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Heat transfer analysis of the Bridgman-Stockbarger configuration for crystal growth
Journal of Crystal Growth 61(1983) 339—354 North-Holland Publishing Company
339
HEAT TRANSFER ANALYSIS OF THE BRIDGMAN-STOCKBARGER CONFIGURATION FOR CRYSTAL GROWTH I. Analytical treatment of the axial temperature profile T. JASINSKI and W.M. ROHSENOW Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
and A.F. WI’T17 Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Received 22 March 1982; manuscript received in final form 6 August 1982
An analytical approach describing the heat transfer during crystal growth in a Bridgman—Stockbarger configuration has been developed. All first order effects on the axial temperature distribution in a solidifying charge are treated on the basis of a one-dimensional model whose validity could be verified through comparison with a published finite difference analysis of a two-dimensional model. The present model includes an insulated region between axially aligned heat pipes and considers the effects of charge diameter, charge motion, thickness and thermal conductivity of a confining crucible, thermal conductivity change at the crystal—melt interface and non-infinite charge length. Results are primarily given in analytical form and can be used without recourse to computer work for both improved furnace design and optimization of growth conditions in a given thermal configuration.
I. Infroduction Conventional techniques for crystal growth from the melt prove increasingly inadequate in meeting property requirements for electronic materials as dictated by advanced device technology. In this context it is of interest that renewed attention is recently being paid to seeded vertical Bridgman growth where the driving force for interfering free melt convection is reduced and the critical axial and radial thermal gradients are at least in principle readily controllable. Motivation for a thorough re-examination of this crystal growth technique is provided by ample evidence in the open literature that basic heat transfer considerations, now recognized to strongly affect both crystalline and chemical perfection of the resulting solid, have been largely ignored in its application, In recent years several thermal analyses of Bridgman-type crystal growth systems have been 0022-0248/83/0000—0000/$03.OO
©
reported in the literature. Noteworthy among these is a series of outstanding publications by Wilcox et al. [1—3]in which the effects of several dimensionless parameters on both axial and radial temperature distributions in the charge were analyzed and significant conclusions drawn. The modeled systems, however, were simplified in several respects and the two-dimensional formulation (except for part of Chang and Wilcox [1]) precluded simple analytical results. Of interest are also the works of Davis [4] and Clyne [5,6] which demonstrate that one-dimensional models can accurately predict experimental axial temperature variations. Their models were not nondimensional and, consequently, significant thermal and geometric parameters which control the system behavior were not identified; their conclusions cannot readily be extended to systems of different parameter values. The present work is part of a combined theoretical and experimental program whose purpose is to
1983 North-Holland
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Heat transfer analysis of Bridgman
provide a basis for the design of an improved vertical Bridgman growth configuration. The cxperimental system (presently in construction) makes use of high temperature heat pipes which
—
Stockbarger growth. I
temperature distribution in the charge *, The parameters of primary concern are: thermal coupling between the charge and the furnace; charge lowering rate; generation of latent heat at the growth interface; thermal conductivity difference between the melt —
—
crucible
—
—
hot heat pipe
-
heating coils
and the crystal; thickness and thermal conductivity of the charge-confining crucible; length of the gradient control region; diameter of the charge. Special attention is given to (1) acquiring a large axial temperature gradient at the growth interface in order to preclude constitutional supercooling and (2) on the axial position of the growth interface (a pertinent factor for the interface curvature [1]). The axial temperature behavior of the charge is predicted on the basis of a steady-state one-dimensional thermal model which, not requiring undue —
Q
— —
D grad~nt region
L D
heating
cold heat
Fig. I. Bridgman—Stockbarger configuration with gradient control region.
provide for axial and axi-symmetric temperature uniformity and thus for thermal boundary conditions which permit a meaningful theoretical analysis. It includes, between aligned heat pipes, a gradient control region (fig. 1) in which the crystal—melt interface is to be located. The purpose of this region, first suggested by Chang and Wilcox [1], is to provide control of heat flow near the growth interface, required for the establishment of desired radial thermal gradients. The design and operation parameters are expected to provide for the system under construction a wide range of critical axial and radial temperature gradients leading to growth interface shapes ranging from convex to concave. Primary attention is focused, however, on the establishment of growth conditions which preclude nucleation at the confining boundary with the simultaneous minimization of radial segregation effects. This study examines the effects of both system and operation parameters on the quasi-steady axial
simplification, yields significant results in analytical form. Its primary asset is the timesaving dcment, since meaningful information can be obtained from the analytical results without resort to computer work. Dimensionless parameters are used throughout.
2. Development of heat transfer model The factors of concern for the development of a quasi-steady one-dimensional heat transfer model of the modified Bridgman growth system depicted in fig. 1 are shown in fig. 2. The hot and cold heat pipes comprise the hot and cold zones; the region between them is called the gradient zone. The length of the charge is broken down into LH, L(~, and L~ within the hot, gradient and cold zones respectively. The charge is lowered through the furnace with a velocity V, has liquid and solid portions with different thermal conductivity, and has a crystal—melt interface which generates latent heat. A crucible provides containment for the charge. *
Radial temperature variations in the charge are the subject of a publication (Part II) presently in preparation.
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T. Jasinski et al.
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gradient zone hot zone
~.—i I
Heat transfer analysis of Bridgman
,
i
r—.. cold zone
~
I
— I
z
I
______
L~ 0cr P~
D h~
I
L ‘Liquid
—
341
Stockbarger growth. I
ature distribution when the thermal coupling between the furnace and the charge is large. These effects are approximated in sections 2.2 and 2.3 where it is shown that they can be included in the
I
___
/
0 ~~Lc
Solid
~
~crucible I
hc
Fig. 2. Thermal model of furnace and charge with crucible.
one-dimensional model as a modification of the thermal coupling between the charge and the furnace. With the assumption that the temperature at each cross-section of the charge is uniform, the charge by Carslaw is analagous and Jaeger to the[7]moving and the thinequation rod treated describing the axial temperature distribution is *: 71D2 d2T ~D2 dT —k————————--pVc —+ITDq
4
The thermal model makes the following assumptions: (1) Hot and cold zone furnace temperatures are uniform, reflecting the heat pipe action. The length of the heat pipes is assumed infinite, (2) The system is at all times in a quasi-steady state, i.e., transients are neglected. (3) No heat transfer by natural convection in the melt. (4) Heat exchange between the furnace and the outer crucible surface is described by a heat transfer coefficient, h, that is constant within each zone. (5) The gradient zone is assumed adiabatic, i.e.,
dZ~
4
~dZ
=0.
(1)
In eq. (1), the first term represents axial conduction of heat within the charge, and the second term represents axial convection of heat due to motion of the charge at velocity V. The factor q” in the third term accounts for radial heat transfer to the surface of the charge, per unit surface area of the charge. According to assumption 4 above (and neglecting the crucible effect), q” is given by: h(T~ T). (2) Expressing eq. (1) in nondimensional form yields, =
d20 —i
—
—
Pe~f+ 4 Bi (O~ 8) —
=
0,
(3a)
h 0 0. (6) Thermal properties are not temperature dependent and the melt and crystal mass densities are equal. (7) The one-dimensional model considers only axial heat transfer within the charge, i.e., the temperature within the charge is not a function of the radius. =
2.1. The basic one-dimensional thermal model of the charge
The thermal model for the axial temperature profile of the charge, in its most basic form, neglects the crucible and radial temperature variations within the charge. The crucible, however, does provide a radial thermal resistance between the charge and the furnace, and radial temperature gradients within the charge affect the axial temper-
or, 28 —Pe’-~+9~—9=0,
(3b)
d where us 2W~, Pc’ us ~Pe/~/~I. (Eqs. (3a) and (3b) are equivalent.) ~‘
2.2. Crucible effect In Bridgman growth a charge is confined in a crucible which, depending on conditions, varies in dimension and composition. Containment of the charge tends to decrease axial temperature gradients (Sen and Wilcox [2]). A crucible of low ther*
For an explanation of the symbols used see nomenclature in appendix D.
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mal conductivity lowers these gradients by adding thermal resistance between the charge and furnace, thereby decreasing the thermal coupling between the charge and the furnace; one of high thermal conductivity lowers these gradients by conducting heat axially within the crucible itself.
transfer The domain within ofthe crucible eq. (3) isapproximates considered toitsbe effect the charge andthe is not explicitly included. only However, a crucible simple model of the heat on q” of eq. (1) which can be expressed through modified Biot and Péclet numbers. The temperature distribution in the crucible obeys the heat conduction equation in cylindrical coordinates:
a
1
“
~
~
2
9Z
V 87~~ —~~:--~-~-
=0.
(4)
The first term of eq. (4) accounts for the radial thermal resistance of the crucible, while the second term accounts for axial conduction of heat within the crucible. At each axial location of the crucible, a7~r/azand d27~r/3Z2 are assumed to be mdcpendent of the radial coordinate (r) and equal to dT/dZ and d2T/dZ2 of the charge (see appendix A). Eq. (4) can thus be integrated to yield a radial distribution of T~rin terms of the axial gradients in the charge. The boundary conditions are: h (T 1—
~~r)
=
kcr
a7~r,/ar,
=
T,
8
.25
0 .01
.8 .6
~
2
10 a K = kcr/k Fig. 3. Effect of the thermal conductivity ratio on the effective Biot number. .10
.01
Bi* ~-~=~l
I +8Bi[~(62_ l)-~ ln(6)] (62
—
1) ~
Bi ln(6)}
(7)
,
I
Pe* =
~l
+
+
~
x
~6
Bi ln(8)
6 Bi
{
K(62
1
~ +
6 Bi
—
1)
{ ~(62
2— l)+
(5a)
at the outer crucible surface, Tcr
Stockbarger growth. I
—
—
—
~
ln(6)]
1)
— ~
—
1)
+
ln(6)]
Bi ln(6)}
— ~.
(8)
+K(6
(Sb)
at the inner crucible surface. Eq. (5b) assumes that the crucible and charge are tn contact. The radial heat flow to the charge in eq. (1), q”, is related to the radial distribution of ~r by: q” kcr a7~r/ar (6) =
at the inner crucible surface. The heat flow given by eq. (6) is substituted for the third term of eq. (1). When nondimensionalized, the resulting expression has the same form as eq. (3) if Bi and Pc are replaced by their respective “effective” values, Bi* and Pe*:
The relationship between the effective Biot number, Bi*, and the conductivity ratio K is shown in 3 for 6 1.25 and reduced various Bi. canlow be seen thatfig.Bi* is significantly by It both and high values of K, especially for larger Bi. (In section 3.2 it will be shown that an increase in Bi* tends to increase the axial gradient at the growth interface.) The conductivity ratio which maximizes Bi* can be obtained from eq. (7): =
K~ = ~~l~6~ii5.
