Surface Tension and Its Temperature Coefficient of Molten Silicon at

In the case of PO2 e PO2,sat (the saturated oxygen partial pressure in the ... increases with the oxygen partial pressure, and the molten silicon drop...
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Langmuir 2002, 18, 2054-2062

Surface Tension and Its Temperature Coefficient of Molten Silicon at Different Oxygen Potentials Zhang Fu Yuan,*,† Kusuhiro Mukai,‡ and Wen Lai Huang† Multi-Phase Reaction Laboratory, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China, and Department of Materials Science and Engineering, Faculty of Engineering, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan Received August 14, 2001. In Final Form: December 13, 2001 Previous investigations on the effects of temperature and impurities on the surface tension of molten silicon, and relevant measurement methods have been reviewed, and the influence of oxygen partial pressure (PO2) in the atmosphere has been analyzed emphatically, on the basis of the results obtained by the sessile drop method and calculations. In the case of PO2 e PO2,sat (the saturated oxygen partial pressure in the Si(l)-O2(g)-SiO2(s) system), the surface tension first remains almost constant and then decreases remarkably with the increase of PO2, the temperature coefficient of surface tension (∂σ/∂T) is negative and increases with the oxygen partial pressure, and the molten silicon drop is very sensitive to outside vibrations. However, in the case of PO2 > PO2,sat, the surface tension increases slightly with the oxygen partial pressure, ∂σ/∂T is higher and also increases with PO2, the molten silicon drop is not influenced by environmental disturbances and remains stable, and EPMA (electron probe microanalyzer) analysis indicates the formation of a thin SiO2(s) film on the surface of the molten silicon drop which might account for the surface tension increase.

1. Introduction During crystal growth by the Czochralski (CZ) or floating zone (FZ) technique, oxygen might be dissociated from quartz crucibles into molten silicon through the melt convection. Fluid flows arising from the surface tension gradient are referred to as Marangoni convection. It is an important flow mode in normal gravity and dominates in low gravity where buoyancy convection is negligible. To achieve a sufficient understanding of the Marangoni convection, the relation between the surface tension and temperature, especially oxygen must be analyzed quantitatively. There exist a few reports on such aspects.1 The driving force for Marangoni flow is the surface tension gradient rather than surface tension itself,2 and thus the flow depends sensitively on the temperature coefficient of surface tension (K ) ∂σ/∂T). The present work aims at clarifying the oxygen or the oxygen partial pressure dependence of the surface tension of molten silicon and its temperature coefficient, quantitatively. The surface tension was measured with the sessile drop method, and the oxygen partial pressure has been controlled conveniently. 2. Experimental Section 2.1. Apparatus. The experimental apparatus consists of a furnace (LaCrO3 as heater), a gas purifier, an oxygen sensor, a photographic system, and a digital system (computer). To maintain the airtightness of the reaction chamber, a doubletube structure is adopted. The reaction tubes are made of highpurity alumina (99.8 mass % Al2O3). The outer tube is 50 mm in external diameter, 42 mm in internal diameter, and 700 mm in length, and the inner tube is 37 mm in external diameter, 30 mm in internal diameter, and 800 mm in length. Both ends of the reaction tube are sealed with water-cooled stainless caps. * Corresponding author. Telephone: 86-10-62554671. Fax: 8610-62527440. E-mail: [email protected]. † Chinese Academy of Sciences. ‡ Kyushu Institute of Technology. (1) Mukai, K.; Niu, Z. J. Jpn. Assoc. Cryst. Growth 1996, 23, 93. (2) Langlois, W. E. J. Cryst. Growth 1982, 56, 15.

The temperature is measured with a 20:40 Pt-Rh thermocouple, which is located directly under the boron nitride substrate. The maximum temperature for the furnace is 1773 K, and the temperature is controlled with a PID digital program controller. A schematic diagram of the apparatus can be seen in our previous work.3 The drop shape of molten silicon is acquired using a camera connected with a telephotographic lens by a bellows. The focal distance is kept constant in all the experiments to fix the magnification of the photographs. The focal distance is preset in the position of infinity, and the focus is adjusted by changing the distance between the molten drop and the camera fixed on a mechanical table, which can be moved three-dimensionally. The argon gas was passed through an Ar gas purifier and further deoxidized by magnesium chips heated at around 823 K. The oxygen partial pressure in argon gas, PO2, was controlled in the range PO2 e PO2,sat (the saturated oxygen partial pressure in the Si(l)-O2(g)-SiO2(s) system) with the aid of the gas deoxidized purifier and the double-tube structure of the furnace. PO2out (PO2 of Ar gas exhausted from the reaction chamber) and PO2in (PO2 of Ar gas before introduction into the heating furnace) were measured with an oxygen sensor of ZrO2-CaO solid electrolytes4 respectively by switching a three-way cock. 2.2. Procedure. The employed cylindrical CZ silicon (about 1.8 g, Si 99.999 mass %, O ≈ 10 mass ppm, B e 1 mass ppm, Sb e 20 mass ppm) was polished with sandpaper to remove oxides on the surface and cleaned with acetone using an ultrasonic automatic washer. The high-purity boron nitride (99.5 mass % BN) plate (25.7 mm × 25.7 mm × 2.5 mm) was used as a substrate after washing with ethanol and drying. The silicon sample was located on the upper surface of the BN substrate, and the horizontality of the substrate was adjusted by two water levels. After the sample was placed into the inner alumina reaction tube, the system was sealed and evacuated with a vacuum pump. Then, the argon gas was introduced into the reaction chamber and the evacuation was repeated three times. After that, the system was heated to experimental temperatures in an argon gas atmosphere. The total flow rate of argon was maintained at 0.16 L/min, and the total pressure was held constant at 0.1 MPa in all the experiments. (3) Mukai, K.; Yuan, Z. Mater. Trans., JIM 2000, 41, 331. (4) Yuan, Z.; Mukai, K.; Takagi, K.; Ohtaka, M. J. Jpn. Inst. Met. 2001, 65, 21.

