Buoyancy and rotation in small-scale vertical Bridgman growth of cadmium zinc telluride using accelerated crucible rotation

Journal of Crystal Growth 233 (2001) 599 608 Buoyancy and rotation in small-scale vertical Bridgman growth of cadmium zinc telluride using accelerated crucible rotation Andrew Yeckel, Jeffrey J. Derby* Department of Chemical Engineering and Materials Science, Army HPC Research Center and Minnesota Supercomputing Institute, University of Minnesota, 151 Amundson Hall, 421 Washington Avenue, S.E., Minneapolis, MN 55455-0132, USA Received 1 January 2001; accepted 29 June 2001 Communicated by M.E. Glicksman Abstract Theoretical simulations of vertical Bridgman growth of cadmium zinc telluride are performed to study the effects of the accelerated crucible rotation technique (ACRT). The results indicate that thermal buoyancy has a dramatic effect on the flow, even in a relatively small system at high rotation rate, contrary to assertions made in recent papers by Liu et al. (J. Crystal Growth 219 (2000) 22). We demonstrate their prior results greatly overstate the effectiveness of ACRT at promoting mixing. Contrary to conventional wisdom, the ACRT rotation cycle considered here for a small-scale growth system actually suppresses mixing in the melt near the ampoule wall, resulting in diffusion-limited mass transport there. r 2001 Elsevier Science B.V. All rights reserved. PACS: 81.10.fq; 81.05.Dz; 47.32. y; 47.20. k Keywords: A1. Computer simulation; A1. Convection; A1. Diffusion; A1. Fluid flows; A2. Accelerated crucible rotation technique; B2. Semiconducting II VI materials 1. Introduction Scheel and Schulz-DuBois introduced the accelerated crucible rotation technique (ACRT) to enhance mixing in solution crystal growth systems [3]. Later, ACRT was applied to several melt growth systems, such as the vertical Bridgman and traveling heater methods. Although numerous *Corresponding author. Tel.: +1-612-625-8881; fax: +1-612-626-7246. E-mail address: derby@tc.umn.edu (J.J. Derby). experimental studies were conducted [4 9], until recently there have been relatively few theoretical studies, most likely because of the complexity that rotation introduces to systems with significant buoyancy-driven flow [10,11]. In the past few years interest in ACRT-modified melt growth has re-emerged, spawning a number of theoretical studies that have been facilitated by the rapid advance in computing power [1,2,12 14]. Although these studies have contributed to better understanding of the effects of accelerated rotation on convection in vertical Bridgman and 0022-0248/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0022-0248(01)01601-3

600 A. Yeckel, J.J. Derby / Journal of Crystal Growth 233 (2001) 599 608 traveling heater growth, many questions remain unanswered. The usual purpose of applying accelerated rotation is to maximize convective flow by generating Ekman boundary layers at solid surfaces [15]. Some of the possible advantages and disadvantages of doing so are detailed in Yeckel and Derby [14]. Yeckel and Derby also confirmed experimental observations of the appearance of a Taylor G.ortler instability at the crucible wall and found that the flow associated with this instability contributes significantly to bulk mixing. The work also demonstrated that, despite strong centrifugal forces induced by ACRT, significant buoyancy effects persist in the system studied, particularly near the interface and crucible wall. This letter was motivated by recent publications of Liu et al. who performed simulations on the effect of accelerated crucible rotation on vertical Bridgman growth of cadmium zinc telluride (CZT) [1] and mercury cadmium telluride (MCT) [2]. In Liu, Jie, and Zhou [1] (referred to hereafter as LJZ), these authors state that Since ACRTforced convection (especially near the melt/crystal interface) resulting mainly from Ekman pumping effect sustains a long time and is frequently several orders of magnitude greater than those due to natural convection [16], the latter can be neglected. On this basis LJZ neglect