Model experiments for the Czochralski crystal growth technique

Transcription

1 Eur. Phys. J. Special Topics 220, (2013) EDP Sciences, Springer-Verlag 2013 DOI: /epjst/e THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS Review Model experiments for the Czochralski crystal growth technique A. Cramer a, J. Pal, and G. Gerbeth Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, Dresden, Germany Received 17 December 2012 / Received in final form 5 February 2013 Published online 26 March 2013 Abstract. A lot of the physical and the numerical modeling of Czochralski crystal growth is done on the generic Rayleigh-Bénard system. To better approximate the conditions in a Czochralski puller, the influences of a rounded crucible bottom, deviations of the thermal boundary conditions from the generic case, crucible and/or crystal rotation, and the influence of magnetic fields are often studied separately. The present contribution reviews some of these topics while concentrating on studies of the flow and related temperature fluctuations in systems where a rotating magnetic field (RMF) was applied. The three-dimensional convective patterns and the resulting temperature fluctuations will be discussed both for the mere buoyant case and for the application of an RMF. It is shown that a system between a Rayleigh-Bénard and a more realistic configuration, which is still cylindrical but whose surface is partially covered by a crystal model, behaves much the same as a Rayleigh-Bénard system. An RMF can be used to damp the temperature fluctuations. Secondly, a more Czochralski-like system is examined. It turns out that the RMF does not provide the desired damping of the temperature fluctutions in the parameter range considered. 1 Introduction A Czochralski (CZ) crystal growth facility is a complex system [1]. It seems impossible to conceive of an analytical model that takes into account the curved crucible bottom, realistic thermal boundary conditions, crucible and/or crystal rotation, and, if present, the influence of magnetic fields. Simplifications are needed, which are often used in the numerical and the physical modeling of CZ systems. The most common one is the Rayleigh-Bénard (RB) configuration. The basics of such cylindrical systems are reviewed in Sect In the following, the restrictions of such generic modeling are described. Our experiments, which overcome some shortcomings are introduced. That natural convection differs, notably for smaller height H to diameter D aspect ratios AR = H/D between cylindrical enclosures and the curved bottom of a CZ crucible, was shown almost a a.cramer@hzdr.de

2 260 The European Physical Journal Special Topics 20 years ago [2]. In a more recent paper, the influence of the crucible bottom on the growth of crystals was investigated [3]. Yet the flow driven by a rotating magnetic field (RMF) may be different in a realistic CZ-like geometry and in a cylinder. To the best of our knowledge, no investigation on that subject exists. Hence, the influence of the crucible bottom will be discussed in Sects. 2.2 and Magnetic fields provide the means for contact-less melt flow control in crystal growth. Static fields can be used to damp the intensity of the mean flow and the fluctuations. This results in a reduced rate of the convective transport and in a suppression of undesired instabilities. Well-defined flow structures to alter the distributions of scalar quantities can be accomplished with the help of alternating (AC) magnetic fields. The swirling flow produced by an RMF may be considered as a replacement for the crucible and/or the crystal rotation. Literature suggests that temperature fluctuations dt = T T, where T is the average value of the temperature signal, issuing from natural convection may also be effectively damped by an RMF [4 7]. The advantage of using an RMF over using a static field is that the required field strengths are significantly lower. Quite a lot of the modeling for CZ crystal growth is done on RB systems, which is true for the experiments in [4,5,8] described in Sect The upper isothermal boundary condition of an RB system can be replaced by a crystal model. This model may cover a centrical part of the surface of the same ratio as the growing crystal in a CZ puller does. The changes affected by this can be found in Sect A further step towards the parameters of an industrial puller is replacing the isothermal boundary condition at the bottom and the adiabatic one at the side wall by a heat flux condition. When implementing this in an experiment, the curved bottom shape was also included. The results of temperature and flow velocity measurements are to be found in Sect Review of previous work 2.1 Rayleigh-Bénard convection An idealized RB system consists of a cylindrical fluid column, a homogeneously heated bottom, adiabatic side walls, and a cover on the colder top that provides an isothermal boundary condition also there. For the vertical temperature difference ΔT v = T b T t between the bottom and the top, an instability threshold exists, below which the fluid is at rest. The relative strength of buoyant convection may be quantified by the Rayleigh number Ra = βgδt vh 3, (1) νχ where β, ν, andχ are the thermal expansion coefficient, the kinematic viscosity, and the thermal diffusivity, and g is the acceleration due to gravity. The critical Rayleigh number Ra c corresponds to a critical value of the driving vertical temperature difference ΔTv c for the onset of convection. An important dimensionless parameter, which has an influence on the developing convective pattern, is the aspect ratio AR. Although there is plenty of work on RB systems, we restrict our discussion to a few that are relevant to the modeling of CZ growth. Because the grown crystal is desirably axisymmetric, the main issue is the axisymmetry of the flow. Variation of the convective patterns in cylindrical containers owing to AR was recently studied numerically and experimentally [9]. For AR 0.63, a single roll corresponding to an azimuthal wave number of m = 1 was found for ΔT slightly above the onset of convection, i.e. fluid rising at one side, moving along the surface to the opposing side,

