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Modelling of directional solidification of BSO
Journal of Crystal Growth 128 (1993) 834-841 North-Holland
,
........ C R Y S T A L GROWTH
Modelling of directional solidification of BSO C h e n t i n g Lin
a
and Shahryar Motakef b
a Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 9 Heat Transfer Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
A thermo-fluid model for vertical Bridgman growth of bismuth silicon oxide (BSO) as model material for semi-transparent, low thermal conductivity oxides is developed. Internal radiative heat transfer, together with convective and conductive heat transfer are considered in this model. Due to the strong internal thermal radiation within the grown crystal, the growth interface is highly convex into the melt, instead of being concave as is the case for opaque materials with the thermal conductivity of the melt larger than that of the solid. Reduction of the growth interface non-planarity through variations in the growth configuration is investigated. A furnace temperature profile consisting of a steep gradient on the melt side and shallow gradient on the solid side of the charge is found to be the most effective approach.
I. Introduction
Bismuth silicon oxide (BSO) is a wide bandgap, high resistivity, optically active semi-insulating material that is photoconductive, and electro-, acousto-, and magneto-optic. This material has wide applications in optical information processing and computing components such as spatial light modulators and volume holographic optical elements, and has also found applications as filters where tailored passbands and phase responses are required [1]. Bi12SiO2o is normally grown by the Czochralski technique. The reported material deficiencies in BSO are, generally, present in all oxides and are related to innate characteristics of the Czochralski growth process. In Cz growth systems, heat is supplied radially to the crucible containing the charge, and removed axially at the solidification front. This sets up destablizing buoyancy forces and an associated unsteady three-dimensional (turbulent) convective structure in the melt [2]. Another inherent shortcoming of the Cz process is the non-axisymmetric heat input and extraction from the crystal, resulting in periodic fluctuation of the growth interface morphology and an associated generation of rota-
tional impurity and point defect striae [1,3,4]. Furthermore, the large melt velocities enhance crucible dissolution, introducing impurity into the matrix. As optical devices generally require property uniformity in volumes with length scales of the order of centimeters, elimination of growth related compositional fluctuations is projected to result in advances in optical device technique. In the present work we numerically study the feasibility of growing BSO by the vertical Bridgman (VB) technique. In this configuration convection in the melt is expected to be much weaker than in the Cz system, and axisymmetric heat input into and extraction from the charge can be more readily achieved. It is, therefore, reasonable to suggest that crystals grown by the VB technique will, as a consequence of a stable and axisymmetric solid liquid interface, exhibit a lower level of micro- and macro-scale compositional variations. We begin with a description of the thermal transport model of the process, including the two-band approach to the modelling of the internal radiative heat transfer in the solid, and briefly describe the numerical method used in the calculations. We will show that semi-transparency of the solid oxide results in a growth interface morphology which is convex into the melt. The effec-
0022-0248/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
C. Lin, S. Motakef / Modelling of directional solidification of BSO
tiveness of various approaches for the reduction of the solid-liquid interface shape curvature are, then, explored. Unlike for most of the oxides, the thermophysical data base for BSO is relatively complete, and thus the present numerical study provides an accurate framework for the design and analysis growth of BSO by the VB technique. A similar study neglecting convection in the melt and the existence of the crucible has been conducted by Brandon and Derby and reported in ref. [5].
2. Model description
2.1. Growth configuration Due to its corrosive nature, BSO melt can be only contained by noble metals such as Pt and Au. However, as the melting point temperatures of Au and BSO are close, Pt has been exclusively used as the crucible material; the thermophysical properties of BSO and Pt are given in table 1. In this study we consider two types of crucibles: (a) standard crucibles made from Pt with the wall thickness varying between 0.5 and 2.0 mm and (b) a composite crucible consisting of a 0.1 mm Pt layer deposited onto a 1.9 mm thick zirconia crucible. In the latter configuration, relatively low axial conduction through the crucible is achieved while the stability of the crucible is preserved; the low axial conduction is later shown to be necessary to reduce the curvature of the solid-liquid interface shape. The present study focuses on a 2 cm diameter and 17 cm long charge. Three growth configurations are studied. In the first configuration, the furnace is modelled to consist of two isothermal zones separated by a gradient region; the temperatures of the isothermal zones are chosen such that the melting point temperature of BSO is at the center of the gradient region (fig. la). The isothermal portions of the furnace are modelled to be "infinitely" long, and the gradient zone length is varied in the analysis. In the second configuration, the gradient zone is split into two regions where, using active heat transfer elements, independently controlled temperature gradients on the solid and liquid sides of the
835
Table 1 Thermophysical properties of materials used in these studies Quantity
Temperature
Value
(K) BSO [6] Melting point Density Solid Liquid Linear expansion coefficient Volume expansion coefficient Specific h e a t / u n i t volume Solid Liquid Thermal conductivity Solid
Liquid Viscosity Young's modulus Breaking strain Latent heat
1168
1168 K
293 1223
9.206 g cm -3 7.63 g c m - 3 16×10 6 K-1
1223
7x10-5 K 1
2.7Jcm 3K 1 3.0 J cm -3 K - 1
973 1073 1168 1168 1173
0.1944Wm -1K i 0.1782 W m - 1 K - I 0.162 W m - i K - 1 0.27 W m - l K - 1 0.22 dyn cm - 2 s 93Gnm z 2 × 10 -4 460 J c m - 3
Pt [71 Thermal conductivity Surface emissivity
900-1300
73.7-79.1 W m a K I
1273
0.15
1073-1923
0.11-0.23 W m -1 K -1
Zirconia, type Z Y C [8] Thermal conductivity
charge are established (fig. lb). In the third configuration, the isothermal zones are abandoned and the furnace is configured to have two nonequal gradients over the entire solid and liquid lengths of the charge (fig. lc).