the effects of thermal and solutal buoyancy forces in their model. The results presented herein will demonstrate that this conclusion greatly oversimplifies the system and leads to erroneous conclusions. However, our results are not presented solely to dispute the findings of LJZ, rather we will discuss some interesting aspects of ACRT applied to the growth of small-diameter CZT crystals which differ significantly from the large-scale systems we have studied previously [14]. Liu et al. refer to an overview paper by Brice et al. [16] to justify their assumption to neglect buoyant flows; however, this reference is largely descriptive on the topic of ACRT effects rather than mathematical. The validity of their conjecture depends strongly on many factors, particularly system size and rotation rate, and LJZ make no attempt to estimate the relative strength of buoyant versus centrifugal forces in the particular system they model. Approaching the topic in a more quantitative manner, dimensionless analysis suggests that rotational flow in their system is reasonably intense, with a Reynolds number of Re ¼ R 2 O=n ¼ 425:8; based on the inner ampoule radius R; the maximum rotation rate O; and the kinematic viscosity of n (thermophysical properties and operating parameters for this system are listed in Tables 1 and 2). In an axisymmetric system, however, convective mass transport occurs only in Table 1 Physical properties Parameter Material Symbol Value Units Ref. Thermal conductivity CdTe solid k s 9:07 10 3 W/cm K [20] CdTe melt k m 1:085 10 2 W/cm K [20] Ampoule k g 2:5 10 2 W/cm K [1] Density CdTe solid r s 5:68 g/cm 3 [21] CdTe melt r m 5:68 g/cm 3 [21] Ampoule r g 2:2 g/cm 3 [1] Heat capacity CdTe solid C p;s 0.160 J/g K [20] CdTe melt C p;m 0.187 J/g K [20] Ampoule C p;g 1.05 J/g K [1] Heat of fusion CdTe DH f 209.2 J/g [22] Melting temperature CdTe T mp 1373 K [22] Thermal expansivity CdTe b 5:0 10 4 K 1 [23] Emissivity Ampoule e g 0.95 [1] Kinematic viscosity CdTe n 4:15 10 3 cm 2 /s [21] Diffusion coefficient Zn in CdTe melt D m 1:0 10 4 cm 2 /s [24] Segregation coefficient Zn in CdTe k 1.35 [25]

A. Yeckel, J.J. Derby / Journal of Crystal Growth 233 (2001) 599 608 601 Table 2 Operating parameters Parameter Value Units Ampoule translation rate 1:0 10 4 cm/s Ampoule rotation rate 0 30 rpm Ampoule inner radius 0.75 cm Ampoule outer radius 1.0 cm Ampoule inner length 8.0 cm Final crystal length 7.6 cm Initial Zn concentration 0.04 mole fraction the meriodonal plane. The intensity of the meriodonal flow is best judged in terms of the Ekman number Ek; from which a Reynolds number representing the intensity of flow is given by Re E ¼ Ek 1 ¼ Re 1=2 ¼ 20:6; indicative of a weak flow. A reckoning of buoyant effects yields a Grashof number estimate of Gr ¼ gbr 3 DT=n 2 E10 4 ; where g is the gravitational constant, b is the thermal expansion coefficient, and DT is a characteristic radial temperature difference (which we estimate here as less than 1 K). This is a rather small Grashof number, also indicative of a weak flow, as reasoned by LJZ, but it is far from certain that the flow due to centrifugal effects dominates flow due to buoyant effects. Moreover, the reversal of rotation direction during the ACRT cycle will necessarily result in periods of time when the rotational flow is very slow and the rotational Reynolds number will approach zero. At times during these periods, i.e., when the rotational flows in the system are slowing and reversing, buoyant flow is likely to dominate rotational flow. Thus, dimensionless analysis strongly suggests that buoyancy effects are important to mass transfer and segregation during much of the ACRT cycle. By neglecting buoyancy effects in their model, LJZ obtain results that greatly overstate the effectiveness of ACRT at promoting mixing within the melt under the conditions of their simulation. The results are especially misleading in that the rotation cycle described by these authors as best proves particularly ineffective at promoting mixing according to our calculations. Another interesting outcome of our study is the observation that if the ACRT cycle is not vigorous enough, the overall effect is to reduce the flow intensity, thus allowing diffusive mass transport to become more important. This result is exactly contrary to the customary wisdom that ACRT will act to better mix a system. 