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Crystallization stability during capillary shaping
Journal of Crystal Growth 50 (1980) 33—44 © North-Holland Publishing Company
CRYSTALLIZATION STABILITY DURING CAPILLARY SHAPING I. General theory of capillary and thermal stability V.A. TATARCHENKO and E.A. BRENER The Solid State Physics Institute of the Academy of Sciences of the USSR, Chernogolovka 142432, Moscow district, USSR Received 22 March 1978; manuscript received in final form 1 June 1979
When shaped crystals are grown from the melt, their cross section is determined by the shape of the melt column. The crystal radius and the height of the crystaffization front may vary independently, i.e. we deal with a system of two degrees of freedom. The stability of this system is investigated by Liapunov’s method. The concepts of capillary and thermal stabifities are introduced. The stability of the system depends on the shape of the pulled crystal, the boundary conditions for the melt column, and the thermal conditions during pulling.
1. Introduction
Some preliminary results on the problem of the process stability were reported earlier [6,9,211. Here, we give a detailed investigation of the problem; the analyses of the capillary and thermal problems will enable us to obtain optimal parameters for the crystal growth process.
There is considerable current interest in methods of growing crystals from the melt using the effect of capillary shaping (fig. la). Various methods of operation are known as the Stepanov method [1,2] or EFG [3]. In all the variants of the method, as well as in the Czochralski technique, the cross-sectional size and shape of the crystal are determined by the shape of the melt column and the height of the crystallization front. As shown earlier [4—6],a coupling of the cross-sectional size and the position of the crystallization front may be obtained from the solution of the Laplace capillary equation with the appropriate boundary conditions. Another relationship between the above parameters can be obtained from the solution of the heat flow problem for the crystal—melt system [7,8]. Therefore, simultaneous solution of the heat flow and capifiary problems enables one to find both the height of the crystallization front and the crystal diameter (or some characteristic dimension of the cross-section), as well as their dependence on the control parameters of the process. The question of uniqueness and stability of the solution is a significant one; this can also be addressed by the above analysis.
N’
t lIE itt — — —
it
—
T±1II
11
-
° .
.
Fig. 1. Schematic of crystal growth conditions at the crystal— melt interface: (a) crystal growth with constant cross section; (b) narrowing crystal. 33
34
V.A. Tatarchenko, E.A. Brener
/ Crystallization stability during capillary shaping.
2. General stability considerations In our case, the crystal radius R and the crystallization front height h may vary independently, i.e. we deal with a system of two degrees of freedom,
I
melt surface at the crystallization front possesses a definite value for a given crystal [11,12]. Let us denote this angle by ~ If the angle between the horizontal and the tangent to the melt surface at the crystallization front is a0 = = ir/2 ~ (fig. la), —
According to Liapunov [10], to investigate the stability of such a system, it is necessary to investigate the system of two linear differential equations for the rates of change of the crystal radius 6R and the position of the crystallization front ~5hdue to small devia. lions ~R and ~ in the system variables from equilib-
the crystal grows with constant cross section. Any change in the height of the crystallization front or an occasional change in crystal radius leads to a0 ~ a~ and causes the crystal radius change according to the relationship (fig. ib):
rium:
~R=~V6ao=—V~J5R—V~6h.
6R
=ARR
6R +ARn ~h,
a
a
(5)
(1) where &io = ae a0 and Vis the pulling rate. Coeffi. cients ARR = ~ a~0/aR and AR,, = V a~0/ah —
~h AhR ~R +Ahh ~h. (2) The solution of the system of equations (1) and (2) is given by: =
C1 exP(ii1t)
+
C2 exP(n2t).
(3)
6h C3 exp(r~jt)+C4exp(02t), where t is time, C, are the integration constants, and the quantities i~ and z~2are the roots of the characteristic equation: —
n
(ARR +A,,~) +
(A RRA hh
—
AR,,AhR)
=
0. (3a)
—
—
can be determined from the solution of the capillary problem [4—6].
4. Capillary coefficients ARR and AR,, An analysis of the signs of the capillary coefficients ARR and AR,, is presented in appendix A. The results are summarized in table 1. Coefficient AR,, is
For the system to be stable, the following necessary and sufficient conditions have to be satisfied [10]: ARR +A,,,, <0, ARRA,,,, AR,,A,,R >0. (4)
negative for all profiles and boundary conditions. The sign of the coefficient ARR depends on the type of the pulled profile and the boundary condition.
The above requires that the real parts of the roots of the characteristic equation (3a) be negative, and hence the perturbations will be damped with time. Coefficients ARR and A,,,, reflect the self-stabiliza~ion of the system parameters, i.e. connections of 6R to ~R, and ~ihto ~ Coefficients AR,, and AhR reflect the mutual stabilization parameters. When A RRA hh < 0, i.e. self-stabilization of one of the parameters is lacking, stability is still possible with mutual stabiization. In order to find stable conditions of growth, it is necessary to derive expressions (1) and (2) in a more recognizable form.
5. Rate of change of crystallization front position
—
A recognizable form of eq. (2) can be obtained as a consequence of solving the non-stationary heat conduction problem for the solid and liquid phases. The temperature distribution in a pulled profile can be rather precisely described by means of the one-dimensional equation of heat conduction [7]. Crystal anisotropy and dependence of thermal characteristics on temperature are neglected in this analysis. The heat flow equation is of the form [13]. 2T, V P aT,8 ~,1 at az ~, az X,R i~ en)’ where i=l, 2 (i1 for the melt, and i2 for the crystal), Z is the coordinate along the pulling direc. tion, iq is the thermal diffusivity, X 1 is the thermal —
3. Rate of change of the crystal radius When pulling crystals from the melt, the angle between the crystal surface and the tangent to the
——
—
‘..
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. I
35
Table 1 Signs of the capillary coefficients No.
Profile
Boundary conditions
Fig.
ARR
ARh
1
Ribbon
Catching
4
<0
<0
2
Ribbon
Wetting r 0—R>m rO—RcEi r0—R
5a,b 5a Sb
<0 >0 <0
<0
3
4 5
Filament; rod
Catching RRm
6
Filament; rod
Wetting
7
>0 <0 <0 >0
Rod; Czochralski method
8
conductivity, p, is the radiation heat coefficient,R is the characteristic size of the crystal (radius for a round crystal or half-thickness for a plate), and Ten is the temperature of the environment. Linearization of the radiation term in the heat conduction equation enables us to use eq. (6) in the presence of radiative heat exchange [14]. The boundary conditions of our system are as fol-
>0
lows (fig. 2): T, ‘zO = Tm
<0
(7)
,
TiIzn(t) = T0 (8) Here Tm is the melt temperature, T0 is the crystallization temperature, and h(t) is the crystallization front coordinate. Various boundary conditions are possible at the far end of the crystal: .
(9)
T2!z~ooTen,
(10)
T2Iz~riTen,
V
aT2IazI~,= 0 Here
1
.
(11)
is the length of the crystal. The condition of
front:
heat balance must be satisfied at the crystallization T
__________
______
h
—X2G2(Z)lz=h + X1G1(Z)Iz—~=Lp(V— (12) Here G~(Z) aT1/&Z,L is the latent heat dh/dt). of crystallization, and p is the density of the solid. The solution
1
form thesolution thermal coefficients inwill eq.determine (2). However, of theofabove heat flow problem the a rigorous meets with mathematical difficul-
— ______________ --
ties. -
--
—-
--
Fig. 2. Schematic of boundary conditions for the heat flow problem: (1) idealized and (2) real form of melt column.
We can apply an important approximation to the above problem which enables us to obtain the coefficients AhR and Ahh in an obvious analytical form and
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. I
36
then to analyze them. This is a quasi-stationary approximation, which assumes that the temperature distribution at each moment satisfies the steady-state heat conduction equation with an instantaneous value of the crystallization front coordinate, This approximation was used by Mullins and Sekerka to study the stability of a planar liquid—solid interface [15]. In order for this approximation to be valid, it is necessary that the characteristic time Trf for the relaxation of the crystallization front to the stationary state be greater than that of the temperature relaxation to the steady state TrT corresponding to the given front position. Temperature gradients in eq. (12) are functions only of the instantaneous value of the crystallization front position h = h0 + 6h. Expanding these functions into Taylor series in the vicinity of the point h0, retaining only linear terms in 6h, and taking into account the fact that h0 is the solution of the equation for dh/dt = 0 (unperturbed problem), we have for ~h dh/dt: ~h =~h(~\Lp 3h
~.
—
h=h0
~8h
~ Lp
\
(13) ‘
For a constant crystal size (for example, during crystallization in the Bridgman method), the expression in eq. (13) can be used to investigate the thermal stability of the process [16]. If the crystal size can change during crystallization, then the following relationship, which takes into account the system reaction to this change, replaces eq. (13): ~ X2 aG2(R0) u~~rwxo 1 = I LLP ~h h=h0 Lp ~h h~h0] X aG ~h ~ X aG (h )~ + ~R[.._~_ 2’ ° Lp ~JR ~R 0 Lp R 1RR0 —
_______
—
—
—
-
(14) Thus, the coefficients AhR and Aim can be found in this approximation from the solution of the steadystate heat flow problem: X aG (h ) 2 0 Lp ~R R=R0
AnR
= ~
A hh
=
~X~ aG2(R0)
X —
h=h~
—
.±
Lp
aG (h )I 1 0 ~R IR=Ro
X1 aG~(R0I1 Lp ~h h=h0
.
