- Email: [email protected]
Development of dislocation density analysis code for annealing process of single-crystal ingot
Journal of Crystal Growth 216 (2000) 6}14
Development of dislocation density analysis code for annealing process of single-crystal ingot N. Miyazaki*, S. Yamamoto, H. Kutsukake Department of Materials Process Engineering, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan Received 26 January 2000; accepted 10 April 2000 Communicated by T. Nishinaga
Abstract Ingot annealing is an indispensable process for a GaAs single crystal to improve its electric characteristics. One of the technical problems of GaAs ingot annealing is the increase of dislocation density during its annealing process which a!ects the performance of electronic devices. A computer code was developed for dislocation density evaluation of a single-crystal ingot during annealing process. In this computer code, temperatures in a single-crystal ingot are used as input data, which were obtained from a transient heat conduction analysis of an ingot. A dislocation kinetics model called the Haasen}Sumino model was used as the constitutive equation. In this model, creep strain rate is related to the dislocation density, and this model which extended to the multiaxial stress state was incorporated into a "nite element elastic creep analysis code for axisymmetric bodies. Dislocation density analyses were performed using this computer code for GaAs ingots of 4- or 6-in diameter, and time variations of the dislocation density and equivalent shear stress and residual stress after the ingot annealing were obtained by this computer code. 2000 Elsevier Science B.V. All rights reserved. Keywords: Ingot annealing; GaAs single crystal; Dislocation density; Finite element method
1. Introduction In recent years, there has been a rapid increase in the demand for mobile communication systems such as cellular phones using high-frequency waves. Devices such as "eld e!ect transistors fabricated on GaAs substrate are key devices in mobile communication systems. There is a need for further cost reduction of GaAs devices to reduce the cost of cellular phones. It is useful for reducing the cost of * Corresponding author. Fax: #81-92-642-3531. E-mail address: [email protected] (N. Miyazaki).
GaAs devices to develop and use large-diameter 6-in wafers instead of using the conventional 3- or 4-in diameter wafers. Manufacturing process of GaAs wafers consists of growth of a bulk single crystal, ingot annealing and wafer processing. Ingot annealing is an indispensable process for a GaAs single crystal to improve its electric characteristics. One of the technical problems of GaAs ingot annealing is the increase of the dislocation density due to the thermal stress induced during its annealing process, which a!ects the performance of devices. This problem would become serious when a large GaAs ingot is annealed. The qualitative relation between dislocation density and thermal
0022-0248/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 0 ) 0 0 4 6 7 - X
N. Miyazaki et al. / Journal of Crystal Growth 216 (2000) 6}14
stress in a GaAs ingot during annealing process was studied by comparing resolved shear stresses obtained from an elastic thermal stress analysis with critical resolved shear stress [1]. At elevated temperatures, solids undergo timedependent inelastic deformation. Hereafter such a deformation is called creep in the present paper. The creep constitutive equation for semiconductor single crystals such as GaAs is known as the Haasen}Sumino model [2,3], in which the creep strain rate is related to the dislocation density. In our previous study [4,5], we developed "nite element codes for a dislocation density analysis during bulk single-crystal growth process. The methodology used in these codes is applicable to a dislocation density analysis during annealing process of semiconductor single crystals such as GaAs. The analysis of annealing process is much simpler than that of crystal growth process, because the shape of a crystal does not change during annealing process. In the present study, we propose a method for the quantitative evaluation of dislocation density during ingot annealing process, and provide numerical results of dislocation density of a GaAs ingot obtained from a "nite element computer code newly developed by the present authors.
7
Fig. 1. Analysis models (left: 4-in diameter GaAs ingot, right: 6-in diameter GaAs ingot).
Fig. 2. Finite element meshes (left: 4-in diameter ingot model, right: 6-in diameter ingot model).
conduction analysis and a dislocation density analysis. The same axisymmetric "nite element models shown in Fig. 2 were used to model half of the ingots for both analyses.
