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Melt-solid interface shape in LEC GaAs crystals: comparison between calculated and experimentally observed shapes
,. . . . . . . . C R Y g T A L OROWTI-I
ELSEVIER
Journal of Crystal Growth 166 (1996) 641-645
Melt-solid interface shape in LEC GaAs crystals" comparison between calculated and experimentally observed shapes Sergio Carrh
a
Sabrina Fogliani
a
Maurizio Masi " Lucio Zanotti b
C l a u d i o M u c c h i n o b Carlo Paorici c,, a Dipartimento di Chimica Fisica Applicata, Politecnico di Milano, Piazza Leonardo da Vinci 32, 1-20133 Milano, Italy b lstituto CNR - Maspec Parma, Via Chiavari 1 8 / A , Parma, Italy c Dipartimento di Fisica, Universith di Parma, Viale delle Scienze, 1-43100 Parma, Italy
Abstract
Two GaAs crystals have been grown by the liquid-encapsulated Czochralski (LEC) technique under different experimental conditions with the aim of revealing the melt-solid interface during growth. The interface shapes have been evidenced through revealing the growth striations in longitudinal cross sections of the ingot, making use of a photoactivated chemical etching. Measurements of the curvature radii of the various striations have been made by means of a Nomarski-equipped metallographic microscope. To investigate the role of the different growth parameters on the interface shape, the growth processes have been simulated through a model based upon the thermal capillary theory, which is able to describe the temperature field and the interface shapes within the LEC system. The calculated interface shapes have been satisfactorily compared with the experimental ones for undoped crystals. The role of the most significant growth parameters on the interface shape, such as the heater temperature, the crucible-bottom insulation, the encapsulant volume and the pull rate, have been finally investigated through a sensitivity analysis.
I. I n t r o d u c t i o n
The crystalline quality and the single crystal yield of HP-LEC GaAs ingots depend on several factors, such as melt stoichiometry, contamination and growth conditions. As the growth conditions could significantly vary depending on the puller and growth station used, the crystal interface shape could be regarded as an index for pulling optimization. It is
* Corresponding author. Fax: + 39 521 905223.
then important to understand the role played by the different growth parameters on the interface shape. Considering the costs and the time required by the experimental growth, a mathematical model of the system can speed-up the understanding of the parameter interplay and, consequently, the whole optimization procedure. Accordingly, a model based upon the thermal capillary theory [1,2] was adopted here. The model was validated upon experimental LEC GaAs growth, where the solid interface was delineated by revealing growth striations in longitudinal cross sections of the ingot by means of photochemical etching
[3].
0022-0248/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PII S0022-0248(96)00062-0
642
S. Carrh et al. / Journal of Crystal Growth 166 (1996) 641-645
heater temperature (K)
2. Experimental procedure
1640
The crystals were grown by LEC in a high pressure puller (LPAI Galaxie Model, Grenoble, France), with a (100) growth direction, by using the direct synthesis technique. The growth parameters were as follows: a B203 thickness of 15 mm, an argon counter pressure of 20 atm, and a pulling rate of 10 m m / h for all runs. Two undoped semi-insulating crystals were grown in a 4 inch diameter crucible. The puller was equipped with an in-house developed automatic diameter controller (ADC) and the obtained crystal diameter was 2 inches corresponding to a total weight of 2 kg. The resistive heating apparatus was driven by the feedback signal given from the ADC system and, meanwhile, the temperature was continuously recorded by a thermocouple located between the heater and the insulation package. The recorded temperature profiles for both the considered growths are reported in Fig. 1. The different interface shapes were obtained by changing the thermal environment around the crucible and crystal. In particular, a graphite shield was located on the top of the crucible and the crucible bottom was insulated by means of a graphite felt. Furthermore, different ratios of the crystal and the crucible rotation were used in the two runs. A summary of the growth conditions is reported in Table 1. The crystals were at first cut in two equal portions along the growth axis. From the first portion a series of longitudinal slices were cut, while the second one was cut orthogonally to the growth direction to obtain half-wafers for defect characterization. After a standard scratchfree polishing procedure, the longitudinal slices were photoetched by the DSL method [3,4] to evidence the interface through revealing the growth striations [5], as well as the structure defects and their evolution during the growth. Etched slices were examined with an optical microscope equipped with a Nomarski
/orowth.1
1620 1600 1580 1560 1540 1520
0 2
4 6 8 10 12 14 16 18 20 growth time (hours)
Fig. 1. Recorded values of the time history of the heater's temperature for the two considered GaAs LEC growths of Table 1.
