Hydrodynamics of Czochralski growth—A review of the effects of rotation and buoyancy force

Prog. Crystal Growth and Charact. 1984, Vol. 9, pp. 139-168 0146-3535/84 $0.00 + .50 Printed in Great Britain. All rights reserved. Copyright © 1984 Pergamon Press Ltd.

HYDRODYNAMICS OF CZOCHRALSKI GROWTH - -

A REVIEW OF THE EFFECTS OF ROTATION AND BUOYANCY FORCE

A. D. W. Jones

School of Physics, The University, Newcastle upon Tyne NE1 7RU, U.K.

(Received 5th December 1983)

ABSTRACT

The aim of this paper is to review work, largely from the hydrodynamic literature, which gives insight into the effects of rotation and buoyancy force on the flow of the melt in Czochralski growth. The importance of this study is first considered and the nature of the hydrodynamic problem is then discussed. Attention is focussed on two flow regimes, the first in which flow is driven by differential rotation, §2, and the second in which flow is driven by buoyancy force, §3. Progress is made towards understanding the physical mechanisms which govern the flows observed in model experiments and numerical simulations, although much remains to be understood. This work also indicates some non-axisymmetric and time-dependent flows which may occur in the melt. The paper ends with a discussion of flows driven by the combined effects of rotation and buoyancy.

I. INTRODUCTION

1.1. The importance of hydrodynamics in Czochralski ~rowth

Large single crystals of semiconductors, oxides, spinels and other electronic materials are required for the manufacture of solid-state devices. The largest single demand is for silicon to be used in integrated circuits. Such crystals are grown from the molten state by several methods [23]. The principal technique is crystal pulling, due originally to Czochralski [14] and commonly known as Czochralski growth.

The essential features of Czochralski growth are shown in fig. 1. The charge material is contained in a crucible, and heated to above its melting temperature. A pull rod with a chuck key at its end containing a seed crystal is lowered so that the seed dips into the melt. It is cooler than the melt so that material crystalises on its end. The crystal is rotated and pulled slowly upwards so that its end face is always in contact with the liquid surface and in this way a cylindrical crystal is grown.

The diameter of the crystal is controlled by varying the heater power and therefore the temperature of the melt. The rotation of the crystal helps to make

139

140 A.D.W. Jones

rotating crystal is drawn from the melt

-....

heat in

J

J

heat in

Fig. I. A schematic diagram of Czochralski growth.

Hydrodynamics of Czochralski growth 141

it axisymmetric and also leads to the formation of a viscous boundary layer under the crystal. Impurities dissolved in the melt will diffuse through it and be incorporated into the crystal as a dopant. If the boundary layer is of uniform thickness the diffusion and incorporation of impurity will also be uniform. Crucible rotation is used in the growth of some materials, notably silicon. The whole is also enclosed to allow the control of ambient gas.

Crystals between 1 cm and lO cm in diameter are commonly grown. Pull rates are varied between lO -~ and lO-Scm sec -I, while the rotation rates of the crystal and crucible are varied between O and 200 r.p.m, and 0 and 20 r.p.m. respectively. The temperature of the melt may be as low as 301K for Gallium and as high as 2408 K for spinel. Through the control of these parameters perfect single crystals may be grown.

Difficulties in growing good crystals may arise, however. Cracks, dislocations, and other microdefects may occur. The impurities grown into the crystal may not be evenly distributed, but occur as bands alternately rich and deficient in impurity, known as striations. Such defects impair the successful operation of devices made from these materials. More serious difficulties arise during growth. The diameter or interface shape may change suddenly and uncontrollably, or the crystal may 'melt-off' or the whole melt suddenly solidify. The crystal grower is concerned to understand why these problems occur and hence discover how to eliminate them. He also wants to know how to scale up the process to grow larger, but still perfect, single crystals.

It has become clear that the hydrodynamic behaviour of the melt greatly affects growth and that the understanding and control of the fluid dynamics of this system is the key to overcoming many of the difficulties described. It has already been mentioned that growth is controlled by maintaining the correct temperature in the melt, and that a uniform boundary layer under the crystal. is desirable to allow a uniform distribution of impurity. It is the temperature distribution and fluid flow in the melt that play an important part in allowing or impairing successful growth.

Burton, Prim and Slichter [5] were the first to analyse the effect of crystal rotation on the diffusion of impurity from the melt into the growing crystal. They calculated an effective segregation coefficient which depends on the flow under the crystal. From experiments by Cockayne, Chesswas, Plant and Vere [13] and analysis due to Carruthers [7] and Hurle, Jakeman and Pike [26] it is found that temperature oscillations in the melt lead to the formation of striations in the crystal. Hurle and Jakeman [25] have shown that temperature oscillations also lead to adecrease in the average degree of segregation. It is clear, therefore, that the nature of the flow under the crystal and the presence of temperature oscillations have a marked effect on the distribution of impurity within the crystal. (For a more complete review of this topic, see Hurle [23]).

Observations during the growth of crystals have demonstrated that the sudden changes in the shape or diameter of the interface are associated with changes in the flow. It was noted by Brice, Bruton, Hill and Whiffin [4] that at a critical rate of rotation of the crystal a wave-like instability appeared on the surface of the melt. At this point the diameter of the crystal varied rapidly and uncontrollably. This instability appears to be associated with a change in flow regimes from flow driven by buoyancy forces to flow driven by the rotation of the crystal [27]. Miller, Valentino and Schick [37] have described how the interface shape changes suddenly as the flow changes from one regime to another; Zydik [53] found that the diameter varied rapidly as the flow changed. The two effects of changing diameter and interface shape are equivalent, and depend on the type of control used.

142 A . D . W . Jones

A knowledge of the flow regimes and temperature fields together with associated instabilities is therefore essential to successful growth. To assist in the production of better crystals, the study of the hydrodynamics of this system should lead to an understanding of the nature of the velocity, temperature and solute fields sufficient to be able to predict and control these through varying parameters such as heater power and the rate of rotation and perhaps by the application of other body forces such as magnetic field. Such an understanding has been sought largely through experimental models and numerical simulations. These investigations show how the melt behaves under the conditions chosen for the experiments but have not yet led to a clear understanding of the physical mechanisms involved. For prediction and control it is important to understand why the melt behaves as it does. This review therefore concentrates on work, mostly in the hydrodynamic literature, which gives insight into the effects of rotation and buoyancy forces. It is complemented by reviews by Schwabe [47] and Langlois [32]. Schwabe discusses the effects of surface tension while Langlois concentrates on numerical simulations.

1.2 The hydrodynamic problem

As a model of crystal growth consider the system shown in fig. 2. The crucible is a flat-bottomed, right circular cylinder of radius R o and height L.

disc

f T I

!

R 0 - -

~ R I ~

melt

I insulating base

"•'•£•o J

right circular cylinder \

To

Fig. 2. The ideaiised model of Czochralski growth.


The temperature of the wall is To. The crystal is modelled by a disc of radius Rz whose temperature is TI and the crucible and crystal rotate with angular velocity ~0 and flz respectively. This model is an idealisation of crystal growth. For example, the detailed shape of the crucible and crystal, the thermal boundary conditions at the free surface and the variation in temperature of the crucible wall have not been included. The detailed boundary conditions are certainly important in determining the flow and temperature field, but this simplified model serves to define characteristic length scales, temperatures and rates of rotation. As a further simplification only the effects of rotation and buoyancy are considered in this review.