2. Method of analysis
2.3. Heat conduction analysis
2.1. Flow of analysis
A transient heat conduction analysis was performed to obtain the temperature distribution of a GaAs ingot during annealing process. In actual ingot annealing, an ingot is subjected to heat radiation from an electric furnace. A heat conduction analysis taking account of heat radiation is required in such a case. In the present analyses, however, we dealt with a simple boundary condition such as prescribed temperatures on the ingot surfaces, because the objective of the present study is to show the e!ectiveness of a computer code for a dislocation density analysis newly developed by the present authors. As shown in Fig. 3, we studied four cases of the analysis where the temperature on a GaAs ingot surface increases from the initial temperature ¹ of 373 K to the maximum temper ature ¹ of 1173 or 1373 K at the rate of temper ature change d¹/dt of 100 or 200 K/h, then keeps at
A single-crystal ingot is placed in an electric furnace during ingot annealing process, and a prescribed temperature history is imposed on the ingot. A transient heat conduction analysis is performed to obtain the temperature distribution of the ingot during annealing process using a suitable computer code. Then we apply a computer code newly developed by the present authors to obtain stress distribution and dislocation density distribution of an ingot during annealing process. 2.2. Analysis model We made analyses of GaAs single-crystal ingots of 4- or 6-in diameter, as shown in Fig. 1. The "nite element method was applied to both a heat
8
N. Miyazaki et al. / Journal of Crystal Growth 216 (2000) 6}14
2.4. Dislocation density analysis
Fig. 3. Time variations of temperature on GaAs ingot surface.
the maximum temperature for 2 h, and decreases from the maximum temperature to the initial temperature at the rate d¹/dt of 100 or 200 K/h. The analysis conditions used in the present study are summarized as follows: Case 1: ¹ "373 K, ¹ "1173 K,
Case 2: ¹ "373 K, ¹ "1173 K,
d¹/dt"200 K/h. Case 3: ¹ "373 K, ¹ "1373 K,
d¹/dt"100 K/h. Case 4: ¹ "373 K, ¹ "1373 K,
d¹/dt"200 K/h. A general purpose "nite element code MARC was used in the transient heat conduction analyses. Thermal conductivity k, speci"c heat C and denN sity o of a GaAs single crystal are given by Ref. [6] k"2.08;10¹\ (W/m K),
o"5200 (kg/m), where ¹ represents absolute temperature (K).
e "f S , GH GH where bk N exp(!Q/k¹)((J !D(N !q )N
, f" 2(J p d S S S "p ! II GH , J " GH GH . GH GH 2 3
d¹/dt"100 K/h.
C "302#8.1;10\¹ (J/kg K), N
Dislocation density in an ingot was found to increase due to thermal stress during annealing process. We describe herein the method for simulating the time variation of dislocation density during this process. Total strain rate e is assumed to be given by GH e "e #e #e , (2) GH GH GH GH where e , e and e are elastic strain rate, thermal GH GH GH strain rate and creep strain rate, respectively. The creep strain rate e can be obtained by extending GH the Haasen}Sumino model [2,3] to a multiaxial stress state in the same way as the isotropic #ow theory of plasticity. Then it can be written as [7,8]
(1)
(3)
(4) (5)
In the above equation, p , S and J are stresses, GH GH deviatoric stresses and the second invariant of the deviatoric stress, respectively. Dislocation density is denoted by N , and its rate is given by
NQ "Kk N e\/ I2((J !D(N !q )N
;((J !D(N )H. (6)
In Eqs. (3) and (6), NQ and e are set equal to zero K GH when (J !D(N !q 60. From Eq. (6), N is
always equal to zero when the initial dislocation density N is zero, so it is necessary to assume
non-zero value of N . In the analysis of ingot
annealing, the dislocation density existing in an as-grown GaAs single-crystal ingot should be used as the value of N . We used an N value of
1.0;10 m\ in the present analyses. Furthermore, q is a drag-stress caused by the interaction be tween dislocations and impurity atoms and set equal to the following equation: log q "4.83#1382/¹,
(7)
N. Miyazaki et al. / Journal of Crystal Growth 216 (2000) 6}14 Table 1 Parameters in Haasen}Sumino model and their values for GaAs single crystal N
B Q k ¹ K D k q p j
density of moving dislocation Magnitude of Burgers vector Peierls potential Boltzmann's constant absolute temperature material constant strain hardening factor material constant drag-stress material constant material constant
m\ 4.0;10\ m 1.5 eV 8.617;10\ eV/K K 7.0;10\ m/N 3.1 N/m 1.80;10\ mN>/NNs Pa 1.7 1.0
which was given by Jordan [6] for a Si-doped GaAs single crystal. Table 1 summarizes the parameters in the Haasen}Sumino model and their values for a GaAs single crystal [9]. In addition to the material parameters for the Haasen}Sumino model, we require the thermal expansion coe$cient a, and the elastic constants C for a GaAs single crystal. They are given by [6] GH E Thermal expansion coe$cient a"4.68;10\#3.82;10\¹.