interference contrast device [5,6]. Three different zones for the two crystals were selected at different growth times (3, 4.5 and 6.5 h since the start of the growth). Each zone was precisely mapped following the striation pattern by using sequential photographs shot with the x, y reference system of the microscope. The Cartesian coordinates of several points along a single growth striation were obtained by direct measurement on the photographs.
3. Modeling The crystal growth was simulated through a series of steady-state calculations using a model based on the thermal capillary theory [1,2]. Such a model describes, under axial symmetry, the temperature field within the system as well as the growth interface shape, the melt-encapsulant and the encapsulant-air menisci. The pseudo-steady-state assumption was justified by the very low value of the relaxation time associated with the temperature per-
Table 1 Summary of the experimental growth conditions Crystal Rotations Bottom insulation Crystal Crucible (counter rotating)
Shield
Compensation of melt level depletion
Singlecrystal yield
1 (164) 2 (195)
None 3 cm cylindrical
Yes No
30% 100%
15 40
-5 - 20
None High
S. Carrh et al. / Journal of C~stal Growth 166 (1996) 641-645
turbations with respect to the one related to the change of the system geometry. The temperature field was obtained by the solution of the energy balance equations within the three phases of the system. The crystal-melt, melt-encapsulant and encapsulant-gas interfaces were obtained by the melting point isotherm and by the solution of the Young-Laplace equations, respectively. A summary of the adopted model equations has already been reported in Refs. [2,7]. A conduction mechanism was considered for the heat transport within the solid, the melt and the encapsulant, while both radiative and convective mechanisms were considered for the heat transport between the previous phases and the surrounding ambient. The contribution of the convective mechanism within the melt was neglected due to the high value of the melt thermal conductivity. In fact, drastic changes in the melt motions have only small effects on the temperature field within the entire growth system [7,8]. Furthermore, an idealized furnace, i.e. with constant crucible wall and surrounding ambient temperatures, was also considered in the calculations. In particular, two values for the ambient temperature, above and below the crucible, were considered. The adopted values were 1133 and 1540 K, respectively. The above temperatures were selected in agreement with previous experiments performed by a system of thermocouples inserted into the growth apparatus [7,9]. For the second growth, the presence of the shield was considered by increasing the ambient temperature by 7%. The crucible wall temperature was estimated by the recorded heater temperature as follows. The system was simulated corresponding to the initial conditions and the crucible temperature was scaled from that of the heater by a constant estimated on these conditions. The model originates a system of partial differential equations that was solved through the finite element method, all the physico-chemical data being taken from the literature [2,10]. The reliability of the thermal capillary theory in predicting the temperature field and the melt-crystal interface has already been verified on silicon [8,11] and on InP growth [9]. 4. Results and discussion
The two sets of growing parameters gave two completely different interface shapes, as illustrated in
643
