Influence of ampoule rotation on three-dimensional convection and segregation in Bridgman crystal growth under imperfect growth conditions
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(2) C.W. Lan et al. / Journal of Crystal Growth 212 (2000) 340}351. azimuthal direction) can induce signi"cant 3D #ows and thus 3D segregation. The situation becomes worse as the buoyancy force increases. On the other hand, the enhanced 3D #ows increases the dopant mixing and may reduce segregation [6], but the dopant uniformity is not axisymmetric. Even in microgravity condition, the situation may not be improved much because the gravity orientation is never "xed during the whole growth period. The weak #ow induced by the residual gravity sometimes induces an even severe segregation [5]. Therefore, how to better control or suppress the 3D e!ects is highly desirable for VB crystal growth. In fact, the situation can become even more complicated as the solutal e!ect is taken into account. The most e!ective way to suppress the 3D e!ects is to damp the convection su$ciently to reach the di!usion-controlled limit. Indeed, the most wellknown approach is the use of magnetic damping (e.g., Refs. [7}9]). The #ow intensity (t ) de creases with the increasing magnetic "eld strength (B or in terms of Hartmann number Ha); t & Ha\ [7]. However, the hardware requirement to provide a su$cient magnetic damping (usually in the order of 0.5 T) is costly. An alternative way to suppress the #ow, maybe other mechanisms like vibration [10] as well, is the use of steady ampoule rotation. As discussed by Lan and Chian [11,12], as well as Yackel et al. [13], a moderate rotation rate may a!ect the buoyancy #ow signi"cantly for the axisymmetric con"guration. In addition, its damping e!ect is similar to that due to the magnetic "eld but less e!ective; t &X\ [14], where X is the rotation speed. For the situation that the buoyancy #ow is weak, the ampoule rotation may be an e!ective tool for the #ow control. More importantly, as the growth is subjected to the imperfect growth conditions as mentioned previously, the 3D #ows may also be suppressed. However, the in#uence of ampoule rotation on the 3D #ows and segregation has not been explored. Furthermore, as the growth front is not axisymmetric, which may be caused by the convection or asymmetric heating, the rotational e!ects are particularly interesting. The rotational growth and melting, which is believed to be the source of rotational striations, can lead to severe segregation and constitutional super-. 341. cooling. Again, no such a numerical study has been reported. Using a centrifuge may also be useful in the #ow control [15], but the convection is not suppressed and the 3D e!ect is introduced. In the present study, a 3D numerical simulation is conducted to investigate the e!ects of ampoule rotation on the 3D #ows and segregation under imperfect growth conditions. The e!ects of both gravity orientation (ampoule tilting) and asymmetric heating are considered. To further illustrate the feasibility of using ampoule rotation for #ow damping, di!erent convection intensities are considered and the reduced gravity situation is discussed. In the next section, the model and its numerical simulation are brie#y described. Section 3 is devoted to the results and discussion, followed by conclusions and comments in Section 4.. 2. Model description and numerical solution A generic Bridgman crystal growth system is illustrated in Fig. 1a. Since axisymmetry is no longer assumed here, the system is described by a Cartesian coordinate (x, y, z). The asymmetry could be introduced by the tilted ampoule, which is represented by the tilt angle c, or the nonuniform ambient thermal pro"le ¹ (x, y, z). In Fig. 1a, each region is characterized by a set of physical properties. Since our major interest in this paper is the e!ects of rotation on the #ows, to simplify the discussion, a pseudo-steady-state growth is further assumed, where the ampoule pulling speed ; is set to be the axial growth rate. The #ow, temperature, and dopant "elds, as well as the melt/crystal interface, are also represented in the Cartesian coordinate (x, y, z). The ampoule is rotated around the growth axis at a constant rotation speed X. Because the coordinate we have chosen is a "xed frame, it is not possible to describe the steady-state dopant "eld in the crystal. Therefore, the dopant segregation calculated in the crystal is only a snap shot of the growth, not a real distribution in the crystal. Accordingly, the dopant boundary condition at the growth front due to the crystal side is only an approximation. Furthermore, it is also assumed that the dopant concentration is very low so that both #ow and liquidus temperature are not.
