- Email: [email protected]
Investigation of the role of the crystal growth zone during silicon carbide crystal growth by the sublimation method
I4AIIIIi5 IIORI&
g ELSEVIER
Materials Science and Engineering B27 (1994) 69-72
Investigation of the role of the crystal growth zone during silicon carbide crystal growth by the sublimation method S.K. Lilov Department of Semiconductor Physics, Faculty of Physics, University of Sofia, Boulevard J. Bourchier 5, 1126 Sofia, Bulgaria Received 6 September 1993
Abstract The dependence of the growth rate of SiC crystals on the diameter of the graphite crystallization sleeve (GCS) forming the crystal growth zone has been investigated. It has been established that at a given crucible construction there is an exactly determined diameter of the GCS for growing of SiC crystals with maximum sizes. Keywords: Semiconductors; Crystallization; Silicon carbide; Evaporation
I. Introduction
Silicon carbide (SIC) is one of the most promising semiconductor materials which are suitable for the creation of the devices capable of operating at temperatures up to 500-700 °C and in conditions of high radiation. Owing to the favourable combination of its optical and electrical properties, at present a number of unique semiconductor devices based on SiC have been created. They have found wide application in various fields of m o d e m electronics, such as quantum, power and microwave electronics, as well as in nuclear and space engineering etc. [1]. There is no doubt that the wide field of utilization of the semiconductor properties of SiC will be enlarged after introducing the highly effective methods for producing SiC crystals of large sizes, high purity and perfect structure. One of the methods for SiC crystal growth most useful in this respect is the method of sublimation. However, insufficient investigations on the growth from the vapour phase of SiC crystals by this method have not provided the possibility up to now of determining all the factors which influence the crystal growth process. Thus for example in the literature there are no data about the influence of the size (in particular the diameter) of the crystal growth zone for growth of SiC crystals on the size and growth rate of the SiC crystals produced. The availability of such data 0921-5107/94/$7.00 © 1994 - Elsevier Science S.A. All rights reserved SSD10921-5107(94)01108-T
would provide the possibility of optimizing the construction of the crucible utilized for growing the SiC crystals. The purpose of the present work is to carry out such an investigation.
2. Experimental details
The single crystals of SiC were grown in a cylindrical graphite crucible (Fig. 1) by the sublimation technique
:1.
2
/ \
/
f
3 4
-
-
i
,5
r/lllllllllllllllllllll
6
Fig. 1. Construction of the graphite crucible for the growth of SiC single crystals: 1, top lid; 2, crystal growth zone; 3, polycrystalline SiC source; 4, graphite crystallization sleeve (GCS); 5, growing crystals; 6, bottom lid.
70
S. K. Lilov
/
Materials Science and Engineering B27 (19941 :~9- 72
first suggested by Lely [2]. The essence of the method consists of the decomposition of the polycrystalline mass of SiC at a temperature of 2300-2600 °C and subsequent crystallization in the form of platelets in the region with a lower temperature. A thin-walled graphite crystallization sleeve (GCS) with holes forms the crystal growth zone and it is simultaneously a substrate for growing the crystals. Powdered SiC synthesized from pure Si and C is used as a starting material. The crucible is heated in a cylindrical resistance oven using a graphite heater. Crystals are grown in an inert gas atmosphere (most frequently argon) at 2400-2600 °C. The nucleation of crystals takes place in the growth zone, mainly in the holes of the GCS. The crystals represent mainly hexagonal platelets with one natural mirror-like smooth face (0001). The other (000i) face is usually "melted" and has a less perfect surface. In many crystals it is even stepped. Some of the crystals obtained have the form of flat-parallel platelets, usually from in the middle part of the crucible. The naturally mirror-like smooth faces of the crystals are oriented in such a way that the normal to their surfaces indicates the coldest places of the crucible: the centre of one of the lids.