The velocity K*, pressure p* and temperature T* are given by the Navier-Stokes, continuity and diffusion equations [2] together with appropriate boundary conditions. Denote by r* t* p, a, g, ~ and < the position vector, time, density, the coefficient of cubical thermal expansion, acceleration due t~gravity, kinematic viscosity and thermal diffusivity respectively, and by T I a reference temperature. In a frame of reference rotating with angular velocity ~, where is a unit vector in the upward vertical direction, the equations are

a at* q* + q*.V*q + 2fl __k ̂ q* = _ ! V ' p * - ~ z ~ ^ (~ ^ r * ) + a g ( T * - T z ) K + vV*2K *

p - - l l

(i.i)

v*.i* = 0 (1.2)

a n d

aT* + K ~ ~ V ' T * = <V*2T * .

at---* (i.3)

Here buoyancy is the only body force included and the Boussinesq approximation [51] has been used. That is, variations in fluid properties other than density are ignored and the density variations are only included insofar as they give rise to gravitational force. Further, the variation of density with temperature is linearised. The momentum equation (1.1) may be simplified by defining a reduced pressure P* by

F* = p* - ½~Z)~2(~ ^ ~*).(~ ^ ~*). (i.4)

Then

i V'p* + ~2 k ^ (k ^ r*) = i VP* + 1 ~2 - ~ _ - ~ ~-~ (k ^ r*).(k ^ r*)V*p. (1.5)

I ~2 ^ However, the term ~ (k ~ r*).(k ^ r*)V*p of order ~(T*-TI)~2R where R is a typical radial dimension. It is small compared with the buoyancy force and may therefore be ignored. (This would not be true in a low gravity environment). The momentum equation therefore becomes

^

8__~q,8 + q*.V*q* + 2~k ^q* = - ~i VP* + ag(T*-T1)k_ + ~V*2q *. (1.6)

144 A . D . W . Jones

Some insight into the nature of the solution of equations (1.2), (1.3) and (1.6) and into the physical mechanisms governing the flow, can be gained using dimensional analysis. Let ~ be the average rate of rotation

= ½ I~o + ~zl , (1.7)

and define a Rossby number Ro, by

Ro = U/~L , (1.8)

where U is a velocity scale yet to be chosen. In order to simplify the analysis it is assumed that L, R0 and Rz are of the same order of magnitude and L is used for the length scale. In fact RI/L is typically 0.25. Dimensionless quantities ~, t, P and T are defined by

R* = U i , ( 1 . 9 )

t* = tL/U , (I.i0)

p* = pU2p , (l.ll)

T* = TAT + T i (1.12) a n d

AT = To - TI . (1.13)

Suppose that the flow is driven by rotation and define the velocity scale by

The dimensionless equations are then

I 1 IRol ~-~ + q. Vq + 2~ ^ il : - VP + GrEZTk + EV2q

and

V.q= 0

rRoJ( T i E ~-t + - Pr V2T '

where the dimensionless parameters E, Gr and Pr are the Ekman, Grashof and Prandtl numbers respectively, which are defined by

and

E = V/gL 2 ,

Gr = ~gL3AT/v 2

PT = V/K

Some values typical in Czochralski growth are

E = i0 -# to 10 -3

(1.14)

(1.15)

(1.16)

(1.17)

(1.18)

(1.19)

(1.2o)

(1.21)


Gr = lO 7 to 5 x lO s , (1.22)

Pr = 0.01 to 23 (1.23)

and

Ro = 2 to 3 (1.24)

This scaling will only be appropriate when GrE 2 << l, since only then will the buoyancy term be negligable. ! In this case there will be thin viscous boundary layers with a thickness 0(LE 2) which would be about 1 mm in a typical puller. Non-linear terms will be important because Ro is O(1) and this makes the analysis difficult.

In the other extreme, the flow is driven by buoyancy forces. scaling, when Gr >> I as in Czochralski growth, is

! U = (agATL) 2 ,

The appropriate

(1.25)

and the dimensionless equations are then

~R 2 1 ~--~ + q.Vq + i ~ ^ q = - VP + Tk + i V2q (1.26)

_ _ Gr~E -- Gr ~

and

V.q = 0 (1.27)

~T 1 (1 .28) ~--~ + ~.VT = I V 2T

PrGr =

It can be seen from equation (1.26) that this is the appropriate scaling when Gr~E >> 1 and that in this case there will be a boundary layer flow with thermal boundary layers of thickness O(LPr-~Gr-¼) which for a low Prandtl number material like silicon would be typically l0 mm. Unlike many problems in fluid mechanics, the fact that this is a boundary layer flow does not lead to a great simplification because the fluid is enclosed.

The two regimes, that is flow driven by rotation and flow driven by buoyancy forc% are discussed in §2 and §3. Each section begins with a description of the flows that have been observed in model experiments and numerical simulations. The hydrodynamic literature is then reviewed, and this gives some understanding of the observations and indicates that non-axisymmetric and time dependent flows may occur. Some conclusions are drawn at the end of each section. The combined influence of rotation and buoyancy force is discussed in §4.

2. FLOWS DRIVEN BY ROTATION

A disc and cylinder system similar to that shown in fig. 2 was used to investigate the flow driven by rotation [27]. With the cylinder stationary it was found that fluid was drawn upwards by the rotating disc and pushed outwards, falling at the cylinder wall and being drawn inwards in the lower half of the tank. With differential rotation the liquid near the disc was seen to be drawn upwards and

146 A . D . W . J o n e s

pushed outwards, the motion being similar to that for disc rotation alone. Further below the disc the direction of the vertical flow depended on the direction of rotation, being towards the disc with co-rotation and away from it with counter-rotation (fig. 3). In this region the fluid circulated in the same sense as the cylinder but was not in solid body rotation. Rather the azimuthal velocity had a marked radial dependence. With counter-rotation liquid spiralled outwards across the base and rose in a boundary layer at the cylinder wall. With co-rotation the flows near the wall and base were similar but with their direction reversed.

Disc L__J l

I

1' I I

I I

i # # i #

I I

I I I I

I

I

I

I

I

_I

• I

i i t i i

Co -rotation Counter-rotation

I

I I I I

I I I I I

Fig. 3. The flow due to differentia[ rotation [27]

Far from the boundaries the azimuthal velocity was found to be approximately independent of height. Similar experiments had earlier been performed by Carruthers and Nassau [I0] but with lower dimensionless rates of rotation. They observed that the region of rising fluid, (with counter-rotation) or falling fluid (with co-rotation) was not at the wall, but some distance from it, and that this was surrounded by fluid in approximately solid body rotation. Good agreement is found between experimental observations and the results of numerical simulations (see [32]). In particular Langlois [31] found that, far from the boundaries, the azimuthal velocity is approximately independent of height, but has a radial dependence.

The general character of these flows may be understood from existing theories. This section begins with a discussion of the Taylor-Proudman theorem, which is a linear theory. It is often suggested in the literature on crystal growth that it is as a consequence of this theorem that the flow is approximately independent of height far from the boundaries, but it is shown that the conditions under which it may be applied are not normally satisfied in crystal growth. The non-linear equations are therefore considered, and most attention is focussed on the similarity solutions for a single infinite rotating disc and for two infinite, coaxially rotating discs. The effect of the finite size of the disc and of radial boundaries is also discussed. Stability is considered since these flows may be unstable to unsteady or non-axisymmetric disturbances. Greenspan [17] gives a fuller account of many of the topics discussed in this section.


