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Effect of temperature gradient in the solution on the growth rate of YBa2Cu3O7−x bulk single crystals
PHYSICA® ELSEVIER
Physica C 264 (1996) 305-310
Effect of temperature gradient in the solution on the growth rate of YBa2Cu307_xbulk single crystals Y. Kanamori *, Y. Shiohara Superconductivity Research Laboratory, 1-10-13 Shinonome, Koto-ku, Tokyo 135, Japan Received 8 February 1996; revised 18 March 1996
Abstract
The bulk single crystals of Y123 are required to clarify the superconductivity phenomenon and to develop electronic devices using unique superconductive properties. Only the solute rich liquid crystal pulling method developed by Yamada et al. has succeeded in continuous growth of the YI23 single crystal. In this paper, we investigated the growth rate of Y123 single crystals under different temperature gradients in the solution in order to understand the growth mechanism of Y 123. It was revealed that Y123 single crystals grow with a spiral growth mode, which is in good agreement with the BCF theory. Keywords: YBa2Cu307_x bulk single crystal; Modified top seeded solution growth method; Growth rate
1. Introduction Since high temperature superconductive oxides of L a - B a - C u - O system have been discovered, new materials possessing a higher critical temperature (Tc) and a higher critical current density (Jc) have been studied actively, as well as the elucidation of superconductivity theory and the application to electronic devices. Bulk single crystals with good crystatlinity are required to clarify the superconductivity phenomena. Larger single crystals, which will be utilized as substrates for thin film deposition, are needed to develop electronic devices using unique superconductive properties. In general, following the synthesis of new materials, single crystals are fabricated by the flux method and then the physical properties are studied. Considering the requirement
* Corresponding author. Fax: + 81 3 3536 5714.
as mentioned above, we attempted to fabricate bulk single crystals of YBa2CU3OT_ x (hereby Y123). One of the useful methods for obtaining large scale single crystals under controlled crystal growth is the crystal pulling method. At the Superconductivity Research Laboratory (SRL), the solute rich liquid crystal pulling (SRL-CP) method was developed by Yamada et al. [1]. So far, only the SRL-CP method has succeeded in continuous growth of Y 123 single crystals. In this paper, the growth of Y123 single crystals under different temperature gradients in the solution was investigated by focusing on the growth rate of Y123.
2. Experimental
The characteristics of the SRL-CP method [1-5] is to use the high temperature stable phase, Y211, as
0921-4534/96/$15.00 Copyright © 1996 Published by Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 3 4 ( 9 6 ) 0 0 2 3 3 - X
306
Y. Kanamori, Y. Shiohara / Physica C 264 (1996)305-310
Y
,tal
B
les
s
crucible Fig. 1. Inside of the crucible used in the SRL-CP method.
the solute supply. Fig. 1 schematically shows the inside of the crucible used in the SRL-CP method. Since the crucible in which Y211 is placed on the
~Tb,
...........!.1!..............,..h "~(2)
Tp
............................ (3) ! +Y123 l
i
: i
Fig. 3. Schematic illustration of the crystal pulling apparatus.
bottom is set under the temperature gradient, the solute is transported from the higher temperature region, which is at the bottom of the melt, to the lower temperature region, which is at the surface of the melt, through the melt by natural and forced convections. Fig. 2 briefly shows the solute transport path during the growth in the phase diagram. Since the SRL-CP method is a modified top seeded solution growth (TSSG) method, an appropriate crucible and solution should be selected. High density yttria is used as crucible material. The ratio of the composition of each metal element (cation) is Y : Ba: Cu = 5 : 36 : 59, which amounts to about 300 g. After melting the solution completely, the temperature at the surface of the melt is controlled to be
/
Y211
Y123 )~
~ 3BaO -5CuO CL(Tb) CL(Ts) Composition
arrow (1) Y211 ~
L(solute rich)
arrow (2) L(solute rich) L(supersaturated) arrow (3) L(supersaturated)
> Y123
Fig. 2. Schematic phase diagram of the Y system.
