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Theoretical model of crystal growth shaping process
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C N Y I T A L
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ELSEVIER
Journal of Crystal Growth 180 (1997)615 626
Theoretical model of crystal growth shaping process V.A. T a t a r c h e n k o a'*, V.S. U s p e n s k i b, E.V. T a t a r c h e n k o b, J . P h . N a b o t ~, T. D u f f a f f , B. R o u x d a Crismatec, BP 52, F-77794 Nemours Cedex, France blnstitute for Mechanics of Moscow State University, Moscow, Russian Federation c Centre d 'Etudes Nuclbaires de Grenoble, Grenoble, France dlnstitut de Recherche sur les Phdnomenes Hors d 'Equilibre, UMR 138, Marseille, Franee
Abstract
A theoretical investigation of the crystal growth shaping process is carried out on the basis of the dynamic stability concept. The capillary dynamic stability of shaped crystal growth processes for various forms of the liquid menisci is analyzed using the mathematical model of the phenomena in the axisymmetric case. The catching boundary condition of the capillary boundary problem is considered and the limits of its application for shaped crystal growth modeling are discussed. The static stability of a liquid free surface is taken into account by means of the JacObi equation analysis. The result is that a large number of menisci having drop-like shapes are statically unstable. A few new non-traditional liquid meniscus shapes (e.g., bubbles and related shapes) are proposed for the case of a catching boundary condition. Keywords:
Shaped crystal growth; Capillarity; Liquid menisci; Catching boundary condition; Stability
I. Introduction
During recent years, much attention has been paid to the growth of shaped single crystals by techniques of pulling from the melt such as the Stepanov Technique [1], Edge Defined Film Fed Growth [2], Capillary Action Shaping Technique [3] and some others. The main idea of all these techniques is to use a shaper (Fig. 1), which limits the area of the liquid free surface and its perturba-
* Corresponding author.
tions occurring in the Czochralski Technique [4]. In this way, it contributes to controlled shaping. According to our analysis [5-9], two approaches can be used to achieve this purpose: shaping by the edge of the shaper (Fig. la-Fig, lc) or by its surface (Fig. ld and Fig. le). All these techniques use the principles of shaping and can be distinguished only by design features. Therefore, it has been suggested in [8] to use a common title (TPS - Techniques of Pulling from Shaper) for the analysis of the shaping conditions. The theory of shaped crystal growth by the TPS is based on the stable-growth concept which means
0022-0248/97/$17.00 Copyright ~5~ 1997 ElsevierScienceB.V. All rights reserved PI1 S0022-0248(97)00293-5
616
IdA. Tatarchenko et al. / Journal q]" Crystal Growth 180 (1997) 615--626
Z
Z
ra z #
i
_
a
_
2R
z/
0
b
c
d
e
Fig. 1. Different schemes of melt growth of a crystalline rod by the technique of pulling from shaper at various melt levels d. R: crystal radius, h: crystallization front height, rd: shaper edge radius, 0: wetting angle, :~d:catching angle of meniscus.
that self-stabilization is present in the system: if growth parameter deviations occur under the action of external perturbations, then they will fade with time. The dynamic stability of shaped crystal growth processes was proposed in Refs. [10, 11], and was studied in detail and reviewed in Refs. [8, 9]. Extensive investigations are now under way for the application of such analysis to different techniques of capillary shaping. This work gives the theoretical study of the influence of melt-shaper contact conditions on the dynamic and static capillary stability of the shaped crystal growth process in axisymmetric cases. We focus our attention on the effect of the catching boundary condition (Fig. la-Fig, lc). The wetting boundary condition (Fig. ld and Fig. le) will be studied in a future paper.
2. Equation for change rate of crystal dimension On the basis of our experience and previous investigations, we assert that the full and complete study of the stability of TPS crystal growth must include at least two independent degrees of freedom: the crystal dimension and a quantity which
affects the value of the contact angle between melt and crystal surface, e.g., the meniscus height h. In the present work, we study only the influence of capillary effects by means of solutions of the Laplace capillary equation. Therefore, we will leave out the heat transfer which affects the meniscus height. The differential equation for the change rate of the crystal radius is based on the phenomenon of growth angle constancy [8, 9]. Let us consider the diagram given in Fig. 2. Vector R lies within the diagram plane and represents the radius of the crystal. The lateral surface of the growing crystal makes an angle ~p with the line tangent to the meniscus (Fig. 2a and Fig. 2b). The angle 0o is called the crystal growth angle and is a physical characteristic of the material. Here, we do not take into consideration the effects of anisotropy, of very high growth rates and of the Gibbs-Thompson energy (for a very large curvature of the crystal surface) on the growth angle. Now, let us introduce the angle :~ made by the tangent to the meniscus at the three-phase point (with positive direction of the horizontal line), and the angle 0 = c~- ~/2 (Fig. 2). A crystal of constant radius R ° grows in vertical direction if ~k = ~o or if :~ = :~0, where eo = ~o + re/2 (Fig. 2a). From the
617
KA. Tatarchenkoet al. / Journal of Crystal Growth 180 (1997) 615-626 z z
oL
\ "t 2
b
[
- ~ arctgz:
Fig. 2. Crystal growth in the cases of capillaryshaping: (a) ~ = ~o: growth of crystals of constant cross-section;(b) :~ < 70: growth of crystals of widening cross-section;(c) c~> ~o: growth of crystals of narrowing cross-section;V is the crystal displacement rate.
geometric diagram given in Fig. 2b, the following algebraic equation governing the change of crystal radius with crystallization rate Vc can be obtained: 6/~ = / ~ = Vc tan(~ - ~o) = Vc tan(~),
lary problem is solved, i.e., the function ,(R,h) has been found, Eq. (3) can be written in the following form:
(1) ~/~= V ~ (
where 6/~ = R - Ro (but here/~o = 0), 8c~ = e - ~o is the angle between the growing crystal lateral surface and the vertical direction (Fig. 2b). The crystallization rate V¢ is equal to the difference between the pulling rate V, and the crystallization front displacement dh/dt: dh V¢ = V - d~-
(4)
Here R ° and h ° are the values of R and h for a = ao. Thus, the equation for the change rate of the crystal cross-section is obtained. From Eq. (4) it follows that for a two-degree of freedom problem the main coefficients of the stability matrix [-7-9] take the form
(2)
We look for the parameters of this equation close to the stationary state, which is characterized by constant values of the radius (i.e., 8~ = 0) and of the meniscus height. Considering the behavior of the system close to the stationary state, we have to expand tan(Sa) into a Taylor series in the vicinity of the stationary point and to keep the first-order infinitesimal term only: V -
°) . /
ARR = V ~a/DR,
R =
-Ro)+~(h-h
tan(Be) ~ V 8~.
(3)
With the angle e being a function of the crystal dimension, the crystallization front position can be determined by the capillary problem which will be discussed and solved later. Assuming that the capil-
ARh = V ~/'~h.
(5)
The explicit form of the coefficients AhR and Ahh can be found in a similar way from the heat-balance conditions at the interface [8, 9], but we do not need their explicit forms for the present analysis since we will focus our attention on the capillary effects only, i.e., we look for conditions of crystallization that give negative values to the coefficient ARR.