2.1 Linear theory (Taylor-Proudman Theorem)

A fluid-filled cylinder rotates with angular velocity ~o and a disc of radius RI, rotates coaxially in the fluid with angular velocity Ql (fig. 2). The equations of motion are non-dimensionalised in the same way as for equations (1.15) to (1.17) but the effects of buoyancy are neglected by setting Gr = O. The dimensionless equations of motion are then

Ro( ~t + ~.V&) + 2k ^ R = -VP + EV2~ (2.1)

and

where

V.~ = 0 , (2.2)

Ro = (~i-~0)/~ , (2.3)

E = ~/£R~ (2.4)

and

Q = (Q1 + ~o ) /2 (2.5)

The Rossby number, Ro, which is a measure of the difference in rates of rotation, may be thought of as the ratio of the (non-linear) inertia terms to the Coriolis force, while the Ekman number E, which is a measure of the mean rate of rotation, is the ratio of the viscous terms to the Coriolis force. In the limit of small Ro and E, equation (2.1) becomes

2k ^ E = -Vp . (2.6)

The curl of this expression gives the Taylor-Proudman thorem

k.Vq = 0 . ( 2 .7 )

Hence, the particle velocity must be independent of the co-ordinate measured along the axis of rotation. Hide and Titman [20] predicted and observed the flow for small E and Ro in the disc and cylinder system described above with the disc totally immersed (fig. 4). They found that above and below the disc there is a region (i) of solid body rotation (a Taylor column) at a rate intermediate between the disc and the tank, while at a larger radius than the disc is a region (ii) rotating with the tank. Between regions (i) and (ii) is a thin vertical shear layer (iii) and there are horizontal viscous boundary layers (iv) at the ends of the tank and on the disc.

When the Rossby number is not small, the conditions required by the Taylor- Proudman theorem are not satisfied and a change in fluid motion is to be expected. Hide and Titman found that above a critical value of the Rossby number Ro (Ro c ~ 0.I at E = I0 -~) the Taylor column is unstable and the flow become~ non- axisymmetric. Interestingly, the flow is still apparently invariant along the axis of rotation in the region far from the end walls and experiments [I0] indicate that, for small values of E, it remains so even for values of Ro which are 0(i). However, in this parameter range, typical of Czochralski growth, the non-linear terms must be included. The rest of this section concerns solutions of these non-linear equations.

148 A . D . W . J o n e s

(ii)

d . . . . . ( ~ . . . . . I I I t I I I I I I I I it t(iii) I

t I I I

I i

(i) (iii~ I t i I I I I I

- - - - - - 4 - I

. . . . . ( i v ) . . . . il i i

l [

I I ( t ¢ i

( i ) I ,

I I i

l id

,/~indricat tank

Fig. 4. VezLicaL shear layers observed by Hide and Titman [20].

2.2. Non-linear theory

2.2.1. Similarity solutions for a sinsle disc. The similarity solution for the flow due to an infinite rotating disc was used by Burton, Prim and Slichter [5] in their calculation of the effective distribution coefficient. Thus the relevance of this area of fluid mechanics to crystal growth has long been recognised. Here the similarity solution is reviewed and the effect of a second coaxial disc and of the finite size of the disc are then considered.

In an intertial frame the dimensional equations of motion for the pressure p and velocity (u,v,w) in the directions (r,@,z) are

au v 2 au 1 ~p + V( a2u a a2u u ar r + w ~z = - par ~ + ~r (u/r) + ~ ) , (2.8)


~2v ~ ~2v ~v __uv + ~w~ = ~ ( ~r-7 z + "~r ( v / r ) + ~ z - - - ~ ) ( 2 . 9 ) U~r + r

~w ~w 1 ~ ~2w 1 ~w ~2w + W~z O + 9( ~ + r~-~r + ~ ) (2.10)

and

~U U ~W ~'-r- + r + ~z = 0 (2.11)

It has been assumed that the motion is axisymmetric so that all quantities are independent of @. If the disc (at z = O) rotates with angular velocity ~ and the fluid far from the disc (z ÷ ~) is stationary then the boundary conditions are

and

U = O, v = r~, w = 0

u = O, v = O, a s z ÷ ~ .

at z=O (2.12)

Von Karman's [29] similarity solution is of the form

! u = r~F(~), v = r~G(~), w = (~Q)2H(~), p = pv~P(~), (2.13)

where

!

= z(~/~) 2. (2.14)

substitution of (2.13) into (2.8) to (2.11) gives

F" = F 2 - G 2 + HF', (2.15)

G" = 2FG + HG', (2.16)

P' = H"- HH' (2.17)

and

H' = - 2F , (2.18)

where dashes indicate differentiation with respect to 6. became

The boundary conditions

F(O) = O, G(0) = I, H(O) = 0 and (2.19)

F ( ~ ) = O, G ( ~ ) = 0 .

Equations (2.8) to (2.11) are thus reduced to a set of ordinary differential equations, (2.15) to (2.18), which have been solved numerically by Cochran [12]. The disc acts as a centrifugal fan, drawing in fluid axially! imparting angular momentum to it in a layer whose thickness is of order (~/~)~ and expelling it tangentially. It is interesting to note that, for Czochralski growth, the value

1 of (~/~)2 is typmcally less than 1 mm. The important property of this solution, noted by Burton, Prim and Slichter is that the axial velocity, w, is independent of r. Thus the region near the disc where w becomes small and diffusion is

150 A . b . W . Jones

important, is of uniform thickness, and according to this model, impurities will be incorporated uniformly into the growing crystal.

Suction at the disc may also be included [50] by modifying the boundary condition on the axial velocity at the disc, so that

H(O) = - a , ( 2 . 2 0 )

where a is the suction velocity sca]edby (vQ) 2, as in (2.13). The effect, for a > O, is to decrease the radial flow at the disc and to increase the axial velocity, and its inclusion may be considered as modelling the effects of mass flux, due to the growth of crystal. However, for typical growth parameters a < 0.I and the velocity profiles will be little different from the case a = O.

Rogers and Lance [42] considered the case when the fluid far from the disc is in solid body rotation with angular velocity sQ. They showed that equations (2.15) to (2.18) apply with the addition of an extra term to (2.15) and with a modified boundary condition on G. The equations become

F" = F 2 _ G2 + HF' + s2~ ( 2 . 2 1 )

G" = 2FG + HG' , (2.22)

and P' = H" - HH' (2.23)

H' = -2F (2.24)

with boundary conditions

and

F ( o ) = o, ¢ ( o ) = 1, H(O) = 0

F ( ~ ) = O , ¢ ( ~ ) = s .

( 2 . 2 5 )

( 2 . 2 5 )

With co-rotation (s > O) and with the disc rotating faster than the far field the flow is qualitatively similar to that described above (s = 0), although velocities decrease as s approaches I. When the disc rotates slower than the far field (s > I) the direction of flow is reversed.

The flow with counter-rotation (s < O) has a more complicated structure which depends on the values of s and a (the suction parameter). Ockendon [39] has shown that finite solutions only exist when there is suction (a ~ O) however small, and has obtained asymptotic solutions for Isl < 1.436 and small a - a parameter range particularly relevant to Czochralski growth. She found that there is a viscous layer on the disc of ~hickness O(a}) which matches with an inviscid region whose thickness is O(a-~). This rotates in the same sense as the disc and flow reversal takes place in a further viscous layer which matches with the flow at infinity. Ockendon also showed that there could exist more than one inviscid layer thus giving other, multiple cell solutions.