2. Governing equations and discretization For the sake of brevity, we refer the reader to our earlier work for a detailed model description [14,17]. We consider flow and mass transfer in the melt and heat transfer through all phases, with the solid liquid interface located along the melting-point isotherm. These governing equations are the same as used by LJZ, except that our model includes thermal buoyancy effects as characterized by the Boussinesq approximation whereas LJZ ignore buoyant flow altogether. The physical and operating parameters, boundary conditions, and ampoule geometry used here are the same as used in LJZ, with two minor exceptions. In LJZ the discretized domain includes a small gas space above the melt; we truncate our model domain at the top of the melt and simply apply their boundary conditions. The second difference is that we can only estimate the furnace temperature profile by approximately fitting the curve shown in Fig. 4 of LJZ. Doing so we obtain the following fourth-order polynomial: T ¼ 0:9937 þ 7:450 10 3 x 4:262 10 4 x 2 1:221 10 5 x 3 1:374 10 6 x 4 ; ð1þ where the dimensionless temperature T is measured in units of the melting temperature. The dimensionless position x (in a reference frame fixed to the furnace, measured in units of the ampoule inner radius) is a function of time and axial position z (in a reference frame fixed to the ampoule) x ¼ z V f t; ð2þ where V f is the furnace translation rate. The equations are solved in the reference frame fixed to the ampoule, with the origin located at the bottom of the computational domain. We employ the Galerkin finite element method [18] to discretize the model equations with a mesh

602 A. Yeckel, J.J. Derby / Journal of Crystal Growth 233 (2001) 599 608 of 1440 elements, the same number of elements used by LJZ. Our model comprises 26 179 mathematical unknowns arising from biquadratic velocity/linear discontinuous pressure elements; this number is approximately four times that resulting from the lower-order finite volume elements used by LJZ. The governing equations are time-integrated using the trapezoid rule for cases with and without buoyancy (equivalent to zero gravity and earth gravity) for 22 360 s without ACRT. In the case without buoyancy, the initial condition is a quiescent state; in the case with buoyancy, the initial condition is steady flow in a stationary furnace. For both, the initial concentration is a uniform 4.0 mol% in the melt. After this initial period, the ACRT cycle identified in LJZ as best, shown in Fig. 1 (Fig. 3(d) of LJZ), is applied. The equations are integrated through 100 ACRT cycles to a final time of 24 738 s; results shown in the figures correspond to various times within the 100th ACRT cycle. Solution accuracy was tested with respect to time step size and mesh refinement for the case without buoyancy. Reducing the time step size of 0.2 s by a factor of four changed maximum and minimum streamfunction values through an ACRT cycle by no more than 0.62%. Solutions obtained using a very fine mesh of 3360 elements and 68 239 degrees of freedom were compared to that computed using the original mesh and found to be within 0.13% for the streamfunction minimum, 0.02% for the streamfunction maximum, and 0.03% for the concentration profile along the melt centerline. We conclude that the mesh of 1440 Fig. 1. ACRT rotation cycle employed here and in Ref. [1]. elements and time step size of 0.2 s used in all calculations well resolves both flow and concentration fields. 