Influence of coil geometry on the induction. heating process in crystal growth systems

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1 Influence of coil geometry on the induction heating process in crystal growth systems Mohammad Hossein Tavakoli, Abdollah Ojaghi, Ebrahim Mohammadi Manesh and Morteza Mansoor Physics Department, Bu-Ali Sina University, Hamedan 65174, Iran ABSTRACT: Different shapes and orientations of an RF-coil turns in oxide Czochralski crystal growth systems are considered and corresponding results of electromagnetic field and volumetric heat generation have been computed using a finite element method (Flex- PDE package). For the calculations, the eddy current in the induction coil (i.e. the self-inductance effect) has been taken into account. The calculation results show the importance of shape, geometry and position of the RF-coil turns with respect to the crucible and afterheater on the heat generation distribution in a CZ growth system. Keywords: A1. Computer simulation, A2. Czochralski method, Induction heating. PACS: Cb; Dh; q; Fq. Tel: Fax: web: 1

2 Influence of coil geometry on the induction heating process in crystal growth systems Mohammad Hossein Tavakoli, Abdollah Ojaghi, Ebrahim Mohammadi Manesh and Morteza Mansoor ABSTRACT: 1 Introduction The principle of induction heating is widely applied to produce the required thermal power in several crystal growth systems. In a basic radiofrequency inductive heating setup such as Czochralski (CZ) furnace (Figure 1 in [1]), a solid state RF-power supply sends an alternative electric current through an induction coil (inductor). The part to be heated (i.e. crucible and active afterheater) is placed inside the coil and circulating eddy currents are induced within the metallic parts. These currents flow against the electrical resistivity of the metal, generating precisely localized heat. Coil design is one of the most important aspects of a CZ crystal growth system [2][3]. A well-designed coil maintains the proper heating pattern and production rate, and maximizes the efficiency of the induction heating power supply. Induction coils are normally made of copper tubing - an extremely good conductor of heat and electricity - and can have single or multiple turns; have a helical, round or square shape. RF-coils are usually cooled by circulating water, and are most often custom-made to fit the shape and size of the crucible. Because of this, coil design is generally based on experience and trial-and-error process. A successful growth of perfect single crystals requires a balance of energy production as well as heat and mass transport through the setup, because the quality of a grown crystal is directly related to its thermal history and the transport phenomena in the 2

3 furnace. The heat transfer processes in crystal growth systems are quite sensitive to the geometry of the furnace, heat generation and orientation of the coil-crucible-afterheaterinsulation [4][5][6]. Therefore, modeling of heat generation as well as heat and mass transfer processes is necessary for achieving a complete knowledge of the growth process in order to improve the quality of the grown crystals [1][7][8]. The goal of this article is to reveal the role of RF-coil geometry on the induction heating process in an oxide Czochralski system and compare the corresponding results of electromagnetic field and heat generation distribution using a 2D FEM numerical approach. To do it, different cross section shapes of a RF-coil and its orientation with respect to the crucible and active afterheater are considered corresponding to the real growth situations. Our challenge is to extend the fundamental understanding of induction heating process used for crystal growth technology, because this understanding will enable the growers to optimize existing procedures, growth new crystals and design new growth methods. 2 Mathematical model 2.1 Governing equations The assumptions and mathematical model of induction heating have been described in detail elsewhere [1]. The set of fundamental equations with boundary conditions have been solved using the finite element method (FlexPDE package [9]). 2.2 The calculation conditions Values of electrical conductivity employed for our calculations are presented in [1] and operating parameters are listed in Table 1. The induction coil has two parts with 6 and 2 hollow copper turns, respectively. Three cross section shapes of the coil turns have been considered for the calculations, rectangular, shielded rectangular and circu- 3

4 lar; corresponding to the real growth setups, Figure 1. In order to compare the results of electromagnetic field and heat generation distribution, we have assumed a driving electrical current with total voltage of 200 v and a frequency of 10 khz in the RF-coil (typical values) for all cases. The results based on this set of parameters will be presented now. 3 Results and discussion We explain the results of electromagnetic field and heat generation distribution in an oxide CZ setup including crucible and active afterheater corresponding to an often used growth situation. The obtained results are presented in the following sections, starting with different cross section shapes of the coil turns and then different radii and turns number of the RF-coil. 3.1 Cross section shapes of the coil turns In the first section, we have considered three shapes for cross section of the coil turns: rectangular, shielded rectangular and circular with unique height and thickness (Figure 1). Figure 2 shows the distribution of in-phase component (right hand side) and out-of-phase component (left hand side) of the magnetic stream function (Ψ B ) for the rectangular cross section. The maximum of in-phase component (C) is located at the lowest and top edges of the RF-coil (C max = W eber) while the minimum (C min = W eber) is located on the middle part of the crucible side wall. Variation of this component is too high in the area close to the maximum and minimum points. In other parts of the system, this component is nearly constant. For the outof-phase component (S), the maximum is located at the outer surfaces of the induction coil turns (with S max = W eber) and its intensity rapidly decreases towards 4