(9)
For typical values of Bi and 6, the conductivity ratio providing the maximum effective Biot is between 0.1 and 1.0.
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Heat transfer analysis of Bridgman
2.3. Effect of radial gradients in the charge Eq. (1) considers that the temperature at each axial location ~ is constant. Presently, radial gradients in the charge are considered insofar as they affect the thermal coupling between the furnace and the charge and thereby the axial temperature distribution, When radial temperature variations in the charge are taken into consideration, eq. (3) assumes the form (see appendix B): dO d20 Pe* m d~ +4 Bi* [(o Om) (0~ 0m)} 0. (10) —
—
—
—
=
The subscript m denotes an area-averaged value, In eq. (10), the term ~s O~accounts for the effect of radial temperature variations within the charge on the q” term of eq. (1). An approximation for 8,~is obtained through the two-dimensional heat conduction equation in the charge: —
—
)
I p ao~+ p dp i9p / —
—
820
—
Pc 80 0. d~’ —
=
(Ii)
Assuming that the Péclet term is negligible, and, as in the crucible effect, that d20/3~2 is only a weak
—
~s
—
8~=
—
~
(12)
(120m/~2,
Combining eqs. (10) and (12) leads to a new effective Biot number, Bi**, in the third term of eq. (3a) which accounts for radial temperature gradients within the charge as well as for the crucible effect:
Stockbarger growth. I
343
The radial resistance of the charge thus limits the degree to which axial temperature gradients can be increased by augmentation of thermal coupling between the charge and the furnace. (A comparison of axial temperature gradients calculated using eq. (13) with those obtained by the two-dimensional model of Fu and Wilcox [3] is provided in section 3.7.) 2.4. Solution
Eq. (3) is applied separately to each region of uniform properties and effective Biot number, Bi**. In this way, for each such region, eq. (3) has constant coefficients. In the present model there are four such regions when the growth interface is located in the gradient zone: the hot and cold zones and the liquid and solid parts of the gradient zone. The solution of eq. (3) within each region yields two exponential terms for the homogeneous solution and a particular solution that depends on 8~.The constants of integration on the exponential terms are found using boundary conditions of equality of temperature and continuity of flux between adjacent regions. At the crystal—melt interface, the flux condition gives: dO [K 5(62
function of the radial coordinate p, eq. (Il) can be 20m/d~2as integrated in the radial direction with d a variable. A parabolic variation for 0 results from which 0s can be determined as:
—
—
1) + RKI
(~
)L
2 1)] (~). (14) 5 RH + [i + K~(6 The Péclet number in eq. (14) is correctly based on the actual growth rate, assumed here to be equal to =
Pc
—
the lowering rate V. (This equality exists in systems with infinite charge length and during growth
(13)
experiments which fix the interface location by appropriate adjustments of the furnace temperatures.) The constants of integration have been determined analytically for systems of infinite charge length; expressions for the axial gradient in the
According to eq. (13), Bi** Bi* for Bi* zg~~ i.e., radial temperature variations within the charge do not affect the axial temperature distribution in the charge by virtue of the effective Biot number. Eq. (13) indicates also that the maximum value that Bi** can attain is 8 when Bi* is very large.
melt and for the axial position of the crystal—melt interface of such systems are given in appendix C. If the complete axial temperature distribution is desired, or if the charge length is not infinite, it is more convenient to determine the constants of integration by computer.
Bi**
=
Bi*/[ 1
+
(Bi*/8)]
.
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3. Analysis
—
Stockbarger growth. I
metric system is: 0(~)+O(—~’)= 1.
This section considers the effects of the thermal parameters of the Bridgman growth system on the axial temperature behavior of the charge. Axial temperature distributions in the charge were calculated by computer using the model described in section 2; analytical expressions for the axial temperature gradient in the melt near the growth interface, in systems of infinite charge length, are taken from the results presented in appendix C. In consideration of the many variables in the thermal model, a system with idealized parameter values, presently called a “symmetric” system, is used as a reference against which the effects of individual parameters will be compared. A symmetric system is defined as follows: (1) Equal heat transfer coefficients in the hot and cold zones, (2) Axial convection of heat due to the charge motion is negligible, i.e., Pc is small. (See section 3.1.) (3) The effect of the generation of latent heat is negligible (PeSRH 0). (See section 3.4.) (4) Equal thermal conductivities of the melt and crystal (RK 1). With condition 1 above, Bi~~* Bi~*. (5) Equal charge lengths in the hot and cold zones (XH Ar). The axial temperature distribution for a sym-
3.1. Péclet number effect
The Péclet number expresses the ratio of axial heat transfer in the charge due to the lowering velocity, V, and due to conduction. The effective Péclet number, Pe*, incorporates the effect of the crucible on the convection term of eq. (3). The unmodified Péclet number is used to describe the generation of latent heat (refer to eq. (14)). The effect of Pe* on the axial temperature distribution in an otherwise symmetric system. as obtained through eq. (3), is shown in fig. 4. It is seen that Pe* tends to increase the temperature of the charge at all locations; the effect is more pronounced as Pe* increases (e.g., high lowering rate or low thermal conductivity of the charge). As a consequence, in systems with fixed hot and cold zone furnace temperatures (i.e., the nondimensional melting point temperature is constant), the growth interface moves toward the cold zone as Pe* increases; alternatively, the nondimensional melting point temperature must increase (i.e., one or both of the furnace temperatures must be lowered) if the interface is to remain at the same axial location as Pe* increases. In typical Bridgman growth experiments, the
—
=
=
=
9 10
B ~
~
I
symmetric
I
I
Hot Zone
_______________
‘Gradient
(15)
Cold Zone
Zone Fig. 4. Effect of charge motion on the axial temperature distribution in a charge.
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/
Heat transfer analysis of Bridgman
Péclet effect is small since the lowering rates are small (e.g., 0,1—10 ~tm/s). A test criterion for its relative magnitude is provided through the characteristic roots of eq. (3b): — t In m ±— 2 ~ e *\‘f ~ 1
L —
I 1 +
4 /~~
e *\~\2l
/2
\ f’
11 ~\)
It can be seen that Pc* disappears from eq. (16) when, (17) Pe*/4(Bi**)~’2~< 1. =
Eq. (17) agrees with the findings of Chang and Wilcox [1] who reported that the Péclet effect is stronger for smaller Biot numbers. If the inequality in eq. (17) holds, the Péclet number is small enough to satisfy the Péclet number criterion for symmetric systems (i.e., Pc 0). Further, when the Péclet effect is negligible, simplified analytical expressions are obtained for the axial gradient within the melt in the gradient zone (appendix C). Since eq. (17) does hold for typical Bridgman growth, these simplified expressions will be used in the remainder of this section.
—
345
Stockbarger growth. 1
of the crucible and radial temperature gradients in the charge on this thermal coupling. Typical values for the effective Biot number range from 0.05 for high conductivity materials such as Ge to 5.0 for low conductivity materials such as CdTe. Axial temperature profiles for several Bi** are shown in fig. 5. In agreement with the results of Chang and Wilcox [1], it is found that the charge temperature follows more closely the furnace temperature and, as a result, the axial temperature gradient in the gradient zone increases, as Bi** increases. The expression for the axial gradient near the crystal—melt interface presented in appendix C for infinite charge length can be simplified for symmetric systems: .
d0/d~
=
[AG
+
(Bi**)
—
1/2
]
—
(18)
.
The dependence of the axial gradient at the growth interface on Bi**, according to eq. (18), is plotted in fig. 6 for various gradient zone lengths A 0. The curves show that the dependence of the axial gradient on A0 becomes stronger with increasing Bi**. It can also be seen (curve A0 0) that there exists a minimum Bi** for any desired nondimensional axial gradient. The heat transfer coefficients of the hot and cold zones are generally not equal due, for examplc, to the temperature dependence of radiative heat transfer. For such conditions, the zone with =
3.2. Bior number effect
The Biot number, through the heat transfer coefficient h, is a direct measure of the thermal coupling between the charge and the furnace. The effective Biot number, Bi**, incorporates the effect
R
Of ~A ~
:
~0I0\
HC
I I
Symmetric system
Hot Zone
I
:
1Gradient
Cold Zone
Zone
Fig. 5. Effect of thermal coupling between furnace and charge, Bi**, on the axial temperature distribution within the charge.
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Heat transfer analysis of Bridgman
Symmetric systems
Stockbarger growth. I
gradient zone for a system with infinite length which is symmetric except hH us h1~-is:
__~
XH
—
~
~
d8/d~= {x0 g
0
(19) Biot numbers has a greater effect on decreasing I
-
~05
+~[(Bi~*)_1/2+(Bi~*)_I/2J~’.
____
Eq. (19) indicates that the smaller of the effective the axial gradient. Efforts to increase the axial gradient by increasing the heat transfer coefficients should therefore first be directed at the zone with the smaller Bi**.
_______
___________________________________________
o.bi
0.1 0 0 Bi~* Fig. 6. Effect of gradient zone length and Bi** on the axial temperature gradient in the melt near the growth interface, 0001
3.3. Effect of unequal thermal conductivity of the crystal and the melt (RK us 1) RK expresses the ratio of melt to crystal thermal conductivity. Semiconductors have RK values greater than unity whereas, for metals, the value of RK is less than unity. The effect of RK on the axial temperature distribution in the charge according to eq. (3) is demonstrated in fig. 8 for systems with 6 1 and which are otherwise symmetric (i.e., BiH Bic/RK). The charge phase with the higher thermal conductivity exhibits in all instances a lower axial gradient because of its lower thermal resistance to heat transfer in the axial direction. The axial gradient in the melt near the interface is therefore less than in the crystal for charges with RK> 1. The expression for the axial gradient in the melt
the larger effective Biot number more strongly influences the overall temperature level (fig. 7). Compared to the symmetric case where Bi~* Bi~.*,the charge temperature for asymmetric systems increases when Bi~*> Bi~*and decreases when Bi~*> Bi~~*.If the location of the crystal—melt interface for each curve of fig. 7 is considered to be at the center of the gradient zone, the effect of unequal Biot numbers appears as a change in nondimensional interface temperature. To retain the interface in a given location requires lowering of the hot and/or cold zone furnace temperatures as Bi~~* increases relative to Bi~*. The expression for the axial gradient in the =
9f~
=
=
a I
1.01 I
.751
hH~hc System otherwise symmetric
XH~XC~co Bic*~a.0I
25
I
I
I
Hot Zone
I
I
of
Bi~**I.0 ______________________________________
Gradient
Cold Zone
Fig. 7. Effect of unequal hot and cold zone Bi** on the axial temperature distribution in the charge.