(3) 342. C.W. Lan et al. / Journal of Crystal Growth 212 (2000) 340}351. where *, P, and C are the dimensionless velocity, pressure, and dopant concentration, respectively. Pr is the Prandtl number (Pr,l /a ), Sc the Schmidt number (Sc,l /D), and h the melting. temperature, which also serves as a reference temperature; l is the kinematic viscosity and D the. dopant di!usivity in the melt. The thermal Rayleigh number Ra in the source term of the 2 momentum equation is de"ned as follows:. Fig. 1. (a) Schematic sketch of Bridgman crystal growth with ampoule rotation; (b) a portion of a sample mesh for calculation.. a!ected. Therefore, the calculated #ow "eld is not a!ected by the dopant distribution. The dimensionless variables are de"ned by scaling length with the ampoule length ¸, velocity with a /¸, pressure with o a /¸, and dopant concen tration with its average concentration C in the crystal, where a is the thermal di!usivity and. o the melt density. The dimensionless temperature. (h) is de"ned as h(x, y, z)"[¹(x, y, z)!¹ ]/(¹ !¹ ), (1) where ¹ and ¹ are the top (hot) and bottom (cold) temperatures, as shown in Fig. 1a. The pseudo-steady-state governing equations describing convection and heat and dopant transport in the melt (m) are as follows:. ) *"0,. (2). * ) *"! P#Pr *!PrRa (h!h )e , 2 * ) h" h,. (3). Pr * ) C" C, Sc. (4) (5). gb *¹¸ Ra , 2 , 2 a l where g is the gravitational acceleration, b 2 the thermal expansion coe$cients, and *¹" ¹ !¹ . Also, if one represents the equation of motion in terms of a rotating frame, the Coriolis force due to steady rotation becomes obvious being 2PrTa e ;* and Ta,XR/l is the Taylor ( number. This term is similar to the Lorentz body force for magnetic damping [14]. The gravity direction e can be decomposed into the Cartesian com ponents (e , e , e ): V W X e "cos(a)e !cos(b)e !cos(c)e , (6) V W X where the a, b, and c are the angles between the gravity direction and the axes of x, y, and z, respectively. For a perfectly aligned ampoule, a"b"p/2, and c"0. In the present study, nonaxisymmetry is introduced only by tilting the ampoule in the y}z plane, i.e., a"p/2. In the crystal (c) and the ampoule (a), only heat transfer needs to be considered: (!Pe e #rPeXe ) ) h"i h, (i"c, a), (7) G ( G X G where Pe ,o Cp ; ¸/k is the translational G G G . Peclect number, PeX,o Cp X¸/k the rotational G G G. Peclect number, e the angular unit vector, r the ( dimensionless radial coordinate, and i ,k /k the G G dimensionless thermal conductivity of crystal or ampoule; k is the thermal conductivity of the melt.. Also, o , Cp , and k are the density, speci"c heat, G G G and thermal conductivity of the phase i (i"c or a), respectively. The no-slip condition is used for the melt velocity on solid boundaries: *"c (!v e #rXHe ), X (. (8).