2.2. Mathematical model The temperature and velocity fields in the melt are obtained through solution of the coupled
C Lin, S. Motakef / Modelling of directional solidification of BSO
836
I It1 Coils in
the melt
)l
.
I
.
Heatpipe
Zone
,n.. in the crystal
i Fig. 1. Schematic diagrams of growth configurations and furnace temperature profiles studied: (a) typical vertical Bridgman furnace, (b) modified Bridgman furnace, with two independently controlled gradients in the gradient zone, (c) modified Bridgman furnace, with two independently controlled temperature gradients along the charge.
axisymmetric momentum and energy conservation equations:
V" 17V=
1
--17P+vV2V+gflT(es'e~), P
-
(1)
1
V" VT= - - I 7 . ( k17T), pc
(2)
where p is density, k thermal conductivity, v kinematic viscosity, c heat capacity,/3 volumetric expansion coefficient, and eu and e~ unit vectors along gravitational and crucible axial directions. In the above, the melt is assumed to be opaque to internal radiation; this is consistent with the results of Gryvnak and Burch [9] who have shown that oxide melts are opaque at near infrared wavelengths. Eq. (1) is subject to no-slip and no-penetration boundary conditions on the velocity field at liquid-solid boundaries. The last term of eq. (1) is the Boussinesq approximation for the buoyancy forces due to density variations. In our study, the release of latent heat of solidification is negligible compared to the other losses at the growth interface and, thus, is neglected. The temperature field in the crystal is influenced by the semitransparency of the BSO solid to infrared radio. Room temperature optical transmission spectra of two BSO samples, 1.5 and 3 mm thick, are shown in fig. 2. The transmittance of the two samples are approximately the
same at wavenumber larger than 1667 cm -1 (wavelengths smaller than about 6 Izm), indicating the loss in transmission at A < 6/zm is due to surface reflection and that negligibly small absorption occurs in this wavelength range. The samples show intermediate absorption at wavelengths between 1597 and 1266 cm-1; in this range blackbody emissive power at the melting temperature of BSO (1168 K) constitutes only 7% of full spectrum black body emissive power. Therefore, this wavelength range is taken to be
80.0
.......S 60,0
o
40.0
20.0 ......... - -
O0 10000
I 20000
1.5 m m somple 3 m m s•mple
I 30000
Wavenumbers ( 1 / cm )
Fig. 2. Infrared transmission spectrum of BSO for two sample thicknesses of 1.5 and 3 mm at room temperature.
C. Lin, S. Motakef /Modelling of directional solidification of BSO
totally opaque. At wavelengths below this range thermal radiation is completely absorbed. The above indicates that a two-band model (splitting the spectral radiation into an opaque and a transparent region separated at the band-edge wavelength of 6 /zm) can be used to model internal radiative heat transfer in the growing crystal. The implicit assumption in this approach is that the room temperature data can be extrapolated to elevated temperatures close to the melting point temperature of BSO. This assumption has been shown to be valid for BzO 3 [10]. In the two-band model, only the fraction of radiant energy (emitted or reflected) which falls within the spectral transparency band of the material is tracked. This fraction can be written as:
l[j:
f - F o _ ~ - o-T 4
]
AS(e C2/aT- 1) dA .
(3)
Assuming the crucible walls are diffuse and gray, heat exchange between inner crucible walls through the crystal can be calculated using the Gebhart formulation. In this approach, the net total heat transfer rate at a surface element with area A k is given by N
Qk = A k e k f k ~rT4 -- ~ Aiejfj°'Tj4Gjk, j=l
(4)
where the f ' s can be calculated from eq. (3), and Gik is the Gebhart factor defined as the fraction of emitted energy from surface element j being absorbed by surface element k [11]. The calculated values of the Qk's are then applied to the inner surface of the crucible in contact with the crystal and the crystal-melt interface. Energy transfer within the crystal and the crucible is by conduction: V( kV7~) = 0.
(5)
The outer surface of the crucible is radiatively coupled with the furnace (separation distance between furnace and crucible is taken to be small). Heat transfer at the outer crucible surface is modelled to obey the following relation which assumes the internal furnace surface to be black:
( q / A ) = o'•(T 4 - T?).
(6)
837
In the above, • is the surface emissivity of the crucible, and is taken to be 0.15 for both inside and outside surfaces of the crucible in our simulations. The emissivity of the crystal-melt interface is taken to be 1 and the index of the crystal is also taken to be 1. The present approach to modelling of internal radiation in BSO is different from that of Brandon and Derby [5], who modelled the internal radiation to obey Beer's law with a constant absorption coefficient. In our work the two-band model exploits the band-edge structure of the spectral transmittance of BSO to simplify the radiative exchange equations within the semitransparent solid; this approach applied to a different growth configuration has yield relatively good agreement with the experimental results [121.