, (15)
(16)
6. Thermal coefficients AhR and Ahh In order to investigate the signs of the coefficients and Ahh, it is necessary to solve the stationary problem of heat conduction for various boundary conditions for the far end of the crystal (i.e., eqs. (9) to (11)). Expressions for the coefficients Ahh and AhR for the different boundary conditions are derived in appendix B. In the following, we examine the results for various heat flow situations. AhR
6.1. Semi-infinite crystal; heat transfer from the crystal surface is neglected
In this case, we have
AhR
=
0 (see eq. (B.l)). In
other words, in the absence of heat transfer from the crystal surface, there is no dependence of the temperature on the crystal radius. In this case, capillary effects stabilize the crystal size, and thermal effects stabilize the front position independently. The latter occurs when ~ <0, i.e. for Tm> T0 (see eq. (B.2)). In an overheated melt, an increase in h leads to a decrease in the absolute value of the gradient in the liquid, and consequently, to a decrease in heat conduction toward the crystallization front. The crystallization rate therefore increases, which results in a decrease in h. 6.2. Semi-infinite crystal; heat transfer from the crystal surface is considered
Here Ahh <0 (eq. (B.4)) and AhR >0 (eq. (B.3)). The physical basis of the latter inequality is the 2, while following: The crystallization heat being supplied for a cylindrical crystal is proportional to R the heat rejection from the crystal surface is proportional to R. Therefore, if the crystal radius increases, .
.
.
the heat being supplied exceeds the heat rejection and the crystal melts. Since in all cases of capillary shaping considered ARn <0, there is a mutual stabiization of parameters (i.e. ARhAhR <0). With an increase of the crystal radius, the crystallization front rises due to thermal effects. The front rising results in the angle a0 increasing, and, consequently, in the crystal narrowing. As a result of this mutual stabiization, the range of stable growth may be wider than that predicted by capillary stability. Stable crystal pulling in the Czochralski method is possible in this
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. I
case, whereas that method is unstable from capillary considerations alone. There is an additional feature of this analysis which should be considered. The coefficient AhR is linked to a perturbation of the crystal size. It is clear, however, that such a perturbation occurs near the crystallization front only. The liquid column, on the other hand, changes its size and shape along the whole length. For simplicity, however, we assumed that, in the solid phase as well, the radius changes along the whole length [9]. Another hmiting case, where the crystal radius can be generally thought to be invariable, is obtained if one considers the terms corresponding to the solid phase to be equal to zero. The intermediate case, where the size of the solid phase changes near the front only, is of particular interest; the appropriate relationship is derived in eq. (B.12). One additional fact which should be noted is that the contribution to the coefficient ~ introduced by the gradient change in the liquid phase is rather small, since, for long crystals, small changes in the front position have a weak influence on the temperature distribution in the crystal.
6.3. Crystal with a finite
37
6.5. Environment temperature is not constant If the environment temperature is a hnear function of the coordinate along the growth direction, there appear additional negative terms in the coefficient Ahh. The contribution of these terms is determined mainly by the value of the gradient of the environment temperature (see eqs. (B.9) and (B.10)).
6.6. Variable coefficient of heat transfer or crystal and melt column diameter In the previous consideration, the crystal and melt column diameters were assumed either to change during perturbations along the whole length or not to change at all. In real situations, the first assumption is valid for the melt column. For the crystal, however, it is necessary to allow the size to change iF .:arrow range A such that ~2 A vi 1. The condition E2 A 1 corresponds to Farlier ~“
case of the crystal size changing along ~ntire length, while the condition E2 A = 0 cor:t:. ~.ds to the case where the crystal diameter does n~ ~ For E2 A vi 1, the term corresponding 1. ~i~i solid phase in the coefficient AhR becomes: .
length 1
Basically, this case differs from the previous ones in that the temperature gradient in the solid phase depends on the crystallization front position, the modulus of the gradient increasing with increasing h. This leads to the result that, in ~ there appears a second negative term which improves the stability (see eq. (B.6)). For small values of 1, this term is similar in form to the first term in the equation. In the coefficient AhR, there appears an additional factor which tends to unity as the crystal length increases (see eq. (B.5)). 6.4. Crystal with a finite length far end of crystal
1; zero
heat transfer at
A positive addition appears in this case in the coefficient ~ which reduces the stability (eq. (B.8)). This is related to the fact that heat transfer from the crystal is realized through the side surface only. The latter decreases with increasing values of h, and this tends to favour instability. For 1 h, the effect of this destabilizing factor is small. ~‘
X2 8G2 j~ ~JR
2(T0
2E Ten) P2 2 ~2A
—
LpR
“~
(17)
.
It is seen that the magnitude of this term is small, its value being proportional to ~ A vi 1. When p changes in steps in a narrow zone A, then even in case of an infinite crystal (for which ~ for ~t = const.) we have: Ahh = —2(7’~ Ap2/LpR <0 —
Ten)
.
=
0
(18)
The cause of a changing heat transfer coefficient may be, for example, a forced blast-cooling of the crystal in a localized zone. As shown by eq. (18), such cooling will favour the puffing stability. More complicated cases of changes in these parameters are considered in appendix B. 6.7. Crystallization temperature depends on crystallization rate In all previous considerations, the crystallization temperature T0 was assumed to be constant and equal to the equilibrium temperature Te. However, taking
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. I
38
into account the kinetic occurrences at the crystallization rates used results in the fact that the true interphase boundary temperature differs from the equilibrium one and is a function of the crystallization rate
—5 T/V. The correction term to Lp is of the order of 3TX/J/h in this approximation. Contribution of this term becomes important when the heat flow due to the supercooling at the front becomes comparable to
T0
the crystallization heat flow LpV.
= T0(V). In other words, in order to realize the given rate of crystallization, it is necessary for supercooling to exist: i.e., öT= Te T0(V). It should be emphasized that cST = Te T0 is the supercooling at the crystallization front but not the supercooling L~T= Tm T0 with respect to the melt volume. In this case, the temperature gradient in eq. (12) depends not only on the instantaneous value of the front position,but on the instantaneous crystallization rate as well. Expanding in a Taylor series, and retaining only the linear terms, the following additional terms, linear with respect to oh, appear: —
—
aG2(h) aT0 ar ~-8h
.
~G~(h)aT0 ~~-Oh .
—
Xi
~-~~—----
0
0
(19)
.
-
That is, in expression (14) for Oh, it is necessary to write instead ofLp the following:
Lp
+
X2
aG2(h0, R0) aT0 aT0 av —
—
X1
aG1(h0, R0) aT0 aT0 av
.
—
7. Crystallization condition for system stability From the calculations given above, it is possible to draw several conclusions regarding crystallization stability during capillary shaping. For an overheated melt, the coefficients ~ <0 and A,~R> 0 (see eqs. (B.l) to (B.8)). We know from the analysis of capil. lary coefficients that ARh <0 and ARR may have different signs (see table 1), in other words, in these systems, there exists thermal stability (Ahh <0) and mutual stabilization (A RhA hR <0), while capillary stability may be absent (ARR > 0). In the case where aa0/aR >0 (i.e. ARR <0), the system is always stable. When capillary stability is absent, the system can be still stable, since there is mutual stabilization. In this case, however, the estimation based on the .
~2O~
signs of the coefficients
“
must consider the actual values of the coefficients
‘
Gradients in the melt and in the crystal, respectively, may be written in the following approximate form:
is not sufficient, but one
and the inequalities in eq. (4) in order to answer the question of the system stability.
