- Email: [email protected]
Computation of stress in Bridgman crystals
Journal of Crystal Growth 69 (1984) 275—280 North-Holland, Amsterdam
275
COMPUTATION OF STRESS IN BRIDGMAN CRYSTALS C.E. HUANG
“,
D. ELWELL and R.S. FEIGELSON
Center for Materials Research, Stanford Unirersity, Stanford, California 94305, USA
Received 25 September 1984
Finite element analysis has been used to calculate the stress distribution in crystals growing by the Bridgman method. It has been found that the temperature distribution, and particularly changes in temperature gradient, are the main factors governing the maximum thermally induced stress. The solid—liquid interface shape is a secondary factor but a concave interface is associated with the highest stress in a uniform temperature gradient. The application of an external stress during growth is also discussed.
1. Introduction Stress is a general problem in crystal growth, and excessive stress will result in dislocation formation or even in cracking if the elastic limit is exceeded. The ideal of a stress-free crystal cannot be realized in practice since crystals are normally grown in a temperature gradient. A reasonable experimental goal is therefore to lower the stress to an acceptable level, Several authors [1—6]have contributed towards an understanding of the origin of stress in crystals, especially those grown by the Czochralski method. A good review of these studies has been presented by Brice [3]. Finite element analysis has been shown to be a powerful method for the study of the temperature distribution in Bridgman systems [7—10].The purpose of this paper is to extend this method to calculate the stress in Bridgman-grown crystals. The results of these calculations will be compared with the earlier theories and with practical daia.
assumed and the potential energy is then minimized to obtain the nodal values of the displacements. Once the displacements are known, it is possible to solve for the strains and stress [11]. The important requirement is that the displacement equations selected must satisfy the boundary conditions. The total potential energy ir of an elastic system consists of two parts, a component resulting from the strain energy A in the body and a component W related to the potential energy of the internal and applied loads, i.e. A+ ~ —
~
—
For a volume V the total strain energy is obtamed by integrating over the constituent elements, giving A
=
f ~( ( ~} Tf a }
—
{ ~o } T{ a } )dV.
(2)
V
In solving problems in the theory of elasticity by finite element analysis, a displacement field is
Here the column vector { r) is the total strain which is related to displacements, and column vector { e~} is an initial strain, which is related to the temperature distribution in the crystal. The stress component a and the strain component are assumed to obey Hooke’s law, so that { a) [D] (~) [D] { ~ (3)
Permanent address: Research Institute of Synthetic Crystals, P.O. Box 733, Beijing, People’s Rep. of China,
where [D] is the property matrix of the material, which is related to Young’s modulus E and Pois-
2. Theoretical analysis
*
=
0022-0248/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
—
276
C.E. Huang et al.
/
Computation of Stress in Bridgman crystals
son’s ratio ~i. In this paper our attention is mainly focussed on Bridgman crystal growth. The radius of the crystals was varied from 0.5 to 25 mm, and the height kept at a value 8 times the radius. For all crystal sizes, the domain was divided into 2485 elements, which were defined by 1346 nodes in the finite element analysis. Thus eq. (1) should be rewritten in summation form as 2485 1T
~ (A~”1—w~’~). e~1
(4)
{ a)
for each element, we
After solving for
2
P
~
g 1
calculated the principal stresses a~,a~and a1 with, as is usual, a~> a2 > a1. Then the largest shear stress Tn~a,(was obtained using ~ ~(a~ a1). The computer program designed for our earlier work [7,8] was used to calculate the temperature distribution inside the sample, and the results were then used here to calculate the thermal stress. Since thermal convection was neglected in our calculations, the temperature profile and hence the stress data will be valid only in systems exhibiting sluggish convection. It is likely that the shear stresses have the most important influence on the quality of a crystal. Therefore our greatest concern is the maximum shear stress which exists in the crystal and its dependence on growth parameters, although the stress and strain patterns can be calculated throughout the crystal using this method. Once the element values are known the nodal values can be obtained by applying the consistent element re—
2
TEMPERATURE Fig. Computer solid-~liquidinterface shape The for various locations1.a—c of interface in Bridgman growth. temperature profile is shown as g1 and 52
sultant theory [12].
