F Dughiero

M Forzan

N Sempreboni

A Tolomio

Preliminary study on molten silicon velocity distribution during the directional solidification into i-DSS furnace

Keywords: Coupled problems, Directional Solidification System, Silicon Casting, mc-silicon

Preliminary study on molten silicon velocity distribution during the directional solidification into i-DSS furnace

F Dughiero, M Forzan, N Sempreboni, A Tolomio

Introduction

The peculiarity of induction directional solidification system is the introduction of this heatingtechnique in the silicon casting. Among other advantages, induction heating allows better control of the process [START_REF] Dughiero | Numerical analysis of ingot casting processes for quasi-silicon crystalline silicon solar cells in the iDSS furnace[END_REF]. The heating system is composed by three different inductors, each one isindependentlyfed byits respective resonant capacitor bankand IGBT converter. The aim of this study is to evaluate if the electromagnetic field, produced by the side coil, has some influenceon the patterns of molten silicon flow. The EM field produced by the side coil has been designed to maximize the power induced in the lateral susceptors and not for controlling the flow motion. Furthermore, the EM field is strongly shielded by susceptors at the working frequency and the field amplitude is relative low in the silicon. A simple Computational Fluid Dynamics (CFD) study,based on the classical k-model,has beendevelopedto predictthe motion of fluid silicon. The model takes into account the buoyancy force and the influence of AMF. The study is appliedto thelab-scale of i-DSS realized to University of Padua.The furnace is able to cast squared ingots up to120kg of silicon load.

The lab-scale i-DSS

The hot-zone of the i-DSS is very similar to the ones of standard DS furnace, fig. 1. In fact, the hot-zone of i-DSS comprisesthe same isostatic graphite plates that contain the commercial quartz crucible like a box. The thermal insulator elements, made of graphite felts, are locatedtoenvelope the graphite box and to cover the top susceptor. The bottom graphite plateis not insulated in order to guarantee the thermal flux required to obtain a directional solidification. Three different inductors, namely the bottom, side and upper inductor,are positioned as shown in Fig. 1.Each inductor has been designed to heat the corresponding susceptorrealized with isostatic graphite plates under and positioned closed to the inductor itself. The different positions of each heating source guarantee an accurate control of the temperature field inside the hot zone during all the process phases. In order to maximize the energy efficiency of the process, thethickness of the lateral insulating graphite has been chosen in order to guarantee an appropriate balance between thermal losses and inductor efficiency. The thickness of graphite felts has an obvious impact on the insulation, but an increasing thickness has also a negative impact on the inductor efficiency.Due to significant distance between the bottom and top inductors from the melt, the AMF generated by these two inductors is negligible in thesilicon. On the contrary, the electromagnetic field generatedby the lateral inductor is not negligible inside the molten silicon. The influence of AMF produced by the side inductor on fluid flow has been investigated in this work.

The simulation model

In order to solve a Navier-Stokes equation for the Newtonian fluid during the solidification process 2D axisymmetric models of the furnace have been developed. The Navier-Stokes equation for Newtonian fluid in vectorial forms can be formulatedas follow: with: 𝜌 densityof molten silicon, 𝑢 velocity vector, p pressure, 𝜂viscosity terms and body force𝐹 .

𝜌 𝜕𝑢 𝜕𝑡 + 𝑢 • ∇ 𝑢 = -grad 𝑝 + 𝜂∆𝑢 + 𝐹 (1)
The body force, in our case, is composed by Lorentz forces (𝐹 𝐿 ),buoyancy forces (𝐹 Buoy ) and Marangoni effect (𝐹 M ). For a liquidmetal, like silicon, whom density and energy surface depend linearly from temperature, the 𝐹 Buoy under gravitational field and the Marangoni effect could be formulated respectively as:

𝐹 Buoy = 𝜌 𝑇 𝑔 = 𝜌 0 1 -𝛽 𝑇 -𝑇 𝑟𝑒𝑓 𝑔 (2)
𝜎 𝜏 = 𝑀 𝑎 • 𝑔𝑟𝑎𝑑 𝜏 𝑇 (3)
The linear thermal expansion is indicated as 𝛽, 𝑀 𝑎 is the coefficient of surface tension variation with temperature. The time harmonic problem has been solved by means of magnetic vector potential formulation, so the 𝐹 𝐿 , in a generic point C, can be described as:

𝐹 𝐿 𝑡 =
The viscosity of molten silicon and the resonant frequency of inductor, above 2 kHz,allow, in first approximation, to neglect the time dependent component of Lorenz Force. In the computational model the onlyconstant component to take into accountis:

𝐹 𝐿𝑎𝑣 = 𝑗 𝐶𝑅𝑒 x𝐵 𝐶𝑅𝑒 + 𝑗 𝐶𝐼𝑚 x𝐵 𝐶𝐼𝑚 2 (5)
To analyze the fluid dynamics,the COMSOL simulation code applies the RANS approach and specifically K-ε formulation. To facilitate the convergence of the problem, the flow pattern has been initially resolved taking into account only thermal distribution. In a second step, the solely electromagnetic force has been introduced. All the data in the models have been reported in the Table 1, in accordance with [START_REF] Dadzis | Modeling of directional solidification of multicrystalline silicon in a traveling magnetic field[END_REF].

[Kg/m 3 ] @ 𝑇 𝑟𝑒𝑓 = 1685 K 𝛽 [K -1 ] 𝜂 [Pa•s] 𝑀𝑎 [N (m•K)-1] K [W (m•K) -1 ] σe-si [S/m] σe-Cu [S/m] σe-C [S/

Geometrical dimensions rSi

[mm] rL-sucep [mm] rbase [mm] rL-ind [mm] hsi-TOT [mm] h1 st -coil [mm] hL-ind [mm] Φind [mm]

The simulation results

In the Fig. 3 the temperature field of molten silicon is presented as result from a steady state thermal model in CGSim code. The distribution of Lorenz force has beensolved by Flux model and imported in COMSOLand it is showed in the Fig. 5. The applied magnitude of current, 875 Arms, is the one that transfers to the system the heating power required during solidification. In this condition, the AMF generated, when the solid part is 10% of the total silicon height, has a strong influence on the fluid flow. The peak of velocity under thermal perturbation is about 13 times lower than theone calculated by imposing Lorentz force. The ratio between the velocities decreases towards the core of molten Silicon, because the skin effect confines the Lorentz forcesto the external layers of silicon. The study of the flow pattern has been done taking into account different positions of the solidification surface that corresponds toheightsof solid partequal to30% (Fig. 7), 50% (Fig. 8), and 70% (Fig. 9) of total height of the molten bath. In the initial steps of solidification, simulations show a pattern with only one main vortex.The velocity near the solidliquid interphase, where the segregation and precipitation of impurity occur,has a relevant effect to obtaining a high quality ingot. A low velocity near the interphase during the growth has a low mixing effect. The pattern of flow during the growth evolves in a double vortex, which is the typical structure taken by anelectrical conductive fluid inside an ideal infinite solenoidal inductor.

Conclusion

The preliminarymodels presented in this paper are strongly simplified and make use of K-e model that has several limitations. Nevertheless, theyallow a preliminary evaluation of the effectsproduced by theelectromagnetic force generated by the lateral inductor on the fluid flow inside the i-DSS furnace. The contribution of Lorenz Force is not negligible on the silicon molten flow and it could havea strong influence in the quality of ingot in terms of defects, like impurity concentration and dislocations. Taking into account the geometry of i-DSS furnace, the use of 3D models is mandatory for achieving more accurate results. The Lorentz force in the molten silicon calculated in 3D model, has a complex 3D distribution as showed in Fig. 10. The real free surfaceis not considered in these models but its evaluation is necessary to improve the result quality, also because Lorentz Force imposes an electromagnetic pressure on free surface modifyingits shape. The numerical results confirmed the necessity of CFD simulations in order to improve the directional solidification process.