(4) C.W. Lan et al. / Journal of Crystal Growth 212 (2000) 340}351. where c ,o /o and v is the dimensionless ampoule pulling speed and XH the dimensionless angular rotation speed; c is assumed to be one in this study. Since the pseudo-steady-state is assumed, the upper open boundary is considered as an arti"cial boundary [5], and its velocity boundary condition is the same. The thermal and solutal boundary conditions at the melt/solid interfaces are set by the heat and mass #ux balances. For example, at the growth front: n ) h" !n ) i h". #c (Pe e !rPeX e )St ) n"0, X ( !n ) C" !c (Pe e !rPeX e ). X ( Sc ; (1!K)C ) n"0, Pr. (9). (10). where n is the unit normal vector at the growth interface pointing to the melt. The Stefan number St,*H/(Cp *¹) scales the heat of fusion (*H). released during solidi"cation to the sensible heat in the melt. The equilibrium segregation coe$cient K of the solute is according to the phase diagram; K,C /C at the growth interface, where C is the dopant concentration in the crystal. It should be pointed out that if the interface is not axisymmetric, (e ) n) is not zero. As a result, for ( a steady-state growth, continuous melting and growth continue even the ampoule is not pulled downward. Accordingly, the heat of fusion also increases the interface shift (an isotherm surface) along the azimuthal direction. Furthermore, for dopant segregation, the growth (e ) n(0) in( creases the concentration in the growth front, while the melting (e ) n'0) decreases the concentration. ( Nevertheless, as mentioned previously, this is not always realistic because we have assumed that at the growth front an equilibrium exists, i.e., C /C"K, even for melting. Therefore, Eq. (10) provides only an approximation. For a detailed simulation, time-dependent calculation including the solid-state di!usion should be included. However, at the present stage, it is still too costly to do so. Another approach assuming that the solid is swallowed by the melt during melting can be used as the feed front boundary condition used in the zone-melting modeling [16], but the result is sim-. 343. ilar to that of Eq. (10). As it will be discussed shortly, for asymmetric heating, severe segregation due to this e!ect is observed, and this is similar to the cause of the rotational striations in Czochralski growth [17]. Furthermore, in our steady-state calculations, since the "xed-frame is chosen, except at the growth interface, it is also not possible to describe the dopant "eld in the crystal. Indeed, if the solid-state di!usion is neglected, a spiral dopant distribution is expected for the cases with 3D #ows. Also, the temperature at the melt/solid interface is assumed to be the equilibrium liquidus temperature of the material. Temperature at the top and bottom surfaces is set to be the furnace temperature there. With the pseudo-steady-state assumption, an arti"cial boundary condition is used at the upper boundary for the consistence of the overall dopant balance [5]: Pe Sc (C!1)(n ) e ). !n ) C" "c. Pr X. (11). The heat exchange between the ampoule and the furnace is by both radiation and convection according to the energy balance along the ampoule surface, !n ) i h" "Bi(h!h )#Rad(h!h),. (12). where n is the unit normal vector on the ampoule surface pointing outwards, Bi,h¸/k the Biot. number, and Rad,pe *¹¸/k the Radiation. number; p is the Stefan Boltzmann constant, while e is the surface emissivity of the ampoule. For simplicity, the e!ective furnace temperature h is assumed to be h (x, y, z)"z#h (x, y, z), (13) where h (x, y, z) is a deviation temperature to de scribe thermal nonuniformity. A Gaussian distribution is used for h : h (x, y, z)"*h exp[![(z!0.5)/aH]] (14) and *h "!*h cos( ), (15) where *h is the dimensionless peak (maxi mum) deviation temperature di!erence along the.
(5) 344. C.W. Lan et al. / Journal of Crystal Growth 212 (2000) 340}351. azimuthal angle ( ) and aH a dimensionless parameter for the width of the distribution. In the present study, the nonaxisymmetry for a perfectly aligned ampoule is introduced by a nonzero *h for nonuniform heating. The complications of computing view factors from the sample to the furnace are neglected here. Furthermore, during ampoule rotation, we also assume that the thermal "eld around the environment is not a!ected, and the rotation should be co-axis. Otherwise, the steadystate temperature distribution is not possible. Another way that may be more convenient for simulation is to use a rotation frame for the whole growth system, including the ampoule and the furnace. However, in the present study, we still stick with the "xed-frame approach, from the observer point of view, for simulation. The above governing equations and their associated boundary conditions can only be solved numerically. We have developed an e$cient "nite volume method (FVM) scheme using the primitive variable formulation [18] and multigrid acceleration [19] for the free boundary problem. This approach is much more e$cient and robust than the previous FVM/Newton's method [5]. A sample converged mesh for calculation is shown in Fig. 1b. As shown, "ner grid spacing is placed near the interfaces to enhance the accuracy of calculation. Detailed description of the numerical method can be found in Refs. [18,19].. 