Experimental study of thermocapillary convection in a germanium melt

Experimental study of thermocapillary convection in a germanium melt

,. . . . . . . . ELSEVIER CRYSTAL GROWTH Journal of Crystal Growth 165 (1996) 351-357 Experimental study of thermocapillary convection in a german...

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Journal of Crystal Growth 165 (1996) 351-357

Experimental study of thermocapillary convection in a germanium melt L e o n i d A. G o r b u n o v * Institute of Physics, Latvian Academy of Sciences, Miera street 32, Salaspils-1, LV-2169, Latvia Received 13 December 1995; accepted 25 January 1996


The present paper is dedicated to the experimental investigation of thermocapillary convection (TCC) in semiconductor melts. The investigation showed that in the process of single crystal growth under terrestrial conditions TCC could be compared to thermogravity convection (TGC) for a number of semiconductor melts such as Ge, Si, GaAs. But in comparatively thin layers with H < 10 mm (at L/H >> l, where L is the container radius) it can dominate over TGC. The experiments were conducted with a Ge melt. Oxide particle tracers were used to measure the melt motion rate. The results obtained emphasize the significance of TCC in the process of single crystal growth under terrestrial conditions.

I. Introduction

The importance of thermocapillary convection was first discovered during some technological experiments in space under reduced gravity conditions. It generated a considerable interest, mainly to its application under weightlessness conditions, when micro accelerations are ( 1 0 - 2 - 1 0 - 5 ) g 0 (go is the free fall acceleration), as well as under conditions of reduced thermogravity convection. Here, thermocapillary convection can determine the melt hydrodynamics in some situations, particularly in crystal growth experiments. Up to recent times thermocapillary convection was ignored in the study of most technological processes on earth. But, as the results show, there are a

* Corresponding author. E-mail: [email protected]

number of processes, where it should be considered under the terrestrial conditions as well [1,2], including the processes of single crystal growth, some processes in chemical technology, vacuum metallurgy, etc. Today a number of works are dedicated to the study of thermocapillary convection. However, they are mostly theoretical; experimental studies are not numerous since investigation of thermocapillary convection is a rather complicated experimental problem. It requires small quantities of fluids and a very thorough check of surface purity and temperature gradients along the surface. The most frequently used fluids are hydrocarbon fluoride, various oils, sodium nitrate, etc. [3-6], i.e. all fluids with Prandtl numbers much larger than for molten metals and semiconductor melts. But experiments on thermocapillary convection modelling on low-temperature liquid metals such as

0022-0248/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PH S 0 0 2 2 - 0 2 4 8 ( 9 6 ) 0 0 2 1 7 - 5


L.A. GorbunoP/ Journal of Cr3,stalGrowth 165 (1996)351-357

Bi, Ga, Sn also have some difficulties dealing with the problem of surface purity control. The first results have been obtained in Ref. [7] on Sn melt with 0.5-3% of Bi. They proved a comparatively high rate of the thermocapillary convection (u = 1-4 c m / s ) for the characteristic numbers of Re~ =

8_~ T ATH 2 ~




A = -~ = 1-10,

(1) and the need to control thermocapillary convection in real processes, not only under weightlessness. Here, a is the surface tension, AT is the characteristic temperature drop, L, H are the length and height of a melt layer, v, P are the kinematic viscosity and melt density. The present paper deals with the study of thermocapillary convection on a germanium melt, By investigating thermocapillary convection under conditions similar to those of single crystal growth, there is, practically, no problem dealing with the surface purity. In this case the needed purity of the surface is obtained in accordance with the technological requirements for single crystal growth.

AT H 3 Ug = f i g T --T ' where

lap p OT

fig . . . .

is the thermal expansion.