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Heat transfer analysis of Bridgman
—
347
Stockbarger growth. I
a II
10
I
Bi~*O.l
I
BiH*Bic/Re System otherwise symmetric
I
XH~XC~O
I
8
a
I25~ Interface Iocation—,~ I
I
Hot Zone
__________________________________
Gradient
Zone
Cold Zone
Fig. 8. Effect of non-equal thermal conductivity of crystal and melt on the axial temperature gradient in the melt near the interface.
near the growth interface for a system of infinite length which is symmetric except RK us I is: /
dO
\
~.)L
_2{R~[(Bi~*Y~2+AG_
2~~] ~BIc
~ (20)a A comparison of eqs. (20) and (18) shows that
0
0 l.a
value of RK greater than unity is detrimental to the establishment of large axial gradients in the melt. The axial gradient as a function of RK, according to eq. (20), is plotted in fig. 9 for several values of Bi0. It is found, for example, that this conductivity effect reduces the gradient in a germanium melt (RK 2.5) by about 50%. 3.4. Latent heat effect (PeSRH ~
The quantity of latent heat liberated at the crystal—melt interface is given in nondimensional form by the product Pe5RH (eq. (14)). The expression for the axial gradient in the melt near the growth interface in a system of infinite length which is symmetric except PeSRH us 0 is: /dO\
_2+PesR~[(Bi**)/2+Ao_2~j] 2[(Bi**)_~2+A0] (21)
0.1 Bi~BIc/Re(hence,b’I)
System otherwise symmetric
~. ~
X~= .0 XH= X 5 =0’
0.01 001
0.01
10
10
RK*kL/ks Fig. 9. Effect of thermal conductivity ratio (RK) and Bi on the axial temperature gradient in the melt near the growth interface.
It can be seen that the effect of latent heat on the axial gradient (and also on the axial temperature profile) is small if: i PesR~[(Bi**)_1/2+AO_2~ij<<1. (22) Eq. (22) demonstrates that the effect of latent heat on the axial temperature behavior is larger for
348
/
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Heat trausfer analysis of Bridgman
—
Stockbarger growth. I
8 1/2 (pesRHd,[(Bi**) I/2~~ XG-2~
I
Pe
5R~I System otherwise symmetric Bi~=0.I XH”Xcas 2/Peus’I 4B/” I
I I I
Interface location—..,
I
I
Hot Zone
_______ _____
Gradient
~
~. I __________________________
zoneCh
I
Cold Zone
C
Fig. 10. Effect of the generation of latent heat at the growth interface on the axial temperature distribution in the charge.
smaller Bi** and for larger lowering rates of the charge (Pc 5 V). Axial temperature profiles for several values of 4) arc given in fig. 10. It is seen that the generation of latent heat both increases the charge temperature and decreases the axial gradient in the melt; the latent heat effect disappears for small values of 4). (The effect of latent heat and the Péclet number effect are coupled through their mutual dependence on the lowering rate, V. In order to isolate the latent heat effect in fig. 10, Pc is chosen sufficiently small so that eq. (17) is satisfied. Such
a small value for Pc would normally also eliminate the latent heat effect; i.e., eq. (22) would also be satisfied. The values of 4) in fig. 10 were therefore obtained by choosing large values of RH.) 3.5. “Infinite” charge length When the charge is infinitely long, the temperature field of the charge does not change during growth and, hence, the interface position remains fixed, Fig. II shows the progression of axial ternperature profiles as the charge moves from the hot
0 A 1.00
System otherwise symmetric (i.e.,high thermal coductivity) =0.05)
0.15
=0.005 (4B,~’°/Pe 79) tal charge length =
0.50 Hot end of charge 0.25
-10
-8
-6
-4
Hot Zone
-2
-0
sO
IGradi Zone I
2
4
6
8
Cold Zone
Fig. 11. Effect of charge position within the furnace on the axial temperature distribution.
0
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/
Heat transfer analysis of Bridgman
zone to the cold zone for a charge length less than infinite. It can be seen that charge temperatures are displaced toward the temperature of the hot zone furnace when most of charge is in the hot zone and vice-versa. Accordingly, to achieve constant interface position for non-infinite charge lengths, the nondimensional melt temperature must be reduced as the experiment proceeds. At constant nondimensional melt temperature, the interface growth velocity will be greater than the lowering rate as the interface moves from the cold zone to the hot zone. The contribution from the positive characteristic root, eq. (16), to the solution of eq. (3) is normally small and is zero for an infinite charge length. The charge thus appears infinite in length when the contribution from the negative root also becomes small, i.e., for large The temperature change within the hot or cold zone reaches approximately 99% of its final value when exp[(mj~’]= 0.01, Using this as a criterion for infinite length: 5/rn (23) ~“.
=
‘
where length of charge within the hot or cold zone for the charge to appear infinitely long. If Pe* is small so that eq. (17) is satisfied, the characteristic roots become: ~
m
±=
±1,
(24)
and by substitution we find: 2, (25) 5 or 5/2(Bi**)” which is a useful expression for determining L.
349
Stockbarger growth. 1
—
propriate boundary conditions must be applied to the ends of the charge. For example, a solid pull rod contacting the crucible at the bottom of the charge can be approximated by treating it as a simple fin exchanging heat with its environment. In this way, a Biot number can be calculated for the end boundary condition. The curves of fig. 11 were calculated using the same Biot number for the ends as for the circumference. 3.6. Charge diameter effect
Any change in the charge diameter, D, affects the dimensionless parameters Bi, Pc, 6, A0, ~ and ~. Further, changes in D may alter the heat transfer coefficients between the furnace and charge as the geometry of the furnace cavity changes. The effect of a change in D can be assessed by reevaluating the necessary parameters and then using the appropriate expression for the axial ternpcrature gradient. In general, increases in D may either increase of decrease the axial temperature gradient in the gradient zone depending on the corresponding changes in the other parameters. As an illustrative example, consider a symmetnc system of infinite charge length. The appropriate expression for the axial temperature gradient in the gradient zone is given by eq. (18). Since, however, changes in diameter are to be analyzed, it is more informative to compare the axial of eq. (18) based on Z, the dirnensionalgradients axial coordinate:
~
dO/dZ
=
—
1 [L 0
Eq. (25) is in agreement with the results of Clyne [5,6] which suggest that longer charges and higher Biot numbers will tend to stabilize the interface position (i.e., the growth rate is equal to the lowering rate). Results from the one-dimensional numerical model of Riquet and Durand [8] suggest that the growth and lowering rates are equal when the length of charge2).This within result the hotagrees and cold is well zones with eq. about the l.5(Bi’~ (25); difference in the constant coefficient is attributed to the choice of criteria defining “infinite” charge length. When the charge length is not infinite, ap-
+ D/(Bi**)~2]
- ~.
(26)
.
Defining the variable: d{D/(Bi**)I/21/dD
(27)
and taking L0 as constant, eq. (26) shows that the axial gradient increases with D when s~is negative and vice-versa. For any increase in D accompanied by a proportional increase Dcr Bi** (6 remains constant), (7) and (13) show inthat does not increaseeqs. as rapidly as D (assuming all other thermal parameters are constant). Consequently (eq. (27)), s~ is positive and the axial gradient according to eq. (26) decreases as D increases.
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T. Jasinski et at.
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Heat transfer analysis of Bridgman
—
Stockbarger growth. I
Table 1 Axial thermal gradients at the crystal—melt interface as obtamed through eqs. (13) and (18), d9m/d~,and average axial thermal gradients (d8/3~)~~, as determined from the two-dimensional model of Fu and Wilcox [31
Si’ .05
so (~O/3~)
d6m/d~
-5
sy~net~c system
SI’
-i 001
.0
05
Bi =2
A~= 0.0
1.310
1.260
Bi
A~ 0.0
0.616
0.617
0.4
=
10
K
5’ kcr/ki
Fig. 12. Effect of Bi and K on the parameter constant D..
4’
(eq. (27)) for
The crucible diameter may, on the other hand, remain constant as D increases. This occurs when increasing the charge diameter within a constant diameter furnace cavity at the expense of the thickness of the crucible. Fig. 12 shows the corresponding behavior of 4’ for various Bi and K5 with 6 1.25. Accordingly, sji is either positive or negative depending on the values of Bi and K5 the axial gradient of eq. (26) may therefore either increase or decrease with an increase in D. =
3.7. Comparison of one- and two-dimensional models
The results of the present one-dimensional model can be compared to the two-dimensional finite difference results of Fu and Wilcox [3]. Fig. 3 of their paper presents the radial variation of the axial gradient at the growth interface of symmetric systems of infinite charge length in the absence of a crucible. Point values, estimated from the curves of their fig. 3, were numerically integrated to obtain the average axial gradient over the cross-section and were compared to the values obtained using eqs. (13) and (18) for several values of Bi and A0 (see table 1). Considering the approximations required in the derivation of eq. (13), the agreement of the data must be taken as excellent. The comparison demonstrates that the onedimensional models presently used provide a meaningful representation of the parameters
governing the axial temperature behavior of a . charge in a Bridgman-type growth configuration. 4. Discussion and conclusions The one-dimensional model developed in section 2 has been shown to correlate well with a corresponding two-dimensional model of Fu and Wilcox [3] concerning the axial temperature distribution in a solidifying charge. It is expected, therefore, that the results derived from this model accurately demonstrate the effects of furnace and material property parameters and can be applied to the optimization of both furnace design and execution of growth experiments. The model is easily applied to machine compution. Its primary asset lies in the fact that it yields useful analytical relationships which are relatively simple. For cxample, criteria could thus be derived which define the conditions under which the effects of charge motion, length of charge in the hot and cold zone furnaces, and the generation of latent heat can be neglected; the effects of thickness and thermal conductivity of a confining crucible could be approximated by a simple modification of the Bi parameter. The model indicates that the axial temperature gradient in the liquid near the interface is adversely affected (decreased) by the following charge properties: large thermal conductivity of the charge; large generation of latent heat; large RK (kL/ks). —
T. Jasinski et at.
/
Heat transfer analysis of Bridgman
—
351
Stockbarger growth. 1
Maximization of axial gradients in the charge can be approached through several operational and design options, each of which, however, has its limitations and potential drawbacks: Biot number. Axial thermal gradients can be increased by increasing the heat transfer coefficients (h) in both the hot and the cold zones (eq.