where t(s) is the time of the whole growth process and V,, (m s ~) is the growth rate of the crystal along the crucible radius. In order to determine the time t for the whole growth process, i.e. the time for decomposition of the whole layer of the polycrystalline SiC source, it is necessary to know the rate of decomposition of this layer, i.e. the rate of the alteration (decrease) in the diameter of this layer. As a result of the investigation it was established that at a given temperature this rate Vd is a constant quantity. It has also been established that this rate is an exponential function of the temperature evaporation, i.e. Va =
A exp -
According to Ref. [3], in the polycrystalline SiC source (Fig. 1 ) a temperature gradient is created. In this way, on increase in the diameter of the GCS, the temperature of its wall and the pressure of the SiC vapours above it are increased. Calculations of the temperature of the crystal growing in the middle part of the GCS have shown that it is approximately equal to half the sum of the temperatures of the hottest (the middle part of GCS) and the coldest (the centre of the crucible lid) points of the crystal growth zone. A simple calculation shows that, if the temperature of the wall of the GCS increases by A ~C, the temperature of the growing crystal will increase by A T/2 °C. The supersaturation and the crystal growth rate will increase on increase in the radius of the GCS because the supersaturation above the growing crystal is determined by the difference between the temperatures of the wall of the GCS and the crystal [4]. The thickness of the layer of the source polycrystalline SiC and the time of the whole growth process will decrease. From this it follows that, at some diameter of the GCS, SiC crystals should be obtained with a maximum size for a given volume of the crucible. The size of the crystal can be presented in the following way: L = t V , , m s -j
(1)
(2)
where A E (J mol-l) is the evaporation energy of the source SiC, R (J mol I K -j) is the universal gas constant, T(K) is the evaporation temperature and A is a parameter which depends on the granulometry and the porosity of the grains of the source SiC. Eq. (2) can be also written as Va=Aexp
3. Results and discussion
ms
-
ms
(3)
where W (J) is the evaporation energy of SiC molecule and k (J k ') is the Boltzmann constant. Taking into account the above mentioned equations, it follows that R--F
t=--
s
(4)
Vd
where R (m) is the outside radius of the layer of the source polycrystalline SiC and r (m) is the radius of the GCS. According to Refs. [5] and [6] the normal growth rate V,1 by vapour deposition of a crystal surface bearing a screw dislocation is given by Vn=flf2novexp
_
a tanh a l a~
ms-t
(5)
\al
where a l =2:WM/2spRT is the intermediate supersaturation, fl is the multiplier taking into account the possibility for insufficiently quick exchange between the molecules of the step and the adsorbed layer on the crystal surface, f2 (m 3) is the volume of the molecule, no (m -2) is the number of the molecules on the unit of area, ?, (J m -2) is the specific free surface energy, v ( -~ 1013 s- 1) is the frequency multiplier, a is the supersaturation, M (kg mol- ') is the molecular weight, p (kg m 3) is the crystal density and 22 (m) is the mean free path of surface diffusion.
S. K. Lilov
/
Materials Science and Engineering B27 (1994) 69- 72
At low supersaturations (a "~ a l) the dependence of the growth rate on the supersaturation is a parabolic function:
Vn-- fifth°a2 exp -
(6)
aj
At high supersaturations (5 >>a l) the dependence is linear:
Vn=fl~novaexp
-
(7)
To carry out the calculation it is necessary to know the specific free surface energy 7. There are no experimental investigations on its determination for SiC. There has been only one attempt to propose a theory for the calculation of 7 for covalent crystals with a small ionic part of the chemical bond [7]. Ormont has approached this problem from the thermodynamic point of view, taking into account the peculiarities of the structure of the substance. The following formula was obtained [7] for the specific total free energy ehklm:
ehklm=694"34
6(C-x)
mnC
a2 ergcm
-2
(8)
where 2 " (kcal mo1-1) is the atomization energy, C is the coordination number of the atom, n is the number of atoms in the formula of the substance, a (A) is the lattice parameter, ma 2 (A2) is the area of the primitive cell in the (hklm) crystallographic plane, 6 is the multiplicity of the primitive cell in the (hklm) crystallographic plane and C-x is the number of broken bonds of one atom on the (hklm) crystallographic plane. As is known the specific free surface energy 7hk~mis determined from the Gibbs-Helmholtz equation [7]