(22)
Thus, when pulling filaments under the boundary condition of catching the system is stable for R >Rm, i.e. for a large crystal radius. The numerical solution of eq. (4) is necessary, however, to answer the question on stability for R
Since, in the usual case, aT0/av< 0, the relaxation time Trf I increases. Actually, while the height of the crystallization front increases, the modulus of the temperature gradient in the melt decreases and thus leads to an increase in the crystallization rate. This increase in the rate results in a decrease in T0 and hence to an increase in the gradient in the melt. This, in turn, leads to a decrease in the rate of crystallization, and hence the front relaxes to its equilibrium position more slowly, When various kinetic mechanisms are involved in the growth process, then different dependence of the growth rate on supercooling may be obtained. In the cases of linear and quadratic laws for the growth rate dependence on supercopling, we find ar0/a V
ing with the wetting boundary condition, ARR >0 (see table 1) and capillary stability is absent. A similar situation exists in the pulling of rods (R/a 1) for which profile curves and their envelopes have been previously calculated [4—61. When producing plates or ribbons (R/a 1) using a boundary condition of catching, the crystallization is stable since ARR <0 (see table 1). With a wetting boundary condition, the system is stable for positive pressures. For negative pressures, ARR >0 and capifiary stability is absent; the conditions in eq. (4) need to be investigated in detail to determine stability. Thus, the main factors affecting the system stability are the choice of boundary regimes for capillary shaping, as well as the equilib-
(Tm Ten). In this case, expression (20) becomes: T0)/h,
Lp
—
G2
(X2~2+ X/h0) aT0/aV. 1/IA,,,,
~2(T0
—
(21)
~‘
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. I
39
In the case where one of the solutions is stable and
h
the other is not, it is clear that the system will be in
the stable state, i.e. only one solution will be realized in practice. Where both solutions are stable (which is possible, in principle), the case is more complicated. Then transitions of the system from one state into another should be observed under some conditions. It is clear that not all sizes of the crystal R in the
I
I
I
range 0 to r I
p
‘
0
R
Fig. 3. Dependence of the melt column height on the crystal
radius: (1) capillary problem; (2), (3) thermal problem, (2) for velocity of pulling V1, and (3) for V2 with (V2 > V1).
rium values of crystal size and the position of the crystallization front during growth. We consider next the problem of the equilibrium values of h and R defined by the simultaneous solution of the stationary heat flow and capillary problems. As a specific example, we consider the problem of growing a fine fiber at zero pressure and the boundary condition of catching. The envelope of the profile curves is obtained by solving the capillary problem; the solution is given by an analytic expression (see eq. (A 3)) and is shown in fig. 3 (curve I). Another relation between the crystallization front position h and the radius R is obtained from the expression for the heat balance at the crystallization front (eq. (12)). After a number of simplifying assumptions (which are supported by numerical esti-
mates), we obtain the following relationship: ~
I
0 —
~I
—
~
\112
en)~~P2/”2”)
Xi(Tm
—
T0)/h
=LpV.
(23)
The dependence in eq. (23) for two different rates of growth V1 and V2 (V2 > V1) is represented by curves 2 and 3 in fig. 3, respectively. The analysis of eqs. (A.3) and (23) shows that such a system may include cases when two solutions are possible (see appendix C); one solution is necessarily in the region where RRm orR
can be obtained in practice. Actually,
for overheatingh assumed eq. (23), setting V = a0 given and substituting from eq.in(A.3), we find the limiting values of the crystal size which can be obtamed. For a finite rate of pulling V> 0, we could,
in principle, obtain crystal sizes within this range. Next, for a given overheating, the critical growth rate can be found beyond which simultaneous solutions of the heat flow and capillary problems do not exist. Not all crystal sizes within the limiting values can be obtained, however, since the solution corresponding to them may be unstable. To find the limits of stability, it is necessary: (1) to set the values for melt overheating and the shaper dimensions; (2) to write down expressions for the coefficients A11 as a function of the crystal size, substituting for h and V from eqs. (A.3) and (23); and (3) to substitute A11(R) into eq. (4) to find the limiting crystal size for which growth is stable. Experimental evidence for the results obtained in this analysis is the following: (a) the difficulties in obtaming crystals with a constant cross-section by the Czochralski method; (b) the ease in growing shaped crystals with a constant cross-section by the Stepanov method when the relation between the crystal and shaper sizes is R/r0 > ~ (c) the development of selfstabilization in shaped growth by the Stepanov method during forced-cooling of a local region of the crystal; and (d) the development of self-stabilization for increasing rates of crystal pulling. Finally, it should be noted that problems of crystallization stability from the melt are also being investigated by other authors [3,17—19].
Acknowledgements The authors would like to thank professor A.A. Chernov of and cussions theProfessor work. B.A. Ljubov for fruitful dis-
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. I
40
Appendix A: Calculation of capillary coefficients A.]. Ribbon growth; boundary condition of catching We write the first integral of the Laplace capillary equation in the form [6]: 2 = cos a 2 = cosa 2 . (A.l) cosa+Z 1 +d 0 +h Here Z is the ordinate of the profile curve, d = Z is the pressure (r 0 is one-half the shaper dimension), h = Z r=R (R is one-half the ribbon thickness), and
a = —arc tan dZ/dr ; a0
=
a1
=
2
h(s) ~7(r)/2
J
r~
.
a
—arc tan dZ/drlr=r0 h(R)
—arc tan dZ/dr!r=R
Fixing
Z
and assuming a1 to be a parameter of the
family of profile curves, we show two profile curves in fig. 4, characterized2)byand angles a~ 1) and a(2)~ a~ >a(2) The respecprofile tively, with ace >aç curve characterized by the angle ace is situated more to
the
Z =
than the
right
r(2) IZ=const).
curve with
ace(r(O >
This is true for any value of
Z. At
h, we have r =R and a = a 0. But we have already
seen that R increases with increasing a0, i.e. aao/ ~R > 0. The derivative aa0/ah is related to aa0/aR through the expression
aa0/aR = —(aa0/ah) a/i/aR
b Fig. 5. Profile curves for a ribbon under the wetting boundary condition: (a) °o > ~ ; (b) s~< O~.
(ah/aR az/aRlaorco~t) In this case,
3h/~R
<0 (fig. 4), and we find that:
A.2. Ribbon growth; boundary condition of wetting
ARR <0 and ARn <0.
S _______
h ________________________________________ ___________
Fig.
4.
_____________________r 5R°1 ~ r
Profile curves for a ribbon under the boundary con-
dition of catching.
The first integral has the form of eq. (A.I); however, for different profile curves, ct1 = const and the parameter for the family of profile curves is the pressure d (fig. 5). The analysis is similar to2~that (fig. pre5a). sented above. We assume that d~’~ > d~ Then, from eq. (A.1), we find that &~~ Further, the profile curve goes more to the left as a increases, i.e. if a(1)U, we thus have aa corresponds to zero pressure 0/aR>0, and a characteristic while for d<0, gap 3a0/aR <0 (fig. 5a, b). In fig. 5a, the minimum on the curve
Z(r)I00~~0~~t = h(R) (m) between the plate and the shaper [6]. Therefore, from the form of the curve h(R) (fig. 5a, b), we have aa 0/ah > 0 for all pressures.
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. 1
41
h5)~g(~) Li ~const
/ / / / / / / / / / / / /
I
1,
_____
—
—
—
—
—
—
I
/ /
Fig. 6. Profile curves for a filament under the boundary condition of catching.
~ I
A.3. Filament growth; boundary condition of catch-
/
ing
We write the first integral of the Laplace capillary equation in the form [6]. r2d — r sin a zr~d— r0 sin a1 R2d — R sin a0. (A.2)
For a fixed value of r, we shall consider two pro2~.From eq. (A.2), it folfileAssume curves characterized by the parameter a1 (fig. 6). that aç~> aç lows that
a(1)
—
//
7. Profile curves for a filament under the wetting boundary condition (00 > 01). Fig.
(
F(h, R, a
0) =R sin a0 arc cosh R sin a0
> a(2) as well. Since for the boundary
condition of catching the various profile curves 1~lies emerge from the same point, the curve with aç 2~ Irconst.) above theaaother curve (i.e. ~ >z~ therefore, 0/ah > 0. The sign of aa0/aR is opposite to the sign of a/i/aR; thus aa0/aR >0 for ‘R >Rm, and aa0/aR <0 forR
—
arc cosh sin .—~—— a ~
—
h = 0.
(A.3)
0/
aa0/ah and aa0/aR as functions of the crystal radius in the range 0 to r0. As Fig. 9 shows the derivatives
The first integral has the form of eq. (A.2). The
appropriate profile parameter is the pressure d
A.4. Filament growth; boundary condition of wetting (fig. 7). If d(1) > ~ then2~ a(1)
below curve 2 (Z~’~ 0, and, since ~h/~R > 0 (fig. 7), it follows that
aa0/aR <0. R
________________________
Filament growth; boundary condition of catching at zero pressure
Fig. 8.
In this case, the function h(R) shown by curve 1 in fig. 8 is described by the analytical expression [6]:
(2) methods.
Rm
A.5.
r.
Variation of the height of the melt column with crystal radius for filament growth at zero pressure (00 = const, ~~1) >00 > ~~2)) in the capillary shaping (1)and Czochralski
42
V.A. Tatarchenko, E.A. Brener
/ Crystallization stability
during capillary shaping. I
I
1(To—Ten)p2p2
2(~Lh) sinh boundary conditions in eqs.(7), (8) and (9):
AhR= ~ °
~J ~M o
~
~
xL
______
~(I>,(
~2
LpR
(Tm Ten)tl/ + 2i~~ V\ jh X [~ 1hcosh(~1h)— sinh(~1h)] + (To - Ten) [~ si~(2~ih)- ~sh]]
r.
Fig. 9. öoo/~h and ~cs0/öR as functions of the filament radius.