3. Shear stress and melt—solid interface shape In our earlier papers [7,8] the convexity was used to describe the interface shape during Bridgman growth. The convexity h~ h/r (h maximum axial displacement of the interface from a plane, r crystal radius) is usually considered to be an important factor affecting the quality of a crystal. Normally a flat interface shape is preferred. If this is not available under experimental conditions, a convex shape is considered to be =
=
=
preferable to a concave, since the latter is associated with a higher incidence of polycrystallinity. However, our study has shown that from the viewpoint of thermally-induced stress, a concave interface shape is not always the worst one. Fig. 1 shows a cross-section through a crucible in a Bridgman system which was used for this study. Curve g gives the temperature gradients on the inner surface of the crucible assumed for the initial calculation. Curves a—e show melt—crystal interface shapes which have been computed from finite element analysis of the freezing isotherms [7,8] and their locations relative to a fixed temper-
CE. Huang et a!.
/
277
Computation of stress in Bridgman crystals
ature profile. Movement of the interface was effected by changing the overall temperature profile, keeping all other parameters constant. For every case the thermal shear stress in every element and at every node can be obtained by a series of calculations by finite element analysis, and then it is easy to find the shear stress distributions in the crystal. As an example, if Young s modulus E is 5 x iO~kg/cm2, the thermal coefficient of expansion a is 5 X iO~ K_i, Poisson’s ratio j.t is 0.2, the steeper temperature gradient g 1 1, the flatter gradient g (see fig. 1) is 25 K cm” 2 is 5 K cm~,and the diameter of the as-grown crystal is 2.5 cm, then the maximum shear stress of crystals with different convexity during growth (corresponding to curves a—e of fig. 1) is shown in fig. 2. The elastic constants chosen are typical of brittle materials (glass, quartz) and were used since values for actual materials of interest (CdTe, AgGaS2) are not available in the literature. In the calculation, equilibrium conditions are assumed, so the results would correspond in practice to crystal growth at very slow rates. The error in applying the conclusions to real crystal growth is probably not serious since our earlier work [7,8]showed that finite growth rates have little influence on the temperature distribution except at rates much faster than those normally used in practice. The approximation was also used that E remains constant up to the melting point. In practice E will fall slowly with temperature and will then drop rapidly to zero over a range just below the melting point. Data for real materials can be introduced into the calculation very simply in cases where measurements have been made. It should be commented that E is used in place of the shear modulus since displacements were first computed for each node and these values were then used to find the local shear stress, From fig. 2 it can be seen that higher maximum shear stresses are associated with convex interface shapes, rather than concave. This result looks very unexpected but in fact it is quite reasonable if we take the heat flow conditions into consideration. Fig. 1 shows that when the interface shape changes from case a, the most convex, to case e, the most concave, the as-grown crystal experiences a progressively more gentle temperature field. For ex-
~
120
-
100
-
~___
~
2”
80
~ ~ ~
60
~ ~
e
20
0.4
I
I
0.3
0.2
I
0.1
I
I
0 0.1 CONVEXITY h~
I
I
0.2
0.3
Fig.2. Maximum shear stress for crystal—liquid interface located at different positions a—c of fig. 1. (Positive convexity indicates convex solid—liquid interface.)
ample in case a, the as-grown crystal is subject to a steeper temperature gradient above the interface and the flatter one in the lower region. In case e, only a relatively flat temperature gradient acts on the crystal, and for this reason the maximum shear stress is much smaller than that in case a. The main point is that the thermal shear stresses of a crystal are essentially determined by the temperature distribution in the crystal. The interface shape also depends on the temperature distribution and so the relationship between interface shape and stress is a mutual dependence on this common factor rather than an interdependence. Fig. 3 is the isostress diagram for case a. This is a typical pattern of the thermal shear stress distribution in a Bridgman grown crystal. For all cases from a to e the maximum shear stress values appear on the outside layers of the crystal, especially near P2 and P~(fig. 1), where the discontinuities in temperature gradient are located. It can be seen from fig. 3 that the outside of the crystal suffers 4—8 times higher shear stresses than the inner part while the crystal is growing. This result is consistent with the observation that when Bridgman-grown crystals crack, the cracks tend to be in the outer layer. A second example will be given to show that the thermal distribution rather than the interface shape is the key factor governing thermal stress.