(8)
E Elastic constants C GH
9
where K, *U, *F, *F and *R denote a sti!ness matrix, an incremental nodal displacement vector, an incremental load vector due to thermal strain, an incremental load vector due to creep strain and a residual load vector, respectively. Solving Eq. (10) at each time step and integrating Eq. (6), we obtain state variables such as displacements, strains, stresses and dislocation density.
3. Results and discussion 3.1. Heat conduction analysis The time variations of temperature distribution in a GaAs ingot were obtained for Cases 1}4 described in Section 2.3. Fig. 4 shows the time variations of the maximum temperature di!erence in the ingot. Temperature distributions in the ingot for Cases 2 and 4 are shown in Fig. 5 at 60 s after annealing starts and at the time when the temperature reaches to ¹ . We can "nd the e!ect of
temperature dependence of thermal conductivity k and speci"c heat C . As shown in Eq. (1), k N decreases and C increases with the increase of N temperature, so thermal di!usivity i("k/C o) deN creases with the increase of temperature. This means that heat does not move well and temperature di!erence in the ingot becomes large when
C "1.217;10!1.44;10¹, C "0.546;10!0.64;10¹, (9) C "0.618;10!0.70;10¹. The units of a, C and ¹ are K\, Pa and K, GH respectively. In the present analyses, a single crystal ingot was assumed to be isotropic, so only Young's modulus and Poisson's ratio in the +1 1 1, plane calculated from C , C and C [10] were used because they are invariant in this plane. The "nite element method for an elastic creep problem is applied to obtain the time variations of dislocation density. The incremental form of a "nite element equilibrium equation is given as follows: K*U"*F#*F#*R,
(10)
Fig. 4. Time variations of maximum temperature di!erence in GaAs ingots.
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N. Miyazaki et al. / Journal of Crystal Growth 216 (2000) 6}14
Fig. 5. Temperature distributions in GaAs ingots (left: 4-in diameter GaAs ingot, right: 6-in diameter GaAs ingot).
N. Miyazaki et al. / Journal of Crystal Growth 216 (2000) 6}14
11
Fig. 7. Time variations of dislocation density (Case 4). Fig. 6. Time variations of equivalent shear stress (J (Case 4).
