropriate for the growth of a single-crystal cylindrical tube with a priori specified inner and outer radii, can be created by the choice of the difference between the inner top level and outer top level of the shaper, or by the choice of the pressure of the gas flows, introduced inside and outside of the tube for release the heat from the tube walls.(ii)
The values of the differences between the inner and outer top level of the shaper, or the pressure in the inner and outer gas flows, depend on the shaper radii. This dependence can be relevant.(iii)
The realization of the thermal conditions, which assure that for the obtained static meniscus the solidification conditions are satisfied at the right places, is easier when we take the shaper inner top at the same level as the shaper outer top, and we use inner and outer gas flows having different pressures (pgi≠pge). The setting of the thermal conditions is not discussed in this paper.



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*Source: 348538-2009-07-29.xml*