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Viscosity of molten silicon and the factors affecting measurement
Journal of Crystal Growth 249 (2003) 404–415
Viscosity of molten silicon and the factors affecting measurement Yuzuru Satoa,*, Yuichi Kamedaa, Toru Nagasawaa, Takashi Sakamotoa, Shinpei Moriguchia, Tsutomu Yamamuraa, Yoshio Wasedab a
Department of Metallurgy, Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan b Institute for Advanced Materials Processing, Tohoku University, Sendai 980-8577, Japan Received 24 July 2002; accepted 28 October 2002 Communicated by T. Hibiya
Abstract Viscosity of molten silicon has been measured to study the behavior, especially around the melting point, in order to supply useful information for the crystal growth of semiconductors. The temperature range was taken as wide as possible from 1891 K down to the supercooled region by using an oscillating viscometer with various materials of crucible to study the effect of the materials. It has been clear that the viscosity of molten silicon showed a good Arrhenian behavior in the entire temperature range including the supercooled region and no abnormal increase around the melting point. The viscosity values were lower than the reported values and the maximum difference reached 50% even in the temperatures sufficiently higher than the melting point. Furthermore, almost no effect on the measurement by the materials of crucible was found. The recommended viscosity is presented by the following equation based on the present results: log Z=mPa s ¼ 0:727 þ 819=T; EZ ¼ 15:7 kJ mol1 : The viscosity and the activation energy were considerably lower than those of other metals with high melting points and the reason was considered to be due to the looser structure of the molten silicon originated from diamond-type solid structure. r 2002 Elsevier Science B.V. All rights reserved. Keywords: A1. Viscosity; A2. Growth from melt; B1. Silicon; B2. Semiconducting silicon
1. Introduction Since silicon is the most important semiconductor material for highly developed information technology and silicon wafer with larger size such as 12 in is now being developed, it is required to simulate the process of crystal growth in the CZ process by means of a numerical model for *Corresponding author. Fax: +81-22-217-7310. E-mail address: [email protected] (Y. Sato).
producing high quality crystal and establishing effective conditions for the production. Therefore, the reliable values of many physicochemical properties of solid and molten silicon are necessary. However, the reliable physicochemical properties, especially in the molten state, are not well known because of experimental difficulties due to the high temperature and the high chemical activity. The viscosity values in the literatures [1–3], which were measured by using also the oscillating viscometer and the crucible made of
0022-0248/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 2 ) 0 2 1 5 3 - X
Y. Sato et al. / Journal of Crystal Growth 249 (2003) 404–415
405
corundum (Al2O3) [1], pyrolytic boron nitride (PBN) [2], and SiC and PBN [3], showed a big discrepancy among them and abnormal increases near the melting point. Furthermore, different values depending on the crucible materials were reported [3]. Such an abnormal behavior has not been found for most of the molten metals and was sometimes considered to be the characteristic feature of molten silicon. In this work, it was planned to obtain the reliable values for viscosity of molten silicon by clarifying the behavior near the melting point, to check the effect of crucible materials and to supply the recommended value for the simulation of the crystal growth.
2. Experimental procedure 2.1. Apparatus Although many methods were developed for viscosity measurements, the application to molten silicon is actually limited to the oscillating method because of its low viscosity and high temperature. In this work, the oscillating viscometer [4] was improved in order to adept to the high temperature, for example, by employing a furnace with excellent uniformity of the temperature profile, by replacing some components refractory materials and by enhancing the cooling system. Various crucibles were used to study the effect of materials on the measurement. The viscometer used in this work consists of a suspension system, an oscillation detection system and a heating system. The suspension and detection system inside the apparatus is shown in Fig. 1. A crucible connected through a tungsten rod, an inertia disk and a mirror block are suspended by a thin torsion wire made of a platinum alloy. The mirror block reflects incident light from a He–Ne laser and the light is focused as a small spot of about 0.1 mm diameter on the detector which consists of three photo-transistors. Then four time intervals between the times when the locus passes through the detector are determined by using a computer with a resolution of 0.1 ms (104s). Rotational force is given to the inertia disk to start the oscillation
Fig. 1. Suspension and detection system in the oscillating viscometer.
electromagnetically with the aid of the coils located above and under the disk. The reproducibility of the time measurement was excellent, for example, the standard deviation of the period of oscillation was typically less than 0.01% even in measurements at high temperatures. The period of the oscillation and the logarithmic decrement were determined from the four time intervals by means of an approximation algorithm described later. Fig. 2 shows the whole apparatus schematically. The torsion wire was kept at 308.270.1 K to maintain the constant rigidity by circulating thermostated water in the copper tube wound around the wire housing. The heating system is also very important because the crucible which is about 85 mm long must be heated uniformly to prevent the convection flow in the crucible and to determine the temperature precisely. The furnace consists of three spiral MoSi2 heaters controlled independently. Additionally, about 25 reflection plates of molybdenum were installed in an outer alumina tube to reduce the vertical thermal radiation loss. The temperature profile was checked and the furnace controller was fine-tuned
Y. Sato et al. / Journal of Crystal Growth 249 (2003) 404–415
406 1 2 3
He inlet
4
5: 6: 7: 8:
5 6 7 8 9
1: 2: 3: 4:
Cooling Water
10 11 12
9: 10: 11: 12: 13: 14: 15:
13
Head corn Gas inlet Torsion wire Water circulation tube Reflection mirror Window Inertia disk Oscillation initiator Water jacket W rod Mo plate Crucible Three divided furnace Zr sponge Thermocouple
14
Cooling Water
9
15
Fig. 2. Schematic diagram of the oscillating viscometer.
carefully during each measurement to obtain the best temperature uniformity. As a result, uniform temperature profile within 0.5 K in the whole length of the crucible was obtained while the temperature of the upper part was set to be slightly higher than that of the lower part. The atmosphere was helium with a purity higher than 99.99%, which is inert chemically and has low viscosity. Convection flow of the atmosphere around the crucible makes the oscillation unstable. Small but irregular movement of the axis of rotation from proper position was sometimes observed in the case of other gases such as argon. However, such a phenomenon was not found for helium and a very stable oscillation was found though other effects by the convection may remain. Therefore, the change of logarithmic decrement and the oscillation at elevated temperatures were measured for the empty crucible as described later. A zirconium sponge was placed under the crucible to absorb the remaining oxygen in the atmosphere.
As mentioned above, Sasaki et al. [3] reported lower viscosity of molten silicon for the PBN crucible than that for the SiC crucible. Sasaki et al. concluded that the reason of lower value was the slipping at the interface between molten silicon and PBN due to the poor wettability between them. However, this conclusion is not likely because the motion of the liquid in the crucible is very slow and an adhesive force such as Van der Waals force is considered to be enough for adhesion. Therefore, it is necessary to study the effect of the material experimentally. Mukai et al. [5] reported that the wettability with molten silicon is very good for SiC, poor for BN and intermediate for SiO2, Si3N4, and Al2O3, respectively. By considering the wettability and the chemical thermodynamic stability against molten silicon, and also the mechanical strength at high temperature, silicon carbide (SiC), graphite (C), highly sintered alumina (Al2O3), silicon nitride (Si3N4), hot pressed boron nitride (BN), quartz (SiO2) and 8 mol% yittria stabilized zirconia (8 mol%-YSZ) were chosen as the crucible materials in this work. Graphite is considered to be in the same group as SiC because it reacts with molten silicon and makes SiC on the surface, and 8 mol%-YSZ is expected to have a good wettability. All the crucibles were specially ordered ones and were precisely machined to obtain the uniform inner diameter of 20 mm whose precision was within 0.05 mm. The shape of the crucible is shown in Fig. 3. The cap has two female screws for connecting with both a crucible body and a tungsten rod with the male screw at its end. 2.2. Determination of logarithmic decrement and period of oscillation In this work, the viscosity is determined based on the absolute method by using Roscoe’s equation [6] in which only physical parameters are required for the calculation. Those are the logarithmic decrement, the period of oscillation, the moment of inertia of the suspension system, the inner radius of the crucible and the density of the melt. The logarithmic decrement and the period of oscillation were obtained from the measurement of time intervals as follows:
Y. Sato et al. / Journal of Crystal Growth 249 (2003) 404–415
407
Fig. 4. Oscillating curve with decay and the time intervals measured in a period.
expressed by Eq. (4). Therefore, an apparent period of the oscillation, t0 is given in Eq. (5):
Fig. 3. Schematic diagram of a typical crucible.