7hklm= 6hklm+ T 07hklm OT
(9)
i.e. 7hkhn and Ehkl,n differ from one another by the entropy term T(OThkhn/OT). Evaluation of the entropy term for binary nitrides and carbides [7] has shown that the upper limit for it does not exceed ehkh,~by 5%. According to the circular process [7, 8], 2 " = Qsic + Ssi + Sc
(10)
where Qsic (kcal mol -l) is the heat of formation of hexagonal SiC, and Ssi (keal (g-atom)-1) and Sc (kcal (g-atom)-1) are the energies of sublimation of silicon and carbon respectively. According to [9-11], Qsic=15.6 keal mo1-1, Ssi=102 keal (g-atom) -j and S c = 1 7 0 kcal (g-
71
atom)-J. Then according to Eq. (10) the atomization energy is obtained as f~* = 287.6 kcal mol-1. Substituting all necessary data in Eq. (8) and introducing the correction for the entropy term we obtain 7hkZm= 2.888 jm-2. According to the calculations of Sears [12], ~.~= a exp
(11)
Using our data for AE [4], 7hkh,,=2.888 J m-:, T = 2 8 0 0 K , we obtain 2 s = 1 9 . 3 x 1 0 -~ m and al = 5 x 10 -4. In [4] it is shown that in the crystal growth zone above the crystals there is a supersaturation a ~ ( 3 - 5 ) x 1 0 -3. In this way it has been established that a ,> 51 and in our conditions for SiC crystal growth the dependence of the normal crystal growth rate of the face (0001) on the supersaturation is linear. In [13, 14] it has been shown that the ratio of the growth rates of the crystal along the c and a directions is constant in a narrow temperature interval and depends mainly on the axial and radial temperature gradients in the crucible. Let this ratio equal B. Therefore the growth rate of the crystal along the a direction, i.e. along the radius of the crucible, will also be a linear function of the supersaturation in the crystal growth zone and can be written according to Eq. (7) as follows:
V.= BV.= Da exp ( - ~ )
(12)
where
D =Bflf2nov
(13)
Let us introduce the following symbols: T~ is the temperature of the outside part of the layer of the source polycrystaUine SiC; TO = [T~ - ( R - r) 0 T/Or] is the temperature of the wall of the GCS in the middle part of the crucible: OT/Or is the temperature gradient in the layer of the polycrystalline SiC source in the middle part of the crucible; T 2 is the temperature of the crystal growing in the middle part of the crucible; T~ is the temperature in the centre of the lid of the crucible. According to the above, the temperature of the crystal is equal to
T, = 0.5 (T,-(R-r)O---~fr+ T~)
(14)
Then the temperature difference between the wall of the GCS and the crystal will be
72
X K. Lilov /
MaterialsScience and Engineering B27 (1994) 69-72
AT=T,-T,
-- T , - ( R - r )
-0.5
TI-(R-r)~+
7"1- T~ 0.5(R_r.OT)
-aTr
The analysis of Eq. (19) shows that it is a parabolic function of r and at s o m e rop t this function has a maximum. Putting OL/Or equal to zero and carrying out the necessary transformations the following expression for ropt has been obtained:
T3
(Is)
The supersaturation above the growing crystal according to [4] and in conformity with the accepted symbols is equal to 6~
P2
P(I -
P0 where P0 and P2 are the pressures of SiC vapours corresponding to the temperatures T0 and T2 and can be written as follows: p , = A* exp( - W ]
,,7)
In [4, 14, 15] it is shown that the optimum conditions for the growth of SiC perfect crystals are when the temperature gradient is a minimum. Usually it is equal to several degrees Celsius per centimetre [3]. That is why the temperatures T0 and T2 differ very little from one another. Taking this into account and expanding the exponents in Eqs. (16) and (17) in a Maclauren series the following expression for a has been obtained: W a =AT--
(18)
kT 2
Because the temperatures T0, T~, T2 and 7"3 differ very little from one another in comparison with their absolute values let us replace T with T1 in Eqs. (3), (12) and (18) for convenience. Putting (18) into (12), (3) into (4), and (12) and (4)into (1)the following expression for the size of the grown crystal can be obtained: R-r D W ( T , - T 3 0.5(R r) OT) A exp~ 7W/kTl) kT, 2 2 - Or]
exp - ~
=A~z(R-r)
(20)
In this way the above analysis has shown indeed that the diameter of the GCS plays a significant role during the growth process of SiC single crystals. On the basis of this analysis it has been established that crystals with maximum sizes from the given crucible construction can be obtained at some determined value of the diameter of the GCS.