278
C.E. Huang et al,
/
Computation of stress in Bridgman crystals
I::
120
‘~
30\ ~~2O
—--25
100
0
25
~4O
~20 tO
( \\
\~
s
0
20 10
I
-0.3
-0.2
I
-0.1
I
0 0.1 CONVEXITY h~
0.2
Fig. 4. Maximum shear stress versus convexity h~when h~is
varied by changing the relative thermal conductivity of solid and liquid. Actual values of these conductivities are listed in table 1.
\~/ 2)for case a of fig. Fig. 3. Isostress pattern (in kg cm~ (interface located just above discontinuity P 1 in temperature gradient).
From our earlier calculations [7] it is known that the difference between melt and solid thermal conductivities can make the melt—solid interface
shape convex or concave. In this second calculation a homogeneous temperature gradient of 25 K cm was assumed and the relative values of the melt and solid thermal conductivities KL and K~ were varied with other parameters held constant. The results are shown in table 1, which lists the maximum thermal shear stress (Tmas), the radial and axial temperature gradient Gr and G~at the point where the maximum shear stress is located, and the convexity h~ (which is negative for a concave interface). The maximum shear stress values are plotted as a function of convexity in fig. 4, which provides a striking contrast with fig. 2. It is clear from fig. 4 that, under the above conditions and from the viewpoint of stress, a flat or almost flat melt—solid interface shape is a favorable growth front. The highest stresses are
Table I Maximum shear stress and radial and axial temperature gradients conductivity
(Gr
KL
K5
Tm~ (kgcm2)
G, (Kcm~1)
(Kcm~)
(Wcm’ K’)
0.09
(Wcm
0.03
—0.232
120.7
34
30
0.05 0.03 0.03 0.03
0.03 0.03 0.05 0.09
—0.108 0 0.106 0.225
55.5 52.8 52.4 76.3
12 0 12 23
30 25 20 24
1 K’~)
h~
and G~)for different values of liquid and solid thermal
C. E. Huang et a!.
/
associated with the concave interface. However, it can also be seen from the table that curved, and especially concave, interface shapes are generally related to a steeper maximum radial temperature gradient as well as a steeper axial temperature gradient. As in the first example, therefore, the interface shape itself is a secondary factor as regards thermal stress. -
4. Applied stress In some cases, crystals may suffer from applied loads on their outer surfaces, for example if a crystal acts against the wall of a crucible. In this
4035
25 20
(~-~
Jj
~““
I) ‘s, , ~ (~~\ \
~
\_
279
Computation of Stress in Bridgman crystals
io ~ ‘5
15 IS
Table 2 Resultant of applied and thermally-induced stress 2) Convexity h~ Tma,, (kg cm Applied Thermal Total loads stresses effect only only — 0.340 106.6 42.5 140.4 —0.232 —0.122 0
31.1 19.2 12.6
77.6 104.4
90.6 116.0
0.122 0.321
13.5 14.2
117.6 119.9
128.9 131.1
case the melt—solid interface shape of crystal plays an important direct role, because of its influence on stress concentration patterns. Fig. 5 shows the isostress pattern when only an applied load of 25 kg cm”2 acts on the surface of the crystal, in the case of a concave interface. The parameters used in the calculation are the same as in section 3. Table 2 lists the convexity of interface shape h~,and the corresponding maximum shear stress found in crystals with different stresses applied through contact with the crucible wall. The first group of data shows the results when only the applied loads are acting on the crystal surfaces. Clearly, the largest stress concentrations occur in the case of a concave interface shape. The second column is taken from fig. 1 and gives the thermal shear stress only. In the last column the data shows the resultant if both applied loads and thermal stresses are acting on the crystal. It can be seen from table 2 that when the applied loads and thermal effects are acting on the crystal at the same time, the total effect is roughly the resultant of their separate effects. 5. Conclusions
~—(j
Fig. 5. Isostress pattern with applied load of 25 kg cm”’2 acting on crystal surface (kg cm 2)~
Finite element analysis has been shown to give isostrtlss patterns in Bridgman-grown crystals which appear reasonable in relation to observations of the incidence of cracks and dislocations. The maximum stress is found to be located around the outer surface of the crystal, and is particularly high near a discontinuity in the temperature gradi-