2.2.2. Similarity solutions for two coaxial discs. For two coaxial discs rotating with angular velocity ~ and s~ and separated by a distance L, define an Ekman number by

E = v/~L 2 (2.26)

With co-rotation (s > 0), when the Ekman number is small (E ~ 6 x 10 -3 with s = 0.5)the region between the discs is in solid body rotation at an intermediate


angular velocity, ~, and there is a drift of fluid towards the faster disc. Each behaves like a single disc with rotation, m, at infinity [i]. Increasing the value of the Ekman number has the effect of increasing the thickness of the region of rapid change in angular velocity at each disc (fig. 5).

disc (~) L

disc (S~)

( i )

disc (~q)

solid body rotation

disc (SQ.)

( i i)

Fig. 5. Flow between co-rotating discs (s > 0), (i) E > O, (ii) E ÷ O, from Batchelor [i]

With counter-rotation, or when one of the discs is stationary (s ~ O) there exist two types of solution. In the Batchelor solution [i] the fluid between the discs is rotating and there is a layerin which the velocity changes sign. As the Ekman number is decreased so the region between the discs becomes increasingly two dimensional (fig. 6). Of this type of flow, there exist other multiple cell solutions, for which the axial velocity vanishes on more than one plane parallel to the discs [22]. The second type is the Stewartson solution [49] in which the main body of fluid is stationary and the flow near each disc is described by the Von Karman solution.

152 A . D . W . Jones

disc (~)

disc (S~) (i)

disc (~)

disc (S~)

(ii)

Fig. 6. Flow between counter-rotating discs (s < 0), (i) E > O, (ii) E ÷ O, from Batchelor [i]

2.2.3. The effect of radial boundaries and of finite disc size. From experimental work [35, 46, 49] with one of the discs stationary it is observed that when the discs are shrouded by a close fitting, cylindrical container, a geometry similar to Czochralski growth, most of the fluid rotates and thus the flow is of the 'Batchelor' type. When the discs are open, then most of the fluid is quiescent, as in the Stewartson solution. This may be explained by the fact that, in the open arrangement, fluid near the rotating disc is pushed outwards to 'infinity', and does not influence the region between the discs, but that with a shroud there is circulation so that the rotational motion in the bounuary layer also influences the interior flow. With counter-rotation (s < O) experiments by Stewartson [49] with an open configuration and by Schultz - Grunow [46] with an enclosure, indicated flow of the Stewartson type. However isothermal simulations of Czochralski growth [I0, 27] show that most of the fluid is rotating in the same sense as the rotating cylinder and that there is flow reversal near the counter-rotating disc. This flow is thus of the Batchelor type. In a recent numerical solution of the Navier-Stokes equations, without the similarity


assumptions, Djikstra and Van Bejist [I0] computed the flow between two counter- rotating, shrouded discs. They found that with weak counter-rotation, as in Czochralski growth, the flow is of the Batchelor type, but with strong counter- rotation, the flow is of the Stewartson type. They also suggested that the inclusion of radial boundary conditions removes the degeneracy found in the similarity solutions.

The effect of the finite size of the disc must also be considered in practical situations. Rogers and Lance [43] computed the flow for a stationary finite disc in a rotating fluid. Their work, together with that by Rott and Lewellen [44], indicates that over the inner half of the disc the similarity profiles are good approximations, but that at larger radii the 'edge' effect becomes increasingly important with the axial velocity changing sign at 0.84 of the radius of the disc.

Experiments with two coaxial shrouded discs have already been mentioned. Maxworthy [36] has measured the angular velocity distribution in the main body of the fluid contained by a rotating disc and cylinder and capped by a stationary disc. As distinct from the flow with two inifinite discs, there is a radial dependence in the angular velocity far from the discs. Using momentum integral methods invented by Von Karman [29], Rott and Lewellen [44] derive an approximate solution which agrees well with Maxworthy'sresults. The principle of the method is to assume that the fluid far from the disc has an unknown radially dependent circulation F(r), and to calculate the flux into or out of the boundary layer on each disc. Since there must be an overall flux balance, these expressions can be equated to give F(r). This method may prove successful in calculating the flows for boundary conditions more appropriate to Czochralski growth.

2.3. Instability

Up to this point the discussion has centred on steady, axisymmetric flows. The transition to non-axisymmetric, time dependent flow, due to instability is now examined.

It is a feature of the flows discussed above that there exists a change in angular velocity in a boundary layer near the disc. With rapid rotation this boundary layer will be thin, and so it is to be expected that at sufficiently large rates of rotation the effects of shear will lead to instability. In fact work on a similar problem, the stability of an Ekman layer [34], has revealed that instability arises from the combined effects of shear and Coriolis force.

For a single, infinite rotating disc define an Ekman number by

E = ~/~r 2, (2.27)

where r is the distance from the centre of the disc. Below a critical value, E , the flow is unstable and horizontal equiangular spiral vortices are formed. I~ a theoretical analysis Malik, Wilkinson and Orszag [35] found E c = 1.2 x I0 -s, although for simplification they neglected some radially dependent terms which may alter the value of E .

c

For Czochralski growth the Ekman number, defined by (2.27) is typically greater than 10 -4 , so that if the crystal is modelled by a single infinite disc present work indicates that instability will not occur. However, the effects of suction, rotation of the fluid or of the finite size of the disc have not yet been investigated and it is expected that the stability characteristics of these flows will be different from the single infinite disc. In particular, the occurrence of regions of reverse flow, as with counter rotation, may increase the value of E c.

154 A.D.W. Jones

I ~O

. . ~ " ' ~ wave like protuberances develop

vertical shear layer

cylinder

Fig. 7. Shear layer instability [20].

f~o ;~o

~ d i r e c t i o n of

~ ~ A precession ~ ~ A

~ . ~ j c y [ i n d e r . V ~ ~

, ,- , \ ~ ,._> / / 7___~°,r,~:,a,o~,o, °

Co-rotation Counter-rotation

Fig. 8. Non-axisymmetric flow observed with ditferent ial total ion [27]. This .~;hows the flow p~ltterns in tile tram~, of reference of the t-ot.at.ing cyiinder.

Hydrodynamics of Czochralski growth

The topic deserves further investigation since this instability is a potential source of temperature fluctuations and therefore of striations in the grown crystal.

Instability may also occur in the region far from the disc. As was mentioned in §2.1, Hide and Titman [20] found that above a critical value of the Rossby number Ro c (Ro c ~ 0.I at E = I0 -~) the flow becomes non-axisymmetric (fig. 7) and they derived an empirical relationship for Ro as a function of E.

c

Theoretical stability calculations [6, 18, 48] are based on a model by Busse [6], who proposed that unstable disturbances grow when vorticity transferred from the shear layer exceeds vorticity diffused in the horizontal Ekman layers. These calculations give moderate agreement with experimental results. Recent experiments [27] show that the flow is non-axisymmetric at Rossby numbers of 0(I) if the Ekman number is sufficiently small (fig. 8), but in this case the mechanism of instability is unclear. If the flow became non-axisymmetric during crystal growth, then it is likely that the heat and mass transport to the solid-liquid interface would change. It is therefore important to investigate this instability.

155

2.4 Conclusions

The Batchelor solution for the flow between two infinite, coaxially rotating discs models some of the important features of the flows expected to occur in Czochralski growth. From this solution it is expected that the Ekman number defined by (2.26) is an important parameter and that it is because E << 1 in crystal growth that the flow is approximately independent of height far from the boundaries. The Batchelor solution also correctly predicts a region of flow reversal close to the counter-rotating disc. The radial dependence of the azimuthal velocity is apparently a consequence of the finite size of the disc, and it is anticipated that the momentum integral method will provide a way of calculating this radial dependence.