},
(B.3)
where
2/4,~ + 2p 2 (V 1/X1R)” Therefore, AhR >0, since To > Ten, Tm > Ten,
=
can be seen from the figure, öa 0/3h > 0 and aa0/
~R >0 for R >Rm. In this case, ARR <0, and we conclude that the system is stable in the capillary sense (point A in fig. 8). This obtains from the fact that the angle a0 increases with the crystal radius; in order to maintain this angle, the crystal begins to 1~ -* A). narrow (cf. fig. lb and fig. 8 point A~ If R
~1h cosh(~1h)sinh(~1h) >0, ~
sinh(2~1h)— ~1h >0;
Ahh =
2(~
—
——~-~-———---—
Lp sinh IV X
~
—
~2K
X (Tm
—
[~h
Ten) +
1h) cosh(~1h)— sinh(~1h)J
~(T0
—
Ten)
< [Tm Ten cosh(~1h) 1] LT0 Ten —
—
)
(B.4)
.
—
Appendix B. Calculation of thermal coefficients
Thus, Ahh <0, if Tm > T0.
B.]. Setni-infinite; heat transfer from the crystal sur-
B.3. Crystal wit/i a finite length I
face is neglected The solution is obtained from the stationary form
of eq. (6), without the last term for the liquid and solid phases, and the boundary conditions in eqs. (7), (8) and (9): AhR=O,
~ X1
The solution is obtained from the stationary form the liquid and solid phases, and the boundary conditions in eqs. (7), (8) and (10): of eq. (6) for
1
AhR
(B.l) Tm —
T0
(B.2)
X fo~—7~~)~ h)] 2~2(l—h)] 2~2inh[2~2(l sin [~2(l—h)] —
2(~h)L[(Tm_Ten)
sinh
B.2. Semi-infinite crystal; heat transfer from the crys-
~
tal surface is considered
X [~1h cosh(~5h)sinh(~1h)]
The solution is obtained from the stationary form of eq. (6) for the liquid and solid phases, and the
+ (To
—
(
—
~1~)2K
Ten)[~siIth(2~1h)— ~ihJ]),
1/
(B.5)
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. I
and
43
B.5. Environment temperature is not constant
<
[
1 X 2(~ih) 2 ~(T0 — Ten) + Lp ~,sinh2[~2(l_h)] sinh
(Tm
Ten)
—
of If thethecoordinate Z along the growth direction, i.e. environment temperature is a linear function
V
= Teno KZ (K > 0), additional negative terms appear in the coefficient Ahh. For the case examined in B.3, the additional terms are:
Ten
2~1
—
X~
A1K V
1[~1hcosh(~1h)—smnh(~1h)]
Ten)]1~.(B.6)
+~[(Tm~Ten) cosh(~1h)- (T0
B.4. Crystal with a finite length 1; zero heat transfer at far end of crystal
This case corresponds to the stationary form of eq. (6) for the liquid and solid phases, and the boundary conditions in eqs. (7), (8) and (11):
1p2(T0~2 AhR=j—~j 1
—
Ten)
V1
— I X2K[ ~2 coth[~2(/ h)I + 2g 2j while, for the case B.4: —
x1K[ v
~2(l—h)~--~
r V
B. 6. Variable coefficient of heat transfer or crystal
and melt column diameters —2
x [—hsinh[~
2(I—
Ii)]
[(Tm
—
+
~2
cosh[~2(l
(
Ten)
+ ~ sinh~(~h)L
X [~1hcosh(E1h)
—
1 +
v —
2g1
h)]1
—
In this case, the stationary form ofeq. (6) is: ~7’. V dT, 2
\
~2
~ih]]
—
J(Z) (T
—
1
h)
—
Ten)
=
0
(13.11)
,
where
1
sinh(~1h)]
Ten)[~ sinh(2Eih)
—
-1-1
(B.l0)
+ 2~sinh
(To
cosh[~2(l-h)]+~2sinh[~2(l_h)I]
XI— sinh[~2(l—h)]+~2cosh[~2(l_h)]j
—
+
coth(~ih)]
~i
L2ic2
Lp
2112
2g2j ~i 2 [~2(l h)] 2K2
+
I sinh[2~(l—h)]
___‘~
L
(B.9)
—
~
+~[~÷(~ —
+~ coth(~ih)]
-~[~
J(Z)
(B.7)
~,
and
1i(Z)/R
(Z).
The theory for obtaining appro~matesolutions of
-
such is presented elsewhere [20] If equations J(Z) is the linear function of the coordinate
r
11 — Lp
l2/12~(T0
Z,
i.e.,J(Z)=J0 +13Z,we have:2)) X1 aG~x~(Ti’~—T1(b2’3
Ten)
/V
XIR(—sinh[~ 2(l—h)]
Lp
ah
—
Lp
L \21C2 ~
L3U
coshf~2(l—h)I)]
[
X12(~ V (TmTen) sinh 1h) X~[~ 1h cosh(~1h) smnh(~1h)]
X
—
+ ~[(Tm
—
Ten)
cosh(~1h)—
f~I
~[2/Ia+bLjI b2/3
(T0 —
Ten)]]
)
I)
\3/2
I .iJ
[2 (Ia + bL~ 3 1)3/211_i 3 b~”
where Uis the Airy function [20],
2/4~, b
.(B.8)
a = 2J0/X1 —
V
=
(B.12)
VA. Tatarchenko, E.A. Brener / Crystallization stability duringcapillary shaping. I
44
1~=_Tm, T~’~=T
L
1h,
T1
Ti2~_T
L21,
0,
2~Ten.
0, T~
f
(x),,
2 r
2
[~Til)3/2
1
2w
+ij<0
(C.6)
has no zeros. This implies that eq. (C.l) has no more than two roots in addition to the root R = 0.
Appendix C. The analysis of simultaneous solutions of thermal and capillary problems
Let us consider eqs. (23) and (A.3) defining the simultaneous solutions of the thermal and capillary problems:
References [1] A.V. Stepanov, Zh. Tekhn. Fiz. 29 (1959) 381. [21Proc. Meeting on Semiconductor Single Crystal Production by the Stepanov’s Method, Izv. Akad. Nauk SSSR, Ser. Fiz., 33 (1969) No. 12; 35 (1971) No. 3; 36 (1972)
Rsina 0
R
sin a0
r0
No.3;37(1973)No.11;40(1976)No7 [3] B. Chalmers, HF. LaBeile, Jr. and Al. Mlavsky, J. Crys-
)
sin a0
\R
—
arc cosh
tal Growth 13/14 (1972) 84.
1
(
[4] V.A. Tatarchenko, A.I. Saet and A.V. Stepanuv, in: Proc. 1st Meeting on Semiconductor Single Crystal Production by the Stepanov’s Method (FTI, Leningrad,
)
sin a0
1 —
r0
/ arc coshf
r0
1 112
1968) p. 83.
yV
—
(C.1)
‘
r0 (r/R)
where
211 F
=
(1967) 255.
2\
2X2/ToTen X~ t,TmTo )
Akad,Tatarchenko, Nauk SSSR, Ser. 33 and (1969) [5] V.A. Al.Fiz. Saet A.V.1954. Stepanov, Izv. [6] V.A. Tatarchenko, J. Crystal Growth 37 (1977) 272. [7] V.A. Tatarchenko and A.T. Saet, Inzh. Phys. Zh. 13
Lp ‘
‘~‘ =
[8] S.V. Tsivinsky, Kristailografiya 12 (1967) 145.
Xs(Tm
—
T
(C.2)
0)
[9] V.A. Tatarchenko, Fizika i Himija Obrabotki Materialov No. 6 (1973) 47.
and 112 — yV~=h1 >0. (C.3) (F/R) R = 0 is one of the roots of the equation. We denote:
r x=(
R
1
1/2
0
)
112 0
___________
Solids (Oxford Univ. Press, 1959) p. 278.
‘
(ro sin ao F) (C.4)
(F sin ao)V2
_____
/ 1 arc cosh(,—)= p
x
—
>0
w
Let us consider the function arc cosh(x2)
—
[14] F.B. Khambatta, P.1. Gielisse, M.P. Wilson, l.A. Adamski and C.H. Sahagian, J. Crystal Growth 13/14 (1972)710. [15] W.W. Muilins and R.F. Sekerka, J. Appi. Phys. 35
Then, the equation will take the following form: —
[12] W. Bardsley, F.C. Frank, G.W. Green and D.T.J. Hurle, J. Crystal Growth 23 (1974) 341. [13] H.W. Carslaw and J.C. Jaeger, Conduction of Heat in
,
sin a
arc cosh(x2)
[10] A.A. Andronov, A.A. Vitt and SE. Haikin, Teoria p.287 (in Russian). Knlebaniy (Theory of Oscillation) (Moscow, 1959) [11] V.V. Vornnkov, Fiz. Tverd. Tela 5 (1963) 571.
arc cosh(’—~—-.-I—
\smn a
0/
(,0X2
x
The second derivative of this function, viz.,
—
w
.
(C.5)
(1964) 444. [16] V.A. Tatarchenko, Inzh. Fiz. Zh. 30 (1976) 532. [17] T. Surek,J. Appl. Phys. 47(1976)4384. [18] T. Surek, CV. Han Ran, J.C. Swartz and L.C. Garone, J. Electrochem. Soc. 124 (1977) 112. [19] T. Surek and SR. Cnrieil, J. Crystal Growth 37 (1977) 253. [20] G.Ch.P. Miller, Tablici Funkci Vebera, Vip. 45 (Tables of Weber Functions) (Moscow, 1968).