3. C a p i l l a r y p r o b l e m f o r m u l a t i o n
3.1. Meniscus surface equation
To define t h e explicit form of the coefficient ARR in Eq. (5) the capillary problem for crystal
V.A. Tatarchenko et al. / Journal of Crystal Growth 180 (1997) 615 626
618
growth by TPS has to be solved. This problem includes the calculation of the melt meniscus shape on the basis of the Laplace capillary equation taking into account the Navier-Stokes equation for dynamic effects. In Refs. [8, 9] it was shown that for real pulling conditions the hydrostatic approximation is applicable and, hence, the equilibrium shape of the liquid surface is described by the Laplace capillary equation [13] only:
~LG RI(M~) + ~
~:LG + pLg~ ~ P"
(6)
Here, PL is the liquid density, 7L~ denotes the liquid surface tension coefficient, 9 relates to the gravity acceleration, the Y-axis is directed vertically upwards, RI(M) and Rz(M ) denote the main radii of liquid surface curvature at the point M of coordinates (?, ~) at the liquid surface (Fig. 2c); R I ( M ) lying in the drawing plane, and R2(M) in the plane perpendicular to the drawing plane. For an axisymmetric meniscus surface, R 2 ( M ) lies on the same line than R I ( M ) and its origin is on the symmetry axis. From differential geometry [12], we have R11 = z'(1 -~- 3'2) -3/2,
substantial constraints: the meniscus surface is assumed to have only one projection onto the horizontal plane at each r, while, for two projections each branch is described by Eq. (7) with different signs in front of the last term, depending on whether the orientation of the normal to the free surface is directed inwards or outwards from the liquid. For example, if we describe the free surface by means of a function: w - z - z(r) = 0, the direction of the normal to the free surface in the plane (r,z) is determined by V(w) = ( - z'(r),l). If this direction is outwards or inwards, we have to choose the positive or the negative sign, respectively, in Eq. (7). In general, when the meniscus has a complicated shape with multiple values of z or r for a given r or z, we highly recommend to use a differential equation written with the natural parameter s (which is the arc length of the curve) as an independent variable. Then, Eq. (7) has to be transformed into the following system: dr --
ds
R2 1 = 0.53'(1 + y,z)-l/z.
dz -
In Eq. (6), the constant value of P depends on the Z-coordinate origin selection and is equal to the pressure on the liquid in the plane Y = 0. In particular, if the Y-coordinate origin (the plane ~O3~) coincides with the plane of the liquid free surface, P=0. Our study is restricted to axisymmetric menisci. Thus, the problem of shape calculation is reduced to find the shape of a profile curve 3 = J ( ~ in the plane (?,~. Now, we introduce the capillary length a, the dimensionless coordinates z and r (based on a as reference length), and the parameter d a = (27LG/PLg) 1/2, Z = Y/a,
r = ?/a; d = aP/(27LC).
Then Eq. (6) takes the form z"r + z'(1 + z '2) _+ 2(d - z) (1 + 2'2)3/2r = 0.
(7)
In Eq. (7) the prime denotes differentiation with respect to r. It is important to note that the transition from Eq. (6) to Eq. (7) is associated with
-
ds d~ ds
~ - C O S O~,
= sin ~, sin -r
_+ 2(d - z)sgn(sin ~).
(8)
The angle ~ is already introduced at the edge of the crystal but we have to define it in a more general way for each point of the meniscus: ~ is the angle between the positive direction of the r-axis and the tangential vector to the meniscus. The direction of this vector coincides with the positive direction of the natural parameter s along the meniscus. The sign in the third term of Eq. (8) must be choosen in accordance with the above discussion. It is useful to note that the direction perpendicular to the meniscus surface changes with the function sin e. Consequently, for the calculation of complicated multivalued menisci, it is only necessary to choose a position of the liquid relatively to the meniscus, by means of a negative or positive sign of the third term of Eq. (8). In this case we do not care about the direction of the normal along the meniscus.
V.A. Tatarchenko et al. / Journal of C~stal Growth 180 (1997) 615-626
3.2. Static stability of a liquid meniscus
619
For an axisymmetric meniscus the function
F(r,r',z) in (10) corresponds to the minimum of It is very important to point out that within the set of liquid menisci which can be obtained from Eq. (8) and a set of boundary conditions, some solutions can be statically unstable and, therefore, cannot be physically realized. The classical Euler theory of integral minimization must be applied in order to access the static stability of a meniscus [12]. The necessary condition for the extremum of the integral of a function F(y(x),y'(x),x) on the interval [Xo, xv]:
potential energy of the system. Therefore, it has the following form:
F(r, r', z) = r(1 + r'2) 1/2 _t_ r22.
(12)
In our case, the Jacobi equation has the following form: d2£
dz 2
d2 r' ) 1 + r '2 (2 + 6rz(1 + r'2) 1/2) + o r2 = 0 dz r (13)
xs
(9)
f F(y(x), y'(x), x) dx Xo
is that every function y(x) corresponding to an extremum of the integral satisfies the Euler differential equation: ~2F ~2F ~2F ~y,2 y" + ~ - ~ ' Y ' + ~x~y'
~F
8y = O.
(10)
Another theorem gives sufficient conditions for a minimum of the integral (9). Let us consider a function y = y(x) and a parameter 2. The function y is an extremum function of Eq. (9) for the given values Xo and xv (with y(xo) = Yo and y(XF) = Yv). The parameter 2 has to satisfy the following equation (where y = y(x)) which is named Jacobi's equation: d (~2F) dx k~y 2 ~'' +
(d ~2F O2F) -dx ~y~)"
e y 2 J )" = O.
(11)
If a solution of Eq. (11) exists for the boundary condition 2(Xo) = 0 and does not take another zero value on the interval Xo < x <~XF it is said that the extremum y = y(x) satisfies the Jacobi condition. If the inequality ~2F/~y'2 > 0 is satisfied for an extremum which also satisfies the Jacobi condition, this extremum is a minimum and, if O2F/~y'2 < O, the extremum is a maximum. In our case, the minimization of the potential energy for a liquid free surface with Eq. (10) leads to the Laplace equation in the form of Eq. (7) or (8). Menisci which do not satisfy the Jacobi condition are statically unstable and have to be excluded from further analysis.
with the initial conditions 2(Zo) = 0, 2%o) = 1. The exact value (excluding 0) on the right-hand side of the second boundary condition does not influence the location of the zero points of the trajectory 2(z), due to the linear form of Eq. (13). The dependence of r(z) upon z in the coefficients of the linear differential Eq. (13) follows from a solution of the Laplace equation. The sign in Eq. (12) and (13) is chosen in accordance with the liquid position relatively to the free surface of the meniscus as it was mentioned above. For a meniscus with a complicated shape, we use the system of Eq. (8) and two additional equations with appropriate boundary conditions (14) determining the static stability of the solution: d2 ds dp ds
p sin :~, 2[s=O = 0,
(14)
(r z sin :¢)- 1). + 2p cos c¢(1 _+ 3rz/sin ~)/r,
pls=O = 1.
Here all the boundary conditions on the left-hand side of the interval are fixed. The parameter p is introduced to reduce the Jacobi second-order differential equation to a system of two equations. For the set of Eq. (14) there is no need to check the orientation of the normal or the position of the liquid relatively to the free surface. Once the position has been chosen, the liquid remains on one side along the curve of the meniscus. Physically, the referring sign ( _+ ) determines the initial position of the liquid relative to the free surface.
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KA. Tatarchenko et al. / Journal o f Crystal Growth 180 (1997) 615-626
The sets of Eq. (8) and (14) are strongly nonlinear. In the following, the results are obtained by solving them numerically. 3.3. Boundary conditions o f the capillary problem
According to ]-8, 9] two types of melt contact with the shaper can be distinguished. The first one is a wetting condition, i.e., the wetting angle 0 is fixed on the surface of the shaper and the contact point can move (Fig. ld and Fig. le) while the second one is a catching boundary condition, i.e., the contact point is fixed, and the angle ~a is variable (Fig. la-Fig, lc). Actually, a wetting condition also applies in this case and the catching condition is just a useful mathematical approach to define a wetting boundary condition on the sharp edges of the shaper. Therefore, we need to determine the limits of this mathematical approach, i.e., to find the maximum value of the radius of curvature of the shaper edges which allows a correct use of this boundary condition. Let us consider a sharp point on the shaper where all the possible starting angles c~a(the catching angles) of the meniscus profile curve could occur (Fig. 3a). We shall compare all these curves, which are solutions of the Laplace equation with those starting from spheres surrounding the sharp point (Fig. 3a). On the smooth surface of a sphere the wetting condition is ~a = n - 0 + arctan(du/dv)[w= wl~,,,,,,
where 0 is the wetting angle, u(v) is the equation of the sphere surface and W is the meniscus contact point on the sphere (Fig. 3a). Generally spoken, we can choose any convex closed surface which is topologically equivalent to a sphere. If the radius of the sphere vanishes, it is intuitively natural that the set of solutions starting from the sphere leads to the set of solutions corresponding to the sharp point of the shaper. Nevertheless we shall give a mathematical estimation of the differences between the solutions, based on a numerical analysis of the problem. We wish to compare the set of solutions of the two problems, which represent the various shapes of the liquid menisci of the growing crystal pulled from a shaper, the first set with a sharp edge and the second set with spheres. The Laplace Eq. (8) is valid
r~
(a)
S=O
Z 1.35
1.3 1.25 1.2
r¢ 2~'c/
\.