2.3. Simulation strategy The solution of the governing equations was achieved by an iterative technique employing two commercially available codes ABAQUS and FIDAP. The iteration technique took advantage of the available flexibility of the two packages and configured them as to provide a time-efficient solution to the governing equations: (i) ABAQUS was used to solve the conduction equation in the crystal, melt, and crucible subject to the internal radiative exchanges in the solid, radiative coupling between the crucible and the furnace, and a solid-liquid interface shape across which the thermal conductivity of the charge experiences a step change. The calculated melting point isotherm is inputted into the next step as the location of the growth interface. (ii) FIDAP is used to solve the momentum and energy equations in the melt. The solid-melt interface shape and melt-crucible interface temperature profiles calculated in step (i) are used as input. The difference between the calculated heat fluxes as the melt-crucible and melt-crystal interface and the conduction solutions of step (i) are fed back to step (i) calculations in form of surface heat fluxes. The grid for the ABAQUS simulations consisted of 10 elements in the radial direction of 99
C. Lin, S. Motakef / Modelfing of directional solidification of BSO
838
elements in the axial direction. The grid for FIDAP simulations consisted of 13 and 63 elements in the radial and axial directions, respectively. Convergent solutions were obtained typically after 15 iterations between steps (i) and (ii) with each iteration taking about 1 CPU hour on a SUN 4-280.
31.8 31.6 31.4 g -~
31
30.8
3. Results
30.6
In absence of rigorous relations between the growth conditions and device sensitive properties of BSO, we use the intensity of convection in the melt, and the melt-crystal interface deflection as the primary figures of merit in identifying optimal growth conditions. The latter is motivated by two considerations. First, the radial temperature gradients associated with the interface non-planarity induce thermal strains close to the growth interface in the crystal; this may result in plastic deformation and residual strain causing birefringence non-uniformity in the grown material. Second, these radial temperature gradients drive the convective intensity in the melt, and may cause flow field instabilities resulting in microscopic remelting and regrowth and an associated microscale property variations in the crystal.
30.4
The influence of crucible thickness on crystal-melt interface shape for the furnace configuration of fig. la is shown in fig. 3, with interface deflection defined as the difference between 1.1
g v
1
g "~ 0.9
•
•
o J~ Composite oe 0 . 8 Crucible ~' 0.7 _E 0
. 0.5
.
1
.
.
1.5
2
Pt CrucibleThickness(mm) Fig. 3. T h e influence of crucible thickness on the deflection of growth interface. The interface deflection decreases for thinner Pt crucibles.
With Convection
31.2
8
13-
<
3.1. The crucible effect
Convection
•
i
L
i
h
0.2
0.4
0.6
0.8
Radial Position (cm) Fig. 4. The influence of convection on the solid-liquid interface shape; convection reduces interface deflection by 20%.
the axial interface location at the center and at the edge of the charge. A typical morphology of the solid-liquid interface using the composite crucible is shown in fig. 4. In fig. 3, the interface deflection is observed to decrease with decreasing crucible thickness; the smallest deflection is observed for the composite crucible. Several factors contribute to the observed solid-liquid interface shape. At the growth interface heat transfer on the crystal side is primarily by radiation (conductivity of solid BSO is very small) to the low-temperature crucible; on the melt side, the molten BSO is opaque to radiation and, thus, heat transfer in the melt is by convection; the large radiative heat transfer in the crystal side results in large axial gradients in the melt. As the radiative view factor between the growth interface and the low-temperature crucible walls is largest at the center of the growth interface, (i.e. has the largest radiative loss) the solid-liquid interface shape becomes convex into the liquid. The second factor controlling the crystal-melt interface shape is the axial conduction through the crucible (thermal conductivity of Pt is about 350 times higher than that of BSO) which leads to a large variation of temperature between the crystal center and its periphery; this factor is, generally, stronger than the former in controlling the growth interface morphology. Thus, the interface deflection associated with growth using the composite (low thermal conductivity) crucible is appreciably reduced.
C. Lin, S. Motakef /Modelling of directional solidification of BSO
Nevertheless, reduction of Pt thickness to less than 0.1 mm is not realistically possible, thus, results of fig. 3 establish the upper limit to the effectiveness of controlling solid-liquid interface shape by varying the thermophysical properties of the crucible #1. In the remainder of the study the composite crucible is used.
3.2. Conuectiue effects in the melt The influence of convection on the solid-liquid interface shape for the furnace profile of fig. la is shown in fig. 4; convective mixing in the melt is found to promote a 20% reduction in the interface deflection. The calculated velocity field in the melt is very weak with a maximum value of 1.36 × 10 -4 m / s (Reynolds number Vr/u = 0.47); the maximum value of V generally occurs close to the centerline of the charge. Low values of Reynolds number were observed for all cases studied here and suggest that flow in the melt is laminar, and flow instabilities should not be expected. The Peclet number (= Vr/a scaling convective to conductive heat transfer) is about 15 for results of fig. 4 indicating presence of convective heat transfer in the melt. The recirculating cell (shown schematically in fig. 1) promotes heat transfer between the crucible and the growth interface resulting in a reduced interface deflection. To first order, however, convection in the melt may be neglected and heat transfer in the system be approximated as conduction-dominated in the melt and radiation-dominated in the crystal. In spite of the limited influence, convection has been included in all simulations reported in this study.
3.3. Influence of gradient zone length Growth interface deflections associated with various lengths of the gradient zone in the furnace configuration of fig. la are shown in fig. 5.
1.1
'"
0.9
"= "~
08
• " 2
0
' ' ' 4 6 8 Gradient Zone Length (cm)
' 10
Fig. 5. The influence of gradient zone length on the deflection of growth interface; optimal gradient zone length is about 4 cm.