3. Results and discussion For comparison purposes, we consider the gallium-doped germanium (GaGe) growth in the Grenoble furnace investigated by Adornato and Brown [20] in this study. The same system was also studied by Lan and Chen [21] as a benchmark problem for comparing the performance of a 2D FVM (stream function-vorticity (t}u) formulation) and a Galerkin "nite element method (primitive variable formulation). The e!ects of ampoule tilting and nonuniform heating for Ra "0}10 were fur2 ther studied by Liang and Lan [5] using a FVM/Newton's method. Therefore, in this paper, we will extend the previous calculations to a much higher Ra (Ra "2.489;10, which corresponds 2 2. Table 1 Physical properties and input parameters [5,11] GaGe o "5.5 g cm\ o "5.5 g cm\. ¹ "937.43C. *H"460 J g\ k "0.17 W cm\ 3C\ k "0.39 W cm\ 3C\. Cp "Cp "0.39 J g\ 3C\ . k"0.00715 g cm\ s\ b "5;10\ 3C\ 2 D"2.1;10\ cm s\ K"0.087 Graphite (ampoule) o "1.8 g cm\ k "3.26 W cm\ 3C\ Cp "1.814 J g\ 3C\ e "0 Other input parameters ¸"7 cm R "0.5 cm R "0.7 cm ¹ "1112.43C ¹ "762.43C *¹"¹ !¹ "3503C ; "4;10\ cm/s h"46.571 W cm\ 3C\ a"1 cm c"0!1.53 *¹ "0!13C X"0!10 RPM (Ta,XR/l "0}4.0556;10) Ra ,gb *¹¸/(a l )"0}2.489;10 2 2 . to a normal gravity condition for *¹"3503C) using the newly developed numerical scheme [19]. Furthermore, we will focus on the study of ampoule rotation on the 3D #ows and segregation induced by imperfect growth conditions for the VB crystal system. Since the dimensionless variables are used for illustration, the calculated results may also be useful for other semiconductor systems with similar #ow structures and dimensionless numbers. For reference, the physical properties of GaGe and some input parameters used in this study are listed in Table 1. Again, the condition of Rad"0 used in.
(6) C.W. Lan et al. / Journal of Crystal Growth 212 (2000) 340}351. Ref. [5] is also adopted here for comparison. The mesh used in the following calculations is similar to Fig. 1b. It contains 84 864 "nite volumes; 22;24;72 (r, h, z) "nite volumes in the melt. This leads to 276 482 nonlinear equations. All computations are performed in an HP/9000-C180 workstation, and one calculation takes about 2 h of CPU time depending on the initial guess. Before the e!ects of ampoule rotation are discussed, some numerical tests are necessary to ensure the correctness of calculations. Besides the extensive numerical tests being performed before [5,11,21], we further consider the comparison with the 2D results for Ra "2.489;10 (normal grav2 ity) and Ra "1;10 (reduced gravity). As shown 2 in Fig. 2a, for Ra "2.489;10, the calculated 2 #ow pattern (on the left-hand side) is in good agreement with the previous 2D result (e.g., Fig. 3c in Ref. [11]); all the detailed #ow structures are revealed. However, still some discrepancy exist for the dopant "eld, which is shown on the right-hand side of Fig. 2a, but it is not signi"cant numerically. Overall speaking, the agreement is quite satisfactory; one may examine the maximum dopant concentration in the melt (C "12.981 versus 13.047 (2D)). In this case, the calculated #ow and solute "elds are still 2D, which can be further examined by the dopant concentration at the growth interface, shown at the bottom of Fig. 2a. The dopant concentration in the crystal phase, C , is obtained by multiplying its corresponding melt concentration with the segregation coe$cient K, i.e., C "KC. The second comparison for Ra "1;10 at 2 5 RPM ampoule rotation (or Ta"1.0139;10) is shown in Fig. 2b, and as shown the agreement is very good due to the much weaker convection. Further mesh re"nement can improve the agreement, but it seems to be unnecessary. For stronger convection considered in this study, although the accuracy of the dopant "elds can be degraded slightly, the error in the maximum dopant concentration is still quite acceptable by using the mesh similar to Fig. 1b. In the following discussion, we will start the discussion on the rotation e!ects for ampoule tilting "rst and then for the asymmetric heating. Conclusions and comments are then given in Section 4.. 345. Fig. 2. Comparison of calculated results by 3D FVM and 2D W}u method [11] for (a) Ra "2.489;10 (0 RPM or Ta"0); 2 (b) Ra "1;10 (5 RPM or Ta"1.0139;10). 2. 3.1. Ampoule tilting As mentioned previously, a perfect ampoule alignment with the gravity may not be possible in practice. During crystal growth, a small tilt (less than 23) from the gravity orientation could be very common, and this may break axisymmetry [5,6]. Figs. 3a and b show the e!ect of 1.53 tilt on the #ow and the dopant segregation for Ra "2.489;10. 2 The same as before for Ra "1;10, the tilt indu2 ces signi"cant 3D #ow and bulk dopant mixing; the convection due to the tilt becomes much stronger,.