Fig. 2a and Fig. 3a. Crystal 1, pulled with a poor insulated environment, showed an "M"-shaped interface, while in the case of crystal 2, pulled with bottom insulation and an upper shield, a convex form was evidenced. As reported by many authors [12-15] the interface shape strongly influences the single crystal yield. In fact, in the first case, only a third of the boule was single, while in the latter a whole monocrystalline ingot (180 mm long) was obtained. The dislocation pattern was different in the two cases. In the M-shaped crystal, grown-in dislocations were arranged in a small-cell structure and, in addition, some stress-induced dislocations were found in the outer ingot rim. In the other case, only grown-in dislocations were present and the cellular structure was considerably larger. A more detailed description can be found in Ref. [13]. The comparisons of the calculated interfaces with the experimental interfaces after 4.5 h, since the start of the growth, for both the above experimental growths are illustrated in detail in Figs. 3b and 4b, where a satisfactory agreement between the two data sets is shown. The growths were simulated considering the idealized crystal shape depicted in the previous figures, which corresponds to a very good approximation of the actual shape. The flow dynamic implications being totally absent in the model considered here, all the differences between the two growth conditions in the simulations were represented only by differences in the thermal conditions. Accordingly, as demonstrated by the agreement obtained between calculations and experiments, the minor effects played by the meltflow dynamic on the system interfaces in the LEC system is confirmed. To investigate which of the growth parameters plays the major role on the solidification interface, a sensitivity analysis was performed [9,16]. The calculation of the normalized sensitivity coefficient (SD. x = O l n D / O l n X ) represents a compact way in which to analyze the relative importance of the several factors affecting the growth. Here D is the growth deflection corresponding to the crystal axis and X is the parameter whose role is tested. Accordingly, the influence of the crucible wall temperature, Tc, of the ambient temperature above the encapsulant, TA, and below the crucible, TB, of the encapsulant volume, Ve, and of the pull rate, Vp, on the growth interface deflection was tested. The values of the calculated
r
644
S. Carrh et al. / Journal o f Crystal Growth 166 (1996) 6 4 1 - 6 4 5
crystal shape (cm) 3.0
crystal shape (cm)
3.0 2.5
)
2.5 2.0 1.5
2.0
1.0
1.5
0.5
1.0
I
0.0 -3
0.5 0 -3
-2
-1
1
0
2
radius (cm) -2
-1
0 1 radius (crn)
2
interface shape (cm)
0.2~
interface shape (cm)
0.08
0.1!
0.06
(b)
w
0.04
0.11
.
0.0~
0.02
0.00 -3
0.00
-2
-1
(b)
0
1
2
3
radius (cm) -0.02 -3
-2
-1
0 1 radius (cm)
2
Fig. 3. Comparison of the calculated ( - - ) and experimental ( * * * *) growth interfaces at 4.5 h since the start of the growth for a GaAs crystal grown with the conditions of crystal 2 of Table 1. (a) Overall crystal shape considered for the simulation; (b) detail of the growth interface. The deflection is given in cm.
normalized sensitivity coefficients, corresponding to 4.5 h since the start of the growth, are reported in the histogram of Fig. 4, for a crystal growth with the
Fig. 2. Comparison of the calculated ( - - ) and experimental ( * * ) growth interfaces at 4.5 h since the start of the growth for a GaAs crystal grown with conditions of crystal 1 of Table 1. (a) Overall crystal shape considered for the simulation. (b) Detail of the growth interface. The deflection is given in cm. (c) Growth striations as evidenced by DSL etching.
TC TA IB
Ve vp
Fig. 4. Normalized sensitivity coefficient of the growth interface deflection with respect to Tc . TA, TB, V~, Vp. Deflection calculated at 4.5 h since the start of growth corresponding to the crystal axis for a GaAs crystal grown with the conditions of crystal 2 of Table 1.