Y ¼ A sin ot þ C;
ð4Þ
t0 ¼ t1 þ t2 þ t3 þ t4 :
ð5Þ
The constants A and C in Eq. (4) are defined by Eqs. (6) and (7), respectively for each period:
An oscillatory curve with decay expressed by Eq. (1) is shown in Fig. 4. L and –L are the positions of the detectors. C is the deviation of the center of oscillation from the center of detectors. The period, t and the logarithmic decrement of oscillation, d are defined by Eqs. (2) and (3): Y ¼ A expðBtÞ sinot þ C;
ð1Þ
t ¼ 2p=o;
ð2Þ
d ¼ 2pB=oBt;
ð3Þ
where o is the angular frequency, A and B are constants related to the amplitude and the decrement, respectively. Four time intervals t1 to t4 measured in a period were used for determining t and d. If C equals 0, which is an insufficient assumption, analytical solutions can be obtained for t and d. However, it is difficult to calculate them analytically in the case of non-zero C. Therefore, the present authors have developed the following approximation method. At first, oscillatory curve without decay is assumed as
A¼ 2L=½sinfpð0:5 t1 =t0 Þg sinfpð1:5 t3 =t0 Þg ;
ð6Þ
C¼ A½sinfpð0:5 t1 =t0 Þg þ sinfpð1:5 t3 =t0 Þg : ð7Þ If it is assumed that the shape of oscillation keeps a regular sine wave without decay in the ith period and the amplitude decreases with keeping also a regular sine wave in the next (i+1)th period, an apparent constant B0 is calculated using Eq. (8) using a series of Ai which are the amplitudes obtained for sequential ith periods. Also, an apparent logarithmic decrement, d0 is given by Eq. (9): B0 ¼ ð1=tÞlnðAi =Ai þ 1Þ;
ð8Þ
d0 ¼ B0 t0 :
ð9Þ 0
This set of first approximate values, d and t0 includes considerable error depending on the logarithmic decrement. The error was estimated by using a sample data of the time intervals which were generated through given parameters: fixed C
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Fig. 5. Oscillating curve with decay for the wave analysis.
of 10 mm, 2L of 100 mm and various B. This condition is relatively worse than that of the real measurement. The deviations of the first approximate values, d0 and t0 from true values were from about 101.5% to 100.5% for the practical range of logarithmic decrement in the measurement. They are only available when the decay is unpractically small. The effect of decay on the oscillation must be taken into account in the next step. The oscillatory curve with decay and the relationship between the apparent periods given by Eq. (5) and the true period is shown in Fig. 5(a). An apparent period, t0 is always longer than a true period, t. The true period can be determined if the difference (t0 t) is estimated precisely. Fig. 5(b) shows the overlapped consecutive half period with decay. The curve is apparently a distorted sine wave and is vertically similar to the consecutive wave. The difference (t0 t) is apparently equal to ta from Fig. 5(a) although ta is not measurable. However, the following relationship in Eq. (10) is helpful: t1;i t1;iþ1 ¼ ta þ tb :
ð10Þ
ta can be determined if the measurable left-hand term, (t1,it1,i+1), is reasonably divided into ta and tb : The times, at which differential coefficient of a wave equals zero, are the same for any period as shown in Fig. 5(b). Therefore, the left-hand term in Eq. (10) was divided at the time mentioned
above to distribute the left-hand term into ta and tb by assuming the ratio ta =tb to be equal to the ratio of divided parts. As a result, a second approximate value of t00 is obtained as in Eq. (11): t00 ¼ t0 ta ¼ t0 ðt1;i t1;iþ1 Þ tan1 ðo=BÞ=p;
ð11Þ
where the parameter t0 is taken from Eq. (5) and o=B ¼ 2pXdis taken from d0 in Eq. (9) which are the first approximate values. The latest period, t00 is used to calculate the second approximate value of logarithmic decrement through Eqs. (6)–(9) again. The second approximate values of the period of oscillation and the logarithmic decrement were sufficiently precise. The error accompanied by the approximation was from about 104.5% to 103.5% in the same conditions above. Therefore, this set of second approximate values was employed for determining the viscosity.
3. Method of viscosity determination Viscosity is determined by means of some physical parameters as mentioned above. The moment of inertia of the suspension system for an empty crucible was determined prior to the viscosity measurement based on the period of
Y. Sato et al. / Journal of Crystal Growth 249 (2003) 404–415
oscillation using an additional inertia disk with known moment of inertia. At elevated temperatures, the moment of inertia and the logarithmic decrement increase mainly due to the thermal expansion of the crucible and the increase in viscosity of helium as the atmosphere, respectively. Therefore, individual empty crucibles were calibrated at various temperatures up to about 1800 K to obtain the changes of these values as a function of temperature. This calibration is useful to compensate the effects by not only the above two factors but also by unknown factors such as the convection flow of helium around the crucible. As the shape and the dimension of the crucible is also very important because most algorithms for calculating viscosity using a cylindrical crucible including Roscoe’s equation do not take the effect of the bottom and free surface into account, the effect should be studied experimentally. The present authors measured the above physical parameters and calculated the viscosity of molten tin for the different ratio of height to radius of liquid column in the crucible, h/r by feeding different amounts of the sample [7]. The apparent viscosity calculated was larger in the lower h/r region and decreased with an increase of h/r. The apparent viscosity converged to an almost constant value in which h=r is higher than six. The apparent value for h=r ¼ 3 was 6–8% larger than the converged value. Namely, it is concluded that a considerably long crucible is required for the reliable measurement. In this work, at least seven of the ratios were achieved by employing the dimensions of 20 mm in the inner diameter and 80 mm in the depth of the crucible as shown in Fig. 3. The crucibles used in previous works [2,3] were too short, may be less than five of the h=r value. For actual measurement, the expansion coefficients for the materials of the crucible were taken from the maker’s technical data and the radius of crucible was corrected at high temperatures. The above method gave good results. The results on the distilled water and mercury at room temperature have agreed with the literature values [8] within 1%. Furthermore, the results on the molten NaCl also has agreed with the recommended value by Janz [9] and the authors’
409
previous result by using a capillary viscometer [10] within 2–3%. Silicon sample was quarried out as a cylindrical shape from a semiconductor grade single crystal made by the CZ method. The purity was almost 8 N.
4. Results All the experiments were carried out both in heating and cooling processes. The measurement was started from a temperature about 20 K higher than melting point of silicon and heated up to highest temperature depending on the crucible, and then cooled down until the solidification was confirmed to check the reproducibility of the measurement. The density of molten silicon, which is the most important and unmeasurable parameter in this work, was taken from the author’s previous work [11]. Fig. 6 shows the viscosities obtained for SiC, graphite, Al2O3, Si3N4, BN, SiO2 and 8 mol%-YSZ as the crucible materials together with the literature values. The viscosities reported by Grazov et al. [1] for corundum crucible and Kakimoto et al. [2] for PBN crucible are relatively high and show non-Arrhenian behavior in the entire temperature range. The result by Sasaki et al. [3] for SiC crucible is the highest and that for PBN crucible is considerably low. Furthermore, a common feature in the literatures is the abnormal increase of the viscosity near the melting point of silicon. In the present work, the maximum temperature was 1891 K, and the measurements could be carried out to the temperatures lower than the melting point of silicon in some cases which was the supercooled temperature region. However, no abnormal increase in the viscosity near the melting point was found in all the measurements and most of them showed a good Arrhenian behavior in the entire temperature range. The present results show very similar viscosities except for graphite and SiC crucibles. The results for graphite crucible showed a poor reproducibility and larger temperature dependence, and that for SiC crucibles showed higher viscosity and larger scatter than for the others.