[21] V.A. Tatarchenkn and E.A. Brener, Izv. Akad. Nauk USSR, Ser. Fiz. 40 (1976) 1456.
CRYSTALLIZATION STABILITY DURING CAPILLARY SHAPING I. General theory of capillary and thermal stability V.A. TATARCHENKO and E.A. BRENER The Solid State Physics Institute of the Academy of Sciences of the USSR, Chernogolovka 142432, Moscow district, USSR Received 22 March 1978; manuscript received in final form 1 June 1979
When shaped crystals are grown from the melt, their cross section is determined by the shape of the melt column. The crystal radius and the height of the crystaffization front may vary independently, i.e. we deal with a system of two degrees of freedom. The stability of this system is investigated by Liapunov’s method. The concepts of capillary and thermal stabifities are introduced. The stability of the system depends on the shape of the pulled crystal, the boundary conditions for the melt column, and the thermal conditions during pulling.
1. Introduction
Some preliminary results on the problem of the process stability were reported earlier [6,9,211. Here, we give a detailed investigation of the problem; the analyses of the capillary and thermal problems will enable us to obtain optimal parameters for the crystal growth process.
There is considerable current interest in methods of growing crystals from the melt using the effect of capillary shaping (fig. la). Various methods of operation are known as the Stepanov method [1,2] or EFG [3]. In all the variants of the method, as well as in the Czochralski technique, the cross-sectional size and shape of the crystal are determined by the shape of the melt column and the height of the crystallization front. As shown earlier [4—6],a coupling of the cross-sectional size and the position of the crystallization front may be obtained from the solution of the Laplace capillary equation with the appropriate boundary conditions. Another relationship between the above parameters can be obtained from the solution of the heat flow problem for the crystal—melt system [7,8]. Therefore, simultaneous solution of the heat flow and capifiary problems enables one to find both the height of the crystallization front and the crystal diameter (or some characteristic dimension of the cross-section), as well as their dependence on the control parameters of the process. The question of uniqueness and stability of the solution is a significant one; this can also be addressed by the above analysis.
N’
t lIE itt — — —
it
—
T±1II
11
-
° .
.
Fig. 1. Schematic of crystal growth conditions at the crystal— melt interface: (a) crystal growth with constant cross section; (b) narrowing crystal. 33
34
V.A. Tatarchenko, E.A. Brener
/ Crystallization stability during capillary shaping.
2. General stability considerations In our case, the crystal radius R and the crystallization front height h may vary independently, i.e. we deal with a system of two degrees of freedom,
I
melt surface at the crystallization front possesses a definite value for a given crystal [11,12]. Let us denote this angle by ~ If the angle between the horizontal and the tangent to the melt surface at the crystallization front is a0 = = ir/2 ~ (fig. la), —
According to Liapunov [10], to investigate the stability of such a system, it is necessary to investigate the system of two linear differential equations for the rates of change of the crystal radius 6R and the position of the crystallization front ~5hdue to small devia. lions ~R and ~ in the system variables from equilib-
the crystal grows with constant cross section. Any change in the height of the crystallization front or an occasional change in crystal radius leads to a0 ~ a~ and causes the crystal radius change according to the relationship (fig. ib):
rium:
~R=~V6ao=—V~J5R—V~6h.
6R
=ARR
6R +ARn ~h,
a
a
(5)
(1) where &io = ae a0 and Vis the pulling rate. Coeffi. cients ARR = ~ a~0/aR and AR,, = V a~0/ah —
~h AhR ~R +Ahh ~h. (2) The solution of the system of equations (1) and (2) is given by: =
C1 exP(ii1t)
+
C2 exP(n2t).
(3)
6h C3 exp(r~jt)+C4exp(02t), where t is time, C, are the integration constants, and the quantities i~ and z~2are the roots of the characteristic equation: —
n
(ARR +A,,~) +
(A RRA hh
—
AR,,AhR)
=
0. (3a)
—
—
can be determined from the solution of the capillary problem [4—6].
4. Capillary coefficients ARR and AR,, An analysis of the signs of the capillary coefficients ARR and AR,, is presented in appendix A. The results are summarized in table 1. Coefficient AR,, is
For the system to be stable, the following necessary and sufficient conditions have to be satisfied [10]: ARR +A,,,, <0, ARRA,,,, AR,,A,,R >0. (4)
negative for all profiles and boundary conditions. The sign of the coefficient ARR depends on the type of the pulled profile and the boundary condition.
The above requires that the real parts of the roots of the characteristic equation (3a) be negative, and hence the perturbations will be damped with time. Coefficients ARR and A,,,, reflect the self-stabiliza~ion of the system parameters, i.e. connections of 6R to ~R, and ~ihto ~ Coefficients AR,, and AhR reflect the mutual stabilization parameters. When A RRA hh < 0, i.e. self-stabilization of one of the parameters is lacking, stability is still possible with mutual stabiization. In order to find stable conditions of growth, it is necessary to derive expressions (1) and (2) in a more recognizable form.
5. Rate of change of crystallization front position
—
A recognizable form of eq. (2) can be obtained as a consequence of solving the non-stationary heat conduction problem for the solid and liquid phases. The temperature distribution in a pulled profile can be rather precisely described by means of the one-dimensional equation of heat conduction [7]. Crystal anisotropy and dependence of thermal characteristics on temperature are neglected in this analysis. The heat flow equation is of the form [13]. 2T, V P aT,8 ~,1 at az ~, az X,R i~ en)’ where i=l, 2 (i1 for the melt, and i2 for the crystal), Z is the coordinate along the pulling direc. tion, iq is the thermal diffusivity, X 1 is the thermal —
3. Rate of change of the crystal radius When pulling crystals from the melt, the angle between the crystal surface and the tangent to the
——
—
‘..
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. I
35
Table 1 Signs of the capillary coefficients No.
Profile
Boundary conditions
Fig.
ARR
ARh
1
Ribbon
Catching
4
<0
<0
2
Ribbon
Wetting r 0—R>m rO—R
5a,b 5a Sb
<0 >0 <0
<0
3
4 5
Filament; rod
Catching R
6
Filament; rod
Wetting
7
>0 <0 <0 >0
Rod; Czochralski method
8
conductivity, p, is the radiation heat coefficient,R is the characteristic size of the crystal (radius for a round crystal or half-thickness for a plate), and Ten is the temperature of the environment. Linearization of the radiation term in the heat conduction equation enables us to use eq. (6) in the presence of radiative heat exchange [14]. The boundary conditions of our system are as fol-
>0
lows (fig. 2): T, ‘zO = Tm
<0
(7)
,
TiIzn(t) = T0 (8) Here Tm is the melt temperature, T0 is the crystallization temperature, and h(t) is the crystallization front coordinate. Various boundary conditions are possible at the far end of the crystal: .
(9)
T2!z~ooTen,
(10)
T2Iz~riTen,
V
aT2IazI~,= 0 Here
1
.
(11)
is the length of the crystal. The condition of
front:
heat balance must be satisfied at the crystallization T
__________
______
h
—X2G2(Z)lz=h + X1G1(Z)Iz—~=Lp(V— (12) Here G~(Z) aT1/&Z,L is the latent heat dh/dt). of crystallization, and p is the density of the solid. The solution
1
form thesolution thermal coefficients inwill eq.determine (2). However, of theofabove heat flow problem the a rigorous meets with mathematical difficul-
— ______________ --
ties. -
--
—-
--
Fig. 2. Schematic of boundary conditions for the heat flow problem: (1) idealized and (2) real form of melt column.
We can apply an important approximation to the above problem which enables us to obtain the coefficients AhR and Ahh in an obvious analytical form and
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. I
36
then to analyze them. This is a quasi-stationary approximation, which assumes that the temperature distribution at each moment satisfies the steady-state heat conduction equation with an instantaneous value of the crystallization front coordinate, This approximation was used by Mullins and Sekerka to study the stability of a planar liquid—solid interface [15]. In order for this approximation to be valid, it is necessary that the characteristic time Trf for the relaxation of the crystallization front to the stationary state be greater than that of the temperature relaxation to the steady state TrT corresponding to the given front position. Temperature gradients in eq. (12) are functions only of the instantaneous value of the crystallization front position h = h0 + 6h. Expanding these functions into Taylor series in the vicinity of the point h0, retaining only linear terms in 6h, and taking into account the fact that h0 is the solution of the equation for dh/dt = 0 (unperturbed problem), we have for ~h dh/dt: ~h =~h(~\Lp 3h
~.
—
h=h0
~8h
~ Lp
\
(13) ‘
For a constant crystal size (for example, during crystallization in the Bridgman method), the expression in eq. (13) can be used to investigate the thermal stability of the process [16]. If the crystal size can change during crystallization, then the following relationship, which takes into account the system reaction to this change, replaces eq. (13): ~ X2 aG2(R0) u~~rwxo 1 = I LLP ~h h=h0 Lp ~h h~h0] X aG ~h ~ X aG (h )~ + ~R[.._~_ 2’ ° Lp ~JR ~R 0 Lp R 1RR0 —
_______
—
—
—
-
(14) Thus, the coefficients AhR and Aim can be found in this approximation from the solution of the steadystate heat flow problem: X aG (h ) 2 0 Lp ~R R=R0
AnR
= ~
A hh
=
~X~ aG2(R0)
X —
h=h~
—
.±
Lp
aG (h )I 1 0 ~R IR=Ro
X1 aG~(R0I1 Lp ~h h=h0
.