3 Electromagnetic Flow Control in Metallurgy, Crystal Growth and Electrochemistry 261 descending there, and closing along the bottom. This flow phenomenon, often termed wind, is frequently observed in RB systems. In the case of smaller AR down to about 0.3, m = 0 was observed, which is a torus and thus axisymmetric. Initially, it seems unclear why the m = 1 mode is realized again for 0.3 A 0.25, and moreover m =1orm = 2 was found near AR 0.25 in different runs. It can be explained by the fact that the marginal stability curves of those modes are close-set and intersect. The Prandtl number Pr = ν/χ of the fluid investigated in [9] was 28.9, whereas semiconductor melts are in the range of Pr 10 2.Itiswell-knownthatRa c for the onset of convection does not depend on this dimensionless number, which quantifies the ratio of the thicknesses of the velocity boundary layer and the thermal boundary layer. To see whether the developing convective pattern depends on Pr, we shall have a look at the numerical stability analysis in [10]. There, Pr = 0.02 was contrasted to the generic case Pr = 1. The curves of neutral stability for the onset of convection are very similar to the findings in [9]. A dependence on Pr is only reported for the second transition at Ra c2, which is either to another m or to time-dependent flow. The dependence of Ra c2 on Pr is stronger the lower AR is, and Ra c2 becomes close to Ra c as AR increases. In [11], it is computed for AR =1and0.02 Pr 6.7 that the secondary instability at Ra c2 is not a transition to another mode, as is often the case for smaller AR, but rather the m = 1 mode becomes anharmonically oscillatory at low frequencies. Similar findings are reported in [12]; the single large-scale recirculation cell prevails for all Ra under investigation. More elevated Rayleigh numbers for AR = 1 are numerically simulated in [13]. While varying Pr, two different flow regimes are distinguished. For Pr > 0.35, the large-scale motion is ineffective for heat transport. Instead, heat transport is dominated by the motion of thermal plumes. The high heat conductivity in the case of low Pr prevents thermal plumes. That is to say, the large-scale recirculation is the most important engine for heat transfer. Ultrasonic flow measurements of the vertical velocity component v z along the center line of a cylindrical cell are reported in [14] forar =0.5, 1, and 2. Mercury with Pr = was the fluid under investigation. Low-frequency peaks were found in the turbulence spectra of v z in the center of the cell. In addition, v z along the vertical center axis of the cell reverses sign in the case of AR =0.5 and 1. From these peaks and the reversal, the authors conclude that the flow structure is a mono-cellular wind. From their data, they explain that the wind is an elliptical pattern where the long axis of the ellipse is tilted against the vertical axis. This finding agrees with the inclined single roll reported in [11,12] in the case of AR = 1. However, an m = 0 azimuthal mode, as predicted by the numerical simulations in [11], can also lead to the observed phenomena. If the flow in a torus, whose axis is aligned with the cylinder axis of the fluid volume, becomes time-dependent in such a manner that the axis of the torus rotates harmonically around the cylinder axis, both phenomena which are reported in [14] will be observed. A critical aspect ratio AR c where the most dangerous mode changes between m =1andm = 0 was considered in [11]. The convective pattern was calculated for AR = 0.5 and 1, aspect ratios that were chosen as representative examples below and above AR c. The results in [11] agree with the stability analysis made earlier in [15,16] as well as with the later one in [10]. We will see the importance of the change at AR c in Sects and Influence of the crucible bottom shape Investigations that include a direct comparison between a cylindrical system and a curved crucible bottom are rare. Low melt heights such that AR 0.25 were investigated numerically in [2]. In this two-dimensional model, a crystal determined