10.1021/la0112920 CCC: $22.00 © 2002 American Chemical Society Published on Web 02/13/2002

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Photographs of the molten silicon drop were taken every 5 min. The measurement time was 60 min for one experimental run. To obtain the values in the equilibrium state, the results measured in the latter 30 min were adopted. According to Rotenberg’s method,5 the surface tension of molten silicon can be calculated from the image contour of the drop, by fitting the numerical solution to the classical Laplace equation with the experimentally measured points. The Laplace equation is usually adopted in the following form:

(

σ

)

sin φn 1 + ) 2σ/b + Fg(h - yn) Rn xn

(1)

where σ is the surface tension, Rn is the radius of the circle which has the same tangent line with the profile at (xn, yn), b and h are the curvature radius and height at the top of the drop, respectively, and φn is the normal angle of the profile at (xn, yn). The magnification was determined by the picture of a standard steel ball (10 mm in diameter) taken with the same camera and at the same focal distance, and the density value required for calculating the surface tension was derived from the report of Mukai and Yuan.6 The measurement error for surface tension is about (2%.3 After the measurement of surface tension, the surface of the solidified silicon samples (embedded in acrylic resin) was examined using an electron probe microanalyzer (JCXA-733, Japan) to investigate the oxygen distribution.

3. Background Review 3.1. Surface Tension of Molten Silicon. To comprehend deeply and simulate the Marangoni flow of molten silicon in CZ and FZ processes, it is primarily necessary to obtain reliable surface tension data, especially its temperature and impurity elements dependence, above all, oxygen dependence. Keene7 reviewed the studies on the surface tension of molten silicon based on 26 previous papers in 1987, and the main quoted results are shown with thin lines in Figure 1.8-22 Except for those of Kostikov et al.8 and Hardy,9 the absolute values of both surface tension and its temperature coefficient are lower than those obtained after Keene’s review (drawn with thick lines in Figure 1), (5) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169. (6) Mukai, K.; Yuan, Z. Mater. Trans., JIM 2000, 41, 323. (7) Keene, B. J. Surf. Interface Anal. 1987, 10, 367. (8) Kostikov, V. I.; Tarabanov, A. S. In Metod IssI i Svoistva Gran Raz Kontakt Faz, AN UKR SSR; Eremenko, V. N., Ed.; Naukova Dumka: Kiev, 1977; p 79. (9) Hardy, S. C. J. Cryst. Growth 1984, 69, 456. (10) Kingery, W. D.; Humenik, M., Jr. J. Phys. Chem. 1953, 57, 359. (11) Naidich, Y. V.; Perevertailo, V. M.; Obushchak, L. P. Russ. J. Phys. Chem. 1975, 49, 917. (12) Dzhemilev, K.; Popel, S. I.; Tsarevski, B. V. Phys. Met. 1964, 18, 77. (13) Popel, S. I.; Shergin, L. M.; Tsarevskii, B. V. Russ. J. Phys. Chem. 1970, 44, 144. (14) Levin, E. S.; Gel’d, P. V.; Baum, B. A. Russ. J. Phys. Chem. 1966, 40, 1455. (15) Shergin, L. M.; Popel, S. I.; Tsarevskii, B. V. Tr. Inst. Fiz. Met. (Akad. Nauk SSSR, Ural’. Nauchn. Tsentr.) 1971, 25, 52. (16) Nizhenko, V. I.; Smirnov, Y. I. In Fiz. Khim. Granits Razdela Kontakt Faz; Eremenko, V. N., Ed.; Naukova Dumka: Kiev, 1976; p 145. (17) Lukin, S. V.; Zhuchkov, V. I.; Vatolin, N. A.; Kozlov, Y. S. J. Less-Common Met. 1979, 67, 407. (18) Shashkov, Y. M.; Kolesnikova, T. P. Russ. J. Phys. Chem. 1963, 37, 747. (19) Tavadze, F. N.; Kekua, M. G.; Khantadze, D. V.; Tsverdvadze, T. G. In Poverkh Yavleniya Rasp; Eremenko, V. N., Ed.; Naukova Dumka: Kiev, 1968; p 159. (20) Khilya, G. P.; Ivashchenko, Y. Dopov Akad. Nauk Ukr. RSR, Ser. B 1973, 35, 69. (21) Keck, P. H.; van Horn, W. Phys. Rev. 1953, 91, 512. (22) Geld, P. V.; Petrushevski, M. S. Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk, Metall. Topl. 1961, 3, 160.