_ - "%










Fig. 1. Diagram of the experiment. (1) Ge melt; (2) experimental container; (3) single crystal; (4) additional seed. It is easy to show that, at a temperature drop of AT = 10-20 K along the melt free surface and characteristic values of L and H of ~ 10 2, the flow strength is already large. So, in order to evaluate the rates of convection under discussion, it is advisable to use expressions of the boundary layer type [1,7,8]:

Uc= ~] p---~vL] ' ug= ( flg----~ ]

First of all let us evaluate the characteristic rates of melt flow for thermocapillary and thermogravity convection in a Ge melt at characteristic values of L and H (Fig. 1). Many publications dedicated to thermocapillary convection use the expressions below for viscous flow to evaluate its characteristic rates [6]. The expressions for thermocapillary and thermogravity convection are:

8-~TT A T H pvL '

lz-- ¢

#~ Z


ATH2 1 I/2

2. Preliminary estimation of basic parameters

uc =




Here, the characteristic Reynolds numbers for u c and Ug are defined by

__]1/3 Re c



= w(Ma)2/3(H)


UgH /}


Lp2v 4 '

(6) (7)

where Ma = Pr[(8~/STXA T ~ H/pv ~{2})] is the Marangoni number, Gr=BgATH4/v2L is the Grashof number, Pr = v/a is the Prandtl number. ATH/L is the characteristic temperature drop across the layer, which has been used to define the Grashof number and estimate the strength of thermogravity convection, which allows one to estimate Ug more precisely.

L.A. Gorbunot / Journal of Co,stal Growth 165 (1996) 351-357 Table 1 Calculated values for the relation of TGC rate to TCC rate in dependence on Ge melt height H H (mm)

uc ( m m / s )



5 7 9 11 13 15


1635.3 2289.4 2943.6 3597.7 4251.8 4905.97

4.84X 10 2 6.78 X 10 -2 8.72 X 10- 2 1.07 x 10-t 1.26X 10 -t 1.453 X 10 -~

44.15 44.15


ties to use a Ge melt for experiments aimed at the investigation of thermocapillary convection. It is obvious that at comparatively small values of L in the melt and a wide range of H change thermocapillary convection dominates. Most important is the possibility to simply and reliably obtain a thermocapillary surface of the melt.

3. Experimental technique and procedure So the condition, under which thermocapillary convection would dominate in the melt, can be written in the following form: Ug


<< 1.



If for a Ge melt we use the constants [9,10] v=l.35X10

-7 mZ/s,

Ga/GT= - 2 . 6 X

10 - 4

/ 3 g = 7 X 1 0 -5 K -1,

p = 5.57 X103 k g / m 3, N / m - K, c~=0.612N/m,

with AT = 20 K (the characteristic temperature drop), L = 75 mm (the characteristic length of the liquid zone), then the dependence Ug/Uc on layer height H can be estimated from to the data presented in Table 1. Thus, it can be stated that thermocapillary convection dominates in a Ge melt over thermogravity for a rather wide range of melt height H change. It is determined by a large value of Goz/GT= - 2 . 6 X 10 - 4 N / m . K and small value of /3g = 7 X 10 -5 K 1. The sign of Goz/GT shows the coincidence of flow directions for thermocapillary and thermogravity convection. Besides, it should be noted that at c~=0.612 N / m , when the melt does not wet the surface of the experimental container, the minimal melt height can be 5 - 6 mm. Otherwise, the melt in the container under the action of the surface tension forces starts gathering into " a large drop" breaking away from the container walls. It reduces the melt height to H > 6 mm for experimental studies. But, as Table 1 shows, this restriction is not significant since in the above range for H thermocapillary convection dominates. Thus, the preliminary estimation reveals possibili-

For the study of thermocapillary convection an experimental (test) bed produced on the base of the facility for single crystal growth has been used. The device comprises a vacuum chamber supplied with heaters as well as with a rod and electric drive for the experimental container rotation and a rod and electric drive for crystal rotation and pulling; an automatic system for the heater temperature control, etc. The device could be operated under vacuum or with inert gases such as helium, argon, etc. The experimental bed was additionally supplied with rods for thermocouple movement along the radius and height of the experimental container as well as for the additional seed crystal movement. A 150 mm diameter crucible made of super-pure graphite was used as the experimental container. Heating was by a standard graphite heater used for crystal growth. A required radial temperature gradient ( ~ 10-20 K) was obtained by choice of cooling conditions for the bottom of the experimental container. Heat was removed partially by radiation from the melt free surface and partially via the lower water-cooled rod. Undoped zone-refined Ge (etched for several minutes in a mixture of HF and HNO 3) was used in the experiments. The melt height H in the experiments was defined with account for the crucible diameter and load weight after the experiment finished. The experiments were conducted in vacuum with a pressure not exceeding 10-5 bar. The velocity of the melt motion on the free surface was defined by the method of tracers especially developed for measuring of the velocity on surfaces of high-temperature and aggressive melts. The procedure is as follows: an additional Ge seed crystal (2-4 mm in diameter and 200-300 mm length, its surface covered with an oxide film) had