Fig. 1 :

 1 Fig. 1: Sketch of the lab-scale of i-DSS furnace.

Fig. 2 :

 2 Fig. 2: 2D axisymmetric model with lateral inductor.

Fig. 3 :

 3 Fig. 3: Temperature field of molten silicon, in K degree.

Fig. 4 :

 4 Fig. 4: Path and velocity of fluid flow due to buoyancy force and Marangoni effect.

Fig. 5 :

 5 Fig. 5: Map of constant component of Lorentz force, 10% of total silicon height.

Fig. 6 :

 6 Fig. 6: Path and velocity of fluid flow EMF, 10% of total silicon height.

Fig. 7 :

 7 Fig. 7: Maps of Lorentz force constant component, 30% of total silicon height.

Fig. 8 :

 8 Fig. 8: Path and velocity of fluid flow EMF, 50% of total silicon height.

Fig. 9 :

 9 Fig. 9: Maps of Lorentz force constant component, 70% of total silicon height.

Fig. 10 :

 10 Fig. 10: 3D Vector plot of constant component of Lorentz force

  𝑗 𝐶𝑅𝑒 x𝐵 𝐶𝑅𝑒 cos 2 𝜔𝑡 + 𝑗 𝐶𝐼𝑚 x𝐵 𝐶𝐼𝑚 sin 2 𝜔𝑡 -𝑗 𝐶𝑅𝑒 x𝐵 𝐶𝐼𝑚 + 𝑗 𝐶𝐼𝑚 x𝐵 𝐶𝑅𝑒 sin 2𝜔𝑡 2

Table 1 : Data implement on computational models

 1

			Silicon proprieties			Copper proprieties	Isostatic Graphite proprieties
	Density at Temperature reference	Linear Thermal expansion coefficient	Viscosity	Marangoni coefficient	Thermal Conductivity	Electric Conductivity	Electric Conductivity	Electric Conductivity
	ρ0

Acknowledgment

Financial support from European project SIliconKErfLOss Recycle (SIKELOR).


A Y Deng
email: dengay@epm.neu.edu.cn

X Q Yin

C Q Yin

E G Wang

Effects of Electromagnetic Stirring on Cast Structure of High-Carbon High-Speed Steel

Keywords: high carbon high speed steel, electromagnetic stirring, cast structure, carbides

This thesis focuses on the effect of electromagnetic stirring on the microstructure of casting and morphology of carbides in high carbon and high speed steel for composite roll. The microstructure of casting and morphology of carbides of are studied using the method of metallographic examination, SEM and EDS. The results show that the carbides of high carbon and high speed steel are mainly the MC, M 2 C and M 6 C carbides. MC carbides distributed mainly by granular, M 2 C carbides distributed by strip and massive, and most of the network carbides are M 6 C type carbides. With electromagnetic stirring, carbides distributed mainly on the boundary become to distribute dispersively in the grain and along the boundary. And with the increase of the applied current, distribution of carbides are more uniform. With electromagnetic stirring, at cast center, the main carbides are MC type carbides. With an optimal electromagnetic stirring current intensity, 250A for our casting system, rotary electromagnetic stirring can tend to form MC carbides, break network carbides and reduce the size of carbides.

Introduction

High-carbon high-speed steel roll containing high carbon and alloy is easy to form a lot of network carbides [1][2][3] . This network carbide is hard and crisp, which seriously reduces the performance of roll. The network carbide is difficult to be eliminated by conventional heat treatment. So it is very crucial to modify the carbide size and distribution in high-carbon high-speed steel composite roll during the casting. Electromagnetic stirring has the role of refining grains, reducing macro-segregation and micro-segregation in crystal, and increasing the solubility of elements in the matrix [4] . Fabrication of high-carbon high-speed steel roll with electromagnetic stirring is expected to obtain better microstructure.

Experimental Method

The schematic of experimental device is shown in Fig. 1. The high-carbon high-speed steel was smelted in a graphite crucible with refractory lining and cast in a magnesia crucible. The pouring temperature is about 1600℃. The diameter of ingot is 60mm and the height is 100mm. At the section about 40mm apart from ingot bottom, three specimens, with size of 15mm×15mm×20mm, 12mm×15mm×20mm, and 10.5mm×15mm×20mm, were selected and tested, which indicates the regions of ingot center, half-radius, and outer surface. A rotational electromagnetic stirrer was applied during casting. The macro-and microstructure of specimens were examined and analyzed by Leica DMI 5000M metalloscope and SSX-550 scanning electron microscope (SEM) respectively.

