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Global temperature field simulation of the vapour pressure controlled Czochralski (VCZ) growth of 3″–4″ gallium arsenide crystals
Journal of Crystal Growth 198/199 (1999) 349—354
Global temperature field simulation of the vapour pressure controlled Czochralski (VCZ) growth of 3 —4 gallium arsenide crystals K. Bo¨ttcher *, P. Rudolph , M. Neubert , M. Kurz, A. Pusztai, G. Mu¨ller Institut fu( r Kristallzu( chtung im Forschungsverbund Berlin e.V., Rudower Chaussee 6, Geb. 19.31, D-12489 Berlin, Germany Institut fu( r Werkstoffwissenschaften VI der Friedrich-Alexander—Universita( t Nu( rnberg-Erlangen, Martensstr. 7, D-91058 Erlangen, Germany
Abstract The vapour pressure controlled Czochralski (VCZ) method belongs to the new methods to provide low-gradient temperature fields during the growth of III—V crystals. For the first time a global two-dimensional model of the VCZ growth of 3 and 4 GaAs crystals is presented. The finite volume code CrysVUN## was used to simulate heat transfer taking into account conduction and radiation in the whole equipment. Thermoelastic stresses are analysed in terms of the von-Mises stress. There is a good agreement between measured and calculated values, e.g., of the convexity of the crystal-melt interface. 1999 Elsevier Science B.V. All rights reserved. PACS: 81.10.Fq; 07.05.Tp; 44.40.#a Keywords: Vapour pressure controlled Czochralski (VCZ) method; Global simulation; Finite volume method; Heat transfer; Thermal stress field; GaAs
1. Introduction There is a rapidly increasing need of high-frequency GaAs devices for the microelectronics. Thus, there is a growing demand of semi-insulating GaAs crystals with diameters in the range from 4 to 6 keeping up the relatively low dislocation density of 3 crystals. In order to achieve this goal
* Corresponding author. Tel.: #49 30 6392 3073; fax: #49 30 6392 3003; e-mail: [email protected].
low-temperature gradient growth methods were developed [1]. The vapour pressure controlled Czochralski (VCZ) method [1,2] is a newer and promising one. In contrast to conventional LEC, the crystal is grown within an additional gas-tight inner chamber that shields the growing crystal and the hot gas from the water-cooled walls of the outer high-pressure resistant vessel. As a result of the low temperature gradient the crystal leaves the boron oxide at a very high temperature. Therefore, to keep the crystal surface in equilibrium with its surrounding gas phase, a certain arsenic partial
0022-0248/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 8 ) 0 0 9 9 6 - 8
350
K. Bo( ttcher et al. / Journal of Crystal Growth 198/199 (1999) 349–354
pressure has to be maintained inside the VCZ chamber by using a temperature-controlled arsenic source. Some recently obtained results are presented in Ref. [3]. However, such a growth arrangement makes the construction very complicate and expensive and requires, therefore, a preceding tailoring by computation. Our heat transfer analysis is aimed at finding out the effect of different heat insulation dispositions on the temperature field, the curvature of the liquid—solid interface and the resulting thermal stresses within the crystal.
2. Model for the simulation of the VCZ growth process In order to catch every important detail within the growth chamber CAD drawings of two real VCZ arrangements, realized in two different furnaces (hereafter called F1 and F2), are used as input data for the preprocessor of the computer code. This type of drawing is a flexible tool for adjusting the correct positions of all moveable parts. To adapt the geometry to a discretized numerical method, a grid generator covers the solution domain with a triangle grid. Room temperature is imposed at all outer surfaces of the vessel as a boundary condition. The control of the heater power ensures a constant crystal radius. An additional heater adjusts the operating temperature of the arsenic source in case of furnace F1. For the global temperature field simulation we have used the finite volume code CrysVUN## [4] that currently comprises heat transfer by conduction and by wall-to-wall radiation within any ensemble of surfaces and the release of the latent crystallization heat. Via solving an inverse problem, the power input of every heater is calculated. The temperature field is assumed to be axially symmetric (two-dimensional) and in a quasi-steady state at any time. The shape of the solid—liquid interface is equiated with the isoline of the melting point temperature. At the present stage, no kind of convection or convective heat transfer has been considered yet, neither in the melt or encapsulant nor in the inert gas. There are not any rotations of crucible and feed through mechanisms included.