S. Carrh et al./ Journal of Crystal Growth 166 (1996) 641-645
conditions reported in the second row of Table 1. It can be readily seen that the crucible temperature is the key parameter (highest sensitivity coefficient, So,rc = - 13) to control the interface deflection, while minor importance is associated to the pull rate (SD vp = -0.26). The deflection value was also significantly influenced by the ambient temperature, above the encapsulant and below the crucible (So,rA = -- 4.0, So,r. = - 8.0). In a radiation-dominated heat transport system, such as the LEC one, this effect seems to be surprising. This apparent anomaly was due to the modeling of the radiative heat transport through a simplified Stefan's approach. In these terms, those temperatures are respectively connected to the top insulation and to the crucible bottom insulation. Accordingly, an increase in these insulations reduces the interface deflection value. The increase of the encapsulant volume slightly reduces the deflection value, reducing the gradients within the crystal and keeping the crystal itself hotter (So,vo = - 1.2). This effect was verified both theoretically, for InP [9] and GaAs [17], and experimentally [18].
5. Conclusions The validity of a model based on the thermal capillary theory associated with the solution of a Stefan problem to simulate the interface growth shape was demonstrated by comparison with experiment. The main consequence is that this model can be used as a powerful and low-cost tool for an efficient optimization of the operative parameters controlling the growth. The use of a sensitivity analysis drove the further experimentation towards a better control of the crucible wall temperature, which was found to be the key parameter, as well as the other thermal parameters, to get the desired deflection of the growth interface.
645
Acknowledgements We would like to thank Dr. Edmondo Giglioli for help in DSL investigations and MURST "Progetto Materiali Innovativi" for financial support.
References [1] J.J. Derby, R.A. Brown, F.T. Geyling, A.S. Jordan and G.A. Nikolakopoulou, J. Electrochem. Soc. 132 (1985) 470. [2] J.J Derby and R.A. Brown, J. Crystal Growth 74 (1986) 605. [3] J.L. Weyher and J. van den Ven, J. Crystal Growth 63 (1983) 285. [4] J. van der Ven, J.L Weyher, J.E.A.M. Meerakker and J.J. Kelly, J. Electrochem. Soc. 133 (1986) 799. [5] J.L. Weyher, P.J. van der Wel, G. Frigerio and C. Mucchino, in: Proc, of 6th Int. Conf. on Semi-insulating III-V Materials, Toronto, Canada, 1990, Eds. A.G. Milnes and C.J. Miner (Hilger, Bristol, 1990) p. 161. [6] J.L. Weyher, C. Frigeri, L. Zanotti, H.Ch. Alt, P. van der Wel and P. Gall in: Proc. of 7th Int. Conf. on Semi-insulating III-V Materials, Ixtapa, Mexico, 1992, Eds. C.J. Miner, W. Ford and E.R. Weber (lOP, Bristol, 1993) p. 97. [7] S. Fogliani, PhD Thesis, Politecnico di Milano (1994). [8] T.A. Kinney, D.E. Bornside, R.A. Brown and K.M. Kim, J. Crystal Growth 126 (1993) 413. [9] S. Fogliani, M. Masi, S. CarrY, G. Guadalupi, B. Smith and L. Meregalli, Mater. Sci. Eng. B 28 (1994) 72. [10] P.D. Thomas, J.J. Derby, L.J. Atherton, R.A. Brown and M.J. Wargo, J. Crystal Growth 96 (1989) 135. [11] T.A. Kinney, D.E. Bomside, R.A. Brown and K.M. Kim, J. Crystal Growth 126 (1993) 413. [12] J.P Tower, R. Tobin and R.M. Ware, paper presented at 9th Int. Conf. on Crystal Growth (ICCG-9), Sendai, Japan, 1989. [13] G. Frigerio, L. Zanotti, J.L. Weyher, C. Paorici, C. Mucchino and C. Bucci, Cryst. Properties Preparation 36-38 (1991) 312. [14] H.D. Marshall, J. Crystal Growth 109 (1991) 218. [15] M. Shibata, T, Suzuki, S. Kuma and T. lnada, J. Crystal Growth 128 (1993) 439. [16] R. Tomovich and M. Vukobrativich, General Sensitivity Theory (Elsevier, New York, 1972). [17] S. Motakef and A.F. Witt, J. Crystal Growth 80 (1987) 37. [18] G. Muller, J. Volk and F. Tomzig, J. Crystal Growth 64 (1983) 40.