Results and Discussion

The cast structure of high-carbon high-speed steel mainly consists of matrix and various carbides. The alloy elements and carbon are apt to form typical alloy carbide, secondary carbide, and eutectoid carbide. As shown in Fig. 2, non-equilibrium structure is formed for high-carbon high-speed steel cast structure. Carbide morphology includes lump forms, fish-bone shape, thin strip and granular shape. Carbides distributed both at grain boundary and in grain. Moreover, according to the observation in the SEM, various carbides show different brightness. The bright white carbides are rather concentrated, and locate near grain boundary. But the distributions of gray carbides are dispersed.

As shown from Fig. 3, the results of energy dispersive spectrum (EDS) analysis show that, as for gray carbide, its major alloy elements are vanadium. In addition, it includes little elements Mo and W. 6-Electromagnetic stirrer; 7-Cables; 8-Low frequency power Therefore, the gray carbide is mainly composed of MC type carbides. As for bright white carbide, its major alloy elements are W and Mo and it is composed of M6C type carbides with larger size. In matrix, there is little alloy elements. The results indicate that the formation of bright white fish-bone carbide is related to W and Mo component in alloy. The M6C carbides would dissolve at 850℃~1150℃ in the subsequent heat treatment, even if at 1350℃, the M6C carbides fail to dissolve wholly. Hence, to improve performance of roller, MC carbides are needed to enhance abrasive resistance and machinability of steel. The results of SEM also indicate that the inhomogeneous distribution of V, W and Mo elements is related to distribution of carbides.

Cray carbide

Bright white carbide Casting matrix Fig. 3: EDS curve of carbides (2.2%C, 5%V, 9%W) Fig. 4 shows the solidification structure of high-carbon high-speed steel without EMS, with forward and backward EMS (250A, 10Hz, mixing time 10s, reversing time 5s), and with rotational EMS (250A, 10Hz). It indicates that without EMS, the carbides are granular and fish-bone shape, and mainly distribute along grain boundary. With forward and backward EMS, the major carbides are granular shape besides a little fish-bone shape carbide. The granular carbides distribute at the whole section. After applying rotational EMS, the major carbides is granular shape, and the distribution of carbides is more uniform.

Without EMS

With forward and backward EMS With rotational EMS Fig. 4: Carbides with and without EMS (2.2%C, 8%V, 5%W)

The SEM and EDS analysis results for carbides of Fig. 4 show that for forward and backward EMS, the major elements of MC type carbides is V, while W and Mo element locate in the fish-bone shape carbides. However, V, W and Mo elements can be found in the MC type carbides in the specimens cast with rotational EMS and without EMS. Moreover, with rotational EMS, the amount of MC type carbides is much more than that without EMS, and there is a lot of W and Mo element in the MC type carbides besides much V element. As a result, the fish-bone shape carbides containing much W and Mo are restrained. That is the reason why there is little fish-bone carbides in the specimens made by rotational EMS.

As we know from literature [5], during solidification of high-carbon high-speed steel, MC type carbides firstly precipitate form molten steel, then with temperature reduction, other carbides would precipitate. Therefore, the EMS enhances steel flow, which accelerates the heat dissipation of molten steel. As a result, during the solidification process that V element combines with carbon to form MC type carbides, the W and Mo elements would also be promoted to participate in to form MC type carbides. Hence, with EMS, there is not enough W and Mo elements to form M6C type carbides, and the fish-bone carbides and strip carbides would be restrained.

With different stirring current, the forms of carbides of specimens are shown in Fig. 5. As shown from Fig. 5, it clearly indicates that with stirring current increasing, the fish-bone shape carbides change to strip and granular carbides. At 250A, the strip carbides decrease, and the granular carbides gradually increase. However, at 350A, some strip carbides appear at the grain boundary once again besides granular carbides. The SEM analysis results also show that without EMS, W and Mo elements locate near the strip carbides, and V mainly distribute on the grain boundary with the forms of reticulation. After imposing EMS, V element distributes in the grain with the forms of granular carbides. The results show that for this casting system, the optimal stirring current is about 250A, and more granular carbides can be obtained. The optimal stirring current maybe relates to the size of casting.