There remains the question of the uniqueness and stability of solutions. The uniqueness of numerical solutions is little discussed in the crystal growth literature, and remains an open question. The regions near the rotating crystal and far from the crystal are both expected to be unstable at sufficiently large rates of rotation, and it will be important to determine whether or not such instabilities can occur in crystal growth.

3. FLOWS DRIVEN BY BUOYANCY FORCE

The buoyancy driven flow observed in a recent experimental simulation of Czochralski growth [27] is shown in fig. 9. Fluid rises at the cylinder wall amd flows inwards in a boundary layer at the top of the "melt". It sinks in a centrai column ot cooler fluid and moves outwards to the wall in a horizontal boundary layer at the base. The region far from the walls and the centre is largely quiescent although there is some entrainment into the regions of rising and descending fluid. It has a small unstable stratification, while the central column has a large unstable stratification. In their experimental model Miller and Pernell [38] used rather different temperature boundary conditions and

156 A . D . W . Jones

Ii

!

! I

! ! l!,,

I I I

!

",,,: , / , I !

; ' entra ined ' I I ""i ,,/--fluid--/,,

I = I I ! I

',Z , / q

boundary layers ;/

T,

Fig. 9. Buoyancy driven flow observed in an experimental simulation of Czochralski growth [27].

observed a stagnant Layer at the bottom of the model crucible. Numerical simulations of the casein which buoyancy forces dominate rotation give qualitatively similar results for the flow field. The temperature field is also similar to that described above when Pr = 0(I), but is very different for Pr << i.

The general character of this flow may be explained by considering the similarity solution for the flow due to a semi-infinite, heated vertical plate which is discussed in §3.1. However, to gain a detailed understanding a difficult internal problem must be solved. The solution of a partially enclosed flow is discussed, since this gives some further insight, and work on fully enclosed flows is briefly reviewed.

Although the details of the velocity and temperature field are not yet known, it is expected that the presence of the cold crystal will lead to the formation of a horizontal boundary layer at the top of the melt, in which temperature decreases with height. If the unstable temperature gradient is sufficiently large then buoyancy driven convection may result. ~n §3.3 convection in a horizontal layer is therefore discussed and, as a first step towards modelling convection under a rotating crystal, the modifying influence of solid body rotation is also considered,

3.1. Convection from a heated vertical surface

Consider a semi-infinite vertical plate at temperature To in a fluid at a lower temperature T . The origin is on the lower edge of the plate with the x axis vertical and ~he y axis perpendicular to the plate. If the plate is wide then the motion will be in the xy plane, in other words two-dimensional. The fluid next to the plate is heated by thermal conduction and rises. If the Grashof number, now defined by

Gr = ag(T0 - T )x3/~ 2 , (3.1)


i s l a r g e enough then t h e speed o f f l o w w i l l be s u f f i c i e n t f o r hea t t o be c a r r i e d o f f i n t h e x d i r e c t i o n b e f o r e i t has p e n e t r a t e d f a r i n the y d i r e c t i o n and convection will occur in a thin boundary layer (fig. I0). The boundary layer

I I I

6 i r"

I I

I

! !

! lL, /

plate

f lu id

Fig. I0. Flow due to a semi-infinite, heated vertical plate. will entrain fluid and increase in thickness with distance up the plate. Outside the boundary layer, with Pr = I, there will be no motion, except for a slow drift of fluid towards the plate, and the temperature will be T . If Pr ~ 1 then the flow will be qualitatively the same but the viscous and thermal boundary layers will be of different thicknesses.

With velocity components (u,v) in the (x,y) direction, the Navier-Stokes, continuity and diffusion equations in an inertial frame are

3u 3u i 3_2 + ~g(T +v ra2u 32u ] U~x + % = - 0 8x - T°°) [~x--~ + 3--7 ' (3.2)

av av i ~p +v(a2v a2v 1 u~+ ray- 0 ay ~ +~ ' (3.3)

and

8u ~v + - 0 (3.4)

3x ~y

3T 3T (32T 32T] U~x + V~y = K ~x---~ + 3~-~ , (3.5)

158


A. D. W. Jones

and

u = v = O, T = T at y = O O

u = O, T = T as y ÷ co . c o

(3.6)

(3.6)

Now make the boundary layer approximation [45] in which it is considered that the flow takes place in a thin layer near the plate. The boundary layer equations then become

3u 3u 32u U~x + V~y = ~g(T - T ) +~--~ , (3.7)

u ~v ~ v + ~ = 0 ( 3 . 8 ) o x oy

and

~T ~T 32T U-~x + V-~y = m~-~-~ (3.9)

Equations (3.7) to (3.9) can be reduced to ordinary differential equations by the similarity transformation

1 1

q = Cyx -~, C = [~g(T ° - T )/4v 2] ~ (3.10)

Then (3.8) gives

!

u = 4vC2x2~'(n), T - T = (T o - Too)O(q ). (3.11)

i

v = vCx-4(n~ ' - 3~), (3.12)

and (3.7) and (3.9) give equations for ~ and @

and


~"' + 3@~" - 2@ 2 + @ = 0 (3.13)

@" + 3 Pr~@' = 0 (3.14)

~ ( o ) = O ' ( o ) = o , ~ ( o ) = i

and (3.15)

4(~) = o, o(~) = o

I t fo l lows from (3.1) and (3.10) that the boundary layer th ickness, 6, i s

!

6 = O(xGr -4) (3.16)


and the boundary layer approximation will be valid if 6 is small compared with the height of the plate. From (3.16), this requires that

Gr 4 >> 1. (3.17)

Equations (3.13) and (3.14) with boundary conditions (3.15) have been solved for various values of the Prandtl number [40]. When the Prandtl number is large the velocity boundary layer is thick compared with the thermal boundary layer since rising heated fluid drags unheated fluid with it, by the action of viscosity. When the Prandtl number is small, the viscous and thermal boundary layers are of similar thickness since any heated fluid will rise, due to buoyancy, even if viscous effects have not spread that far from the wall [51].

The effect of wall curvature may be considered by comparing the above results for a flat plate with calculations for the flow due to a heated/ vertical cylinder [40]. If it is decided that the flat plate results are sufficiently accurate if they give errors of five per cent or less in calculating heat transfer parameters for the cylinder, then the flat plate results may be used if

1

2 s/2 Gr -~ x/R ~ 0. II for Pr = 0.72 (3.18)

and 1

23 /2 Gr-4 x/R _-< 0.13 for Pr = 1 (3.19)

where R is the radius of the cylinder.

As may be expected conditions (3.18) and (3.19) require that the boundary layer thickness is small compared with the radius of the cylinder and these conditions are normally satisfied for values typical in Czochralski growth.

The effect of a non-isothermal condition has also been investigated [40]. The similarity solution exists if the wall temperature has a power law or exponential variation with distance up the plate or if the heat flux has a power law variation. Particularly relevant to crystal growth is the case of constant heah flux, where it is found that the temperature of the plate increases with distance up the plate and is proportional to x I/s

There also exist similarity solutions for the boundary layer flow on axisyn~netric bodies [40]. The details of these flows depend on the body shape, but they are qualitatively the same as the boundary layer flow on a vertical plate. These solutions are also applicable to the flow inside axisymmetric shells if the fluid outside the boundary layer is quiescent and isothermal. This raises the problem of the analysis of flows in enclosed regions (internal flows) which is now discussed.