temperature becomes high. Such an increase in temperature di!erence leads to the increase in thermal stress in the ingot which causes the multiplication of dislocations. The temperature di!erence becomes maximum at the end of the heating-up process, and it becomes larger as the rate of temperature change d¹/dt and the ingot diameter become larger. 3.2. Dislocation density analysis Distributions of the dislocation density and stresses in the GaAs ingot and their time variations were obtained using the temperature data obtained in Section 3.1. For Case 4, time variations of equivalent shear stress (J due to thermal stress and q at the observation point depicted in Fig. 1 are shown in Fig. 6 and that of the dislocation density at the same point is shown in Fig. 7. During the heating-up process, temperature gradient in the ingot increases due to the temperature dependence of the thermal material properties, so thermal stress or (J increases during the heating-up process. Around 10 000 s, (J exceeds a threshold value for dislocation multiplication and the dislocation density increases after 10 000 s as shown in Fig. 7. Exactly speaking, the threshold value is q #D(N , but it can be appro
ximated by q because D(N is much smaller
than q . After exceeding the threshold value,
(J starts decreasing due to the e!ect of stress relaxation caused by dislocation multiplication. When kept at the maximum temperature, the stress which relaxed during the heating-up process remains in the ingot as residual stress. The direction of the residual shear stress is reverse to that of the shear stress during the heating-up process. The equivalent shear stress (J should be a positive value. Because of these reasons we can "nd a turnup of (J at the time of 18 000 s in Fig. 6. The equivalent shear stress (J exceeds q during the turnup, and the dislocation density jumps up at the time of 18 000 s, as shown in Fig. 7. At the beginning of the cooling-down process, (J in creases due to the increase of thermal stress induced by the increase of temperature gradient in the ingot, and the dislocation density jumps up again. During the cooling-down process, (J decreases and a turnup of (J caused by a change in the direc tion of the equivalent shear stress is also found. (J changes from a decrease to an increase at the turnup point, and residual stress again remains in the ingot after annealing process. The same behaviors can be seen in the equivalent shear stress and the dislocation density for Cases 1}3. Figs. 8 and 9 show the distributions of the equivalent shear stress (J and the dislocation density after ingot annealing, respectively. We can regard Fig. 8 as the results of residual stress distributions.
12
N. Miyazaki et al. / Journal of Crystal Growth 216 (2000) 6}14
Fig. 8. Distributions of equivalent shear stress (J after ingot annealing (left: 4-in diameter GaAs ingot, right: 6-in diameter GaAs ingot).
N. Miyazaki et al. / Journal of Crystal Growth 216 (2000) 6}14
13
Fig. 9. Distributions of dislocation density after ingot annealing (left: 4-in diameter GaAs ingot, right: 6-in diameter GaAs ingot).
14
N. Miyazaki et al. / Journal of Crystal Growth 216 (2000) 6}14
If creep strain does not occur during the ingot annealing process, there exists only an elastic strain and no residual stress occurs. When large dislocation multiplication causes large stress relaxation, large residual stress remains in the ingot after annealing process. We can con"rm such a relation between the residual stress and the dislocation density by making a comparison between Figs. 8 and 9. As for the e!ect of ingot diameter, the 6-in diameter ingot has a higher dislocation density than the 4-in diameter ingot. It is found from a comparison between Cases 1 and 2 or Cases 3 and 4 that the larger the rate of temperature change d¹/dt is, the higher the dislocation density is. On the other hand, the e!ect of the maximum temperature ¹ is some what complicated. Although it is not clear in the case of d¹/dt of 100 K/h, we can "nd the following in the case of d¹/dt of 200 K/h by making a comparison between Cases 2 and 4, where Case 2 has lower ¹ than Case 4. Case 2 has a larger peak
value of dislocation density but a smaller area of high dislocation density than Case 4. So Case 4, which has higher ¹ , has a higher dislocation
density than Case 2 as a whole.
4. Concluding remarks A new computer code was developed for calculating dislocation density during annealing of a single-crystal ingot. We applied this computer
code to a GaAs ingot annealing process and showed that this computer code can be successfully applied to obtain the time variations of dislocation density and stresses in the ingot and also residual stress. It is expected that this computer code can be applied to an optimization of the annealing process of a single-crystal ingot.
Acknowledgements This study was partially supported by a Grantin-Aid for Science Research from the Ministry of Education and Culture.