0A

150A 250A 350A Fig. 6: EDS curve of granular carbide in specimens in Fig. 5 (1.6%C, 5%V, 5%W) 0A 150A 250A 350A Fig. 7: EDS curve for strip carbide in specimens in Fig. 5 (1.6%C, 5%V, 5%W) Fig. 6 and Fig. 7 are the EDS analysis results for granular carbides and strip carbides respectively. Although the main composition of granular carbides is V element, as show from Fig. 6 and Fig. 7, with stirring current increasing, the component of V element decrease and the component of W and Mo increase in the granular carbides. In the strip or fish-bone shape carbides, the component of Mo element has the trend to decrease firstly and then increase with stirring current. However, the component of W element has reverse tendency. Without EMS, Fig. 8 is the EDS analysis results of granular carbides in specimens located at center, half-radius, and the outer surface of casting. The results show that at the half-radius of casting, the component of V element in granular carbides is higher than that in center or at the outer surface. In the conventional solidification process, solidification rate at outer surface of casting is higher than that in inner region. MC type carbides precipitate firstly at outer surface region, and the V element supplements continuously from inner due to the steel flow and diffusion. As a result, the component of vanadium in inner region decreases. Because of sharp cooling rate at outer surface region, the solidification is too rapid to supplement of V element. At the half-radium region, the solidification rate slows down, and there is enough time for V element supplement from inner. Therefore, in half-radium region, the component of V element in MC type carbides is higher than that at outer surface and in center region.

With EMS, the EDS analysis for granular carbides at different positions is shown in Fig. 9. It clearly shows that in the center region of ingot, the component of V element is very higher in MC type carbides. In the outer surface region, the component of elements is almost unchanged for the casting condition with EMS and without EMS. Comparing with Fig. 8, the distribution of V element indicates that the EMS maybe changes the temperature distribution and the MC type carbides firstly precipitate in the center region of specimen. Therefore, controlling the temperature distribution and cooling rate would be appropriate measures to get more MC type carbides. center half-radius outer surface Fig. 9: EDS curve for granular carbides at different positions with EMS (1.6% C, 8% V, 5% W)

Conclusions

With electromagnetic stirring, carbides distributed mainly on the boundary become to dispersively distribute in the grain and along the boundary. And with the increase of the applied current, distribution of carbides are more uniform. Rotary electromagnetic stirring tends to promote to form MC type carbides, break reticulation or strip carbides and reduce the size of carbides. There exists an optimal electromagnetic stirring current intensity. EMS can affect the type of precipitated carbides. With electromagnetic stirring, at cast center, the main carbides are MC type carbides.

Fig. 1 :

 1 Fig. 1: Schematic of experimental apparatus 1-Power supply; 2-Smelting furnace; 3-Induction coil; 4-Platform; 5-Casting crucible; 6-Electromagnetic stirrer; 7-Cables; 8-Low frequency power

Fig. 2 :

 2 Fig.2: Carbides of high-carbon high-speed steel (2.2%C, 5%V, 9%W)

5 :

 5 Forms of carbides with different stirring current (1.6%C, 5%V, 5%W) center half-radius outer surface Fig. 8: EDS curve for granular carbide at different positions without EMS (1.6%C, 8%V, 5%W)

Acknowledge:

Financial supports from National Natural Science Foundation of China (51474065) and "111" Project (B07015), are gratefully acknowledge.


N Dropka

T Ervik

F M Kiessling

Scale-up of DS-silicon growth process under TMF

Keywords: directional solidification, simulation, silicon, travelling magnetic field, scale-up

Despite many research studies of DS-Si growth under TMF at lab scale, transfer of technology to industrial scale is still empirical and limited to low scale-up ratios. To assure safe upscaling, we derived a set of similarity principles for numerical modeling of DS-Si processes in TMF and verified the obtained numerical results with experimental data gained in G1-, G2-and G5-sized furnaces provided with KRISTMAG ® heater magnet modules. Results pointed out that shared parameters with various exponentiation in various dimensionless numbers hindered accurate upscaling. Nevertheless, with selection of crucial dimensionless numbers and correct definition of the characteristic parameters, the fundamental upscaling from G1 to G5 is feasible.