3. Results 3.1. Flow and concentration field without thermal buoyancy This series of computations is meant to replicate the conditions considered in LJZ. Here, no buoyant flows are considered; all flows are driven by solely by ampoule rotation. Fig. 2 shows streamlines and concentration contours at three times within the 100th ACRT cycle. These times correspond to the indicated solid points on the ACRT cycle shown in Fig. 1, specifically at the end of the acceleration period (marked in both figures as A and D for the streamlines and concentration contours, respectively), at the end of the deceleration period (B and E), and at the end of the stationary period (C and F). These results correspond to the results shown in Figs. 5 7 of LJZ. In many respects the two sets of results are similar, but there are several qualitative and quantitative differences. A comparison of the solid liquid interface shape shows that the interface deflection in LJZ is nearly double that in our results. Convective heat transport cannot be responsible, since our results differ even at the start of ACRT, when the liquid is quiescent. The likely reason for the different deflections is the difference in the furnace profiles used. We note that Eq. (1) was fitted to a handful of points from Fig. 4 of LJZ; a discrepancy of several K/cm in the temperature gradient at the interface is certainly possible. Another explanation may be the different numerical algorithms employed to solve the governing mathematical equations. The cell-and-marker model used by LJZ does not allow for the direct implementation of interfacial flux balance conditions (the last two equations given on page 24 in LJZ), rather a fictitious local source term accounts for latent heat release and the jump in thermal conductivity at the interface within the partially filled cells. Our approach, known as the isotherm method,

A. Yeckel, J.J. Derby / Journal of Crystal Growth 233 (2001) 599 608 603 explicitly represents the solidification interface and allows for the accurate specification of fluxes [19]. A comparison of streamlines shows that our results are qualitatively similar to those of LJZ in terms of shape and location of vortices. Our maximum and minimum streamfunction values deviate from those of LJZ by less than 20%, with one notable exception: the maximum value at the end of the acceleration period differs by nearly a factor of three. However, the vortex to which this value corresponds is quite weak, and, when the difference in maximum values is compared relative to the overall strength of the flow, the discrepancy is only 5%. The source of the discrepancy between our results and those of LJZ is not clear, but certainly the difference in interface deflection contributes to some extent. The largest discrepancy in the two sets of results occurs in the concentration field predictions. The LJZ concentration contours in the upper part of the melt are similar to our results, but dramatic differences occur in the lower part of the melt. There are several possible causes. First, it is not clear at what time LJZ compute the results shown in their paper. Whereas the exact time introduces little uncertainty to the streamline comparison (because the flow changes very little from cycle to cycle), it introduces considerable uncertainty to the comparison of concentration contours. Due to segregation and mixing, the contours undergo constant change throughout the entire ACRT period. Another cause of the discrepancy is the time integration procedure of LJZ, which cannot be accurate. LJZ use a time step of 4 s when integrating the concentration field. For the cycle considered here, each ACRT component period (acceleration, deceleration, and stationary) is 4 s long. The detailed mechanics of flow transitions, in b Fig. 2. Streamfunction and concentration contours in 100th ACRT cycle without thermal buoyancy, at times indicated in Fig. 1. Dashed streamfunction contours indicate counter-clockwise rotation at intervals Dc ¼ c max =10; and solid contours indicate clockwise rotation at intervals Dc ¼ c min =10; in cm 3 /s. A: c min ¼ 8:72 10 3 ; c max ¼ 3:13 10 8 : B: c min ¼ 5:01 10 4 ; c max ¼ 5:88 10 3 : C: c min ¼ 5:45 10 4 ; c max ¼ 3:46 10 3 : Concentration contours at intervals Dc ¼ ðc max c min Þ=20; with c max ¼ 3:692; c min ¼ 3:328; in mol%.