. O
_
\,\
(O
1.15 1.1 1.05 1 0.95 0.9 (b) 0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
r
Fig. 3. Shape of menisci, starting from spheres of different radii r~: 0.003; 0.06; 0.09 and from a sharp edge of the shaper: r~ = 0. Wetting angle: 0 = 10c. Angle of crystal growth: ~0 = 10'L Pressure value: d = 1.0. Shaper radius: ra = 1.0. (a) Scheme of calculation and definitions. (b) Different solutions obtained for various sphere radii r~ (a:0; b:0.03; c:0.06; d:0.09) and various catching angles ~a (1: 210~; 2: 180'; 3: 150°).
for both problems. We will study the case of middle-range Bond numbers (i.e., Bo ~ 1), which takes place in most practical applications. Let us formulate the referring boundary condition.
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KA. Tatarchenko et al. / Journal of C~stal Growth 180 (1997) 615 626 •
It follows after calculations, that the main term is always linear only if C~ois not too close to ~a. For further analysis, we need to determine a value of the parameter G~, which is introduced as a m a x i m u m of the function gl(~b):
The catching boundary condition is r[,=o=ra=
1,
Zls=o= 1,
~ls=o=:Xa,
~[s-s~ ~-- ~0"
The 2D problem has a free surface with a length s = sl, whose position is determined by the value ~ = ~o. Changing the catching angle ~a gives a set of solutions, starting from a shaper sharp point. • The wetting boundary condition is rls=o=l+rscos~b, =1.,=0 = 6 + 0,
zl~=o=l+r~sin~,
~1. . . . --=o.
The additional term rscosqS, 0 ~< ~ ~< rt, determines the wetting point W on the sphere of radius rs. Without any limitation, 7d = ~b + n/2 - 0 can be chosen to compare two solutions (Fig. 3b, curves b-d). We need to obtain the function g(rs,qS) which allows to estimate the difference between two solutions at the end points s = Sl and s = s2. The angle ~o is considered as a constant and taken equal to rt/2 + ~'o, with ~o = 10 °. We define a norm for the difference between two solutions in the space of solutions of (8): II'l[ .....
=
[(r 1
--
r2) 2 -4-(z
1 --
z2)2]
1/2
~<
gl(~b) ~
g~(qS) - G1.
max
(17)
The solution of Eq. (8) does not exist over the entire region of ~b values. The m a x i m u m values of g l were achieved for values of q5 ~ 0. An example of convergence process of two solutions for various rs and ~b is shown in Fig. 4. The curve of convergence which corresponds to the m a x i m u m tangent angle 7d in the vicinity of rs = 0 gives a value of the linear convergence rate G1 = 6.3. The convergence rate takes a linear profile if rs -< 0.1. That means that the linear term in the Taylor series is the main one
G 0.035
I
!
I
1
0.03
9(rs,(a).
(15) The explicit form of the function g(rs,~) can hardly be obtained even in simple cases. Normally it should be represented in the form of a Taylor series near the zero point r~ = 0. g(r~, ~b) = gl(qS)r~ + g2(~b)r2 + g3(q~)r3 + ....
(16)
In this case, the term go - 0, because both problems coincide. To estimate the difference between the solutions we restrict ourself to the first non-zero term in the series (16). Here, we would like to mention that our solutions are most probably not close to the boundary lines which divide the space of solutions into several regions with different character of parameter dependence. In other cases, a general analysis cannot be provided and any case must to be studied separately for each particular set of boundary conditions.
0.025
0.02
o
0.015
0.01
0.005
r
I
I
I
I
I
0
0.001
0.002
0.003
0.004
0.005
rs
Fig. 4. Convergence lines representing the difference G between the solutions of the Laplace capillary equation with a catching boundary and with wetting conditions for different angles 0: 30°, 60 "~,9 0 , 120 °, 150 ' .
622
V.A. Tatarchenko et al. / Journal o f Crystal Growth 180 (1997) 615- 626
for r~ less than 0.1 times the radius of linear convergence. For this r-region all the solutions on the wetting sphere converge linearly to the solution determined by the catching boundary condition on the sharp edge. Therefore, the difference between the two solutions can be estimated by the inequality:
IIll ..... ~< 6.3r~ ifrs <~ 0.1.
(18)
This conclusion was obtained using dimensionless parameters (with the capillary length a as reference length). Thus, linear convergence of wetting to catching conditions takes place if the radius of curvature of the shaper is less than 0.1a. Moreover, we would like to stress that it is necessary to estimate the sharpness of the shaper before applying a catching boundary condition. It can be shown for particular cases with specially chosen boundary conditions, that a little displacement of the meniscus on the edge of a shaper with high curvature crucially changes the shape of the meniscus and its properties of static stability.
4. Pressure influence on meniscus shape and capillary stability In this paragraph, we would like to present a set of possible shapes of liquid menisci, depending on the catching angle and the pressure level. For low and large Bond numbers, an analytical study of the meniscus shape has already been presented [8, 9]. Now our main interest is to investigate the meniscus shape for middle-range Bond numbers (i.e., Bo = 1), which most frequently occur in practice and have been left out of the regular study. We must use a numerical approach in this case in order to solve Eq. (8). We use the catching boundary condition on the shaper edge, i.e., the radius ra and height of the shaper ha are fixed and are determined by the particular features of the installation. As was mentioned above, the value of the constant d in Eq. (8) is equal to the pressure in the liquid on the plane z = 0. It is more convenient to choose z = 0 at the level of the melt free-surface yielding d = 0. The contact angle between meniscus and shaper,
~d, is a parameter of the problem. The boundary conditions are written as follows: r[s=0 = rj,
z[~-0 = ha,
~[~-o = ~a,
(19)
O~[s=s2 = 3~O.
The crystal radius ro and the height of meniscus ho must be determined as the result of the calculations. A boundary curve is built as the set of end points of the menisci starting from the shaper. This boundary curve determines the crystal radius and the meniscus height for a fixed growth angle and for all possible catching angles ~a. From a mathematical point of view Eq. (8) are an example of a nonlinear system which generates strange attractors. From a physical point of view, the curve in the (r,z) plane can describe a physical solution only up to its self-intersection in this plane. The liquid is on the left of the meniscus curve (if the orientation has been chosen along the positive direction of the curve natural parameter). The boundary curves marked on Figs. 5 and 6 with thick lines (continuous, dashed or dotted) give a final position of the meniscus curve which corresponds to the crystal growth angle ~0. The coordinates (r, z) of the boundary curve are the crystal radius and meniscus height. We have to choose the stable menisci (marked on Figs. 5 and 6 with thick continuous or dashed lines) among all possible ones. Some statically unstable menisci according to Eq. (14) are marked in the figures with dashed thin lines, the boundary curves which correspond to these menisci being marked with thick dotted line. Conditions (4) determine the capillary dynamic stability of crystal growth. The boundary curves marked with thick dashed lines represent capillary unstable states of crystal growth. Only the menisci corresponding to the thick continuous boundary curve are stable (from both dynamic and static points of view) and could physically be realized for the crystal growth process. As it was mentioned above, there are various kinds of TPS crystal growth installations where the edges of the shaper could be positioned at different levels: (i) below the plane surface of the melt (positive hydrostatic pressure in the meniscus), (ii) above the surface of the melt (negative hydrostatic
IdA. Tatarchenko et al. / Journal ~?[Crystal Growth 180 (1997) 615-626
623
Z 06
I
I
I
I
"1:~
2 : a 1=161
0.5
I
.I
I
I
I
=150 °
"
~
'~J=1~°°
f =170 ° d
#
t,: %
~
=80 °
/
0.4
=:%:7o o'~
,/
:120° 03
, f / i
e:
~ =50 e
f
t':g
0.2
=40 °
mi,aF ~
"# /" f ,
g : (~o
I
O.I
I, : % =~_c¢'
or
0
i
J
~
~
L
i
i
i
i
0.1
0.2
0.3
0.4
0.5
0.6
0,7
0.8
09
~
t
r
Fig. 5. Shapes of menisci (thin lines 1-8) for different catching angles c~d and b o u n d a r y curves (thick lines a i) for different angles of crystal growth a0 (zero value of pressure). The parts of b o u n d a r y curves m a r k e d with c o n t i n u o u s thick lines correspond to the dynamically stable capillary states.