(16)
exp( )
L
T~r,,pt = R - 20T/Or
2
or/
(19)
Acknowledgment This work has been performed with the financial support of the Bulgarian Ministry of Science and High Education under Contract 277.
References [1] Yu. M. Tairov and V. E Tsvetkov, Trudi H Vses. Soveshch. po Shirokozonnim Poluprovodnikam (Proc. 2nd USSR Conf. on Wide-Gap Semiconductors), Nauka, Leningrad, 1980, p. 122. [2] J.A. Lely, Ber. Dtsch. Keram. Ges.,32(1955)229. [3] Yu. M. Tairov, Silicon Carbide, Naukova Dumka, Kiev, 1966, p. 189. [4] S. K. Lilov and I. Y. Yanchev, Cryst. Res. Technol., 28 (1993) 495. [5] W. Burton, N. Cabrera and F. Frank, Philos. Trans. Roy'. Soc. London, Ser. A,243 ( 1951 ) 299. [6] A.A. Chernov, Usp. Fiz. Nauk, 73(1961) 279. [7] B. E Ormont, Dokl. Akad. Nauk SSSR, 106 (1956) 687. [8] B. F. Ormont, Zh. Fiz. Khim., 31 (1957) 509. [9] W. Knippenbergand G. Verspui, Mater. Res. Bul., 4(1969) 45. [10] R. Rein and J. Chipman, J. Phys. Chem., 67(1963) 839. [11] W. Good and J. Lacina, J. Phys. Chem., 68(1964) 579. [12] G.W. Sears, J. Chem. Phys., 25(1956) 154. [13] D. Hamilton, Silicon Carbide, Pergamon, Oxford, 1960, p. 43. [14] k. Kroko, J. Electrochem. Soc., 113 (1966) 301. [15] W. Knippenberg, PhilipsRes. Reps., 18(1963) 161.
g ELSEVIER
Materials Science and Engineering B27 (1994) 69-72
Investigation of the role of the crystal growth zone during silicon carbide crystal growth by the sublimation method S.K. Lilov Department of Semiconductor Physics, Faculty of Physics, University of Sofia, Boulevard J. Bourchier 5, 1126 Sofia, Bulgaria Received 6 September 1993
Abstract The dependence of the growth rate of SiC crystals on the diameter of the graphite crystallization sleeve (GCS) forming the crystal growth zone has been investigated. It has been established that at a given crucible construction there is an exactly determined diameter of the GCS for growing of SiC crystals with maximum sizes. Keywords: Semiconductors; Crystallization; Silicon carbide; Evaporation
I. Introduction
Silicon carbide (SIC) is one of the most promising semiconductor materials which are suitable for the creation of the devices capable of operating at temperatures up to 500-700 °C and in conditions of high radiation. Owing to the favourable combination of its optical and electrical properties, at present a number of unique semiconductor devices based on SiC have been created. They have found wide application in various fields of m o d e m electronics, such as quantum, power and microwave electronics, as well as in nuclear and space engineering etc. [1]. There is no doubt that the wide field of utilization of the semiconductor properties of SiC will be enlarged after introducing the highly effective methods for producing SiC crystals of large sizes, high purity and perfect structure. One of the methods for SiC crystal growth most useful in this respect is the method of sublimation. However, insufficient investigations on the growth from the vapour phase of SiC crystals by this method have not provided the possibility up to now of determining all the factors which influence the crystal growth process. Thus for example in the literature there are no data about the influence of the size (in particular the diameter) of the crystal growth zone for growth of SiC crystals on the size and growth rate of the SiC crystals produced. The availability of such data 0921-5107/94/$7.00 © 1994 - Elsevier Science S.A. All rights reserved SSD10921-5107(94)01108-T
would provide the possibility of optimizing the construction of the crucible utilized for growing the SiC crystals. The purpose of the present work is to carry out such an investigation.