, (15)
(16)
6. Thermal coefficients AhR and Ahh In order to investigate the signs of the coefficients and Ahh, it is necessary to solve the stationary problem of heat conduction for various boundary conditions for the far end of the crystal (i.e., eqs. (9) to (11)). Expressions for the coefficients Ahh and AhR for the different boundary conditions are derived in appendix B. In the following, we examine the results for various heat flow situations. AhR
6.1. Semi-infinite crystal; heat transfer from the crystal surface is neglected
In this case, we have
AhR
=
0 (see eq. (B.l)). In
other words, in the absence of heat transfer from the crystal surface, there is no dependence of the temperature on the crystal radius. In this case, capillary effects stabilize the crystal size, and thermal effects stabilize the front position independently. The latter occurs when ~ <0, i.e. for Tm> T0 (see eq. (B.2)). In an overheated melt, an increase in h leads to a decrease in the absolute value of the gradient in the liquid, and consequently, to a decrease in heat conduction toward the crystallization front. The crystallization rate therefore increases, which results in a decrease in h. 6.2. Semi-infinite crystal; heat transfer from the crystal surface is considered
Here Ahh <0 (eq. (B.4)) and AhR >0 (eq. (B.3)). The physical basis of the latter inequality is the 2, while following: The crystallization heat being supplied for a cylindrical crystal is proportional to R the heat rejection from the crystal surface is proportional to R. Therefore, if the crystal radius increases, .
.
.
the heat being supplied exceeds the heat rejection and the crystal melts. Since in all cases of capillary shaping considered ARn <0, there is a mutual stabiization of parameters (i.e. ARhAhR <0). With an increase of the crystal radius, the crystallization front rises due to thermal effects. The front rising results in the angle a0 increasing, and, consequently, in the crystal narrowing. As a result of this mutual stabiization, the range of stable growth may be wider than that predicted by capillary stability. Stable crystal pulling in the Czochralski method is possible in this
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. I
case, whereas that method is unstable from capillary considerations alone. There is an additional feature of this analysis which should be considered. The coefficient AhR is linked to a perturbation of the crystal size. It is clear, however, that such a perturbation occurs near the crystallization front only. The liquid column, on the other hand, changes its size and shape along the whole length. For simplicity, however, we assumed that, in the solid phase as well, the radius changes along the whole length [9]. Another hmiting case, where the crystal radius can be generally thought to be invariable, is obtained if one considers the terms corresponding to the solid phase to be equal to zero. The intermediate case, where the size of the solid phase changes near the front only, is of particular interest; the appropriate relationship is derived in eq. (B.12). One additional fact which should be noted is that the contribution to the coefficient ~ introduced by the gradient change in the liquid phase is rather small, since, for long crystals, small changes in the front position have a weak influence on the temperature distribution in the crystal.
6.3. Crystal with a finite
37
6.5. Environment temperature is not constant If the environment temperature is a hnear function of the coordinate along the growth direction, there appear additional negative terms in the coefficient Ahh. The contribution of these terms is determined mainly by the value of the gradient of the environment temperature (see eqs. (B.9) and (B.10)).
6.6. Variable coefficient of heat transfer or crystal and melt column diameter In the previous consideration, the crystal and melt column diameters were assumed either to change during perturbations along the whole length or not to change at all. In real situations, the first assumption is valid for the melt column. For the crystal, however, it is necessary to allow the size to change iF .:arrow range A such that ~2 A vi 1. The condition E2 A 1 corresponds to Farlier ~“
case of the crystal size changing along ~ntire length, while the condition E2 A = 0 cor:t:. ~.ds to the case where the crystal diameter does n~ ~ For E2 A vi 1, the term corresponding 1. ~i~i solid phase in the coefficient AhR becomes: .
length 1
Basically, this case differs from the previous ones in that the temperature gradient in the solid phase depends on the crystallization front position, the modulus of the gradient increasing with increasing h. This leads to the result that, in ~ there appears a second negative term which improves the stability (see eq. (B.6)). For small values of 1, this term is similar in form to the first term in the equation. In the coefficient AhR, there appears an additional factor which tends to unity as the crystal length increases (see eq. (B.5)). 6.4. Crystal with a finite length far end of crystal
1; zero
heat transfer at
A positive addition appears in this case in the coefficient ~ which reduces the stability (eq. (B.8)). This is related to the fact that heat transfer from the crystal is realized through the side surface only. The latter decreases with increasing values of h, and this tends to favour instability. For 1 h, the effect of this destabilizing factor is small. ~‘
X2 8G2 j~ ~JR
2(T0
2E Ten) P2 2 ~2A
—
LpR
“~
(17)
.
It is seen that the magnitude of this term is small, its value being proportional to ~ A vi 1. When p changes in steps in a narrow zone A, then even in case of an infinite crystal (for which ~ for ~t = const.) we have: Ahh = —2(7’~ Ap2/LpR <0 —
Ten)
.
=
0
(18)
The cause of a changing heat transfer coefficient may be, for example, a forced blast-cooling of the crystal in a localized zone. As shown by eq. (18), such cooling will favour the puffing stability. More complicated cases of changes in these parameters are considered in appendix B. 6.7. Crystallization temperature depends on crystallization rate In all previous considerations, the crystallization temperature T0 was assumed to be constant and equal to the equilibrium temperature Te. However, taking
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. I
38
into account the kinetic occurrences at the crystallization rates used results in the fact that the true interphase boundary temperature differs from the equilibrium one and is a function of the crystallization rate
—5 T/V. The correction term to Lp is of the order of 3TX/J/h in this approximation. Contribution of this term becomes important when the heat flow due to the supercooling at the front becomes comparable to
T0
the crystallization heat flow LpV.
= T0(V). In other words, in order to realize the given rate of crystallization, it is necessary for supercooling to exist: i.e., öT= Te T0(V). It should be emphasized that cST = Te T0 is the supercooling at the crystallization front but not the supercooling L~T= Tm T0 with respect to the melt volume. In this case, the temperature gradient in eq. (12) depends not only on the instantaneous value of the front position,but on the instantaneous crystallization rate as well. Expanding in a Taylor series, and retaining only the linear terms, the following additional terms, linear with respect to oh, appear: —
—
aG2(h) aT0 ar ~-8h
.
~G~(h)aT0 ~~-Oh .
—
Xi
~-~~—----
0
0
(19)
.
-
That is, in expression (14) for Oh, it is necessary to write instead ofLp the following:
Lp
+
X2
aG2(h0, R0) aT0 aT0 av —
—
X1
aG1(h0, R0) aT0 aT0 av
.
—
7. Crystallization condition for system stability From the calculations given above, it is possible to draw several conclusions regarding crystallization stability during capillary shaping. For an overheated melt, the coefficients ~ <0 and A,~R> 0 (see eqs. (B.l) to (B.8)). We know from the analysis of capil. lary coefficients that ARh <0 and ARR may have different signs (see table 1), in other words, in these systems, there exists thermal stability (Ahh <0) and mutual stabilization (A RhA hR <0), while capillary stability may be absent (ARR > 0). In the case where aa0/aR >0 (i.e. ARR <0), the system is always stable. When capillary stability is absent, the system can be still stable, since there is mutual stabilization. In this case, however, the estimation based on the .
~2O~
signs of the coefficients
“
must consider the actual values of the coefficients
‘
Gradients in the melt and in the crystal, respectively, may be written in the following approximate form:
is not sufficient, but one
and the inequalities in eq. (4) in order to answer the question of the system stability.