3.2. Internal Flows

A simplified approach to the analysis of internal flows was used by Lighthill [33] who studied convection in a cylindrical tube. Consider a vertical cylinder, radius R and length L, with the lower end closed (fig. ll). The wall and base are at a temperature To while the temperature at the orifice, on the axis of the cylinder, is TI(Tz < To). Further, it is required that mass is conserved across the cross section of the tube. In other words, the upward flow in the boundary layer at the wall must be balanced by a downward flow in the interior. Lighthill found that if the thickness of the vertical boundary layer at the wall is small compared with the radius of the cylinder then the flow can be described by the

160 A. D. W. Jones

no mass flux

i I ' 'l i". I I I L/, f I I I I

I \ T, / / I I

I I

To

2R

F i g . I 1 . Flow in a p a r t i a l l y e n c l o s e d tube when t ~ 3 ,400 and Pr >> 1 [33]

s o l u t i o n f o r a s e m i - i n f i n i t e v e r t i c a l p l a t e , bu t i f t h e t h i c k n e s s o f t h e boundary l a y e r i s s u c h t h a t i t f i l l s a s i g n i f i c a n t a r e a o f t h e t u b e , t h e f l o w i s a p p r e c i a b l y a l t e r e d . He d e f i n e d a p a r a m e t e r t by

t = ~gR~(T0 - T x ) / L ~ < , ( 3 . 2 0 )

and found t h a t w i t h t ~ 3400 (wh ich i s t h e c a s e i f t h i s i s c o n s i d e r e d a s a model o f C z o c h r a l s k i g r o w t h ) t h e s i m i l a r i t y s o l u t i o n f o r t h e i n f i n i t e p l a t e i s a p p l i c a b l e w i t h t h e r e g i o n o u t s i d e t h e bounda ry l a y e r a t t h e t e m p e r a t u r e Tz. I t s h o u l d be n o t e d t h a t t h i s a n a l y s i s i s r e s t r i c t e d to t h e c a s e Pr >> 1.

The a n a l y s i s o f c o n v e c t i o n i n a t o t a l l y e n c l o s e d r e g i o n p r e s e n t s a r a t h e r more d i f f i c u l t p r o b l e m , p a r t i c u l a r l y when t h e G r a s h o f number i s l a r g e . The f low w i l l t h e n be of a bounda ry l a y e r t y p e , but t h e b e h a v i o u r of t h e i n t e r i o r r e g i o n , f a r from t h e b o u n d a r i e s i s n o t d e t e r m i n e d by e x t e r n a l bounda ry c o n d i t i o n s ; r a t h e r i t d e p e n d s on t h e bounda ry l a y e r f l o w and t h e r e f o r e t h e e q u a t i o n s f o r t h e i n t e r i o r and bounda ry l a y e r r e g i o n s a r e c o u p l e d . A t t e n t i o n has been f o c u s s e d on two p r o b l e m s [ 4 1 ] . The f i r s t i s t h a t . o f c o n v e c t i o n be tween two w a l l s a t d i f f e r e n t c o n s t a n t t e m p e r a t u r e s , and t h e s e c o n d c o n c e r n s c o n v e c t i o n ~n a h o r i z o n t a l c y l i n d e r whose w a l l has a c o s i n e v a r i a t i o n in t e m p e r a t u r e . In t h e e a r l i e r s t u d i e s t h e c o r e r e g i o n , o u t s i d e t h e boundary l a y e r s , was assumed t o be i s o t h e r m a l , h o w ev e r , i t now seems t h a t when h e a t i n g i s from t h e s i d e t h e c o r e i s s t r a t i f i e d . The bounda ry c o n d i t i o n s f o r t h e s e two p r o b l e m s a r e r a t h e r d i f f e r e n t f rom C z o c h r a s l k i g rowth and so t h e d e t a i l s o f t h e s e f l o w s do no t seem p a r t i c u l a r l y r e l e v a n t . However , i t i s i n t e r e s t i n g to n o t e t h a t i n an e x p e r i m e n t a l s i m u l a t i o n o f c r y s t a l g rowth [27] i n which t h e c r u c i b l e w a l l was i s o t h e r m a l , t h e c o r e was found to be weakly s t r a t i f i e d a s i n t h e s e two p r o b l e m s .

3 .3 C o n v e c t i o n i n a h o r i z o n t a l l a y e r

3 . 3 . 1 , C o n y e c t i o n i n a n o n - r o t a t i n g l a y e r . In t h e s o l u t i o n s d e s c r i b e d i n §3 .1 and § 3 . 2 , t h e r e a r e h o r i z o n t a l t e m p e r a t u r e g r a d i e n t s which l e a d t o buoyancy d r i v e n m o t i o n however s raa l l t h e t e m p e r a t u r e d i f f e r e n c e s . However , i n a l a y e r o f f l u i d i n wh ich t h e r e i s an u n s t a b l e v e r t i c a l t e m p e r a t u r e g r a d i e n t , t h a t i s one f o r wh ich d e n s i t y i n c r e a s e s w i t h h e i g h t , but i n which t h e r e i s no t e m p e r a t u r e v a r i a t i o n i n t h e h o r i z o n t a l d i r e c t i o n , mo t ion o n l y t a k e s p l a c e above a c e r t a i n c r i t i c a l v a l u e o f t h e t e m p e r a t u r e g r a d i e n t . T h i s i s b e c a u s e v i s c o s i t y and t h e r m a l d i f f u s i v i t y have a s t a b i l i s i n g e f f e c t . The p ro b l em i s t h e r e f o r e t o d e t e r m i n e


when motion begins and then how it develops, and this is discussed in detail by Drazin and Reid [16]. Only the onset of motion is considered here, for which linear stability theory is applicable.

Consider a layer of fluid of depth d in which the temperature decreases uniformly with increasing height with a temperature gradient of -8. Now suppose this basic state is perturbed by a small amount and look for conditions under which this perturbation will grow with time. In cartesian co-ordinates (x', y', z') in which z' is in the direction of the upwards vertical, let the perturbation velocity have components (u', v', w') and let Q' and ~' be the perturbation temperature and pressure. Non-dimensionalise using d, d2/< and 8d as scales for length, time and temperature so that

(x, y, z) = (x'/d, y'/d, z'/d), t = t'</d 2,

(u, v, w) = (u'd/<, v'd/<, w'd/<), O = O'/Sd, (3.21)

p = p'd2/O0 <2,

where unprimed quantities are dimensionless, and where P0 is the density at z = O. Seek normal mode solutions by writing all perturbation quantities in the form

w = W(z) exp(i(kxX + k y) + ct), (3.22) Y

and substitute the perturbed quantities, that is the basic state plus its perturbation, into the equationsof motion. Elimination gives

(D 2 _ £2)(D2 _ £2 - c)(D 2 _ £2_ c/Pr)W = -£2RaW, (3.23)

where £, Ra, Pr are the wavenumber, Rayleigh number and Prandtl number respectively, defined by

1

= + , ( 3 . 2 4 )

Ra = ~gBd4/v< (3.25)

and Pr = v/<, (3.26)

and where D is the operator d/dz.