ELSEVIER
Journal of Crystal Growth 166 (1996) 641-645
Melt-solid interface shape in LEC GaAs crystals" comparison between calculated and experimentally observed shapes Sergio Carrh
a
Sabrina Fogliani
a
Maurizio Masi " Lucio Zanotti b
C l a u d i o M u c c h i n o b Carlo Paorici c,, a Dipartimento di Chimica Fisica Applicata, Politecnico di Milano, Piazza Leonardo da Vinci 32, 1-20133 Milano, Italy b lstituto CNR - Maspec Parma, Via Chiavari 1 8 / A , Parma, Italy c Dipartimento di Fisica, Universith di Parma, Viale delle Scienze, 1-43100 Parma, Italy
Abstract
Two GaAs crystals have been grown by the liquid-encapsulated Czochralski (LEC) technique under different experimental conditions with the aim of revealing the melt-solid interface during growth. The interface shapes have been evidenced through revealing the growth striations in longitudinal cross sections of the ingot, making use of a photoactivated chemical etching. Measurements of the curvature radii of the various striations have been made by means of a Nomarski-equipped metallographic microscope. To investigate the role of the different growth parameters on the interface shape, the growth processes have been simulated through a model based upon the thermal capillary theory, which is able to describe the temperature field and the interface shapes within the LEC system. The calculated interface shapes have been satisfactorily compared with the experimental ones for undoped crystals. The role of the most significant growth parameters on the interface shape, such as the heater temperature, the crucible-bottom insulation, the encapsulant volume and the pull rate, have been finally investigated through a sensitivity analysis.
I. I n t r o d u c t i o n
The crystalline quality and the single crystal yield of HP-LEC GaAs ingots depend on several factors, such as melt stoichiometry, contamination and growth conditions. As the growth conditions could significantly vary depending on the puller and growth station used, the crystal interface shape could be regarded as an index for pulling optimization. It is
* Corresponding author. Fax: + 39 521 905223.
then important to understand the role played by the different growth parameters on the interface shape. Considering the costs and the time required by the experimental growth, a mathematical model of the system can speed-up the understanding of the parameter interplay and, consequently, the whole optimization procedure. Accordingly, a model based upon the thermal capillary theory [1,2] was adopted here. The model was validated upon experimental LEC GaAs growth, where the solid interface was delineated by revealing growth striations in longitudinal cross sections of the ingot by means of photochemical etching
[3].
0022-0248/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PII S0022-0248(96)00062-0
642
S. Carrh et al. / Journal of Crystal Growth 166 (1996) 641-645
heater temperature (K)
2. Experimental procedure
1640
The crystals were grown by LEC in a high pressure puller (LPAI Galaxie Model, Grenoble, France), with a (100) growth direction, by using the direct synthesis technique. The growth parameters were as follows: a B203 thickness of 15 mm, an argon counter pressure of 20 atm, and a pulling rate of 10 m m / h for all runs. Two undoped semi-insulating crystals were grown in a 4 inch diameter crucible. The puller was equipped with an in-house developed automatic diameter controller (ADC) and the obtained crystal diameter was 2 inches corresponding to a total weight of 2 kg. The resistive heating apparatus was driven by the feedback signal given from the ADC system and, meanwhile, the temperature was continuously recorded by a thermocouple located between the heater and the insulation package. The recorded temperature profiles for both the considered growths are reported in Fig. 1. The different interface shapes were obtained by changing the thermal environment around the crucible and crystal. In particular, a graphite shield was located on the top of the crucible and the crucible bottom was insulated by means of a graphite felt. Furthermore, different ratios of the crystal and the crucible rotation were used in the two runs. A summary of the growth conditions is reported in Table 1. The crystals were at first cut in two equal portions along the growth axis. From the first portion a series of longitudinal slices were cut, while the second one was cut orthogonally to the growth direction to obtain half-wafers for defect characterization. After a standard scratchfree polishing procedure, the longitudinal slices were photoetched by the DSL method [3,4] to evidence the interface through revealing the growth striations [5], as well as the structure defects and their evolution during the growth. Etched slices were examined with an optical microscope equipped with a Nomarski
/orowth.1
1620 1600 1580 1560 1540 1520
0 2
4 6 8 10 12 14 16 18 20 growth time (hours)
Fig. 1. Recorded values of the time history of the heater's temperature for the two considered GaAs LEC growths of Table 1.