ature changes, the nondimensional interface temperature is altered and results in the interface being shifted into an undesirable region within the gradient zone. Gradient zone length. For a given temperature difference between the hot and cold zones, a decrease in the gradient zone length, A0, will in-
(19)). Since heat transfer in typical high temperature growth experiments is largely controlled by radiation, it is imperative that furnace emissivities be kept high. (Within the gradient zone, however, undesirable radial heat transfer can be reduced through the installation of highly reflecting radiation shields.) Given the approximate third power temperature dependence of the radiation heat transfer coefficient, a desirable increase of Bi may be achieved through an increase of the furnace temperatures. This approach, however, has its limitations in systems which develop high vapor pressures and thus require special crucible materials and construction which, in turn, may reduce the effective Biot number, Heat conduction across the gap between the furnace and the crucible can significantly contribute to the overall heat transfer coefficient if the gap width is small and the intcrjaccnt gas has a high thermal conductivity. For example, at 900°C, He in a one millimeter gap will transfer about the same quantity of heat by conduction as is transferred by radiation. Small gap widths, however, accentuate errors in centering the charge within the furnace cavity and thus may prevent the establishment of axi-symmetric boundary conditions in systems with significant conductive heat transfer in the gap. Hot and cold zone furnace temperatures. An obvious alternative to increasing the Biot number for achieving larger axial gradients is to increase the temperature difference between the hot and cold zones since:
crease the axial gradient, especially for large Bi* * (eq. (18)). A smaller A0, however, will also produce larger radial gradients in the gradient zone [3], and the requirement for the precision with which the interface must be localized in order to satisfy radial gradient criteria will increase. Crucible. To prevent a severe reduction in Bi*, the thermal conductivity of the crucible should be close to that of the charge. In systems with large conductivity differences between the melt and the crystal, the conductivity of the crucible should be chosen so that the nondimensional axial gradient is maximized according to eq. (20). A thin crucible (6 close to unity) also serves to keep Bi* high. If vapor pressure considerations are important, the resultant decrease in crucible strength can be overcome by pressurizing the furnace system. An alternative approach to pressurizing the system when the crucible thickness is small is the use of a metallic crucible coated internally with a chemically inert substance. Charge diameter. An increase in the charge diameter in a given system may either increase or decrease the axial gradient in the liquid in the gradient zone depending on the values of the prevailing thermal parameters. Growth rate. If the generation of latent heat is significant, any decrease in the growth rate (smaller PcSRH) will serve to increase the axial gradient within the melt in the gradient zone (eq. (21)).
dT/d~’=(TfH—Tfc)dO/d~’.
(28)
This approach is contingent on the absence of adverse side effects such as the development of excessive vapor pressures associated with increases of the hot zone temperature; it is also not a viable alternative if, as a consequence of furnace temper-
Acknowledgments .
.
.
The authors wish to express their appreciation to the National Aeronautics and Space Administration (Materials Processing in Space Division, Grant No. NSG 7645) for their financial support and cooperation in this study.
352
T. Jasinski et a!.
/
Heat transfer analysis of Bridgman
Appendix A
—
Stockbarger growth. I
.~H
averaged quantity, eq. (11) becomes: 2O dOm d
In order to obtain a simple analytical cxpression for the effect of the crucible thickness and conductivity on the axial temperature distribution in the charge, the crucible model assumes that the first and second axial temperature derivatives in the crucible arc independent of the radial coordinate, p, and equal to the respective one-dimensional values of the charge. This appendix discusses the validity of these assumptions. Sen and Wilcox [2] determined isotherm shapes in a charge/crucible system for various K and Bi by a two-dimensional finite difference computer model under the following simplified conditions: 6 1.25, A 0 =0, X~1=A~ 0.75, Pc 0, and RK 1.0. The shapes of their computed isotherrns show that the assumptions made in the crucible model arc very reasonable for values of K greater than about 1/2. As K decreases to below 1/2. increasing radial resistance of the crucible produces a pronounced radial temperature distribution in the crucible and the axial gradients in the crucible are larger than those in the charge near the ends of the hot and cold zone. It is therefore expected that the crucible effect, as described by eqs. (7) and (8) will be most in error for small values of K. (It should also be noted that the absence of a gradient zone serves to accentuate the radial gradients in the crucible and therefore the error in the crucible model for small values of K.)
—~
—
Pc
+
—
d~2
4
d~
ép Ip1/2
The boundary condition at the surface of the charge is, in nondimensional form, aO I Bi (9, 6~). (B.3) p 1/2 -~
=
=
—
where 9~is the surface temperature of the charge. Substituting eq. (B.3) into eq. (B.2) yields: d20 ~
—
dO Pe_~j-~
d~2
=
=
(B.2)
=~.
+4 Bi{(Of— ~m)~
=
~
—
On] =0.
(B.4)
=
The crucible effect (section 2.2) is included by using the effective Biot and Péclet numbers in eq. (B.4), thus leading to eq. (10).
Appendix C For an infinite charge length and crystal—melt interface position within the gradient zone, the model of section 2 yields the following analytical expression for the axial gradient in the melt near the growth interface:
F
/dO\ ~~JL.I
=
Appendix B
~
This appendix presents the derivation of eq. (10). In nondimensional form, the heat conduction equation in the charge is given by eq. (11). Integrating each term in eq. (11) according to the formula:
where:
=
[jD/2O(r
8
f
—
m4 [1[i ++(m4/m2— m3 (m3/m1 — 1)1) e~i~h] e~42)]
z) 2~rdrJ/~D2
J
--
(C.l)
m1
=
~Pet{ 1
+ [1 +
16
Bi~*/(Pet)2J
l/2~
2), m2=~Pe~{l —[1 rn, PeL, =
+
16 Bi~*/(Pe~)2]~
m
1/2 =
*
m4 1 [1 + (m4/m2 — 1) cm4~) + Pc~R*Hj
O(p,
~)p
dp,
(B.l)
4=Pe5,
-‘0
where subscript m in eq. (B. 1) denotes an area-
and ~
=
dimensionless interface position with re-
T Jasinski et a!.
/
Heat transfer analysis of Bridgman
spect to the center of the gradient zone;
—
k
~t <
+~t.
The relation between the interface temperature (9) and position (~)is given by: —
o.
m
=
4
1
I
+ (m4/m2
x
—
+
1)
Pc5 R~,
+ (m4/m2
—
—
1
Lowering rate of the charge Axial coordinate
r RH
-I
+ (rn
RKm3 3/m1 — 1)
.
(C.2)
e_m3~) Note that an iterative solution is required if desired to determine given O~ For small Pe*, eq. (17) is satisfied and m
is
it
~,
1 2and m2 can berespectively, approximated 2(Bi~*)l~” 2(Bi~.*)I/2, leadingbyto simplified forms
Thermal conductivity of charge (liquid or solid) kcr/k Length of charge within the hot, gradient or cold zones
zV
q” 1) em4~)
353
R~ RK R R~ T
Pc Pe*
*
Stockbarger growth. 1
Characteristic roots of eq. (3) Péclet number us VD/a Effective Péclet number, eq. (8) Heat flux to the surface of the charge Radial coordinate 2 1)] ~H,,I/cPS(TfH—TfC) RH/[l + K5(6 kL/ks 2 l)]/[l + K 2 1)] ~. [RK/~ + K~(6 5(6 Temperature
m+
em4~1
m4
1
K L
—
—
—
—
of the above expressions: Greek symbols ~
_2+[(Bi*c*)_2+Ac_2~]pesR~}
d~’ L.i
a
xfR* I1B.**~/2+A —2~1 1,
°C 1
K[\
6
ij
0
+A 0 +1/22~’ ~ Pe~R~,[(Bi~*) + ]}1 A0
Thermal diffusivity of charge (liquid or solid) Dcr/D
+ [Bi~*y~2
—0
=
{l
—
I
—
(C.3) 2~~]} -
{ [~~*)
x l+R~ (Bi~Y 1/2 + A0 -
1/2
+
—
2~’II
x0 + 2~i
I
.
(C 4)
Appendix D. Nomenclature
English symbols
Bi Bi* Bi** c,,, D
h
Biot number us hD/k Effective Biot number, eq. (7) Effective Biot number, eq. (13) Specific heat Charge diameter Heat transfer coefficeint between the furnace and outer crucible surface
~H,,1 Latent Dimensionless of axial solidification coordinate, Z/D Length heat of charge in the hot or cold zone for the charge to appear infinitely long ~‘ ~
0
T~/(T~—T~) Dimensionless temperature,
(T
pA 4)
Dimensionless Mass dial coordinate, density in zone r/D eq. length, (1); dimensionless L/D Latent heat parameter defined in eq. (22)ra-
—
Subscripts
cr C f G H i L m s S
Crucible Cold zone Furnace Gradient zone Hot zone Interface Liquid portion of the charge (i.e., the melt) Mean charge temperature Charge surface temperature Solid portion of the charge (i.e., the crystal)
354
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/
Heat transfer analysis of Bridgnnan — Stockbarger growth. I
References
[41KG.
[I] C.E. Chang and W.R. Wilcox, J. Crystal Growth 21(1974)
[61 T.W. Clyne, J. Crystal Growth 50 (1980) 691. [71H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 1st ed., (Oxford Univ. Press, London, 1947) p. 128. [81 J.P. Riquet and F. Durand, J. Crystal Growth 33 (1976) 303.
135.
[21S. Sen and W.R. Wilcox, J. Crystal Growth 28 (1975) 36. [31T.-W. Fu and W.R. Wilcox, J. Crystal Growth 48 (1980) 416.
Davis, Can. Met. Quart. 11(1972)317. [5] T.W. Clyne, J. Crystal Growth 50 (1980) 684.
