physical modelling of the melt flow during large-diameter silicon single crystal growth

Journal of Crystal Growth 257 (2003) 7–18

Physical modelling of the melt flow during large-diametersilicon single crystal growth

L. Gorbunova,*, A. Pedchenkoa, A. Feodorova, E. Tomzigb, J. Virbulisb,1,W.V. Ammonb

a Institute of Physics, University of Latvia,32 Miera street, Salaspils 1, LV-2169, LatviabWacker Siltronic AG, Germany

Received 17 April 2003; accepted 19 May 2003

Communicated by D.T.J. Hurle

Abstract

The reported investigations concern physical modelling of Czochralski growth of silicon large-diameter single

crystals. InGaSn eutectic was used as a modelling liquid, employing actual criteria of the real process (Prandtl,

Reynolds, Grashof numbers, etc.) and geometric similarity. A multi-channel measuring system was used to collect and

process the temperature and flow velocity data. The investigations were focused on the study of heat transfer, in

particular, the instability of the ‘‘cold zone’’ of the melt at the crystallization front.

r 2003 Elsevier B.V. All rights reserved.

Keywords: A1. Fluid flows; A1. Heat transfer; A1. Modelling; A2. Czochralski method; A2. Single crystal growth; B2. Semiconducting

silicon

1. Introduction

One of the principal tendencies in the produc-tion of semiconductor single crystals, in particularsilicon single crystals, is the constant increase ofdiameters of grown crystals and, hence, the needfor enlargement of diameters of crucibles andcharge weight. This allows to reduce essentially theproduction cost of microelectronic devices. On the

other hand, the development of new super sizefacilities for large-diameter single crystal growthrequires a lot of expensive experimental studies toimprove the pulling conditions. A considerableproportion of these researches can be carried outby implementing physical and numerical model-ling of single crystal growth processes.

Direct observation of the melt flow in moltensilicon is very complicated. Real insight into thebulk of molten silicon is given by Watanabe et al.[1] using tracer-based X-ray radiography in a smallcrucible (75mm diameter). Some informationabout the melt flow can be derived from thetemperature measurements in molten silicon dur-ing the crystal growth [2].

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*Corresponding author. Tel.: +371-7944829; fax: +371-

7901214.

E-mail address: [email protected] (L. Gorbunov).1Presently at Center for Processes’ Analysis and Research,

Riga, Latvia.

0022-0248/03/$ - see front matter r 2003 Elsevier B.V. All rights reserved.

doi:10.1016/S0022-0248(03)01376-9

Most investigations of the melt flow in largecrucibles are based on numerical simulation. Theadvantage of numerical simulation is not onlylower costs, but also the good visualizationcapability, which helps to understand the flowstructure better. The disadvantages are the simpli-fications made during the modelling procedure,e.g. restriction to 2D or the use of turbulencemodels. The simplest and most suitable ones forparameter studies are 2D turbulence models [3–5].The 3D time-dependent calculations with LargeEddy model [6] and without any turbulencemodels [7,8] show better agreement with experi-ments than 2D simulations.

Physical modelling is more expensive thannumerical modelling, but has fewer simplificationsand, therefore, shows better agreement with theprocesses in industrial facilities. Modelling at lowtemperatures and in non-aggressive melts ispossible. Most works on physical modelling aredone in very small crucibles. Lee and Chun presenttheir results of model experiments on oscillatoryconvection in mercury in a Czochralski (CZ)configuration with the cusp magnetic field [9] andthe axial magnetic field [10] in a small, 160mmdiameter, crucible.

