Convection in melts and crystal growth

.~.Vo1.3,No.5,pp.51—60,l983 Printed in Great Britain.A11 rights reserved.

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CONVECTION IN MELTS AND CRYSTAL GROWTH G. Muller instirut für Werkstoffwissenschaften, Kristallabor, Universität Er/angen-Nurnberg, D-8520 Er/angen, F. R. G.

ABSTRACT

Convection plays a significant role in determining the electronic properties of semiconductor crystals grown from the melt. Inhomogeneities (doping striations) in these crystals can be caused by unsteady natural convection. Therefore, the origins of unsteady natural convection, which means fluctuating temperatures in a fluid, are investigated in this paper. Several effects have been found at increasing Rayleigh numbers Ra which cause fluctuating temperatures. But it was also observed, that unsteady convection became again steady if Ra was increased to higher values. This effect, which is called relaminari— zation, is very interesting for crystal growth and is discussed in this paper. INTRODUCTION

Convection plays a significant role in determining the electronic properties of semiconductor crystals grown from the melt. This role is due to two effects: —

—

The heat and mass transfer in melt growth is strongly influenced by the intensity and type of convective flows in the melt. In many cases of crystal growth especially in solution growth this effect is beneficial. Convective flows have basically a tendency to become unsteady at higher flow intensities. Unsteady convection causes temperature fluctuations at the solid—liquid phase boundary of a growing crystal. This leads to fluctuations of the growth rate. Growth rate fluctuations cause compositional inhomogeneities in the growing crystal because the segregation coefficients of the most constituents (e.g. dopants) are dependent on the growth rate. In semiconductor crystals this effect is well known as doping st,~iations.*

ir ths contribution we treat only this disadvantageous effect of convection: the formation of doping striations in semiconductor single crystals, caused by unsteady convective flows. This problem has become an important economic one, since the reliable performance of the microelectronic semiconductor devices is strongly dependent on the quality of the semiconductor single crystals from which they are fabricated. Therefore many investigations have been made in this field in the past 1—41. But the problem of inhomogeneous electrical resistivity in the microscale. which is equivalent to doping striations, is still unsolved. A variety of driving forces for convective flows exists in common configurations of crystal growth from melts. They can be classified into three types: — density gradients within the fluid in a gravitational field (free’ or “natural convection”) *

The expression “striations’ comes from the striation—’Iike etch pattern which can be observed on sections of crystals. For review articles on this topic see 11—41

52

— —

G. Hiller

pressure gradients created by external forces (forced convection”) thermocapillary effects (“Marangoni effect”).

In this contribution only natural convection is treated because there is a strong relation to crystal growth under microgravity. CRYSTAL GROWTH UNDER MICROGRAVITY A lot of proposals for crystal growth experiments in SPACELAB or other space missions have the aim to grow more homogeneous crystals under microgravity conditions as it could be .grown on earth, at least to grow striation—free crystals in space. This assumption arises from the fact that gravity is the driving force for natural convection in crystal growth melts and has therefore to be decreased to avoid unsteady convective flows. This point of view is fairly well illustrated by fig. 1, which shows the results of a crystal growth experiment (Te—doped InSb, Bridgman technique on a centrifuge I 4, 5~at stepwise increase of the acceleration. On the left hand side is depicted a recorder diagram of the temperatures in the melt. On the right hand side is shown an etched section through a piece of InSb crystal which was grown during that time when the temperature was measured. Both parts of the figure have the same time scale. Two important results can be observed in fig. 1: 1) Increasing acceleration changes convection in the melt from steady to unsteady state. ii) Unsteady convection causes inhomogeneous crystals. Each doping striation in the crystal can clearly be attributed to a temperature fluctuation in the melt.

‘ui •

1,50

.c

155

LC)

E E E

~

‘~

~

‘—

1,60

. •

.•

I,U

Fig.l Recording trace of temperatures in the melt (left) and interference microphotograph of a longitu— dinal section through a Te— doped InSb crystal (right). that time when temperatures have been measured. Growth direction is from top to bottom in this figure. The time scale is equal for both sides

1,70x9,8lms -2

of the

figure.

The

numbers on the giye left the margin of the figure values changed at the dotted lines.