LA. Gorbunot,/ Journal of Crystal Growth 165 (1996) 351-357


been grown, for example, at the step of the melt surface cleaning. It was placed in the vacuum chamber (Fig. 1), and by a special device the crystal was moved to the melt surface near the side wall of the container, where the temperature was maximal. After touching the surface of the melt overheated by 10-20 K, a small part (0.5 mm) was melted. When the crystal broke away from the melt surface, there remained an oxide fraction - a tracer (or several fractions - tracers), driven by the flowing melt. The characteristic size of the fractions, 0.5-1 mm, can be clearly observed on the melt surface and allows one to measure the velocity of the convective flow. For that purpose, the motion of the tracers was video-tape recorded. Simultaneously with it, the time of the process was registered by a timer with an accuracy of about 10 -2 s. At decoding of the video records, the path of every particle was divided in 2 0 - 5 0 intervals, their initial and final coordinates as well as time t being recorded. The distortion of the pictures filmed by the video camera, placed at an angle to the melt surface, was taken into account. Fig. 2a depicts the characteristic paths of the tracer motion for three values of rotation velocity of the experimental crucible. Fig. 2b depicts the pictures of a TCC investigating process, when a Ge crystal grows. A tracer, obtained by melting of an additional seed crystal near the side wall of the crucible (see the picture (right); the thermocouple is placed opposite (left)) and its further separation, can be clearly observed on the melt surface. So the velocity of the tracer motion consists of a radial velocity linked to TCC, and an azimuthal velocity linked to the rotation of the experimental container. The above figure allows one to clearly trace the path of the particle. The radial and axial components of the flow velocity were defined as






r=~/x 2+y2.

(Further below, melt flow velocity u corresponds to its radial component.) The temperature distribution was measured together with the thermocapillary convection rate. Here, a tungsten-rhenium thermocouple (50 /zm in diameter) sealed in a quartz covering (1-1.5 mm in diameter) was used. The thermocouple covering was of letter L form, its horizontal part was 15 mm. Work





Fig. 2. (a) Characteristic data for tracer motion paths at the rotation velocity of the experimental container: (1) n~ = 0; (2) nc = 1.5 rot/m; (3) nc = 7.5 rot/m. (b) Pictures of the tracer motion on the melt surface. A tracer, obtained by melting of an additional seed crystal (right) near the side wall of the crucible (the thermocouple is placed opposite (left)) and its further separation, can be clearly observed on the melt surface. with semiconductor high-temperature melts proves that the above form provides a high accuracy of temperature measuring. By the particular coordinating device the thermocouple was moved along the crucible radius with its height unchanged. In all the experiments conducted, the temperature was measured under the melt surface - 1.5 mm depth. At a melt height of H > 10 mm the temperature distribution was also measured near the crucible bottom. The thermocouple moved rather slowly; the measurements were done in 400 points along the radius. H was defined from measurements of the weight of the Ge at the completion of the experiment.