interference contrast device [5,6]. Three different zones for the two crystals were selected at different growth times (3, 4.5 and 6.5 h since the start of the growth). Each zone was precisely mapped following the striation pattern by using sequential photographs shot with the x, y reference system of the microscope. The Cartesian coordinates of several points along a single growth striation were obtained by direct measurement on the photographs.
3. Modeling The crystal growth was simulated through a series of steady-state calculations using a model based on the thermal capillary theory [1,2]. Such a model describes, under axial symmetry, the temperature field within the system as well as the growth interface shape, the melt-encapsulant and the encapsulant-air menisci. The pseudo-steady-state assumption was justified by the very low value of the relaxation time associated with the temperature per-
Table 1 Summary of the experimental growth conditions Crystal Rotations Bottom insulation Crystal Crucible (counter rotating)
Shield
Compensation of melt level depletion
Singlecrystal yield
1 (164) 2 (195)
None 3 cm cylindrical
Yes No
30% 100%
15 40
-5 - 20
None High
S. Carrh et al. / Journal of C~stal Growth 166 (1996) 641-645
turbations with respect to the one related to the change of the system geometry. The temperature field was obtained by the solution of the energy balance equations within the three phases of the system. The crystal-melt, melt-encapsulant and encapsulant-gas interfaces were obtained by the melting point isotherm and by the solution of the Young-Laplace equations, respectively. A summary of the adopted model equations has already been reported in Refs. [2,7]. A conduction mechanism was considered for the heat transport within the solid, the melt and the encapsulant, while both radiative and convective mechanisms were considered for the heat transport between the previous phases and the surrounding ambient. The contribution of the convective mechanism within the melt was neglected due to the high value of the melt thermal conductivity. In fact, drastic changes in the melt motions have only small effects on the temperature field within the entire growth system [7,8]. Furthermore, an idealized furnace, i.e. with constant crucible wall and surrounding ambient temperatures, was also considered in the calculations. In particular, two values for the ambient temperature, above and below the crucible, were considered. The adopted values were 1133 and 1540 K, respectively. The above temperatures were selected in agreement with previous experiments performed by a system of thermocouples inserted into the growth apparatus [7,9]. For the second growth, the presence of the shield was considered by increasing the ambient temperature by 7%. The crucible wall temperature was estimated by the recorded heater temperature as follows. The system was simulated corresponding to the initial conditions and the crucible temperature was scaled from that of the heater by a constant estimated on these conditions. The model originates a system of partial differential equations that was solved through the finite element method, all the physico-chemical data being taken from the literature [2,10]. The reliability of the thermal capillary theory in predicting the temperature field and the melt-crystal interface has already been verified on silicon [8,11] and on InP growth [9]. 4. Results and discussion
The two sets of growing parameters gave two completely different interface shapes, as illustrated in
643
Fig. 2a and Fig. 3a. Crystal 1, pulled with a poor insulated environment, showed an "M"-shaped interface, while in the case of crystal 2, pulled with bottom insulation and an upper shield, a convex form was evidenced. As reported by many authors [12-15] the interface shape strongly influences the single crystal yield. In fact, in the first case, only a third of the boule was single, while in the latter a whole monocrystalline ingot (180 mm long) was obtained. The dislocation pattern was different in the two cases. In the M-shaped crystal, grown-in dislocations were arranged in a small-cell structure and, in addition, some stress-induced dislocations were found in the outer ingot rim. In the other case, only grown-in dislocations were present and the cellular structure was considerably larger. A more detailed