5 the crucible and afterheater wall. The S distribution has a linear gradient in the space between the coil and the crucible and afterheater side wall. The intensity of S max is 8 times greater than C min (absolute value). The expulsion of the C-field component from the RF-coil and the S-field component from within the crucible is particularly evident in the space between the crucible and the induction coil, as is the fact that the C and S components tend to follow the crucible and the induction coil, respectively. In the crucible and afterheater S/C 3 where the required heat generation occurs. It means that within the conductors the S component contributes to the heat generation much more than the C component, according to Eq. (4) in [1]. For this reason, the spatial distribution of heat generation in the conductors is similar to the S component distribution in that region. The volumetric heat generation rate (q) in the crucible and afterheater has been shown in Figure 3. The maximum value of energy deposition in the crucible is q C A max = W/m 3 and is located at the middle portion of the outer surface of the crucible wall. Also, there is a local maximum of energy production in the afterheater wall which its position depends strongly on the afterheater location and orientation with respect to the crucible and the RF-coil [1]. The total heat generation rate in the crucible and afterheater is Q C A total = 7609 W by using integral over their volume. The spatial distribution of heat generation in the induction coil is mostly uniform with local hot spots at the lowest and upper edges. These hot spots are not singularities in the solution for the electromagnetic field at the coil corners. They are a result of the fields curvature (especially C component) in tracing with the coil geometry (i.e. the cross section shape of turns). It shows the corner and edge effect which is a common occurrence in induction heating applications [1][8]. The total heat generation of the RF-coil is Q coil total = 574 W. It means that about 7% of the total power of the system is produced in the RF-coil and 93% in the crucible and afterheater. Figures 4-7 show the distribution of electromagnetic field components in the system 5

6 and heat generation in the crucible and afterheater with the shielded rectangular and the circular cross section shape of the coil turns, respectively. The most important differences compared to the rectangular cross section shape are as follow: The distribution of C component is similar except close to the RF-coil which markedly modified by the coil cross section shape. There is no corner singularities at the lowest and upper edges of the RF-coil in the case of circular shape because of absence sharp corners. The intensity of this component is similar for all cases. The distribution of S component is clearly similar in all cases but its intensity is stronger in the case of circular shape than other cases. Because of modification in the intensity and spatial distribution of the electromagnetic field components, the volumetric heating distribution in the crucible and afterheater with circular cross section shape is more effective compared to other cases considered here, Table 2. It means that in this case, the coil can be coupled to the crucible and afterheater as closely as feasible for maximum energy transfer. It should be mentioned that, although the C intensity is similar in all cases but S is larger in the circular case than rectangular and shielded rectangular case, as has already been explained. This leads to greater energy deposition in the system. 3.2 Coil radius In the second section, the inner radius of the RF-coil is increased from r co = 78 mm to 85.8 mm (i.e. r coil = 10%) for the case of rectangular cross section shape of the coil turns. On the other hand the distance between crucible and coil is increased about %. Figure 9 shows the distribution of in-phase component (right hand side) and out-of-phase component (left hand side) of the magnetic stream function for this configuration. By comparison with the smaller coil radius (Figure 2), we can find that C component has a reduction of 10% but S component has not been changed. It is important to mention that the distribution of the in-phase component depends on 6