LA. Gorbunov / Journal of Co,sial Growth 165 (1996) 351-357

4. Results and discussion

T, g

As was stated above, the velocity of the melt motion was defined by the velocity of the tracers' motion. The distribution of u along the crucible radius p * = p/L was calculated after video recording of the paths of the tracers' motion and their analysis. Fig. 3 shows the characteristic data for u obtained after the analysis of one of the paths. The obtained fluctuations of the tracers' velocity are connected with the fluctuation of the melt flow velocity (similar fluctuations can be observed in the motion of separate particles on the melt surface, e.g. in the absence of a crystal), as with the procedure of the paths treatment and some difficulties of accurate definition of the particle coordinates while it is moving. The absolute velocity values here are 10-20 m m / s , which determine characteristic Reynolds numbers Re c = uH/~, = 103-2 X 103. Thus, this experiment proves a turbulent character of the flow. The temperature distributions along the crucible diameter were also obtained. On all the diagrams these data are given with account for the shift by the Ge solidification temperature (Tcr = 938 K). Fig. 4 shows three characteristic curves for the temperature distribution across the container radius. Curve 1 was obtained for the case without crystal, curve 2 was obtained for a growing crystal with a small radius (R = 5 - 7 mm), curve 3 for a crystal with radius R c = 15 mm and rotation velocity n c = 6 rot/min. The measurements were performed along the melt free surface and under the crystal as well. They evidenced that without a crystal, the temperature gradients in the melt were smaller compared to the case with a crystal. At the same time, for the case






Fig. 3. Dependence of velocity u of the tracer motion on the container radius, obtained after analyzing of a motion path.






0 ~........

I 'I



Fig. 4. Characteristic data for the temperature distribution in a Ge melt at z = 1.5 m m without crystal (curve l), with crystal at R~ = 5 mm (curve 2) and at R c = 15 mm, n c = 6 r o t / m i n (curve


without crystal, the experiments can be carried out at a much higher mean temperature (T) of the melt. These results allow one to obtain mean values for the temperature distribution in the melt as well as for temperature pulsations. A series of experiments on measuring of the temperature pulsations has been conducted. Fig. 5a and Fig. 5b present the main results for flow rate measuring on the melt surface for thermocapillary convection (a) and for the temperature distribution under the melt surface (1.5 mm deep) (b). The measurements were performed in a 150 mm diameter graphite crucible. There was a thorough check of the surface purity and axisymmetry of the temperature field. For most experiments the measurements were carried out at a growing Ge crystal, 5-15 mm in diameter. In order to provide an unchanged height of the melt, the velocity of crystal pulling was practically zero. The experiments were performed without crystal and crucible rotation. The tracers obtained as a result of the additional seed crystal melting, gathered in the crystal vicinity (Fig. 1). In order to get one curve, the velocity measuring results for 4 - 5 tracers were analyzed, the obtained data being averaged. The characteristic tern-

L.A. GorbunoL,/ Journal of Co'stal Growth 165 (1996) 351-357


perature drop was A T = 10-20 K on the crucible radius (see Fig. 5b). In such a way the data for thermocapillary convection were obtained for H = 6.12, 11.18, and 13.42 mm, respectively. The analogous data for H = 7.15 mm were obtained in the absence of a crystal. In this case the tracers gathered in the thermal center of the melt. Here, the experiment was conducted at a much higher mean temperature of the melt. All the experimental data for the velocity of the tracers' motion were scaled to u / u c, where uc was defined according to Eq. (4). This approach allows


'°I T,K






Fig. 6. Temperature distribution in the melt with H = 13.42 m m and R~ = 5 m m at z = 1.5 m m (curve 1) and z = 12 m m (curve








, o (b)



lo j



9* I 1

Fig. 5. (a) A v e r a g e d values for the rate o f thermocapillary convection at H = 6.12 m m (curve 1); H = 7.15 m m (curve 2), H = 11. l 8 m m (curve 3) and H = 13.42 m m (curve 4). (b) A v e r a g e d values for the temperature distribution at z = 1.5 m m u n d e r the melt surface at H = 6 . 1 2 m m (curve 1); H = 7 . 1 5 m m (curve 2), H = 11.18 m m (curve 3) and H = 13.42 m m (curve 4).