604 A. Yeckel, J.J. Derby / Journal of Crystal Growth 233 (2001) 599 608 which vortices form, decay, and reverse direction, cannot be accurately computed using one time step. Furthermore, it is even less likely that the dynamics of convective mass transport are computed accurately, since the subtle balance between convection and diffusion during the periods of strengthening and weakening flows would be unresolved. This may explain features in each of the concentration contour plots in LJZ where two contours merge together in the lower part of the melt near the ampoule wall. The merging of two contours is unphysical in this system and may indicate the presence of an extremely sharp, but unresolved, concentration gradient in their computations. Mechanistically, our computations show the complicated evolution of secondary flows driven by the ACRT rotation cycle. Fig. 2(A) shows the flow resulting after spin-up. In addition to a strong azimuthal flow (not shown directly in the figure), the meridional flow exhibits a fairly strong clockwise vortex near the solid liquid interface due to the effects of Ekman pumping at the solid surface. The counter-clockwise vortex above this and the extremely weak clockwise region of circulation occupying the top two-thirds of the melt are remnants of the periodic flow reversals driven by the ACRT cycle. The evolution of this flow structure is evident in Fig. 2(B), where flows just after ampoule deceleration are shown. Here, the deceleration of ampoule rotation acts to reverse the direction of Ekman pumping, and a new counter-clockwise vortex grows out from the corner. As the ampoule is held stationary in the cycle, the fluid continues to spin down, continuing the forces which strengthen this counter-clockwise vortex. It grows and displaces the original clockwise vortex formed during spin-up; see Fig. 2(C). The cycle is repeated, whereby meridional circulations of opposing direction are spawned at the solid liquid interface while displacing the older structures upward, which eventually weaken and decay. The effect of these flows on mass transfer is quite complicated. In general, the middle portion of the melt is reasonably well mixed; however, there are significant gradients in concentration near both ends of the melt column. The more significant region is near the liquid solid interface, where the effects of flow reversal across the interface are clearly evident by the bending of iso-concentration contours in Figs. 2(D), (E), and (F). A more detailed discussion on the effects of the ACRTdriven flows on axial mass transfer is presented in Section 3.3 3.2. Flow and concentration field with thermal buoyancy In this section, we present results for the identical case considered in the prior section, except that flows driven by thermal buoyancy are included. These effects were not considered in LJZ. The dimensionless Grashof and Reynolds numbers computed a posteriori from the case of buoyant flow alone (no ACRT) yield Gr ¼ 9:9 10 3 ; where DT is computed as the maximum radial temperature difference across the melt, and Re ¼ 5:8; where the characteristic velocity is taken to be the computed maximum velocity in the buoyant flow. The Reynolds number computed in the same manner for ACRT without buoyant flow (the results in the previous section) varies from 11.2 to 46.8, depending on the time within the ACRT cycle. The upper limit is an order of magnitude greater than in the buoyant flow case, but this upper limit is achieved only briefly near the peak rotation rate in the cycle. The Reynolds number remains below 15 for most of the ACRT cycle, comparable to the buoyant flow case, which indicates that buoyant effects are indeed important in this system. This importance is demonstrated in the case discussed below, which reveals that the flow and concentration fields differ dramatically from those discussed in the previous section. The entire flow structure changes when thermal buoyancy is included, and there are several features of particular interest in the streamlines of Fig. 3. First, the lower vortex near the solid liquid interface reverses direction of circulation much sooner in the ACRT deceleration than for the case without buoyancy; Fig. 3(B) shows this vortex to be well established, whereas the counter-clockwise vortex in Fig. 2(B) is still growing out from the ampoule wall. This observation can be understood by considering the underlying

