______ 1 Crystal Reewrch and Technology I 16 I 11 I 1981 11239- 1245 1

Gurnam SINGH, B. GHOSH, R. Y. DESHPANDE

Bhabha Atomic Research Centre, Technical Physics Division, Bombay, India

On the Shape of Crystals Grown by Kyropoulos Technique

The shape of the crystal grown by Kyropoulos technique has been evaluated on the basis of simple maas transfer approach. For some specific cases we have computed the shapes which are resaoneble and generally in conformity with physical requirements. To verify the results crystals of KC1 were grown under two specific conditions. The shapes of the resulting crystals agreeing with the calculated shapes within limits of error lend support to the velidity of msss transfer approach.

The shape of crystals grown by the Kyropoulos technique depends on the temperature distribution inside the melt which is governed by the design of the furnace, rate of cooling of the seed and the furnace, thermal characteristics of the melt and the crystal etc. Determination of the shape of the crystal based on the heat transfer taking into considerations all these parameters is difficult and has not been attempted so far. It is our aim in this paper to show that the general shape of crystals can be deduced simply by using mass balance equation. We have taken the solidpiquid inter- face as convex towards the melt as it is normally observed. SCHONHERR has evaluated as well as shown the shape of the solid liquid interface to be an ellipsoid of rotation. However, the shape of the interface reproduced might as well be approximated as spherical. It wi l l be obvious in this paper that the shape of the crystal would not deviate very much as to whether solid liquid interface is assumed spherical or ellipsoid of rotation or of any other shape.

The level of the melt inside the crucible falls as the crystal grows around the station- ary seed crystal. The solid liquid interface moves down without any discontinuity at any point - change of volume on solidification being reflected in the diameter of the growing crystal. Shape acquired by the growing crystal is governed by the rate of fall of melt with reference to the radius of the crystal which is strongly dependent on the shape of the interface. We have worked out the shapes of the growing crystals under various conditions based on the mass transfer approach with the following simplifying assumptions : (1) There is no bending of melt surface due to surface tension. (2) There is no loss of melt by evaporation during growth. Figure 1 depicts the dynamic conditions for a growing crystal and its solidlliquid interfaces a t depths z and z + dz, from the end of the seed crystal. Figure 2 represents the interfaces corresponding to (2, z ) and (x + dx, z + dz) positions of the growing crystal. The two circles represent the two spheres of which interfaces are parts. When the level of the melt drops from z to z + dz the radius of the interface changes from r to r + dr, the angle subtended by the interface at the centre of the sphere changes from 8 to 8 + d8 and the centre is displaced by dc along the z axis.

Referring to Figure 2, we obtain, dx = dr sin e + r cos e de (1)

and COB 0 dr - r sin 8 d0 + dc = dz .

Gurnam SINGE et al. 1240

++

,dz

Fig. 1 Fig. 2

Fig. 1. Schematic representation of crystal growth by Kyropoulos teohnique

Fig. 2. Geometrical representation of interlaces

Decrease in the mass of melt when the level falls from z to z + dz is given by nr3 n 3 3

[ n ~ a az- - (2 - 3 cos e + cos3 e ) + - (r + ar)3 ( 2 - 3 cos (e + ae) + + C0S3 (6 + a w l el

or

where el is the density of the melt and R is the radius of the crucible. Similarly, increase in the mass of the crystal when melt level falls from z to z + dz is given by

Where e8 is the density of crystal.

riel [RB dz + (2 - 3 COB 8 + C O S ~ 8) ra dr + 1.9 sin3 8 do] (3)

neE [za dz + (2 - 3 COB 8 + C O S ~ 8 ) ra dr + r3 sin3 8 a81 . (4)

Equating (4) and (3) on the basis of conservation of mass, we get, (2 - 3 cos 8 + C O S ~ 8) (eE - e l ) dr + r3 sin3 8(eE - e l ) a8 =

= (Rael - %'@a) dZ . (5) Solving for dr, dc and d8 in terms of dx and dz using equations (l), (2) and (5) we obtain :

-2ra(e, - el)sin8( 1 - coS0)dz + (Rael - %'el) + (el -pl)r8(l - cos8)adz

(7 )

ac = - e l ) (1 - cos e)a

t

- e l ) (2 - 3 cos 8 + C O S ~ 8 ) dz - (Rael - Z e,) sin 8 dz d o = - - - e l ) (1 - cos 8)s - (8)

A number of situations may be obtained for the convexity of the interface as the crystal grows. We start with a special case when the convexity remains constant.

On the Shape of Crystals Grown by Kyropoulos Technique 1241

This would require the radial temperature gradient inside the furnace (melt) invariant with respect to z. The sit,uation might be obtained by suitably winding the furnace. Shape of the crystal for the case is obtained by putting dr = 0 in (6) and solving the resulting equation

Plot of z against x gives the shape of the crystal. Maximum value of r might be infinite corresponding to the planar interface.

In another situation the crystal may grow uniformly around the seed crystal as shown in Figure 3. This happens when the seed crystal is strongly cooled and most of

Fig.3. Crystal interface

growing uniformly around the starting

the heat is extracted through it. This requires dc = 0 as the crystal grows. The shape of the crystal is obtained by equating equation (7) to zero and solving the following equation :

(Wete, - 9) + (1. - edeJ x

(10) Where r, is the starting radius of the interface and Ro is the radius of the seed crystal. Minimum ro is equal to the radius of the seed crystal.