Perturbations will grow with time if Re(c) > 0 and the boundary between stability and instability, known as marginal stability, is given by Re(c) = O. It can be shown that on this boundary Im(c) = O, so that at the onset of motion

(D 2 - £2)3W =-£2RaW. (3.27)

The stability problem is given by equation (3.27) together with boundary conditions at the top and bottom of the layer. For two free-surface isothermal boundaries these counditions are

W = D2W = D~W = 0 at z = 0,I, (3.28)

162 A D W Jones

while for two rigid isothermal boundaries they are

W = DW = (D 2 - E2)2W = 0 at z = 0,I. (3.29)

The resulting eigenvalue problem is solved to give Ra as a function of %, and there is a critical value, Rac, which is the lowest eigenvalue Ra. For Rayleigh numbers above Ra c the layer will be unstable and convection will take place. With free-surface, isothermal boundary conditions Ra = 657.5 and %c = 2.22, while for rigid isothermal boundaries Ra c = 1708 and %c = 3.12. These results agree well with experiments. Convection takes place in the form of cells in which, for many liquids, fluid rises at the centre of each cell and falls at the edge. In air the motion is reversed, and it seems that the direction of motion depends on whether viscosity increases or decreases with temperature. In an infinite layer the cells are regular hexagons, but their structure is affected by the presence and shape of lateral boundaries and many patternsareobserved in experiments. As the Rayleigh number is increased above Ra c the motion becomes time-dependent and eventually turbulent, and the route to 'chaos' is a subject of much study, both theoretical and experimental.

Convection may also occur when there is a basic flow. In this case the cells take the form of rolls aligned with the flow. It has been proposed [28] that spoke lines, sometimes observed on the surface of the melt in Czochralski growth, are boundaries between pairs of convection rolls aligned with the buoyancy driven flow from the crucible wall towards the crystal.

3.3.2 Convection in a rotating layer. Convection in a rotating layer has also been studied [ii] and it is found that rotation has some important effects. The first is to stabilise the motion, so that the value of Rac increases as the rate of rotation is increased. The second is that under certain conditions, which are discussed below, the motion is oscillatory at the onset of convection.

Suppose that the angular velocity of the layer is ~, so that the axis of rotation is parallel to the direction of gravity. The method of solution is the same as before, although this time the equations of motion are written in a frame of reference rotating with angular velocity ~k. An additional dimensionless parameter is introduced which is a measure of the rate of rotation. It is the Taylor number, T, defined by

T = 4~2d~/~ 2. (3.30)

It is found that Ra c is a function of T and increases as T increases. For example, with two isothermal, free-surface Boundaries, Ra c = 2.131 x I0 ~ at T = lO s . When the Prandtl number is less than a certain critical value Pr * the motion is oscillatory, that is Im(c) ~ O, if the Taylor number is greater than a value T Pr. The value of Pr* depends on the nature of the bounding surfaces, with Pr* = 0.6766 for two free boundaries and by inference, Pr ~ 1 with other boundary conditions. The occurrence of oscillatory convection due to the influence of rotation is therefore a possible source of temperature oscillations in the region of the solid/liquid interface in low Prandtl number melts such as silicon [8].

3.4. Conclusions

It is expected that the buoyancy flow is driven principally by horizontal temperature gradients at the heated crucible wall. Lighthill's solution suggests that the flow near the crucible wall may be approximated by the similarity solution for a semi-infinite heated vertical plate. However this theory does


not account for important details of the observed flow; namely that there is an unstable stratification, that the downwards flow is concentrated in a column below the crystal, and that there is a boundary layer at the base of the crucible. Work on other internal flow problems indicates that the analyst must begin with an approximate idea of the nature of the flow and temperature fields far from the boundaries. Further data from numerical and experimental simulations would therefore be particularly useful.

The topic of buoyancy driven convection in a horizontal layer has also been discussed. This has already been proposed as the mechanism by which spoke patterns are formed, and may also be a cause of the temperature fluctuation observed in some melts. Time-dependent convection would be expected in all melts at sufficiently high Rayleigh numbers, while in materials for which Fr ~ 1 the coupling of rotation and buoyancy may also lead to temperature fluctuations.

4. FLOWS DRIVEN BY ROTATION AND BUOYANCY FORCES

It is expected that a knowledge of the separate effects of rotation and buoyancy discussed in §2 and §3, will give insight into the behaviour of melts which are normally subject to both these forces. There are also flows which arise from the coupling of rotation and buoyancy, and one of these is the time-dependent flow discussed in §3.3.2. Another is baroclinic instability [19] which was proposed [3] as a possible cause of the swirl pattern associated with interface inversion in oxide melts. However, baroclinic instability occurs under conditions of stable stratification whereas in Czochralski growth the melt is unstably stratified. Existing theories are therefore not applicable, and in fact it is suggested that baroclinic instability will not occur with unstable stratification. Rather, the wave pattern seems to arise when fluid moving in from the crucible wall under the action of buoyancy forces, meets fluid pushed outwards by the rotating crystal [27]. It is therefore likely that interface inversion occurs when there is a transition from buoyancy driven to rotationally driven flow under the crystal.

Using dimensional analysis it was shown in §1.2 that GrE 2 is an important parameter in determining when rotation or buoyancy forces can be neglected. Kobayashi [30] had demonstrated this in a numerical simulation in which he shows that the velocity field has the characteristics of buoyancy driven flow with "high" values of GrE 2 and of rotationally driven flow with "low" values of GrE 2. (In his notation GrE 2 is written Gr/Re2). From a study by Hignett, Ibbetson and Killworth [21] of the flow in a differentially heated, rotating annulus it is anticipated that as the parameter GrE 2 is varied there will be a complicated transition from buoyancy driven to rotationally driven flow. Hignett et al., used boundary layer scaling analysis to show the existence of six flow regimes which depended on the value of a parameter Q defined by

Q = 2~L~/S <2zs/(agAT ) 2/s ~3rs, (4.11)

Where L is the distance between the inner and outer ~¢ylinders and AT is a characteristic temperature difference. This work suggests that a detailed analysis of the configuration of Czochralski growth would reveal many possible regimes, since there is the additional complication of differential rotation. Even if the details of the flow in each regime were not known, it would be of considerable help to clarify the possible flow regimes and to know the values of the dimensionless parameters which define the boundaries between regimes. If the crystal grower were then able to measure and control the values of these dimensionless parameters he would be able to control the hydrodynamic behaviour of the melt.

164 A . D . W . Jones

ACKNOWLEDGEMENTS

I would like to thank Dr. D.T.J. Hurle for allowing me to make use of his unpublished review of flow due to a rotating disc, and for his comments on the first draft. This work was supported by the S.E.R.C. and by a Wilfred Hall Fellowship from Newcastle University.

I.

2.

3.

4.

5.

6.

7.

8.

9.

i0.

ii.

12.

13.

14.

REFERENCES

G.K. Batchelor, Note on a class of solutions of the Navier-Stokes equations representing steady rotationally-symmetric flow, Quart. J. Mech. Appl. Math., i, 29 (1951).

G.K. Batchelor, An Introduction to ~luid Dynamics, Cambridge University Press (1967).

C.D. Brandle, Flow transitions in Czochralski oxide melts, J. Crystal Growth, 57, 65 (1982).

J.C. Brice, T.M. Bruton, D.F. Hill and P.A.C. Whiffin, The Czochralski growth of Bil2SiOz0 crystals, J. Crystal Growth, 24/25, 429 (1974).

J.A. Burton, R.C. Prim and W.P. Slichter, The distribution of solute in crystals grown from the melt. Part I. Theoretical, J. Chem. Phys., 21, 1987 (1953).