A. Yeckel, J.J. Derby / Journal of Crystal Growth 233 (2001) 599 608 605 buoyant forces in this system near the solid liquid interface. Due to latent heat effects, the solid liquid interface deflects downward and the radial thermal gradients are reversed with respect to the overlying melt. Near the interface, warmer fluid lies toward the centerline and cooler fluid lies near the ampoule wall, therefore, the underlying buoyant forces drive a flow which rises at the centerline and descends near the ampoule wall, producing a counter-clockwise vortex. Thus buoyancy reinforces the reversal of Ekman pumping upon spindown of the fluid. Second, the vertical flow structure does not change as appreciably as the flows in Fig. 2 during the ACRT cycle, namely the overall flow comprises two stacked vortices. As discussed above, the lower vortex reverses direction through the ACRT cycle; however, underlying buoyant effects in the bulk of the melt (where warmer fluid rises near to the ampoule wall and cooler fluid descends along the system centerline) tend to keep a single, clockwise vortex spinning. The notable change to this structure occurs during spin-up, when the flow in the upper part of the melt is nearly stagnant, as shown in Fig. 3(A), at the end of the acceleration period. At this stage in the rotation cycle, the effect of thermal buoyancy is to nearly cancel the effect of ACRT. This effect is not seen at all in the prior case with no buoyancy. The situation for mass transfer in the melt is also quite different from the prior case. Since the twocell flow structure persists for most of the ACRT cycle, there is a separation streamline where the meridional component of flow is nearly stationary; this is clearly visible in Figs. 3(B) and (C). Mass transport across this separation streamline is virtually unaided by convection. As a consequence, a sharp internal concentration layer is formed in the vicinity of the separation streamline, b Fig. 3. Streamfunction and concentration contours in 100th ACRT cycle with thermal buoyancy, at times indicated in Fig. 1. Dashed streamfunction contours indicate counter-clockwise rotation at intervals Dc ¼ c max =10; and solid contours indicate clockwise rotation at intervals Dc ¼ c min =10; in cm 3 /s. A: c min ¼ 1:14 10 2 ; c max ¼ 9:52 10 4 : B: c min ¼ 4:63 10 3 ; c max ¼ 3:83 10 3 : C: c min ¼ 2:20 10 3 ; c max ¼ 4:54 10 3 : Concentration contours at intervals Dc ¼ ðc max c min Þ=20; with c max ¼ 3:964; c min ¼ 3:676; in mol%.

606 A. Yeckel, J.J. Derby / Journal of Crystal Growth 233 (2001) 599 608 across which transport is nearly diffusion-limited. This situation is much like classical vertical Bridgman growth with thermal buoyancy when ACRT is not used. Also, the weakening of the upper flow cell on spin-up limits the effect of convection near the ampoule wall, and mass transport to the interface near the wall is aided very little by convection at any time in the rotaion cycle. 3.3. Assessing the effects of ACRT on segregation In order to assess the degree of mixing caused by ACRT and its likely effect on radial and axial segregation, we plot axial zinc concentration profiles in the melt along the ampoule centerline and side wall in Fig. 4. Figs. 4(A) and (B) are plotted from model computations with no buoyancy, whereas Figs. 4(C) and (D) show results for our model with buoyant effects included. We first consider predictions from the model which does not include buoyancy (i.e., the model employed by LJZ). Fig. 4(A) shows the initial concentration profiles through the melt, before the start of the ACRT cycles. Here, the flow is everywhere quiescent, so the concentration profiles are determined by segregation at the growth interface and diffusion alone throughout the domain. This is the classical scenario of diffusion-limited growth. For long enough ampoules, axial segregation will be nearly uniform along most of the crystal, but a small amount of radial Fig. 4. Axial concentration profiles in melt along ampoule centerline and side wall. A: at start of ACRT without thermal buoyancy. B: after 100 ACRT cycles without thermal buoyancy. C: at start of ACRT with thermal buoyancy. D: after 100 ACRT cycles with thermal buoyancy.