pressure), (iii) in coincidence with the surface of the melt (zero hydrostatic pressure in the down part of meniscus). Therefore, we are going to study these three different situations, because the shape of the meniscus strongly depends on the level of pressure in the system. • The edge of the shaper is placed at the level of the melt free-surface. The boundary lines are built for a few values of crystal growth angles (Fig. 5). All the menisci are statically stable. Capillary stable states (where ARR < 0) for the point (ro,ho) on the boundary curve here correspond to their decreasing parts which are marked on the figure with a thick continuous line. Stable crystal
growth occurs if the diameter of the crystal is larger than one-half of the shaper diameter. All the menisci have a convex form. The catching angle is always in the interval ~/2 < :~e < r~. In the case of a capillary unstable state corresponding to a given pressure and crystal diameter, we can stabilize the process by decreasing the shaper diameter as it has been shown by our analysis. The shaper is placed below the level of the melt free-surface. The menisci have a concave-convex shape and the tangential angle of the meniscus curve changes non-monotonically along the curve (Fig. 6a). There are two branches of the
V.A. Tatarchenko et al. / Journal o/Crystal Growth 180 (1997) 615 626
624
2:
ga
II°~
~
h %
- "10°
0.5
. ..----a: %
=
-./ ......
90 °
<".'-7 ,%
.......... ~,.I . . . . .
0
-,;;
.......................... ............... ~'~
\
-0.5
1/ t \ \ \ W
-I
-
,, % ' "
,;'""'-i... D
~
-I ,5
a
~
.
.
6: ~ =
"2
(a)
0
I
J
I
0.2
04
0.6
.,
~
180 °
_
?;
~
=
\
/"
19D °
" 8:
9: % =
~d
.
260 °
230 °
210°
I
I
I
I
I
i
I
0.8
1
1.2
1.4
1.6
1.8
2
r
z 1 l: _
o
_.
% = o° - .........
o5
-0,5
;7
"
"~-.
,
.\./:
-1
/. . . . . .
><
~ , . . . - , ,,..--
o
~
90*
"\,/
-i .5
~
O° I
(b)
~
0.2
_ I
0,4
L
0,6
--~-~. i
O.S
~ - - ~ = - ~ . . . . . i.... i .
I
1.2
7; ~ = 210 ~ ~
1.4
I
,
t ,6
1.8
r
Fig. 6. Shapes of menisci (thin lines 1-11) for different values of ~d and boundary curves (thick lines a,b) for ~0 = 90'. The pressures are: (a) p = 1.0; (b) p = 1.5; (c) p = -3.0. The portions of the menisci marked with a thin dashed line are statically unstable. The portions of the boundary curve marked with a thick dotted line correspond to the statically unstable menisci. The ones marked with a thick continuous line correspond to the dynamically stable capillary states of crystal growth (r is growing crystal radius, h is meniscus height). The boundary curves marked with a thick dashed line are dynamically capillary unstable.
KA. Tatarchenko et al. /Journal of Co'stal Growth 180 (1997) 615-626
625
3.1
t
2.9
..',
\
\
2.8
........
/',,
\%
\
..,..,Y><, ) ',-" "'-. 1
\
)/
/
1/
A I
2.7
" " - ~ - " - ' " W-.~4_.
' - ><
-
- ) 7 / / '"- ~ "
2.6
2.5 0.6
(c)
L
f
0.7
0.8
I
I
0.9
I
f
f
I
i
1.1
1.2
1.3
14
Fig. 6. Continued.
boundary curve which are proposed for the growth angle @o = 0. The first one is formed by the menisci which have a catching angle on the shaper in the interval ~/4 < ee < 3~/4. All these menisci are statically stable. The states which correspond to the decreasing parts of the boundary curve are capillary stable. The size of a growing crystal can be larger or smaller than the shaper diameter. The second part of the boundary curve is formed by menisci which have the form of a drop. Some of these menisci are statically unstable. The diameter of the grown crystal for all these states is larger than the shaper size and the catching angle is in the interval < ed < 7~/4. All the states corresponding to the decreasing part of the boundary curve are capillary stable. Increasing the pressure leads to a topological bifurcation of the boundary curve (Fig. 6b). Then, only a meniscus having a drop shape represents a capillary stable state. Some menisci are statically unstable, which is common for a droplike meniscus curve. There are two families of
stable menisci. The first one represents states where the crystal diameter is close to the shaper size; the meniscus height in this case corresponds approximately to the applied pressure. The capillary stable state corresponds to the decreasing part of the boundary curve. The crystallization front of the growing crystal is placed at the level of the melt free-surface. The second family represents menisci with crystal diameter larger than the shaper size, and menisci with small heights. Here, the capillary stable states correspond to the increasing parts of the boundary curve. The shaper is placed above the melt free-surface. Some menisci are statically unstable (Fig. 6c). The boundary curves are built for a zero value of the crystal growth angle. Some menisci have a bubble form. There are two different families of stable states on the boundary curves (marked with a thick continuous line). The first one represents a positive meniscus height and crystal diameters which are somewhat lower than the shaper size. All the menisci have a convex form. The second family of menisci corresponds to
626
KA. Tatarchenko et al. / Journal o f C~_ stal Growth 180 (1997) 615--626
a negative meniscus height and a crystal diameter significantly less than the shaper size. All the menisci in this case have a bubble shape. In the case of a negative pressure in the meniscus and a capillary unstable state, it is possible to stabilize the process by increasing the pressure by means of shaper displacement towards the plane free-surface of the melt.
5. Conclusions In the present work, the dynamic stability properties of the possible shapes of menisci for axisymmetric crystal growth were studied. The case of a catching b o u n d a r y condition on the sharp edge of the shaper was analyzed. The limits of application of this b o u n d a r y condition were estimated. A few non-traditional stable shapes of menisci with drop and bubble shapes were obtained, which could be realized in practice. The bubble shape with negative level of pressure and negative height of meniscus is very interesting. It seems that the temperature gradient at the crystallization front can be significantly smaller c o m p a r e d to the cases where the lateral surface of the crystal is cooled by radiation. This configuration could be applied for the growth of high-quality crystals.
Acknowledgements This work was supported by the Direction G6n6rale de l'Armement ( D G A - D R E T ) within the frame of contract no. 92.525 with the C E A / C E R E M .