(22)
Thus, when pulling filaments under the boundary condition of catching the system is stable for R >Rm, i.e. for a large crystal radius. The numerical solution of eq. (4) is necessary, however, to answer the question on stability for R
Since, in the usual case, aT0/av< 0, the relaxation time Trf I increases. Actually, while the height of the crystallization front increases, the modulus of the temperature gradient in the melt decreases and thus leads to an increase in the crystallization rate. This increase in the rate results in a decrease in T0 and hence to an increase in the gradient in the melt. This, in turn, leads to a decrease in the rate of crystallization, and hence the front relaxes to its equilibrium position more slowly, When various kinetic mechanisms are involved in the growth process, then different dependence of the growth rate on supercooling may be obtained. In the cases of linear and quadratic laws for the growth rate dependence on supercopling, we find ar0/a V
ing with the wetting boundary condition, ARR >0 (see table 1) and capillary stability is absent. A similar situation exists in the pulling of rods (R/a 1) for which profile curves and their envelopes have been previously calculated [4—61. When producing plates or ribbons (R/a 1) using a boundary condition of catching, the crystallization is stable since ARR <0 (see table 1). With a wetting boundary condition, the system is stable for positive pressures. For negative pressures, ARR >0 and capifiary stability is absent; the conditions in eq. (4) need to be investigated in detail to determine stability. Thus, the main factors affecting the system stability are the choice of boundary regimes for capillary shaping, as well as the equilib-
(Tm Ten). In this case, expression (20) becomes: T0)/h,
Lp
—
G2
(X2~2+ X/h0) aT0/aV. 1/IA,,,,
~2(T0
—
(21)
~‘
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. I
39
In the case where one of the solutions is stable and
h
the other is not, it is clear that the system will be in
the stable state, i.e. only one solution will be realized in practice. Where both solutions are stable (which is possible, in principle), the case is more complicated. Then transitions of the system from one state into another should be observed under some conditions. It is clear that not all sizes of the crystal R in the
I
I
I
range 0 to r I
p
‘
0
R
Fig. 3. Dependence of the melt column height on the crystal
radius: (1) capillary problem; (2), (3) thermal problem, (2) for velocity of pulling V1, and (3) for V2 with (V2 > V1).
rium values of crystal size and the position of the crystallization front during growth. We consider next the problem of the equilibrium values of h and R defined by the simultaneous solution of the stationary heat flow and capillary problems. As a specific example, we consider the problem of growing a fine fiber at zero pressure and the boundary condition of catching. The envelope of the profile curves is obtained by solving the capillary problem; the solution is given by an analytic expression (see eq. (A 3)) and is shown in fig. 3 (curve I). Another relation between the crystallization front position h and the radius R is obtained from the expression for the heat balance at the crystallization front (eq. (12)). After a number of simplifying assumptions (which are supported by numerical esti-
mates), we obtain the following relationship: ~
I
0 —
~I
—
~
\112
en)~~P2/”2”)
Xi(Tm
—
T0)/h
=LpV.
(23)
The dependence in eq. (23) for two different rates of growth V1 and V2 (V2 > V1) is represented by curves 2 and 3 in fig. 3, respectively. The analysis of eqs. (A.3) and (23) shows that such a system may include cases when two solutions are possible (see appendix C); one solution is necessarily in the region where R
can be obtained in practice. Actually,
for overheatingh assumed eq. (23), setting V = a0 given and substituting from eq.in(A.3), we find the limiting values of the crystal size which can be obtamed. For a finite rate of pulling V> 0, we could,
in principle, obtain crystal sizes within this range. Next, for a given overheating, the critical growth rate can be found beyond which simultaneous solutions of the heat flow and capillary problems do not exist. Not all crystal sizes within the limiting values can be obtained, however, since the solution corresponding to them may be unstable. To find the limits of stability, it is necessary: (1) to set the values for melt overheating and the shaper dimensions; (2) to write down expressions for the coefficients A11 as a function of the crystal size, substituting for h and V from eqs. (A.3) and (23); and (3) to substitute A11(R) into eq. (4) to find the limiting crystal size for which growth is stable. Experimental evidence for the results obtained in this analysis is the following: (a) the difficulties in obtaming crystals with a constant cross-section by the Czochralski method; (b) the ease in growing shaped crystals with a constant cross-section by the Stepanov method when the relation between the crystal and shaper sizes is R/r0 > ~ (c) the development of selfstabilization in shaped growth by the Stepanov method during forced-cooling of a local region of the crystal; and (d) the development of self-stabilization for increasing rates of crystal pulling. Finally, it should be noted that problems of crystallization stability from the melt are also being investigated by other authors [3,17—19].
Acknowledgements The authors would like to thank professor A.A. Chernov of and cussions theProfessor work. B.A. Ljubov for fruitful dis-
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. I
40
Appendix A: Calculation of capillary coefficients A.]. Ribbon growth; boundary condition of catching We write the first integral of the Laplace capillary equation in the form [6]: 2 = cos a 2 = cosa 2 . (A.l) cosa+Z 1 +d 0 +h Here Z is the ordinate of the profile curve, d = Z is the pressure (r 0 is one-half the shaper dimension), h = Z r=R (R is one-half the ribbon thickness), and
a = —arc tan dZ/dr ; a0
=
a1
=
2
h(s) ~7(r)/2
J
r~
.
a
—arc tan dZ/drlr=r0 h(R)
—arc tan dZ/dr!r=R
Fixing
Z
and assuming a1 to be a parameter of the
family of profile curves, we show two profile curves in fig. 4, characterized2)byand angles a~ 1) and a(2)~ a~ >a(2) The respecprofile tively, with ace >aç curve characterized by the angle ace is situated more to
the
Z =
than the
right
r(2) IZ=const).
curve with
ace(r(O >
This is true for any value of
Z. At
h, we have r =R and a = a 0. But we have already
seen that R increases with increasing a0, i.e. aao/ ~R > 0. The derivative aa0/ah is related to aa0/aR through the expression
aa0/aR = —(aa0/ah) a/i/aR
b Fig. 5. Profile curves for a ribbon under the wetting boundary condition: (a) °o > ~ ; (b) s~< O~.
(ah/aR az/aRlaorco~t) In this case,
3h/~R
<0 (fig. 4), and we find that:
A.2. Ribbon growth; boundary condition of wetting
ARR <0 and ARn <0.
S _______
h ________________________________________ ___________
Fig.
4.
_____________________r 5R°1 ~ r
Profile curves for a ribbon under the boundary con-
dition of catching.
The first integral has the form of eq. (A.I); however, for different profile curves, ct1 = const and the parameter for the family of profile curves is the pressure d (fig. 5). The analysis is similar to2~that (fig. pre5a). sented above. We assume that d~’~ > d~ Then, from eq. (A.1), we find that &~~ Further, the profile curve goes more to the left as a increases, i.e. if a(1)
Z(r)I00~~0~~t = h(R) (m) between the plate and the shaper [6]. Therefore, from the form of the curve h(R) (fig. 5a, b), we have aa 0/ah > 0 for all pressures.
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. 1
41
h5)~g(~) Li ~const
/ / / / / / / / / / / / /
I
1,
_____
—
—
—
—
—
—
I
/ /
Fig. 6. Profile curves for a filament under the boundary condition of catching.
~ I
A.3. Filament growth; boundary condition of catch-
/
ing
We write the first integral of the Laplace capillary equation in the form [6]. r2d — r sin a zr~d— r0 sin a1 R2d — R sin a0. (A.2)
For a fixed value of r, we shall consider two pro2~.From eq. (A.2), it folfileAssume curves characterized by the parameter a1 (fig. 6). that aç~> aç lows that
a(1)
—
//
7. Profile curves for a filament under the wetting boundary condition (00 > 01). Fig.
(
F(h, R, a
0) =R sin a0 arc cosh R sin a0
> a(2) as well. Since for the boundary
condition of catching the various profile curves 1~lies emerge from the same point, the curve with aç 2~ Irconst.) above theaaother curve (i.e. ~ >z~ therefore, 0/ah > 0. The sign of aa0/aR is opposite to the sign of a/i/aR; thus aa0/aR >0 for ‘R >Rm, and aa0/aR <0 forR
—
arc cosh sin .—~—— a ~
—
h = 0.
(A.3)
0/
aa0/ah and aa0/aR as functions of the crystal radius in the range 0 to r0. As Fig. 9 shows the derivatives
The first integral has the form of eq. (A.2). The
appropriate profile parameter is the pressure d
A.4. Filament growth; boundary condition of wetting (fig. 7). If d(1) > ~ then2~ a(1)
below curve 2 (Z~’~
aa0/aR <0. R
________________________
Filament growth; boundary condition of catching at zero pressure
Fig. 8.
In this case, the function h(R) shown by curve 1 in fig. 8 is described by the analytical expression [6]:
(2) methods.
Rm
A.5.
r.
Variation of the height of the melt column with crystal radius for filament growth at zero pressure (00 = const, ~~1) >00 > ~~2)) in the capillary shaping (1)and Czochralski
42
V.A. Tatarchenko, E.A. Brener
/ Crystallization stability
during capillary shaping. I
I
1(To—Ten)p2p2
2(~Lh) sinh boundary conditions in eqs.(7), (8) and (9):
AhR= ~ °
~J ~M o
~
~
xL
______
~(I>,(
~2
LpR
(Tm Ten)tl/ + 2i~~ V\ jh X [~ 1hcosh(~1h)— sinh(~1h)] + (To - Ten) [~ si~(2~ih)- ~sh]]
r.
Fig. 9. öoo/~h and ~cs0/öR as functions of the filament radius.
},
(B.3)
where
2/4,~ + 2p 2 (V 1/X1R)” Therefore, AhR >0, since To > Ten, Tm > Ten,
=
can be seen from the figure, öa 0/3h > 0 and aa0/
~R >0 for R >Rm. In this case, ARR <0, and we conclude that the system is stable in the capillary sense (point A in fig. 8). This obtains from the fact that the angle a0 increases with the crystal radius; in order to maintain this angle, the crystal begins to 1~ -* A). narrow (cf. fig. lb and fig. 8 point A~ If R
~1h cosh(~1h)sinh(~1h) >0, ~
sinh(2~1h)— ~1h >0;
Ahh =
2(~
—
——~-~-———---—
Lp sinh IV X
~
—
~2K
X (Tm
—
[~h
Ten) +
1h) cosh(~1h)— sinh(~1h)J
~(T0
—
Ten)
< [Tm Ten cosh(~1h) 1] LT0 Ten —
—
)
(B.4)
.