Fluctuations in the curvature of the interface arising due to fluctuation in the temperature of furnace or surroundings, rate of cooling of the seed crystal or furnace can be studied from equation (6) rewritten as:

' (11)

Equations (9), (10) and (11) giving the shapes of the crystals under various conditions of constraints have been computationally solved by Runge-Kutta method. For the numerical solutions the parameters ro and R have been assigned the values 1.0 Cms and 5.0 Cms, respectively. The value of has been taken as 0.8 for KC1 (LONDON et al.). Figure 4 shows that the curvature of the upper surface of the resulting crystal decreases with increase in the radius of the interface. However, the crystal takes a limiting cylindrical shape of radius R in all the three cases after a length which decreases with r. Curvature of the top surface of the crystal arises due to convexity of the interface. When the interface is convex the amount of the material to be solidified

ax - (Raeg/es - za) r/.a - za + r (1 - el/e,) ( r - l/r2 - za)~ (dr/dz) dz x v - ede4 ---

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0-

7 -

2 -

3 -

4 -

5-

6 - 7 -

8-

Gurnclm SINQE et al.

0- 7 -

2 -

3 - 4- 5- 6 -

7 -

8-

L 1 I 1 1 I I I I I I , 5 4 3 2 7 0 7 2 3 4 5

Fig. 4. Shape of the cryStSl for dr = 0. (1) r = 5 G: (2) r = 10.00 and (3) r = 40.00 cm

per unit increase in the diameter of the crystal is more than that for a planar interface. This causes the level of melt to fall a t a higher rate which is significant compared to radi- al growth rate and therefore the resultant of the two produces a curved upper surface of the crystal. Similar shapes can also be obtained from equation (8) for 8 = constant case as illustrated in Figure 5. The shapes of crystals corresponding to dc = 0 is shown in Figure 6. Considering Figure 6 we find the curvature of the upper surface of the crystal to decrease with increasing r,, the general shapes of the curves agreeing

I I I I I I I I I l l

5 4 3 2 7 0 7 2 3 4 5 Fig. 5. Shape of the crgstal for d0 = 0. (1) 0 - 4 2 . (2) e = 4 4 and (3) 8

On the Shape of Crystals Grown by Kyropoulos Technique

3 - 4- 5 -

6 - 7 - 8 - 9 -

1243

1 1 l 1 1 1 1 1 1 1 1

5 4 3 2 7 0 . 7 2 3 4 5 Fig. 6. Shape of the crystal for dc = 0. (1) t o = 1.00: (2) ro = 2.00 and (3) ro = 50.00 cm

with those shown in Figures 4 and 6. However, there is a difference in this case, the radius of the growing crystals goes on increasing and shoots beyond R by a finite but a very small amount, the magnitude of the overshoot decreasing with T,.

On attaining the maximum value it approaches the limiting value asymptotically. The overshoot takes care of the loss of mass due to constant decrease in the interface curvature.

Is is evident from Figures 4,6 and 6 that the extreme changes in the convexity of the interface give rise to small curvature on the top surface of the crystal, there being no apparent deviation in the shape of the crystal; therefore it can be concluded that the shape of the interface whether assumed spherical or ellipsoidal will have little influence on the shape of the crystal.

To study the effect of interface fluctuations on the shape of the crystal we have

cos 2nfz where rmal and taken T varying sinusoidally as r =

r d n are the maximum and minimum values of r which oscillates with frequency f . rmpx and rdn are so chosen that the radius of the crystal does not increase beyond the ra- dius of the crucible. Results obtained are graphically shown in Figure 7 for two diffe- rent values of f . From Figure 7 it is seen that as the radius of the interface fluctuates sinusoidally the radius of the resulting crystal oscillates with the same frequency. However, it is should be appreciated that when the variation of the radius of interface is small the variation in the radius of the crystal would also be small. As a preliminary check a few KC1 crystals were grown using an alumina crucible

of 10 cms id. and 10 cms ht in a furnace consisting of a vertical muffle heater and a bottom plate heater. Figure 8 is a photograph of a typical crystal grown with spherical interface obtained through minimization in the variation of the radial temperature gradient. This is realised by the proper choice of furnace windings and power supplies to the heaters. Of course, it is difficult to obtain dr =O during the growth of the crys- tal. However, the shape of the crystal is seen to agree with the calculated shape within

rmax + rmin rmax - rmin - 2 2

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0- 7 -

2 -

3 -

4- 5 -

6 - 7 -

8 -

Gurnem SINGE et el.

I I I I I I I I I I I 5 4 3 2 7 0 7 2 3 4 5

Fig. 7 . Shape of the crystal for r varying sinusoidally (1) f = 1 cyclelcm; (2) f = 0.5 cyclelcm, in both ca8es Tmax = 9.00 cm and rmin = 5.00 cm

Fig. 8. Photograph of the crystal grown with nearly spherical interface of constant curvature

Fig. 9. Photograph o! the crystal grown by fluctuating the temperature at the interface

On the Shape of Crystals Grown by Kyropoulos Technique 1346

limits of error. Figure 9shows the crystal grown by varying the temperature of the interface by means of the bottom heater. The fluctuations were not sinusoidal, even then the shape indicates the trend of the shape given in Figure 7. The shapes of the crystals agreeing with the calculated shapes within limits of error lend support to the validity of the approach.

Our thanks are due to Drs. Deepak Dhar and A. C. Biswas of Tata Institute of Funda- mental Research for critically going through the manuscript and valuable suggestions.

Beterenoes

LONDON, G. J., UBBELOHDE, A. R.: Trans. Faraday SOC. 647 (1966) SOHONEERR, E.: J. Cryst. Growth 8-4, 266 (1968)

(Received March 27, 1981)

Authors’ addr-8 :

c/o Dr. B. GHOSE Bhabha Atomic Research Centre, Technical Physics Division Bombay 400 085, India