The result of fig.l is qualitatively in confirmity with the results of hydro— dynamic investigations of the BENARD cell 16,71 which has been used up to now as the basic model for discussion of convective effects in crystal growth. The BENARD cell consists of a fluid layer between a heated plate on the bottom and a cooled one on the top in a distance h which is small compared to lateral dimensions. The convective effects in this cell can be well described by using the dimensionless Rayleigh number Ra Ra

3

=

b~/~•.~T . h

(1)

where b is the acceleration, e.g. earth gravity g; ,.~ is the volumetric thermal expansion coefficient; iT is the temperature difference between the plates; i)is the kinematic viscosity;~is the thermal diffusivity. Fig.2 is taken from a paper of Busse 161. The figure shows a transition diagram of convective states which Krishnamurti 171 used to present her data and those of other investigators in a comprehensive form. Ra is plotted versus the Prandtl number P = ~ /~t. The different convective states are denoted by numbers, which were explained in the figure caption. It follows from fig.2 that convective flows occur, if Ra exceeds a critical value RaC1, which is independent

Convection in Melts and Crystal Growth

53

from P anti is given by the lower horizontal line (~. Convection remains steady until Ha exceeds a second critical value Ha . According to fig.2, it is not possible to grow a striation-free crystal as long as Ra is larger than Ra~. Therefore, ~he only possibility to grow crystals in steady state is to keep Ha below RaC . This can only be done by a reduction of gravity level, if h and LT cannot be decreased because of crystal growth conditions. 106

I

/ / / V/

,•

Fig.2 Transitions in thermal convection as a function of Ra and P after Busse [6] and Krishnamurti 17].

lHb/

// i.., / 1 /

10~

1118

The curves indicate the onset of

SteD4ybimodal convection

steady rolls (I), 3—dimensional con—

11

10~

/

(lila) and uniformly throughout the .

in

layer (11Th), turbulent convection

Steadyrolls

No motion 103 102

I

10—1

1

10

I

102

I

10~

10~

P

Experimental results which have recently been obtained in our laboratory have

shown that this assumption is not valid for crystal growth in containers w~h large aspect ratio (height/diameter). We found steady convection at Ra> Ra This effect which we call

r e 1 a m I n a r i z a t i o n

will be discussed

in the next section. RELAMINARIZATION

2. If Ha was further increased In Ra just exceeds critical value RaC in fig.l that experiment by an the increase of the centrifugal acceleration, we found a result which is shown in fig.3. On the left hand side again the recording trace of a thermocouple is depicted, showing the temperatures in the melt during growth of the crystal which itself is shown on the right hand side. The upper part of the crystal has been grown at an acceleration b 2.3 g in an unsteady convective state, showing temperature oscillations in the melt and doping striations in the grown crystal. If b was increased to 2.7 g the tem-

perature oscillations disappeared, which means that convection was now in a steady state. The crystal which has been grown during that period (at b=2.7 g) is free of doping striations. This result was obtained for a Bridgman—configu— ration with temperature gradient parallel to centrifugal acceleration [4, 5]. But relaminarization effects are not comstricted on this configuration. We have obtained a similar result also in horizontal zone melting of InSb at increased acceleration [8j. In these experiments the resulting acceleration (centrifugal acceleration plus earth gravity) was directed perpendicular to growth direction. As in the Bridgman experiments a thermocouple was positioned in the melt during crystal growth and acceleration b was varied. The result of a typical experimental run is shown in fig.4. This figure consists of several parts of an original recorder diagram of temperature measure— ments at various values of b. Three ranges of b were obtained in all experimental runs, which show different temperature behavior: a) 1 g The temperature is steady if the resulting acceleration b is below a critical value bC2 (the index C2 is in correspondance to RaC2). The absolute value of temperature decreases with increasing b. b) bC 2~b;bC3 The temperature becomes unsteady if b exceeds b

which can be determined

with an accuracy of ~ 0.5 g. The amplitudes of ~mperature in the range of 0.3 to 0.8 K.