References [1] S. Sawada, K. Hashio, M. Tatsumi, K. Fujita, M. Shiozaki, H. Miyazima, R. Nakai, Y. Nishida, Sumitomo Electric Technical Review, No. 149, 1996, p. 108. [2] P. Haasen, Z. Phys. 167 (1962) 461. [3] I. Yonenaga, K. Sumino, J. Appl. Phys. 50 (1978) 685. [4] N. Miyazaki, S. Okuyama, J. Crystal Growth 183 (1998) 81. [5] N. Miyazaki, Y. Kuroda, JSME Int. J. Ser. A 42 (1999) 485. [6] A.S. Jordan, J. Crystal Growth 49 (1980) 631. [7] C.T. Tsai, O.W. Dillion Jr, R.J. DeAngelis, J. Eng. Mater. Technol. 112 (1990) 183. [8] C.T. Tsai, M.W. Yao, A. Chait, J. Crystal Growth 125 (1992) 62. [9] D. Maroudas, R.A. Brown, J. Crystal Growth 108 (1991) 399. [10] W.A. Brantley, J. Appl. Phys. 44 (1973) 534.
Development of dislocation density analysis code for annealing process of single-crystal ingot N. Miyazaki*, S. Yamamoto, H. Kutsukake Department of Materials Process Engineering, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan Received 26 January 2000; accepted 10 April 2000 Communicated by T. Nishinaga
Abstract Ingot annealing is an indispensable process for a GaAs single crystal to improve its electric characteristics. One of the technical problems of GaAs ingot annealing is the increase of dislocation density during its annealing process which a!ects the performance of electronic devices. A computer code was developed for dislocation density evaluation of a single-crystal ingot during annealing process. In this computer code, temperatures in a single-crystal ingot are used as input data, which were obtained from a transient heat conduction analysis of an ingot. A dislocation kinetics model called the Haasen}Sumino model was used as the constitutive equation. In this model, creep strain rate is related to the dislocation density, and this model which extended to the multiaxial stress state was incorporated into a "nite element elastic creep analysis code for axisymmetric bodies. Dislocation density analyses were performed using this computer code for GaAs ingots of 4- or 6-in diameter, and time variations of the dislocation density and equivalent shear stress and residual stress after the ingot annealing were obtained by this computer code. 2000 Elsevier Science B.V. All rights reserved. Keywords: Ingot annealing; GaAs single crystal; Dislocation density; Finite element method
1. Introduction In recent years, there has been a rapid increase in the demand for mobile communication systems such as cellular phones using high-frequency waves. Devices such as "eld e!ect transistors fabricated on GaAs substrate are key devices in mobile communication systems. There is a need for further cost reduction of GaAs devices to reduce the cost of cellular phones. It is useful for reducing the cost of * Corresponding author. Fax: #81-92-642-3531. E-mail address: [email protected] (N. Miyazaki).
GaAs devices to develop and use large-diameter 6-in wafers instead of using the conventional 3- or 4-in diameter wafers. Manufacturing process of GaAs wafers consists of growth of a bulk single crystal, ingot annealing and wafer processing. Ingot annealing is an indispensable process for a GaAs single crystal to improve its electric characteristics. One of the technical problems of GaAs ingot annealing is the increase of the dislocation density due to the thermal stress induced during its annealing process, which a!ects the performance of devices. This problem would become serious when a large GaAs ingot is annealed. The qualitative relation between dislocation density and thermal
0022-0248/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 0 ) 0 0 4 6 7 - X
N. Miyazaki et al. / Journal of Crystal Growth 216 (2000) 6}14
stress in a GaAs ingot during annealing process was studied by comparing resolved shear stresses obtained from an elastic thermal stress analysis with critical resolved shear stress [1]. At elevated temperatures, solids undergo timedependent inelastic deformation. Hereafter such a deformation is called creep in the present paper. The creep constitutive equation for semiconductor single crystals such as GaAs is known as the Haasen}Sumino model [2,3], in which the creep strain rate is related to the dislocation density. In our previous study [4,5], we developed "nite element codes for a dislocation density analysis during bulk single-crystal growth process. The methodology used in these codes is applicable to a dislocation density analysis during annealing process of semiconductor single crystals such as GaAs. The analysis of annealing process is much simpler than that of crystal growth process, because the shape of a crystal does not change during annealing process. In the present study, we propose a method for the quantitative evaluation of dislocation density during ingot annealing process, and provide numerical results of dislocation density of a GaAs ingot obtained from a "nite element computer code newly developed by the present authors.