References [1] A.V. Stepanov, Zh. Tech. Fiz. 29 (1959) 339 [Sov. Phys.JETP]. [2] B. Chalmers, H.E. LaBell, A.J. Mlavsky, J. Crystal Growth 13/14 (1972) 84. [3] G.H. Schwuttke, Phys. Status Solidi (a) 43 (19771 43. [4] J. Czochralski, Zs. Phys. Chem. 92 (1917) 219. [5] V.A. Tatarchenko, A.I. Saet, A.V. Stepanov, Proc. 1st Conf. on Stepanov's Growth of Semiconductor Single Crystals, Ioffe Phys.-Tech. Inst, Leningrad, 1968, p. 83. [6] V.A. Tatarchenko, A.I. Saet, A.V. Stepanov, Bull. Acad. Sci. USSR, Phys. Set. 33 t1969) 1782. [7] V.A. Tatarchenko, J. Crystal Growth 37 (1977} 272. [8] V.A. Tatarchenko, Shaped Crystal Growth~ Kluwer, Dordrecht, 1993. [9] V.A. Tatarchenko, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth, vol. 2, part 2, North-Holland, Amsterdam, 1994~pp. 1011 1111. [10] V.A. Tatarchenko, Fiz. Khim. Obrabotki Mater. 6 (1973) 47 [Soy. Phys. Chem. Mater. Treatment]. [11] V.A. Tatarchenko, 4th Int. Conf. on Crystal Growth, Thesis of Reports, Tokyo, 1974, pp. 521 522. [12] G.A. Korn, T.M. Korn, Mathematical Handbook for Scientists and Engineers, Mc Graw-Hill, New York, 1961,p. 282. [13] L.D. Landau, E.M. Lifchits, Mecanique des Fluides, Mir, Moscow, 1971, p. 289.
C N Y I T A L
QNow'rH
ELSEVIER
Journal of Crystal Growth 180 (1997)615 626
Theoretical model of crystal growth shaping process V.A. T a t a r c h e n k o a'*, V.S. U s p e n s k i b, E.V. T a t a r c h e n k o b, J . P h . N a b o t ~, T. D u f f a f f , B. R o u x d a Crismatec, BP 52, F-77794 Nemours Cedex, France blnstitute for Mechanics of Moscow State University, Moscow, Russian Federation c Centre d 'Etudes Nuclbaires de Grenoble, Grenoble, France dlnstitut de Recherche sur les Phdnomenes Hors d 'Equilibre, UMR 138, Marseille, Franee
Abstract
A theoretical investigation of the crystal growth shaping process is carried out on the basis of the dynamic stability concept. The capillary dynamic stability of shaped crystal growth processes for various forms of the liquid menisci is analyzed using the mathematical model of the phenomena in the axisymmetric case. The catching boundary condition of the capillary boundary problem is considered and the limits of its application for shaped crystal growth modeling are discussed. The static stability of a liquid free surface is taken into account by means of the JacObi equation analysis. The result is that a large number of menisci having drop-like shapes are statically unstable. A few new non-traditional liquid meniscus shapes (e.g., bubbles and related shapes) are proposed for the case of a catching boundary condition. Keywords:
Shaped crystal growth; Capillarity; Liquid menisci; Catching boundary condition; Stability
I. Introduction
During recent years, much attention has been paid to the growth of shaped single crystals by techniques of pulling from the melt such as the Stepanov Technique [1], Edge Defined Film Fed Growth [2], Capillary Action Shaping Technique [3] and some others. The main idea of all these techniques is to use a shaper (Fig. 1), which limits the area of the liquid free surface and its perturba-
* Corresponding author.
tions occurring in the Czochralski Technique [4]. In this way, it contributes to controlled shaping. According to our analysis [5-9], two approaches can be used to achieve this purpose: shaping by the edge of the shaper (Fig. la-Fig, lc) or by its surface (Fig. ld and Fig. le). All these techniques use the principles of shaping and can be distinguished only by design features. Therefore, it has been suggested in [8] to use a common title (TPS - Techniques of Pulling from Shaper) for the analysis of the shaping conditions. The theory of shaped crystal growth by the TPS is based on the stable-growth concept which means
0022-0248/97/$17.00 Copyright ~5~ 1997 ElsevierScienceB.V. All rights reserved PI1 S0022-0248(97)00293-5
616
IdA. Tatarchenko et al. / Journal q]" Crystal Growth 180 (1997) 615--626
Z
Z
ra z #
i
_
a
_
2R
z/
0
b
c
d
e
Fig. 1. Different schemes of melt growth of a crystalline rod by the technique of pulling from shaper at various melt levels d. R: crystal radius, h: crystallization front height, rd: shaper edge radius, 0: wetting angle, :~d:catching angle of meniscus.
that self-stabilization is present in the system: if growth parameter deviations occur under the action of external perturbations, then they will fade with time. The dynamic stability of shaped crystal growth processes was proposed in Refs. [10, 11], and was studied in detail and reviewed in Refs. [8, 9]. Extensive investigations are now under way for the application of such analysis to different techniques of capillary shaping. This work gives the theoretical study of the influence of melt-shaper contact conditions on the dynamic and static capillary stability of the shaped crystal growth process in axisymmetric cases. We focus our attention on the effect of the catching boundary condition (Fig. la-Fig, lc). The wetting boundary condition (Fig. ld and Fig. le) will be studied in a future paper.
2. Equation for change rate of crystal dimension On the basis of our experience and previous investigations, we assert that the full and complete study of the stability of TPS crystal growth must include at least two independent degrees of freedom: the crystal dimension and a quantity which
affects the value of the contact angle between melt and crystal surface, e.g., the meniscus height h. In the present work, we study only the influence of capillary effects by means of solutions of the Laplace capillary equation. Therefore, we will leave out the heat transfer which affects the meniscus height. The differential equation for the change rate of the crystal radius is based on the phenomenon of growth angle constancy [8, 9]. Let us consider the diagram given in Fig. 2. Vector R lies within the diagram plane and represents the radius of the crystal. The lateral surface of the growing crystal makes an angle ~p with the line tangent to the meniscus (Fig. 2a and Fig. 2b). The angle 0o is called the crystal growth angle and is a physical characteristic of the material. Here, we do not take into consideration the effects of anisotropy, of very high growth rates and of the Gibbs-Thompson energy (for a very large curvature of the crystal surface) on the growth angle. Now, let us introduce the angle :~ made by the tangent to the meniscus at the three-phase point (with positive direction of the horizontal line), and the angle 0 = c~- ~/2 (Fig. 2). A crystal of constant radius R ° grows in vertical direction if ~k = ~o or if :~ = :~0, where eo = ~o + re/2 (Fig. 2a). From the
617
KA. Tatarchenkoet al. / Journal of Crystal Growth 180 (1997) 615-626 z z
oL
\ "t 2
b
[
- ~ arctgz:
Fig. 2. Crystal growth in the cases of capillaryshaping: (a) ~ = ~o: growth of crystals of constant cross-section;(b) :~ < 70: growth of crystals of widening cross-section;(c) c~> ~o: growth of crystals of narrowing cross-section;V is the crystal displacement rate.
geometric diagram given in Fig. 2b, the following algebraic equation governing the change of crystal radius with crystallization rate Vc can be obtained: 6/~ = / ~ = Vc tan(~ - ~o) = Vc tan(~),
lary problem is solved, i.e., the function ,(R,h) has been found, Eq. (3) can be written in the following form:
(1) ~/~= V ~ (
where 6/~ = R - Ro (but here/~o = 0), 8c~ = e - ~o is the angle between the growing crystal lateral surface and the vertical direction (Fig. 2b). The crystallization rate V¢ is equal to the difference between the pulling rate V, and the crystallization front displacement dh/dt: dh V¢ = V - d~-
(4)
Here R ° and h ° are the values of R and h for a = ao. Thus, the equation for the change rate of the crystal cross-section is obtained. From Eq. (4) it follows that for a two-degree of freedom problem the main coefficients of the stability matrix [-7-9] take the form
(2)
We look for the parameters of this equation close to the stationary state, which is characterized by constant values of the radius (i.e., 8~ = 0) and of the meniscus height. Considering the behavior of the system close to the stationary state, we have to expand tan(Sa) into a Taylor series in the vicinity of the stationary point and to keep the first-order infinitesimal term only: V -
°) . /
ARR = V ~a/DR,
R =
-Ro)+~(h-h
tan(Be) ~ V 8~.
(3)
With the angle e being a function of the crystal dimension, the crystallization front position can be determined by the capillary problem which will be discussed and solved later. Assuming that the capil-
ARh = V ~/'~h.
(5)
The explicit form of the coefficients AhR and Ahh can be found in a similar way from the heat-balance conditions at the interface [8, 9], but we do not need their explicit forms for the present analysis since we will focus our attention on the capillary effects only, i.e., we look for conditions of crystallization that give negative values to the coefficient ARR.