Table 1 Material properties of GaAs, boron oxide and argon GaAs Heat conductivity (s) Heat conductivity (l) Emissivity Latent heat Elastic stiffness C C C Thermal expansion Boron oxide Heat conductivity Emissivity Argon Heat conductivity
7.12 W/m K 17.8 W/m K 0.5 668.5 J/g 1.007;10 Pa 4.58;10Pa 5.14;10Pa 1.03;10\ K\ 0.237#0.0011 T W/m K 0.75 0.0018 #5.8854;10\ T !2.2487;10\ T #4.9213;10\ T W/m K
The temperature field within the crystal is used to calculate the thermo-elastic stress distribution in terms of the von-Mises stress p " + (((p !p )#(p !p )#(p !p )#6p ), XX (( PP (( XX PX PP where p , (i"r, z, ) are the normal stresses and GG p is the real shear stress. From that definition the PX von-Mises stress is scalar positive and it does not contain any information whether the tensile or compressive stress contributes to the von-Mises stress. The material properties of GaAs, boron oxide and argon are summarized in Table 1. The data for heat conductivity and emissivity of all other technical parts consisting of boron nitride, graphite, quartz, steel, felt, ceramics, etc., are used according to the data delivered by the supplying companies. None of these data was varied for better fitting the experimental results. 3. Results and discussion 3.1. Comparison between calculated and measured data First of all the reliability of the model was evaluated by comparing the results of modelling with
K. Bottcher et al. / Journal of Crystal Growth 198/199 (1999) 349–354 Table 2 Comparison for items inside the chamber
Temp. gradient at Axis Surface
Simulation 8.7 mm
Experiment 8—12 mm
21—33 K/cm 23—47 K/cm
+40 K/cm
Table 3 Comparison for items outside the chamber As heater power
Temperature at Crucible bottom Main heater
Simulation 239 W
Experiment 312 W
1215°C 1242°C
1174°C 1200°C
351
the chamber. Concerning the convective heat transfer the situation can be estimated by characteristic values of the gas phase Grashof number (+1.5;10) and both the Newtonian and linearized radiation heat transfer coefficient being identically about 0.03 W/cm K. Obviously, the reason for such a good agreement arises from the focussing of the heater control on the maintenance of the melting point temperature at the threephase-boundary compensating the effect of neglecting the heat losses from crystal and encapsulant to the gas by calculating a relativly low necessary heat input. Indeed, the simulated heat input is essentially lower than the experimental one. 3.2. 3 and 4 crystal growth
experimental values of the furnace F1. Tables 2 and 3 comprise data according to a growing 3 crystal just having a length of 30 mm. We have found that the calculated convexity of the crystal—melt interface agrees satisfactorily with the curvature of experimentally ascertained growth striations. The simulated temperature gradients match also roughly the experimental values measured by using a graphite dummy. The desired temperature of the arsenic source, however, needs in the model a lower power consumption. In fact, it has to be assumed that in reality a part of the heater power is dissipated due to the gas convection and somewhat more energy input is required. The comparison of the temperature values outside of the inner chamber reveals evident differences. This can be attributed to the direct contact of this region to the main heater and the enhanced gas streaming through the chinks of the insulating insertions. The gas convection distributes heat and, thus, decreases the temperature peak in the vicinity of the heater. As a result, lower experimental values than calculated ones will be observed. Nevertheless, there is a good accordance of the simulated and the measured difference between the temperatures at the crucible bottom and the main heater. One can say that there is a good agreement between simulation and experiments for those items measured inside