280
C. E. Huang et a!.
/
Computation of stress in Bridgman crystals
ent. It is the temperature distribution which determines the stress, and the solid—liquid interface shape is a secondary factor, although the temperature distribution also determines the interface shape. In a uniform temperature gradient, higher stresses occur in association with a concave interface, but the greatest stresses may be associated with a convex interface where this is located near steep temperature gradients. Stress concentration in the presence of a uniform applied stress is greatest for a concave interface. The combined effects of thermally induced and applied stress (e.g. from the crucible wall) are found to be roughly additive.
13] 141
J.C. Brice, J. Crystal Growth 42 (1977) 427. iC. Brice, Acta Electron. 16 (1973) 291. [5] v.L. Indenbom, IS. Zhitominskii and T.S. Chebanova, in: Growth of Crystals, Vol. 8 (Consultants Bureau, New York. 1969) p. 249. 161 E. Billig, Brit. J. AppI. Phys. 7 (1956) 375. [7] CE. Huang, D. Elwell and R. Feigelson, J. Crystal Growth 64 (1983) 441. [8] CE. Huang, D. Elwell and R. Feigelson, presented at ICCG-7, Stuttgart, September 1983. [9] L.-Y. Chin and F.M. Carlson, J. Crystal Growth 62 (1983) 561. [10] R.A. Brown, C.J. Cheng and P.M. Adornato, paper presented at ICCG-7, Stuttgart, September 1983. [Ii] S.P. Timoshenko and J.N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1970). [12] UT. Oden and Hi. Branchli, Numerical Methods Eng. 3 (1971) 317.
References [1] SB. Tsivinskii, Fiz. Metal. Metalloved. 25 (1968) 1013. [2] AS. Jordan, R. Caruso, AR. Von Neida aiid J.W. Nielsen, J. Appl. Phys. 52 (1981) 3331.
275
COMPUTATION OF STRESS IN BRIDGMAN CRYSTALS C.E. HUANG
“,
D. ELWELL and R.S. FEIGELSON
Center for Materials Research, Stanford Unirersity, Stanford, California 94305, USA
Received 25 September 1984
Finite element analysis has been used to calculate the stress distribution in crystals growing by the Bridgman method. It has been found that the temperature distribution, and particularly changes in temperature gradient, are the main factors governing the maximum thermally induced stress. The solid—liquid interface shape is a secondary factor but a concave interface is associated with the highest stress in a uniform temperature gradient. The application of an external stress during growth is also discussed.
1. Introduction Stress is a general problem in crystal growth, and excessive stress will result in dislocation formation or even in cracking if the elastic limit is exceeded. The ideal of a stress-free crystal cannot be realized in practice since crystals are normally grown in a temperature gradient. A reasonable experimental goal is therefore to lower the stress to an acceptable level, Several authors [1—6]have contributed towards an understanding of the origin of stress in crystals, especially those grown by the Czochralski method. A good review of these studies has been presented by Brice [3]. Finite element analysis has been shown to be a powerful method for the study of the temperature distribution in Bridgman systems [7—10].The purpose of this paper is to extend this method to calculate the stress in Bridgman-grown crystals. The results of these calculations will be compared with the earlier theories and with practical daia.
assumed and the potential energy is then minimized to obtain the nodal values of the displacements. Once the displacements are known, it is possible to solve for the strains and stress [11]. The important requirement is that the displacement equations selected must satisfy the boundary conditions. The total potential energy ir of an elastic system consists of two parts, a component resulting from the strain energy A in the body and a component W related to the potential energy of the internal and applied loads, i.e. A+ ~ —
~
—
For a volume V the total strain energy is obtamed by integrating over the constituent elements, giving A
=
f ~( ( ~} Tf a }
—
{ ~o } T{ a } )dV.