7 the coil-crucible-afterheater orientation and for the out-of-phase component depends strongly on the RF-coil geometry. For this reason by increasing the distance between the conductors (crucible and afterheater) and the induction coil, the influence of their interaction becomes less effective and as a result there is an intensity reduction for the C component and no visible change for the S component. The volumetric heat generation rate (q) in the crucible and afterheater has been shown in Figure 10. The maximum value and total energy deposition in the crucible and afterheater are q C A max = 5.0 W/m 3 and Q C A total = 5096 kw, respectively. The decrease in the total power generation is 33% (2511 W att). Also noteworthy is that the change in the coil diameter has not any effective influence on the spatial distribution of heat generation in the crucible and afterheater. 3.3 Gap between the coil turns In the final simulation reported here, we considered a case in which the gap between the coil turns was increased to 37.5 mm from 3 mm. To do it, The number of coil turns was changed to 3+1 instead of 6+2 with the fixed position of the lowest and upper turns of the main coil. In order to compare the obtained results, we have assumed a unique voltage for every turn, i.e. the total voltage of the RF-coil is reduced to 100 v proportional to the turns number. The resulting fields of electromagnetic field and heat generation are shown in the Figures The distribution of C and S components as well as q has been modified in the second configuration especially around the induction coil compared to the first configuration. The most important features are as follow: Although C min has been decreased (5%) in the second configuration (which is predictable) but C max has been increased (6%). This surprising result arises from the electromagnetic end and edge effects (i.e. the distortion of electromagnetic filed in its end and edge areas) [1][2][8]. Because of large distance between the coil turns these 7

8 effects are more effective in the second configuration compared to first one. As a result, there is an increase for C max which has been located at the coil turns. The intensity of S component is the same. It means that any change in the distance between the coil turns has not any influence on the intensity of this component. In the first configuration, a uniform gradient of both components exists between the coil and the crucible side wall while in the second one both components have a wavy shape that indicates a non-uniform gradient of these components in that area. These non-uniform gradients make a non-uniform distribution of heat production in the crucible and afterheater which is shown in Figure 15(b). There are two maxima in the crucible side wall with an approximately the same intensity. Also notably is that in this configuration, the location of q max has been lifted to the upper part of the crucible wall. These important features affect directly the temperature and flow field of the system. Finally the total conductor power input is now down to 6108 W from 6890 kw. It indicates that in the second configuration, the induction coil can not deposit enough power into the crucible to get the job done, i.e. to melt oxide materials such as sapphire (melting point 2050 o C). 4 Conclusions We have presented and demonstrated some numerical calculation results of induction heating process for an oxide Czochralski crystal growth system with different geometries of an RF-coil by using a finite element method. The obtained results indicate that, inductive coupling of the crucible and active afterheater with the induction coil is quite important and depends strongly on their geometry, location and orientation with respect to each other. The mathematical model - described in the present work and [1] - can accurately predict the impact of changing the RF-coil geometry on the structure of heat generation 8

9 within the furnace. It can be useful for correct choice of the coil style, design purposes of large and modern growth systems and advanced manufacturing techniques. 9

10 References [1] Tavakoli, M. H.; Samavat, F. and Babaiepour, M. Influence of active afterheater on the induction heating process in oxide Czochralski systems. Cryst. Res. Technol , [2] Leatherman, A.F. and Stutz, D.E. Induction heating advances; National Aeronautics and Space Administration, [3] Rudnev, V.; Loveles, D.; Cook, R. and Black, M. Handbook of induction heating; New York NY., [4] Tavakoli, M. H. and Wilke, H. Numerical study of heat transport and fluid flow of melt and gas during the seeding process of sapphire Czochralski crystal growth. Cryst. Growth Des. 2007, 7, [5] Tavakoli, M. H. and Wilke, H. Numerical investigation of heat transport and fluid flow during the seeding process of oxide Czochralski crystal growth - Part 1: nonrotating seed. Cryst. Res. Technol. 2007, 42, [6] Tavakoli, M. H. and Wilke, H. Numerical investigation of heat transport and fluid flow during the seeding process of oxide Czochralski crystal growth - Part 2: rotating seed. Cryst. Res. Technol. 2007, 42, [7] Gresho, P.M. and Derby, J.J. A finite element model for induction heating of a metal crucible J. Crystal Growth 1987, 85, [8] Tavakoli, M. H. Modeling of induction heating in oxide Czochralski systems - advantages and problems. Cryst. Growth Des. 2007, 8, [9] 10