description can be found in Ref. [13]. The comparisons of the calculated interfaces with the experimental interfaces after 4.5 h, since the start of the growth, for both the above experimental growths are illustrated in detail in Figs. 3b and 4b, where a satisfactory agreement between the two data sets is shown. The growths were simulated considering the idealized crystal shape depicted in the previous figures, which corresponds to a very good approximation of the actual shape. The flow dynamic implications being totally absent in the model considered here, all the differences between the two growth conditions in the simulations were represented only by differences in the thermal conditions. Accordingly, as demonstrated by the agreement obtained between calculations and experiments, the minor effects played by the meltflow dynamic on the system interfaces in the LEC system is confirmed. To investigate which of the growth parameters plays the major role on the solidification interface, a sensitivity analysis was performed [9,16]. The calculation of the normalized sensitivity coefficient (SD. x = O l n D / O l n X ) represents a compact way in which to analyze the relative importance of the several factors affecting the growth. Here D is the growth deflection corresponding to the crystal axis and X is the parameter whose role is tested. Accordingly, the influence of the crucible wall temperature, Tc, of the ambient temperature above the encapsulant, TA, and below the crucible, TB, of the encapsulant volume, Ve, and of the pull rate, Vp, on the growth interface deflection was tested. The values of the calculated
r
644
S. Carrh et al. / Journal o f Crystal Growth 166 (1996) 6 4 1 - 6 4 5
crystal shape (cm) 3.0
crystal shape (cm)
3.0 2.5
)
2.5 2.0 1.5
2.0
1.0
1.5
0.5
1.0
I
0.0 -3
0.5 0 -3
-2
-1
1
0
2
radius (cm) -2
-1
0 1 radius (crn)
2
interface shape (cm)
0.2~
interface shape (cm)
0.08
0.1!
0.06
(b)
w
0.04
0.11
.
0.0~
0.02
0.00 -3
0.00
-2
-1
(b)
0
1
2
3
radius (cm) -0.02 -3
-2
-1
0 1 radius (cm)
2
Fig. 3. Comparison of the calculated ( - - ) and experimental ( * * * *) growth interfaces at 4.5 h since the start of the growth for a GaAs crystal grown with the conditions of crystal 2 of Table 1. (a) Overall crystal shape considered for the simulation; (b) detail of the growth interface. The deflection is given in cm.
normalized sensitivity coefficients, corresponding to 4.5 h since the start of the growth, are reported in the histogram of Fig. 4, for a crystal growth with the
Fig. 2. Comparison of the calculated ( - - ) and experimental ( * * ) growth interfaces at 4.5 h since the start of the growth for a GaAs crystal grown with conditions of crystal 1 of Table 1. (a) Overall crystal shape considered for the simulation. (b) Detail of the growth interface. The deflection is given in cm. (c) Growth striations as evidenced by DSL etching.
TC TA IB
Ve vp
Fig. 4. Normalized sensitivity coefficient of the growth interface deflection with respect to Tc . TA, TB, V~, Vp. Deflection calculated at 4.5 h since the start of growth corresponding to the crystal axis for a GaAs crystal grown with the conditions of crystal 2 of Table 1.
S. Carrh et al./ Journal of Crystal Growth 166 (1996) 641-645
conditions reported in the second row of Table 1. It can be readily seen that the crucible temperature is the key parameter (highest sensitivity coefficient, So,rc = - 13) to control the interface deflection, while minor importance is associated to the pull rate (SD vp = -0.26). The deflection value was also significantly influenced by the ambient temperature, above the encapsulant and below the crucible (So,rA = -- 4.0, So,r. = - 8.0). In a radiation-dominated heat transport system, such as the LEC one, this effect seems to be surprising. This apparent anomaly was due to the modeling of the radiative heat transport through a simplified Stefan's approach. In these terms, those temperatures are respectively connected to the top insulation and to the crucible bottom insulation. Accordingly, an increase in these insulations reduces the interface deflection value. The increase of the encapsulant volume slightly reduces the deflection value, reducing the gradients within the crystal and keeping the crystal itself hotter (So,vo = - 1.2). This effect was verified both theoretically, for InP [9] and GaAs [17], and experimentally [18].