(2)
V
In solving problems in the theory of elasticity by finite element analysis, a displacement field is
Here the column vector { r) is the total strain which is related to displacements, and column vector { e~} is an initial strain, which is related to the temperature distribution in the crystal. The stress component a and the strain component are assumed to obey Hooke’s law, so that { a) [D] (~) [D] { ~ (3)
Permanent address: Research Institute of Synthetic Crystals, P.O. Box 733, Beijing, People’s Rep. of China,
where [D] is the property matrix of the material, which is related to Young’s modulus E and Pois-
2. Theoretical analysis
*
=
0022-0248/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
—
276
C.E. Huang et al.
/
Computation of Stress in Bridgman crystals
son’s ratio ~i. In this paper our attention is mainly focussed on Bridgman crystal growth. The radius of the crystals was varied from 0.5 to 25 mm, and the height kept at a value 8 times the radius. For all crystal sizes, the domain was divided into 2485 elements, which were defined by 1346 nodes in the finite element analysis. Thus eq. (1) should be rewritten in summation form as 2485 1T
~ (A~”1—w~’~). e~1
(4)
{ a)
for each element, we
After solving for
2
P
~
g 1
calculated the principal stresses a~,a~and a1 with, as is usual, a~> a2 > a1. Then the largest shear stress Tn~a,(was obtained using ~ ~(a~ a1). The computer program designed for our earlier work [7,8] was used to calculate the temperature distribution inside the sample, and the results were then used here to calculate the thermal stress. Since thermal convection was neglected in our calculations, the temperature profile and hence the stress data will be valid only in systems exhibiting sluggish convection. It is likely that the shear stresses have the most important influence on the quality of a crystal. Therefore our greatest concern is the maximum shear stress which exists in the crystal and its dependence on growth parameters, although the stress and strain patterns can be calculated throughout the crystal using this method. Once the element values are known the nodal values can be obtained by applying the consistent element re—
2
TEMPERATURE Fig. Computer solid-~liquidinterface shape The for various locations1.a—c of interface in Bridgman growth. temperature profile is shown as g1 and 52
sultant theory [12].
3. Shear stress and melt—solid interface shape In our earlier papers [7,8] the convexity was used to describe the interface shape during Bridgman growth. The convexity h~ h/r (h maximum axial displacement of the interface from a plane, r crystal radius) is usually considered to be an important factor affecting the quality of a crystal. Normally a flat interface shape is preferred. If this is not available under experimental conditions, a convex shape is considered to be =
=
=
preferable to a concave, since the latter is associated with a higher incidence of polycrystallinity. However, our study has shown that from the viewpoint of thermally-induced stress, a concave interface shape is not always the worst one. Fig. 1 shows a cross-section through a crucible in a Bridgman system which was used for this study. Curve g gives the temperature gradients on the inner surface of the crucible assumed for the initial calculation. Curves a—e show melt—crystal interface shapes which have been computed from finite element analysis of the freezing isotherms [7,8] and their locations relative to a fixed temper-
CE. Huang et a!.
/
277
Computation of stress in Bridgman crystals
ature profile. Movement of the interface was effected by changing the overall temperature profile, keeping all other parameters constant. For every case the thermal shear stress in every element and at every node can be obtained by a series of calculations by finite element analysis, and then it is easy to find the shear stress distributions in the crystal. As an example, if Young s modulus E is 5 x iO~kg/cm2, the thermal coefficient of expansion a is 5 X iO~ K_i, Poisson’s ratio j.t is 0.2, the steeper temperature gradient g 1 1, the flatter gradient g (see fig. 1) is 25 K cm” 2 is 5 K cm~,and the diameter of the as-grown crystal is 2.5 cm, then the maximum shear stress of crystals with different convexity during growth (corresponding to curves a—e of fig. 1) is shown in fig. 2. The elastic constants chosen are typical of brittle materials (glass, quartz) and were used since values for actual materials of interest (CdTe, AgGaS2) are not available in the literature. In the calculation, equilibrium conditions are assumed, so the results would correspond in practice to crystal growth at very slow rates. The error in applying the conclusions to real crystal growth is probably not serious since our earlier work [7,8]showed that finite growth rates have little influence on the temperature distribution except at rates much faster than those normally used in practice. The approximation was also used that E remains constant up to the melting point. In practice E will fall slowly with temperature and will then drop rapidly to zero over a range just below the melting point. Data for real materials can be introduced into the calculation very simply in cases where measurements have been made. It should be commented that E is used in place of the shear modulus since displacements were first computed for each node and these values were then used to find the local shear stress, From fig. 2 it can be seen that higher maximum shear stresses are associated with convex interface shapes, rather than concave. This result looks very unexpected but in fact it is quite reasonable if we take the heat flow conditions into consideration. Fig. 1 shows that when the interface shape changes from case a, the most convex, to case e, the most concave, the as-grown crystal experiences a progressively more gentle temperature field. For ex-
~
120
-
100
-
~___
~
2”
80
~ ~ ~
60
~ ~
e
20
0.4
I
I
0.3
0.2
I
0.1
I
I
0 0.1 CONVEXITY h~
I
I
0.2
0.3
Fig.2. Maximum shear stress for crystal—liquid interface located at different positions a—c of fig. 1. (Positive convexity indicates convex solid—liquid interface.)
