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Solutions of this equation candescribe the outer or inner free surface of a static meniscusthe static liquid bridge free surfacebetween the shaper and the crystal surface occurring in si

Volume 2009, Article ID 732106, 28 pages

doi:10.1155/2009/732106

Research Article

Inequalities for Single Crystal Tube Growth by

Edge-Defined Film-Fed Growth Technique

1 Department of Computer Science, Faculty of Mathematics and Computer Science,

West University of Timisoara, Blv V.Parvan 4, 300223 Timisoara, Romania

2 Faculty of Physics, West University of Timisoara, Blv V.Parvan 4, 300223 Timisoara, Romania

Correspondence should be addressed to Agneta M Balint,balint@physics.uvt.ro

Received 3 January 2009; Accepted 29 March 2009

Recommended by Yong Zhou

The axi-symmetric Young-Laplace differential equation is analyzed Solutions of this equation candescribe the outer or inner free surface of a static meniscusthe static liquid bridge free surfacebetween 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 p which represents the

controllable part of the pressure difference across the free surface Inequalities are established for pwhich are necessary or sufficient conditions for the existence of solutions which represent a stableand convex outer or inner free surfaces of a static meniscus The analysis is numerically illustratedfor the static menisci occurring in silicon tube growth by edge-defined film-fed growthEFGs

technique This kind of inequalities permits the adequate choice of the process parameter p With

this aim this study was undertaken

Copyrightq 2009 S Balint and A M Balint This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited

1 Introduction

The first successful Si tube growth was reported in1 Also a theory of tube growth by E.F.G.process is developed there to show the dependence of the tube wall thickness on the growthvariables The theory concerns the calculation of the shape of the liquid-vapor interfacei.e.,the free surface of the meniscus and of the heat flow in the system The inner and the outerfree surface shapes of the meniscusFigure 1 were calculated from Young-Laplace capillaryequation, in which the pressure difference Δp across a point on the free surface is considered

to be Δp ρ · g · Heff constant, where Heffrepresents the effective height of the growthinterface Figure 1 The above approximation of Δp is valid when Heff  h, where h is

the height of the growth interface above the shaper top Another approximation used in

1 is that the outer and inner free surface 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 in1 has been shown to beuseful tool understanding the feasible limits of the wall thickness control A more accuratepredictive model would require an increase of the acceptable tolerance range introduced byapproximation.

The growth process was scaled up by Kaljes et al in2 to grow 15 cm diameter silicontubes It has been realized that theoretical investigations are necessary for the improvement

of the technology Since the growth system consists of a small die type1 mm width and

a thin tubeorder of 200 μm wall thickness, the width of the melt/solid interface and the

meniscus are accordingly very small Therefore, it is essential to obtain accurate solution forthe free surface of the meniscus, the temperature, and the liquid-crystal interface position inthis tinny region

For single crystal tube growth by edge-defined film-fed growthE.F.G. technique, inhydrostatic approximation the free surface of a static meniscus is described by the Young-Laplace capillary equation3:

Here γ is the melt surface tension, ρ denotes the melt density, g is the gravity acceleration,

surface, z is the coordinate of M with respect to the Oz axis, directed vertically upwards, and p is the pressure difference across the free surface For the outer free surface, p p e

and inner walls of the tube, respectively, and H 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 H > 0, and when the crucible melt level is above the shaper top level, then H < 0seeFigure 1

To calculate the outer and inner free surface shapes of the static meniscus, it isconvenient to employ the Young-Laplace1.1 in its differential form This form of the 1.1can be obtained as a necessary condition for the minimum of the free energy of the meltcolumn3

For the growth of a single crystal tube of inner radius r i ∈ R gi , R gi ge /2 and outer radius r e ∈ R gi ge /2, R ge the axi-symmetric differential equation of the outer freesurface is given by

Inner gas flow

Outer gas flow

r i

r e

Tube

Outer free surface Inner

free surface Meniscus melt Shaper Capillary channel

Figure 1: Axisymmetric meniscus geometry in the tube growth by E.F.G method.

The axi-symmetric differential equation of the inner free surface is given by

of a liquid meniscus, there are no complete analysis and solution For the general equationonly numerical integrations were carried out for a number of process parameter values thatwere 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 authorspresent theoretical and numerical study of meniscus dynamics under axi-symmetric andasymmetric configurations In 6 the meniscus free surface is approximated by an arc of

constant curvature, and a meniscus dynamics model is developed to consider meniscusshape and its dynamics, heat and mass transfer around the die-top and meniscus Analysisreveals the correlations among tube wall thickness, effective melt height, pull-rate, die-toptemperature, 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 p, 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 notapproximated as in1,6, by an arc with constant curvature, and the pressure of the gas flowintroduced 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

Consider the differential equation

and α c , α g such that 0 < α c < π/2 − α g , α g ∈ 0, π/2.

meniscus on the interval r e , R ge  R gi ge /2 < r e < R ge if possesses the followingproperties:

a zr e  − tanπ/2 − α g ,

b zR ge  − tan α c , and

c zR ge  0 and zr is strictly decreasing on r e , R ge

The described outer free surface is convex onr e , R ge if in addition the following inequalityholds:

d zr > 0 ∀r ∈ r e , R ge .

