Optimization of the design of a crucible for a SiC sublimation growth system using a global model

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1 Journal of Crystal Growth 310 (2008) Optimization of the design of a crucible for a SiC sublimation growth system using a global model X.J. Chen a,, L.J. Liu a, H. Tezuka b, Y. Usuki b, K. Kakimoto a a Research Institute for Applied Mechanics, Kyushu University, 6-1 Kasuga-koen, Kasuga, Fukuoka , Japan b R&D Division, Furukawa Co. Ltd , Kannondai 1-chome, Tsukuba-shi, Ibaraki , Japan Available online 12 November 2007 Abstract Induction heating, temperature field and growth rate for a sublimation growth system of silicon carbide were calculated by using a global simulation model. The effects of shape of the crucible on temperature distribution and growth rate were investigated. It was found that thickness of the substrate holder, distance between the powder and substrate, and angle between the crucible wall and powder free surface are important for growth rate and crystal quality. Finally, a curved powder free surface was also studied. The results indicate that the use of a curved powder free surface is also an effective method for obtaining a higher growth rate. r 2007 Elsevier B.V. All rights reserved. PACS: Cb; i; a Keywords: A1. Computer simulation; A1. Heat transfer; A1. Substrates; A2. Growth from vapor 1. Introduction Silicon carbide (SiC) is a promising material for semiconductors to be used in electronic and optoelectronic devices operated under extreme conditions, such as high temperature, high frequency, high power and intensive radiation environments, owing to its stable chemical and thermal properties. The sublimation growth technique based on the modified Lely method has been widely applied to grow SiC single crystals since 1970s [1,2]. To grow SiC crystals of high quality and to control the crystal growth process, it is necessary to know the temperature distribution inside the crucible. However, observation and measurements are difficult because the temperature inside the crucible is more than 2000 K, and numerical simulation is therefore used for investigating the induction heating and thermal field of SiC crystal sublimation growth systems. Many researchers, including Pons et al. [3,4], Hofmann et al. [5,6], Karpov et al. [7] and Chen et al. [8,9], have successfully applied numerical Corresponding author. Tel.: ; fax: address: xjchen@riam.kyushu-u.ac.jp (X.J. Chen) /$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi: /j.jcrysgro analysis to compute the temperature distribution and growth rate in a SiC sublimation growth system. Kitou et al. [10] investigated the effect of an inner guide-tube inside the crucible on sublimation growth of bulk SiC. They found that crystal quality was improved by using the guide-tube. Kulik et al. [11] demonstrated advantages of inverse computations in the optimization of the crucible design for SiC sublimation growth. Dramatic improvement of temperature uniformity throughout the powder charge was achieved by only variation of the outer crucible wall profile. However, there has been no detailed study on the effects of shape of powder in a crucible on temperature distribution and growth rate. Thus, there is a lack of information on better design of a crucible for achieving a high growth rate and growing SiC crystals of high quality. Such information is needed for commercial production. The main objective of the present study was to determine the effects of the shape of the crucible on growth rate and temperature distribution. Several parameters for crucible design, including thickness of the substrate holder, distance between the powder and substrate, angle between the crucible wall and SiC powder free surface, were studied. A two-dimensional axisymmetric numerical global model 转载