2. Experimental details
The single crystals of SiC were grown in a cylindrical graphite crucible (Fig. 1) by the sublimation technique
:1.
2
/ \
/
f
3 4
-
-
i
,5
r/lllllllllllllllllllll
6
Fig. 1. Construction of the graphite crucible for the growth of SiC single crystals: 1, top lid; 2, crystal growth zone; 3, polycrystalline SiC source; 4, graphite crystallization sleeve (GCS); 5, growing crystals; 6, bottom lid.
70
S. K. Lilov
/
Materials Science and Engineering B27 (19941 :~9- 72
first suggested by Lely [2]. The essence of the method consists of the decomposition of the polycrystalline mass of SiC at a temperature of 2300-2600 °C and subsequent crystallization in the form of platelets in the region with a lower temperature. A thin-walled graphite crystallization sleeve (GCS) with holes forms the crystal growth zone and it is simultaneously a substrate for growing the crystals. Powdered SiC synthesized from pure Si and C is used as a starting material. The crucible is heated in a cylindrical resistance oven using a graphite heater. Crystals are grown in an inert gas atmosphere (most frequently argon) at 2400-2600 °C. The nucleation of crystals takes place in the growth zone, mainly in the holes of the GCS. The crystals represent mainly hexagonal platelets with one natural mirror-like smooth face (0001). The other (000i) face is usually "melted" and has a less perfect surface. In many crystals it is even stepped. Some of the crystals obtained have the form of flat-parallel platelets, usually from in the middle part of the crucible. The naturally mirror-like smooth faces of the crystals are oriented in such a way that the normal to their surfaces indicates the coldest places of the crucible: the centre of one of the lids.
where t(s) is the time of the whole growth process and V,, (m s ~) is the growth rate of the crystal along the crucible radius. In order to determine the time t for the whole growth process, i.e. the time for decomposition of the whole layer of the polycrystalline SiC source, it is necessary to know the rate of decomposition of this layer, i.e. the rate of the alteration (decrease) in the diameter of this layer. As a result of the investigation it was established that at a given temperature this rate Vd is a constant quantity. It has also been established that this rate is an exponential function of the temperature evaporation, i.e. Va =
A exp -
According to Ref. [3], in the polycrystalline SiC source (Fig. 1 ) a temperature gradient is created. In this way, on increase in the diameter of the GCS, the temperature of its wall and the pressure of the SiC vapours above it are increased. Calculations of the temperature of the crystal growing in the middle part of the GCS have shown that it is approximately equal to half the sum of the temperatures of the hottest (the middle part of GCS) and the coldest (the centre of the crucible lid) points of the crystal growth zone. A simple calculation shows that, if the temperature of the wall of the GCS increases by A ~C, the temperature of the growing crystal will increase by A T/2 °C. The supersaturation and the crystal growth rate will increase on increase in the radius of the GCS because the supersaturation above the growing crystal is determined by the difference between the temperatures of the wall of the GCS and the crystal [4]. The thickness of the layer of the source polycrystalline SiC and the time of the whole growth process will decrease. From this it follows that, at some diameter of the GCS, SiC crystals should be obtained with a maximum size for a given volume of the crucible. The size of the crystal can be presented in the following way: L = t V , , m s -j
(1)
(2)
where A E (J mol-l) is the evaporation energy of the source SiC, R (J mol I K -j) is the universal gas constant, T(K) is the evaporation temperature and A is a parameter which depends on the granulometry and the porosity of the grains of the source SiC. Eq. (2) can be also written as Va=Aexp
3. Results and discussion
ms
-
ms
(3)
where W (J) is the evaporation energy of SiC molecule and k (J k ') is the Boltzmann constant. Taking into account the above mentioned equations, it follows that R--F
t=--
s
(4)
Vd
where R (m) is the outside radius of the layer of the source polycrystalline SiC and r (m) is the radius of the GCS. According to Refs. [5] and [6] the normal growth rate V,1 by vapour deposition of a crystal surface bearing a screw dislocation is given by Vn=flf2novexp
_
a tanh a l a~
ms-t
(5)
\al
where a l =2:WM/2spRT is the intermediate supersaturation, fl is the multiplier taking into account the possibility for insufficiently quick exchange between the molecules of the step and the adsorbed layer on the crystal surface, f2 (m 3) is the volume of the molecule, no (m -2) is the number of the molecules on the unit of area, ?, (J m -2) is the specific free surface energy, v ( -~ 1013 s- 1) is the frequency multiplier, a is the supersaturation, M (kg mol- ') is the molecular weight, p (kg m 3) is the crystal density and 22 (m) is the mean free path of surface diffusion.