3. C a p i l l a r y p r o b l e m f o r m u l a t i o n
3.1. Meniscus surface equation
To define t h e explicit form of the coefficient ARR in Eq. (5) the capillary problem for crystal
V.A. Tatarchenko et al. / Journal of Crystal Growth 180 (1997) 615 626
618
growth by TPS has to be solved. This problem includes the calculation of the melt meniscus shape on the basis of the Laplace capillary equation taking into account the Navier-Stokes equation for dynamic effects. In Refs. [8, 9] it was shown that for real pulling conditions the hydrostatic approximation is applicable and, hence, the equilibrium shape of the liquid surface is described by the Laplace capillary equation [13] only:
~LG RI(M~) + ~
~:LG + pLg~ ~ P"
(6)
Here, PL is the liquid density, 7L~ denotes the liquid surface tension coefficient, 9 relates to the gravity acceleration, the Y-axis is directed vertically upwards, RI(M) and Rz(M ) denote the main radii of liquid surface curvature at the point M of coordinates (?, ~) at the liquid surface (Fig. 2c); R I ( M ) lying in the drawing plane, and R2(M) in the plane perpendicular to the drawing plane. For an axisymmetric meniscus surface, R 2 ( M ) lies on the same line than R I ( M ) and its origin is on the symmetry axis. From differential geometry [12], we have R11 = z'(1 -~- 3'2) -3/2,
substantial constraints: the meniscus surface is assumed to have only one projection onto the horizontal plane at each r, while, for two projections each branch is described by Eq. (7) with different signs in front of the last term, depending on whether the orientation of the normal to the free surface is directed inwards or outwards from the liquid. For example, if we describe the free surface by means of a function: w - z - z(r) = 0, the direction of the normal to the free surface in the plane (r,z) is determined by V(w) = ( - z'(r),l). If this direction is outwards or inwards, we have to choose the positive or the negative sign, respectively, in Eq. (7). In general, when the meniscus has a complicated shape with multiple values of z or r for a given r or z, we highly recommend to use a differential equation written with the natural parameter s (which is the arc length of the curve) as an independent variable. Then, Eq. (7) has to be transformed into the following system: dr --
ds
R2 1 = 0.53'(1 + y,z)-l/z.
dz -
In Eq. (6), the constant value of P depends on the Z-coordinate origin selection and is equal to the pressure on the liquid in the plane Y = 0. In particular, if the Y-coordinate origin (the plane ~O3~) coincides with the plane of the liquid free surface, P=0. Our study is restricted to axisymmetric menisci. Thus, the problem of shape calculation is reduced to find the shape of a profile curve 3 = J ( ~ in the plane (?,~. Now, we introduce the capillary length a, the dimensionless coordinates z and r (based on a as reference length), and the parameter d a = (27LG/PLg) 1/2, Z = Y/a,
r = ?/a; d = aP/(27LC).
Then Eq. (6) takes the form z"r + z'(1 + z '2) _+ 2(d - z) (1 + 2'2)3/2r = 0.
(7)
In Eq. (7) the prime denotes differentiation with respect to r. It is important to note that the transition from Eq. (6) to Eq. (7) is associated with
-
ds d~ ds
~ - C O S O~,
= sin ~, sin -r
_+ 2(d - z)sgn(sin ~).
(8)
The angle ~ is already introduced at the edge of the crystal but we have to define it in a more general way for each point of the meniscus: ~ is the angle between the positive direction of the r-axis and the tangential vector to the meniscus. The direction of this vector coincides with the positive direction of the natural parameter s along the meniscus. The sign in the third term of Eq. (8) must be choosen in accordance with the above discussion. It is useful to note that the direction perpendicular to the meniscus surface changes with the function sin e. Consequently, for the calculation of complicated multivalued menisci, it is only necessary to choose a position of the liquid relatively to the meniscus, by means of a negative or positive sign of the third term of Eq. (8). In this case we do not care about the direction of the normal along the meniscus.
V.A. Tatarchenko et al. / Journal of C~stal Growth 180 (1997) 615-626
3.2. Static stability of a liquid meniscus
619
For an axisymmetric meniscus the function
F(r,r',z) in (10) corresponds to the minimum of It is very important to point out that within the set of liquid menisci which can be obtained from Eq. (8) and a set of boundary conditions, some solutions can be statically unstable and, therefore, cannot be physically realized. The classical Euler theory of integral minimization must be applied in order to access the static stability of a meniscus [12]. The necessary condition for the extremum of the integral of a function F(y(x),y'(x),x) on the interval [Xo, xv]:
potential energy of the system. Therefore, it has the following form:
F(r, r', z) = r(1 + r'2) 1/2 _t_ r22.
(12)
In our case, the Jacobi equation has the following form: d2£
dz 2
d2 r' ) 1 + r '2 (2 + 6rz(1 + r'2) 1/2) + o r2 = 0 dz r (13)
xs
(9)
f F(y(x), y'(x), x) dx Xo
is that every function y(x) corresponding to an extremum of the integral satisfies the Euler differential equation: ~2F ~2F ~2F ~y,2 y" + ~ - ~ ' Y ' + ~x~y'
~F
8y = O.
(10)
Another theorem gives sufficient conditions for a minimum of the integral (9). Let us consider a function y = y(x) and a parameter 2. The function y is an extremum function of Eq. (9) for the given values Xo and xv (with y(xo) = Yo and y(XF) = Yv). The parameter 2 has to satisfy the following equation (where y = y(x)) which is named Jacobi's equation: d (~2F) dx k~y 2 ~'' +
(d ~2F O2F) -dx ~y~)"
e y 2 J )" = O.
(11)
If a solution of Eq. (11) exists for the boundary condition 2(Xo) = 0 and does not take another zero value on the interval Xo < x <~XF it is said that the extremum y = y(x) satisfies the Jacobi condition. If the inequality ~2F/~y'2 > 0 is satisfied for an extremum which also satisfies the Jacobi condition, this extremum is a minimum and, if O2F/~y'2 < O, the extremum is a maximum. In our case, the minimization of the potential energy for a liquid free surface with Eq. (10) leads to the Laplace equation in the form of Eq. (7) or (8). Menisci which do not satisfy the Jacobi condition are statically unstable and have to be excluded from further analysis.
with the initial conditions 2(Zo) = 0, 2%o) = 1. The exact value (excluding 0) on the right-hand side of the second boundary condition does not influence the location of the zero points of the trajectory 2(z), due to the linear form of Eq. (13). The dependence of r(z) upon z in the coefficients of the linear differential Eq. (13) follows from a solution of the Laplace equation. The sign in Eq. (12) and (13) is chosen in accordance with the liquid position relatively to the free surface of the meniscus as it was mentioned above. For a meniscus with a complicated shape, we use the system of Eq. (8) and two additional equations with appropriate boundary conditions (14) determining the static stability of the solution: d2 ds dp ds
p sin :~, 2[s=O = 0,
(14)
(r z sin :¢)- 1). + 2p cos c¢(1 _+ 3rz/sin ~)/r,
pls=O = 1.
Here all the boundary conditions on the left-hand side of the interval are fixed. The parameter p is introduced to reduce the Jacobi second-order differential equation to a system of two equations. For the set of Eq. (14) there is no need to check the orientation of the normal or the position of the liquid relatively to the free surface. Once the position has been chosen, the liquid remains on one side along the curve of the meniscus. Physically, the referring sign ( _+ ) determines the initial position of the liquid relative to the free surface.