one to compare data obtained at different values of AT (see Fig. 5a). The obtained data show that the mean velocity of the tracers and, consequently, the flow velocity of the melt near the surface are close to the values in Eq. (4) for thermocapillary convection. Eq. (4) yields a value for thermocapillary convection 2 - 3 times larger for all the experimental values of H. It should be noted here that in the range H = 6.12-13.42 mm the u / u c ratio does not change significantly. It proves the accuracy of the above estimates for thermocapillary and thermogravity convection. Thermogravity convection in the melt seems small comparing to the thermocapillary one, it does not strongly affect the convective flow at H = 13.42 mm (see Table 1). Fig. 5a shows that curve 4 lies lower than curve 1. To explain this, consider the experimental data for the temperature distribution in the melt at H = 13.42 mm (Fig. 6). Here, curve 1 is got just under the melt surface (z = 1.5 mm), but curve 2 is at H = 12 mm (z = 12 mm). It means that the melt surface temperature near the container walls and in its middle part ( p* = p / L > 0.2) is higher than the melt temperature near the crucible bottom. The zone in the center of the crucible is an exception, because the temperature here is lower than near the bottom due to the crystal presence (R e = 5 mm) and to heat radiation from the melt surface. The existence of such a zone in the crucible suggests instability of the melt flow as well as elimination of the thermogravity convection action because of the hot melt near the bottom of the crucible, which generates a buoyancy force directed opposite to the melt motion.

L.A. Gorbunot' / Journal of Crystal Growth 165 (1996) 351-357

The experiment shows that enlargement of the crystal radius Rc = 15 mm on the melt surface (Fig. 4) causes a much more considerable cooling of the melt in the subcrystal zone and increase of the temperature gradients in the melt at p > R~.

5. Conclusion The investigations conducted prove that in a Ge melt (with a small value of the coefficient of volumetric expansion and a large value of 8 a / ~ T ) thermocapillary convection dominates over thermogravity convection even at comparatively large values of H. The experimental data obtained show the flow velocity u to be high due to thermocapillary convection. It is 30-50% of the u c velocity (found from Eq. (4)) for boundary layer flow. The experiments prove that the ratio u/u c does not depend on H for the investigated range 6-13 mm. The absolute velocity values obtained in the experiments manifest a turbulent flow regime at temperature gradients of 100-150 K / m and H > 6 mm ( A > I 2 ) ( R e = 1500-5000, Ma = 2500-8000). The studies of temperature pulsations did not allow us to obtain values for the critical numbers of Rear and Macr. It could be connected with the experimental procedure with a minimal temperature (of the crystal) on the melt surface in the center of the crucible, which induced instabilities at comparatively small Reynolds numbers, as well as with the investigated range for Re and Ma, for which "overcritical" regimes seemed to be characteristic. In general, the data obtained for u, prove a much


more significant role of thermocapillary convection for single crystal growth, than is considered at present. Accounting for the fact that estimation of u c and the experiments have been carried out under conditions similar to those for Ge single crystal growth, the obtained results allow one to conclude that thermocapillary convection can be comparable to thermogravity convection in real processes of single crystal growth, but at small values of H it can dominate.

References [1] S. Ostrach, Teoreticheskije Osnovy 105 (1983) 1 [Russian translation from Fluid Mechanics in Crystal Growth]. [2] Yu.M. Gelfgat and L.A. Gorbunov, in: MHD Metallurgical Technologies, Energy Conversion. and Magnetohydrodynamic Flows, Progress in Astronautics and Aeronautics, Vol. 148, Eds. H. Branover and Y. Unger (Ben-Gurion University of Negev, Israel, 1990) p. 138. [3] D. Schwabe, A. Scharmann, F. Preisser and R.J. Oeder, J. Crystal Growth 43 (1978) 305. [4] Ch.-H. Chun and W. Wuest, Acta Astronaut. 5 (1978) 681. [5] D. Schwabe, A. Scharmann and F. Preisser, Int. Astronautics Federation, IAF-80-c- 1'40 (Pergamon, Oxford, 1980). [6] M. Vargas, S. Ostrach and Y. Kamotani, Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, Ohio, 1982. [7] P. Tison, D. Camel, 1. Tosello and J.-J. Favier, in: Hydromechanics and Heat/Mass Transfer in Microgravity (Gordon and Breach, New York, 1991) p. 121. [8] H. Schlichting, Grenzschicht - Theorie, Russian Translation, Moscow. 1974. [9] V.M. Glazov, S.N. Chizhevskaya and N.N. Glagoleva, Liquid Semiconductors, USSR Academy of Sciences (Nauka, Moscow, 1967). [10] R. Rupp and G. Muller, J. Crystal Growth 113 (1991) 131.