11 Figure captions: Figure 1. Three cross section shapes of the coil turns: rectangular, shielded rectangular and circular with unique diameter and thickness. Figure 2. Components of the magnetic stream function (Ψ B ) calculated for the rectangular cross section shape of the coil turns. The right hand side shows the in-phase component (C) with C max = weber on the lowest and top edges of the RF-coil and C min = weber on the crucible wall. The left hand side shows the out-of-phase component (S) with S max = weber on the outer surfaces of the induction coil turns. Figure 3. Volumetric power distribution (q) in the crucible and afterheater calculated for the rectangular cross section shape of the coil (for a better demonstration the wall, bottom and afterheater top cover part are separately magnified). The maximum value of energy deposition is q rec max = W/m 3 and the total heat generation rate is Q rec total = 7609 W. Figure 4. Components of the magnetic stream function (Ψ B ) calculated for the shielded rectangular cross section shape of the coil turns. The right hand side shows the in-phase component (C) with C max = weber on the lowest and top edges of the RF-coil and C min = weber on the crucible wall. The left hand side shows the out-of-phase component (S) with S max = weber on the outer surfaces of the induction coil turns. Figure 5. Volumetric power distribution (q) in the crucible and afterheater calculated for the shielded rectangular cross section shape of the coil. The maximum value of 11

12 energy deposition is q s rec max Q s rec total = 7514 W. = W/m 3 and the total heat generation rate is Figure 6. Components of the magnetic stream function (Ψ B ) calculated for the circular cross section shape of the coil turns. The right hand side shows the in-phase component (C) with C max = weber on the lowest and top surfaces of the RF-coil turns and C min = weber on the crucible wall. The left hand side shows the out-of-phase component (S) with S max = weber on the outer surfaces of the induction coil turns. Figure 7. Volumetric power distribution (q) in the crucible and afterheater calculated for the circular cross section shape of the coil. The maximum value of energy deposition is q C A max = W/m 3 and the total heat generation rate is Q C A total = 8050 W. Figure 8. Components of the magnetic stream function (Ψ B ) calculated for the r co = cm and rectangular cross section shape of the coil turns. The right hand side shows the in-phase component (C) with C max = weber on the lowest and top edges of the RF-coil and C min = weber on the crucible wall. The left hand side shows the out-of-phase component (S) with S max = weber on the outer surfaces of the induction coil turns. Figure 9. Volumetric power distribution (q) in the crucible and afterheater calculated for the r co = cm and rectangular cross section shape of the coil. The maximum value of energy deposition is q max = W/m 3 and the total heat generation rate is Q total = 5096 W. Figure 10. Components of the magnetic stream function (Ψ B ) calculated for the con- 12

13 figuration with 6+1 rectangular coil turns. The right hand side shows the in-phase component (C) with C max = weber on the lowest and top edges of the RF-coil and C min = weber on the crucible wall. The left hand side shows the out-of-phase component (S) with S max = weber on the outer surfaces of the induction coil turns. Figure 11. Volumetric power distribution (q) in the crucible and afterheater calculated for the configuration with 6+1 rectangular coil turns. The maximum value of energy deposition is q max = W/m 3 and the total heat generation rate is Q total = 6890 W. Figure 12. Components of the magnetic stream function (Ψ B ) calculated for the configuration with 3+1 rectangular coil turns. The right hand side shows the in-phase component (C) with C max = weber on the lowest and top edges of the RF-coil and C min = weber on the crucible wall. The left hand side shows the out-of-phase component (S) with S max = weber on the outer surfaces of the induction coil turns. Figure 13. Volumetric power distribution (q) in the crucible and afterheater calculated for the configuration with 3+1 rectangular coil turns. The maximum value of energy deposition is q max = W/m 3 and the total heat generation rate is Q total = 6108 W. Figure 14. Profiles of the heat generated along the outer surface of the crucible and afterheater side wall for the configuration with (a) 6+1 and (b) 3+1 rectangular coil turns. 13

14 Table 1. Operating parameters used for calculations. Description (units) Symbol Value Crucible inner radius (mm) r c 50 Crucible wall thickness (mm) l c 2 Crucible inner height (mm) h c 100 Afterheater inner height (mm) h af 100 Afterheater hole (mm) r af 10 Distance between the crucible and afterheater (mm) D ca 30 Coil inner radius (mm) r co 78 Coil width (mm) l co 13 Coil wall thickness (mm) l co 1.5 Height of coil turns (mm) h co 20 Distance between coil turns (mm) d co 3 Distance between two coils (mm) D co 55 14

15 Table 2. Detail information about the heat generated in the CZ system, calculated for different cross section of the RF-coil. Efficiency = Q C A total /Q total Cross section shape Q C A total (W att) Q coil total (W att) Efficiency (%) Rectangular Shielded rectangular Circular

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