Figure 1. Surface tension of molten silicon measured by various investigators.

although they are in good agreement with each other. Keene7 indicated that these low values might be attributed to the contamination of molten silicon, above all, with oxygen, which can be substantiated qualitatively with the corresponding results.9,10 Kingery and Humenik10 obtained a 100 mN‚m-1 higher surface tension under hydrogen than under helium with the sessile drop method using MgO substrates, though afterward we found that surface tension measurement using a MgO substrate under argon atmosphere was very difficult because the silicon drop vibrated continuously on the MgO substrate.1 Hardy9 reported that the surface tension of molten silicon and its temperature coefficient progressively decreased as oxygen leakage into the system increased, although the evaluation of the oxygen concentration in his system was qualitative. The surface tension values obtained by Hardy9 under the most favorable conditions were in good agreement with those of Kingery and Humenik10 using hydrogen. On the basis of the above-mentioned results, Keene7 deduced that the low absolute values of surface tension and its temperature coefficient shown in Figure 1 were possibly achieved under oxygen-saturated conditions. He also pointed out the importance of future measurements under various controlled partial pressures of oxygen. The effects of the impurity concentration on the surface tension have been investigated for fairly many silicon binary systems and summarized in Table 1. It is found that most impurities increase the surface tension of molten silicon. Elements with much higher melting points than that of silicon, such as B, Mo, Ta, and W, increase the surface tension of molten silicon to a larger extent, while those with melting points lower than that of silicon, such as Ca, Mg, and Sn, reduce the surface tension.7 To clarify the effect of other impurity elements on the surface tension, the effect of oxygen should be separated, which requires precise controlling of oxygen partial pressure. However, the oxygen partial pressures in these measurement systems had never been determined. Our previous results on the effects of B,3 C,23 and Sb24 under precisely controlled oxygen partial pressures, and the effect of oxygen investigated in the present work are also listed in Table 1. 3.2. Measurement Methods. Almost all the surface tension measurements quoted by Keene7 were conducted (23) Mukai, K.; Yuan, Z. J. Jpn. Soc. Simul. Technol. 2000, 19, 112. (24) Yuan, Z.; Mukai, K.; Huang, W. L. J. Colloid Interface Sci., in press.

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with the sessile drop method. After Keene’s review, several other methods have been applied and developed. Przyborowski et al.25 measured the surface tension of Sb- or B-doped molten silicon with an oscillating drop method (using electromagnetic levitation) over a wide temperature range from 1373 to 1773 K (including the supercooled condition of ∆T ≈ 300 K). This fully containerless technique provides a high-purity environment and avoids contamination of the surface of molten silicon except for doping elements necessary for levitating a drop by the electromagnetic force. Also, the mass, not the density of the drop, is required in this method, which warrants its possible higher accuracy. The results are given in eq 2 and shown in Figure 1.

is, the ring method,26 the dynamic hanging method,27 and the sessile drop method.28 The surface tension data obtained using a SiC ring and a SiC-coated graphite crucible showed a linear temperature dependence from 1703 to 1823 K. The surface tension in a quartz glass crucible also showed a linear temperature dependence. The absolute values of the surface tension and its temperature coefficient in SiC and quartz glass crucibles belong to the group of lower values shown in Figure 1, though the value obtained in the SiC crucible is relatively higher than that in quartz glass crucible. They reported that the temperature coefficient of the surface tension became positive at about 1698 K and returned to a negative value just above the melting point. Although the abnormal temperature dependence of the surface tension seems very attractive, more careful interpretation of the results taking account of the experimental error and further measurements are required because the surface tension change which caused the abnormal dependence is smaller than only 3 mN‚m-1 and succeeding investigators including the authors did not find the abnormal behavior any more. The research group of Kimura27 developed a new measuring method using the rotation of a hanging drop. The measurement was performed by observing the oscillation of a silicon drop hanging from a SiC-coated carbon rod under 53 kPa of argon atmosphere. The oscillation was induced by a sudden high rotation speed above 570 rpm. The surface tension was estimated as 819 mN‚m-1 at the melting point of 1688 K, and its temperature coefficient was -0.308 mN‚m-1‚K-1. The absolute values of the surface tension and its temperature coefficient belong to the high value group shown in Figure 1, though the maximum scattering of the surface tension data is (6%. The research group of Kimura28 also employed the sessile drop method to investigate the surface tension of molten silicon in relation to the oxygen partial pressure PO2 of the argon atmosphere. ∂σ/∂T ) -0.093 mN‚m-1‚K-1, which belongs to the lower value group in Figure 1, was obtained, and the surface tension decreased by 10 mN‚m-1 in the range of PO2 from 4 × 10-6 to 3 × 10-4 MPa. The PO2 values in the systems are 1012 to 1016 times higher than PO2,sat, and SiO2(s) can form thermodynamically. Nishio et al.29 applied a surface laser-light scattering method to the surface tension measurement of molten silicon. The result is given by