2. Basic principles of physical modelling

Some basic conditions should be considered forphysical modelling of the hydrodynamics and heattransfer in the processes of single crystal growth.First of all, these conditions include geometricalsimilarity, i.e., maintaining the ratios of thecrucible diameter, melt height and radius of singlecrystal. The second important condition is thechoice of liquid that simulates the melt. Thethermophysical properties of the modelling liquidshould be close to those of a semiconductor melt,particularly, the Prandtl number Pr that relatesmolecular and convective thermal conductivity.Another parameter determining the hydrodynamicsimilarity is the Reynolds number Re for forcedconvection driven by crystal and crucible rotation.The Grashof number Gr is the crucial parameterfor modelling buoyancy. The process of singlecrystal growth is greatly affected by the heat loss

from the melt-free surface. As elementary estima-tions show, this heat loss makes B80–90% of thetotal heat flow through the melt and in manyrespects determines the temperature field in themelt. To simulate the melt flow with highaccuracy, the ratio between the heat flux fromthe melt surface QS and that throughout thecrystal QC should be close to the real, i.e.,QS=QCB4� 10:

The electrical conductivity s of a modellingliquid is of great importance for the simulation ofmagnetohydrodynamic processes. As the conduc-tivity of most semiconductor melts is close to thatof liquid metals and is about 106O�1m�1, theHartmann number Ha ¼ BRðs=rnÞ1=2 (B is theinduction of magnetic field, R is the typicaldimension, s is the electric conductivity, r is thedensity and n is the kinematic viscosity) and theparameter of MHD influence N ¼ Ha2=Re areusually larger than unity (Ha > 1 and N > 1).Besides, when model single crystal growth in analternating magnetic field, the value of the para-meter e ¼ som0R

2 (characterizing the penetrationof the alternating magnetic field into the melt),where o is the angular frequency and m0 is themagnetic permeability of the vacuum, shouldalso be maintained to keep the distributionsimilarity of electromagnetic forces and heat lossesin the melt.

All this necessitates simulating the hydrody-namics and heat transfer on liquid metal alloyssuch as mercury, gallium, eutectic InGaSn,Wood’s alloy, etc. The analysis demonstrates thatthe most expedient melts for simulating the melthydrodynamics for single crystal growth areeutectic InGaSn or a gallium melt. To performthe investigations, we took eutectic InGaSn thatsolidifies at 10.5�C and stays liquid at roomtemperature.

For the modelling of industrial crystal growthprocess in a 600-mm-diameter crucible for a 200-mm-diameter crystal, we used a 500-mm-diametercrucible filled with InGaSn. As can be seen inTable 1, the simulation of this process on InGaSncan reach a very good correspondence practicallyin all parameters. As an example, let us examine asilicon single crystal of 200-mm-diameter pulled ata rotation rate of the crystal nc ¼ 12 rpm, that of

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L. Gorbunov et al. / Journal of Crystal Growth 257 (2003) 7–188

the crucible—n ¼ �4 rpm; a temperature dropbetween the crystallization front and the cruciblewall—DT ¼ 15�C, under the impact of a staticmagnetic field of induction B ¼ 0:01T, and thencalculate it in appropriate criteria (Re; Gr; Ha) foran InGaSn melt. The results of such calculationdemonstrate that when simulated in a 500-mm-diameter crucible, we obtain a very close corre-spondence in crystal/crucible rotation rates, intemperature drop DT in the melt, in magnetic fieldinduction, and other parameters (see Table 1). Itshould be stressed that the experimental setupallows physical simulation for even larger dia-meters of the crystal and the crucible.

3. Experimental setup. Method and procedure of

experiments

The experimental setup for simulation of thehydrodynamics and heat transfer was producedbased on an older facility for single crystal growth(Fig. 1). As main units for the experimental setup,we used the drives, both rotating and moving thecrystal and the crucible, the power supply andautomated control devices. As a crucible, we tooka standard 500-mm quartz crucible, used to growsilicon single crystals. A nickel-chrome heater wasarranged directly on the crucible to investigate