7
Fig. 1. Analysis models (left: 4-in diameter GaAs ingot, right: 6-in diameter GaAs ingot).
Fig. 2. Finite element meshes (left: 4-in diameter ingot model, right: 6-in diameter ingot model).
conduction analysis and a dislocation density analysis. The same axisymmetric "nite element models shown in Fig. 2 were used to model half of the ingots for both analyses.
2. Method of analysis
2.3. Heat conduction analysis
2.1. Flow of analysis
A transient heat conduction analysis was performed to obtain the temperature distribution of a GaAs ingot during annealing process. In actual ingot annealing, an ingot is subjected to heat radiation from an electric furnace. A heat conduction analysis taking account of heat radiation is required in such a case. In the present analyses, however, we dealt with a simple boundary condition such as prescribed temperatures on the ingot surfaces, because the objective of the present study is to show the e!ectiveness of a computer code for a dislocation density analysis newly developed by the present authors. As shown in Fig. 3, we studied four cases of the analysis where the temperature on a GaAs ingot surface increases from the initial temperature ¹ of 373 K to the maximum temper ature ¹ of 1173 or 1373 K at the rate of temper ature change d¹/dt of 100 or 200 K/h, then keeps at
A single-crystal ingot is placed in an electric furnace during ingot annealing process, and a prescribed temperature history is imposed on the ingot. A transient heat conduction analysis is performed to obtain the temperature distribution of the ingot during annealing process using a suitable computer code. Then we apply a computer code newly developed by the present authors to obtain stress distribution and dislocation density distribution of an ingot during annealing process. 2.2. Analysis model We made analyses of GaAs single-crystal ingots of 4- or 6-in diameter, as shown in Fig. 1. The "nite element method was applied to both a heat
8
N. Miyazaki et al. / Journal of Crystal Growth 216 (2000) 6}14
2.4. Dislocation density analysis
Fig. 3. Time variations of temperature on GaAs ingot surface.
the maximum temperature for 2 h, and decreases from the maximum temperature to the initial temperature at the rate d¹/dt of 100 or 200 K/h. The analysis conditions used in the present study are summarized as follows: Case 1: ¹ "373 K, ¹ "1173 K,
Case 2: ¹ "373 K, ¹ "1173 K,
d¹/dt"200 K/h. Case 3: ¹ "373 K, ¹ "1373 K,
d¹/dt"100 K/h. Case 4: ¹ "373 K, ¹ "1373 K,
d¹/dt"200 K/h. A general purpose "nite element code MARC was used in the transient heat conduction analyses. Thermal conductivity k, speci"c heat C and denN sity o of a GaAs single crystal are given by Ref. [6] k"2.08;10¹\ (W/m K),
o"5200 (kg/m), where ¹ represents absolute temperature (K).
e "f S , GH GH where bk N exp(!Q/k¹)((J !D(N !q )N
, f" 2(J p d S S S "p ! II GH , J " GH GH . GH GH 2 3
d¹/dt"100 K/h.