4 262 The European Physical Journal Special Topics the homogeneous temperature boundary condition, while the remainder of the surface was either adiabatic or released energy by radiation. At the bottom and at the side wall, the temperature was homogeneous. Horizontal temperature gradients T h thus render this model different from an RB system. The crucible bottom curvature was found to have a damping effect on the buoyancy-induced oscillations. As AR is lowered, the mode of convection changes from a vertical wall-dominated recirculating flowtobénard convection. It is noteworthy that for the highest AR of only 0.25, the bottom shape was found to have no pronounced influence on the flow pattern and the heat flux. Two growth experiments in D = 120 mm, AR= 1 geometries are reported in [3]. Induction heating by a coil axially surrounding the crucible was used, implying horizontal temperature gradients. These gradients drive natural convection from the hot crucible wall to the colder crystal at the surface, opposed by the forced convection from the rotation of the crystal. For the same power of the heating system, the temperature drop ΔT between the crucible wall and the melting point were 59 K and K for the flat and the curved bottom, respectively. The consequence of the increased ΔT in the case of the curved bottom is a more vigorous natural convection. This contradicts the findings in [2]. However, the interface shape of the crystal was nearly flat for a flat-bottomed crucible, and convex in the case of the curved bottom. The difference in deflection between 1 mm and 11 mm supports the assumption of higher flow velocities in the case of the curved bottom. The obvious contradiction between [2] and [3] leaves the question about the influence of a curved bottom on the vigor of the flow open. A damping and thus stabilizing effect as in [2] would be the desired situation for crystal growth. We shall return to this question in Sect Application of a rotating magnetic field Background of RMF-driven flows If a volume of electrically conducting matter is subjected to an AC magnetic field B, an electric current j is induced. j circulating in the conductor creates an induced magnetic field, which may be neglected if S = μσωl 2 < 1. (2) μ is the magnetic permeability, σ the electrical conductivity, ω = 2πf the angular frequency of the applied magnetic field B, and L a characteristic length. Equation (2) is commonly called the low-frequency approximation. If the shielding parameter S is small, i.e. ω is sufficiently low, B penetrates the whole volume and j interacts only with the applied field that produced it, resulting in a Lorentz force f L = j B. Provided f L 0, a flow is driven if the conducting matter is liquid. Maxwell s first equation states that the electric field E comprises the time variation of the magnetic field and the gradient of the electric potential φ: E = Ȧ φ, B = A. (3) A is the vector potential. Note that the applied field B canbeusedifs < 1. In most applications and laboratory experiments, it can be reasonably assumed that v/l ω, with v being the maximum value of the fluid velocity v. This condition is frequently called the low-induction approximation. Its equivalent in Ohm s law, j = σ(e + v B), is that v B becomes negligibly small compared to the electric field. Thus, with the help of Eq. (3), j = σ(ȧ + φ). (4)

5 Electromagnetic Flow Control in Metallurgy, Crystal Growth and Electrochemistry 263 Fig. 1. Flow driven by an RMF B. The 7 parallel arrows indicate a snapshot in time of a homogeneous field B 0 rotating with frequency ω about the vertical axis. B causes a timeindependent volume force acting in the azimuthal direction. This force drives a primary swirling flow with the rotation rate ω F (left). In the low-induction approximation, ω F ω. The weaker secondary flow in the meridional plane is depicted on the right-hand side. This figure was taken from [19]. For an RMF applied to a finite length cylindrical liquid column, an analytical expression for f L with the current in Eq. (4) was first derived in [17]: f L = σωb2 0 rs(r, z)e ϕ, (5) 2 where r, z, ande ϕ denote the radial and axial coordinates, and the unit vector in the azimuthal direction. s(r, z) is a shape function that reflects the influence of the finite length of the cylinder; it depends on R and H and vanishes at the bottom and top walls. The dimensionless form of Eq. (5) can be obtained by the usual transformations r = r/r, t =(ν/r 2 )t, and m = m/(ρr 3 ), where m is the mass and the tilde is used for the dimensionless quantities: fl =Ta rs( r, z)e ϕ, Ta = σωb2 0R 4 2ρν 2 (6) The Taylor number in Eq. (6) is hereafter used as the relative force characterizing the vigor of a flow driven by an RMF. It is readily seen from Eqs. (5) and (6) that the Lorentz force acts in the azimuthal direction, i.e. it drives a swirling flow. The left hand side of Fig. 1 sketches the RMF rotating with the frequency ω. The gray arrows indicate a snapshot in time of B in a plane at half the cylinder s height; an ideal RMF would be invariant along the vertical z-axis across the container. It is well-known that the swirling flow is also almost constant along z, except of course at the bottom and at the top, even if the azimuthal force depends on z. This is depicted with the four circular arrows one on top of another. That ṽ ϕ Ta 2/3 over a wide range of Ta has also been known for a long time [18]. For very small Ta, ṽ ϕ Ta. According to theory, the changeover from this Stokes regime to the 2/3-law laminar boundary layer regime should occur in the range 10 3 Ta For the first time measurements of very small velocities below 0.1 mm/s with an electric potential difference probe showed that the changeover took place at Ta 10 3 [19]. The experimentally determined exponents of 1.09 and 0.659