In these calculations, the temperature gradient is held at 25 K/cm. Results indicate that increasing the gradient zone length from 0.5 to 4 cm produces a 10% reduction in interface deflection. Beyond 4 cm the gradient zone length has a minimal effect on interface deflection. Overall, the gradient zone length does not have an appreciable influence.
3.4. Two temperature gradient furnaces Investigation of the furnace configurations shown in figs. lb and lc is motivated by the following consideration: The crystal-melt interface deflection is controlled by the radiative loss at this interface through the crystal to the crucible with the driving force proportional to the difference between the fourth power of the temperatures. Thus, by reducing the furnace axial gradient along the portion of the crucible containing the crystal, radiative heat loss at the interface center should be reduced. Also, with a large axial gradient on the melt side, radial non-uniformity in radiative loss at the growth interface
E
0.9
~ 0.8 E~
• l It must be noted that depending on the surface finish the emissivity of zirconia (crucible) may be larger than the assumed emissivity value of 0.15. Increasing the emissivity of the outer surface of the crucible modestly reduces the interface deflection.
839
0 7.
i
25/25
i
i
i
50/16.7 Gradient Ratio GH/GC
100/14.2
Fig. 6. The growth interface deflection for the configuration of fig. lb; interface deflection decreases with increasing gradient ratio.
C. Lin, S. Motakef / Modelling of directional solidification of BSO
840
g 0'8 -~ 0.7 I 0.6 i
E
100/100
"1
i
L
100/5 Gradient Ratio GH/GC
i
100/2.5
Fig. 7. The growth interface deflection for the configuration of fig lb; in this configuration, reduction of the cold zone gradient does not result in much smaller interface deflections.
should result in a smaller interface deflection. These considerations suggest investigation of a VB growth configuration capable of promoting distinctly different axial temperature gradients above and below the melting point of BSO. In the first study the isothermal zones are retained and the gradient zone temperature distribution is taken to consist of two axial gradients as shown in fig. lb. Three cases of G H:G c of 25:25 (the same as in fig. la), 50:16.7 and 100:14.2 K / c m are studied (fig. 6); G H and G c are axial gradients on the hot and cold sides of the gradient zones, respectively. In these calculations the total gradient zone length is held constant at 4 cm. Results indicate a nearly 10% reduction in the interface deflection. The limited effectiveness of this approach is related to the short length of gradient zone relative to the charge length. In the second study, the isothermal zones in the furnace are abandoned and the furnace profile is taken to consist of two axial gradients as shown in fig. lc. In this configuration (fig. 7), the growth interface morphology is reduced by nearly 40% for the G H : G c ratios used in the calculations. The G H : G c ratios 100:100, 100:5, and 100:2.5 are relatively large; these values are, however, necessary to balance the large radiative losses at the crystal-melt interface.
4. Conclusion
The present study indicates that VB growth of BSO is associated with relatively weak, and therefore stable, convection in the melt. The growth interface morphology is calculated to be convex
into the melt with interface deflection of the order of 1 cm in a 2 cm diameter crystal. This large interface deflection is shown to be cause by two factors: (a) low thermal conductivity of BSO relative to the PT crucible resulting in a large radial temperature variation and (b) large radiative losses at the crystal-melt interface through the semi transparent crystal to the crucible wails. The former can be controlled by reducing the thickness of the platinum crucible; a composite crucible consisting of a thin layer of Pt deposited onto a low conductivity ceramic is suggested. The latter can be best controlled through modification of the VB configuration to allow for establishment of drastically different thermal gradients along the melt and crystal sides of the charge. In this configuration, the growth rate is expected to vary along the crystal length and not be always equal to the charge lowering rate. Forced convective mixing of the melt was not studied here, yet non-turbulent melt mixing may be a powerful tool in controlling the interface shape deflection. Further investigation of this issue appears warranted.
Note added in proof
The value for the index of refraction used in the present calculations was one. Simulations with a value of 2.54, measured by others in the visible wavelength range, were found to result in increased interface non-planarity by as much as a factor of two and relocation of the interface towards the colder region. The relative effectiveness of various approaches studied in this paper to control the interface shape is not influenced by the change in the value of n.
References [1] M. Garret, Doctoral Dissertation, Department of Electrical Engineering, University of South Carolina (1989). [2] G. Muller, G. Neumann and H. Matz, J. Crystal Growth 84 (1987) 36. [3] A.R. Tanguay, Jr., Doctoral Dissertation, Department of Physics, Yale University (1977). [4] J.R. Carruthers, J. Crystal Growth 32 (1976) 13. 15] S. Brandon and J.J. Derby, J. Crystal Growth 110 (1991) 481.
C. Lin, S. Motakef / Modelling of directional solidification of BSO [6] J.C. Brice, M.J. High and P.A.C. Whiffin, Philips Tech. Rev. 9/10 (1977) 250. [7] CRC Handbook of Chemistry and Physics, 69th ed. (CRC Press, 1988-89). [8] Zircar Products, Inc., Technical Data, Bulletin No. ZPI 205, 10/1/1984. [9] D.A. Gryvnak and D.E. Burch, J. Opt. Soc. Am. 55 (1965) 625.
841
[10] A.G. Ostrogorsky, K.H. Yao and A.F. Witt, J. Crystal Growth 84 (1987) 460. [11] B. Gebhart, Heat Transfer (McGraw-Hill, New York, 1971). [12] K.W. Kelly, K. Koai and S. Motakef, J. Crystal Growth 113 (1991) 254.