Both F1 and F2 are supplied with inner chamber constructions but considerably different heat shield arrangements. While furnace F1 is suitable to grow 3 crystals only furnace F2 is designed for 4 crystals, too. Via numerical simulation of conductive and radiative heat transfer, the suitability of the crystal growth arrangements were analyzed, whereby the efforts were concentrated on (i) a reduction of the temperature gradients along the growth axis and crystal surface, (ii) a slightly convex curvature of the crystal—melt interface, and (iii) minimizing the thermal stress resulting from the nonlinearities of the temperature distribution within the crystal. 3.2.1. Furnace F1 The thermal situation generated by furnace F1 is demonstrated in the case of a growing crystal of 80 mm in diameter having an already crystallized cylindrical part of 50 mm in length. Due to the crucible diameter of 125 mm and a charge of 300 g boron oxide the thickness of the encapsulant layer is 28 mm. The total GaAs charge is 3 kg and the growth velocity is about 5 mm/h. The temperature distribution in the crucible region and the adjacent parts, accompanied by the thermal stress distribution in terms of the von-Mises stress, is presented in Fig. 1. For a better illustration of the isotherm morphology this region is enlarged while further construction details are omitted. As Fig. 1 shows, the temperature decrease towards the crystal top is less than in conventional LEC crystals [1]. In the
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K. Bo( ttcher et al. / Journal of Crystal Growth 198/199 (1999) 349–354
Fig. 2. Radial distributions of the von-Mises thermal stress, at the z-coordinate of the intersection line of crystal, melt and boron oxide. All crystals are about 50 mm in length. The labels F1 and F2 refer to the type of crystal growth furnace.
Fig. 1. Growth situation of an 80 mm crystal in the furnace F1. Left: thermal stress in terms the von-Mises stress, gap between isolines: 0.5 MPa. Right: cut-out of the global temperature field around the crucible; convexity of the interface: 9.6 mm; gap between isolines: 10 K.
VCZ case the crystal has a temperature of about 1145°C when it emerges from the boron oxide. The temperature gradients derived from the axial profile at r"0 are permanently below 30 K/cm. The gradients at the surface vary between 30 and 40 K/cm for the submerged and between 20 and 30 K/cm for the emerged part. The latter is influenced by the radiation from the free crucible wall, while the boron oxide is considered to be non-transparent [5]. These data identify furnace F1 as a low gradient arrangement. However, in contrast to the temperature field in the central part of the crystal, the field in the outer part is strongly nonlinear. Both the temperature field near and at the surface and the relatively large curvature of the crystal—melt interface cause still too large thermal stresses (Fig. 1), though the tendency toward decreasing values is obvious in comparison with conventional LEC growth [6]. The local stress peaks are (i) at the intersection line of crystal, melt and boron oxide, (ii) where the crystal
Fig. 3. Radial distributions of the etch pit density of various samples. The labels F1 and F2 refer to the type of crystal growth furnace.