6 264 The European Physical Journal Special Topics for the two regimes support theoretical predictions. [19] contains a detailed description of the sensitive velocity measuring technique. When turbulent friction becomes the dominating retarding force, ṽ ϕ Ta 1/2 [5,18,20]. For mild shielding S 7.5, Ta was experimentally determined for this second changeover [5]. In Fig. 1 the secondary flow is sketched at the right hand side. It consists of two toroidal vortices one on top of another with a common radially outwards directed flow at half the height of the cylinder. This secondary flow results from an imbalance between the centrifugal force and a radial pressure gradient at non-vertical boundaries [18,21]. The case of equal in size upper and lower vortices sketched in Fig. 1 corresponds to no-slip boundary conditions at the top and the bottom of the cylinder. The influence of an RMF on RB convection is discussed in [4] in terms of the ratio of the electromagnetic force to the buoyant force. For this force ratio, the authors introduce the parameter N rot =Ta/Gr, where Gr = Ra/Pr is the Grashof number. As we are mainly interested in reducing temperature fluctuations by the RMF, it is noticeable that no thermodynamical property appears in Gr. Therefore, the ratio N =Ta/Ra is used further on to characterize the influence of the RMF on the temperature field. This choice of relevant parameters is also supported by the numerical study in [22] Model experiments on RMF-driven flows In [8] four flow regimes are distinguished by means of temperature signals measured by azimuthally spaced thermistors at the cylinder wall in a small cell filled with gallium. The first three regimes are quasi-periodic and differ in frequency and phase shift between the thermistors. The fourth one, for which Ta Ta c, is characterized by neither the autocorrelation function of a thermistor nor the correlation function between thermistors shows a significant decay, but the time series still exhibit a high degree of periodicity. The authors call this flow regime the Ω 4 regime. In the investigated low range of Ra , it was found that the RMF is the dominating force for the Ω 4 oscillations, appearing as soon as Ta overcomes the linear stability threshold Ta c independent of Ra. Ta c was determined to be As the first regime corresponds to a thermal roll driven around the cylinder by the RMF, the authors concede that, based on the experimental data, a definite physical model of the flow in the other three regimes is not obvious. However, since the thermistors are mounted at the perimeter and the signals are either quasi-harmonic or in a more complex kind still periodic in the Ω 4 regime, it might be said that the thermal roll or likewise other global thermal structures extend up to the wall, i.e. they span the whole container, even when an RMF is applied. It is explicitly mentioned in [8] that the average azimuthal velocity v ϕ and the azimuthal velocity of the thermal roll rotation vϕ thr are generally in the same order of magnitude, but not identical. While referring to their previous work [22], the authors state that vϕ thr depends not only on Ta, but on the skin depth and the Prandtl number as well. The skin depth is closely related to the shielding parameter in Eq. (2) and can be regarded as a dimensionless measure of the penetration depth of an AC magnetic field into an electrically conducting medium. Damping of dt by an RMF in an RB experiment also done on gallium is reported in [4]. Results are presented for (B,ω) = const. with varying ΔT,(ΔT,B) = const. with varying ω, and (ΔT,ω) = const. with varying B. These three series are referred to as Cases 1 to 3 henceforth. Quasi-harmonic temperature variations, again recorded with thermistors mounted azimuthally spaced at the cylinder wall, were observed for N 1 in all experiments, just as in one of the first three regimes in [8]. Since Ta = < Ta c in Case 1, it is not a surprise that quasi-harmonic time series of T, with the frequency of the roll rotation increasing with ΔT, were observed.