,
........ C R Y S T A L GROWTH
Modelling of directional solidification of BSO C h e n t i n g Lin
a
and Shahryar Motakef b
a Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 9 Heat Transfer Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
A thermo-fluid model for vertical Bridgman growth of bismuth silicon oxide (BSO) as model material for semi-transparent, low thermal conductivity oxides is developed. Internal radiative heat transfer, together with convective and conductive heat transfer are considered in this model. Due to the strong internal thermal radiation within the grown crystal, the growth interface is highly convex into the melt, instead of being concave as is the case for opaque materials with the thermal conductivity of the melt larger than that of the solid. Reduction of the growth interface non-planarity through variations in the growth configuration is investigated. A furnace temperature profile consisting of a steep gradient on the melt side and shallow gradient on the solid side of the charge is found to be the most effective approach.
I. Introduction
Bismuth silicon oxide (BSO) is a wide bandgap, high resistivity, optically active semi-insulating material that is photoconductive, and electro-, acousto-, and magneto-optic. This material has wide applications in optical information processing and computing components such as spatial light modulators and volume holographic optical elements, and has also found applications as filters where tailored passbands and phase responses are required [1]. Bi12SiO2o is normally grown by the Czochralski technique. The reported material deficiencies in BSO are, generally, present in all oxides and are related to innate characteristics of the Czochralski growth process. In Cz growth systems, heat is supplied radially to the crucible containing the charge, and removed axially at the solidification front. This sets up destablizing buoyancy forces and an associated unsteady three-dimensional (turbulent) convective structure in the melt [2]. Another inherent shortcoming of the Cz process is the non-axisymmetric heat input and extraction from the crystal, resulting in periodic fluctuation of the growth interface morphology and an associated generation of rota-
tional impurity and point defect striae [1,3,4]. Furthermore, the large melt velocities enhance crucible dissolution, introducing impurity into the matrix. As optical devices generally require property uniformity in volumes with length scales of the order of centimeters, elimination of growth related compositional fluctuations is projected to result in advances in optical device technique. In the present work we numerically study the feasibility of growing BSO by the vertical Bridgman (VB) technique. In this configuration convection in the melt is expected to be much weaker than in the Cz system, and axisymmetric heat input into and extraction from the charge can be more readily achieved. It is, therefore, reasonable to suggest that crystals grown by the VB technique will, as a consequence of a stable and axisymmetric solid liquid interface, exhibit a lower level of micro- and macro-scale compositional variations. We begin with a description of the thermal transport model of the process, including the two-band approach to the modelling of the internal radiative heat transfer in the solid, and briefly describe the numerical method used in the calculations. We will show that semi-transparency of the solid oxide results in a growth interface morphology which is convex into the melt. The effec-
0022-0248/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
C. Lin, S. Motakef / Modelling of directional solidification of BSO
tiveness of various approaches for the reduction of the solid-liquid interface shape curvature are, then, explored. Unlike for most of the oxides, the thermophysical data base for BSO is relatively complete, and thus the present numerical study provides an accurate framework for the design and analysis growth of BSO by the VB technique. A similar study neglecting convection in the melt and the existence of the crucible has been conducted by Brandon and Derby and reported in ref. [5].
2. Model description
2.1. Growth configuration Due to its corrosive nature, BSO melt can be only contained by noble metals such as Pt and Au. However, as the melting point temperatures of Au and BSO are close, Pt has been exclusively used as the crucible material; the thermophysical properties of BSO and Pt are given in table 1. In this study we consider two types of crucibles: (a) standard crucibles made from Pt with the wall thickness varying between 0.5 and 2.0 mm and (b) a composite crucible consisting of a 0.1 mm Pt layer deposited onto a 1.9 mm thick zirconia crucible. In the latter configuration, relatively low axial conduction through the crucible is achieved while the stability of the crucible is preserved; the low axial conduction is later shown to be necessary to reduce the curvature of the solid-liquid interface shape. The present study focuses on a 2 cm diameter and 17 cm long charge. Three growth configurations are studied. In the first configuration, the furnace is modelled to consist of two isothermal zones separated by a gradient region; the temperatures of the isothermal zones are chosen such that the melting point temperature of BSO is at the center of the gradient region (fig. la). The isothermal portions of the furnace are modelled to be "infinitely" long, and the gradient zone length is varied in the analysis. In the second configuration, the gradient zone is split into two regions where, using active heat transfer elements, independently controlled temperature gradients on the solid and liquid sides of the
835
Table 1 Thermophysical properties of materials used in these studies Quantity
Temperature
Value
(K) BSO [6] Melting point Density Solid Liquid Linear expansion coefficient Volume expansion coefficient Specific h e a t / u n i t volume Solid Liquid Thermal conductivity Solid
Liquid Viscosity Young's modulus Breaking strain Latent heat
1168
1168 K
293 1223
9.206 g cm -3 7.63 g c m - 3 16×10 6 K-1
1223
7x10-5 K 1
2.7Jcm 3K 1 3.0 J cm -3 K - 1
973 1073 1168 1168 1173
0.1944Wm -1K i 0.1782 W m - 1 K - I 0.162 W m - i K - 1 0.27 W m - l K - 1 0.22 dyn cm - 2 s 93Gnm z 2 × 10 -4 460 J c m - 3
Pt [71 Thermal conductivity Surface emissivity
900-1300
73.7-79.1 W m a K I
1273
0.15
1073-1923
0.11-0.23 W m -1 K -1
Zirconia, type Z Y C [8] Thermal conductivity
charge are established (fig. lb). In the third configuration, the isothermal zones are abandoned and the furnace is configured to have two nonequal gradients over the entire solid and liquid lengths of the charge (fig. lc).