(7) 346. C.W. Lan et al. / Journal of Crystal Growth 212 (2000) 340}351. which can be seen from the velocity "elds in Fig. 3b. Accordingly, C is increased to 9.1484 (1.0569 in Fig. 3a). Furthermore, the radial segregation (Cc !Cc ) is reduced due to better local mix ing. In fact, tilting the ampoule to enhance dopant mixing was also reported by Xiao et al. [6]. However, as shown in the dopant "eld at the interface in Fig. 3b, it is no longer axisymmetric. Although such a 3D segregation in the crystal can be further reduced in practice by solid-state di!usion. Still, it is in general not favored. On the other hand, before the ampoule is tilted, if we apply 10 RPM rotation (Ta"4.0556;10) to the growth, as illustrated in Fig. 3c, the convection can be damped signi"cantly and stretched axially as discussed by Lan before [11]. More importantly, the secondary cell disappears and the elongated main #ow cell penetrates further into the bulk melt. As a result, the global mixing is enhanced by the 10 RPM rotation even though the convection is signi"cantly weaker. The radial segregation is reduced slightly, but this is not always true for ampoule rotation [11]. Again, tilting the ampoule 1.53 induces 3D #ows as shown in Fig. 3d. However, as compared with Fig. 3b, the convection is much weaker due to rotation. The bulk dopant mixing is reduced as well, which can be seen from the lower C (6.876 versus 9.1484 in Fig. 3c) and the iso concentration lines. In addition, due to rotation, the dopant "eld at the interface is distorted along the azimuthal direction. Unfortunately, the radial segregation is signi"cantly increased by rotation; C is increased to 14.13 as compared with 12.166 in Fig. 3b and 12.519 in Fig. 3c. Such an opposite e!ect may be caused by the secondary #ow cells near the growth interface. Clearly, from the results of Fig. 3, the 10 RPM ampoule rotation, beside damping the #ow, does not improve the growth. Therefore, for further study, one may need to use a higher rotation speed or use it for weaker buoyancy convection. We have chosen the later case for discussion. As the buoyancy convection is reduced, which can be achieved by reducing temperature di!erence *¹ or gravity g, the rotation becomes more signi"cant. Figs. 4a and b are still the e!ects of 1.53 tilt for 3D #ows and dopant segregation, but now for Ra "1;10. As discussed before by Liang and 2. Fig. 3. E!ects of ampoule tilting and rotation on the calculated #ow patterns (LHS) and dopant "elds (RHS) for Ra " 2 2.489;10: (a) c"03, X"0 RPM (Ta"0); (b) c"1.53, X"0 RPM (Ta"0); (c) c"03, X"10 RPM (Ta"4.0556; 10); (d) c"1.53, X"10 RPM (Ta"4.0556;10); *C" (C !C )/10. .