Y. Sato et al. / Journal of Crystal Growth 249 (2003) 404–415
410
Temperature, T / K
0.10
log (Viscosity, / mPa.s
0.00
1850
1800
1750
1700
1650 1.50
Glazov (ref.1) Kakimoto (ref.2) Sasaki-SiC (ref.3) Sasaki-PBN(ref.3) Graphite Silicon Carbide Alumina Silicon Nitride Boron Nitride Quartz 8mol% YSZ
1.40 1.30 1.20 1.10 1.00 0.90
-0.10
0.80
0.70
Viscosity, / mPa.s
0.20
1900
-0.20 0.60
-0.30
0.50
↑ Melting point -0.40 0.52
0.54
0.56
0.58
0.60
0.40 0.62
T -1/10-3K-1 Fig. 6. Viscosities of molten silicon obtained for various materials of crucibles with literature values.
It is clear that the viscosities obtained in the present work for Al2O3, Si3N4, BN, SiO2 and 8 mol%-YSZ crucibles are concentrated in the narrow range of about 3% and show almost the same activation energy. On the other hand, the results for SiC and graphite crucibles are different from others although the reason why such a difference appeared is still unclear. Although the measurements were typically carried out 2 or 3 times to confirm the reproducibility for each crucible except SiO2 and 8 mol%-YSZ, the results with the widest temperature range for each material are shown in the figure. The reproducibility for SiC and graphite was poorer in contrast with good reproducibility for BN, Al2O3 and Si3N4 within 1% of standard deviation for the scatter of data points independent from the heating and cooling processes. The numerical results obtained by using Al2O3, Si3N4, BN,
SiO2 and 8 mol%-YSZ crucibles are shown in Table 1. To study the uncertainty of the measurement, the error propagation was simulated by using the data obtained in this work for Al2O3 crucible. The input parameters used in Roscoe’s equation to calculate the viscosity are logarithmic decrement, period of oscillation, moment of inertia, inner radius of the crucible, total mass of the melt and density of the melt. The first two parameters are obtained from the experiment and others are separately given. The height of the melt is derived from given parameters. These parameters were independently deviated from actual input data and compared with actually obtained viscosity as shown in Table 2. The manner of the error propagation was made clear and the quantitative evaluation was possible. The reproducibility of the logarithmic decrement and the period of
Y. Sato et al. / Journal of Crystal Growth 249 (2003) 404–415
411
Table 1 Viscosity values obtained by using various materials of crucible Alumina
Silicon nitride
Boron nitride
Quartz
8 mol%-YSZ
T (K)
Z (mPa s)
T (K)
Z (mPa s)
T (K)
Z (mPa s)
T (K)
Z (mPa s)
T (K)
Z (mPa s)
1877.9 1868.8 1858.7 1849.8 1838.6 1829.8 1820.2 1810.6 1799.5 1791.2 1781.3 1761.2 1752.5 1742.8 1722.9 1713.2 1703.6 1689.5 1684.6 1675.0 1669.7 1665.6 1660.9 1655.6
0.5100 0.5136 0.5191 0.5200 0.5196 0.5259 0.5260 0.5327 0.5328 0.5292 0.5418 0.5453 0.5488 0.5565 0.5555 0.5623 0.5774 0.5647 0.5591 0.5756 0.5780 0.5815 0.5831 0.5865
1856.7 1844.7 1828.2 1818.2 1798.9 1789.5 1770.5 1760.6 1741.6 1731.9 1713.5 1704.2 1694.3 1684.9
0.5284 0.5285 0.5318 0.5344 0.5411 0.5477 0.5562 0.5567 0.5658 0.5675 0.5729 0.5789 0.5817 0.5852
1856.0 1839.6 1820.2 1800.8 1785.7 1773.0 1767.1 1752.2 1745.8 1732.8 1724.1 1720.0 1713.2 1705.3 1694.3 1688.3 1684.9 1679.8
0.5220 0.5264 0.5342 0.5352 0.5466 0.5450 0.5466 0.5583 0.5568 0.5585 0.5685 0.5702 0.5664 0.5729 0.5820 0.5826 0.5840 0.5849
1790.2 1786.4 1780.6 1776.2 1771.2 1766.2 1761.5 1756.5 1752.5 1746.4 1742.2 1736.4 1732.5 1727.7 1722.7 1717.6 1714.1 1707.4 1703.9 1698.7 1694.6 1688.3 1683.2 1678.4 1673.6 1668.9 1664.2
0.5323 0.5344 0.5360 0.5357 0.5400 0.5396 0.5421 0.5420 0.5473 0.5464 0.5512 0.5473 0.5532 0.5493 0.5594 0.5579 0.5613 0.5586 0.5645 0.5637 0.5673 0.5694 0.5689 0.5711 0.5768 0.5727 0.5768
1891.8 1873.0 1851.9 1832.8 1794.0 1737.9
0.5021 0.5074 0.5148 0.5188 0.5284 0.5579
Table 2 Error of viscosity caused by the deviation of physical parameters used for the calculation of the viscosity Deviation of parameter
+0.1%
+0.3%
+1.0%
Logarithmic decrement Period of oscillation Moment of inertia Radius of crucible Mass of the melt Density of the melt
0.44% +0.18% 0.36% +0.44% +0.36% 0.14%
1.35% +0.52% 1.11% +1.31% +1.07% 0.44%
4.45% +1.71% 3.70% +4.33% +3.58% 1.57%
oscillation were 0.1–0.2% and 0.002–0.005%, respectively. The uncertainty in the moment of inertia and in the radius of crucible at high temperature were estimated to be less than 0.2% for both. The change of the mass between after
and before the measurement was less than 0.02%. In general, it is difficult to estimate the error contained in the density because it depends on the separate experiment. Although the density used in this work was obtained by the authors [11] and the scatter in the data was lesser than 0.1%, but unknown factors might affect the data and the uncertainty was estimated to be less than 0.3%. The error caused by the individual factor does not exceed 1%. As the result, the total error may be about 1% by considering the theory of statistics. This value is excellent for the measurement at high temperature. However, the total error including the error caused by unknown factors should be larger although very careful efforts were paid to reduce the effects of the factors. The maximum discrepancy in viscosities obtained in this work shown in Fig. 6, except for graphite and SiC
Y. Sato et al. / Journal of Crystal Growth 249 (2003) 404–415
crucible, is about 3% and also 2–3% of difference was found between the results of molten NaCl by the oscillating method and the capillary method mentioned above. So it is considered that the uncertainty accompanied by the measurement at high temperature is estimated to be less than 3%. It is extremely small compared with the discrepancy found in the literature values.