F.H. Busse, Shear flow instabilities in rotating systems, J. Fluid Mech., 33, 577 (1968).

J.R. Carruthers, Solute incorporation during the cyclic solidification of silicon, Can. Met. Quart., ~, 55 (1966).

J.R. Carruthers, Temperature oscillations in Czochralski crystal growth, J. Electrochem. Soc., 114, 1077 (1967).

J.R. Carruthers, Origin of convective temperature oscillations in crystal growth melts, J. Crystal Growth, 32, 13 (1976).

J.R. Carruthers and K. Nassau, Nonmixing cells due to crucible rotation during Czochralski crystal growth, J. appl. Phys., 39, 5205 (1968).

S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Ch.3 Oxford at the Clarendon Press (1961).

W.G. Cochran, The flow due to a rotating disc, Proc. Cambridge Phil. Soc., 30, 365 (1934).

B. Cockayne, M. Chesswas, J.G. Plant and A.W. Vere, Ferroelectric domains and growth striae in barium sodium niobate single crystals, J. Mater. Sci., i, 565 (1969)

J. Czochralski, Ein neues Verfahren zur Messung der Kristallisationsgesh- windigkeit der Metalle, Z. Physik. Chem., (Leipzig), 92, 219 (1917).


15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

D. Dijkstra and G.J.F. Van Heist, The flow between rotating discs enclosed by a cylinder, J. Fluid Mech. 128, 123 (1983).

P.G. Drazin and W.H. Reid, Hydrodynamic Stability Ch. 2, Cambridge University Press (1983).

H.P. Greenspan, The Theory of Rotating Fluids, Cambridge University Press (1968).

K. Hashimoto, On the stability of the Stewartson layer, J. Fluid Mech., 76, 289 (1976).

R. Hide and P.J. Mason, Sloping convection, Adv. Phys., 24, 47 (1975).

R. Hide and C.W. Titman, Detached shear layers in a rotating fluid, J. Fluid Mech., 29, 39-60 (1967).

P. Hignett, A. Ibbetson and P.D. Killworth, On rotating thermal convection driven by non-uniform heating from below, J. Fluid Mech., 109, 161 (1981).

M. Hoiodniok, M. Kubicek and V. Hlavacek, Computation of the flow between two rotating coaxial discs, J. Fluid Mech., 81, 689 (1977).

D.T.J. Hurle, Hydrodynamics, convection and crystal growth, J. Crystal Growth, 13/14, 39 (1972).

D.T.J. Hurle, Hydrodynamics in crystal growth, in Crystal Growth and Materials, ed. Kaldis, E. and Scheel, H.J., pp. 550-569, North-Holland (1977)

D.T.J. Hurle and E. Jakeman, Effects of fluctuations on the measurement of distribution coefficients by directional solidification, J. Crystal Growth, ~, 227 (1969).

D.T.J. Hurle, E. Jakeman and E.R. Pike, Striated solute distributions produced by temperature oscillations during crystal growth from the melt, J. Crystal Growth, 3_/_4, 633 (1968).

A.D.W. Jones, An experimental model of the flow in Czochralski growth, J. Crystal Growth, 6.1, 235 (1983).

A.D.W. Jones, Spoke patterns, J. Crystal Growth, 63, 70 (1983).

T. Von Karman, Uber laminare und turbulente Reibung, Z. angew. Math. Mech., i, 233 (1921).

N. Kobayashi, Hydrodynamics in crystal growth-computer analysis and experiments, J. Crystal Growth, 52, 425 (1983).

W.E. Langlois, Digital simulations of Czochralski bulk flow in a parameter range appropriate for liquid semiconductors, J. Crystal Growth, 42, 386 (1977).

W.E. Langlois, Convection in Czochralski growth melts, Physico-Chem. Hydrodynamics, ~, 245 (1981).

M.J. Lighthill, Theorectical considerations on free convection in tubes, Quart, J. Mech. Appl. Math., 6, 398 (1953).

166 A.D.W. Jones

34. D.K. Lilly, On the instability of the Ekman boundary layer, J. Atmos. Sci., 23, 481 (1966).

35. M.R. Malik, S.P. Wilkinson and S.A. Orszag, Instability and transition in rotating disc flow, AIAA Journal, 19, 1131 (1981).

36. T. Maxworthy, The flow between a rotating disc and a coaxial, stationary disc, Space Programs Summary, i, sec. 327, 37, J.P.L., Cal. Tech. (1964).

37. D.C. Miller, A.J. Valentino and L.K. Schick, The effect of melt flow phenomena of the perfection of Czochralski grown gadolinium galium garnet, J. Crystal Growth, 44, 121 (1978).

38. D.C. Miller and T.L. Pernell, Fluid flow patterns in a simulated garnet melt, J. Crystal Growth, 57, 253 (1983).

39. H. Ockendon, An asymptotic solution for steady flow above an infinite rotating disc with suction, Quart. J. Mech. Appl. Math., 2_5, 291 (1972).

40. S. Ostrach, Laminar flows with body forces, in Theory of Laminar Flows, ed. Moore, F.K., Oxford University Press (1964).

41. S. Ostrach, Natural convection in enclosures, Advances in Heat Transfer, 8, 161 (1972).

42 M.H. Rogers and G.N. Lance, The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disc, J. Fluid Mech., ~, 617 (1960).

43. M.H. Rogers and G.N. Lance, The boundary layer on a disc of finite radius in a rotating fluid, Quart. J. Mech. Appl. Math., 17, 319 (1964).

44. N. Rott and W.S. Lewellen, Boundary Layers and their interaction in rotating flows, in Progress in Aeronautical Sciences, Vol. 7, ed. Kuchemann, Pergamon Press (1966).

45. H. Schlichting, Boundary Layer Theory, Ch. 7, McGraw-Hill Book Company (1960).

46. F. Schultz-Grunow, Der Riebungswiderstand rotierenden Schieben in GehNusen, Z. angew. Math. Mech., 15, 191 (,935).

47. D. Schwabe, Marangoni effects in crystal growth melts, Physico-Chem. Hydrodynamics, ~, 263 (1981).

48. W.L. Siegman, Evolution of unstable shear layers in a rotating fluid, J. Fluid Mech., 64, 289 (1974).

49. K. Stewartson, On the flow between two rotating coaxial discs, Proc. Cambridge Phil. Soc., 49, 333 (1953).

50. J.T. Stuart, On the effects of uniform suction on the steady flow due to a rotating disc, Quart. J. Mech. Appl. Math., Z, 446 (1954).

51. D.J. Tritton, Physical Fluid Dynamics Ch. 14, Cambridge University Press (1977).


52.

53.

P.A.C. Whiffin, T.M. Bruton and J.C. Brice, Simulated rotational instabilities in molten bismuth silicon oxide, J. Crystal Growth, 32, 205 (1976).

G. Zydik, Interface transitions in Czochralski growth of garnets, Mat. Res. Bull., i0, 701 (1975).

168 A . D . W . Jones

THE AUTHOR

ALISTAIR JONES

Alistair Jones received his Ph.D. degree in 1982 at the University of Bristol. He now holds Wilfred Hall and S.E.R.C. Fellowships at the University of Newcastle upon Tyne where he is working on an experimental and analytical study of the fluid flow in Czochralski growth.

School of Physics, The University, Newcastle upon Tyne NEI 7RU, U.K.

Documents

Hydrodynamics of Czochralski growth—A review of the effects of rotation and buoyancy force