2.2. Mathematical model The temperature and velocity fields in the melt are obtained through solution of the coupled
C Lin, S. Motakef / Modelling of directional solidification of BSO
836
I It1 Coils in
the melt
)l
.
I
.
Heatpipe
Zone
,n.. in the crystal
i Fig. 1. Schematic diagrams of growth configurations and furnace temperature profiles studied: (a) typical vertical Bridgman furnace, (b) modified Bridgman furnace, with two independently controlled gradients in the gradient zone, (c) modified Bridgman furnace, with two independently controlled temperature gradients along the charge.
axisymmetric momentum and energy conservation equations:
V" 17V=
1
--17P+vV2V+gflT(es'e~), P
-
(1)
1
V" VT= - - I 7 . ( k17T), pc
(2)
where p is density, k thermal conductivity, v kinematic viscosity, c heat capacity,/3 volumetric expansion coefficient, and eu and e~ unit vectors along gravitational and crucible axial directions. In the above, the melt is assumed to be opaque to internal radiation; this is consistent with the results of Gryvnak and Burch [9] who have shown that oxide melts are opaque at near infrared wavelengths. Eq. (1) is subject to no-slip and no-penetration boundary conditions on the velocity field at liquid-solid boundaries. The last term of eq. (1) is the Boussinesq approximation for the buoyancy forces due to density variations. In our study, the release of latent heat of solidification is negligible compared to the other losses at the growth interface and, thus, is neglected. The temperature field in the crystal is influenced by the semitransparency of the BSO solid to infrared radio. Room temperature optical transmission spectra of two BSO samples, 1.5 and 3 mm thick, are shown in fig. 2. The transmittance of the two samples are approximately the
same at wavenumber larger than 1667 cm -1 (wavelengths smaller than about 6 Izm), indicating the loss in transmission at A < 6/zm is due to surface reflection and that negligibly small absorption occurs in this wavelength range. The samples show intermediate absorption at wavelengths between 1597 and 1266 cm-1; in this range blackbody emissive power at the melting temperature of BSO (1168 K) constitutes only 7% of full spectrum black body emissive power. Therefore, this wavelength range is taken to be
80.0
.......S 60,0
o
40.0
20.0 ......... - -
O0 10000
I 20000
1.5 m m somple 3 m m s•mple
I 30000
Wavenumbers ( 1 / cm )
Fig. 2. Infrared transmission spectrum of BSO for two sample thicknesses of 1.5 and 3 mm at room temperature.
C. Lin, S. Motakef /Modelling of directional solidification of BSO
totally opaque. At wavelengths below this range thermal radiation is completely absorbed. The above indicates that a two-band model (splitting the spectral radiation into an opaque and a transparent region separated at the band-edge wavelength of 6 /zm) can be used to model internal radiative heat transfer in the growing crystal. The implicit assumption in this approach is that the room temperature data can be extrapolated to elevated temperatures close to the melting point temperature of BSO. This assumption has been shown to be valid for BzO 3 [10]. In the two-band model, only the fraction of radiant energy (emitted or reflected) which falls within the spectral transparency band of the material is tracked. This fraction can be written as:
l[j:
f - F o _ ~ - o-T 4
]
AS(e C2/aT- 1) dA .
(3)
Assuming the crucible walls are diffuse and gray, heat exchange between inner crucible walls through the crystal can be calculated using the Gebhart formulation. In this approach, the net total heat transfer rate at a surface element with area A k is given by N
Qk = A k e k f k ~rT4 -- ~ Aiejfj°'Tj4Gjk, j=l
(4)
where the f ' s can be calculated from eq. (3), and Gik is the Gebhart factor defined as the fraction of emitted energy from surface element j being absorbed by surface element k [11]. The calculated values of the Qk's are then applied to the inner surface of the crucible in contact with the crystal and the crystal-melt interface. Energy transfer within the crystal and the crucible is by conduction: V( kV7~) = 0.
(5)
The outer surface of the crucible is radiatively coupled with the furnace (separation distance between furnace and crucible is taken to be small). Heat transfer at the outer crucible surface is modelled to obey the following relation which assumes the internal furnace surface to be black:
( q / A ) = o'•(T 4 - T?).
(6)
837
In the above, • is the surface emissivity of the crucible, and is taken to be 0.15 for both inside and outside surfaces of the crucible in our simulations. The emissivity of the crystal-melt interface is taken to be 1 and the index of the crystal is also taken to be 1. The present approach to modelling of internal radiation in BSO is different from that of Brandon and Derby [5], who modelled the internal radiation to obey Beer's law with a constant absorption coefficient. In our work the two-band model exploits the band-edge structure of the spectral transmittance of BSO to simplify the radiative exchange equations within the semitransparent solid; this approach applied to a different growth configuration has yield relatively good agreement with the experimental results [121.