5. Discussion As mentioned above, the problems previously pointed out on the viscosity of molten silicon are abnormal temperature dependence near the melting point and the effect of the material of crucible on the measurement. In the present work, various refractory materials with different wettability with molten silicon were used as the crucible materials to study the effect of the materials. The results indicated no difference among the materials with poor and good wettability against the molten silicon. After the experiments, the interface between the solidified silicon and the crucible wall was observed visually. The silicon has adhered strongly to the SiC, graphite, SiO2, Si3N4 and 8 mol%-YSZ. The contact angle was nearly 901 for SiO2 and was lesser than for others. These materials were decided to be well wettable to molten silicon. On the other hand, solidified silicon was easily removed from Al2O3 and BN and the contact angles were larger than 901. In this case, wettability was decided to be poor. Therefore, it is concluded that the wettability almost does not affect the viscosity measurement as predicted by taking almost the same viscosity values obtained except SiC and graphite into account. This suggests that there is no slipping between crucible wall and molten silicon. However, the wettability should affect the measurement due to the different shape of meniscus of molten silicon. This effect was probably minimized by using the long crucible with large h/r value. On the abnormal behavior of the viscosity near the melting point, such a phenomenon was not observed in all the measurements in this study although the viscosity measurement was carried
out both above and below the melting point. The lowest temperature in the present work was 1656 K, which was almost 30 K lower than the melting point of silicon, with the Al2O3 crucible. This supercooling occurred due to probably excellent temperature uniformity. At this temperature, interesting phenomena were observed as it happened during the measurement. Sudden increases of the temperature and the logarithmic decrement occurred. Temperature rose above 1670 K in a few seconds due to the latent heat of solidification. The reason why the temperature was still lower than the melting point was that the thermocouple was kept away from the crucible as shown in Fig. 2. Simultaneously, logarithmic decrement also increased rapidly almost twice as demonstrated in Fig. 7. After that, the temperature decreased slowly until the melt solidified completely and the logarithmic decrement also decreased to a very small value which corresponds to the solid. The twice value of the logarithmic decrement means almost one order large in the viscosity. These phenomena suggest that the larger value in the viscosity can be obtained if the melt solidified partially near the melting point. If the melt solidifies partially and a liquid–solid mixture like slurry is formed in the crucible, it makes the apparent viscosity larger. Partial solidification may occur if the temperature profile around the crucible is not sufficiently good. In this work, a
0.030
0.025
logarithmic decrement
412
0.020
0.015
0.010
0.005
0.000
0
5
10
15
20
25
30
35
40
Time, t / min
Fig. 7. Change in the logarithmic decrement due to the rapid solidification during the measurement of molten silicon supercooled at 1655.6 K.
Y. Sato et al. / Journal of Crystal Growth 249 (2003) 404–415
very good temperature profile was obtained by carefully performing the experiment. It is concluded that, at least, the phenomenon of the abnormal increase in the viscosity may not be a peculiar property of molten silicon. As mentioned above, the results obtained with the aid of Al2O3, Si3N4, BN, SiO2 and 8 mol%YSZ crucibles were almost the same within the uncertainty of the measurement, and also the reproducibility was sufficiently good. The reliable viscosity of molten silicon can be derived based on the present results. Fig. 8 shows the results obtained except by SiC and graphite, and the regression line calculated by using all the values for Al2O3, Si3N4, BN, SiO2 and 8 mol%-YSZ crucibles. The recommended viscosity of molten silicon and its activation energy are shown in Eqs. (12) and (13), respectively: log Z=mPa s ¼ 0:727 þ 819=T;
ð12Þ
EZ ¼ 15:7 kJ mol1 :
ð13Þ
The properties of molten silicon are low viscosity and low activation energy for viscous flow in spite of its high melting point although there is a tendency, in general, that a metal with a
413
higher melting point shows higher viscosity and activation energy. The viscosity of 0.575 mPa s at the melting point of 1683 K is surprisingly one order lower than that of molten iron, 5.9 mPa s, which is the author’s data, which will be published, at the melting point of 1811 K. This suggests that the mechanism of viscous flow of the molten semiconductors is different from those of conventional molten metals although the molten semiconductors were assumed to be metallic. Indeed, not only molten silicon but also the molten germanium [12], molten InSb and GaSb [13] show low viscosities. It is well known that the viscosity is closely related to the free volume or the excess volume in the melt. The semiconductors including III–V compounds such as antimonides, in general, show a decrease in volumes on melting. This is due to the breaking of the diamond-type solid structure, which is not a close-packed structure, and the melts form the nearly metallic-like melts. However, it is considered that the melts still keep a looser structure than molten metals; for example, molten silicon has almost 40% larger atomic volume than that of molten iron at the melting point although the volume of silicon decreases by about 10% on melting.
Temperature, T / K 1850
1800
1750
1700
1650 0.62
log (Viscosity, / mPa.s
Average -0.22
Alumina Silicon Nitride
-0.24
Quartz 8mol% YSZ
0.60
Boron Nitride
0.58 0.56
-0.26 0.54
↑
-0.28
Melting point -0.30
-0.32 0.52
Viscosity, / mPa.s
-0.20
1900
0.52
0.50
0.53
0.54
0.55
0.56
0.57
0.58
0.59
0.60
0.48 0.61
T -1/10-3K-1 Fig. 8. Recommended average viscosity regressed for Al2O3, Si3N4, BN, SiO2 and 8 mol%-YSZ crucibles.
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The result of X-ray diffraction on molten silicon [14] demonstrated that a very minor structural change occurred on melting based on the coordination number determination. The first and second coordination numbers were 3.7 and 2.0, respectively at 1733 K and showed almost no temperature dependence between 1693 and 1768 K. These values are not so different from the solid, 4 and 2, respectively. A similar result was obtained for molten germanium where they were 3.95 and 1.97 at 1253 K [15]. It is considered that the molten semiconductors keep looser structure than the metallic melts. This may be a reason for the molten semiconductors to show lower viscosity than the molten metals. The activation energy obtained, 15.7 kJ mol1 is comparable to that of molten zinc or aluminum and considerably smaller than that of the metal with high melting temperature, for example, about one-third of molten iron from the author’s data. By considering a rough tendency for molten metals that the activation energy increases by an increase of the melting point, the value of molten silicon is considerably low in spite of the high melting temperature. The diffusion coefficient of molten silicon at the melting point estimated theoretically is several times higher than for other molten metals, not only low melting point metals but also silver and copper [16]. The activation energy for the diffusion of molten silicon also estimated is similar to molten zinc and lower than molten silver. Thus the viscosity and the diffusion coefficient show similar behavior and the reason is considered to be due to the liquid structure of molten silicon which is similar to that of the solid in short range and looser than conventional molten metals.
6. Conclusion The viscosity of molten silicon has been measured up to 1891 K including the supercooled region with the aid of the modified high temperature oscillating viscometer with various materials of crucible which were long enough melt heights in the crucible for neglecting the meniscus effect and the furnace with excellent temperature uniformity.
All the results showed a good Arrhenian behavior over the entire temperature range and no abnormal increase in the viscosity was found near the melting point. Furthermore, it was found that the viscosity measurement was almost not affected by the material of the crucibles except SiC and graphite. The recommended value of the viscosity is presented based on the results in this study. Molten silicon showed very low viscosity and also low activation energy compared to conventional molten metals. It is considered that the feature is due to the loose melt structure originated from and similar to the solid structure in the short range.
Acknowledgements This work is the results of ‘‘Technology for Production of High Quality Crystal’’ which is supported by the New Energy and Industrial Technology Development Organization (NEDO) through the Japan Space Utilization Promotion Center (JSUP) in the program of the Ministry of Economy, Trade and Industry (METI). The authors also thank SUMITOMO SITIX Co. who kindly supplied the single crystal of semiconductor silicon.
References [1] V.M. Glazov, S.N. Chizhevskaya, N.N. Glagoleva, Liquid Semiconductors, Plenum Press, New York, 1969, p. 55. [2] K. Kakimoto, M. Eguchi, H. Watanabe, T. Hibiya, J. Crystal Growth 94 (1989) 412. [3] H. Sasaki, E. Tokizaki, X-M. Huang, K. Terashima, S. Kimura, Jpn. J. Appl. Phys. 34 (1995) 3432. [4] T. Ejima, Y. Sato, S. Yaegashi, T. Kijima, E. Takeuchi, K. Tamai, J. Jpn. Inst. Met. 51 (1987) 328. [5] K. Mukai, Z. Yuan, Mater. Trans. JIM 41 (2000) 338. [6] R. Roscoe, Proc. Phys. Soc. London 72 (1958) 576. [7] Y. Sato, T. Yamamura, Kinzoku 70 (2000) 577. [8] R.C. Weast, M.J. Astle, CRC Handbook of Chemistry and Physics, 61st Edition, CRC Press, Florida, 1981, pp. F-51–F-55. [9] G.J. Janz, Final Report of Molten Salts Standards Program Project, Rensselaer Polytechnic Institute, New York, 1979, p. 97.