2 X.J. Chen et al. / Journal of Crystal Growth 310 (2008) was employed to simulate the SiC growth system. Finally, the effectiveness of a curved SiC powder free surface for controlling the growth rate of SiC sublimation growth was also studied. 2. Simulation model and method To study the induction heating and heat transfer in the SiC growth furnace, an axisymmetric SiC crystal growth system shown schematically in Fig. 1 is simulated. The growth system consists of induction coils, SiC powder, graphite crucible, pedestal, insulation shield, reflectors and furnace walls. The substrate is located at the lid of the crucible. One monitoring point for temperature is defined as T m at the top of the lid of the crucible. First, the heater power in the crucible is computed in the present global analysis. Then the temperature distribution in the furnace is calculated using this power as a source term. Finally, the growth rate is obtained from the temperature field. For the calculation of induction heating, it is assumed that all of the media in the furnace have linear and isotropic electrical properties and that their electrical properties are temperature-independent. It is also assumed that the system is axisymmetric, and the displacement current is neglected. Based on these assumptions, from Maxwell s equations and Ohm s Law, it is shown in Ref. [12] that the governing equations in cylindrical coordinates can be solved: q 1 q qr r qr ðc BÞ þ 1 r q 2 c B qz 2 ¼ mj y, (1) substrate 1 SiC powder 2 crucible 3 insulation shield 4 pedestal 5 reflector 6 furnace walls 7 coils Fig. 1. Configuration of the SiC crystal growth furnace. T m c B ¼ Cðr; zþ cos ot þ Sðr; zþ sin ot, (2) Q H ðr; zþ ¼so 2 Cðr; zþ2 þ Sðr; zþ 2 2r 2, (3) where Cðr; zþ and Sðr; zþ are the in-phase and out-of-phase components of the magnetic stream function c B, respectively, o is angular frequency of the electrical current in the induction coil, Q H ðr; zþ is the heater power distribution, J is the charge current density, m is the magnetic permeability and t is time. The boundary conditions is c B ¼ 0, for both r ¼ 0andðr; zþ!1. For the calculation of heat transfer, the following major assumptions are made: (1) the geometry of the furnace configuration is axisymmetric, (2) radiative transfer is modeled as diffuse-gray surface radiation, and (3) the effect of gas flow in the furnace is negligible. Thus, the conduction and radiation equations can be expressed as follows: rðk h rtþþq H ¼ 0, (4) Z qð~xþ eð~xþ Kð~x; ~x Þ 1 eð~x Þ ~x 2qV eð~x qð~x Þ ds Þ Z ¼ s s T 4 ð~xþ Kð~x; ~x Þs s T 4 ð~x Þ ds, ð5þ ~x 2qV where ~x and ~x are infinitesimal radiative surface elements on qv, ds is the area of the infinitesimal surface element, and Kð~x; ~x Þ is the surface view factor between ~x and ~x, which was described in detail in Ref. [13]. The boundary conditions are Tð~rÞ ¼T cnd ð~rþ and k i ðqt=qnþ ¼q rad ð~rþ at all the interfaces. As explained in Ref. [8], only the reaction, 2SiC ðsþ Ð SiC 2ðgÞ þ Si ðl;gþ, is more important near the seed and should be considered. The SiC 2 and Si vapor pressure are assumed to have the same transport rate, so based on the Hertz Knudsen equation and the supersaturation of the ratedetermining species SiC 2, the growth rate of SiC crystal will be G SiC ¼ 2M SiC w r SiC2 ½p SiC2 ðlþ p SiC 2 ðlþš, (6) SiC pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where w SiC2 ¼ 1= 2pM SiC2 RT and p SiC 2 ðlþ is the equilibrium vapor pressure of SiC 2. The advective velocities U of the SiC 2 and Si are also assumed to be same. From the one-dimensional mass transfer equation for Stefan flow [14], U and p SiC2 ðlþ can be obtained by the following coupled equations [9] in an iterative calculation: p SiC2 ðlþ ¼p SiC 2 ðlþþup=ð2w SiC2 RTÞ, (7) U ¼ D=L lnððp 2p SiC2 ðlþþ=ðp 2p SiC2 ð0þþþ. (8) Then the growth rate is obtained from Eq. (6). To take the two-dimensional effect of temperature distribution on growth rate into account, the temperature distribution TðrÞ along the seed surface is used to calculate the growth rate.