under non-isothermal conditions. The heater wascomposed of two sections, one of which providedheating of the cylindrical part of the crucible wall(a near-wall heater), another section heated itsbottom (a bottom heater). To eliminate theinfluence of the magnetic field of the heater onthe melt, its sections have bifilar winding. Thedrive of the crucible provided stable rotationwithin the range of 1–20 rpm. As a crystal modela cylindrical vessel of 165mm diameter, made ofstainless steel, was used (the bottom of the vessel incontact with the melt was 1mm thick). The crystalmodel was water-cooled through the pull rod ofthe facility. During the experiments, the water flowrate through the crystal model was controlled.This provided a constant temperature on thecrystal model inner surface. Its temperaturepractically was equal to that of the cooling water.In the experiments, we measured the value of theheat loss from the melt through the crystal usingdifferential thermocouples at inlet and outlet andthe flow meter as well. The drive rotating thecrystal ensured its stable rotation rate rangednc ¼ 1250 rpm.

During the experiments, the free surface ofInGaSn was protected from oxidation by a weakaqueous solution of HCl (B5%). As the density of

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Table 1

Comparison of growth parameters of industrial and experi-

mental systems

Industrial

(Si)

Experiment

(InGaSn)

Prandtl number 0.013 0.015–0.025

Crystal diameter, m 0.20 0.165

Crucible diameter, m 0.60 0.50

Melt height, m 0.18 0.15

Crystal rotation rate, rpm 12 17

Crucible rotation rate, rpm 4 5.6

Reynolds number for crystal 3.6� 104 3.6� 104

Reynolds number for

crucible

105 105

Temperature drop, K 15 22

Grashof number 3.9� 109 3.9� 109

Induction, T 0.1 0.09

Hartmann number 103 103

34

5

1

InGaSn - melt

HCl+H2O

nc

nc

2

6

Fig. 1. Principal schematic of the experimental facility. 1—

melt, 2—crucible, 3—crystal model, 4—heat exchanger, 5—

heater, and 6—magnetic system.

L. Gorbunov et al. / Journal of Crystal Growth 257 (2003) 7–18 9

HCl solution is much lower than the density ofInGaSn, the mobility of the melt surface wasinduced.

A distinctive feature of the experimental setup isthe possibility to simulate the heat loss from themelt-free surface. The radiation heat loss, dom-inating over the heat transfer in the crystal growthprocess, is negligible at room temperature, there-fore, a special cooler near the free surface isnecessary. So a water-cooled spiral was installed inthe electrolyte above the melt-free surface (Fig. 1).The copper spiral completely sank in the electro-lyte layer poured above the melt-free surface. Thedepth of this layer of electrolyte in the experimentswas kept about 20mm, an interval between theheat exchanger and the melt-free surface wasabout 5mm. Thus, the presence of the heatexchanger had no effect on the free surfacemobility. The value of the heat loss is controlledby the flow rate of cooling water in the heatexchanger. A differential thermocouple and a flowmeter device measured the heat loss.

The experimental setup (Fig. 1) was equippedwith a magnet system generating steady verticaland axial–radial (CUSP) magnetic fields and aone-phase or multi-phase alternating magneticfield, used in further experiments not discussed inthis paper.

A multi-channel measurement system was usedto collect and process the temperature in 32 pointssimultaneously. The system is PC-based andemploys a DAS-1800 acquisition board. The datawere supplied by 32 probes arranged either in themeridional plane (r2z) of the crucible or in itsazimuthal (r2j) plane in a sector with a 90� angle.The system quickly collects and processes the data,i.e., the sampling rate of each probe was 4000Hz.This permits to obtain practically instant tempera-ture distributions either in the meridional orazimuthal planes of the crucible. Each probe is a0.1-mm diameter electrode made of constantanand in contact with the melt. One of the electrodesarranged in the middle of the crucible as far as5mm from the crystal surface was a referencepoint, and another electrode made of copper wasalso arranged at this point. So alongside with thetemperature measurements, the thermo-electromo-tive (t.e.m.) force induced at the contact of InGaSn

with the constantan electrodes was directly mea-sured in experiments, too. This allowed measuringthe temperature at certain selected points withregard to the reference point on the crucible axis inthe sub-crystal zone. The presence of both copperand constantan electrodes permitted to measurethe absolute temperature as well.