Y. Sato et al. / Journal of Crystal Growth 249 (2003) 404–415 [10] T. Ejima, K. Shimakage, Y. Sato, H. Okuda, N. Kumada, A. Ishigaki, J. Chem. Soc. Jpn. (1982, No. 6) 961. [11] Y. Sato, T. Nishizuka, K. Hara, T. Yamamura, Y. Waseda, Int. J. Thermophys. 21 (2000) 1463. [12] Y. Sato, T. Nishizuka, T. Tachikawa, M. Hoshi, T. Yamamura, Y. Waseda, High Temp. High Press. 32 (2000) 253.
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[13] Y. Sato, T. Nishizuka, T. Takamizawa, T. Yamamura, Y. Waseda, Int. J. Thermophys. 23 (2002) 235. [14] S. Takeda, Jpn. J. Appl. Phys. 34 (1995) 4889. [15] V. Petkov, S. Takeda, Y. Waseda, K. Sugiyama, J. NonCryst. Solids 168 (1994) 97. [16] P.D. Mitev, M. Saito, Y. Waseda, Y. Sato, High Temp. Mater. Processes 19 (2000) 307.
Viscosity of molten silicon and the factors affecting measurement Yuzuru Satoa,*, Yuichi Kamedaa, Toru Nagasawaa, Takashi Sakamotoa, Shinpei Moriguchia, Tsutomu Yamamuraa, Yoshio Wasedab a
Department of Metallurgy, Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan b Institute for Advanced Materials Processing, Tohoku University, Sendai 980-8577, Japan Received 24 July 2002; accepted 28 October 2002 Communicated by T. Hibiya
Abstract Viscosity of molten silicon has been measured to study the behavior, especially around the melting point, in order to supply useful information for the crystal growth of semiconductors. The temperature range was taken as wide as possible from 1891 K down to the supercooled region by using an oscillating viscometer with various materials of crucible to study the effect of the materials. It has been clear that the viscosity of molten silicon showed a good Arrhenian behavior in the entire temperature range including the supercooled region and no abnormal increase around the melting point. The viscosity values were lower than the reported values and the maximum difference reached 50% even in the temperatures sufficiently higher than the melting point. Furthermore, almost no effect on the measurement by the materials of crucible was found. The recommended viscosity is presented by the following equation based on the present results: log Z=mPa s ¼ 0:727 þ 819=T; EZ ¼ 15:7 kJ mol1 : The viscosity and the activation energy were considerably lower than those of other metals with high melting points and the reason was considered to be due to the looser structure of the molten silicon originated from diamond-type solid structure. r 2002 Elsevier Science B.V. All rights reserved. Keywords: A1. Viscosity; A2. Growth from melt; B1. Silicon; B2. Semiconducting silicon
1. Introduction Since silicon is the most important semiconductor material for highly developed information technology and silicon wafer with larger size such as 12 in is now being developed, it is required to simulate the process of crystal growth in the CZ process by means of a numerical model for *Corresponding author. Fax: +81-22-217-7310. E-mail address: [email protected] (Y. Sato).
producing high quality crystal and establishing effective conditions for the production. Therefore, the reliable values of many physicochemical properties of solid and molten silicon are necessary. However, the reliable physicochemical properties, especially in the molten state, are not well known because of experimental difficulties due to the high temperature and the high chemical activity. The viscosity values in the literatures [1–3], which were measured by using also the oscillating viscometer and the crucible made of
0022-0248/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 2 ) 0 2 1 5 3 - X
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corundum (Al2O3) [1], pyrolytic boron nitride (PBN) [2], and SiC and PBN [3], showed a big discrepancy among them and abnormal increases near the melting point. Furthermore, different values depending on the crucible materials were reported [3]. Such an abnormal behavior has not been found for most of the molten metals and was sometimes considered to be the characteristic feature of molten silicon. In this work, it was planned to obtain the reliable values for viscosity of molten silicon by clarifying the behavior near the melting point, to check the effect of crucible materials and to supply the recommended value for the simulation of the crystal growth.
2. Experimental procedure 2.1. Apparatus Although many methods were developed for viscosity measurements, the application to molten silicon is actually limited to the oscillating method because of its low viscosity and high temperature. In this work, the oscillating viscometer [4] was improved in order to adept to the high temperature, for example, by employing a furnace with excellent uniformity of the temperature profile, by replacing some components refractory materials and by enhancing the cooling system. Various crucibles were used to study the effect of materials on the measurement. The viscometer used in this work consists of a suspension system, an oscillation detection system and a heating system. The suspension and detection system inside the apparatus is shown in Fig. 1. A crucible connected through a tungsten rod, an inertia disk and a mirror block are suspended by a thin torsion wire made of a platinum alloy. The mirror block reflects incident light from a He–Ne laser and the light is focused as a small spot of about 0.1 mm diameter on the detector which consists of three photo-transistors. Then four time intervals between the times when the locus passes through the detector are determined by using a computer with a resolution of 0.1 ms (104s). Rotational force is given to the inertia disk to start the oscillation
Fig. 1. Suspension and detection system in the oscillating viscometer.
electromagnetically with the aid of the coils located above and under the disk. The reproducibility of the time measurement was excellent, for example, the standard deviation of the period of oscillation was typically less than 0.01% even in measurements at high temperatures. The period of the oscillation and the logarithmic decrement were determined from the four time intervals by means of an approximation algorithm described later. Fig. 2 shows the whole apparatus schematically. The torsion wire was kept at 308.270.1 K to maintain the constant rigidity by circulating thermostated water in the copper tube wound around the wire housing. The heating system is also very important because the crucible which is about 85 mm long must be heated uniformly to prevent the convection flow in the crucible and to determine the temperature precisely. The furnace consists of three spiral MoSi2 heaters controlled independently. Additionally, about 25 reflection plates of molybdenum were installed in an outer alumina tube to reduce the vertical thermal radiation loss. The temperature profile was checked and the furnace controller was fine-tuned
Y. Sato et al. / Journal of Crystal Growth 249 (2003) 404–415
406 1 2 3
He inlet
4
5: 6: 7: 8:
5 6 7 8 9
1: 2: 3: 4:
Cooling Water
10 11 12
9: 10: 11: 12: 13: 14: 15:
13
Head corn Gas inlet Torsion wire Water circulation tube Reflection mirror Window Inertia disk Oscillation initiator Water jacket W rod Mo plate Crucible Three divided furnace Zr sponge Thermocouple
14
Cooling Water
9
15
Fig. 2. Schematic diagram of the oscillating viscometer.
carefully during each measurement to obtain the best temperature uniformity. As a result, uniform temperature profile within 0.5 K in the whole length of the crucible was obtained while the temperature of the upper part was set to be slightly higher than that of the lower part. The atmosphere was helium with a purity higher than 99.99%, which is inert chemically and has low viscosity. Convection flow of the atmosphere around the crucible makes the oscillation unstable. Small but irregular movement of the axis of rotation from proper position was sometimes observed in the case of other gases such as argon. However, such a phenomenon was not found for helium and a very stable oscillation was found though other effects by the convection may remain. Therefore, the change of logarithmic decrement and the oscillation at elevated temperatures were measured for the empty crucible as described later. A zirconium sponge was placed under the crucible to absorb the remaining oxygen in the atmosphere.