oscillations are

The frequencies of temperature oscillations

54

C. Muller vary between 0.3 and 0.5 $~l The frequency of centrifuge rotation is in the range of 1.3 to 3.3

5

1K

_

C

_____

2,3

E

-2

2,7x9.81 ms

•“.~.;‘‘~

~•~‘•

‘

•~~• ~ —

‘‘‘•‘‘~I ~

Fig.3

•“S

•~ SS

Recording trace of temperatures in the melt

(left) and interference microphotograph of a longitudinal section through a Te—doped InSb crystal (right). The crystal is grown during that time when temperatures have been measured. Growth direction is from up to down in this figure. The time scale is equal for both sides of the figure. The numbers on the left margin of the figure give the values of the acceleration b. b is changed at the temperature spike. al

1~(b(b~

2.1O2g20.5g

2 b’~52

56 x9Slrn/s

5.~.

bI b~~=1O.2g2O5g
b~tO

Fig.4 Recording trace of tern— peratures measured in an InSb melt during horizontal zone melting at different values of acceleration b.

112

The dotted lines denote the times

when b was changed. b is given by the numbers between the dotted lines.

______

~ 118

120x981rn/s2

a) 1 g~b~b~ 2= (lO.210.5)

b) ~ c) b03 ~b

c~bc3~1~4g~OSg
g

(14.4t0.5) g ~

30

g

2 b~16O

c) bC

3b<3O

162 x981 rn/s

g

The temperature becomes again steady (relaminarization) if b exceeds b and remains steady in the acceleration range up to b = 30 g. b00 is de~ined as that value where the amplitudes of temperature oscillations have been decreased below the detection limit of about 0L02 K. bC3 can be determined with an accuracy of±0.5 g.

It has been proved experimentally that the melt h~ight of the zone is constant and independent from b for b> 4 g.

Convection in Melts and

Crystal Growth

55

1 nan 1KJ

2

02mm I

b~160x981rns

1mm lx!

b

1.~’

~

:

‘~•‘

Ufl1,2i~1~~~I b~~7Ux~.61 rns2

I

0.2 mm

1 mm,,

c~

b 29.0 x 9.81 ms2

I

-

02mm

Fig.5 Recording traces of temperatures in the melt zone and interference microphotographs of longitudinal sections through Te—doped InSb crystals grown by horizontal zone melting at different accelerations. b = 16.0 g (above), b = 27.0 g (middle) and b = 29.0 g. The growth direction is from left to right side. Fig.5

shows

some pieces of the crystal (grown at different accelerations)

to-

gether with measured melt temperatures.It can clearly be seen that unsteady convection

(bC

2. b~b03) causes doping striations (fig.5b), and steady convection (b~bc2 and b> bC3) gives homogeneous crystals (fig.Sa,c). In the two examples of fig.3 and fig.5 we found a relaminarization b applying increased acceleration. This might suggest that relaminarization is a centrifuge effect. But we found relaminarization also at normal earth gravity in experiments with H20. Results of these experiments, which were obtained with

C. Muller

transparent test cells by using flow visualization techniques and temperature measurements, will be treated in detail in the next section. INVESTIGATION OF FLOW—CONFIGURATIONS IN CONFINED CONTAINERS WITH LARGE ASPECT RATIO (HEIGHT/DIAMETER) In this section we report on results of flow investigations in containers with large aspect ratio (height h over diameter d). Such containers are often used in crystal growth (e.g. vertical Bridgman technique). Convection in containers with h/d, 1 has been investigated in several papers in the past [2, 3, 9—11]. These papers have investigat&d the onset 2). of convective flows (RaCl) and the occurrence of unsteady convection (RaC RaC2 has been found to increase with increasing aspect ratio, up to h/d = 5 and to be constant for h/d>5 (see fig.6). This result is due to the damping influence of the container walls. But up to now there has been no systematic experimental investigation of the origins of temperature fluctuations in these configurations. Relaminarization has not been reported up to now.

Rd~(conducting boundaries) iü6

..~,

theoretical after Corruthers

2 5

A

E

Ra this work

•

--

C D~ 10

0

a conducting boundaries

/ / o

1o~ -‘

°

•o~Dxinsulating boundaries

Rd~1(insulating boundaries) theoretical of ter Corruthers

b

102

1

0

•2

3

I

I

4

5

I

6

7

aspect ratio h/d Fig.6 Rayleigh numbers Ra~ and RaC2 for metallic melts versus aspect ratio h/d. Both curves give theoretical values of Ra~. The different signs give our experimental values of Ra obtained for different experimental configurations described in [4].