The data were sampled with a frequency of4000Hz and averaged over 500–2000 time mo-ments, it results in a time step of 0.125–0.5 sbetween two saved data points. A standardfrequency of the instant pictures illustrating thetemperature distribution was 1Hz, the time of onemeasurement cycle—1000 s. The choice of theseparameters was determined mainly by a compara-tively low frequency of temperature patternsvariation in time. The mean temperature distribu-tions (isotherms) and the root-mean square scatterof temperature (RMS) were plotted based on thesedata. Besides, most of the experiments have videoanimations, each composing 200 pictures. Suchanimation allows to learn the nature and structureof disturbances more illustratively.

A conductive anemometer (probe) with a localmagnetic field measured the velocity of the meltflow. The local magnetic field was induced by asamarium–cobalt cylindrical magnet of 2–3mmdiameter and about 2–3mm long. The magnet wasfixed on a holder (a pipe made of stainless steel) of1–2mm diameter. Copper and tin electrodes(0.1mm diameter) were arranged on the edgesurface of the magnet. These electrodes wereconnected to the registering equipment by isolatedcopper and tin wires.

4. Results and discussion

In most regimes of the reported investigations, aconductive anemometer was first used to measurethe hydrodynamic flows, then the temperaturefields were measured using temperature sensors.For each regime, the processed experimental datawere illustrated in figures as the isolines ofmeridional flows, the isotherms of mean tempera-ture distribution and the RMS isolines of tem-perature in the meridional and azimuthal planes.Besides, every version of the temperature field has

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instant distributions of temperature (200 picturesobtained with 1 s interval), which were analyzed toillustrate the process evolution in time.

All experiments discussed in this paper wereperformed for the melt level being 150mm, thepower of the bottom heater 2500W and that of thenear-wall heater 500W. The total heat flow fromthe free surface and from the crystal wasQ ¼ 2600W with the ratio of heat loss from themelt-free surface QS to that through the crystal QC

being QS=QC ¼ 4:

4.1. Without rotation

All the results of investigation obtained for theregime with no crystal and crucible rotation, i.e.,nc ¼ 0; n ¼ 0; are discussed below. Under thisregime, the melt hydrodynamics is governedmostly by buoyancy. Fig. 2 illustrates the mea-surements of the flow velocity pattern presented asisolines of the melt meridional flow. The flow has asingle vortex, which rotates anti-clockwise. Themelt at the free surface moves from the cruciblewall to the center. The flow intensity is relativelysmall (the maximum absolute value of the streamfunction is �4.4� 10�5m3/s). The value of theradial velocity component is �10mm/s at the pointwith r ¼ 200mm and z ¼ 10mm that correspondsto the Reynolds number Re ¼ 8� 103:

In Fig. 3a, one can see the isotherms of the meantemperature field in the meridional plane. Thesedata were obtained by averaging 1000 instanttemperature fields for the total time of theexperiments being 1000 s. In this and other figures,the temperature in the sub-crystal zone is assumedzero at the reference point with r ¼ 0 andz ¼ 5mm. So the temperature drop DT in themelt is a little smaller than the real one, if onemeasures the temperature directly on the crystalsurface, but this difference is insufficient. In thiscase, the temperature maximum in the melt for thebuoyancy convection is relatively small—DT ¼ 7�C. The zone of a ‘‘cold’’ melt, as expected,lies in the sub-crystal zone, and that of the ‘‘hot’’melt at the wall near the free surface. Thetemperature gradient is maximum in the sub-crystal zone and minimum in the crucible zone.The Grashof number, calculated referring to the

maximum temperature drop DT ¼ 7�C, is Gr ¼8:4� 108: So using the above value in theestimation, the Reynolds number will be Re ¼Gr0:5 ¼ 2:9� 104: This value exceeds approxi-mately 3.5 times the one obtained in the experi-ment. At the same time, if one also uses the valueof melt level H as a characteristic length scale inthe above estimation, then the correlation, ob-viously, will be better, as also shown in Ref. [11].