As mentioned above, Sasaki et al. [3] reported lower viscosity of molten silicon for the PBN crucible than that for the SiC crucible. Sasaki et al. concluded that the reason of lower value was the slipping at the interface between molten silicon and PBN due to the poor wettability between them. However, this conclusion is not likely because the motion of the liquid in the crucible is very slow and an adhesive force such as Van der Waals force is considered to be enough for adhesion. Therefore, it is necessary to study the effect of the material experimentally. Mukai et al. [5] reported that the wettability with molten silicon is very good for SiC, poor for BN and intermediate for SiO2, Si3N4, and Al2O3, respectively. By considering the wettability and the chemical thermodynamic stability against molten silicon, and also the mechanical strength at high temperature, silicon carbide (SiC), graphite (C), highly sintered alumina (Al2O3), silicon nitride (Si3N4), hot pressed boron nitride (BN), quartz (SiO2) and 8 mol% yittria stabilized zirconia (8 mol%-YSZ) were chosen as the crucible materials in this work. Graphite is considered to be in the same group as SiC because it reacts with molten silicon and makes SiC on the surface, and 8 mol%-YSZ is expected to have a good wettability. All the crucibles were specially ordered ones and were precisely machined to obtain the uniform inner diameter of 20 mm whose precision was within 0.05 mm. The shape of the crucible is shown in Fig. 3. The cap has two female screws for connecting with both a crucible body and a tungsten rod with the male screw at its end. 2.2. Determination of logarithmic decrement and period of oscillation In this work, the viscosity is determined based on the absolute method by using Roscoe’s equation [6] in which only physical parameters are required for the calculation. Those are the logarithmic decrement, the period of oscillation, the moment of inertia of the suspension system, the inner radius of the crucible and the density of the melt. The logarithmic decrement and the period of oscillation were obtained from the measurement of time intervals as follows:
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Fig. 4. Oscillating curve with decay and the time intervals measured in a period.
expressed by Eq. (4). Therefore, an apparent period of the oscillation, t0 is given in Eq. (5):
Fig. 3. Schematic diagram of a typical crucible.
Y ¼ A sin ot þ C;
ð4Þ
t0 ¼ t1 þ t2 þ t3 þ t4 :
ð5Þ
The constants A and C in Eq. (4) are defined by Eqs. (6) and (7), respectively for each period:
An oscillatory curve with decay expressed by Eq. (1) is shown in Fig. 4. L and –L are the positions of the detectors. C is the deviation of the center of oscillation from the center of detectors. The period, t and the logarithmic decrement of oscillation, d are defined by Eqs. (2) and (3): Y ¼ A expðBtÞ sinot þ C;
ð1Þ
t ¼ 2p=o;
ð2Þ
d ¼ 2pB=oBt;
ð3Þ
where o is the angular frequency, A and B are constants related to the amplitude and the decrement, respectively. Four time intervals t1 to t4 measured in a period were used for determining t and d. If C equals 0, which is an insufficient assumption, analytical solutions can be obtained for t and d. However, it is difficult to calculate them analytically in the case of non-zero C. Therefore, the present authors have developed the following approximation method. At first, oscillatory curve without decay is assumed as
A¼ 2L=½sinfpð0:5 t1 =t0 Þg sinfpð1:5 t3 =t0 Þg ;
ð6Þ
C¼ A½sinfpð0:5 t1 =t0 Þg þ sinfpð1:5 t3 =t0 Þg : ð7Þ If it is assumed that the shape of oscillation keeps a regular sine wave without decay in the ith period and the amplitude decreases with keeping also a regular sine wave in the next (i+1)th period, an apparent constant B0 is calculated using Eq. (8) using a series of Ai which are the amplitudes obtained for sequential ith periods. Also, an apparent logarithmic decrement, d0 is given by Eq. (9): B0 ¼ ð1=tÞlnðAi =Ai þ 1Þ;
ð8Þ
d0 ¼ B0 t0 :
ð9Þ 0
This set of first approximate values, d and t0 includes considerable error depending on the logarithmic decrement. The error was estimated by using a sample data of the time intervals which were generated through given parameters: fixed C
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Fig. 5. Oscillating curve with decay for the wave analysis.
of 10 mm, 2L of 100 mm and various B. This condition is relatively worse than that of the real measurement. The deviations of the first approximate values, d0 and t0 from true values were from about 101.5% to 100.5% for the practical range of logarithmic decrement in the measurement. They are only available when the decay is unpractically small. The effect of decay on the oscillation must be taken into account in the next step. The oscillatory curve with decay and the relationship between the apparent periods given by Eq. (5) and the true period is shown in Fig. 5(a). An apparent period, t0 is always longer than a true period, t. The true period can be determined if the difference (t0 t) is estimated precisely. Fig. 5(b) shows the overlapped consecutive half period with decay. The curve is apparently a distorted sine wave and is vertically similar to the consecutive wave. The difference (t0 t) is apparently equal to ta from Fig. 5(a) although ta is not measurable. However, the following relationship in Eq. (10) is helpful: t1;i t1;iþ1 ¼ ta þ tb :
ð10Þ
ta can be determined if the measurable left-hand term, (t1,it1,i+1), is reasonably divided into ta and tb : The times, at which differential coefficient of a wave equals zero, are the same for any period as shown in Fig. 5(b). Therefore, the left-hand term in Eq. (10) was divided at the time mentioned
above to distribute the left-hand term into ta and tb by assuming the ratio ta =tb to be equal to the ratio of divided parts. As a result, a second approximate value of t00 is obtained as in Eq. (11): t00 ¼ t0 ta ¼ t0 ðt1;i t1;iþ1 Þ tan1 ðo=BÞ=p;
ð11Þ
where the parameter t0 is taken from Eq. (5) and o=B ¼ 2pXdis taken from d0 in Eq. (9) which are the first approximate values. The latest period, t00 is used to calculate the second approximate value of logarithmic decrement through Eqs. (6)–(9) again. The second approximate values of the period of oscillation and the logarithmic decrement were sufficiently precise. The error accompanied by the approximation was from about 104.5% to 103.5% in the same conditions above. Therefore, this set of second approximate values was employed for determining the viscosity.
3. Method of viscosity determination Viscosity is determined by means of some physical parameters as mentioned above. The moment of inertia of the suspension system for an empty crucible was determined prior to the viscosity measurement based on the period of
Y. Sato et al. / Journal of Crystal Growth 249 (2003) 404–415
oscillation using an additional inertia disk with known moment of inertia. At elevated temperatures, the moment of inertia and the logarithmic decrement increase mainly due to the thermal expansion of the crucible and the increase in viscosity of helium as the atmosphere, respectively. Therefore, individual empty crucibles were calibrated at various temperatures up to about 1800 K to obtain the changes of these values as a function of temperature. This calibration is useful to compensate the effects by not only the above two factors but also by unknown factors such as the convection flow of helium around the crucible. As the shape and the dimension of the crucible is also very important because most algorithms for calculating viscosity using a cylindrical crucible including Roscoe’s equation do not take the effect of the bottom and free surface into account, the effect should be studied experimentally. The present authors measured the above physical parameters and calculated the viscosity of molten tin for the different ratio of height to radius of liquid column in the crucible, h/r by feeding different amounts of the sample [7]. The apparent viscosity calculated was larger in the lower h/r region and decreased with an increase of h/r. The apparent viscosity converged to an almost constant value in which h=r is higher than six. The apparent value for h=r ¼ 3 was 6–8% larger than the converged value. Namely, it is concluded that a considerably long crucible is required for the reliable measurement. In this work, at least seven of the ratios were achieved by employing the dimensions of 20 mm in the inner diameter and 80 mm in the depth of the crucible as shown in Fig. 3. The crucibles used in previous works [2,3] were too short, may be less than five of the h=r value. For actual measurement, the expansion coefficients for the materials of the crucible were taken from the maker’s technical data and the radius of crucible was corrected at high temperatures. The above method gave good results. The results on the distilled water and mercury at room temperature have agreed with the literature values [8] within 1%. Furthermore, the results on the molten NaCl also has agreed with the recommended value by Janz [9] and the authors’
409
previous result by using a capillary viscometer [10] within 2–3%. Silicon sample was quarried out as a cylindrical shape from a semiconductor grade single crystal made by the CZ method. The purity was almost 8 N.