ample in case a, the as-grown crystal is subject to a steeper temperature gradient above the interface and the flatter one in the lower region. In case e, only a relatively flat temperature gradient acts on the crystal, and for this reason the maximum shear stress is much smaller than that in case a. The main point is that the thermal shear stresses of a crystal are essentially determined by the temperature distribution in the crystal. The interface shape also depends on the temperature distribution and so the relationship between interface shape and stress is a mutual dependence on this common factor rather than an interdependence. Fig. 3 is the isostress diagram for case a. This is a typical pattern of the thermal shear stress distribution in a Bridgman grown crystal. For all cases from a to e the maximum shear stress values appear on the outside layers of the crystal, especially near P2 and P~(fig. 1), where the discontinuities in temperature gradient are located. It can be seen from fig. 3 that the outside of the crystal suffers 4—8 times higher shear stresses than the inner part while the crystal is growing. This result is consistent with the observation that when Bridgman-grown crystals crack, the cracks tend to be in the outer layer. A second example will be given to show that the thermal distribution rather than the interface shape is the key factor governing thermal stress.
278
C.E. Huang et al,
/
Computation of stress in Bridgman crystals
I::
120
‘~
30\ ~~2O
—--25
100
0
25
~4O
~20 tO
( \\
\~
s
0
20 10
I
-0.3
-0.2
I
-0.1
I
0 0.1 CONVEXITY h~
0.2
Fig. 4. Maximum shear stress versus convexity h~when h~is
varied by changing the relative thermal conductivity of solid and liquid. Actual values of these conductivities are listed in table 1.
\~/ 2)for case a of fig. Fig. 3. Isostress pattern (in kg cm~ (interface located just above discontinuity P 1 in temperature gradient).
From our earlier calculations [7] it is known that the difference between melt and solid thermal conductivities can make the melt—solid interface
shape convex or concave. In this second calculation a homogeneous temperature gradient of 25 K cm was assumed and the relative values of the melt and solid thermal conductivities KL and K~ were varied with other parameters held constant. The results are shown in table 1, which lists the maximum thermal shear stress (Tmas), the radial and axial temperature gradient Gr and G~at the point where the maximum shear stress is located, and the convexity h~ (which is negative for a concave interface). The maximum shear stress values are plotted as a function of convexity in fig. 4, which provides a striking contrast with fig. 2. It is clear from fig. 4 that, under the above conditions and from the viewpoint of stress, a flat or almost flat melt—solid interface shape is a favorable growth front. The highest stresses are
Table I Maximum shear stress and radial and axial temperature gradients conductivity
(Gr
KL
K5
Tm~ (kgcm2)
G, (Kcm~1)
(Kcm~)
(Wcm’ K’)