σ ) 783.5 - 0.65(T - 1410) mN‚m-1

σ ) 801 - 0.24(T - 1683) mN‚m-1

Table 1. Surface Tension Change of Molten Silicon with the Contents of Some Impurities

element6

surface tension change per unit atom % (mN‚m-1‚(atom %)-1)

content range of impurity (atom %)

W Zr

+27 (-11)a +5 +0.5 +0.4 0 (-13)a +0.4 or +5.5 +60 +4.5 or +2.0 +20 ? +8 +3 +5 +200 -17 +63 +10 +163 +20

B C Sb O

Values Reported by the Present Authors max: +30 mN‚m-1‚(mass %)-1 0-2.09 mass %3 0 0-84 mass ppm23 max: -65 mN‚m-1‚(mass %)-1 0-0.9 mass %24 present work as shown in eqs 17-20

B Ca Co Cu Au Fe Mg Mn Mo Ni Nb O Pd Pt Rh Ta Sn Ti

0-1 0-10 0-10 0-10 0-10 0-10 0-10 0-10 0-0.5 0-10 0-1 0-10 0-10 0-10 0.36-0.53 0-10 0-2.7 2.7-5 0.14-0.33 3-10

a Based on extrapolation of curve at temperatures below 1683 K.

(2)

The oxygen concentration in the silicon specimen was less than 2 × 1016 atoms‚cm-3 (0.23 mass ppm) after the measurement, indicating that the measurements were performed under fairly low oxygen partial pressures. This little contamination by oxygen might contribute to the high absolute values of surface tension and its temperature coefficient. Although this method possesses the above-mentioned eminent advantages, the following tasks still remain: (1) At the present stage, scattering of the measured value is pretty large, that is, around (4%. (2) A doping element is required for levitating a silicon drop under normal gravity (1 G). (3) It is difficult to determine precisely the oxygen partial pressure equilibrated with the silicon drop during the surface tension measurement. The research group of Kimura measured the surface tension of molten silicon by three different methods, that (25) Przyborowski, M.; Hibiya, T.; Eguchi, M.; Egry, I. J. Cryst. Growth 1995, 151, 60.

(3)

The oxygen concentration in the silicon sample after the measurement was 1.66 × 1018 atoms‚cm-3 (19 mass ppm). Although the scattering of the measured values is (7%, this method is pretty interesting because it enables the simultaneous measurement of the surface tension at different points on the molten silicon surface, which is valuable for clarifying the Marangoni convection of molten silicon in CZ and FZ processes. Rhim et al.30 developed a high-temperature electrostatic levitation for measuring the surface tension of molten (26) Sasaki, H.; Anzai, Y.; Huang, X.; Terashima, K.; Kimura, S. Jpn. J. Appl. Phys. 1995, 34, 414. (27) Chung, S.; Izunome, K.; Yokotani, A.; Kimura, S. Jpn. J. Appl. Phys. 1995, 34, L631. (28) Huang, X.; Togawa, S.; Chung, S.; Terashima, K.; Kimura, S. 15th Jpn. Symp. Thermophys. Prop. 1994, 15, 223. (29) Nishio, T.; Tanaka, S.; Kawasaki, N.; Nagasaka, Y. Proc. 4th Asian Thermophysical Properties Conference; JSTP: Tokyo, 1995; p 69. (30) Rhim, W.-K.; Chung, S. K.; Rulison, A. J.; Spjut, R. E. Proc. 4th Asian Thermophysical Properties Conference; JSTP: Tokyo, 1995; p 353.

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silicon and obtained the following result in the supercooled temperature range:

σ ) 875 - 0.22(T - 1687) mN‚m-1

(4)

The apparatus can melt pure silicon samples during levitating under normal gravity. The levitated drop is quiescent and maintains an axisymmetric shape along a vertical direction. The density value is required in this method to obtain surface tension. Fujii et al.31 have analyzed the levitated drop method under microgravity (10-5 G) for the surface tension measurement of molten silicon, and pointed out that the surface oscillation of a drop is much simpler under microgravity than under normal gravity. All the surface tension data obtained by various methods after Keene’s review lie in the higher surface tension area of Figure 1, except those by Sasaki et al.26 These higher surface tension values may be attributed to the progress in purity of the silicon sample and the controlling technique of decreasing the oxygen potential in the atmosphere. However, there exist pretty large differences among the surface tension data obtained by different measuring methods. The differences may be caused by the different systematic errors of the methods, the difference of the density data required to obtain the surface tension, except for the oscillating drop method,25 and the difference in the oxygen potentials of the systems.

Figure 2. Equilibrium relation between oxygen and temperature in the Si(l)-O2(g)-SiO2(s)-OSi system.