Fig. 3b shows the isolines for the RMStemperature scatter T�; which was defined byanalyzing 1000 sequential temperature fields dur-ing 1000 s of the experiment. T� characterizes theamplitude of temperature pulsations at corre-sponding points in the melt and is defined by theformula

T ¼1

N

XN

i¼1

Ti; T� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N

XN

i¼1ðTi � TÞ2

r; ð1Þ

where Ti is the instant temperature value; T is themean value of temperature at a given point. Thezone of maximum T� for the buoyancy convectionis in the sub-crystal zone.

Figs. 3c and d illustrate the results of measure-ments in the azimuthal plane when the thermo-couples were arranged 10mm deep under the meltlevel. Here, one can see the data on the meantemperature field (c) and the RMS scatter oftemperature (d). The point with r ¼ 0; z ¼ 10mmwas accepted as a reference point with T ¼ 0:Since this point is 10mm below the crystal surface,the temperature there is higher than at the point

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Fig. 2. Isolines of the melt meridional flow governed by

buoyancy. cmax ¼ 6� 10�6 m3/s, cmin ¼ �4:4� 10�5 m3/s.


with r ¼ 0; z ¼ 5mm, which was the referencepoint for the measurements in the meridionalplane. So the temperature drop DT in Fig. 3cshould be less if compared to Fig. 3a. One canclearly see that without rotation there is no axialsymmetry of the temperature field and the thermalcenter of the crucible does not match its geome-trical center. The analysis of the instant distribu-tions of the temperature field in the azimuthalplane also shows that the melt flow is not axiallysymmetric. This flow exists as separate ‘‘tongue’’-shaped flows streaming from the crucible wall tothe sub-crystal zone. Apparently, in the experi-ment (as in an actual process) there is always asmall axial misalignment of the crystal and thecrucible, an incomplete axial symmetry of the

boundary conditions on the crucible wall, etc.These values are likely small, but they can becomethe reason why buoyancy becomes axially asym-metric.

The analysis of the instant distributions of thetemperature field in the meridional plane hasrevealed a specific hydrodynamic instability. Theinstability of the first phase of the process is relatedto the instability of the ‘‘cold’’ melt zone with ahigher density at the crystallization front. Move-ment of the ‘‘cold’’ melt zone onto the crystalresults in ‘‘drops’’ or ‘‘jets’’ of the ‘‘cold’’ melt,streaming down to the crucible bottom. Such‘‘drops’’ or ‘‘jets’’ appear either in the cruciblecenter or near the crystal edge. When the ‘‘colddrops’’ leave the ‘‘cold’’ zone and approach the

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Fig. 3. Isotherms (a,c) and isolines of temperature RMS T� (b,d) for TGC in the meridional and azimuthal planes (maximum

DT ¼ 7�C, T� ¼ 1:73�C).


crucible bottom, they warm up and vanish (Fig.4a–d). The occurrence and break-off of the dropsare likely casual and quasi-periodic. After the firstphase of the process (the reduction of the ‘‘cold’’melt zone at the crystallization front), the ‘‘hot’’zone near the crucible wall expands (Fig. 5a–d).During the second phase this zone spreadspractically to the crucible axis, and to the ‘‘cold’’zone on the crystal contracts. Then the wholeprocess is repeated (Figs. 3 and 4).