A. Yeckel, J.J. Derby / Journal of Crystal Growth 233 (2001) 599 608 607 segregation will occur due to interface deflection [17]. Such a result is never attainable under earth gravity and would be difficult to obtain even in microgravity experiments conducted in near-earth orbit. Fig. 4(B) shows axial zinc concentration profiles through the melt after 100 ACRT cycles; it is evident that considerable mixing has occurred, but the concentration is by no means uniform throughout the melt. The zinc concentration at the solid liquid interface is relatively uniform, as seen by comparing the level of the profiles at their leftmost extent (they begin at different positions due to the considerable interface deflection caused by the ACRT flows). LJZ also make note of this and interpret it to mean that radial segregation is very small. However, a more careful consideration of radial uniformity is to compare the concentration level in the crystal at a fixed axial location. Using this measure, the radial compositional variation is significant, the cause being interface deflection. Indeed, it is clear that, in the case without buoyancy, ACRT makes radial segregation worse, since ACRT causes interface deflection to increase. We now consider our model predictions when buoyancy is included by plotting axial zinc composition in the melt for the initial condition (buoyant flow with no ACRT) in Fig. 4(C) and after the application of 100 ACRT cycles in Fig. 4(D). The results shown in Fig. 4(C) are typical of vertical Bridgman systems, with the axial composition in the melt exhibiting an upper zone that is fairly well mixed, a lower zone that is less well mixed, and a transition between the zones in which mass transport is dominated by diffusion across a shear layer. A large degree of radial inhomogeneity is present across the solid liquid interface due to incomplete mixing in the lower zone. The situation is changed after the application of 100 ACRT cycles, as shown in Fig. 4(D). Interestingly, upon the application of ACRT, the melt is less mixed than the initial state with only buoyant flow. This is evident by the large radial variation of concentration throughout most of the melt and by the steeper axial gradients of composition compared to those in Fig. 4(C). Application of ACRT destroys the two zone structure, and axial mass transport near the wall is mainly determined by diffusion; here, the ACRT forces act to cancel much of the mixing in the upper portion of the melt. In addition, a large radial variation of zinc concentration along the interface persists, which, coupled with the predicted increase in interface deflection, indicates substantial radial segregation. Our predictions show that the ACRT cycle described by LJZ as best, in the sense of minimizing radial segregation, most likely will result in greater radial segregation compared to the case of growth without application of ACRT. 4. Conclusions We have presented results which refute those of the recent publication of Liu et al. [1]. While we do not intend to indict these researchers, we wish to emphasize that the effective modeling of crystal growth systems requires a careful consideration of the coupled effects of many nonlinear phenomena. LJZ chose to simplify a mathematical model by neglecting the seemingly small effects of buoyant flow; however, they obtained results which are notably contrary to a more complicated (and realistic) model. Indeed, our model predicts that the rotation cycle described by these authors as best will likely result in a less desirable outcome (i.e., greater radial segregation) than doing nothing at all. By themselves, the new results presented here show some very interesting effects. First, it is instructive to compare these results to those in our earlier work on ACRT effects on larger-scale systems [14]. Our prior calculations showed a case where ACRT resulted in much better mixing within the melt. A key difference in the two sets of results is the appearance of the Taylor G.ortler instability in the system modeled in [14]. This instability triggered the formation and growth of many small, but intense, vortices stacked axially along the ampoule inner wall during the spindown portion of the ACRT cycle. This intense flow greatly enhanced mixing in the upper part of the melt. We found no tendency for the flow in the system considered here to exhibit this instability; likely its occurrence would require a more intense spin-down cycle. The behavior of the larger scale