C "302#8.1;10\¹ (J/kg K), N
Dislocation density in an ingot was found to increase due to thermal stress during annealing process. We describe herein the method for simulating the time variation of dislocation density during this process. Total strain rate e is assumed to be given by GH e "e #e #e , (2) GH GH GH GH where e , e and e are elastic strain rate, thermal GH GH GH strain rate and creep strain rate, respectively. The creep strain rate e can be obtained by extending GH the Haasen}Sumino model [2,3] to a multiaxial stress state in the same way as the isotropic #ow theory of plasticity. Then it can be written as [7,8]
(1)
(3)
(4) (5)
In the above equation, p , S and J are stresses, GH GH deviatoric stresses and the second invariant of the deviatoric stress, respectively. Dislocation density is denoted by N , and its rate is given by
NQ "Kk N e\/ I2((J !D(N !q )N
;((J !D(N )H. (6)
In Eqs. (3) and (6), NQ and e are set equal to zero K GH when (J !D(N !q 60. From Eq. (6), N is
always equal to zero when the initial dislocation density N is zero, so it is necessary to assume
non-zero value of N . In the analysis of ingot
annealing, the dislocation density existing in an as-grown GaAs single-crystal ingot should be used as the value of N . We used an N value of
1.0;10 m\ in the present analyses. Furthermore, q is a drag-stress caused by the interaction be tween dislocations and impurity atoms and set equal to the following equation: log q "4.83#1382/¹,
(7)
N. Miyazaki et al. / Journal of Crystal Growth 216 (2000) 6}14 Table 1 Parameters in Haasen}Sumino model and their values for GaAs single crystal N
B Q k ¹ K D k q p j
density of moving dislocation Magnitude of Burgers vector Peierls potential Boltzmann's constant absolute temperature material constant strain hardening factor material constant drag-stress material constant material constant
m\ 4.0;10\ m 1.5 eV 8.617;10\ eV/K K 7.0;10\ m/N 3.1 N/m 1.80;10\ mN>/NNs Pa 1.7 1.0
which was given by Jordan [6] for a Si-doped GaAs single crystal. Table 1 summarizes the parameters in the Haasen}Sumino model and their values for a GaAs single crystal [9]. In addition to the material parameters for the Haasen}Sumino model, we require the thermal expansion coe$cient a, and the elastic constants C for a GaAs single crystal. They are given by [6] GH E Thermal expansion coe$cient a"4.68;10\#3.82;10\¹.
(8)
E Elastic constants C GH
9
where K, *U, *F, *F and *R denote a sti!ness matrix, an incremental nodal displacement vector, an incremental load vector due to thermal strain, an incremental load vector due to creep strain and a residual load vector, respectively. Solving Eq. (10) at each time step and integrating Eq. (6), we obtain state variables such as displacements, strains, stresses and dislocation density.
3. Results and discussion 3.1. Heat conduction analysis The time variations of temperature distribution in a GaAs ingot were obtained for Cases 1}4 described in Section 2.3. Fig. 4 shows the time variations of the maximum temperature di!erence in the ingot. Temperature distributions in the ingot for Cases 2 and 4 are shown in Fig. 5 at 60 s after annealing starts and at the time when the temperature reaches to ¹ . We can "nd the e!ect of
temperature dependence of thermal conductivity k and speci"c heat C . As shown in Eq. (1), k N decreases and C increases with the increase of N temperature, so thermal di!usivity i("k/C o) deN creases with the increase of temperature. This means that heat does not move well and temperature di!erence in the ingot becomes large when
C "1.217;10!1.44;10¹, C "0.546;10!0.64;10¹, (9) C "0.618;10!0.70;10¹. The units of a, C and ¹ are K\, Pa and K, GH respectively. In the present analyses, a single crystal ingot was assumed to be isotropic, so only Young's modulus and Poisson's ratio in the +1 1 1, plane calculated from C , C and C [10] were used because they are invariant in this plane. The "nite element method for an elastic creep problem is applied to obtain the time variations of dislocation density. The incremental form of a "nite element equilibrium equation is given as follows: K*U"*F#*F#*R,
(10)
Fig. 4. Time variations of maximum temperature di!erence in GaAs ingots.
10
N. Miyazaki et al. / Journal of Crystal Growth 216 (2000) 6}14
Fig. 5. Temperature distributions in GaAs ingots (left: 4-in diameter GaAs ingot, right: 6-in diameter GaAs ingot).