4.2. Crystal rotation

The situation changes significantly, when thecrystal rotation is switched on. The crystalrotation under the isothermal regime (withoutbuoyancy) also drives a practically single-vortexflow in the melt. In this case, its pattern is verysimilar to that driven by buoyancy, but differs indirection. Thus, in Fig. 6 one can see the isolines ofthe meridional melt flow when the crystal rotateswith 15 rpm, the crucible does not rotate and Gr ¼

0: The melt in the main vortex rotates clockwise,the maximum value of the stream function isc ¼ 2:4� 10�5 m3/s, which is a bit smaller thanthe intensity in the case of buoyancy alone in Fig.2. Therefore, if buoyancy is switched on, at smallcrystal rotation rates the buoyancy convectiondominates. Increasing of the crystal rotation ratefirst decreases the strength of the meridional flowand increases the temperature drop in the melt.For the crystal rotation rate nc ¼ 15 rpm, themaximum temperature drop of 10�C is achieved.At further increasing of the crystal rotation rate,the forced convection driven by the crystalrotation dominates in the melt. This changes themelt flow direction on the melt-free surface as wellas decreases the maximum temperature drop.

4.3. Crystal and crucible rotation

Additional rotation of the crucible (fornc ¼ 15 rpm, n ¼ �5 rpm, Fig. 7) strengthensslightly the vortex within the crucible volume, if

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Fig. 4. Instant pictures of temperature distribution (isotherms) in the meridional plane without rotation. Distributions are plotted with

a 2 s time interval.


compared to the case without rotations (Fig. 2).This is related to buoyancy and differentialrotation of the crystal and the crucible. In thiscase the melt rotates anti-clockwise; the maximumvalue of the stream function is c ¼ 8:8� 10�5 m3/s, which is slightly higher compared to the abovesituations with buoyancy and forced convection

driven only by crystal rotation. The explanation ofthis phenomenon can be as follows: on the onehand, the crucible rotation makes weaker thestrength of the melt flow in the meridional plane,on the other hand, it increases the maximumtemperature drop and, consequently, increases thestrength of the melt flow driven by buoyancy. So

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Fig. 5. Instant pictures of temperature distribution (isotherms) in the meridional plane without rotation. Distributions are plotted with

a 2 s time interval.

Fig. 6. Isolines of the meridional flow under isothermal regime

at nc ¼ 15 rpm, n ¼ 0; Gr ¼ 0; (cmax ¼ 2:4� 10�5 m3/s,

cmin ¼ �2:6� 10�6 m3/s).

Fig. 7. Isolines of the meridional flow at nc ¼ 15 rpm,

n ¼ �5 rpm, Gr ¼ 2:5� 109; cmax ¼ 2:3� 10�5 m3/s,

cmin ¼ �9� 10�5 m3/s.


the increase of the maximum temperature dropfrom DT ¼ 7�C for buoyancy to, as shown,DT ¼ 20:5�C for nc ¼ 15 rpm, n ¼ �5 rpm hasresulted in a certain increase of the flow strength.

The crucible rotation suppresses the melt flow inthe meridional plane and decreases the role of thehydrodynamic instability (characteristic of buoy-ancy). This manifests itself in increasing both themaximum temperature drop DT and the axialsymmetry of temperature distribution in the melt.Thus, at a relatively small rate of crystal andcrucible rotation nc ¼ 10 rpm and n ¼ �3 rpm,respectively, the maximum temperature drop inthe melt is already DT ¼ 16�C, and the meandistribution of temperature in the azimuthal planepractically turns out axially symmetric. Figs. 8aand b demonstrate the results of the investigation

for the closest to the actual regime of Si singlecrystal growth, the rates of crystal and cruciblerotation being nc ¼ 15 rpm, n ¼ �5 rpm. Fig. 8ashows that the hottest zone of the melt is located inthe right bottom part of the crucible and thecoldest one at the crystallization front; thetemperature drop in the melt is DT ¼ 20:5�C.The crystal/crucible rotation has decreased thetemperature pulsations in the melt volume and inthe sub-crystal zone. The analysis of instanttemperature distributions in the meridional planetestifies that at nc ¼ 15 rpm and n ¼ �5 rpm, thepattern of isotherms in the melt changes insignif-icantly with time. They just oscillate about theirmean position. The instant temperature distribu-tions show that the ‘‘cold’’ liquid streams from thecrystal down to the crucible bottom along the