(8) C.W. Lan et al. / Journal of Crystal Growth 212 (2000) 340}351. Lan [5], even for such a weak #ow, the small tilt is signi"cant. However, due to the poorer mixing, the radial segregation for both cases increases; C "14.13 in a and 15.03 in b. Interestingly, the 3D #ow induced by the tilt increases the radial segregation even though the global mixing in enhanced. However, as 5 RPM ampoule rotation is introduced, the convection is damped signi"cantly for both cases; we have purposely used a much smaller vector length scale to represent the velocity "elds. The radial segregation in Fig. 4c is reduced slightly by the rotation. Interestingly, although the convection in Fig. 4d is still 3D, the dopant "led is nearly axisymmetric. Obviously, the di!usion predominates in the dopant segregation. As a result, as compared with Fig. 4b, the 3D e!ect of the ampoule tilting is signi"cantly suppressed. The rotation pushes the segregation towards the di!usion-controlled regime and thus reduces the e!ect of ampoule tilting. The results in Fig. 4 also implies that the rotation can be quite e!ective in reduced gravity, which in turn indicates that the faster rotation is needed for normal-gravity growth. However, during crystal growth in space, the gravity orientation is quite arbitrary and maybe changing with time due to orbiting or astronaut maneuver. Therefore, the situation is somewhat di!erent from that due to the small tilt. To illustrate that, we further reduce Ra 2 to 10, and the e!ect of gravity orientation is shown in Fig. 5. As shown in Fig. 5a, if the gravity is the same as the growth orientation, the dopant "eld is almost the same with that for Ra "0, i.e., di!u2 sion-controlled limit (more precisely, the convection contribution is only by the ampoule translation). Its velocity "eld (now shown here) is almost due to the ampoule translation only. However, as the gravity orientation is 453 from the growth direction, signi"cant 3D #ow is induced. One may compare the velocity magnitude with that in Fig. 4a (Ra "10, c"03, and 0 RPM) and "nd 2 that their convection levels are comparable. Accordingly, the radial segregation is thus severe being about the same order as that in Fig. 4a. Further increasing the gravity orientation to 903 does not change much the convection and the segregation (Fig. 5c). Interestingly, the result for 1353 in Fig. 5d is the same as that for 453. This is because the radial. 347. Fig. 4. E!ects of ampoule tilting and rotation on the calculated #ow patterns (LHS) and dopant "elds (RHS) for Ra "1;10; (a) c"03, X"0 RPM (Ta"0); (b) c"1.53, 2 X"0 RPM (Ta"0); (c) c"03, X"5 RPM (Ta" 1.0139;10); (d) c"1.53, X"5 RPM (Ta"1.0139;10); *C" (C !C )/10. .
(9) 348. C.W. Lan et al. / Journal of Crystal Growth 212 (2000) 340}351. Fig. 5. E!ects of gravity orientation on the calculated #ow patterns (LHS) and dopant "elds (RHS) for Ra "1;10: (a) c"03; 2 (b) c"453; (c) c"903; (d) c"1353; (e) c"1803.. temperature di!erences for convection in both cases are the same. Due to the same reason, the results for the inverted growth in Fig. 5e are almost the same as that in Fig. 5a. The 3D #ows induced in Figs. 5b and c are similar to the 2D results by Arnold et al. [22] in a microgravity condition. The feasibility of using magnetic "elds to suppress the induced convection was investigated by Yao et al. [9] for the growth of PbSnTe, and more than 30 kG of magnetic "eld was found necessary to achieve a nearly di!usion-controlled segregation. In fact, using steady ampoule rotation, which is much simpler, can be e!ective as well. As shown in Fig. 6, the 5 RPM steady ampoule rotation can almost eliminate the gravitational e!ect. The dopant "elds imply that the di!usion-controlled limit is achieved. Although the use of ampoule rotation to damp 3D #ows has not been adopted in space experiments, it is believed that it may be an e!ective way for better growth control. In fact, a recent study by Polovko et al. [23] pointed out that the Coriolis force due to spacecraft rotation has signi"cant in#uences on the. thermal convection and thus the segregation; a cubic #uid system was investigated. Indeed, steady ampoule rotation could be a much easier way for growth control. On the other hand, we have assumed that the heating is perfectly axisymmetric here. However, if the heating is not uniform, timedependent #ows and growth striations can be introduced by the rotation, which makes the growth even worse. Therefore, a feasible way to minimize such an e!ect is to rotate the furnace and the ampoule together. 3.2. Nonuniform heating In addition to the ampoule tilting or gravity orientation, asymmetric heating can also induce 3D #ow as illustrated by Liang and Lan [5]. As shown in Fig. 7a, even the nonuniformity is only 23C (*¹ "13C), signi"cant 3D #ow can be induced; the total temperature di!erence *¹"3503C. Unlike the tilting e!ect, the 3D #ow is more localized to the growth interface, where the heating nonuniformity is imposed. Meanwhile, due to the.