4. Results and Discussion 4.1. Effects of Temperature and Oxygen on Surface Tension. Aiming at clarifying the temperature and PO2 dependences of the surface tension quantitatively, we developed the sessile drop method with precisely controlled oxygen partial pressure,32,33 and adopted it in the present experiments. The affinity of silicon for oxygen is very strong. The thermodynamic system becomes more complicated when SiO(g) forms. The thermodynamic details and corresponding reactions are described elsewhere.34 The oxygen concentration in silicon, CO, which is in equilibrium with PO2 in the atmosphere, and PO2,sat are presented in Figure 2. The results are calculated from the standard free energy change of reactions 5 and 7.

Si(l) + O2(g) ) SiO2(s)

(5)

∆G5° ) -952700 + 203.80T J/mol35

(6)

/2O2(g) ) OSi (mass %, in molten silicon)

(7)

∆G7° ) -306888 + 48.26T J/mol36

(8)

K7 ) aO/PO21/2 ) fOCO/PO21/2

(9)

1

PO2,sat is the saturated value of the oxygen partial pressure for equilibrium reaction 5 in a Si(l)-SiO2(s)O2(g) system. The oxygen concentration in CZ silicon is equal to 8.5 × 1017 atoms/cm3 (around 0.001 mass %), and thus, PO2 equilibrated with the above oxygen concentration is lower than PO2,sat. (31) Fujii, H.; Matsumoto, T.; Nogi, K. Acta Mater. 2000, 48, 2933. (32) Niu, Z.; Mukai, K.; Shiraishi, Y.; Hibiya, T.; Kakimoto, K.; Koyama, M. J. Jpn. Assoc. Cryst. Growth 1996, 23, 374. (33) Niu, Z.; Mukai, K.; Shiraishi, Y.; Hibiya, T.; Kakimoto, K.; Koyama, M. J. Jpn. Assoc. Cryst. Growth 1997, 24, 369. (34) Mukai, K.; Sako, T.; Yuan, Z.; Su, Z. Mater. Trans., JIM 2000, 41, 639.

Figure 3. Surface tension of molten silicon as a function of logarithmic oxygen partial pressure in argon.

As shown in Figure 3, with the oxygen partial pressure increasing, the surface tension remains almost constant in the range PO2 e 10-22 MPa, decreases remarkably in the range PO2 ) 10-22 to 10-20 MPa, and increases slightly in the range PO2 > PO2,sat. 4.2. Case of PO2 e PO2,sat. 4.2.1. Relation between Surface Tension and Oxygen. Belton37 introduced the relation between surface tension and the concentration of component i in a more usable form, by combining the Gibbs adsorption isotherm with the Langmuir isotherm:

σ ) σ° - RT Γi° ln(1 + Kiai)

(10)

where σ is the surface tension (mN/m), σ° is the surface tension of pure metal (mN/m), R is the gas constant (8.3145 J‚K-1‚mol-1), Γi° is the saturated surface excess concentration of surface active component i (mol/cm2), Ki is the adsorption coefficient of component i, ai is the activity of component i in solution, and i is oxygen in this paper. (35) Kubaschewiki, O.; Alcock, C. B. Metallurgical Thermochemistry, 5th ed.; Pergamon Press: New York, 1985. (36) Hirata, H.; Hoshikawa, K. J. Cryst. Growth 1990, 106, 657. (37) Belton, G. B. Met. Trans. 1976, 7B, 35.

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Γ° can be determined by calculating the slope of the surface tension versus the logarithm of the solute activity curve at high activities of the solute according to the Gibbs adsorption equation:

Γ° ) -

1 ∂σ RT ∂ ln aO

(11a)

Γ° ) -

∂σ 2 RT ∂ ln PO2

(11b)

where Γ° is the surface excess concentration (mol/cm2). Then, eq 12 for the Si-O system in the range PO2 e PO2,sat is given by combining eq 9 with eq 10 under the assumption of a Langmuir isotherm and oxygen equilibrium between gas and liquid silicon phases, as for reaction 7.

σ ) σ° - RTΓ° ln(1 + KPPO21/2)

(12)

where KP ) KOK7 and K7 ()aO/PO21/2 ) fOCO/PO21/2) is the equilibrium constant of reaction 7. aO is the oxygen activity in molten silicon, and the standard state for the oxygen activity is taken as 1 mass % in silicon. fO is the activity coefficient of oxygen in molten silicon. It is assumed that oxygen in silicon obeys Henry’s law (fO ) 1) because its concentration (CO) is very low. σ° is achieved by extrapolating the measured values to PO2 ) 0 or CO ) 0. Γ° ) 2.09 × 10-6 mol/m2 at 1693 K is obtained from eq 11. Substituting Γ°, σ°, T, and actual data into eq 12, KP can be determined using the least-squares method. The relation between σ and PO2 can be described with eqs 13-16.

Figure 4. Relation between the surface tension of molten silicon and the oxygen concentration.