Introduction

A common approach to improve the process economy of directionally solidified silicon (DS-Si) is to scale-up the ingot size. To achieve that, larger ingot cross sections and higher aspect ratios are favorable. In industrial DS-Si, the control of the s/l interface in G4-G6 geometry (G: generation; 4-6: obtained brick number per ingot width for standard wafers 156x156 mm 2 ) is challenging. Further scaling-up to G7 was demonstrated [1], but due to low crystal quality, this process still hasn't reached industrial maturity. To control morphology and shape of the s/l interface, travelling magnetic fields (TMFs) were proposed and successfully applied [2]. Despite many research studies at lab scale G1, the transfer of technology between different scales is still empirical and limited to low scale-up ratios. In crystal growth practice, the scale up ratio is defined as the ratio of ingot widths i.e. diameters at two scales. Rare fundamental scale-up studies based on 3D modeling were verified in liquid metal set-ups with either small cylindrical D = H = 14 cm [3] or small rectangular crucibles L = W = H = 10 cm [START_REF] Dadzis | Modeling of directional solidification of multicrystalline silicon in a traveling magnetic field[END_REF], often under isothermal conditions. Bearing in mind: i) the significant difference in heat conductivity and generated latent heat between e.g. GaInSn and Si, and ii) a nonaxisymmetric DS-Si furnace geometry, the higher scale-up ratios based on liquid metal experiments easily become speculative. The aim of this paper was: (i) to derive a set of similarity principles [START_REF] Kline | Similitude and Approximation Theory[END_REF] for modeling of DS-Si processes in TMF so that a numerical model can exactly reproduce the coupled transport phenomena and (ii) to verify the obtained numerical results with experimental data from real set-ups. For that purpose, it was necessary to reappraise the definition of dimensionless numbers that are relevant for DS process. Particularly, we studied mc-Si growth in real G1-, G2-and G5-sized furnaces equipped with KRISTMAG ® heater magnet modules (HMMs) [2,[START_REF] Lange | [END_REF] shown in Fig. 1.

Model and Methodology

The numerical model described DS-Si growth under TMF. Heat and Lorentz forces F L were generated by the side KRISTMAG ® HMM with several heating coils positioned upon each other and supplied with a combination of ac and dc. Supplementary heat originated from top and bottom resistance heaters in meander-form. The transport phenomena taking place in G1-, G2-and G5-sized furnaces were governed by the equations of continuity, Navier Stokes, energy and induction together with the Ohms law given elsewhere [7]. Due to the complexity of 3D models, we solved them stepwise, performing global 3D simulations for each furnace neglecting Si melt flow and local 3D simulations for Si flows driven by buoyancy and TMF. The global CFD analysis provided thermal boundary conditions (BCs) for the local simulations. Furthermore, the global 3D electro-magnetic (EM) analysis provided the Lorentz force density fields for the local simulations. Pure buoyancy driven melt flows were used for initiation of the magnetic driven flows and as a benchmark.

The 3D CFD and EM simulations were performed using the commercial software ANSYS CFX 14.0 and Ansys Emag.

The scale-up was discussed using dimensionless quantities such as Reynolds Re (1), Grashof Gr (2), Stefan Ste (3), magnetic forcing F (4), magnetic Reynolds Re m [START_REF] Kline | Similitude and Approximation Theory[END_REF] and Shielding S (6) number. Skin depth  of TMF was defined in (7). The notation corresponded to the ones in [11].

  5 0 0 2 0 2 3 2 3 2 . el r * * el m * L sl S growth l L s S * y , x * f ; ) L ( S ; L u Re ; L F F ; H r z T z T Ste ; L T g Gr ; L u Re                                                             (1-7)
The characteristic length L * was defined individually for each transport phenomenon, e.g. in case of flow

L * = (V l ) 1/3
while for thermal and EM simulations, half ingot width was used L * = L/2. The symbols V l , r growth and T x,y corresponded to the melt volume, the crystal growth rate and the horizontal temperature difference, respectively. Tab.1: TMF parameters used in verification experiments.

size The influence of ingot size on F L was studied numerically in G1-, G2-and G5-sized furnaces in the wide range of EM parameters: frequency f = 10 -600 Hz, phase shift  = 0° ± 110° and ac amplitude I o = 0 -500 A.