2.3. Simulation strategy The solution of the governing equations was achieved by an iterative technique employing two commercially available codes ABAQUS and FIDAP. The iteration technique took advantage of the available flexibility of the two packages and configured them as to provide a time-efficient solution to the governing equations: (i) ABAQUS was used to solve the conduction equation in the crystal, melt, and crucible subject to the internal radiative exchanges in the solid, radiative coupling between the crucible and the furnace, and a solid-liquid interface shape across which the thermal conductivity of the charge experiences a step change. The calculated melting point isotherm is inputted into the next step as the location of the growth interface. (ii) FIDAP is used to solve the momentum and energy equations in the melt. The solid-melt interface shape and melt-crucible interface temperature profiles calculated in step (i) are used as input. The difference between the calculated heat fluxes as the melt-crucible and melt-crystal interface and the conduction solutions of step (i) are fed back to step (i) calculations in form of surface heat fluxes. The grid for the ABAQUS simulations consisted of 10 elements in the radial direction of 99
C. Lin, S. Motakef / Modelfing of directional solidification of BSO
838
elements in the axial direction. The grid for FIDAP simulations consisted of 13 and 63 elements in the radial and axial directions, respectively. Convergent solutions were obtained typically after 15 iterations between steps (i) and (ii) with each iteration taking about 1 CPU hour on a SUN 4-280.
31.8 31.6 31.4 g -~
31
30.8
3. Results
30.6
In absence of rigorous relations between the growth conditions and device sensitive properties of BSO, we use the intensity of convection in the melt, and the melt-crystal interface deflection as the primary figures of merit in identifying optimal growth conditions. The latter is motivated by two considerations. First, the radial temperature gradients associated with the interface non-planarity induce thermal strains close to the growth interface in the crystal; this may result in plastic deformation and residual strain causing birefringence non-uniformity in the grown material. Second, these radial temperature gradients drive the convective intensity in the melt, and may cause flow field instabilities resulting in microscopic remelting and regrowth and an associated microscale property variations in the crystal.
30.4
The influence of crucible thickness on crystal-melt interface shape for the furnace configuration of fig. la is shown in fig. 3, with interface deflection defined as the difference between 1.1
g v
1
g "~ 0.9
•
•
o J~ Composite oe 0 . 8 Crucible ~' 0.7 _E 0
. 0.5
.
1
.
.
1.5
2
Pt CrucibleThickness(mm) Fig. 3. T h e influence of crucible thickness on the deflection of growth interface. The interface deflection decreases for thinner Pt crucibles.
With Convection
31.2
8
13-
<
3.1. The crucible effect
Convection
•
i
L
i
h
0.2
0.4
0.6
0.8
Radial Position (cm) Fig. 4. The influence of convection on the solid-liquid interface shape; convection reduces interface deflection by 20%.
the axial interface location at the center and at the edge of the charge. A typical morphology of the solid-liquid interface using the composite crucible is shown in fig. 4. In fig. 3, the interface deflection is observed to decrease with decreasing crucible thickness; the smallest deflection is observed for the composite crucible. Several factors contribute to the observed solid-liquid interface shape. At the growth interface heat transfer on the crystal side is primarily by radiation (conductivity of solid BSO is very small) to the low-temperature crucible; on the melt side, the molten BSO is opaque to radiation and, thus, heat transfer in the melt is by convection; the large radiative heat transfer in the crystal side results in large axial gradients in the melt. As the radiative view factor between the growth interface and the low-temperature crucible walls is largest at the center of the growth interface, (i.e. has the largest radiative loss) the solid-liquid interface shape becomes convex into the liquid. The second factor controlling the crystal-melt interface shape is the axial conduction through the crucible (thermal conductivity of Pt is about 350 times higher than that of BSO) which leads to a large variation of temperature between the crystal center and its periphery; this factor is, generally, stronger than the former in controlling the growth interface morphology. Thus, the interface deflection associated with growth using the composite (low thermal conductivity) crucible is appreciably reduced.
C. Lin, S. Motakef /Modelling of directional solidification of BSO
Nevertheless, reduction of Pt thickness to less than 0.1 mm is not realistically possible, thus, results of fig. 3 establish the upper limit to the effectiveness of controlling solid-liquid interface shape by varying the thermophysical properties of the crucible #1. In the remainder of the study the composite crucible is used.
3.2. Conuectiue effects in the melt The influence of convection on the solid-liquid interface shape for the furnace profile of fig. la is shown in fig. 4; convective mixing in the melt is found to promote a 20% reduction in the interface deflection. The calculated velocity field in the melt is very weak with a maximum value of 1.36 × 10 -4 m / s (Reynolds number Vr/u = 0.47); the maximum value of V generally occurs close to the centerline of the charge. Low values of Reynolds number were observed for all cases studied here and suggest that flow in the melt is laminar, and flow instabilities should not be expected. The Peclet number (= Vr/a scaling convective to conductive heat transfer) is about 15 for results of fig. 4 indicating presence of convective heat transfer in the melt. The recirculating cell (shown schematically in fig. 1) promotes heat transfer between the crucible and the growth interface resulting in a reduced interface deflection. To first order, however, convection in the melt may be neglected and heat transfer in the system be approximated as conduction-dominated in the melt and radiation-dominated in the crystal. In spite of the limited influence, convection has been included in all simulations reported in this study.
3.3. Influence of gradient zone length Growth interface deflections associated with various lengths of the gradient zone in the furnace configuration of fig. la are shown in fig. 5.
1.1
'"
0.9
"= "~
08
• " 2
0
' ' ' 4 6 8 Gradient Zone Length (cm)
' 10
Fig. 5. The influence of gradient zone length on the deflection of growth interface; optimal gradient zone length is about 4 cm.