7 Electromagnetic Flow Control in Metallurgy, Crystal Growth and Electrochemistry 265 Up to N =0.435, dt/δt was not affected. At N =0.867, dt/δt went down to less than a third. Finally, at N =1.09, dt/δt becomes very small. The second case, in which only quasi-harmonic time series of T were recorded, is Case 2. Ra was fixed to , which was higher than the maximum Ra max in [8]. This measurement series includes the time series at Ta = 0. dt approximately quadruples when the RMF is applied, so that N = At N =0.279 the frequency has about tripled, whereas dt is not yet affected. Strong damping was observed at N = The difference from Case 1 may be explained by shielding, since ω was higher there. The only case in which non-harmonic oscillations were observed and dt did not significantly decrease, even for N =3.37, is Case 3. These oscillations most probably belong to an Ω 4 regime as identified in [8]. The authors argue that, although the measured signals show no regular oscillations for N 0.4, the Fourier spectra of the time series of T exhibit a small band of frequencies, or even a clear mean frequency. They attribute this peak frequency to the azimuthal velocity of the flow since vϕ thr Ta 2/3. In a model experiment also done on gallium, the suppression of dt in an RB system was studied in a larger AR = 1 cell 140 mm in diameter at higher values of the Grashof and Taylor numbers [5]. Maximum values of Gr and Ta were and , respectively. In series of measurements varying Gr and Ta independently, the relation Ta tr 1.63 Gr 0.8 for a significant suppression of the buoyancy driven temperature fluctuations was obtained. This relation implies that N is not a direct criterion for suppression due to the exponent of Gr being unequal to unity. Insertion of the lowest Gr min = and the highest Gr max = under investigation into this relation yields an N of 1.4 and 0.9, respectively. Applying the relation to the highest field strength in Case 3 in [4], which is the most reasonable one to compare here since this is the only case with RMF driven Ω 4 oscillations, yields N =3.79. Fair agreement to the value of N =3.37 measured in [4] can be stated. It is noteworthy that in a recent work the transition between turbulent magnetically driven flow states was investigated for Gr requiring Ta in the v ϕ Ta 1/2 -range for a significant suppression of dt [7]. It is shown there that the exponent in the relation Ta tr vs. Gr changes (see Fig. 7 in [7], it becomes higher in the elevated range of Gr). As this high Gr-range is above the reachable values of the present experiments in Sect. 3, these findings are beyond the scope of this paper. In [5], the velocity of the thermal structures transported by the flow were also measured by means of correlation analysis from azimuthally displaced sensors mounted on the side wall. As in [4], it was found that this velocity follows the Ta 2/3 rule up to Ta = Above, the scaling changes to Ta 1/2. From this analysis, the authors conclude that vϕ thr measures v ϕ. If we assume that the buoyant convection and the secondary flow produced by the RMF, which are both meridional recirculations, are true linear superpositions to v ϕ driven by the RMF, then this can be compared to the mere RMF driven case. This was done in [23], where it was found that vϕ thr from [5] fits perfectly into the flow measurements of v ϕ by conductive probes over a wide range of Ta. 3 More Czochralski-like model experiments Czochralski crystal growth on the industrial scale is done from both high and low Prandtl number melts. Flow in oxidic melts, an example of the first, is easily modeled physically. Numerous experiments done on water, silicone oil, alcohols, ether, acetone etc. are reported in the literature. However, the key industrial sector of growth of semiconductor crystals from the melt belongs exclusively in the low Prandtl number range. Substantially fewer investigations have been carried out for Pr 1 because

8 266 The European Physical Journal Special Topics all relevant substances are opaque and/or difficult to handle. The usual measuring techniques, such as laser Doppler anemometry or particle image velocimetry, are not applicable to opaque substances. An example in which much effort was devoted to accommodating the difficulties of molten salts, including chemical aggressiveness and high temperatures can be found in [24] and yet only Pr 2.3 could be achieved. The present work is concerned with two examples in which the model fluid GaInSn was used. This ternary alloy is liquid at room temperature and has become the quasistandard for experiments on low Prandtl number fluids. Working at moderate temperatures allows the use of a variety of measuring techniques, in particular ultrasonic Doppler velocimetry. The primary objective of these two experiments was to move from the RB system step by step towards the configuration of a Czochralski puller. 3.1 Cylindrical geometry with a crystal model The first step from an RB system to a CZ puller was preserving the cylindrical geometry and replacing the upper isothermal boundary condition by one more similar to that of an industrial puller. Only a centrical region covering about one third of the diameter was cooled. This introduces a primary T h,whichbatchelor has shown in his celebrated work that it immediately drives flow, however small T h [25]. The first question was whether the instability mechanism with its growth of infinitesimal disturbances would lead to RB convection when there is initially a developed flow due to the primary T h. The second objective was damping the fluctuations by the application of a rotating magnetic field. The cylindrical fluid volume was 90 mm in height and 90 mm in diameter. As this is unlikely to be the situation in a CZ puller, one may argue that it would be desirable to grow crystals out of a melt at this initial filling height of the crucible. However, AR = 1 allows comparison to the majority of experiments described above. A detailed description of the experimental setup can be found in [6] The mere buoyant case The flow due to the presence of T h would be axisymmetric in the form of a torus penetrating the whole fluid volume. It was stated in [6] that the Rayleigh-Bénard convection dominates this flow. The monocellular wind was detected by means of temperature measurements; it was however not quantified by broad ultrasonic flow measurements. Figure 2 shows such flow measurements that were conducted since that time. The ultrasonic transducer was attached to the side wall close to the bottom of the cylinder such that the direction of measurement pointed across the axial center towards the opposing cylinder wall. Since the velocimeter measures the projection of the fluid velocity onto the direction of measurement, the measurements consist of the radial velocity component at various values of the radial coordinate. Integrating from one position at the side wall to the opposing side will result in zero in the direction perpendicular to the wind direction. Measuring along several azimuthal positions, the maximum is found in the wind direction. Thus, a sinusoidal dependence on the azimuthal angle is expected for the integral velocity, which is nicely reproduced in Fig Damping of temperature fluctuations with a rotating magnetic field Since the flow in the modified Rayleigh-Bénard system with the partially cooled surface behaves quite similar to that in a traditional RB configuration, it suggests that