339
HEAT TRANSFER ANALYSIS OF THE BRIDGMAN-STOCKBARGER CONFIGURATION FOR CRYSTAL GROWTH I. Analytical treatment of the axial temperature profile T. JASINSKI and W.M. ROHSENOW Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
and A.F. WI’T17 Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Received 22 March 1982; manuscript received in final form 6 August 1982
An analytical approach describing the heat transfer during crystal growth in a Bridgman—Stockbarger configuration has been developed. All first order effects on the axial temperature distribution in a solidifying charge are treated on the basis of a one-dimensional model whose validity could be verified through comparison with a published finite difference analysis of a two-dimensional model. The present model includes an insulated region between axially aligned heat pipes and considers the effects of charge diameter, charge motion, thickness and thermal conductivity of a confining crucible, thermal conductivity change at the crystal—melt interface and non-infinite charge length. Results are primarily given in analytical form and can be used without recourse to computer work for both improved furnace design and optimization of growth conditions in a given thermal configuration.
I. Infroduction Conventional techniques for crystal growth from the melt prove increasingly inadequate in meeting property requirements for electronic materials as dictated by advanced device technology. In this context it is of interest that renewed attention is recently being paid to seeded vertical Bridgman growth where the driving force for interfering free melt convection is reduced and the critical axial and radial thermal gradients are at least in principle readily controllable. Motivation for a thorough re-examination of this crystal growth technique is provided by ample evidence in the open literature that basic heat transfer considerations, now recognized to strongly affect both crystalline and chemical perfection of the resulting solid, have been largely ignored in its application, In recent years several thermal analyses of Bridgman-type crystal growth systems have been 0022-0248/83/0000—0000/$03.OO
©
reported in the literature. Noteworthy among these is a series of outstanding publications by Wilcox et al. [1—3]in which the effects of several dimensionless parameters on both axial and radial temperature distributions in the charge were analyzed and significant conclusions drawn. The modeled systems, however, were simplified in several respects and the two-dimensional formulation (except for part of Chang and Wilcox [1]) precluded simple analytical results. Of interest are also the works of Davis [4] and Clyne [5,6] which demonstrate that one-dimensional models can accurately predict experimental axial temperature variations. Their models were not nondimensional and, consequently, significant thermal and geometric parameters which control the system behavior were not identified; their conclusions cannot readily be extended to systems of different parameter values. The present work is part of a combined theoretical and experimental program whose purpose is to
1983 North-Holland
340
T. Jasinski et al.
/
Heat transfer analysis of Bridgman
provide a basis for the design of an improved vertical Bridgman growth configuration. The cxperimental system (presently in construction) makes use of high temperature heat pipes which
—
Stockbarger growth. I
temperature distribution in the charge *, The parameters of primary concern are: thermal coupling between the charge and the furnace; charge lowering rate; generation of latent heat at the growth interface; thermal conductivity difference between the melt —
—
crucible
—
—
hot heat pipe
-
heating coils
and the crystal; thickness and thermal conductivity of the charge-confining crucible; length of the gradient control region; diameter of the charge. Special attention is given to (1) acquiring a large axial temperature gradient at the growth interface in order to preclude constitutional supercooling and (2) on the axial position of the growth interface (a pertinent factor for the interface curvature [1]). The axial temperature behavior of the charge is predicted on the basis of a steady-state one-dimensional thermal model which, not requiring undue —
Q
— —
D grad~nt region
L D
heating
cold heat
Fig. I. Bridgman—Stockbarger configuration with gradient control region.
provide for axial and axi-symmetric temperature uniformity and thus for thermal boundary conditions which permit a meaningful theoretical analysis. It includes, between aligned heat pipes, a gradient control region (fig. 1) in which the crystal—melt interface is to be located. The purpose of this region, first suggested by Chang and Wilcox [1], is to provide control of heat flow near the growth interface, required for the establishment of desired radial thermal gradients. The design and operation parameters are expected to provide for the system under construction a wide range of critical axial and radial temperature gradients leading to growth interface shapes ranging from convex to concave. Primary attention is focused, however, on the establishment of growth conditions which preclude nucleation at the confining boundary with the simultaneous minimization of radial segregation effects. This study examines the effects of both system and operation parameters on the quasi-steady axial
simplification, yields significant results in analytical form. Its primary asset is the timesaving dcment, since meaningful information can be obtained from the analytical results without resort to computer work. Dimensionless parameters are used throughout.
2. Development of heat transfer model The factors of concern for the development of a quasi-steady one-dimensional heat transfer model of the modified Bridgman growth system depicted in fig. 1 are shown in fig. 2. The hot and cold heat pipes comprise the hot and cold zones; the region between them is called the gradient zone. The length of the charge is broken down into LH, L(~, and L~ within the hot, gradient and cold zones respectively. The charge is lowered through the furnace with a velocity V, has liquid and solid portions with different thermal conductivity, and has a crystal—melt interface which generates latent heat. A crucible provides containment for the charge. *
Radial temperature variations in the charge are the subject of a publication (Part II) presently in preparation.
‘
T. Jasinski et al.
I
gradient zone hot zone
~.—i I
Heat transfer analysis of Bridgman
,
i
r—.. cold zone
~
I
— I
z
I
______
L~ 0cr P~
D h~
I
L ‘Liquid
—
341
Stockbarger growth. I
ature distribution when the thermal coupling between the furnace and the charge is large. These effects are approximated in sections 2.2 and 2.3 where it is shown that they can be included in the
I
___
/
0 ~~Lc
Solid
~
~crucible I
hc
Fig. 2. Thermal model of furnace and charge with crucible.
one-dimensional model as a modification of the thermal coupling between the charge and the furnace. With the assumption that the temperature at each cross-section of the charge is uniform, the charge by Carslaw is analagous and Jaeger to the[7]moving and the thinequation rod treated describing the axial temperature distribution is *: 71D2 d2T ~D2 dT —k————————--pVc —+ITDq
4
The thermal model makes the following assumptions: (1) Hot and cold zone furnace temperatures are uniform, reflecting the heat pipe action. The length of the heat pipes is assumed infinite, (2) The system is at all times in a quasi-steady state, i.e., transients are neglected. (3) No heat transfer by natural convection in the melt. (4) Heat exchange between the furnace and the outer crucible surface is described by a heat transfer coefficient, h, that is constant within each zone. (5) The gradient zone is assumed adiabatic, i.e.,
dZ~
4
~dZ
=0.
(1)
In eq. (1), the first term represents axial conduction of heat within the charge, and the second term represents axial convection of heat due to motion of the charge at velocity V. The factor q” in the third term accounts for radial heat transfer to the surface of the charge, per unit surface area of the charge. According to assumption 4 above (and neglecting the crucible effect), q” is given by: h(T~ T). (2) Expressing eq. (1) in nondimensional form yields, =
d20 —i
—
—
Pe~f+ 4 Bi (O~ 8) —
=
0,
(3a)
h 0 0. (6) Thermal properties are not temperature dependent and the melt and crystal mass densities are equal. (7) The one-dimensional model considers only axial heat transfer within the charge, i.e., the temperature within the charge is not a function of the radius. =
2.1. The basic one-dimensional thermal model of the charge
The thermal model for the axial temperature profile of the charge, in its most basic form, neglects the crucible and radial temperature variations within the charge. The crucible, however, does provide a radial thermal resistance between the charge and the furnace, and radial temperature gradients within the charge affect the axial temper-
or, 28 —Pe’-~+9~—9=0,
(3b)
d where us 2W~, Pc’ us ~Pe/~/~I. (Eqs. (3a) and (3b) are equivalent.) ~‘
2.2. Crucible effect In Bridgman growth a charge is confined in a crucible which, depending on conditions, varies in dimension and composition. Containment of the charge tends to decrease axial temperature gradients (Sen and Wilcox [2]). A crucible of low ther*
For an explanation of the symbols used see nomenclature in appendix D.
342
T. Jasinski et a!.
/
Heat transfer analysis of Bridgman
mal conductivity lowers these gradients by adding thermal resistance between the charge and furnace, thereby decreasing the thermal coupling between the charge and the furnace; one of high thermal conductivity lowers these gradients by conducting heat axially within the crucible itself.
transfer The domain within ofthe crucible eq. (3) isapproximates considered toitsbe effect the charge andthe is not explicitly included. only However, a crucible simple model of the heat on q” of eq. (1) which can be expressed through modified Biot and Péclet numbers. The temperature distribution in the crucible obeys the heat conduction equation in cylindrical coordinates:
a
1
“
~
~
2
9Z
V 87~~ —~~:--~-~-
=0.
(4)
The first term of eq. (4) accounts for the radial thermal resistance of the crucible, while the second term accounts for axial conduction of heat within the crucible. At each axial location of the crucible, a7~r/azand d27~r/3Z2 are assumed to be mdcpendent of the radial coordinate (r) and equal to dT/dZ and d2T/dZ2 of the charge (see appendix A). Eq. (4) can thus be integrated to yield a radial distribution of T~rin terms of the axial gradients in the charge. The boundary conditions are: h (T 1—
~~r)
=
kcr
a7~r,/ar,
=
T,
8
.25
0 .01
.8 .6
~
2
10 a K = kcr/k Fig. 3. Effect of the thermal conductivity ratio on the effective Biot number. .10
.01
Bi* ~-~=~l
I +8Bi[~(62_ l)-~ ln(6)] (62
—
1) ~
Bi ln(6)}
(7)
,
I
Pe* =
~l
+
+
~
x
~6
Bi ln(8)
6 Bi
{
K(62
1
~ +
6 Bi
—
1)
{ ~(62
2— l)+
(5a)
at the outer crucible surface, Tcr
Stockbarger growth. I
—
—
—
~
ln(6)]
1)
— ~
—
1)
+
ln(6)]
Bi ln(6)}
— ~.
(8)
+K(6
(Sb)
at the inner crucible surface. Eq. (5b) assumes that the crucible and charge are tn contact. The radial heat flow to the charge in eq. (1), q”, is related to the radial distribution of ~r by: q” kcr a7~r/ar (6) =
at the inner crucible surface. The heat flow given by eq. (6) is substituted for the third term of eq. (1). When nondimensionalized, the resulting expression has the same form as eq. (3) if Bi and Pc are replaced by their respective “effective” values, Bi* and Pe*:
The relationship between the effective Biot number, Bi*, and the conductivity ratio K is shown in 3 for 6 1.25 and reduced various Bi. canlow be seen thatfig.Bi* is significantly by It both and high values of K, especially for larger Bi. (In section 3.2 it will be shown that an increase in Bi* tends to increase the axial gradient at the growth interface.) The conductivity ratio which maximizes Bi* can be obtained from eq. (7): =
K~ = ~~l~6~ii5.