4. Results All the experiments were carried out both in heating and cooling processes. The measurement was started from a temperature about 20 K higher than melting point of silicon and heated up to highest temperature depending on the crucible, and then cooled down until the solidification was confirmed to check the reproducibility of the measurement. The density of molten silicon, which is the most important and unmeasurable parameter in this work, was taken from the author’s previous work [11]. Fig. 6 shows the viscosities obtained for SiC, graphite, Al2O3, Si3N4, BN, SiO2 and 8 mol%-YSZ as the crucible materials together with the literature values. The viscosities reported by Grazov et al. [1] for corundum crucible and Kakimoto et al. [2] for PBN crucible are relatively high and show non-Arrhenian behavior in the entire temperature range. The result by Sasaki et al. [3] for SiC crucible is the highest and that for PBN crucible is considerably low. Furthermore, a common feature in the literatures is the abnormal increase of the viscosity near the melting point of silicon. In the present work, the maximum temperature was 1891 K, and the measurements could be carried out to the temperatures lower than the melting point of silicon in some cases which was the supercooled temperature region. However, no abnormal increase in the viscosity near the melting point was found in all the measurements and most of them showed a good Arrhenian behavior in the entire temperature range. The present results show very similar viscosities except for graphite and SiC crucibles. The results for graphite crucible showed a poor reproducibility and larger temperature dependence, and that for SiC crucibles showed higher viscosity and larger scatter than for the others.
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410
Temperature, T / K
0.10
log (Viscosity, / mPa.s
0.00
1850
1800
1750
1700
1650 1.50
Glazov (ref.1) Kakimoto (ref.2) Sasaki-SiC (ref.3) Sasaki-PBN(ref.3) Graphite Silicon Carbide Alumina Silicon Nitride Boron Nitride Quartz 8mol% YSZ
1.40 1.30 1.20 1.10 1.00 0.90
-0.10
0.80
0.70
Viscosity, / mPa.s
0.20
1900
-0.20 0.60
-0.30
0.50
↑ Melting point -0.40 0.52
0.54
0.56
0.58
0.60
0.40 0.62
T -1/10-3K-1 Fig. 6. Viscosities of molten silicon obtained for various materials of crucibles with literature values.
It is clear that the viscosities obtained in the present work for Al2O3, Si3N4, BN, SiO2 and 8 mol%-YSZ crucibles are concentrated in the narrow range of about 3% and show almost the same activation energy. On the other hand, the results for SiC and graphite crucibles are different from others although the reason why such a difference appeared is still unclear. Although the measurements were typically carried out 2 or 3 times to confirm the reproducibility for each crucible except SiO2 and 8 mol%-YSZ, the results with the widest temperature range for each material are shown in the figure. The reproducibility for SiC and graphite was poorer in contrast with good reproducibility for BN, Al2O3 and Si3N4 within 1% of standard deviation for the scatter of data points independent from the heating and cooling processes. The numerical results obtained by using Al2O3, Si3N4, BN,
SiO2 and 8 mol%-YSZ crucibles are shown in Table 1. To study the uncertainty of the measurement, the error propagation was simulated by using the data obtained in this work for Al2O3 crucible. The input parameters used in Roscoe’s equation to calculate the viscosity are logarithmic decrement, period of oscillation, moment of inertia, inner radius of the crucible, total mass of the melt and density of the melt. The first two parameters are obtained from the experiment and others are separately given. The height of the melt is derived from given parameters. These parameters were independently deviated from actual input data and compared with actually obtained viscosity as shown in Table 2. The manner of the error propagation was made clear and the quantitative evaluation was possible. The reproducibility of the logarithmic decrement and the period of
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Table 1 Viscosity values obtained by using various materials of crucible Alumina
Silicon nitride
Boron nitride
Quartz
8 mol%-YSZ
T (K)
Z (mPa s)
T (K)
Z (mPa s)
T (K)
Z (mPa s)
T (K)
Z (mPa s)
T (K)
Z (mPa s)
1877.9 1868.8 1858.7 1849.8 1838.6 1829.8 1820.2 1810.6 1799.5 1791.2 1781.3 1761.2 1752.5 1742.8 1722.9 1713.2 1703.6 1689.5 1684.6 1675.0 1669.7 1665.6 1660.9 1655.6
0.5100 0.5136 0.5191 0.5200 0.5196 0.5259 0.5260 0.5327 0.5328 0.5292 0.5418 0.5453 0.5488 0.5565 0.5555 0.5623 0.5774 0.5647 0.5591 0.5756 0.5780 0.5815 0.5831 0.5865
1856.7 1844.7 1828.2 1818.2 1798.9 1789.5 1770.5 1760.6 1741.6 1731.9 1713.5 1704.2 1694.3 1684.9
0.5284 0.5285 0.5318 0.5344 0.5411 0.5477 0.5562 0.5567 0.5658 0.5675 0.5729 0.5789 0.5817 0.5852
1856.0 1839.6 1820.2 1800.8 1785.7 1773.0 1767.1 1752.2 1745.8 1732.8 1724.1 1720.0 1713.2 1705.3 1694.3 1688.3 1684.9 1679.8
0.5220 0.5264 0.5342 0.5352 0.5466 0.5450 0.5466 0.5583 0.5568 0.5585 0.5685 0.5702 0.5664 0.5729 0.5820 0.5826 0.5840 0.5849
1790.2 1786.4 1780.6 1776.2 1771.2 1766.2 1761.5 1756.5 1752.5 1746.4 1742.2 1736.4 1732.5 1727.7 1722.7 1717.6 1714.1 1707.4 1703.9 1698.7 1694.6 1688.3 1683.2 1678.4 1673.6 1668.9 1664.2
0.5323 0.5344 0.5360 0.5357 0.5400 0.5396 0.5421 0.5420 0.5473 0.5464 0.5512 0.5473 0.5532 0.5493 0.5594 0.5579 0.5613 0.5586 0.5645 0.5637 0.5673 0.5694 0.5689 0.5711 0.5768 0.5727 0.5768
1891.8 1873.0 1851.9 1832.8 1794.0 1737.9
0.5021 0.5074 0.5148 0.5188 0.5284 0.5579
Table 2 Error of viscosity caused by the deviation of physical parameters used for the calculation of the viscosity Deviation of parameter
+0.1%
+0.3%
+1.0%
Logarithmic decrement Period of oscillation Moment of inertia Radius of crucible Mass of the melt Density of the melt
0.44% +0.18% 0.36% +0.44% +0.36% 0.14%
1.35% +0.52% 1.11% +1.31% +1.07% 0.44%
4.45% +1.71% 3.70% +4.33% +3.58% 1.57%
oscillation were 0.1–0.2% and 0.002–0.005%, respectively. The uncertainty in the moment of inertia and in the radius of crucible at high temperature were estimated to be less than 0.2% for both. The change of the mass between after
and before the measurement was less than 0.02%. In general, it is difficult to estimate the error contained in the density because it depends on the separate experiment. Although the density used in this work was obtained by the authors [11] and the scatter in the data was lesser than 0.1%, but unknown factors might affect the data and the uncertainty was estimated to be less than 0.3%. The error caused by the individual factor does not exceed 1%. As the result, the total error may be about 1% by considering the theory of statistics. This value is excellent for the measurement at high temperature. However, the total error including the error caused by unknown factors should be larger although very careful efforts were paid to reduce the effects of the factors. The maximum discrepancy in viscosities obtained in this work shown in Fig. 6, except for graphite and SiC
Y. Sato et al. / Journal of Crystal Growth 249 (2003) 404–415
crucible, is about 3% and also 2–3% of difference was found between the results of molten NaCl by the oscillating method and the capillary method mentioned above. So it is considered that the uncertainty accompanied by the measurement at high temperature is estimated to be less than 3%. It is extremely small compared with the discrepancy found in the literature values.