9 Electromagnetic Flow Control in Metallurgy, Crystal Growth and Electrochemistry 267 Fig. 2. Integration of the radial velocity component over the diameter as a function of the azimuthal angle. The measurement series were conducted on different days; in between the heating was switched off. Hence, the wind direction is stable and reproducible. Fig. 3. Dependence of the normalized standard deviation sd T/ΔT on the temperature time series on the strength of the RMF. the damping of dt by an RMF should be examined. To measure the temperature over time, small thermocouples were introduced into the melt through the cover lid. They protruded a few millimeters below the surface. The location was in the vicinity of the perimeter of the crystal model. Measuring durations were sufficiently long to cover several rotations of the thermal roll in the case of lower Ta where the rotation of the temperature field showed periodicity. Four series for different temperature differences between the bottom and the crystal model were measured. The range of ΔT from 20 to 80 K corresponds to Grashof numbers from to The amplitude of the fluctuations decreased to less than 10% when the RMF field strength was increased about 50 times. A detailed representation of the damping depending on Ta is seen in Fig. 3. The fitted gray lines are intended to serve as a guide to the eye. Closer examination of the range of low Ta may suggest that the amplitudes of the fluctuations increase first, before they are damped at higher Ta. Such a tendency is also reported in [4]. That the normalized standard deviation sd T /ΔT decreases to about 20% is a feature common to all Gr under investigation. sd T /ΔT also asymptotically approaches the same value for all four measurement series.

10 268 The European Physical Journal Special Topics Fig. 4. Photo and sketch of the double-walled crucible. The sketch to the right shows that AR is calculated with an effective height which is determined by the volume V of the melt and the area A at a height where the crucible is cylindrical. 3.2 Experiments in a crucible with a rounded bottom The crucible shown in Fig. 4 has a bottom shape similar to those used in Czochralski pullers. AR = 0.59 was chosen such that, according to Sect. 2.1 in conjunction with the strong primary horizontal temperature gradient, an axisymmetric flow structure could be expected. Provided dt can be effectively damped by an RMF, the accordingly high initial filling height would be beneficial for the yield of the growth process. The liquid metal was heated by passing thermostated silicone oil through the double-walled crucible. The choice of this fluid was motivated by the need for higher temperatures to reach high Rayleigh numbers. The experiments reported below were conducted at an inlet temperature of about 145 C. Three outlets close to the upper end were sufficient for a fairly homogeneous azimuthal temperature distribution. Because of the high flow rates, it may be reasonably assumed that the temperature gradients within the silicone oil are small. At least, that they are small compared to the gradient across the inner crucible wall. The isothermal boundary condition in RB systems is thus replaced by a heat flux condition. A remaining difference regarding the heat flux conditions at the crucible wall from an industrial device is that the heat fluxes in the bottom and side regions cannot be controlled separately. With respect to the entity of all boundary conditions, it is pointed out that the experiments in the crucible are modeling effects that are not included in the generic RB configuration. However, such phenomena as radiation of heat and Marangoni convection at the surface, and the shape of the solidification front are not yet addressed Flow structure of the natural convection That the expected axisymmetric flow structure is not realized can be seen in the plot of surface velocities in Fig. 5. A small ultrasonic transducer 10 mm in axial length and 8 mm in diameter was immersed directly at the rim so that it is just covered by the melt. The measuring direction was across the crucible center towards the opposing wall. Due to T h the flow adjacent to the rim is directed away from the wall over the entire circumference, which is depicted by the inwards directed arrows. Although a series of measurements in the azimuthal range from 0 to would have been sufficient to characterize the convective pattern, measurements were done from 0 to