(9)
For typical values of Bi and 6, the conductivity ratio providing the maximum effective Biot is between 0.1 and 1.0.
T. Jasinski et a!.
/
Heat transfer analysis of Bridgman
2.3. Effect of radial gradients in the charge Eq. (1) considers that the temperature at each axial location ~ is constant. Presently, radial gradients in the charge are considered insofar as they affect the thermal coupling between the furnace and the charge and thereby the axial temperature distribution, When radial temperature variations in the charge are taken into consideration, eq. (3) assumes the form (see appendix B): dO d20 Pe* m d~ +4 Bi* [(o Om) (0~ 0m)} 0. (10) —
—
—
—
=
The subscript m denotes an area-averaged value, In eq. (10), the term ~s O~accounts for the effect of radial temperature variations within the charge on the q” term of eq. (1). An approximation for 8,~is obtained through the two-dimensional heat conduction equation in the charge: —
—
)
I p ao~+ p dp i9p / —
—
820
—
Pc 80 0. d~’ —
=
(Ii)
Assuming that the Péclet term is negligible, and, as in the crucible effect, that d20/3~2 is only a weak
—
~s
—
8~=
—
~
(12)
(120m/~2,
Combining eqs. (10) and (12) leads to a new effective Biot number, Bi**, in the third term of eq. (3a) which accounts for radial temperature gradients within the charge as well as for the crucible effect:
Stockbarger growth. I
343
The radial resistance of the charge thus limits the degree to which axial temperature gradients can be increased by augmentation of thermal coupling between the charge and the furnace. (A comparison of axial temperature gradients calculated using eq. (13) with those obtained by the two-dimensional model of Fu and Wilcox [3] is provided in section 3.7.) 2.4. Solution
Eq. (3) is applied separately to each region of uniform properties and effective Biot number, Bi**. In this way, for each such region, eq. (3) has constant coefficients. In the present model there are four such regions when the growth interface is located in the gradient zone: the hot and cold zones and the liquid and solid parts of the gradient zone. The solution of eq. (3) within each region yields two exponential terms for the homogeneous solution and a particular solution that depends on 8~.The constants of integration on the exponential terms are found using boundary conditions of equality of temperature and continuity of flux between adjacent regions. At the crystal—melt interface, the flux condition gives: dO [K 5(62
function of the radial coordinate p, eq. (Il) can be 20m/d~2as integrated in the radial direction with d a variable. A parabolic variation for 0 results from which 0s can be determined as:
—
—
1) + RKI
(~
)L
2 1)] (~). (14) 5 RH + [i + K~(6 The Péclet number in eq. (14) is correctly based on the actual growth rate, assumed here to be equal to =
Pc
—
the lowering rate V. (This equality exists in systems with infinite charge length and during growth
(13)
experiments which fix the interface location by appropriate adjustments of the furnace temperatures.) The constants of integration have been determined analytically for systems of infinite charge length; expressions for the axial gradient in the
According to eq. (13), Bi** Bi* for Bi* zg~~ i.e., radial temperature variations within the charge do not affect the axial temperature distribution in the charge by virtue of the effective Biot number. Eq. (13) indicates also that the maximum value that Bi** can attain is 8 when Bi* is very large.
melt and for the axial position of the crystal—melt interface of such systems are given in appendix C. If the complete axial temperature distribution is desired, or if the charge length is not infinite, it is more convenient to determine the constants of integration by computer.
Bi**
=
Bi*/[ 1
+
(Bi*/8)]
.
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/
Heat transfer analysis of Bridgman
3. Analysis
—
Stockbarger growth. I
metric system is: 0(~)+O(—~’)= 1.
This section considers the effects of the thermal parameters of the Bridgman growth system on the axial temperature behavior of the charge. Axial temperature distributions in the charge were calculated by computer using the model described in section 2; analytical expressions for the axial temperature gradient in the melt near the growth interface, in systems of infinite charge length, are taken from the results presented in appendix C. In consideration of the many variables in the thermal model, a system with idealized parameter values, presently called a “symmetric” system, is used as a reference against which the effects of individual parameters will be compared. A symmetric system is defined as follows: (1) Equal heat transfer coefficients in the hot and cold zones, (2) Axial convection of heat due to the charge motion is negligible, i.e., Pc is small. (See section 3.1.) (3) The effect of the generation of latent heat is negligible (PeSRH 0). (See section 3.4.) (4) Equal thermal conductivities of the melt and crystal (RK 1). With condition 1 above, Bi~~* Bi~*. (5) Equal charge lengths in the hot and cold zones (XH Ar). The axial temperature distribution for a sym-
3.1. Péclet number effect
The Péclet number expresses the ratio of axial heat transfer in the charge due to the lowering velocity, V, and due to conduction. The effective Péclet number, Pe*, incorporates the effect of the crucible on the convection term of eq. (3). The unmodified Péclet number is used to describe the generation of latent heat (refer to eq. (14)). The effect of Pe* on the axial temperature distribution in an otherwise symmetric system. as obtained through eq. (3), is shown in fig. 4. It is seen that Pe* tends to increase the temperature of the charge at all locations; the effect is more pronounced as Pe* increases (e.g., high lowering rate or low thermal conductivity of the charge). As a consequence, in systems with fixed hot and cold zone furnace temperatures (i.e., the nondimensional melting point temperature is constant), the growth interface moves toward the cold zone as Pe* increases; alternatively, the nondimensional melting point temperature must increase (i.e., one or both of the furnace temperatures must be lowered) if the interface is to remain at the same axial location as Pe* increases. In typical Bridgman growth experiments, the
—
=
=
=
9 10
B ~
~
I
symmetric
I
I
Hot Zone
_______________
‘Gradient
(15)
Cold Zone
Zone Fig. 4. Effect of charge motion on the axial temperature distribution in a charge.
T. Jasinski et at.
/
Heat transfer analysis of Bridgman
Péclet effect is small since the lowering rates are small (e.g., 0,1—10 ~tm/s). A test criterion for its relative magnitude is provided through the characteristic roots of eq. (3b): — t In m ±— 2 ~ e *\‘f ~ 1
L —
I 1 +
4 /~~
e *\~\2l
/2
\ f’
11 ~\)
It can be seen that Pc* disappears from eq. (16) when, (17) Pe*/4(Bi**)~’2~< 1. =
Eq. (17) agrees with the findings of Chang and Wilcox [1] who reported that the Péclet effect is stronger for smaller Biot numbers. If the inequality in eq. (17) holds, the Péclet number is small enough to satisfy the Péclet number criterion for symmetric systems (i.e., Pc 0). Further, when the Péclet effect is negligible, simplified analytical expressions are obtained for the axial gradient within the melt in the gradient zone (appendix C). Since eq. (17) does hold for typical Bridgman growth, these simplified expressions will be used in the remainder of this section.
—
345
Stockbarger growth. 1
of the crucible and radial temperature gradients in the charge on this thermal coupling. Typical values for the effective Biot number range from 0.05 for high conductivity materials such as Ge to 5.0 for low conductivity materials such as CdTe. Axial temperature profiles for several Bi** are shown in fig. 5. In agreement with the results of Chang and Wilcox [1], it is found that the charge temperature follows more closely the furnace temperature and, as a result, the axial temperature gradient in the gradient zone increases, as Bi** increases. The expression for the axial gradient near the crystal—melt interface presented in appendix C for infinite charge length can be simplified for symmetric systems: .
d0/d~
=
[AG
+
(Bi**)
—
1/2
]
—
(18)
.
The dependence of the axial gradient at the growth interface on Bi**, according to eq. (18), is plotted in fig. 6 for various gradient zone lengths A 0. The curves show that the dependence of the axial gradient on A0 becomes stronger with increasing Bi**. It can also be seen (curve A0 0) that there exists a minimum Bi** for any desired nondimensional axial gradient. The heat transfer coefficients of the hot and cold zones are generally not equal due, for examplc, to the temperature dependence of radiative heat transfer. For such conditions, the zone with =
3.2. Bior number effect
The Biot number, through the heat transfer coefficient h, is a direct measure of the thermal coupling between the charge and the furnace. The effective Biot number, Bi**, incorporates the effect
R
Of ~A ~
:
~0I0\
HC
I I
Symmetric system
Hot Zone
I
:
1Gradient
Cold Zone
Zone
Fig. 5. Effect of thermal coupling between furnace and charge, Bi**, on the axial temperature distribution within the charge.
346
T. Jasinski et a!.
/
Heat transfer analysis of Bridgman
Symmetric systems
Stockbarger growth. I
gradient zone for a system with infinite length which is symmetric except hH us h1~-is:
__~
XH
—
~
~
d8/d~= {x0 g
0
(19) Biot numbers has a greater effect on decreasing I
-
~05
+~[(Bi~*)_1/2+(Bi~*)_I/2J~’.
____
Eq. (19) indicates that the smaller of the effective the axial gradient. Efforts to increase the axial gradient by increasing the heat transfer coefficients should therefore first be directed at the zone with the smaller Bi**.
_______
___________________________________________
o.bi
0.1 0 0 Bi~* Fig. 6. Effect of gradient zone length and Bi** on the axial temperature gradient in the melt near the growth interface, 0001
3.3. Effect of unequal thermal conductivity of the crystal and the melt (RK us 1) RK expresses the ratio of melt to crystal thermal conductivity. Semiconductors have RK values greater than unity whereas, for metals, the value of RK is less than unity. The effect of RK on the axial temperature distribution in the charge according to eq. (3) is demonstrated in fig. 8 for systems with 6 1 and which are otherwise symmetric (i.e., BiH Bic/RK). The charge phase with the higher thermal conductivity exhibits in all instances a lower axial gradient because of its lower thermal resistance to heat transfer in the axial direction. The axial gradient in the melt near the interface is therefore less than in the crystal for charges with RK> 1. The expression for the axial gradient in the melt
the larger effective Biot number more strongly influences the overall temperature level (fig. 7). Compared to the symmetric case where Bi~* Bi~.*,the charge temperature for asymmetric systems increases when Bi~*> Bi~*and decreases when Bi~*> Bi~~*.If the location of the crystal—melt interface for each curve of fig. 7 is considered to be at the center of the gradient zone, the effect of unequal Biot numbers appears as a change in nondimensional interface temperature. To retain the interface in a given location requires lowering of the hot and/or cold zone furnace temperatures as Bi~~* increases relative to Bi~*. The expression for the axial gradient in the =
9f~
=
=
a I
1.01 I
.751
hH~hc System otherwise symmetric
XH~XC~co Bic*~a.0I
25
I
I
I
Hot Zone
I
I
of
Bi~**I.0 ______________________________________
Gradient
Cold Zone
Fig. 7. Effect of unequal hot and cold zone Bi** on the axial temperature distribution in the charge.