f 1 ; f 2 [Hz]   ;   [°] I 1 / I 2 [1] F L direction
Verification experiments were performed in G1-, G2-and G5-sized furnaces loaded with 14 kg, 75 kg and 640 kg Si respectively. Corresponding ingot aspect ratios L / H were 1.7, 1.9 and 2.1. Average experimental growth rate was ca. r growth = 1 cm/h. EM parameters of double frequency 2f-TMF that were used in experiments are given in Tab. 1. Double frequency 2f-TMFs were used in order to protect crucible coatings by shifting the maximal velocities towards the melt bulk [8].

Results and Discussions

The influence of ingot size on F L magnitude, direction, and distribution in the melt is given in Fig. 2 and Fig. 3.

The results showed that the magnitude of F L at all scales depends nearly linearly on frequency, quadratically on ac magnitude and exhibits a maximum at phase shift Δϕ = 0° (Fig. 2). The reversal of F L maxima between G1-and G2case (Fig. 2b) was a consequence of their unequal number of heating coils. For the same number of heating coils, the reversal didn't appear, i.e. the highest F L maximum was obtained in G1-case followed by G2-and G5-cases, respectively. For the same EM parameters, upscaling of the melt decreased the F L magnitude significantly. Conversely e.g. to achieve the same magnitude of F L at higher scales by constant f and , ac has to increase. Considering F L direction and spatial distribution at all scales, our findings revealed the crucial role of frequency. With an increase in frequency, the slope of F L vectors decreased (Fig. 3a,d). Moreover, by the increase of frequency from e.g. 10 to 50 Hz, TMF penetration depth δ decreased from 145 mm to 65 mm. Full penetration of G5 melt i.e.  = L * was achieved for f ≤ 1.2 Hz. Nevertheless, very low frequencies are not feasible for practical applications due to ac demand above the thermal requirements of the process. For the same EM parameters, the slope of F L vectors remained the same during upscaling (Fig. 3a-c). The small discrepancy observed at the bottom region originated from the axially off centered melt position relative to the HMM in G2 and G5 furnaces. If common three-phase ac (f = 50 Hz, = -120°) was used for generation of TMF, F L vectors would have axially non-uniform low slope and magnitude. For practical application, TMF has to be accurately adjusted with a time to both: cooling program and to the position of s/l interface relative to the heaters that is a challenge. Therefore, ac with lower frequency (e.g. f  20 Hz) and negative phase shift of ca.= -90° is a better choice for interface shaping due to higher δ, higher F L slope and relatively uniform axial magnitude distribution. If scale-up from G1-to G5-size was analyzed using dimensionless magnetic numbers, the following results were obtained: for constant f = 20 Hz, Shielding number S increased as expected from S G1 = 2.4 to S G2 = 7.0 and S G5 = 32.9.