3 1812 X.J. Chen et al. / Journal of Crystal Growth 310 (2008) The electromagnetic equations, conduction equations and radiation equations are solved using the finite volume method. A multi-block method is applied to generate a discrete system. In electromagnetic calculation, a computational domain of 600 mm 1700 mm ðdr DzÞ is used. The reasonability of selection of the computational domain was checked by comparing the current results with results obtained by using a domain of 750 mm 2100 mm. The computed results are almost the same with a maximum error less than 2%. For temperature field simulation, to obtain the given temperature, T m ¼ 2250 K, at the monitoring point, the heater power was adjusted iteratively in the global iteration [13]. 3. Results and discussion Several important parameters of the crucible were numerically studied. The definitions of parameters are given in Fig. 2. is thickness of the substrate holder, H p2c is distance between the SiC powder and substrate, and a is angle between the crucible wall and SiC powder Crucible Wall R 0 H cur α SiC Powder Fig. 2. Definitions of some parameters for the crucible. free surface. A curved powder free surface is defined as follows: zðrþ ¼Ar 2 þ B and H cur ¼ br 2 0, (9) where R 0 is the radius of the SiC powder free surface and H cur is the center height of the curved surface, which are shown in Fig Effects of thickness of the substrate holder The temperature difference between the powder and substrate DT p2c and growth rates for three different thicknesses of the substrate holder ( ¼ 8, 15 and 20 mm) are shown in Fig. 3(a), and the temperature distribution along the surface of the substrate is shown in Fig. 3(b). When the thickness of the substrate holder increases, temperature difference DT p2c slightly decreases, but growth rate increases. The reason for this is that the growth rate is not only a linear function of DT p2c but also an exponential function of the inverse growth temperature, which is proved by the experimental data in Ref. [15]. It can be seen in Fig. 3(b) that the temperature along the surface of the substrate increases with increase in the thickness of the substrate holder and that the temperature gradient in the radial direction along the surface of the substrate decreases. In the case of larger thickness of the substrate holder, higher growth rate and small temperature gradient along the substrate surface can be obtained, which means lower thermal stress in the grown crystal can be obtained. However, in the case of greater thickness of substrate holder, the maximum temperature in the crucible increases because the thicker substrate holder increases the heat resistance in the crucible while the temperature is fixed at monitoring point T m. Due to the limitation of working temperature of some materials, such as graphite, the maximum temperature should be limited in the crucible, and the thickness of the substrate holder should also (mm/hr) T = 8 mm =15mm =20mm Fig. 3. (a) Temperature difference between the powder and substrate, DT p2c and growth rate versus thickness of substrate holder,. (b) Temperature distribution along the substrate surface for different thicknesses of substrate holder,.

4 X.J. Chen et al. / Journal of Crystal Growth 310 (2008) ( mm/hr) T =45mm =40mm =35mm Fig. 4. (a) Temperature difference between the powder and substrate, DT p2c and growth rate versus distance, H p2c. (b) Temperature distribution along the substrate surface for different distances, H p2c. therefore be limited. The calculated result of growth rate shown in Fig. 3(a) agrees semi-qualitatively with the experiment result [16] Effects of distance between the powder and substrate H p2c Calculations with different distances between the powder and substrate (H p2c ¼ 35, 40 and 45 mm) were carried out by changing the position of the SiC powder free surface while the substrate is kept at the same position. The results of temperature difference DT p2c, growth rate and temperature distribution along the substrate surface are shown in Fig. 4. Both the temperature difference DT p2c and the growth rate are increased when the distance H p2c is increased. Moreover, the shape of the temperature distribution along the substrate surface becomes flat with increase in the distance H p2c. Higher growth rate and lower thermal stress in the grown crystal can be achieved with a larger distance H p2c in the crucible. However, the effects of the convection inside the crucible would be increased if the distance H p2c is greatly increased, and the distance H p2c should therefore be limited. Trend of the growth rate obtained from the calculation shown in Fig. 4(a) is almost the same as that obtained from the experiment [16] Effects of angle between the crucible wall and SiC free surface a In the present crucible for SiC crystal sublimation growth, the angle between the crucible wall and SiC free surface is one of the important parameters for crucible design. Several angles (a ¼ 60, 65 and 70 ) were calculated while the diameter of the substrate is kept constant. Fig. 5(a) shows the temperature difference between the powder and substrate DT p2c and the growth rates for three different angles. Fig. 5(b) shows the effects of the angle on temperature distribution along the substrate surface. In the case of a ¼ 60, there is a larger temperature difference between the powder and substrate and a higher growth rate than those in case of larger angles ða ¼ 65 ; 70 Þ. The shape of the temperature distribution along the substrate surface is flatter for a ¼ 60 than for a ¼ 65 and a ¼ 70. This means that the temperature gradient in the radial direction along the substrate surface is small in the case of a ¼ 60, ensuring lower thermal stress in the grown crystal Effects of curved powder free surface The curved powder free surface is defined in Fig. 2 and by Eq. (9). If the position of the edge of the powder free surface in the z direction is z p, the curve can be expressed as zðrþ ¼bðr 2 R 2 0 Þþz p. (10) Here, the surface would be a convex curve and the center height of the surface H cur is minus if the parameter bo0, while it is a concave curve and H cur is positive if b40. The results of temperature difference DT p2c, growth rate and temperature difference along the substrate surface DT sub are shown in Fig. 6. From the figure, both the temperature difference DT p2c and growth rate increase as the height H cur increases. To obtain a high growth rate using the curved powder free surface, the curve should be concave. If the average distance between the curved surface and the substrate H p2c is considered, a concave curve implies that the distance H p2c is increased, while a convex curve implies that the distance H p2c is decreased. Similar results have been shown in Section 3.2. The result about DT sub in Fig. 6 shows that the concaveshaped surface can get more flat shape of temperature distribution along the substrate surface, but the effect is not significant.