σ ) 831 - 29.5 ln(1 + (1.72 × 1010)PO21/2), 1693 K (13) σ ) 814 - 30.1 ln(1 + (9.54 × 109)PO22),

1723 K (14)

σ ) 793 - 30.6 ln(1 + (5.87 × 109)PO21/2), 1753 K (15) σ ) 774 - 31.0 ln(1 + (3.32 × 109)PO21/2), 1773 K (16) The results calculated from eqs 13-16 are shown with the dashed lines in Figure 3, which are in good agreement with the measured values. The relations between the surface tension of molten silicon and the oxygen concentration in silicon are obtained as follows:

σ ) 831 - 29.5 ln(1 + 1944CO), 1693 K

(17)

σ ) 814 - 30.1 ln(1 + 1573CO), 1723 K

(18)

σ ) 793 - 30.6 ln(1 + 1395CO), 1753 K

(19)

σ ) 774 - 31.0 ln(1 + 1001CO), 1773 K

(20)

Equations 17-20 can describe the results shown in Figure 4 fairly well. The surface tension decreases by 50 mN/m with increasing CO from 0.0005 to 0.0032 mass % (the saturated oxygen concentration, CO,sat) at 1693 K, which indicates that oxygen is a strong surface active element for molten silicon. According to the obtained Γ° ) 2.09 × 10-6 mol/m2 at 1693 K, the area occupied by one oxygen atom on the silicon surface is calculated to be 7.9 × 10-19 m2. Since the occupying area of one silicon

Figure 5. Temperature dependence of the surface tension of molten silicon in the range PO2 e PO2,sat.

atom on the surface is equal to 8.4 × 10-19 m2, it can be deduced that there exists one oxygen atom per 10 silicon atoms on the molten silicon surface. 4.2.2. Temperature Dependence of Surface Tension. Figure 5 shows the temperature dependence of the surface tension of molten silicon with various PO2 values in the range PO2 e PO2,sat. The surface tension decreases linearly with increasing temperature, and the decreasing rate reduces with increasing oxygen partial pressure. At PO2 ) 3.9 × 10-25 MPa, the relation between σ and T is given by eq 21.

σ ) 835 - 0.74(T - 1687) mN·m-1

(21)

The absolute value of (∂σ/∂T)PO2 is the highest among those of the previous investigations, and the surface tension at the melting point belongs to the highest group (see Figure 1). 4.2.3. Effects of Oxygen and Temperature on ∂σ/∂T. As shown in Figure 6, the temperature coefficient of surface tension, ∂σ/∂T, is negative and increases with the oxygen

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Langmuir, Vol. 18, No. 6, 2002 2059

The relation between KO and T is given by eq 29 from eqs 17-20.

ln KO ) 2.28 × 104/T - 5.90

(29)

The standard free energy change of adsorption reaction 28 can be calculated from eq 29 and given as

∆G28° ) -189559 + 49.05T J/mol

(30)

Then, KC and KP can be acquired by integrating eq 24 between 0 and CO, and eq 25 between 0 and PO2, respectively.

∫0C

KC ) K° +

O

∂KC dCO ) ∂CO

K° - RΓ° ln(1 + KOCO) Figure 6. Relation between the temperature coefficient of surface tension and the oxygen partial pressure.

partial pressure. It is about -0.74 mN‚m-1‚K-1 at PO2 ) 10-25 MPa and increases gradually with the oxygen partial pressure up to 10-22 MPa. In the range of PO2 from 10-22 to 10-20 MPa, it increases steeply with the increasing oxygen partial pressure. And with further increase of PO2 up to 10-15 MPa, it increases progressively again up to about -0.15 mN‚m-1‚K-1. 4.2.4. Relation between ∂σ/∂T and PO2 or CO. In the following differentials, eqs 22 and 23 lead to eqs 24 and 25 by the use of eqs 11, 26, and 27 under the assumption that the temperature changes of Γ° and ∆H° are negligibly small.

∫0P

KP ) K° +

O2

∆H°Γ°KOCO T(1 + KOCO)

(31)

∂KP dPO2 ∂PO2

) K° - RΓ° ln(1 + KPPO21/2) (∆H° + ∆H7°)Γ°KPPO21/2 T(1 + KPPO21/2)

(32)

where K° ) (∂σ/∂T)CO ) 0 or (∂σ/∂T)PO2 ) 0. The mean KP ) (∂σ/∂T)PO2 and KC ) (∂σ/∂T)CO in T1 to T2 can also be deduced as follows: T KP dT ∫ T P K ave ) ∫TT dT 2

1

∂KC ∂(∂σ/∂CO) ) ∂CO ∂T

2

(22)

) K° -

∂KP ∂(∂σ/∂PO2) ) ∂PO2 ∂T

(23)

(

)

RΓO ∆H° ∂KC )1+ ∂CO CO (1 + KOCO)RT

(

ΓO ) Γ° ΓO ) Γ°

O2

KOCO 1 + KOCO KPPO21/2

1 + KPPO21/2

RΓ° (T ln(1 + KP2PO21/2) T2 - T1 2 1/2

T1 ln(1 + KP1PO2 )) -

)

(

(25) ≈ K° (26a)

(∆H° + ∆H7°)Γ°PO21/2

× T2 - T1 KP1 ln T1 KP2 ln T2 1/2 1 + KP2PO2 1 + KP1PO21/2

(24)

RΓO ∆H° + ∆H7° ∂KP )1+ ∂PO2 2PO2 (1 + K P 1/2)RT P

1

)

RΓ° (T ln(1 + KP1PO21/2) T2 - T1 1 T2 ln(1 + KP2PO21/2)) (33)

∫TT KC dT K ave ) ∫TT dT 2

C

1

2

(26b)

1

) K° -

KO ) Ae-∆H°/RT

(27a)

K7 ) A7e-∆H7°/RT

(27b)

T1 ln(1 + KO1CO)) -

(

∆H°Γ°CO KO ln T2 2

where KC ) (∂σ/∂T)CO, KP ) (∂σ/∂T)PO2, A in eq 27 corresponds to the entropy term, eq 26 is derived from the Langmuir isotherm, and ∆H° is the standard heat of adsorption in reaction 28.