5. Conclusions The validity of a model based on the thermal capillary theory associated with the solution of a Stefan problem to simulate the interface growth shape was demonstrated by comparison with experiment. The main consequence is that this model can be used as a powerful and low-cost tool for an efficient optimization of the operative parameters controlling the growth. The use of a sensitivity analysis drove the further experimentation towards a better control of the crucible wall temperature, which was found to be the key parameter, as well as the other thermal parameters, to get the desired deflection of the growth interface.
645
Acknowledgements We would like to thank Dr. Edmondo Giglioli for help in DSL investigations and MURST "Progetto Materiali Innovativi" for financial support.
References [1] J.J. Derby, R.A. Brown, F.T. Geyling, A.S. Jordan and G.A. Nikolakopoulou, J. Electrochem. Soc. 132 (1985) 470. [2] J.J Derby and R.A. Brown, J. Crystal Growth 74 (1986) 605. [3] J.L. Weyher and J. van den Ven, J. Crystal Growth 63 (1983) 285. [4] J. van der Ven, J.L Weyher, J.E.A.M. Meerakker and J.J. Kelly, J. Electrochem. Soc. 133 (1986) 799. [5] J.L. Weyher, P.J. van der Wel, G. Frigerio and C. Mucchino, in: Proc, of 6th Int. Conf. on Semi-insulating III-V Materials, Toronto, Canada, 1990, Eds. A.G. Milnes and C.J. Miner (Hilger, Bristol, 1990) p. 161. [6] J.L. Weyher, C. Frigeri, L. Zanotti, H.Ch. Alt, P. van der Wel and P. Gall in: Proc. of 7th Int. Conf. on Semi-insulating III-V Materials, Ixtapa, Mexico, 1992, Eds. C.J. Miner, W. Ford and E.R. Weber (lOP, Bristol, 1993) p. 97. [7] S. Fogliani, PhD Thesis, Politecnico di Milano (1994). [8] T.A. Kinney, D.E. Bornside, R.A. Brown and K.M. Kim, J. Crystal Growth 126 (1993) 413. [9] S. Fogliani, M. Masi, S. CarrY, G. Guadalupi, B. Smith and L. Meregalli, Mater. Sci. Eng. B 28 (1994) 72. [10] P.D. Thomas, J.J. Derby, L.J. Atherton, R.A. Brown and M.J. Wargo, J. Crystal Growth 96 (1989) 135. [11] T.A. Kinney, D.E. Bomside, R.A. Brown and K.M. Kim, J. Crystal Growth 126 (1993) 413. [12] J.P Tower, R. Tobin and R.M. Ware, paper presented at 9th Int. Conf. on Crystal Growth (ICCG-9), Sendai, Japan, 1989. [13] G. Frigerio, L. Zanotti, J.L. Weyher, C. Paorici, C. Mucchino and C. Bucci, Cryst. Properties Preparation 36-38 (1991) 312. [14] H.D. Marshall, J. Crystal Growth 109 (1991) 218. [15] M. Shibata, T, Suzuki, S. Kuma and T. lnada, J. Crystal Growth 128 (1993) 439. [16] R. Tomovich and M. Vukobrativich, General Sensitivity Theory (Elsevier, New York, 1972). [17] S. Motakef and A.F. Witt, J. Crystal Growth 80 (1987) 37. [18] G. Muller, J. Volk and F. Tomzig, J. Crystal Growth 64 (1983) 40.