leaves the boron oxide and (iii) where the conus transforms to the cylinder. The radial profile of the von-Mises stress at the z-coordinate of location (i) is drawn in Fig. 2. The graph shows a slightly developed W-shape. In the part of the crystal near the interface, exhibiting a temperature more than 1200°C, the stress values are larger than the critical resolved shear stress of about 0.7 MPa [7]. The shape of the von-Mises stress profile can roughly be compared with the etch pit density (epd) graph measured along cut crystal slices. Fig. 3 shows the epd values along two different traces. Similarly to the von-Mises graph
K. Bottcher et al. / Journal of Crystal Growth 198/199 (1999) 349–354
353
the epd values increase markedly towards the periphery of the crystal. 3.2.2. Furnace F2 Furnace F2 is equipped with an improved arrangement of heat insulations. The resulting heat input is roughly 1/3 of that of furnace F1. Thus, the heat insulation of F2 can be clearly estimated as more effective. As a result, the heat transfer analysis of an 80 mm crystal (same material charge but 25 mm thickness of the boron oxide layer) yields improved thermo-mechanical parameters. The bending of the interface is reduced to about 2.9 mm, the maximum von-Mises stress is about 4.9 MPa, the axial temperature gradient along the centerline is permanently below 25 K/cm. All these values are markedly below those of conventional LEC arrangements (see Refs. [1,6], for example). Due to the small bending of the interface, the radial profile of the von-Mises stress at the z-coordinate of location (i) is degenerated to a V-shape (Fig. 2). In Fig. 4, the stress and temperature fields of an 80 mm-long 4 crystal are sketched. Because the same amount of boron oxide was used as in the 3 case the 4 crystal is up to 3/4 encapsulated. The interface bending (6.7 mm) lies nearly between that of both other 3 crystals. But, as for the maximum thermal von-Mises stress and the temperature gradient along the growth axis, the 4 crystal shows nearly the same behaviour as the 3 F2 one. The values are 5 MPa and below 25 K/cm, respectively. Compared with the 3 crystal of furnace F1 the radial distribution of the von-Mises stress shows even a more slightly developed W-shape. Besides this, the von-Mises stress profile of the 4 F2 crystal (Fig. 2) are roughly equal or lower than that of the 3 F1 crystal. The corresponding epd values behave similar (Fig. 3) Note, for better meeting the crystal geometry the transformation from shoulder conus to cylinder is scetched more softly. However, as the comparison between Fig. 1 and Fig. 4 shows the soft conus edge does not prevent the rise of a local stress peaks. In both the cases (soft and sharp edge) the temperature field at the edge is distorted due to the rapid change of the temperature when going from the top of the crystal into the gas atmosphere above of the crystal.
Fig. 4. Growth situation of an 105 mm crystal in the furnace F2. Left: thermal stress in terms the von-Mises stress, gap between isolines: 0.5 MPa. Right: cut-out of the global temperature field around the crucible; convexity of the interface: 6.7 mm; gap between isolines: 10 K.
4. Conclusions The tailoring of the temperature field design within the whole crystal growth furnace is an essential task to provide favourable low temperature gradient conditions. Using the global heat transfer code CrysVUN##, two VCZ furnaces with different heat insulations were analyzed and compared. Though the analysis does not cover convective phenomena, there is a good agreement between calculated and measured values, e.g., of the bending of the crystal-melt interface. The analysis reveals that the furnace F2, with an improved heat insulation around the inner VZC chamber, leads to a more advantageous temperature field resulting in a smaller bending of the solid—liquid interface and lower values of the thermal stresses. Although the VCZ developments are not completed yet improved temperature and stress fields, leading to markedly reduced epd values, compared with conventional 3 and 4 GaAs LEC crystals, have been obtained.
Acknowledgements This work was supported by the German Ministry for Education, Science, Research and Technique under contract No. 01 BM 501/0.
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K. Bo( ttcher et al. / Journal of Crystal Growth 198/199 (1999) 349–354
References [1] P. Rudolph, M. Neubert, S. Arulkumaran, M. Seifert, Crys. Res. Technol. 32 (1997) 35. [2] M. Tatsumi, T. Kawase, Y. Iguchi, K. Fujita, M. Yamada, in: M. Godlewski (Ed.), Semi-insulating III—V Materials, World Scientific, Singapore, 1994, p. 11. [3] M. Neubert, P. Rudolph, Seifert, in: M. Melloch, M.A. Reed (Eds.), 1997 IEEE Int. Symp. on Compound Semiconductors, Inst. Phys. Publ., Bristol, 1998, p. 53.
[4] M. Kurz, A. Pusztai, G. Mu¨ller, J. Crystal Growth 198/199 (1999) 101. [5] A.G. Ostrogorsky, K.H. Yao, A.F. Witt, J. Crystal Growth 84 (1987) 460. [6] A.S. Jordan, A.R. von Neida, R. Caruso, J. Crystal Growth 76 (1986) 243. [7] A.S. Jordan, J. Crystal Growth 49 (1980) 631.