(10) C.W. Lan et al. / Journal of Crystal Growth 212 (2000) 340}351. 349. Fig. 6. E!ects of 5 RPM rotation (Ta"1.0139;10) on the calculated #ow patterns (LHS) and dopant "elds (RHS) for Ra "1;10 2 at di!erent gravity orientations: (a) c"03; (b) c"453; (c) c"903; (d) c"1353; (e) c"1803.. better local dopant mixing, the radial segregation is reduced slightly. If one examines the interface shape carefully, the interface is no longer axisymmetric; the left side is lower due to the higher ambient temperature there. With 10 RPM rotation, the convection is damped signi"cantly. Especially, as shown by the dopant "eld away from the interface, it becomes almost axisymmetric. However, severe abnormal dopant segregation is found near the growth interface; Cc is increased to 2.0111 and Cc is reduced to only 0.4425. In fact, due to the asymmetric heating, the asymmetric interface is a problem. When the ampoule is rotating, continuous melting (n ) e '0) and growth (n ) e (0) pro( ( ceed. As a result, the melting part leads to the lower dopant concentration due to the dilution from the crystal, while the growth part to a higher concentration. This can be further explained by the interface dopant balance in Eq. (10), where the rotation term becomes important when (n ) e ) is not trivial. ( As mentioned previously, this boundary condition is not exact, because the interface dopant equilibrium may not be achieved. However, even (1!K) C in Eq. (10) is replaced by C!C , the di!erence is . still not much. Therefore, such a severe rotational segregation is believed to be realistic. Even without buoyancy convection, the rotational segregation is still signi"cant. Fig. 8 shows the e!ects of 5 RPM rotation for the asymmetric heating at Ra "0 and 10. At Ra "0 and X"0, 2 2 the asymmetric segregation in Fig. 8a is due to the asymmetric interface shape. As the ampoule is rotated at 5 RPM, as shown in Fig. 8b, the dopant "eld near the interface is a!ected signi"cantly due to the rotational segregation at the interface, and the e!ect increases with the increasing radial coordinate due to the increasing azimuthal velocity (rX). With the buoyancy convection at Ra "1;10 2 (Fig. 8c), the rotational segregation is further enhanced. The dopant "eld is further distorted by the convection in the rotational direction. Away from the interface, the dopant "eld is still nearly axisymmetric. One may easily imagine that the rotational segregation is also the cause of the rotation striations due to the continuous growth and remelting. The rotational striations in Czochralski growth have been known for a long time [17], but to our best knowledge no numerical work has been.