/
T. Jasinski et at.
Heat transfer analysis of Bridgman
—
347
Stockbarger growth. I
a II
10
I
Bi~*O.l
I
BiH*Bic/Re System otherwise symmetric
I
XH~XC~O
I
8
a
I25~ Interface Iocation—,~ I
I
Hot Zone
__________________________________
Gradient
Zone
Cold Zone
Fig. 8. Effect of non-equal thermal conductivity of crystal and melt on the axial temperature gradient in the melt near the interface.
near the growth interface for a system of infinite length which is symmetric except RK us I is: /
dO
\
~.)L
_2{R~[(Bi~*Y~2+AG_
2~~] ~BIc
~ (20)a A comparison of eqs. (20) and (18) shows that
0
0 l.a
value of RK greater than unity is detrimental to the establishment of large axial gradients in the melt. The axial gradient as a function of RK, according to eq. (20), is plotted in fig. 9 for several values of Bi0. It is found, for example, that this conductivity effect reduces the gradient in a germanium melt (RK 2.5) by about 50%. 3.4. Latent heat effect (PeSRH ~
The quantity of latent heat liberated at the crystal—melt interface is given in nondimensional form by the product Pe5RH (eq. (14)). The expression for the axial gradient in the melt near the growth interface in a system of infinite length which is symmetric except PeSRH us 0 is: /dO\
_2+PesR~[(Bi**)/2+Ao_2~j] 2[(Bi**)_~2+A0] (21)
0.1 Bi~BIc/Re(hence,b’I)
System otherwise symmetric
~. ~
X~= .0 XH= X 5 =0’
0.01 001
0.01
10
10
RK*kL/ks Fig. 9. Effect of thermal conductivity ratio (RK) and Bi on the axial temperature gradient in the melt near the growth interface.
It can be seen that the effect of latent heat on the axial gradient (and also on the axial temperature profile) is small if: i PesR~[(Bi**)_1/2+AO_2~ij<<1. (22) Eq. (22) demonstrates that the effect of latent heat on the axial temperature behavior is larger for
348
/
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—
Stockbarger growth. I
8 1/2 (pesRHd,[(Bi**) I/2~~ XG-2~
I
Pe
5R~I System otherwise symmetric Bi~=0.I XH”Xcas 2/Peus’I 4B/” I
I I I
Interface location—..,
I
I
Hot Zone
_______ _____
Gradient
~
~. I __________________________
zoneCh
I
Cold Zone
C
Fig. 10. Effect of the generation of latent heat at the growth interface on the axial temperature distribution in the charge.
smaller Bi** and for larger lowering rates of the charge (Pc 5 V). Axial temperature profiles for several values of 4) arc given in fig. 10. It is seen that the generation of latent heat both increases the charge temperature and decreases the axial gradient in the melt; the latent heat effect disappears for small values of 4). (The effect of latent heat and the Péclet number effect are coupled through their mutual dependence on the lowering rate, V. In order to isolate the latent heat effect in fig. 10, Pc is chosen sufficiently small so that eq. (17) is satisfied. Such
a small value for Pc would normally also eliminate the latent heat effect; i.e., eq. (22) would also be satisfied. The values of 4) in fig. 10 were therefore obtained by choosing large values of RH.) 3.5. “Infinite” charge length When the charge is infinitely long, the temperature field of the charge does not change during growth and, hence, the interface position remains fixed, Fig. II shows the progression of axial ternperature profiles as the charge moves from the hot
0 A 1.00
System otherwise symmetric (i.e.,high thermal coductivity) =0.05)
0.15
=0.005 (4B,~’°/Pe 79) tal charge length =
0.50 Hot end of charge 0.25
-10
-8
-6
-4
Hot Zone
-2
-0
sO
IGradi Zone I
2
4
6
8
Cold Zone
Fig. 11. Effect of charge position within the furnace on the axial temperature distribution.
0
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/
Heat transfer analysis of Bridgman
zone to the cold zone for a charge length less than infinite. It can be seen that charge temperatures are displaced toward the temperature of the hot zone furnace when most of charge is in the hot zone and vice-versa. Accordingly, to achieve constant interface position for non-infinite charge lengths, the nondimensional melt temperature must be reduced as the experiment proceeds. At constant nondimensional melt temperature, the interface growth velocity will be greater than the lowering rate as the interface moves from the cold zone to the hot zone. The contribution from the positive characteristic root, eq. (16), to the solution of eq. (3) is normally small and is zero for an infinite charge length. The charge thus appears infinite in length when the contribution from the negative root also becomes small, i.e., for large The temperature change within the hot or cold zone reaches approximately 99% of its final value when exp[(mj~’]= 0.01, Using this as a criterion for infinite length: 5/rn (23) ~“.
=
‘
where length of charge within the hot or cold zone for the charge to appear infinitely long. If Pe* is small so that eq. (17) is satisfied, the characteristic roots become: ~
m
±=
±1,
(24)
and by substitution we find: 2, (25) 5 or 5/2(Bi**)” which is a useful expression for determining L.
349
Stockbarger growth. 1
—
propriate boundary conditions must be applied to the ends of the charge. For example, a solid pull rod contacting the crucible at the bottom of the charge can be approximated by treating it as a simple fin exchanging heat with its environment. In this way, a Biot number can be calculated for the end boundary condition. The curves of fig. 11 were calculated using the same Biot number for the ends as for the circumference. 3.6. Charge diameter effect
Any change in the charge diameter, D, affects the dimensionless parameters Bi, Pc, 6, A0, ~ and ~. Further, changes in D may alter the heat transfer coefficients between the furnace and charge as the geometry of the furnace cavity changes. The effect of a change in D can be assessed by reevaluating the necessary parameters and then using the appropriate expression for the axial ternpcrature gradient. In general, increases in D may either increase of decrease the axial temperature gradient in the gradient zone depending on the corresponding changes in the other parameters. As an illustrative example, consider a symmetnc system of infinite charge length. The appropriate expression for the axial temperature gradient in the gradient zone is given by eq. (18). Since, however, changes in diameter are to be analyzed, it is more informative to compare the axial of eq. (18) based on Z, the dirnensionalgradients axial coordinate:
~
dO/dZ
=
—
1 [L 0
Eq. (25) is in agreement with the results of Clyne [5,6] which suggest that longer charges and higher Biot numbers will tend to stabilize the interface position (i.e., the growth rate is equal to the lowering rate). Results from the one-dimensional numerical model of Riquet and Durand [8] suggest that the growth and lowering rates are equal when the length of charge2).This within result the hotagrees and cold is well zones with eq. about the l.5(Bi’~ (25); difference in the constant coefficient is attributed to the choice of criteria defining “infinite” charge length. When the charge length is not infinite, ap-
+ D/(Bi**)~2]
- ~.
(26)
.
Defining the variable: d{D/(Bi**)I/21/dD
(27)
and taking L0 as constant, eq. (26) shows that the axial gradient increases with D when s~is negative and vice-versa. For any increase in D accompanied by a proportional increase Dcr Bi** (6 remains constant), (7) and (13) show inthat does not increaseeqs. as rapidly as D (assuming all other thermal parameters are constant). Consequently (eq. (27)), s~ is positive and the axial gradient according to eq. (26) decreases as D increases.
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Heat transfer analysis of Bridgman
—
Stockbarger growth. I
Table 1 Axial thermal gradients at the crystal—melt interface as obtamed through eqs. (13) and (18), d9m/d~,and average axial thermal gradients (d8/3~)~~, as determined from the two-dimensional model of Fu and Wilcox [31
Si’ .05
so (~O/3~)
d6m/d~
-5
sy~net~c system
SI’
-i 001
.0
05
Bi =2
A~= 0.0
1.310
1.260
Bi
A~ 0.0
0.616
0.617
0.4
=
10
K
5’ kcr/ki
Fig. 12. Effect of Bi and K on the parameter constant D..
4’
(eq. (27)) for
The crucible diameter may, on the other hand, remain constant as D increases. This occurs when increasing the charge diameter within a constant diameter furnace cavity at the expense of the thickness of the crucible. Fig. 12 shows the corresponding behavior of 4’ for various Bi and K5 with 6 1.25. Accordingly, sji is either positive or negative depending on the values of Bi and K5 the axial gradient of eq. (26) may therefore either increase or decrease with an increase in D. =
3.7. Comparison of one- and two-dimensional models
The results of the present one-dimensional model can be compared to the two-dimensional finite difference results of Fu and Wilcox [3]. Fig. 3 of their paper presents the radial variation of the axial gradient at the growth interface of symmetric systems of infinite charge length in the absence of a crucible. Point values, estimated from the curves of their fig. 3, were numerically integrated to obtain the average axial gradient over the cross-section and were compared to the values obtained using eqs. (13) and (18) for several values of Bi and A0 (see table 1). Considering the approximations required in the derivation of eq. (13), the agreement of the data must be taken as excellent. The comparison demonstrates that the onedimensional models presently used provide a meaningful representation of the parameters
governing the axial temperature behavior of a . charge in a Bridgman-type growth configuration. 4. Discussion and conclusions The one-dimensional model developed in section 2 has been shown to correlate well with a corresponding two-dimensional model of Fu and Wilcox [3] concerning the axial temperature distribution in a solidifying charge. It is expected, therefore, that the results derived from this model accurately demonstrate the effects of furnace and material property parameters and can be applied to the optimization of both furnace design and execution of growth experiments. The model is easily applied to machine compution. Its primary asset lies in the fact that it yields useful analytical relationships which are relatively simple. For cxample, criteria could thus be derived which define the conditions under which the effects of charge motion, length of charge in the hot and cold zone furnaces, and the generation of latent heat can be neglected; the effects of thickness and thermal conductivity of a confining crucible could be approximated by a simple modification of the Bi parameter. The model indicates that the axial temperature gradient in the liquid near the interface is adversely affected (decreased) by the following charge properties: large thermal conductivity of the charge; large generation of latent heat; large RK (kL/ks). —
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/
Heat transfer analysis of Bridgman
—
351
Stockbarger growth. 1
Maximization of axial gradients in the charge can be approached through several operational and design options, each of which, however, has its limitations and potential drawbacks: Biot number. Axial thermal gradients can be increased by increasing the heat transfer coefficients (h) in both the hot and the cold zones (eq.