11 Electromagnetic Flow Control in Metallurgy, Crystal Growth and Electrochemistry 269 Fig. 5. Visualization of the flow structure at the melt surface. The innermost circle (thick black line) marks the region covered by the crystal model. Positions where measurements of v ϕ were taken are depicted by crossed squares. In the region around the ellipses, hot melt coming from the rim descends, i.e. the convective pattern is stagnated flow (clockwise direction in Fig. 5). Dropping the quantitative information from the measured radial velocity component v r, a line is drawn from the rim towards the center with a length corresponding to the measuring depths R r, for which v r stays negative. The convention used here is that v r < 0 if the flow is in the measuring direction and R r R. Joints of lines from opposing sides are enclosed in ellipses. The sizes of these ellipses correspond to the gap or the overlap between measurements from opposing sides. Lines for which v r 0atR r>0 are drawn in light gray, the thinner black lines visualize the case v r 0atR r<0. This representation of v r suggests that an axisymmetric m = 0 mode is superimposed to the m = 1 wind. The direction of the wind is from between 180 and towards between 0 and The stagnation point in the center in the case of an axisymmetric flow degenerates into an arc through the center of the ellipses. Such a complex non-axisymmetric flow is observed for any Rayleigh number such that Ra Ra c. The measurement in Fig. 5 was taken at Ra = RMF-driven flow in the Czochralski crucible Before applying the RMF to the buoyant convection, the flow evoked by the field was measured under isothermal conditions to find out the influence of the rounded crucible bottom. A transducer was put on the melt surface such that the measuring direction pointed downwards. Traversing the sensor from one side at the crucible wall to the opposing side while measuring v z (h) showed that the upper torus of the secondary recirculation was significantly larger than the lower one compared to the case of a cylindrical melt volume (c.f. Fig. 1) with a rigid lid on top. The velocity was also higher in the upper part. Before addressing whether the rounded bottom has a damping ([2]) or enhancing ([3]) effect on the flow, we shall examine the slip boundary conditions at the top. The lid on top in the cylindrical configuration implies v r = v ϕ =0ath = H. For the measurement in the crucible, the crystal model was removed. This poses the question whether the whole surface is free in the sense of a free-slip boundary condition. Section referenced [18, 21] regarding the secondary

12 270 The European Physical Journal Special Topics Fig. 6. Left: temperature vs. time at r = 0.43R of mere buoyant convection. The three curves correspond to different azimuthal positions of the thermocouples. Right: with RMF, the signals of the sensors became identical with a phase shift corresponding to the azimuthal location. Thus, only one signal set is shown. The low frequency spikes were measured at N = 0.23, and the higher frequency irregular fluctuations at N = flow as the result of an imbalance between the centrifugal force and a radial pressure gradient. If the surface were actually free, such an imbalance should not be present, since there should be no viscous friction. Consequently, an upper vortex should not exist. Numerical simulations for both no-slip and free-slip boundary conditions at the surface of a cylinder can be found in [26]. The vector plots of the meridional velocity therein do not show an upper vortex when the surface is free. Transfer of this result to the measurement in the crucible implies that there should be at least some friction causing the radial imbalance and the related upper vortex. An explanation may be that the GaInSn melt oxidizes and the oxides form a more or less immobile film. The boundary condition then would be something in between free-slip and that for a rigid wall. Even for the no-slip condition, the fact that the velocity in the upper part was higher means that the crucible bottom has a damping influence. The more the actual slip condition deviates from that at a wall, the more the rounded bottom damps the flow in the lower part. However, there is a contradiction also to [2]. AR was 0.59 in the present experiment, and the damping in the lower part was significant. It is stated in [2] that the damping influence of the rounded bottom vanishes towards higher AR. For the highest AR max =0.25 in [2], which is much smaller than the present AR, hardly any damping was found Does a rotating magnetic field damp the temperature fluctuations? One can visualize the swirling RMF flow smoothing out azimuthal gradients, thus rendering the flow more axisymmetric. Up to a certain Ta tr, the RMF can be expected to primarily rotate the wind. The interest here is focussed on whether the beneficial reduction of dt, as found in RB systems and in the cylindrical geometry with partially cooled surface, also appears in the present model. Temperature vs. time plots measured close to the perimeter of the crystal model at r = 38 mm are presented on the left side of Fig. 6. These measurements were conducted in the pure buoyant case. The dark gray curve shows much less variation than the other two. The azimuthal position of the corresponding thermocouple was at 180 (c.f. Fig. 5), almost directly into the wind (i.e. in the direction from which the wind comes). The light gray curve represents the opposing azimuthal position, and the