Theorem 2.2 If there exists a solution of 2.1, which describes a convex outer free surface of a static

Proof Let z r be a solution of 2.1, which describes a convex outer free surface of a staticmeniscus on the closed intervalr e , R ge  and αr − arctan zr The function αr verifies

Since zr > 0 on r e , R ge , zr is strictly increasing and αr − arctan zr is strictly

decreasing onr e , R ge , therefore the values of αr are in the range α c , π/2 − α g and thefollowing inequalities hold:

Equality2.5 and inequalities 2.6 imply inequalities 2.2

Corollary 2.3 If r e R ge /n with 1 < n < 2 · R ge / R gi ge , then inequalities 2.2 become

Corollary 2.4 If n → 2 · R ge / R gi ge , then r e → R gi ge /2 and 2.7 becomes

interval on which the function zr exists and by αr the function αr − arctan zr defined on I Remark that for αr the equality 2.3 holds

It is clear that r∗≥ 0 and for any r ∈ r∗, R ge inequalities 2.12 hold

From 2.12 and 2.13 it follows that zr is strictly increasing and bounded on

r∗, R ge  Therefore zr∗ r → r∗, r>r∗zr exists and satisfies

− tan

2 − α g



Moreover, since zr is strictly decreasing and possesses bounded derivative on r∗, R ge,

z r∗ r → r∗, r>r∗z r exists too, it is finite, and satisfies

We will show now that r∗ > R ge /n and zr∗ g In order to show that

for some r ∈ r∗, R ge  Hence αr∗ g This last inequality is impossible, since

according to the inequality2.14, we have αr∗ g Therefore, r∗, defined by

2.14, satisfies r∗> R ge /n.

In order to show that zr∗ g we remark that from the definition

2.14 of r∗it follows that at least one of the following three equalities holds:

2 − α g



, zr∗ 2.17

Since zr∗ r ≤ − tan α c for any r ∈ r∗, R ge  it follows that the equality zr∗

− tan α c is impossible Hence, we obtain that at r∗at least one of the following two equalities

holds: zr∗ r∗ g  We show now that the equality zr∗

is impossible For that we assume the contrary, that is, zr∗

In this way we obtain that the equality zr∗ g holds.

For r e r∗ the solution of the initial value problem 2.8 on the interval r e , R gedescribes a convex outer free surface of a static meniscus

Corollary 2.6 If for p e the following inequality holds:

Corollary 2.7 If for 1 < n< n < 2 · R ge / R gi ge  the following inequalities hold:

2.10 on the interval r e , R ge  describes a convex outer free surface of a static meniscus The existence

That is because zR ge  > 0 if and only if αR ge  −zR ge · cos2α c < 0, that is,

implies the existence of r∈ I, 0 < r< R ge such that αr > α c , z r > 0 and p e ρ · g · zr

γ/r· sin αr > γ/R ge · sin α c, what is impossible

Theorem 2.9 If a solution z1 z1r of 2.1 describes a convex outer free surface of a static

energy functional of the melt column:

and Jacobi conditions are satisfied in this case

Denote by Fr, z, z the function defined as

r, z, z

r ·

1

Hence, the Legendre condition is satisfied

The Jacobi equation

For2.27 the following inequalities hold:

is a Sturm type upper bound for2.27 7

Since every nonzero solution of2.29 vanishes at most once on the interval r e , R ge,

the solution ηr of the initial value problem

d dr

has only one zero on the intervalr e , R ge 7 Hence the Jacobi condition is satisfied

meniscus is said to be stable if it is a weak minimum of the energy functional of the meltcolumn

static meniscus on the intervalr e , R ge, then it is stable

Theorem 2.12 If the solution z zr of the initial value problem 2.10 is concave (i.e., z”r < 0)

Proof zr < 0 on r e , R ge  implies that zr is strictly decreasing on r e , R ge  Hence zr e  >

zR ge  − tan α c > − tanπ/2 − α g

Theorem 2.13 If p e > γ/R ge ·sin α c and there exists r e ∈ R gi ge /2, R ge  such that the solution

the following inequalities hold:

represent the outer free surface of a static meniscus on the closed intervalr e , R ge  Let αr

be defined as inTheorem 2.2 for r ∈ r e , R ge  Since p e > γ/R ge · sin α c , we have zR ge