S. K. Lilov
/
Materials Science and Engineering B27 (1994) 69- 72
At low supersaturations (a "~ a l) the dependence of the growth rate on the supersaturation is a parabolic function:
Vn-- fifth°a2 exp -
(6)
aj
At high supersaturations (5 >>a l) the dependence is linear:
Vn=fl~novaexp
-
(7)
To carry out the calculation it is necessary to know the specific free surface energy 7. There are no experimental investigations on its determination for SiC. There has been only one attempt to propose a theory for the calculation of 7 for covalent crystals with a small ionic part of the chemical bond [7]. Ormont has approached this problem from the thermodynamic point of view, taking into account the peculiarities of the structure of the substance. The following formula was obtained [7] for the specific total free energy ehklm:
ehklm=694"34
6(C-x)
mnC
a2 ergcm
-2
(8)
where 2 " (kcal mo1-1) is the atomization energy, C is the coordination number of the atom, n is the number of atoms in the formula of the substance, a (A) is the lattice parameter, ma 2 (A2) is the area of the primitive cell in the (hklm) crystallographic plane, 6 is the multiplicity of the primitive cell in the (hklm) crystallographic plane and C-x is the number of broken bonds of one atom on the (hklm) crystallographic plane. As is known the specific free surface energy 7hk~mis determined from the Gibbs-Helmholtz equation [7]
7hklm= 6hklm+ T 07hklm OT
(9)
i.e. 7hkhn and Ehkl,n differ from one another by the entropy term T(OThkhn/OT). Evaluation of the entropy term for binary nitrides and carbides [7] has shown that the upper limit for it does not exceed ehkh,~by 5%. According to the circular process [7, 8], 2 " = Qsic + Ssi + Sc
(10)
where Qsic (kcal mol -l) is the heat of formation of hexagonal SiC, and Ssi (keal (g-atom)-1) and Sc (kcal (g-atom)-1) are the energies of sublimation of silicon and carbon respectively. According to [9-11], Qsic=15.6 keal mo1-1, Ssi=102 keal (g-atom) -j and S c = 1 7 0 kcal (g-
71
atom)-J. Then according to Eq. (10) the atomization energy is obtained as f~* = 287.6 kcal mol-1. Substituting all necessary data in Eq. (8) and introducing the correction for the entropy term we obtain 7hkZm= 2.888 jm-2. According to the calculations of Sears [12], ~.~= a exp
(11)
Using our data for AE [4], 7hkh,,=2.888 J m-:, T = 2 8 0 0 K , we obtain 2 s = 1 9 . 3 x 1 0 -~ m and al = 5 x 10 -4. In [4] it is shown that in the crystal growth zone above the crystals there is a supersaturation a ~ ( 3 - 5 ) x 1 0 -3. In this way it has been established that a ,> 51 and in our conditions for SiC crystal growth the dependence of the normal crystal growth rate of the face (0001) on the supersaturation is linear. In [13, 14] it has been shown that the ratio of the growth rates of the crystal along the c and a directions is constant in a narrow temperature interval and depends mainly on the axial and radial temperature gradients in the crucible. Let this ratio equal B. Therefore the growth rate of the crystal along the a direction, i.e. along the radius of the crucible, will also be a linear function of the supersaturation in the crystal growth zone and can be written according to Eq. (7) as follows:
V.= BV.= Da exp ( - ~ )
(12)
where
D =Bflf2nov
(13)