ature changes, the nondimensional interface temperature is altered and results in the interface being shifted into an undesirable region within the gradient zone. Gradient zone length. For a given temperature difference between the hot and cold zones, a decrease in the gradient zone length, A0, will in-
(19)). Since heat transfer in typical high temperature growth experiments is largely controlled by radiation, it is imperative that furnace emissivities be kept high. (Within the gradient zone, however, undesirable radial heat transfer can be reduced through the installation of highly reflecting radiation shields.) Given the approximate third power temperature dependence of the radiation heat transfer coefficient, a desirable increase of Bi may be achieved through an increase of the furnace temperatures. This approach, however, has its limitations in systems which develop high vapor pressures and thus require special crucible materials and construction which, in turn, may reduce the effective Biot number, Heat conduction across the gap between the furnace and the crucible can significantly contribute to the overall heat transfer coefficient if the gap width is small and the intcrjaccnt gas has a high thermal conductivity. For example, at 900°C, He in a one millimeter gap will transfer about the same quantity of heat by conduction as is transferred by radiation. Small gap widths, however, accentuate errors in centering the charge within the furnace cavity and thus may prevent the establishment of axi-symmetric boundary conditions in systems with significant conductive heat transfer in the gap. Hot and cold zone furnace temperatures. An obvious alternative to increasing the Biot number for achieving larger axial gradients is to increase the temperature difference between the hot and cold zones since:
crease the axial gradient, especially for large Bi* * (eq. (18)). A smaller A0, however, will also produce larger radial gradients in the gradient zone [3], and the requirement for the precision with which the interface must be localized in order to satisfy radial gradient criteria will increase. Crucible. To prevent a severe reduction in Bi*, the thermal conductivity of the crucible should be close to that of the charge. In systems with large conductivity differences between the melt and the crystal, the conductivity of the crucible should be chosen so that the nondimensional axial gradient is maximized according to eq. (20). A thin crucible (6 close to unity) also serves to keep Bi* high. If vapor pressure considerations are important, the resultant decrease in crucible strength can be overcome by pressurizing the furnace system. An alternative approach to pressurizing the system when the crucible thickness is small is the use of a metallic crucible coated internally with a chemically inert substance. Charge diameter. An increase in the charge diameter in a given system may either increase or decrease the axial gradient in the liquid in the gradient zone depending on the values of the prevailing thermal parameters. Growth rate. If the generation of latent heat is significant, any decrease in the growth rate (smaller PcSRH) will serve to increase the axial gradient within the melt in the gradient zone (eq. (21)).
dT/d~’=(TfH—Tfc)dO/d~’.
(28)
This approach is contingent on the absence of adverse side effects such as the development of excessive vapor pressures associated with increases of the hot zone temperature; it is also not a viable alternative if, as a consequence of furnace temper-
Acknowledgments .
.
.
The authors wish to express their appreciation to the National Aeronautics and Space Administration (Materials Processing in Space Division, Grant No. NSG 7645) for their financial support and cooperation in this study.
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T. Jasinski et a!.
/
Heat transfer analysis of Bridgman
Appendix A
—
Stockbarger growth. I
.~H
averaged quantity, eq. (11) becomes: 2O dOm d
In order to obtain a simple analytical cxpression for the effect of the crucible thickness and conductivity on the axial temperature distribution in the charge, the crucible model assumes that the first and second axial temperature derivatives in the crucible arc independent of the radial coordinate, p, and equal to the respective one-dimensional values of the charge. This appendix discusses the validity of these assumptions. Sen and Wilcox [2] determined isotherm shapes in a charge/crucible system for various K and Bi by a two-dimensional finite difference computer model under the following simplified conditions: 6 1.25, A 0 =0, X~1=A~ 0.75, Pc 0, and RK 1.0. The shapes of their computed isotherrns show that the assumptions made in the crucible model arc very reasonable for values of K greater than about 1/2. As K decreases to below 1/2. increasing radial resistance of the crucible produces a pronounced radial temperature distribution in the crucible and the axial gradients in the crucible are larger than those in the charge near the ends of the hot and cold zone. It is therefore expected that the crucible effect, as described by eqs. (7) and (8) will be most in error for small values of K. (It should also be noted that the absence of a gradient zone serves to accentuate the radial gradients in the crucible and therefore the error in the crucible model for small values of K.)
—~
—
Pc
+
—
d~2
4
d~
ép Ip1/2
The boundary condition at the surface of the charge is, in nondimensional form, aO I Bi (9, 6~). (B.3) p 1/2 -~
=
=
—
where 9~is the surface temperature of the charge. Substituting eq. (B.3) into eq. (B.2) yields: d20 ~
—
dO Pe_~j-~
d~2
=
=
(B.2)
=~.
+4 Bi{(Of— ~m)~
=
~
—
On] =0.
(B.4)
=
The crucible effect (section 2.2) is included by using the effective Biot and Péclet numbers in eq. (B.4), thus leading to eq. (10).
Appendix C For an infinite charge length and crystal—melt interface position within the gradient zone, the model of section 2 yields the following analytical expression for the axial gradient in the melt near the growth interface:
F
/dO\ ~~JL.I
=
Appendix B
~
This appendix presents the derivation of eq. (10). In nondimensional form, the heat conduction equation in the charge is given by eq. (11). Integrating each term in eq. (11) according to the formula:
where:
=
[jD/2O(r
8
f
—
m4 [1[i ++(m4/m2— m3 (m3/m1 — 1)1) e~i~h] e~42)]
z) 2~rdrJ/~D2
J
--
(C.l)
m1
=
~Pet{ 1
+ [1 +
16
Bi~*/(Pet)2J
l/2~
2), m2=~Pe~{l —[1 rn, PeL, =
+
16 Bi~*/(Pe~)2]~
m
1/2 =
*
m4 1 [1 + (m4/m2 — 1) cm4~) + Pc~R*Hj
O(p,
~)p
dp,
(B.l)
4=Pe5,
-‘0
where subscript m in eq. (B. 1) denotes an area-
and ~
=
dimensionless interface position with re-
T Jasinski et a!.
/
Heat transfer analysis of Bridgman
spect to the center of the gradient zone;
—
k
~t <
+~t.
The relation between the interface temperature (9) and position (~)is given by: —
o.
m
=
4
1
I
+ (m4/m2
x
—
+
1)
Pc5 R~,
+ (m4/m2
—
—
1
Lowering rate of the charge Axial coordinate
r RH
-I
+ (rn
RKm3 3/m1 — 1)
.
(C.2)
e_m3~) Note that an iterative solution is required if desired to determine given O~ For small Pe*, eq. (17) is satisfied and m
is
it
~,
1 2and m2 can berespectively, approximated 2(Bi~*)l~” 2(Bi~.*)I/2, leadingbyto simplified forms
Thermal conductivity of charge (liquid or solid) kcr/k Length of charge within the hot, gradient or cold zones
zV
q” 1) em4~)
353
R~ RK R R~ T
Pc Pe*
*
Stockbarger growth. 1
Characteristic roots of eq. (3) Péclet number us VD/a Effective Péclet number, eq. (8) Heat flux to the surface of the charge Radial coordinate 2 1)] ~H,,I/cPS(TfH—TfC) RH/[l + K5(6 kL/ks 2 l)]/[l + K 2 1)] ~. [RK/~ + K~(6 5(6 Temperature
m+
em4~1
m4
1
K L
—
—
—
—
of the above expressions: Greek symbols ~
_2+[(Bi*c*)_2+Ac_2~]pesR~}
d~’ L.i
a
xfR* I1B.**~/2+A —2~1 1,
°C 1
K[\
6
ij
0
+A 0 +1/22~’ ~ Pe~R~,[(Bi~*) + ]}1 A0
Thermal diffusivity of charge (liquid or solid) Dcr/D
+ [Bi~*y~2
—0
=
{l
—
I
—
(C.3) 2~~]} -
{ [~~*)
x l+R~ (Bi~Y 1/2 + A0 -
1/2
+
—
2~’II
x0 + 2~i
I
.
(C 4)
Appendix D. Nomenclature
English symbols
Bi Bi* Bi** c,,, D
h
Biot number us hD/k Effective Biot number, eq. (7) Effective Biot number, eq. (13) Specific heat Charge diameter Heat transfer coefficeint between the furnace and outer crucible surface
~H,,1 Latent Dimensionless of axial solidification coordinate, Z/D Length heat of charge in the hot or cold zone for the charge to appear infinitely long ~‘ ~
0
T~/(T~—T~) Dimensionless temperature,
(T
pA 4)
Dimensionless Mass dial coordinate, density in zone r/D eq. length, (1); dimensionless L/D Latent heat parameter defined in eq. (22)ra-
—
Subscripts
cr C f G H i L m s S
Crucible Cold zone Furnace Gradient zone Hot zone Interface Liquid portion of the charge (i.e., the melt) Mean charge temperature Charge surface temperature Solid portion of the charge (i.e., the crystal)
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Heat transfer analysis of Bridgnnan — Stockbarger growth. I
References
[41KG.
[I] C.E. Chang and W.R. Wilcox, J. Crystal Growth 21(1974)
[61 T.W. Clyne, J. Crystal Growth 50 (1980) 691. [71H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 1st ed., (Oxford Univ. Press, London, 1947) p. 128. [81 J.P. Riquet and F. Durand, J. Crystal Growth 33 (1976) 303.
135.
[21S. Sen and W.R. Wilcox, J. Crystal Growth 28 (1975) 36. [31T.-W. Fu and W.R. Wilcox, J. Crystal Growth 48 (1980) 416.
Davis, Can. Met. Quart. 11(1972)317. [5] T.W. Clyne, J. Crystal Growth 50 (1980) 684.