−1/cos2α c  · αR ge  < 0 Hence αR ge  > 0 and therefore αr < αR ge  α c for r < R ge , r

close to R ge Taking into account the fact that αr e  π/2 − α e > α cit follows that there exists

r∈ r e , R ge  such that αr 0

Therefore p e ρ · g · zr · sin αr Since 0 ≤ αr ≤ π/2 − α g and r e < r, the

following inequality holds: γ/r· sin αr < γ/r e · cos α g On the other hand zr < zr e ≤

R ge − r e  · tanπ/2 − α g Using the above evaluations we obtain inequalities 2.31

2· R ge / R gi ge, then inequality 2.31 becomes

Hence: zr −1/cos2α r · αr ≤ 0 for r ∈ I ∩ R ge /n, R ge

3 Meniscus Inner Free Surface Shape Analysis in

the Case of Tube Growth

Consider now the differential equation

Definition 3.1 A solution z zx of 3.1 describes the inner free surface of a static meniscus

on the intervalR gi , r i  R gi < r i < R gi ge /2 if possesses the following properties:

a zR gi  tan α c ,

b zr i  tanπ/2 − α g , and

c zR gi  0 and zr is strictly increasing on R gi , r i

The described inner free surface is convex onR gi , r i if in addition the following inequalityholds:

d zr > 0 ∀r ∈ Rgi, r i

Theorem 3.2 If there exists a solution of 3.1, which describes a convex inner free surface of a static

meniscus on the closed intervalR gi , r i  and αr arctan zr The function αr verifies the

equation

αr ρ · g · zr − p i

γ ·cos αr1 −1

r · tan αr 3.3and the boundary conditions

Since zr > 0 on R gi , r i , zr is strictly increasing and αr arctan zr is strictly

increasing onR gi , r i, therefore the following inequalities hold:

Corollary 3.3 If r i m · R gi with 1 < m < R gi ge /2 · R gi , then inequalities3.2 become

interval on which the function zr exists and by αr the function αr arctan zr defined

on I Remark that for αr the following equality holds:

It is clear that r∗≤ R gi ge /2 and for any r ∈ R gi , r∗ inequalities 3.13 hold Moreover,

zr∗− 0 limr → r∗, r<r∗zr exists and satisfies, zr∗− 0 > 0 and zr∗− 0 ≤ tanπ/2 − α g

Hence zr∗− 0 limr → r∗, r<r∗z r is finite, it is strictly positive, and for every r ∈ R gi , r∗ thefollowing inequalities hold:

We will show now that r∗≤ m · R gi and zr∗− 0 tanπ/2 − α g

In order to show that r∗≤ m · R gi , we assume the contrary, that is, r∗ > m · R gi Underthis hypothesis we have

for some r ∈ R gi , r∗ Hence αr∗− 0 > π/2 − α g and it follows that there exists r1 such

that R gi < r1 < r∗ and αr1 π/2 − α g This last inequality is impossible according to the

definition of r∗

Therefore, r∗defined by3.14 satisfies r∗< m · R gi.

In order to show that zr∗− 0 tanπ/2 − α g we remark that from the definition

3.14 of r∗it follows that at least one of the following three equalities holds:

We show now that the equality zr∗− 0 0 is impossible For this we assume the contrary,

that is, zr∗− 0 0 Under this hypothesis, from 3.11, we have

In this way we obtain that the equality zr∗− 0 tanπ/2 − α g holds

For r i r∗ the solution of the initial value problem 3.10 on the interval R gi , r idescribes a convex inner free surface of a static meniscus

Corollary 3.6 If for p i the following inequality holds

Corollary 3.7 If for 1 < m< m < R gi ge /2 · R gi the following inequalities hold:

then there exists r i in the interval m· R gi , m · R gi  such that the solution of the initial value problem

3.10 on the interval R gi , r i  describes a convex inner free surface of a static meniscus.

The existence of r i and the inequality r i ≤ m · R gi follows from Theorem 3.5 The

inequality r i ≥ m· R gifollows from theCorollary 3.3

Theorem 3.8 If a solution z1 z1r of 3.1 describes a convex inner free surface of a static

energy functional of the melt column:

meniscus is said to be stable if it is a weak minimum of the energy functional of the meltcolumn

static meniscus on the intervalR gi , r i, then it is stable

if and only if

p i <− γ

R gi · sin α c 3.23

Theorem 3.12 If zr represents the inner free surface of a static meniscus on the closed interval

R gi , r i  r i < R gi ge /2 which possesses the following properties:

a zr is convex at R gi , and

b the shape of zr changes once on the interval R gi , r i , that is, there exists a point r ∈

R gi , r i  such that zr > 0 for r ∈ R gi , r i , zr 0 and zr < 0 for r ∈ r, r i ,

then there exists r i1 ∈ R gi , r i  such that zr1

i  tanπ/2 − α g  and for p i the following inequality holds:

i  π/2 − α g The maximum value αr

is less than π/2 From3.11 we have

p i ρ · g · zr

− γ

r · sin αr