Let us introduce the following symbols: T~ is the temperature of the outside part of the layer of the source polycrystaUine SiC; TO = [T~ - ( R - r) 0 T/Or] is the temperature of the wall of the GCS in the middle part of the crucible: OT/Or is the temperature gradient in the layer of the polycrystalline SiC source in the middle part of the crucible; T 2 is the temperature of the crystal growing in the middle part of the crucible; T~ is the temperature in the centre of the lid of the crucible. According to the above, the temperature of the crystal is equal to
T, = 0.5 (T,-(R-r)O---~fr+ T~)
(14)
Then the temperature difference between the wall of the GCS and the crystal will be
72
X K. Lilov /
MaterialsScience and Engineering B27 (1994) 69-72
AT=T,-T,
-- T , - ( R - r )
-0.5
TI-(R-r)~+
7"1- T~ 0.5(R_r.OT)
-aTr
The analysis of Eq. (19) shows that it is a parabolic function of r and at s o m e rop t this function has a maximum. Putting OL/Or equal to zero and carrying out the necessary transformations the following expression for ropt has been obtained:
T3
(Is)
The supersaturation above the growing crystal according to [4] and in conformity with the accepted symbols is equal to 6~
P2
P(I -
P0 where P0 and P2 are the pressures of SiC vapours corresponding to the temperatures T0 and T2 and can be written as follows: p , = A* exp( - W ]
,,7)
In [4, 14, 15] it is shown that the optimum conditions for the growth of SiC perfect crystals are when the temperature gradient is a minimum. Usually it is equal to several degrees Celsius per centimetre [3]. That is why the temperatures T0 and T2 differ very little from one another. Taking this into account and expanding the exponents in Eqs. (16) and (17) in a Maclauren series the following expression for a has been obtained: W a =AT--
(18)
kT 2
Because the temperatures T0, T~, T2 and 7"3 differ very little from one another in comparison with their absolute values let us replace T with T1 in Eqs. (3), (12) and (18) for convenience. Putting (18) into (12), (3) into (4), and (12) and (4)into (1)the following expression for the size of the grown crystal can be obtained: R-r D W ( T , - T 3 0.5(R r) OT) A exp~ 7W/kTl) kT, 2 2 - Or]
exp - ~
=A~z(R-r)
(20)
In this way the above analysis has shown indeed that the diameter of the GCS plays a significant role during the growth process of SiC single crystals. On the basis of this analysis it has been established that crystals with maximum sizes from the given crucible construction can be obtained at some determined value of the diameter of the GCS.
(16)
exp( )
L
T~r,,pt = R - 20T/Or
2
or/
(19)
Acknowledgment This work has been performed with the financial support of the Bulgarian Ministry of Science and High Education under Contract 277.
References [1] Yu. M. Tairov and V. E Tsvetkov, Trudi H Vses. Soveshch. po Shirokozonnim Poluprovodnikam (Proc. 2nd USSR Conf. on Wide-Gap Semiconductors), Nauka, Leningrad, 1980, p. 122. [2] J.A. Lely, Ber. Dtsch. Keram. Ges.,32(1955)229. [3] Yu. M. Tairov, Silicon Carbide, Naukova Dumka, Kiev, 1966, p. 189. [4] S. K. Lilov and I. Y. Yanchev, Cryst. Res. Technol., 28 (1993) 495. [5] W. Burton, N. Cabrera and F. Frank, Philos. Trans. Roy'. Soc. London, Ser. A,243 ( 1951 ) 299. [6] A.A. Chernov, Usp. Fiz. Nauk, 73(1961) 279. [7] B. E Ormont, Dokl. Akad. Nauk SSSR, 106 (1956) 687. [8] B. F. Ormont, Zh. Fiz. Khim., 31 (1957) 509. [9] W. Knippenbergand G. Verspui, Mater. Res. Bul., 4(1969) 45. [10] R. Rein and J. Chipman, J. Phys. Chem., 67(1963) 839. [11] W. Good and J. Lacina, J. Phys. Chem., 68(1964) 579. [12] G.W. Sears, J. Chem. Phys., 25(1956) 154. [13] D. Hamilton, Silicon Carbide, Pergamon, Oxford, 1960, p. 43. [14] k. Kroko, J. Electrochem. Soc., 113 (1966) 301. [15] W. Knippenberg, PhilipsRes. Reps., 18(1963) 161.