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KA. Tatarchenko et al. / Journal o f Crystal Growth 180 (1997) 615-626
The sets of Eq. (8) and (14) are strongly nonlinear. In the following, the results are obtained by solving them numerically. 3.3. Boundary conditions o f the capillary problem
According to ]-8, 9] two types of melt contact with the shaper can be distinguished. The first one is a wetting condition, i.e., the wetting angle 0 is fixed on the surface of the shaper and the contact point can move (Fig. ld and Fig. le) while the second one is a catching boundary condition, i.e., the contact point is fixed, and the angle ~a is variable (Fig. la-Fig, lc). Actually, a wetting condition also applies in this case and the catching condition is just a useful mathematical approach to define a wetting boundary condition on the sharp edges of the shaper. Therefore, we need to determine the limits of this mathematical approach, i.e., to find the maximum value of the radius of curvature of the shaper edges which allows a correct use of this boundary condition. Let us consider a sharp point on the shaper where all the possible starting angles c~a(the catching angles) of the meniscus profile curve could occur (Fig. 3a). We shall compare all these curves, which are solutions of the Laplace equation with those starting from spheres surrounding the sharp point (Fig. 3a). On the smooth surface of a sphere the wetting condition is ~a = n - 0 + arctan(du/dv)[w= wl~,,,,,,
where 0 is the wetting angle, u(v) is the equation of the sphere surface and W is the meniscus contact point on the sphere (Fig. 3a). Generally spoken, we can choose any convex closed surface which is topologically equivalent to a sphere. If the radius of the sphere vanishes, it is intuitively natural that the set of solutions starting from the sphere leads to the set of solutions corresponding to the sharp point of the shaper. Nevertheless we shall give a mathematical estimation of the differences between the solutions, based on a numerical analysis of the problem. We wish to compare the set of solutions of the two problems, which represent the various shapes of the liquid menisci of the growing crystal pulled from a shaper, the first set with a sharp edge and the second set with spheres. The Laplace Eq. (8) is valid
r~
(a)
S=O
Z 1.35
1.3 1.25 1.2
r¢ 2~'c/
\.
. O
_
\,\
(O
1.15 1.1 1.05 1 0.95 0.9 (b) 0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
r
Fig. 3. Shape of menisci, starting from spheres of different radii r~: 0.003; 0.06; 0.09 and from a sharp edge of the shaper: r~ = 0. Wetting angle: 0 = 10c. Angle of crystal growth: ~0 = 10'L Pressure value: d = 1.0. Shaper radius: ra = 1.0. (a) Scheme of calculation and definitions. (b) Different solutions obtained for various sphere radii r~ (a:0; b:0.03; c:0.06; d:0.09) and various catching angles ~a (1: 210~; 2: 180'; 3: 150°).
for both problems. We will study the case of middle-range Bond numbers (i.e., Bo ~ 1), which takes place in most practical applications. Let us formulate the referring boundary condition.
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KA. Tatarchenko et al. / Journal of C~stal Growth 180 (1997) 615 626 •
It follows after calculations, that the main term is always linear only if C~ois not too close to ~a. For further analysis, we need to determine a value of the parameter G~, which is introduced as a m a x i m u m of the function gl(~b):
The catching boundary condition is r[,=o=ra=
1,
Zls=o= 1,
~ls=o=:Xa,
~[s-s~ ~-- ~0"
The 2D problem has a free surface with a length s = sl, whose position is determined by the value ~ = ~o. Changing the catching angle ~a gives a set of solutions, starting from a shaper sharp point. • The wetting boundary condition is rls=o=l+rscos~b, =1.,=0 = 6 + 0,
zl~=o=l+r~sin~,
~1. . . . --=o.
The additional term rscosqS, 0 ~< ~ ~< rt, determines the wetting point W on the sphere of radius rs. Without any limitation, 7d = ~b + n/2 - 0 can be chosen to compare two solutions (Fig. 3b, curves b-d). We need to obtain the function g(rs,qS) which allows to estimate the difference between two solutions at the end points s = Sl and s = s2. The angle ~o is considered as a constant and taken equal to rt/2 + ~'o, with ~o = 10 °. We define a norm for the difference between two solutions in the space of solutions of (8): II'l[ .....
=
[(r 1
--
r2) 2 -4-(z
1 --
z2)2]
1/2
~<
gl(~b) ~
g~(qS) - G1.
max
(17)
The solution of Eq. (8) does not exist over the entire region of ~b values. The m a x i m u m values of g l were achieved for values of q5 ~ 0. An example of convergence process of two solutions for various rs and ~b is shown in Fig. 4. The curve of convergence which corresponds to the m a x i m u m tangent angle 7d in the vicinity of rs = 0 gives a value of the linear convergence rate G1 = 6.3. The convergence rate takes a linear profile if rs -< 0.1. That means that the linear term in the Taylor series is the main one
G 0.035
I
!
I
1
0.03
9(rs,(a).
(15) The explicit form of the function g(rs,~) can hardly be obtained even in simple cases. Normally it should be represented in the form of a Taylor series near the zero point r~ = 0. g(r~, ~b) = gl(qS)r~ + g2(~b)r2 + g3(q~)r3 + ....
(16)
In this case, the term go - 0, because both problems coincide. To estimate the difference between the solutions we restrict ourself to the first non-zero term in the series (16). Here, we would like to mention that our solutions are most probably not close to the boundary lines which divide the space of solutions into several regions with different character of parameter dependence. In other cases, a general analysis cannot be provided and any case must to be studied separately for each particular set of boundary conditions.
0.025
0.02
o
0.015
0.01
0.005
r
I
I
I
I
I
0
0.001
0.002
0.003
0.004
0.005
rs
Fig. 4. Convergence lines representing the difference G between the solutions of the Laplace capillary equation with a catching boundary and with wetting conditions for different angles 0: 30°, 60 "~,9 0 , 120 °, 150 ' .
622
V.A. Tatarchenko et al. / Journal o f Crystal Growth 180 (1997) 615- 626
for r~ less than 0.1 times the radius of linear convergence. For this r-region all the solutions on the wetting sphere converge linearly to the solution determined by the catching boundary condition on the sharp edge. Therefore, the difference between the two solutions can be estimated by the inequality:
IIll ..... ~< 6.3r~ ifrs <~ 0.1.
(18)
This conclusion was obtained using dimensionless parameters (with the capillary length a as reference length). Thus, linear convergence of wetting to catching conditions takes place if the radius of curvature of the shaper is less than 0.1a. Moreover, we would like to stress that it is necessary to estimate the sharpness of the shaper before applying a catching boundary condition. It can be shown for particular cases with specially chosen boundary conditions, that a little displacement of the meniscus on the edge of a shaper with high curvature crucially changes the shape of the meniscus and its properties of static stability.
4. Pressure influence on meniscus shape and capillary stability In this paragraph, we would like to present a set of possible shapes of liquid menisci, depending on the catching angle and the pressure level. For low and large Bond numbers, an analytical study of the meniscus shape has already been presented [8, 9]. Now our main interest is to investigate the meniscus shape for middle-range Bond numbers (i.e., Bo = 1), which most frequently occur in practice and have been left out of the regular study. We must use a numerical approach in this case in order to solve Eq. (8). We use the catching boundary condition on the shaper edge, i.e., the radius ra and height of the shaper ha are fixed and are determined by the particular features of the installation. As was mentioned above, the value of the constant d in Eq. (8) is equal to the pressure in the liquid on the plane z = 0. It is more convenient to choose z = 0 at the level of the melt free-surface yielding d = 0. The contact angle between meniscus and shaper,
~d, is a parameter of the problem. The boundary conditions are written as follows: r[s=0 = rj,
z[~-0 = ha,
~[~-o = ~a,
(19)
O~[s=s2 = 3~O.