—
Appendix B. Calculation of thermal coefficients
Thus, Ahh <0, if Tm > T0.
B.]. Setni-infinite; heat transfer from the crystal sur-
B.3. Crystal wit/i a finite length I
face is neglected The solution is obtained from the stationary form
of eq. (6), without the last term for the liquid and solid phases, and the boundary conditions in eqs. (7), (8) and (9): AhR=O,
~ X1
The solution is obtained from the stationary form the liquid and solid phases, and the boundary conditions in eqs. (7), (8) and (10): of eq. (6) for
1
AhR
(B.l) Tm —
T0
(B.2)
X fo~—7~~)~ h)] 2~2(l—h)] 2~2inh[2~2(l sin [~2(l—h)] —
2(~h)L[(Tm_Ten)
sinh
B.2. Semi-infinite crystal; heat transfer from the crys-
~
tal surface is considered
X [~1h cosh(~5h)sinh(~1h)]
The solution is obtained from the stationary form of eq. (6) for the liquid and solid phases, and the
+ (To
—
(
—
~1~)2K
Ten)[~siIth(2~1h)— ~ihJ]),
1/
(B.5)
V.A. Tatarchenko, E.A. Brener / Crystallization stability during capillary shaping. I
and
43
B.5. Environment temperature is not constant
<
[
1 X 2(~ih) 2 ~(T0 — Ten) + Lp ~,sinh2[~2(l_h)] sinh
(Tm
Ten)
—
of If thethecoordinate Z along the growth direction, i.e. environment temperature is a linear function
V
= Teno KZ (K > 0), additional negative terms appear in the coefficient Ahh. For the case examined in B.3, the additional terms are:
Ten
2~1
—
X~
A1K V
1[~1hcosh(~1h)—smnh(~1h)]
Ten)]1~.(B.6)
+~[(Tm~Ten) cosh(~1h)- (T0
B.4. Crystal with a finite length 1; zero heat transfer at far end of crystal
This case corresponds to the stationary form of eq. (6) for the liquid and solid phases, and the boundary conditions in eqs. (7), (8) and (11):
1p2(T0~2 AhR=j—~j 1
—
Ten)
V1
— I X2K[ ~2 coth[~2(/ h)I + 2g 2j while, for the case B.4: —
x1K[ v
~2(l—h)~--~
r V
B. 6. Variable coefficient of heat transfer or crystal
and melt column diameters —2
x [—hsinh[~
2(I—
Ii)]
[(Tm
—
+
~2
cosh[~2(l
(
Ten)
+ ~ sinh~(~h)L
X [~1hcosh(E1h)
—
1 +
v —
2g1
h)]1
—
In this case, the stationary form ofeq. (6) is: ~7’. V dT, 2
\
~2
~ih]]
—
J(Z) (T
—
1
h)
—
Ten)
=
0
(13.11)
,
where
1
sinh(~1h)]
Ten)[~ sinh(2Eih)
—
-1-1
(B.l0)
+ 2~sinh
(To
cosh[~2(l-h)]+~2sinh[~2(l_h)I]
XI— sinh[~2(l—h)]+~2cosh[~2(l_h)]j
—
+
coth(~ih)]
~i
L2ic2
Lp
2112
2g2j ~i 2 [~2(l h)] 2K2
+
I sinh[2~(l—h)]
___‘~
L
(B.9)
—
~
+~[~÷(~ —
+~ coth(~ih)]
-~[~
J(Z)
(B.7)
~,
and
1i(Z)/R
(Z).
The theory for obtaining appro~matesolutions of
-
such is presented elsewhere [20] If equations J(Z) is the linear function of the coordinate
r
11 — Lp
l2/12~(T0
Z,
i.e.,J(Z)=J0 +13Z,we have:2)) X1 aG~x~(Ti’~—T1(b2’3
Ten)
/V
XIR(—sinh[~ 2(l—h)]
Lp
ah
—
Lp
L \21C2 ~
L3U
coshf~2(l—h)I)]
[
X12(~ V (TmTen) sinh 1h) X~[~ 1h cosh(~1h) smnh(~1h)]
X
—
+ ~[(Tm
—
Ten)
cosh(~1h)—
f~I
~[2/Ia+bLjI b2/3
(T0 —
Ten)]]
)
I)
\3/2
I .iJ
[2 (Ia + bL~ 3 1)3/211_i 3 b~”
where Uis the Airy function [20],
2/4~, b
.(B.8)
a = 2J0/X1 —
V
=
(B.12)
VA. Tatarchenko, E.A. Brener / Crystallization stability duringcapillary shaping. I
44
1~=_Tm, T~’~=T
L
1h,
T1
Ti2~_T
L21,
0,
2~Ten.
0, T~
f
(x),,
2 r
2
[~Til)3/2
1
2w
+ij<0
(C.6)
has no zeros. This implies that eq. (C.l) has no more than two roots in addition to the root R = 0.
Appendix C. The analysis of simultaneous solutions of thermal and capillary problems
Let us consider eqs. (23) and (A.3) defining the simultaneous solutions of the thermal and capillary problems:
References [1] A.V. Stepanov, Zh. Tekhn. Fiz. 29 (1959) 381. [21Proc. Meeting on Semiconductor Single Crystal Production by the Stepanov’s Method, Izv. Akad. Nauk SSSR, Ser. Fiz., 33 (1969) No. 12; 35 (1971) No. 3; 36 (1972)
Rsina 0
R
sin a0
r0
No.3;37(1973)No.11;40(1976)No7 [3] B. Chalmers, HF. LaBeile, Jr. and Al. Mlavsky, J. Crys-
)
sin a0
\R
—
arc cosh
tal Growth 13/14 (1972) 84.
1
(
[4] V.A. Tatarchenko, A.I. Saet and A.V. Stepanuv, in: Proc. 1st Meeting on Semiconductor Single Crystal Production by the Stepanov’s Method (FTI, Leningrad,
)
sin a0
1 —
r0
/ arc coshf
r0
1 112
1968) p. 83.
yV
—
(C.1)
‘
r0 (r/R)
where
211 F
=
(1967) 255.
2\
2X2/ToTen X~ t,TmTo )
Akad,Tatarchenko, Nauk SSSR, Ser. 33 and (1969) [5] V.A. Al.Fiz. Saet A.V.1954. Stepanov, Izv. [6] V.A. Tatarchenko, J. Crystal Growth 37 (1977) 272. [7] V.A. Tatarchenko and A.T. Saet, Inzh. Phys. Zh. 13
Lp ‘
‘~‘ =
[8] S.V. Tsivinsky, Kristailografiya 12 (1967) 145.
Xs(Tm
—
T
(C.2)
0)
[9] V.A. Tatarchenko, Fizika i Himija Obrabotki Materialov No. 6 (1973) 47.
and 112 — yV~=h1 >0. (C.3) (F/R) R = 0 is one of the roots of the equation. We denote:
r x=(
R
1
1/2
0
)
112 0
___________
Solids (Oxford Univ. Press, 1959) p. 278.
‘
(ro sin ao F) (C.4)
(F sin ao)V2
_____
/ 1 arc cosh(,—)= p
x
—
>0
w
Let us consider the function arc cosh(x2)
—
[14] F.B. Khambatta, P.1. Gielisse, M.P. Wilson, l.A. Adamski and C.H. Sahagian, J. Crystal Growth 13/14 (1972)710. [15] W.W. Muilins and R.F. Sekerka, J. Appi. Phys. 35
Then, the equation will take the following form: —
[12] W. Bardsley, F.C. Frank, G.W. Green and D.T.J. Hurle, J. Crystal Growth 23 (1974) 341. [13] H.W. Carslaw and J.C. Jaeger, Conduction of Heat in
,
sin a
arc cosh(x2)
[10] A.A. Andronov, A.A. Vitt and SE. Haikin, Teoria p.287 (in Russian). Knlebaniy (Theory of Oscillation) (Moscow, 1959) [11] V.V. Vornnkov, Fiz. Tverd. Tela 5 (1963) 571.
arc cosh(’—~—-.-I—
\smn a
0/
(,0X2
x
The second derivative of this function, viz.,
—
w
.
(C.5)
(1964) 444. [16] V.A. Tatarchenko, Inzh. Fiz. Zh. 30 (1976) 532. [17] T. Surek,J. Appl. Phys. 47(1976)4384. [18] T. Surek, CV. Han Ran, J.C. Swartz and L.C. Garone, J. Electrochem. Soc. 124 (1977) 112. [19] T. Surek and SR. Cnrieil, J. Crystal Growth 37 (1977) 253. [20] G.Ch.P. Miller, Tablici Funkci Vebera, Vip. 45 (Tables of Weber Functions) (Moscow, 1968).
[21] V.A. Tatarchenkn and E.A. Brener, Izv. Akad. Nauk USSR, Ser. Fiz. 40 (1976) 1456.