5. Discussion As mentioned above, the problems previously pointed out on the viscosity of molten silicon are abnormal temperature dependence near the melting point and the effect of the material of crucible on the measurement. In the present work, various refractory materials with different wettability with molten silicon were used as the crucible materials to study the effect of the materials. The results indicated no difference among the materials with poor and good wettability against the molten silicon. After the experiments, the interface between the solidified silicon and the crucible wall was observed visually. The silicon has adhered strongly to the SiC, graphite, SiO2, Si3N4 and 8 mol%-YSZ. The contact angle was nearly 901 for SiO2 and was lesser than for others. These materials were decided to be well wettable to molten silicon. On the other hand, solidified silicon was easily removed from Al2O3 and BN and the contact angles were larger than 901. In this case, wettability was decided to be poor. Therefore, it is concluded that the wettability almost does not affect the viscosity measurement as predicted by taking almost the same viscosity values obtained except SiC and graphite into account. This suggests that there is no slipping between crucible wall and molten silicon. However, the wettability should affect the measurement due to the different shape of meniscus of molten silicon. This effect was probably minimized by using the long crucible with large h/r value. On the abnormal behavior of the viscosity near the melting point, such a phenomenon was not observed in all the measurements in this study although the viscosity measurement was carried
out both above and below the melting point. The lowest temperature in the present work was 1656 K, which was almost 30 K lower than the melting point of silicon, with the Al2O3 crucible. This supercooling occurred due to probably excellent temperature uniformity. At this temperature, interesting phenomena were observed as it happened during the measurement. Sudden increases of the temperature and the logarithmic decrement occurred. Temperature rose above 1670 K in a few seconds due to the latent heat of solidification. The reason why the temperature was still lower than the melting point was that the thermocouple was kept away from the crucible as shown in Fig. 2. Simultaneously, logarithmic decrement also increased rapidly almost twice as demonstrated in Fig. 7. After that, the temperature decreased slowly until the melt solidified completely and the logarithmic decrement also decreased to a very small value which corresponds to the solid. The twice value of the logarithmic decrement means almost one order large in the viscosity. These phenomena suggest that the larger value in the viscosity can be obtained if the melt solidified partially near the melting point. If the melt solidifies partially and a liquid–solid mixture like slurry is formed in the crucible, it makes the apparent viscosity larger. Partial solidification may occur if the temperature profile around the crucible is not sufficiently good. In this work, a
0.030
0.025
logarithmic decrement
412
0.020
0.015
0.010
0.005
0.000
0
5
10
15
20
25
30
35
40
Time, t / min
Fig. 7. Change in the logarithmic decrement due to the rapid solidification during the measurement of molten silicon supercooled at 1655.6 K.
Y. Sato et al. / Journal of Crystal Growth 249 (2003) 404–415
very good temperature profile was obtained by carefully performing the experiment. It is concluded that, at least, the phenomenon of the abnormal increase in the viscosity may not be a peculiar property of molten silicon. As mentioned above, the results obtained with the aid of Al2O3, Si3N4, BN, SiO2 and 8 mol%YSZ crucibles were almost the same within the uncertainty of the measurement, and also the reproducibility was sufficiently good. The reliable viscosity of molten silicon can be derived based on the present results. Fig. 8 shows the results obtained except by SiC and graphite, and the regression line calculated by using all the values for Al2O3, Si3N4, BN, SiO2 and 8 mol%-YSZ crucibles. The recommended viscosity of molten silicon and its activation energy are shown in Eqs. (12) and (13), respectively: log Z=mPa s ¼ 0:727 þ 819=T;
ð12Þ
EZ ¼ 15:7 kJ mol1 :
ð13Þ
The properties of molten silicon are low viscosity and low activation energy for viscous flow in spite of its high melting point although there is a tendency, in general, that a metal with a
413
higher melting point shows higher viscosity and activation energy. The viscosity of 0.575 mPa s at the melting point of 1683 K is surprisingly one order lower than that of molten iron, 5.9 mPa s, which is the author’s data, which will be published, at the melting point of 1811 K. This suggests that the mechanism of viscous flow of the molten semiconductors is different from those of conventional molten metals although the molten semiconductors were assumed to be metallic. Indeed, not only molten silicon but also the molten germanium [12], molten InSb and GaSb [13] show low viscosities. It is well known that the viscosity is closely related to the free volume or the excess volume in the melt. The semiconductors including III–V compounds such as antimonides, in general, show a decrease in volumes on melting. This is due to the breaking of the diamond-type solid structure, which is not a close-packed structure, and the melts form the nearly metallic-like melts. However, it is considered that the melts still keep a looser structure than molten metals; for example, molten silicon has almost 40% larger atomic volume than that of molten iron at the melting point although the volume of silicon decreases by about 10% on melting.
Temperature, T / K 1850
1800
1750
1700
1650 0.62
log (Viscosity, / mPa.s
Average -0.22
Alumina Silicon Nitride
-0.24
Quartz 8mol% YSZ
0.60
Boron Nitride
0.58 0.56
-0.26 0.54
↑
-0.28
Melting point -0.30
-0.32 0.52
Viscosity, / mPa.s
-0.20
1900
0.52
0.50
0.53
0.54
0.55
0.56
0.57
0.58
0.59
0.60
0.48 0.61
T -1/10-3K-1 Fig. 8. Recommended average viscosity regressed for Al2O3, Si3N4, BN, SiO2 and 8 mol%-YSZ crucibles.
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Y. Sato et al. / Journal of Crystal Growth 249 (2003) 404–415
The result of X-ray diffraction on molten silicon [14] demonstrated that a very minor structural change occurred on melting based on the coordination number determination. The first and second coordination numbers were 3.7 and 2.0, respectively at 1733 K and showed almost no temperature dependence between 1693 and 1768 K. These values are not so different from the solid, 4 and 2, respectively. A similar result was obtained for molten germanium where they were 3.95 and 1.97 at 1253 K [15]. It is considered that the molten semiconductors keep looser structure than the metallic melts. This may be a reason for the molten semiconductors to show lower viscosity than the molten metals. The activation energy obtained, 15.7 kJ mol1 is comparable to that of molten zinc or aluminum and considerably smaller than that of the metal with high melting temperature, for example, about one-third of molten iron from the author’s data. By considering a rough tendency for molten metals that the activation energy increases by an increase of the melting point, the value of molten silicon is considerably low in spite of the high melting temperature. The diffusion coefficient of molten silicon at the melting point estimated theoretically is several times higher than for other molten metals, not only low melting point metals but also silver and copper [16]. The activation energy for the diffusion of molten silicon also estimated is similar to molten zinc and lower than molten silver. Thus the viscosity and the diffusion coefficient show similar behavior and the reason is considered to be due to the liquid structure of molten silicon which is similar to that of the solid in short range and looser than conventional molten metals.
6. Conclusion The viscosity of molten silicon has been measured up to 1891 K including the supercooled region with the aid of the modified high temperature oscillating viscometer with various materials of crucible which were long enough melt heights in the crucible for neglecting the meniscus effect and the furnace with excellent temperature uniformity.
All the results showed a good Arrhenian behavior over the entire temperature range and no abnormal increase in the viscosity was found near the melting point. Furthermore, it was found that the viscosity measurement was almost not affected by the material of the crucibles except SiC and graphite. The recommended value of the viscosity is presented based on the results in this study. Molten silicon showed very low viscosity and also low activation energy compared to conventional molten metals. It is considered that the feature is due to the loose melt structure originated from and similar to the solid structure in the short range.
Acknowledgements This work is the results of ‘‘Technology for Production of High Quality Crystal’’ which is supported by the New Energy and Industrial Technology Development Organization (NEDO) through the Japan Space Utilization Promotion Center (JSUP) in the program of the Ministry of Economy, Trade and Industry (METI). The authors also thank SUMITOMO SITIX Co. who kindly supplied the single crystal of semiconductor silicon.
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