RΓ° (T ln(1 + KO2CO) T2 - T1 2

T2 -T1 1 + KO2CO

-

KO1 ln T1 1 + KO1CO

)

(28)

RΓ° (T ln(1 + KO2CO) T2 - T1 2 T1 ln(1 + KO1CO)) (34)

where OS and VS represent adsorbed and unadsorbed oxygen, respectively.

where KP1, KP2, KO1, and KO2 are the values of KP and KO at T1 and T2, respectively.

OSi + VS ) OS

≈ K° -

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Yuan et al.

Figure 7. Temperature dependence of the surface tension of molten silicon in the range PO2 > PO2,sat.

Figure 8. Equilibrated partial pressures of gas phases with condensed phases of Si(l) and SiO2(s).

To reveal the relation between KP (KC) and PO2 (CO), according to eqs 31-34, ∆H°, ∆H7°, and KP (KO) should be known. Γ° ) 2.09 × 10-6 mol/m2 at 1693 K is adopted in the temperature range 1693-1818 K. ∆H7° ) -306 888 J/mol is known from eq 8, ∆H° ) -189 559 J/mol is obtained from the slope of ln KO versus 1/T plots in the present work, and the values of KP (KO) are known from eqs 13-16 (17-20). K° ) -0.75 mN‚m-1‚K-1 is the mean temperature coefficient of surface tension obtained by extrapolating the solid lines in Figures 3 and 4 to PO2 (CO) ) 0. Hence, KPave and KCave in the temperature range from 1693 to 1773 K are given as

KCave ) -0.75 + 0.370 ln(1 + 1944CO) 0.388 ln(1 + 1001CO) (35) KPave ) -0.75 + 0.370 ln(1 + (1.72 × 1010)PO21/2) 0.388 ln(1 + (3.32 × 109)PO21/2) (36) The KP values of eq 32 at 1693, 1723, 1753, and 1773 K are also given as

KP1693 ) -0.75 - 0.0175 ln(1 + (1.72 × 1010)PO21/2) + (1.06 × 1010)PO21/2/(1 + (1.72 × 1010)PO21/2) (37) KP1723 ) -0.75 - 0.0175 ln(1 + (9.54 × 109)PO21/2) + 9

1/2

9

1/2

(5.77 × 10 )PO2 /(1 + (9.54 × 10 )PO2 ) (38) KP1753 ) -0.75 - 0.0175 ln(1 + (5.87 × 109)PO21/2) + (3.49 × 109)PO21/2/(1 + (5.87 × 109)PO21/2) (39) KP1773 ) -0.75 - 0.0175 ln(1 + (3.32 × 109)PO21/2) + (1.95 × 109)PO21/2/(1 + (3.32 × 109)PO21/2) (40) The results calculated from eqs 36-40 are also shown in Figure 6. These calculated results follow the trend of observed results, and KPave appears to be the best fitting.

Figure 9. Illustrations of the molten drops at different PO2: (a) PO2 e PO2,sat; (b) PO2 > PO2,sat.

4.3. Case of Po2 > Po2,sat. Figure 7 illustrates the temperature dependence of the surface tension of molten silicon with various PO2 values in the range PO2 > PO2,sat. The surface tension also decreases linearly with increasing temperature, but the decreasing rate is smaller than that in the range PO2 e PO2,sat (Figure 5). ∂σ/∂T is from -0.25 to -0.15 mN‚m-1‚K-1 and increases with the oxygen partial pressure in the range from 5.9 × 10-19 to 3.1 × 10-15 MPa, which is higher than that in the range of PO2 e PO2,sat (Figure 5). The lower absolute values of (∂σ/∂T)PO2 in the range of PO2 > PO2,sat are in good agreement with the lower group,1

Surface Tension of Molten Silicon

Langmuir, Vol. 18, No. 6, 2002 2061

Figure 10. EPMA analyses on the surface of solidified silicon in the case of PO2 (≈10-15 MPa) > PO2,sat.