Global temperature field simulation of the vapour pressure controlled Czochralski (VCZ) growth of 3 —4 gallium arsenide crystals K. Bo¨ttcher *, P. Rudolph , M. Neubert , M. Kurz, A. Pusztai, G. Mu¨ller Institut fu( r Kristallzu( chtung im Forschungsverbund Berlin e.V., Rudower Chaussee 6, Geb. 19.31, D-12489 Berlin, Germany Institut fu( r Werkstoffwissenschaften VI der Friedrich-Alexander—Universita( t Nu( rnberg-Erlangen, Martensstr. 7, D-91058 Erlangen, Germany
Abstract The vapour pressure controlled Czochralski (VCZ) method belongs to the new methods to provide low-gradient temperature fields during the growth of III—V crystals. For the first time a global two-dimensional model of the VCZ growth of 3 and 4 GaAs crystals is presented. The finite volume code CrysVUN## was used to simulate heat transfer taking into account conduction and radiation in the whole equipment. Thermoelastic stresses are analysed in terms of the von-Mises stress. There is a good agreement between measured and calculated values, e.g., of the convexity of the crystal-melt interface. 1999 Elsevier Science B.V. All rights reserved. PACS: 81.10.Fq; 07.05.Tp; 44.40.#a Keywords: Vapour pressure controlled Czochralski (VCZ) method; Global simulation; Finite volume method; Heat transfer; Thermal stress field; GaAs
1. Introduction There is a rapidly increasing need of high-frequency GaAs devices for the microelectronics. Thus, there is a growing demand of semi-insulating GaAs crystals with diameters in the range from 4 to 6 keeping up the relatively low dislocation density of 3 crystals. In order to achieve this goal
* Corresponding author. Tel.: #49 30 6392 3073; fax: #49 30 6392 3003; e-mail: [email protected].
low-temperature gradient growth methods were developed [1]. The vapour pressure controlled Czochralski (VCZ) method [1,2] is a newer and promising one. In contrast to conventional LEC, the crystal is grown within an additional gas-tight inner chamber that shields the growing crystal and the hot gas from the water-cooled walls of the outer high-pressure resistant vessel. As a result of the low temperature gradient the crystal leaves the boron oxide at a very high temperature. Therefore, to keep the crystal surface in equilibrium with its surrounding gas phase, a certain arsenic partial
0022-0248/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 8 ) 0 0 9 9 6 - 8
350
K. Bo( ttcher et al. / Journal of Crystal Growth 198/199 (1999) 349–354
pressure has to be maintained inside the VCZ chamber by using a temperature-controlled arsenic source. Some recently obtained results are presented in Ref. [3]. However, such a growth arrangement makes the construction very complicate and expensive and requires, therefore, a preceding tailoring by computation. Our heat transfer analysis is aimed at finding out the effect of different heat insulation dispositions on the temperature field, the curvature of the liquid—solid interface and the resulting thermal stresses within the crystal.
2. Model for the simulation of the VCZ growth process In order to catch every important detail within the growth chamber CAD drawings of two real VCZ arrangements, realized in two different furnaces (hereafter called F1 and F2), are used as input data for the preprocessor of the computer code. This type of drawing is a flexible tool for adjusting the correct positions of all moveable parts. To adapt the geometry to a discretized numerical method, a grid generator covers the solution domain with a triangle grid. Room temperature is imposed at all outer surfaces of the vessel as a boundary condition. The control of the heater power ensures a constant crystal radius. An additional heater adjusts the operating temperature of the arsenic source in case of furnace F1. For the global temperature field simulation we have used the finite volume code CrysVUN## [4] that currently comprises heat transfer by conduction and by wall-to-wall radiation within any ensemble of surfaces and the release of the latent crystallization heat. Via solving an inverse problem, the power input of every heater is calculated. The temperature field is assumed to be axially symmetric (two-dimensional) and in a quasi-steady state at any time. The shape of the solid—liquid interface is equiated with the isoline of the melting point temperature. At the present stage, no kind of convection or convective heat transfer has been considered yet, neither in the melt or encapsulant nor in the inert gas. There are not any rotations of crucible and feed through mechanisms included.