608 A. Yeckel, J.J. Derby / Journal of Crystal Growth 233 (2001) 599 608 system differed in another key manner as well: the lower vortex increased and then decreased in size during the ACRT cycle, causing a vertical pumping of liquid from the upper part of the melt into the lower part, causing substantial mixing to also occur near the solid liquid interface. Here, the lower vortex of the system reversed during the ACRT cycle but did not appreciably change in size or form. Our computations indicate that the applied ACRT cycle for the system considered in the current paper is not vigorous enough to promote good mixing in the melt. In fact, the overall effect of the ACRT cycle applied here is to reduce the overall flow intensity, thus allowing diffusive mass transport to become more important and the system to become less well-mixed than a system not subject to any rotation. This result is exactly contrary to the customary wisdom that ACRT will act to better mix a system. The results presented here and in our prior study [14] show that the interaction of buoyancy and centrifugal forces is quite complicated and highly dependent on system operating and design parameters. Clearly it is not possible to draw broad generalizations regarding the effect of ACRT on the vertical Bridgman system. This conclusion is consistent with the experiments of Capper et al. [5] who found that seemingly minor changes in the rotation cycle sometimes have a large effect on the quality of crystal grown. We believe that an appropriate level of detailed modeling is absolutely essential to understand such results. Acknowledgements This work was supported in part by NASA, the Minnesota Supercomputing Institute, and the Army HPC Research Center under the auspices of the Department of the Army, Army Research Laboratory cooperative agreement DAAH04-95- 2-0003/contract DAAH04-95-C-0008, the content of which does not necessarily reflect the position or policy of the government, and no official endorsement should be inferred. References [1] X. Liu, W. Jie, Y. Zhou, J. Crystal Growth 219 (2000) 22. [2] X. Liu, W. Jie, Y. Zhou, J. Crystal Growth 209 (2000) 751. [3] H.J. Scheel, E.O. Schulz-Dubois, J. Crystal Growth 8 (1971) 304. [4] P. Capper, J.J.G. Gosney, C.L. Jones, J. Crystal Growth 70 (1984) 356. [5] W.G. Coates, P. Capper, C.L. Jones, J.J.G. Gosney, C.K. Ard, I. Kenworthy, A. Clark, J. Crystal Growth 94 (1989) 959. [6] P. Capper, J.J.G. Gosney, C.L. Jones, I. Kenworthy, J. Electron. Mater. 15 (1986) 371. [7] P. Capper, W.G. Coates, C.L. Jones, J.J.G. Gosney, C.K. Ard, I. Kenworthy, J. Crystal Growth 83 (1987) 69. [8] P. Capper, J.C. Brice, C.L. Jones, W.G. Coates, J.J.G. Gosney, C.K. Ard, I. Kenworthy, J. Crystal Growth 89 (1988) 171. [9] R.U. Bloedner, P. Gille, J. Crystal Growth 130 (1993) 181. [10] J.R. Carruthers, J. Electrochem. Soc. 114 (1967) 959. [11] H.P. Greenspan, The Theory of Rotating Fluids, Cambridge University Press, London, 1968. [12] R.U. Barz, P. Sabhapathy, M. Salcudean, J. Crystal Growth 180 (1997) 566. [13] C.W. Lan, J.H. Chian, J. Crystal Growth 203 (1999) 286. [14] A. Yeckel, J.J. Derby, J. Crystal Growth 209 (2000) 734. [15] E.O. Schulz-Dubois, J. Crystal Growth 12 (1972) 81. [16] J.C. Brice, P. Capper, C.L. Jones, J.J.G. Gosney, Prog. Crystal Growth Charact. 13 (1986) 197. [17] A. Yeckel, F.P. Doty, J.J. Derby, J. Crystal Growth 203 (1999) 87. [18] P.M. Gresho, R.L. Sani, Incompressible Flow and the Finite Element Method, Wiley, West Sussex, England, 1998. [19] H.M. Ettouney, R.A. Brown, J. Comput. Phys. 49 (1983) 118. [20] S. Sen, W.H. Konkel, S.J. Tighe, L.G. Bland, S.R. Sharma, R.E. Taylor, J. Crystal Growth 86 (1988) 111. [21] V.M. Glazov, S.N. Chizhevskaya, N.N. Glagoleva, Liquid Semiconductors, Plenum, New York, 1969. [22] K. Zanio, in: R.K. Willardson, A.C. Beer (Eds.) Semiconductors and Semimetals, Vol. 13, Academic Press, New York, 1978. [23] P. Rudolph, M. M.uhlberg, Mater. Sci. Eng. B 16 (1993) 8. [24] K. Schwenkenbecher, P. Rudolph, Crystal Res. Tech. 20 (1985) 1609. [25] K. Mochizuki, K. Masumoto, K. Miyazaki, Mater. Lett. 6 (1988) 119.

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