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Fig. 8. Isotherms for nc ¼ 15 rpm, n ¼ �5 rpm in the meridional (a) and azimuthal (b) planes (maximum DT ¼ 22�C).


crucible axis only either like ‘‘jets’’ or ‘‘drops’’ of‘‘cold’’ melt. The ‘‘cold’’ zones on the crystalsurface and on the crucible bottom are linked, andperiodically, at some instants of time, a warmerfluid separates them.

Fig. 8b illustrates similar data for the tempera-ture field in the azimuthal plane, when the probesare arranged 10mm below the melt surface. Onecan see that the isotherms of the mean temperatureare distributed precisely symmetrically in the melt.The temperature has its minimum in the cruciblecenter and maximum on its side surface.

The analysis of instant pictures of temperaturedistributions in the azimuthal plane demonstratesthat the pattern of the temperature field at smallrates of crystal/crucible rotation differs greatlyfrom an axially symmetric one. The isotherms inthe melt resemble an irregular polygon, or ‘‘aflower’’. Such isotherm pattern, while varying intime, rotates together with the melt with a rate,which is slightly less than the crucible rotationrate. As the rates of crystal and crucible rotationincrease, the temperature field pattern tends toaxially more symmetric. Some illustrative data onthe instant distributions of temperature in theazimuthal plane at nc ¼ 15 rpm, n ¼ �5 rpm arepresented in Fig. 9. These data show that anunstable and axially asymmetric melt flow regimeis kept at nc ¼ 15 rpm and n ¼ �5 rpm as well.

The results of similar experiments for a moreintensive melt rotation at nc ¼ 25 rpm, n ¼ �7 rpmtestify that the increasing of the crystal andcrucible rotation rate has resulted in a compara-

tively small decrease of DT and T�; even thoughthe distributions of T andT� remain close to thosein Fig. 8. In their turn, the instant distributions oftemperature in the melt show less oscillations ofthe isotherms about their mean position with timeand enhanced process stability. High axial sym-metry of temperature distribution in the azimuthalplane is characteristic of both the mean tempera-ture field and the instant temperature distribu-tions. So the axial symmetry increases as thecrystal/crucible rotation rates increase.

Let us discuss the importance of the cruciblerotation. The investigations have shown thatincreasing of the crucible rotation rate significantlyincreases the temperature drop DT in the melt andthe axial symmetry of T distribution, if comparedto the situation with nc ¼ 0; n ¼ 0: At the sametime, the role of the instability, characterizing thecase of nc ¼ 0; n ¼ 0; turns out to be lessimportant. In general, the experimental dataobtained, when the crucible rotates alone atn ¼ 3; 5; 7 rpm, are very close to the data obtainedat nc ¼ 10 rpm and n ¼ �3 rpm, nc ¼ 15 rpm andn ¼ �5 rpm, nc ¼ 25 rpm and n ¼ �7 rpm. Thisemphasizes a determining influence of the cruciblerotation, or, in more precise words, a determininginfluence of the melt rotation.

4.4. Comparison of different regimes

The results of the performed investigations forthe maximum temperature drop DT in the meltdependent on the rate of crystal and crucible

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Fig. 9. Instant distributions of temperature (isotherms) in the azimuthal plane at nc ¼ 15 rpm, n ¼ �5 rpm. Distributions are plotted

with a 1 s time interval.


rotation are illustrated in Fig. 10a. Its vertical axisshows the crystal rotation rate nc; and thehorizontal axis shows the crucible rotation rate n:One can clearly see that the rotation of the cruciblealways results in increasing of the maximumtemperature drop DT ; but that of the crystal atnc > 10215 rpm in its decreasing. The maximumDT is achieved at nc ¼ 15 rpm (see Fig. 10a).