(11) 350. C.W. Lan et al. / Journal of Crystal Growth 212 (2000) 340}351. Fig. 7. E!ects of asymmetric heating and ampoule rotation on the calculated #ow patterns (LHS) and dopant "elds (RHS) for Ra "2.489;10: (a) *¹ "13C, X"0 RPM (Ta"0); 2 (b) *¹ "13C, X"10 RPM (Ta"4.0556;10). . reported for that. For VB growth, the numerical simulation of the rotational segregation is conducted here for the "rst time. By considering the axial growth and crystal rotation, an outcome from this e!ect is the rotational striations proposed by Gatos [17]. Furthermore, from Fig. 8, it is clear that even for only 5 RPM rotation, the continuous growth and melting rate may be much higher than the axial growth rate (; ) to induce the severe segregation. Therefore, in such a case, rotation is believed to be harmful. Constitutional supercooling due to rotation can be an important issue as well.. 4. Conclusions The e!ects of steady ampoule rotation on the 3D #ows and dopant segregation induced by ampoule tilting or asymmetric heating in Bridgman crystal growth are investigated numerically for the "rst time. It is found that rotation can damp the 3D. Fig. 8. E!ects of 5 RPM ampoule rotation on dopant "elds for *¹ "13C: (a) Ra "0, X"0 RPM; (b) Ra "0, X" 2 2 5 RPM; (c) Ra "1;10, X"5 RPM. 2. #ows; however, it is still hard to achieve an axisymmetric convection. Nevertheless, if the convection can be suppressed su$ciently, the di!usion becomes dominant in the segregation. We still have a good chance to have a nearly axisymmetric growth. This is particularly true for the growth in a reduced-gravity environment using low-speed rotation, even through the gravity orientation is arbitrary. The e!ects of ampoule rotation for asymmetric heating are more complicated. In addition to the 3D #ows, due to the asymmetric growth interface, the continuous melting and growth can lead to severe radial segregation. This rotational segregation is still signi"cant even at low-speed rotation and without thermal convection. Although the present simulation uses a pseudo-steady-state approximation, which may not describe accurately the dopant "eld, the calculated results here may still provide a useful insight to the detailed transport phenomena in Bridgman crystal growth. Again,.
(12) C.W. Lan et al. / Journal of Crystal Growth 212 (2000) 340}351. further crystal growth experiments are necessary to verify the results proposed here.. Acknowledgements The authors are grateful for the support from the National Science Council and the National Center for High Performance Computing of the Republic of China under Grant No. 88-2214-E-002-006.. References [1] W.A. Gault, E.M. Monberg, J.E. Clemans, J. Crystal Growth 74 (1986) 491. [2] K. Hoshikawa, H. Nakanishi, H. Kohda, M. Sasaura, J. Crystal Growth 94 (1989) 643. [3] E.M. Monberg, W.A. Gault, F. Simchock, F. Domingguez, J. Crystal Growth 83 (1987) 174. [4] E.M. Monberg, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth 2a: Basic Techniques, North-Holland, Amsterdam, 1994, p. 52. [5] M.C. Liang, C.W. Lan, J. Crystal Growth 167 (1996) 320.. 351. [6] Q. Xiao, S. Kuppurao, A. Yeckel, J.J. Derby, J. Crystal Growth 167 (1996) 292. [7] D.H. Kim, P.M. Adornato, R.A. Brown, J. Crystal Growth 89 (1988) 339. [8] S. Motakef, J. Crystal Growth 104 (1990) 833. [9] M. Yao, A. Chait, A.L. Fripp, W.J. Debnam, J. Crystal Growth 173 (1997) 467. [10] V. Uspenskii, J.J. Favier, Int. J. Heat Mass Transfer 37 (1994) 691. [11] C.W. Lan, J. Crystal Growth 197 (1999) 983. [12] C.W. Lan, J.H. Chian, J. Crystal Growth 203 (1999) 286. [13] A. Yeckel, F.P. Doty, J.J. Derby, J. Crystal Growth 203 (1999) 87. [14] C.W. Lan, J. Crystal Growth, in preparation. [15] W.A. Arnold, W.R. Wilcox, F. Carlson, A. Chait, L.L. Regel, J. Crystal Growth 119 (1992) 24. [16] C.W. Lan, M.C. Liang, J. Crystal Growth 186 (1998) 203. [17] H.C. Gatos, J. Electrochem. Soc. 122 (1975) 287. [18] M.C. Liang, C.W. Lan, J. Comp. Phys. 127 (1996) 330. [19] C.W. Lan, M.C. Liang, J. Comp. Phys. 152 (1999) 55. [20] P.M. Adornato, R.A. Brown, J. Crystal Growth 80 (1987) 155. [21] C.W. Lan, F.C. Chen, Comput. Methods Appl. Mech. Engng. 131 (1996) 191. [22] W.A. Arnold, D.A. Jacqmin, R.L. Gaug, A. Chait, Spacecraft Rockets 28 (1991) 238. [23] A.Y. Polovko, V.V. Sazonov, V.S. Yuferev, J. Crystal Growth 198 (1999) 182..
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