The crystal radius ro and the height of meniscus ho must be determined as the result of the calculations. A boundary curve is built as the set of end points of the menisci starting from the shaper. This boundary curve determines the crystal radius and the meniscus height for a fixed growth angle and for all possible catching angles ~a. From a mathematical point of view Eq. (8) are an example of a nonlinear system which generates strange attractors. From a physical point of view, the curve in the (r,z) plane can describe a physical solution only up to its self-intersection in this plane. The liquid is on the left of the meniscus curve (if the orientation has been chosen along the positive direction of the curve natural parameter). The boundary curves marked on Figs. 5 and 6 with thick lines (continuous, dashed or dotted) give a final position of the meniscus curve which corresponds to the crystal growth angle ~0. The coordinates (r, z) of the boundary curve are the crystal radius and meniscus height. We have to choose the stable menisci (marked on Figs. 5 and 6 with thick continuous or dashed lines) among all possible ones. Some statically unstable menisci according to Eq. (14) are marked in the figures with dashed thin lines, the boundary curves which correspond to these menisci being marked with thick dotted line. Conditions (4) determine the capillary dynamic stability of crystal growth. The boundary curves marked with thick dashed lines represent capillary unstable states of crystal growth. Only the menisci corresponding to the thick continuous boundary curve are stable (from both dynamic and static points of view) and could physically be realized for the crystal growth process. As it was mentioned above, there are various kinds of TPS crystal growth installations where the edges of the shaper could be positioned at different levels: (i) below the plane surface of the melt (positive hydrostatic pressure in the meniscus), (ii) above the surface of the melt (negative hydrostatic
IdA. Tatarchenko et al. / Journal ~?[Crystal Growth 180 (1997) 615-626
623
Z 06
I
I
I
I
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2 : a 1=161
0.5
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.I
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I
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i
i
i
0.1
0.2
0.3
0.4
0.5
0.6
0,7
0.8
09
~
t
r
Fig. 5. Shapes of menisci (thin lines 1-8) for different catching angles c~d and b o u n d a r y curves (thick lines a i) for different angles of crystal growth a0 (zero value of pressure). The parts of b o u n d a r y curves m a r k e d with c o n t i n u o u s thick lines correspond to the dynamically stable capillary states.
pressure), (iii) in coincidence with the surface of the melt (zero hydrostatic pressure in the down part of meniscus). Therefore, we are going to study these three different situations, because the shape of the meniscus strongly depends on the level of pressure in the system. • The edge of the shaper is placed at the level of the melt free-surface. The boundary lines are built for a few values of crystal growth angles (Fig. 5). All the menisci are statically stable. Capillary stable states (where ARR < 0) for the point (ro,ho) on the boundary curve here correspond to their decreasing parts which are marked on the figure with a thick continuous line. Stable crystal
growth occurs if the diameter of the crystal is larger than one-half of the shaper diameter. All the menisci have a convex form. The catching angle is always in the interval ~/2 < :~e < r~. In the case of a capillary unstable state corresponding to a given pressure and crystal diameter, we can stabilize the process by decreasing the shaper diameter as it has been shown by our analysis. The shaper is placed below the level of the melt free-surface. The menisci have a concave-convex shape and the tangential angle of the meniscus curve changes non-monotonically along the curve (Fig. 6a). There are two branches of the
V.A. Tatarchenko et al. / Journal o/Crystal Growth 180 (1997) 615 626
624
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,
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Fig. 6. Shapes of menisci (thin lines 1-11) for different values of ~d and boundary curves (thick lines a,b) for ~0 = 90'. The pressures are: (a) p = 1.0; (b) p = 1.5; (c) p = -3.0. The portions of the menisci marked with a thin dashed line are statically unstable. The portions of the boundary curve marked with a thick dotted line correspond to the statically unstable menisci. The ones marked with a thick continuous line correspond to the dynamically stable capillary states of crystal growth (r is growing crystal radius, h is meniscus height). The boundary curves marked with a thick dashed line are dynamically capillary unstable.
KA. Tatarchenko et al. /Journal of Co'stal Growth 180 (1997) 615-626
625
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2.9
..',
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\
2.8
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L
f
0.7
0.8
I
I
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I
f
f
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i
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1.2
1.3
14
Fig. 6. Continued.
boundary curve which are proposed for the growth angle @o = 0. The first one is formed by the menisci which have a catching angle on the shaper in the interval ~/4 < ee < 3~/4. All these menisci are statically stable. The states which correspond to the decreasing parts of the boundary curve are capillary stable. The size of a growing crystal can be larger or smaller than the shaper diameter. The second part of the boundary curve is formed by menisci which have the form of a drop. Some of these menisci are statically unstable. The diameter of the grown crystal for all these states is larger than the shaper size and the catching angle is in the interval < ed < 7~/4. All the states corresponding to the decreasing part of the boundary curve are capillary stable. Increasing the pressure leads to a topological bifurcation of the boundary curve (Fig. 6b). Then, only a meniscus having a drop shape represents a capillary stable state. Some menisci are statically unstable, which is common for a droplike meniscus curve. There are two families of
stable menisci. The first one represents states where the crystal diameter is close to the shaper size; the meniscus height in this case corresponds approximately to the applied pressure. The capillary stable state corresponds to the decreasing part of the boundary curve. The crystallization front of the growing crystal is placed at the level of the melt free-surface. The second family represents menisci with crystal diameter larger than the shaper size, and menisci with small heights. Here, the capillary stable states correspond to the increasing parts of the boundary curve. The shaper is placed above the melt free-surface. Some menisci are statically unstable (Fig. 6c). The boundary curves are built for a zero value of the crystal growth angle. Some menisci have a bubble form. There are two different families of stable states on the boundary curves (marked with a thick continuous line). The first one represents a positive meniscus height and crystal diameters which are somewhat lower than the shaper size. All the menisci have a convex form. The second family of menisci corresponds to
626
KA. Tatarchenko et al. / Journal o f C~_ stal Growth 180 (1997) 615--626
a negative meniscus height and a crystal diameter significantly less than the shaper size. All the menisci in this case have a bubble shape. In the case of a negative pressure in the meniscus and a capillary unstable state, it is possible to stabilize the process by increasing the pressure by means of shaper displacement towards the plane free-surface of the melt.
5. Conclusions In the present work, the dynamic stability properties of the possible shapes of menisci for axisymmetric crystal growth were studied. The case of a catching b o u n d a r y condition on the sharp edge of the shaper was analyzed. The limits of application of this b o u n d a r y condition were estimated. A few non-traditional stable shapes of menisci with drop and bubble shapes were obtained, which could be realized in practice. The bubble shape with negative level of pressure and negative height of meniscus is very interesting. It seems that the temperature gradient at the crystallization front can be significantly smaller c o m p a r e d to the cases where the lateral surface of the crystal is cooled by radiation. This configuration could be applied for the growth of high-quality crystals.
Acknowledgements This work was supported by the Direction G6n6rale de l'Armement ( D G A - D R E T ) within the frame of contract no. 92.525 with the C E A / C E R E M .
References [1] A.V. Stepanov, Zh. Tech. Fiz. 29 (1959) 339 [Sov. Phys.JETP]. [2] B. Chalmers, H.E. LaBell, A.J. Mlavsky, J. Crystal Growth 13/14 (1972) 84. [3] G.H. Schwuttke, Phys. Status Solidi (a) 43 (19771 43. [4] J. Czochralski, Zs. Phys. Chem. 92 (1917) 219. [5] V.A. Tatarchenko, A.I. Saet, A.V. Stepanov, Proc. 1st Conf. on Stepanov's Growth of Semiconductor Single Crystals, Ioffe Phys.-Tech. Inst, Leningrad, 1968, p. 83. [6] V.A. Tatarchenko, A.I. Saet, A.V. Stepanov, Bull. Acad. Sci. USSR, Phys. Set. 33 t1969) 1782. [7] V.A. Tatarchenko, J. Crystal Growth 37 (1977} 272. [8] V.A. Tatarchenko, Shaped Crystal Growth~ Kluwer, Dordrecht, 1993. [9] V.A. Tatarchenko, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth, vol. 2, part 2, North-Holland, Amsterdam, 1994~pp. 1011 1111. [10] V.A. Tatarchenko, Fiz. Khim. Obrabotki Mater. 6 (1973) 47 [Soy. Phys. Chem. Mater. Treatment]. [11] V.A. Tatarchenko, 4th Int. Conf. on Crystal Growth, Thesis of Reports, Tokyo, 1974, pp. 521 522. [12] G.A. Korn, T.M. Korn, Mathematical Handbook for Scientists and Engineers, Mc Graw-Hill, New York, 1961,p. 282. [13] L.D. Landau, E.M. Lifchits, Mecanique des Fluides, Mir, Moscow, 1971, p. 289.