Table 1 Material properties of GaAs, boron oxide and argon GaAs Heat conductivity (s) Heat conductivity (l) Emissivity Latent heat Elastic stiffness C C C Thermal expansion Boron oxide Heat conductivity Emissivity Argon Heat conductivity
7.12 W/m K 17.8 W/m K 0.5 668.5 J/g 1.007;10 Pa 4.58;10Pa 5.14;10Pa 1.03;10\ K\ 0.237#0.0011 T W/m K 0.75 0.0018 #5.8854;10\ T !2.2487;10\ T #4.9213;10\ T W/m K
The temperature field within the crystal is used to calculate the thermo-elastic stress distribution in terms of the von-Mises stress p " + (((p !p )#(p !p )#(p !p )#6p ), XX (( PP (( XX PX PP where p , (i"r, z, ) are the normal stresses and GG p is the real shear stress. From that definition the PX von-Mises stress is scalar positive and it does not contain any information whether the tensile or compressive stress contributes to the von-Mises stress. The material properties of GaAs, boron oxide and argon are summarized in Table 1. The data for heat conductivity and emissivity of all other technical parts consisting of boron nitride, graphite, quartz, steel, felt, ceramics, etc., are used according to the data delivered by the supplying companies. None of these data was varied for better fitting the experimental results. 3. Results and discussion 3.1. Comparison between calculated and measured data First of all the reliability of the model was evaluated by comparing the results of modelling with
K. Bottcher et al. / Journal of Crystal Growth 198/199 (1999) 349–354 Table 2 Comparison for items inside the chamber
Temp. gradient at Axis Surface
Simulation 8.7 mm
Experiment 8—12 mm
21—33 K/cm 23—47 K/cm
+40 K/cm
Table 3 Comparison for items outside the chamber As heater power
Temperature at Crucible bottom Main heater
Simulation 239 W
Experiment 312 W
1215°C 1242°C
1174°C 1200°C
351
the chamber. Concerning the convective heat transfer the situation can be estimated by characteristic values of the gas phase Grashof number (+1.5;10) and both the Newtonian and linearized radiation heat transfer coefficient being identically about 0.03 W/cm K. Obviously, the reason for such a good agreement arises from the focussing of the heater control on the maintenance of the melting point temperature at the threephase-boundary compensating the effect of neglecting the heat losses from crystal and encapsulant to the gas by calculating a relativly low necessary heat input. Indeed, the simulated heat input is essentially lower than the experimental one. 3.2. 3 and 4 crystal growth
experimental values of the furnace F1. Tables 2 and 3 comprise data according to a growing 3 crystal just having a length of 30 mm. We have found that the calculated convexity of the crystal—melt interface agrees satisfactorily with the curvature of experimentally ascertained growth striations. The simulated temperature gradients match also roughly the experimental values measured by using a graphite dummy. The desired temperature of the arsenic source, however, needs in the model a lower power consumption. In fact, it has to be assumed that in reality a part of the heater power is dissipated due to the gas convection and somewhat more energy input is required. The comparison of the temperature values outside of the inner chamber reveals evident differences. This can be attributed to the direct contact of this region to the main heater and the enhanced gas streaming through the chinks of the insulating insertions. The gas convection distributes heat and, thus, decreases the temperature peak in the vicinity of the heater. As a result, lower experimental values than calculated ones will be observed. Nevertheless, there is a good accordance of the simulated and the measured difference between the temperatures at the crucible bottom and the main heater. One can say that there is a good agreement between simulation and experiments for those items measured inside