Figure 11. EPMA analyses on the surface of solidified silicon in the case of PO2 (≈10-22 MPa) e PO2,sat.

which suggests that the previously reported lower absolute values of (∂σ/∂T)PO2 might be obtained under PO2 > PO2,sat, as indicated by Keene.7 In the measurement at PO2 ) 3.1 × 10-15 MPa (Figure 7), Ar gas was not treated by the Mg deoxidized furnace, and PO2in was from 1.52 × 10-6 to 8.87 × 10-8 MPa. After the Ar gas passed through the reaction chamber where the surface tension was measured, PO2out was 10-14 to 10-16 MPa, just in the range of stable SiO2(s) formation, as shown in Figure 8 (drawn from thermodynamic data35) and Figure 1. When a thin SiO2(s) film forms on the surface of a molten silicon drop, the PSiO value shown in Figure 8 decreases and the stability of the SiO2(s) film increases. In addition, the mass loss of the silicon sample decreases with increasing oxygen partial pressure in argon gas. It can be estimated that the formation and evaporation of SiO(g) gas may be retarded because of the formation of a thin SiO2(s) film. Figure 9 illustrates the characteristics of molten silicon drops for different cases of PO2. Under the condition PO2 < 10-22 MPa, the molten silicon drop on the BN substrate was very sensitive to outside vibration according to camera observation. Especially at temperatures higher than 1753 K, it would oscillate continuously for several seconds even if someone were walking nearby. The large central reflection speckle in the photograph indicates its highly lustrous surface. The corresponding solidified sample also exhibits a bright and smooth mirror-like surface. However, when PO2 was about 10-15 MPa, higher than PO2,sat, the molten silicon drop was not influenced by environmental

disturbances any more and kept stable. The central speckle in the photograph is much smaller, indicating the lower luster of the drop surface. The surface of the solidified sample can also be observed to have partially lost metallic luster. The surface EPMA (electron probe microanalyzer) analyses of the solidified silicon samples in the cases of PO2 > PO2,sat and PO2 e PO2,sat are presented in Figures 10 and 11, respectively. A sharp increase of silicon concentration is observed in both cases and represents the sample surfaces, but the oxygen distributions are different. It is found that, in the case of PO2 > PO2,sat, there appears an apparent peak of oxygen concentration (about 10 µm in width) near the surface which is much higher than that in the bulk, which strongly suggests the formation of a thin SiO2(s) film on the surface of the molten silicon drop. The corresponding photographs are also given in Figure 10. The left image represents the morphology near the sample surface, the upper-right image shows the silicon distribution, and the lower-right image gives the oxygen distribution. It is obvious that the sample surface is regular and the oxygen is more concentrated on the surface than in the bulk. However, in the case of PO2 e PO2,sat, as shown in Figure 11, the oxygen concentration near the sample surface is the same as that in the bulk and there is no oxygen concentration peak. From the corresponding photographs, it is also observed that there is no concentrated oxygen near the sample surface (the lower-right image), and the sample surface is relatively irregular (the left image). These results are consistent with the preceding

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discussions and indicate that, in the case of PO2 > PO2,sat, a thin SiO2(s) film might form on the drop surface and the molten drop might be more stable. According to above analyses, it might be considered that the thin SiO2(s) film formed on the surface of the molten silicon drop under the condition PO2 > PO2,sat leads to the formation of a composite interface consisting of the SiO2(s) surface and the SiO2(s)-Si(l) interface, as illustrated in Figure 9, and results in a certain increase of the surface tension, as shown on the right side of Figure 3. 5. Conclusions Recent studies on the surface tension of molten silicon and the measuring methods have been reviewed, and the influence of oxygen potential has been investigated with the sessile drop method under precisely controlled oxygen partial pressures. In the case of PO2 e PO2,sat, the surface tension of molten silicon remains almost constant in the range PO2 e 10-22 MPa, and decreases remarkably with the oxygen partial pressure increase in the range PO2 ) 10-22 to 10-20 MPa, which indicates that oxygen is a strong surface active element for molten silicon. The surface tension decreases linearly with increasing temperature, and the decreasing rate reduces with the oxygen partial pressure increase. The temperature coefficient of surface tension, ∂σ/∂T, is negative and increases with the oxygen partial pressure. The molten silicon drop was very sensitive to outside vibration under the condition PO2 < 10-22 MPa. The solidified silicon samples exhibited a bright and smooth mirror-like surface.

Yuan et al.

In the case of PO2 > PO2,sat, the surface tension increases slightly with the oxygen partial pressure, and decreases linearly with increasing temperature, but the decreasing rate is smaller than that in the range PO2 e PO2,sat. ∂σ/∂T is from -0.25 to -0.15 mN‚m-1‚K-1 and increases with PO2 in the range from 5.9 × 10-19 to 3.1 × 10-15 MPa, which is higher than ∂σ/∂T in the range PO2 e PO2,sat. When PO2 was about 10-15 MPa, higher than PO2,sat, the molten silicon drop was not influenced by environmental disturbances and remained stable, and oscillation could not be observed anymore. The surface of solidified silicon samples partially lost metallic luster. EPMA analysis indicates that the oxygen concentration on the surface of solidified silicon samples (about 10 µm) is higher than that of the silicon bulk, and suggests the formation of a thin SiO2(s) film on the surface of the molten silicon drop in the case of PO2 > PO2,sat, which might result in some increase of the surface tension. Acknowledgment. This work is the result 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 International Trade and Industry (MITI). The authors also appreciate the financial support of the Opening Foundation of the National Microgravity Laboratory of China during paper preparation. LA0112920