In these calculations, the temperature gradient is held at 25 K/cm. Results indicate that increasing the gradient zone length from 0.5 to 4 cm produces a 10% reduction in interface deflection. Beyond 4 cm the gradient zone length has a minimal effect on interface deflection. Overall, the gradient zone length does not have an appreciable influence.
3.4. Two temperature gradient furnaces Investigation of the furnace configurations shown in figs. lb and lc is motivated by the following consideration: The crystal-melt interface deflection is controlled by the radiative loss at this interface through the crystal to the crucible with the driving force proportional to the difference between the fourth power of the temperatures. Thus, by reducing the furnace axial gradient along the portion of the crucible containing the crystal, radiative heat loss at the interface center should be reduced. Also, with a large axial gradient on the melt side, radial non-uniformity in radiative loss at the growth interface
E
0.9
~ 0.8 E~
• l It must be noted that depending on the surface finish the emissivity of zirconia (crucible) may be larger than the assumed emissivity value of 0.15. Increasing the emissivity of the outer surface of the crucible modestly reduces the interface deflection.
839
0 7.
i
25/25
i
i
i
50/16.7 Gradient Ratio GH/GC
100/14.2
Fig. 6. The growth interface deflection for the configuration of fig. lb; interface deflection decreases with increasing gradient ratio.
C. Lin, S. Motakef / Modelling of directional solidification of BSO
840
g 0'8 -~ 0.7 I 0.6 i
E
100/100
"1
i
L
100/5 Gradient Ratio GH/GC
i
100/2.5
Fig. 7. The growth interface deflection for the configuration of fig lb; in this configuration, reduction of the cold zone gradient does not result in much smaller interface deflections.
should result in a smaller interface deflection. These considerations suggest investigation of a VB growth configuration capable of promoting distinctly different axial temperature gradients above and below the melting point of BSO. In the first study the isothermal zones are retained and the gradient zone temperature distribution is taken to consist of two axial gradients as shown in fig. lb. Three cases of G H:G c of 25:25 (the same as in fig. la), 50:16.7 and 100:14.2 K / c m are studied (fig. 6); G H and G c are axial gradients on the hot and cold sides of the gradient zones, respectively. In these calculations the total gradient zone length is held constant at 4 cm. Results indicate a nearly 10% reduction in the interface deflection. The limited effectiveness of this approach is related to the short length of gradient zone relative to the charge length. In the second study, the isothermal zones in the furnace are abandoned and the furnace profile is taken to consist of two axial gradients as shown in fig. lc. In this configuration (fig. 7), the growth interface morphology is reduced by nearly 40% for the G H : G c ratios used in the calculations. The G H : G c ratios 100:100, 100:5, and 100:2.5 are relatively large; these values are, however, necessary to balance the large radiative losses at the crystal-melt interface.
4. Conclusion
The present study indicates that VB growth of BSO is associated with relatively weak, and therefore stable, convection in the melt. The growth interface morphology is calculated to be convex
into the melt with interface deflection of the order of 1 cm in a 2 cm diameter crystal. This large interface deflection is shown to be cause by two factors: (a) low thermal conductivity of BSO relative to the PT crucible resulting in a large radial temperature variation and (b) large radiative losses at the crystal-melt interface through the semi transparent crystal to the crucible wails. The former can be controlled by reducing the thickness of the platinum crucible; a composite crucible consisting of a thin layer of Pt deposited onto a low conductivity ceramic is suggested. The latter can be best controlled through modification of the VB configuration to allow for establishment of drastically different thermal gradients along the melt and crystal sides of the charge. In this configuration, the growth rate is expected to vary along the crystal length and not be always equal to the charge lowering rate. Forced convective mixing of the melt was not studied here, yet non-turbulent melt mixing may be a powerful tool in controlling the interface shape deflection. Further investigation of this issue appears warranted.
Note added in proof
The value for the index of refraction used in the present calculations was one. Simulations with a value of 2.54, measured by others in the visible wavelength range, were found to result in increased interface non-planarity by as much as a factor of two and relocation of the interface towards the colder region. The relative effectiveness of various approaches studied in this paper to control the interface shape is not influenced by the change in the value of n.
References [1] M. Garret, Doctoral Dissertation, Department of Electrical Engineering, University of South Carolina (1989). [2] G. Muller, G. Neumann and H. Matz, J. Crystal Growth 84 (1987) 36. [3] A.R. Tanguay, Jr., Doctoral Dissertation, Department of Physics, Yale University (1977). [4] J.R. Carruthers, J. Crystal Growth 32 (1976) 13. 15] S. Brandon and J.J. Derby, J. Crystal Growth 110 (1991) 481.
C. Lin, S. Motakef / Modelling of directional solidification of BSO [6] J.C. Brice, M.J. High and P.A.C. Whiffin, Philips Tech. Rev. 9/10 (1977) 250. [7] CRC Handbook of Chemistry and Physics, 69th ed. (CRC Press, 1988-89). [8] Zircar Products, Inc., Technical Data, Bulletin No. ZPI 205, 10/1/1984. [9] D.A. Gryvnak and D.E. Burch, J. Opt. Soc. Am. 55 (1965) 625.
841
[10] A.G. Ostrogorsky, K.H. Yao and A.F. Witt, J. Crystal Growth 84 (1987) 460. [11] B. Gebhart, Heat Transfer (McGraw-Hill, New York, 1971). [12] K.W. Kelly, K. Koai and S. Motakef, J. Crystal Growth 113 (1991) 254.