13 Electromagnetic Flow Control in Metallurgy, Crystal Growth and Electrochemistry 271 Fig. 7. Measurements of the azimuthal velocity and the rotation rate of the thermal structures. The two measurements at Ta = 10 7 depicted by gray symbols were done at the positions labeled 1 in Fig. 5. All other measurements were taken at positions labeled 2. black curve represents 67.5, which is almost perpendicular to the wind direction. The amplitude of the temperature fluctuations in the direction the wind blows and perpendicular to it are almost the same, possibly because the two measuring locations are much closer to the stagnant region than the upstream measuring station. It was reported in [4] that the amplitude of dt quadrupled when the RMF was switched on. The black curve in the right panel of Fig. 6 shows that dt at N =0.23 is almost the same as without magnetic field. When increasing N to 1.72, depicted by the grey curve, no damping of dt is observed for this relatively high value. Ta as high as were measured with the result that dt cannot be effectively damped. Although N = 230 at the highest Ta, the amplitude of dt decreased to only about half its value at N =0.23, 1.72, or without RMF. The benefit of reduced temperature fluctuations by an RMF suggested by most of the investigations done on RB systems is not achieved in these first measurements in a more Czochralski-like crucible. An explanation for this behavior is given towards the end. Besides this result, other not yet fully understood phenomena were observed. The plots of angular velocities in Fig. 7 show pronounced lagging between the rotation of the thermal roll and the fluid, ω T ω F. ω T even decreases before the rotation of the thermal structure loses periodicity at Ta a little less than 10 7.ForTa< 10 8, ω F (r =36mm)>ω F (r = 73 mm). The measurements of the mere RMF-driven flow in Sect do not exhibit a core rotating distinctly quicker than the outer region. For the axial and meridional forcing exerted by a combination of an RMF and a travelling magnetic field (TMF), swirl accumulation was observed, which was most pronounced when the forcing by the TMF was about 100 times that of the RMF. The maximum speed of the swirl was mainly governed by the meridional force [27]. Swirl accumulation might be an explanation also here, albeit it was not possible to detect acycloneeye.sincev ϕ v r, small deviations of the measuring direction introduce a fraction of v ϕ in the readings of v r. However, core speed and rotation of the outer regions differ from each other the most at a force ratio of about 50. The dent in the plot of the core speed below Ta = 10 7 may indicate that the center of the cyclone eye had moved radially across the measuring location. That dt could not be damped effectively is reasonably explained in the numerical study [22]. While referring to the cylinder rotation treated in [28], the authors found that the application of an RMF has no influence on the thermo-convective stability of the axisymmetric flow modes. This means that the m = 0 mode in the CZ crucible

14 272 The European Physical Journal Special Topics does not couple to the RMF-induced rotation as the wind does. In addition, since it cannot be effectively damped, the m = 0 mode increasingly dominates the flow with increasing N. The axisymmetric mode, as it predominantly determines the convective heat flux directed inwards along the surface, may be seen as the reason for ω T ω F. Loss of periodicity is readily explained by the diminishing influence of the wind and by the increasing dominance of instability over the flow due to the RMF for increasing N. 4 Conclusions and perspectives Convection in a Rayleigh-Bénard (RB) cell represents a simple model for the more complex case of CZ crystal growth. Related results for the RB convection are, therefore, reviewed as basis. Moving step by step towards a realistic CZ configuration is the primary objective of the present work. In a first step, the upper homogeneous temperature boundary condition was replaced by a crystal model. This introduces a primary horizontal, radially inwards directed temperature gradient ΔT h. It was found that the flow in this system is, despite ΔT h, very similar to the generic RB case. Consequently, the interesting feature of a significant reduction of temperature fluctuations caused by an RMF has been also observed in such a modified RB configuration. The second stage towards a CZ puller additionally contains a close-to-reality shaped crucible made of glass. Hence, there is no more a clear division between bottom and sidewall. Temperature boundary conditions are changed from adiabatic/constant (sidewall/bottom) to a heat flux condition everywhere at the crucible wall. A remaining difference to a real CZ process is that the cooling agent circulating through the double-walled crucible has approximately the same temperature everywhere, whereas in a CZ puller the heat flux in the bottom region and in the side region can be controlled separately. Because the melt is heated at the side up to the surface, ΔT h may be expected to be stronger than in the first setup. Flow measurements in the first experiment showed the well-known single convection cell, which corresponds to a mode with an azimuthal wave number m = 1. On account of the stronger ΔT h, axisymmetric flow (m = 0) was expected in the second setup. The result of the measurements was a superposition of m =0andm = 1 modes. The application of an RMF did not lead to the strong reduction of temperature fluctuations as observed in the RB and modified RB cases. This result needs further systematic analysis, e.g., for varying aspect ratios. Though the second setup is geometrically obviously closer to the CZ case, its main difference to the real growth of high-melting materials such as silicon or gallium-arsenide consists in the heat transfer at the free melt surface. The strong heat radiation at, e.g., a free silicon melt surface gives rise to a much lower ΔT h as in the present model [29]. Hence, the installation of a surface cooler [29] will be a next step in the further qualification of the model setup. The second experimental setup was put into operation in summer 2011, so the results presented here are relatively new. The facility also offers independent crucible and/or crystal rotation, but investigations on this feature have not been done so far. Revealing more insight into the complex flow phenomena of coupled buoyant and RMF-driven convection needs more sophisticated measuring techniques. Ultrasonic flow diagnostics capable of multi-plane multi-component measurements are available [30]. These have so far not being qualified for the elevated temperature range of the present experiments. It also needs further technical development to adapt the ultrasonic arrays to the geometry of the crucible. This will, in addition to investigating other AR and installing a surface cooler, be the subject of ongoing work in the near future. On a longer time scale, it is an interesting question whether an RMF can replace crystal or crucible rotation in CZ crystal growth. The data of the model

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