~,observation

1.

1 I

thermocouples II l~

lig~cut

J~tsink(C~ I

HH— glass

3 thermocouples

1 ________

~e~er(C~ lOmm

water

Fig.7

Sketch of the test cell for

flow visualization by light cut technique. Above: horizontal section. The light cut is in a plane perpendicular to this section. Below: vertical section The numbered lines denote the four thermocouples.

Convection

in Melts and Crystal Growth

57

In this section experimental results will be reported, which we have obtained recently with water. The flows in the different test cells with circular, square and rectangular cross sections have been visualized by using the light cut technique. A small amount of A1 powder was added for that purpose to the water. Temperatures were measured in the fluid by coated thermocouples. Fig.7 shows the principle of the test cell with square cross section. The Rayleigh number has been increased in this experiments by an increase of the temperature difference ‘IT between the two copper plates. The rate of the increase of~T has been given by the heating rate of the lower plate, which has been varied between 1K/h to 8K/h. The variation of this rate has only been of secondary influence on the results presented here. Typical results of experiments obtained with the test cell of fig.7 are described for the different ranges of Ra. Within each experiment the values of Ra for transitions are very sharp (- 0.1 RaC~). The errors given in the brackets relate to differences between different experimental runs.

—

__________

__________

~

Figs.8-ll

Photographs of flow patterns

in the test cell of fig.7 made by the light cut technique. Beneath figs.9—ll is a sketch of the flow as it has been

_____

observed during the experiment.

__________

I

j 11

Ra~Ra~

=

io6

01 = 106 no convective motion is observed in the fluid Up to a value Ra The temperatures are steady. -

(fig.8)

.

58

C. Muller

Ra~Ra~(1.7±O.3)xRaCl A single cellular, none axisymmetric, steady state basic flow occurs in that range

(fig.9).

The measured temperatures are steady

(1.7±0.3) xRaC~Ra<(2.9±l) xRaCl In this range of Ha convective flows are unsteady. The flow configurations change continuously between basic flow (fig.9), double roll cell (fig.1O) and triple roll cell (fig.11). As one can see in figs. 10 and 11 the double and triple rolls are not clearly separated but are connected by streamlines. For example the flow pattern of the double cell looks like an “eight” and seems to be created by twisting the basic flow. The temperatures measured by the thermocouples are unsteady in this range of Ha which is illustrated by the recording traces of the four thermocouples in fig.12. (2.9~1) xRa~dRa~(4.O:l) xRa~ Convective flows are steady. Either a double roll cell (fig.lO) or a triple roll cell as shown in fig.l1 stays steady during that range of Ha. The temper-

atures are steady. This range of Ha was very small (<0.1 x RaCl) in some experiments. Racc(4.3 tl.5) x Ra~ Near that value of Ha several events occurred: mostly convective flows became 1 changing flow configurations again a small of 0.3 RaC as in unsteady the first within unsteady range.range In one case x sinusoidal temperature oscillations have been observed which are shown in fig.13. These oscillations are created by a periodical up and down movement of the roll centers as shown in fig.14. A similar effect has been described by Carruthers [7]. In some experiments with heating rates ~ 8K/h another kind of instability was observed: Isolated fluid spots were ejected from the fluid layer near the bottom (heated copper plate) of the cell and moved very quickly to the top. This effect, which is very similar to the thermals described by Carruthers [71, creates temperature fluctuations which are shown in fig.15. (4.3±1.5) x Ra01ZRa’1l5 x Ha01 Convective flows are again steady, in most cases with double roll cells. The temperatures are steady.

12

Federriyersatz

13

1~L~ 3~ 21

34

Federnversotz Fig.l2,l3

Recording

traces

of temperatures

measured at thermocouples 1—4

Convection

59

in ~ielts and Crystal Growth

~ 1; ~

__

__

__

I~

Fig.14 A sequence of light cut photographs of the periodic changes of the roll sizes in a triple roll configuration. The two white lines giving the borders between the rolls are synchronous to the measured temperature below.