Both F1 and F2 are supplied with inner chamber constructions but considerably different heat shield arrangements. While furnace F1 is suitable to grow 3 crystals only furnace F2 is designed for 4 crystals, too. Via numerical simulation of conductive and radiative heat transfer, the suitability of the crystal growth arrangements were analyzed, whereby the efforts were concentrated on (i) a reduction of the temperature gradients along the growth axis and crystal surface, (ii) a slightly convex curvature of the crystal—melt interface, and (iii) minimizing the thermal stress resulting from the nonlinearities of the temperature distribution within the crystal. 3.2.1. Furnace F1 The thermal situation generated by furnace F1 is demonstrated in the case of a growing crystal of 80 mm in diameter having an already crystallized cylindrical part of 50 mm in length. Due to the crucible diameter of 125 mm and a charge of 300 g boron oxide the thickness of the encapsulant layer is 28 mm. The total GaAs charge is 3 kg and the growth velocity is about 5 mm/h. The temperature distribution in the crucible region and the adjacent parts, accompanied by the thermal stress distribution in terms of the von-Mises stress, is presented in Fig. 1. For a better illustration of the isotherm morphology this region is enlarged while further construction details are omitted. As Fig. 1 shows, the temperature decrease towards the crystal top is less than in conventional LEC crystals [1]. In the
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K. Bo( ttcher et al. / Journal of Crystal Growth 198/199 (1999) 349–354
Fig. 2. Radial distributions of the von-Mises thermal stress, at the z-coordinate of the intersection line of crystal, melt and boron oxide. All crystals are about 50 mm in length. The labels F1 and F2 refer to the type of crystal growth furnace.
Fig. 1. Growth situation of an 80 mm crystal in the furnace F1. Left: thermal stress in terms the von-Mises stress, gap between isolines: 0.5 MPa. Right: cut-out of the global temperature field around the crucible; convexity of the interface: 9.6 mm; gap between isolines: 10 K.
VCZ case the crystal has a temperature of about 1145°C when it emerges from the boron oxide. The temperature gradients derived from the axial profile at r"0 are permanently below 30 K/cm. The gradients at the surface vary between 30 and 40 K/cm for the submerged and between 20 and 30 K/cm for the emerged part. The latter is influenced by the radiation from the free crucible wall, while the boron oxide is considered to be non-transparent [5]. These data identify furnace F1 as a low gradient arrangement. However, in contrast to the temperature field in the central part of the crystal, the field in the outer part is strongly nonlinear. Both the temperature field near and at the surface and the relatively large curvature of the crystal—melt interface cause still too large thermal stresses (Fig. 1), though the tendency toward decreasing values is obvious in comparison with conventional LEC growth [6]. The local stress peaks are (i) at the intersection line of crystal, melt and boron oxide, (ii) where the crystal
Fig. 3. Radial distributions of the etch pit density of various samples. The labels F1 and F2 refer to the type of crystal growth furnace.