The dependencies for the radial temperaturegradient, when the crystal edge is 5mm deep underthe melt level, are different (see Fig. 10b). One cansee that in this case the crystal rotation rate nc is adetermining parameter. The radial temperaturegradient here increases as nc increases, and atnc ¼ 15 rpm the gradient achieves its maximum

value. The further increase of nc decreases theradial gradient of temperature at the crystal edge,Fig. 10c. Such dependencies are likely related tothe fact that the strength of the flow driven bybuoyancy is close to that driven by forcedconvection at nc ¼ 15 rpm. This changes the flowpattern and gives rise to a two-vortex flow (Fig. 7)and, consequently, decreases the convective heattransfer that, in turn, results in increasing themaximum temperature drop DT and temperaturegradients in the melt.

Similar dependencies of the amplitude of tem-perature pulsations are characteristic both of thecrucible wall (Fig. 10c) and of the sub-crystal zone(Fig. 10d).

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25

20

15

10

5

00 2 4 6

n, rpm

nc, r

pm

8 10 12 14

25

20

15

10

5

00 2 4 6

n, rpm

nc, r

pm

8 10 12 14

25

20

15

10

5

00 2 4 6

n, rpm

nc, r

pm

8 10 12 14

25

20

15

10

5

00 2 4 6

n, rpm

nc, r

pm

8 10 12 14

(a) (b)

(c) (d)

Fig. 10. Isolines of temperature maximum drop (a), radial temperature gradient on the crystal edge (b), temperature pulsations on the

crucible wall (c), and temperature pulsations in the sub-crystal zone (d); nc is the crystal rotation rate, n is the crucible rotation rate.


So the data of Fig. 10 testify that the tempera-ture drop DT and the amplitude of temperaturepulsations on the crucible wall can greatly dependon the rate of crucible rotation, but the tempera-ture gradients and the amplitude of temperaturepulsations in the sub-crystal zone depend on therate of crystal rotation. The presence of suchdependencies allows a more purposeful control ofthe hydrodynamics and heat transfer in actualprocesses of single crystal growth.

5. Concluding remarks

The investigations deal with physical simulationof the CZ-growth of large-diameter silicon singlecrystals. The investigations have shown that theeutectic melt InGaSn can be used for modellingthe process at the actual values of Prandtl,Reynolds and Grashof numbers, considering alsothe radiation heat losses from the melt-freesurface.

The presence of instability in the zone of ‘‘cold’’melt at the crystallization front has been foundexperimentally. It is shown that such instabilitymanifests itself most strongly in the absence ofcrystal/crucible rotation (for buoyancy only). Therotation of the crucible decreases the influence ofthis instability and stabilizes the melt hydrody-namics as well as heat and mass transfer. This isachieved by suppressing the meridional convectiveflows when the melt rotates.

There is practically no axial symmetry of thetemperature field in the melt when neither thecrystal nor the crucible rotates. At small rates ofcrystal and crucible rotation, the pattern of the

temperature field greatly differs from the axiallysymmetric one. The isotherms in the melt resemblean irregular polygon, or ‘‘a flower’’. The rotationof isotherms enhances the axial symmetry of thetemperature field under all regimes. Such anisotherm pattern, while varying in time, rotatestogether with the melt with a rate, which is slightlyless than the crucible rotation rate. As the rates ofcrystal and crucible rotation increase, the tem-perature field pattern tends to become more axiallysymmetric.

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Crystal Growth Charact. Mater. 38 (1999) 215.

[2] O. Gr.abner, A. M .uhe, G. M .uller, E. Tomzig, J. Virbulis,

W. von Ammon, Mater. Sci. Eng. B 73 (2000) 130.

[3] Yu.E. Egorov, Yu.N. Makarov, E.A. Rudinsky, E.M.

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physical modelling of the melt flow during large-diameter silicon single crystal growth

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