N. Miyazaki et al. / Journal of Crystal Growth 216 (2000) 6}14
11
Fig. 7. Time variations of dislocation density (Case 4). Fig. 6. Time variations of equivalent shear stress (J (Case 4).
temperature becomes high. Such an increase in temperature di!erence leads to the increase in thermal stress in the ingot which causes the multiplication of dislocations. The temperature di!erence becomes maximum at the end of the heating-up process, and it becomes larger as the rate of temperature change d¹/dt and the ingot diameter become larger. 3.2. Dislocation density analysis Distributions of the dislocation density and stresses in the GaAs ingot and their time variations were obtained using the temperature data obtained in Section 3.1. For Case 4, time variations of equivalent shear stress (J due to thermal stress and q at the observation point depicted in Fig. 1 are shown in Fig. 6 and that of the dislocation density at the same point is shown in Fig. 7. During the heating-up process, temperature gradient in the ingot increases due to the temperature dependence of the thermal material properties, so thermal stress or (J increases during the heating-up process. Around 10 000 s, (J exceeds a threshold value for dislocation multiplication and the dislocation density increases after 10 000 s as shown in Fig. 7. Exactly speaking, the threshold value is q #D(N , but it can be appro
ximated by q because D(N is much smaller
than q . After exceeding the threshold value,
(J starts decreasing due to the e!ect of stress relaxation caused by dislocation multiplication. When kept at the maximum temperature, the stress which relaxed during the heating-up process remains in the ingot as residual stress. The direction of the residual shear stress is reverse to that of the shear stress during the heating-up process. The equivalent shear stress (J should be a positive value. Because of these reasons we can "nd a turnup of (J at the time of 18 000 s in Fig. 6. The equivalent shear stress (J exceeds q during the turnup, and the dislocation density jumps up at the time of 18 000 s, as shown in Fig. 7. At the beginning of the cooling-down process, (J in creases due to the increase of thermal stress induced by the increase of temperature gradient in the ingot, and the dislocation density jumps up again. During the cooling-down process, (J decreases and a turnup of (J caused by a change in the direc tion of the equivalent shear stress is also found. (J changes from a decrease to an increase at the turnup point, and residual stress again remains in the ingot after annealing process. The same behaviors can be seen in the equivalent shear stress and the dislocation density for Cases 1}3. Figs. 8 and 9 show the distributions of the equivalent shear stress (J and the dislocation density after ingot annealing, respectively. We can regard Fig. 8 as the results of residual stress distributions.
12
N. Miyazaki et al. / Journal of Crystal Growth 216 (2000) 6}14
Fig. 8. Distributions of equivalent shear stress (J after ingot annealing (left: 4-in diameter GaAs ingot, right: 6-in diameter GaAs ingot).
N. Miyazaki et al. / Journal of Crystal Growth 216 (2000) 6}14
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Fig. 9. Distributions of dislocation density after ingot annealing (left: 4-in diameter GaAs ingot, right: 6-in diameter GaAs ingot).
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If creep strain does not occur during the ingot annealing process, there exists only an elastic strain and no residual stress occurs. When large dislocation multiplication causes large stress relaxation, large residual stress remains in the ingot after annealing process. We can con"rm such a relation between the residual stress and the dislocation density by making a comparison between Figs. 8 and 9. As for the e!ect of ingot diameter, the 6-in diameter ingot has a higher dislocation density than the 4-in diameter ingot. It is found from a comparison between Cases 1 and 2 or Cases 3 and 4 that the larger the rate of temperature change d¹/dt is, the higher the dislocation density is. On the other hand, the e!ect of the maximum temperature ¹ is some what complicated. Although it is not clear in the case of d¹/dt of 100 K/h, we can "nd the following in the case of d¹/dt of 200 K/h by making a comparison between Cases 2 and 4, where Case 2 has lower ¹ than Case 4. Case 2 has a larger peak
value of dislocation density but a smaller area of high dislocation density than Case 4. So Case 4, which has higher ¹ , has a higher dislocation
density than Case 2 as a whole.
4. Concluding remarks A new computer code was developed for calculating dislocation density during annealing of a single-crystal ingot. We applied this computer
code to a GaAs ingot annealing process and showed that this computer code can be successfully applied to obtain the time variations of dislocation density and stresses in the ingot and also residual stress. It is expected that this computer code can be applied to an optimization of the annealing process of a single-crystal ingot.
Acknowledgements This study was partially supported by a Grantin-Aid for Science Research from the Ministry of Education and Culture.
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