leaves the boron oxide and (iii) where the conus transforms to the cylinder. The radial profile of the von-Mises stress at the z-coordinate of location (i) is drawn in Fig. 2. The graph shows a slightly developed W-shape. In the part of the crystal near the interface, exhibiting a temperature more than 1200°C, the stress values are larger than the critical resolved shear stress of about 0.7 MPa [7]. The shape of the von-Mises stress profile can roughly be compared with the etch pit density (epd) graph measured along cut crystal slices. Fig. 3 shows the epd values along two different traces. Similarly to the von-Mises graph
K. Bottcher et al. / Journal of Crystal Growth 198/199 (1999) 349–354
353
the epd values increase markedly towards the periphery of the crystal. 3.2.2. Furnace F2 Furnace F2 is equipped with an improved arrangement of heat insulations. The resulting heat input is roughly 1/3 of that of furnace F1. Thus, the heat insulation of F2 can be clearly estimated as more effective. As a result, the heat transfer analysis of an 80 mm crystal (same material charge but 25 mm thickness of the boron oxide layer) yields improved thermo-mechanical parameters. The bending of the interface is reduced to about 2.9 mm, the maximum von-Mises stress is about 4.9 MPa, the axial temperature gradient along the centerline is permanently below 25 K/cm. All these values are markedly below those of conventional LEC arrangements (see Refs. [1,6], for example). Due to the small bending of the interface, the radial profile of the von-Mises stress at the z-coordinate of location (i) is degenerated to a V-shape (Fig. 2). In Fig. 4, the stress and temperature fields of an 80 mm-long 4 crystal are sketched. Because the same amount of boron oxide was used as in the 3 case the 4 crystal is up to 3/4 encapsulated. The interface bending (6.7 mm) lies nearly between that of both other 3 crystals. But, as for the maximum thermal von-Mises stress and the temperature gradient along the growth axis, the 4 crystal shows nearly the same behaviour as the 3 F2 one. The values are 5 MPa and below 25 K/cm, respectively. Compared with the 3 crystal of furnace F1 the radial distribution of the von-Mises stress shows even a more slightly developed W-shape. Besides this, the von-Mises stress profile of the 4 F2 crystal (Fig. 2) are roughly equal or lower than that of the 3 F1 crystal. The corresponding epd values behave similar (Fig. 3) Note, for better meeting the crystal geometry the transformation from shoulder conus to cylinder is scetched more softly. However, as the comparison between Fig. 1 and Fig. 4 shows the soft conus edge does not prevent the rise of a local stress peaks. In both the cases (soft and sharp edge) the temperature field at the edge is distorted due to the rapid change of the temperature when going from the top of the crystal into the gas atmosphere above of the crystal.
Fig. 4. Growth situation of an 105 mm crystal in the furnace F2. Left: thermal stress in terms the von-Mises stress, gap between isolines: 0.5 MPa. Right: cut-out of the global temperature field around the crucible; convexity of the interface: 6.7 mm; gap between isolines: 10 K.
4. Conclusions The tailoring of the temperature field design within the whole crystal growth furnace is an essential task to provide favourable low temperature gradient conditions. Using the global heat transfer code CrysVUN##, two VCZ furnaces with different heat insulations were analyzed and compared. Though the analysis does not cover convective phenomena, there is a good agreement between calculated and measured values, e.g., of the bending of the crystal-melt interface. The analysis reveals that the furnace F2, with an improved heat insulation around the inner VZC chamber, leads to a more advantageous temperature field resulting in a smaller bending of the solid—liquid interface and lower values of the thermal stresses. Although the VCZ developments are not completed yet improved temperature and stress fields, leading to markedly reduced epd values, compared with conventional 3 and 4 GaAs LEC crystals, have been obtained.
Acknowledgements This work was supported by the German Ministry for Education, Science, Research and Technique under contract No. 01 BM 501/0.
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K. Bo( ttcher et al. / Journal of Crystal Growth 198/199 (1999) 349–354
References [1] P. Rudolph, M. Neubert, S. Arulkumaran, M. Seifert, Crys. Res. Technol. 32 (1997) 35. [2] M. Tatsumi, T. Kawase, Y. Iguchi, K. Fujita, M. Yamada, in: M. Godlewski (Ed.), Semi-insulating III—V Materials, World Scientific, Singapore, 1994, p. 11. [3] M. Neubert, P. Rudolph, Seifert, in: M. Melloch, M.A. Reed (Eds.), 1997 IEEE Int. Symp. on Compound Semiconductors, Inst. Phys. Publ., Bristol, 1998, p. 53.
[4] M. Kurz, A. Pusztai, G. Mu¨ller, J. Crystal Growth 198/199 (1999) 101. [5] A.G. Ostrogorsky, K.H. Yao, A.F. Witt, J. Crystal Growth 84 (1987) 460. [6] A.S. Jordan, A.R. von Neida, R. Caruso, J. Crystal Growth 76 (1986) 243. [7] A.S. Jordan, J. Crystal Growth 49 (1980) 631.