According to the similarity theory, the same S values assure similar penetration behavior of TMF at various scales. In our case S G5 = 32.9 in G5 furnace will be also obtained in G1 furnace if f increases to f = 272 Hz. Nevertheless, such increased frequency brought significant loss in the slope of F L vectors (Fig. 3a,d) that is crucial for interface shaping. Therefore, Shielding number S seems to be less relevant for DS-Si growth if the goal of TMF application is to provide slightly convex s/l interface shape. Further on, magnetic forcing F increased by upscaling even if EM parameters were kept constant i.e. if F L decreased. For the cases shown in Fig. 3a-c, F had a values of F G1 = 1.36•10 8 , F G2 = 4.46•10 8 and F G5 = 8.36•10 8 , respectively. For preserving the F value constant by upscaling from G1-to G5-size, Lorentz force density F L should decrease 50.6 times down to the negligible value of F L = 0.57 N/m 3 . Obviously magnetic forcing F alone is not able to describe magnetic similarity. The s/l interface shapes obtained in verification experiments in G1-, G2-and G5-sized furnaces in form of Lateral Photovoltage Scanning (LPS) images [9] of ingot vertical cuts are shown in Fig. 4. As already presented in our previous papers e.g. [7,8,10], upwards directed TMF promoted s/l interface concavity (Fig. 4b), while downwards directed TMF promoted convex interface (Fig. 4a,c). Similar temperature fields and interface deflections were obtained in our 3D simulations. An example of simulated temperature and velocity distributions obtained during Si growth in G2-sized furnace exposed to upwards 2f-TMF is shown in Fig. 5. The Lorentz force density streamlines in the melt corresponded to EM parameters given in Tab.1. The Lorentz force density maximum was located at the middle of the crucible side walls near s/l interface (Fig. 5a). The melt flow was in laminar regime with a velocity maximum located at the melt diagonals near the free surface (Fig. 5c). The flow exhibited a complex 3D multi-vortex pattern with dominantly upward directed velocities at the melt side peripheries. Particularly the velocities in this peripheral region determined the concave interface shape. In downwards directed TMF in G1 and G5 case studies, again the 3D multi-vortex velocity pattern was obtained with dominantly downward directed velocities near the three-phase junction that induced convex interface [10]. In all considered cases, laminar regime was obtained. Simulation results showed that in upscaled cases, stronger F L were needed to counteract buoyancy force and flatten the interface. In dimensionless numbers, upscaled G2 case with ca. Obviously, neither of these numbers, if considered separately was able to relate F L with buoyancy force that is crucial for interface shaping. As already shown for VGF-GaAs growth [11], a combined dimensionless number F•Gr -1 •Ste -1 contained all relevant variables for crystal growth in TMF: temperature gradients, growth rate and Lorentz force density.

If the aim of TMF application is exclusively the interface shaping, then only axial F L component in the vicinity of threephase junction should be used for the calculation of F. Please note that L* didn't appear in this combined number.

Nevertheless, the increase of ingot size was accounted indirectly by the increased T and needed F L to compensate the buoyancy force.

Conclusions

Our 3D simulation results showed that TMF, if its EM parameters were carefully selected is a very promising tool for the control of s/l interface shape in DS-Si growth at various scales. Particularly, we studied scale-up in G1-, G2-and G5-sized furnaces that are using KRISTMAG ® technology for the generation of TMF. Based on the simulation findings, we proposed a method for the theoretical, i.e. numerical scale-up using a combination of dimensionless numbers Grashof Gr, Stephan Ste and magnetic forcing number F. Obtained numerical results were in good agreement with experiments.

Fig. 1 :

 1 Fig.1: Silicon ingots (a) and crucial parts of G1-, G2-and G5sized DS-furnaces with KRISTMAG ® HMMs (b -d).

Fig. 2 :

 2 Fig. 2: Maximal Lorentz force density in G1-, G2-and G5-sized furnaces as a function of: a) ac amplitude, b) phase shift and c) frequency.

Fig. 3 :

 3 Fig. 3: Lorentz force density streamlines in Si melt at various scales for: a-c) f = 20 Hz, = -90° and I 0 = 200 A, d) f = 272 Hz, = -90° and I 0 = 200 A.

Fig. 4 :

 4 Fig. 4: LPS images of G1-(a), G2-(b) and G5-sized (c) partial (right-half) vertical cuts of ingots grown in downwards (a, c) and upwards (b) 2f-TMFs; interface curvature and crucible right side wall were marked red and black, respectively; image (c) was reprinted from [10] with permission of Elsevier.

  half crystallized Si was characterized by F = 9.6•10 7 , Re = 1.04•10 4 , Gr = 1.75•10 8 , Re m = 4.16•10 -3 and Ste = 2.21•10 -1 . At G5 scale with 20% crystallized Si, dimensionless numbers reached following values: F = 1.9•10 9 , Re = 1.41•10 4 , Gr = 1.78•10 9 , Re m = 5.23•10 -3 and Ste = 7.5•10 -2 .

Fig. 5 :

 5 Fig. 5: Simulation results for Si growth in G2-sized furnace exposed to upwards 2f-TMF defined in Tab.1: a) Lorentz force density streamlines in the melt, b) temperature distribution in Si in vertical mid plane, c) velocity streamlines in the melt.

Acknowledgment

Experimental support of Ch. Frank-Rotsch, R. Menzel and D. Linke during the growth of G2 crystal and T. Turschner and S. Kayser for providing LPS images is gratefully acknowledged.