Federr,versotz

1 2

Fig.l5 Recording traces of’ temperatures measured at thermocouples 1-4.

1K~

Results of experiments with other test cells In the experim.ents with other test cells (circular and1 recta~gular cross sec= 10 . But the regions tion) the onset convection also roll observed RaC observed are larger than of Ha where the of basic flow andwas double cells at were those obtained for the cell with square cross section. Unsteady behavior has been observed only for Ha (6—10) x Ha01. DISCUSSION AND CONCLUSIONS The aim of this paper is to give a contribution to two basic problems of crystal growth; i) ii)

What are the origins of temperature fluctuations in containers with h/d,l? Is it possible get r e 1 a m i n a r i z a t 1 o n at large Rayleigh numbers (Ha numbers?

‘

Ra

)

and to grow striation—free crystals at those Rayleigh

60

C. MOller

•

Although it is difficult and not proved up to now if we can transfer the results obtained with H~,O (P 7) to the low P fluids, we can make the following conclusions from the ~xperimental results presented here: —

Flows in confined containers with h/d

=

5 are basically three dimensional.

Also in cylindrical containers we have not found flow configurations with

—

—

rotational symmetry. Therefore, efforts have to be made to get three dimensional theoretical calculations of convective flows. Temperature fluctuations in the fluid can have different origins: i) changes of the flow configuration (aperiodic) ii) oscillations of a stable flow configuration (periodic) iii) instabilities of a boundary layer (periodic). The occurrence of temperature fluctuations is limited on special ranges of the Rayleigh number. The effects i, ii and iii occur in different ranges of Ra and are separated by ranges of steady convection. That means: relaminari— zation can be well understood if unsteady convection originates from effects i, ii, iii. In that cases the growth of striation—free crystals, as shown in fig.3, can be well understood.

The shape of the fluid container is of great influence on the flow transi1. tions in the range Ha Ha° The results presented in this paper show that convective effects in confined containers as they are used in crystal growth are very complex. But it seems to be useful for crystal growers to continue the investigations of basic fluid flow phenomena, because there seem to be chancts in getting knowledge for the growth of better crystals. —

ACKNOWLEDGEMENT The author would like to thank Dipi. Ing. G. Neumann for discussions and Dipi. Ing. W. Weber for contributing experimental results of his diploma work to this paper.

The German Bundesministerium für Forschung und Technologie has supported this investigations financially. References 1.

J.R. Carruthers, A.F. Witt, in; Crystal Growth and Characterization, eds. H. Ueda and J.B. Mullin North Holland Publ. Comp., Amsterdam, 1975, pp. 107—154.

2.

S.M. Pimputkar,

3.

J.R. Carruthers, in: Preparation and Properties of Solid State Materials, eds. W.R.

S. Ostrach, J. Crystal Growth, 55, 614 (1981)

Wilcox and R.A.

Levever, Dekker, New York, 1977, pp.

1—121.

4.

G. Muller, in: Convective Transport and Instability Phenomena,

5.

eds. J. Zierep and H. Oertel jun., Braun Verlag, Karisruhe, 1982 G. Muller, E. Schmidt, P. Kyr, J. Crystal Growth, 49, 387 (1980).

6.

F.H. Busse, in: Hydrodynamic Instabilities and the Transition to Turbulence (Topics in Applied Physics Vol. 45), eds. H.L. Swinney and J.P. Gollub, Springer—Verlag Berlin — Heidelberg, 1981, 119.

7.

H. Krishnamurti, J. Fluid Mech.,

8.

G. MUller, 0. Neumann, J. Crystal Growth, 58 (1982).

9.

J.R. Carruthers, J. Crystal Growth, 32, 13 (1976).

10. J.N.

Koster, U. Muller,

60, 285 (1973).

Recent Developments in Theoretical and

Experimental Fluid Mechanics, eds. U. Muller, K.G. Roesn~-r, B. Schmidt, Springer—Verlag, Berlin — Heidelberg, 1979, pp. 367—375. 11. F. Rosenberger,

Physico Chemical Hydrodynamics.

North Ireland, 1980, pp. 3—26. See also F. Rosenberger,

this conference.

Vol. 1.

Pergamon Press.