0.09
(Wcm
0.03
—0.232
120.7
34
30
0.05 0.03 0.03 0.03
0.03 0.03 0.05 0.09
—0.108 0 0.106 0.225
55.5 52.8 52.4 76.3
12 0 12 23
30 25 20 24
1 K’~)
h~
and G~)for different values of liquid and solid thermal
C. E. Huang et a!.
/
associated with the concave interface. However, it can also be seen from the table that curved, and especially concave, interface shapes are generally related to a steeper maximum radial temperature gradient as well as a steeper axial temperature gradient. As in the first example, therefore, the interface shape itself is a secondary factor as regards thermal stress. -
4. Applied stress In some cases, crystals may suffer from applied loads on their outer surfaces, for example if a crystal acts against the wall of a crucible. In this
4035
25 20
(~-~
Jj
~““
I) ‘s, , ~ (~~\ \
~
\_
279
Computation of Stress in Bridgman crystals
io ~ ‘5
15 IS
Table 2 Resultant of applied and thermally-induced stress 2) Convexity h~ Tma,, (kg cm Applied Thermal Total loads stresses effect only only — 0.340 106.6 42.5 140.4 —0.232 —0.122 0
31.1 19.2 12.6
77.6 104.4
90.6 116.0
0.122 0.321
13.5 14.2
117.6 119.9
128.9 131.1
case the melt—solid interface shape of crystal plays an important direct role, because of its influence on stress concentration patterns. Fig. 5 shows the isostress pattern when only an applied load of 25 kg cm”2 acts on the surface of the crystal, in the case of a concave interface. The parameters used in the calculation are the same as in section 3. Table 2 lists the convexity of interface shape h~,and the corresponding maximum shear stress found in crystals with different stresses applied through contact with the crucible wall. The first group of data shows the results when only the applied loads are acting on the crystal surfaces. Clearly, the largest stress concentrations occur in the case of a concave interface shape. The second column is taken from fig. 1 and gives the thermal shear stress only. In the last column the data shows the resultant if both applied loads and thermal stresses are acting on the crystal. It can be seen from table 2 that when the applied loads and thermal effects are acting on the crystal at the same time, the total effect is roughly the resultant of their separate effects. 5. Conclusions
~—(j
Fig. 5. Isostress pattern with applied load of 25 kg cm”’2 acting on crystal surface (kg cm 2)~
Finite element analysis has been shown to give isostrtlss patterns in Bridgman-grown crystals which appear reasonable in relation to observations of the incidence of cracks and dislocations. The maximum stress is found to be located around the outer surface of the crystal, and is particularly high near a discontinuity in the temperature gradi-
280
C. E. Huang et a!.
/
Computation of stress in Bridgman crystals
ent. It is the temperature distribution which determines the stress, and the solid—liquid interface shape is a secondary factor, although the temperature distribution also determines the interface shape. In a uniform temperature gradient, higher stresses occur in association with a concave interface, but the greatest stresses may be associated with a convex interface where this is located near steep temperature gradients. Stress concentration in the presence of a uniform applied stress is greatest for a concave interface. The combined effects of thermally induced and applied stress (e.g. from the crucible wall) are found to be roughly additive.
13] 141
J.C. Brice, J. Crystal Growth 42 (1977) 427. iC. Brice, Acta Electron. 16 (1973) 291. [5] v.L. Indenbom, IS. Zhitominskii and T.S. Chebanova, in: Growth of Crystals, Vol. 8 (Consultants Bureau, New York. 1969) p. 249. 161 E. Billig, Brit. J. AppI. Phys. 7 (1956) 375. [7] CE. Huang, D. Elwell and R. Feigelson, J. Crystal Growth 64 (1983) 441. [8] CE. Huang, D. Elwell and R. Feigelson, presented at ICCG-7, Stuttgart, September 1983. [9] L.-Y. Chin and F.M. Carlson, J. Crystal Growth 62 (1983) 561. [10] R.A. Brown, C.J. Cheng and P.M. Adornato, paper presented at ICCG-7, Stuttgart, September 1983. [Ii] S.P. Timoshenko and J.N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1970). [12] UT. Oden and Hi. Branchli, Numerical Methods Eng. 3 (1971) 317.
References [1] SB. Tsivinskii, Fiz. Metal. Metalloved. 25 (1968) 1013. [2] AS. Jordan, R. Caruso, AR. Von Neida aiid J.W. Nielsen, J. Appl. Phys. 52 (1981) 3331.