5 1814 X.J. Chen et al. / Journal of Crystal Growth 310 (2008) ( mm/hr ) T α =60 o α =65 o α =70 o α ( 0 ) 2264 Fig. 5. (a) Temperature difference between the powder and substrate, DT p2c and growth rate versus angle, a. (b) Temperature distribution along the substrate surface for different angles, a. (mm/hr) H cur 4. Conclusion ΔT sub Fig. 6. Temperature difference between the powder and substrate, DT p2c, growth rate and temperature difference along substrate surface, DT sub versus center height of curved surface, H cur. An inductively heated SiC growth system was computed and analyzed by using a global simulation method, and the effects of crucible shape and curved powder free surface on temperature distribution and growth rate were investigated. The temperature distribution in the crucible and growth rate were greatly affected by crucible shape and the curved surface. To obtain a high growth rate and low thermal stress in the crystal, the results suggest that the thickness of the substrate holder should be increased, the distance between the powder and substrate should be slightly increased, and the angle between the crucible wall and powder free surface should be decreased. In the case of (K ) ΔT sub a curved surface, a concave surface should be used to achieve a high growth rate. References [1] Y.M. Tairov, V.F. Tsvetkov, J. Crystal Growth 43 (1978) 209. [2] Y.M. Tairov, V.F. Tsvetkov, J. Crystal Growth (1981) 146. [3] M. Pons, E. Blanquet, J.M. Dedulle, I. Garcon, R. Madar, C. Bernard, J. Electrochem. Soc. 143 (1996) [4] M. Pons, E. Blanquet, J.M. Dedulle, R. Madar, C. Bernard, Mater. Sci. Eng. B 46 (1997) 308. [5] D. Hofmann, M. Heinze, A. Winnacker, F. Durst, L. Kadinski, P. Kaufmann, Y. Makarov, M. Schafer, J. Crystal Growth 146 (1995) 214. [6] D. Hofmann, R. Eckstein, M. Ko lbl, Y. Makarov, S.G. Mu ller, E. Schmitt, A. Winnacker, R. Rupp, R. Stein, J. Vo lkl, J. Crystal Growth 174 (1997) 669. [7] S.Yu. Karpov, A.V. Kulik, I.A. Zhmakin, Yu.N. Makarov, E.N. Mokhov, M.G. Ramm, M.S. Ramm, A.D. Roenkov, Yu.A. Vodakov, J. Crystal Growth 211 (2000) 347. [8] Q.-S. Chen, H. Zhang, V. Prasad, C.M. Balkas, N.K. Yushin, S. Wang, J. Crystal Growth 224 (2001) 101. [9] Q.-S. Chen, H. Zhang, V. Prasad, C.M. Balkas, N.K. Yushin, J. Heat Transfer 123 (6) (2001) [10] Y. Kitou, W. Bahng, T. Kato, S. Nishizawa, K. Arai, Mater. Sci. Forum (2002) 83. [11] A.V. Kulik, et al., Mater. Res. Soc. Symp. Proc. 640 (2001) H [12] J.J. Derby, L.J. Atherton, P.M. Gresho, J. Crystal Growth 97 (1989) 792. [13] L.-J. Liu, K. Kakimoto, Int. J. Heat Mass Transfer 48 (2005) [14] D.T.J. Hurle, Handbook of Crystal Growth, vol. 2, Elsevier, Amsterdam, 1996, p [15] C.M. Balkas, A.A. Maltsev, M.D. Roth, N.K. Yushin, Mater. Sci. Forum (2000) 79. [16] H. Tezuka, Y. Usuki, Private communication, 2006.