Table 1 Crystal growth conditions Solute Solvent Crucible Rotation speed (rpm) Pulling rate ( m m / h ) Temp. at the surface (Ts) (°C) Temp. at the bottom (T b) (°C) Atmosphere Growth time (h)
Y211 BaCuO 2 + CuO 2 (Ba: Cu = 3 : 5) Y203 60, 120, 180 0.02 ~ 1000 1005-1015 Air ~ 5
Y. Kanamori, Y. Shiohara / Physica C 264 (1996) 305-310
307
about 1000°C. A YBCO thin film deposited on a MgO single crystal is used as a seed crystal, which is pulled with relatively high rotation speed. Fig. 3 shows the schematic illustration of the crystal pulling apparatus. In order to investigate the effect of temperature gradients in the solution on the crystal growth, the growth rates, R, were measured at each condition by dividing the length along the c-axis by the growth time of about 5 h. The growth conditions are shown in Table 1. Furthermore, the growth pattern of the as-grown a-b-plane of the Y123 single crystal was observed by AFM (atomic forced microscopy) to measure the terrace width for estimating the interface supersaturation.
3. Results
Fig. 4. AFM image of the as-grown a - b - p l a n e of the Y123 single crystal.
f
E x p e r i m e n t a l El: to= 60rpm O: Co= 120rpm
--- 5.0 "~
A: co= 180rpm
Calculated
~ 2.5
O t_
~
0 0
0.05 Supersaturation
0.1 G
Fig. 5. Supersaturation dependence of the growth rate of Y 123.
Because of experimental difficulties, for instance, particles floating on the surface of the melt, which prevent the continuous growth of Y123 single crystals, we could obtain Y 123 single crystals only under limited conditions. Y 123 single crystals fabricated by the SRL-CP method have an a-b-plane of about 5 mm × 5 ram. First of all, to investigate the growth pattern of Y123 single crystals, observation of the as-grown a-b-plane of Y123 single crystals by AFM was carried out. As shown in the top of Fig. 4, the spiral growth has a squared symmetry and the terrace widths are almost the same. It is noted that the white inclusions appearing on the surface originated from the wetted solution onto the as-grown single crystal which was solidified during taking the single crystal out of the fumace. Furthermore, whether the step height of the spiral is equal to the lattice parameter of the c-axis of Y123 was observed in detail to calculate the growth rate of the Y123 single crystal based on the BCF theory [6]. As shown in the bottom of Fig. 4, the spiral has a step height of 1.178 nm, which is almost the same as the lattice constant along the c-axis of the Y123 single crystal. Secondly, the growth rate of the Y123 single crystal was measured by dividing the length along the c-axis by the growth time. Provided that under low supersaturation single crystals grow with a spiral mode, which have to possess a step height equal to
Y. Kanamori, Y. Shiohara / Physica C 264 (1996)305-310
308
one unit cell, the BCF theory [6] predicts the growth rate of a single crystal is in proportion to the squared supersaturation. Though there are to some extent errors, the calculated growth rate, which represents a parabolic law as depicted in Fig. 5, is in good agreement with experiment.
4. Discussion In the SRL-CP method, as mentioned above, the supersaturation in the solution necessary to grow Y 123 single crystals is given from the high temperature stable phase Y211 placed on the bottom of the crucible and transported toward the surface of the melt by both natural convection caused by the temperature gradient in the solution and forced convection caused by the rotation of the seed crystal (or the growing Y123 single crystal). The Y concentration in the melt is assumed to be uniform except in the thin solute layer adjacent to the interface between the crystal and the melt. At the surface of the melt, a thin diffusion boundary layer exists in front of the interface due to the high speed rotation of the growing Y123 single crystal. The thickness of the layer, 6, is approximated by the Cochran's analysis [7], as follows:
6 = 1.6DI/3pl/6to - I/2 where D is the diffusivity, v the kinematic viscosity and w the angular rotation speed of the crystal, respectively. Since the interface supersaturation as a driving force is necessary to grow facetted crystals, the Y concentration at the interface must be higher than the equilibrium concentration. In the boundary layer, the transport of the solute is limited by diffusion. The diffusion flux at the interface of the growing Y123 single crystal for the steady state is given by the Fick's first law, as follows:
CL(Tb)C j, = D
6
'
(2)
where CL(Tb) is the Y concentration in the melt given by the equilibrium liquidus composition at the bottom of the crucible which is equal to the liquid concentration at a distance 6 from the crystal/solution interface due to the assumption of homogeneous Y concentration, C~ is the Y concentration at the
interface. On the other hand, the solute flux necessary to grow Y123 at growth rate R in the steady state is given as the product of the growth rate and the difference of the Y concentration between the solid phase and the liquid phase, as follows: j2 = R ( C ~ 2 3 - C i )
(3)
,
where C~23 is the Y concentration of solid Y123. Provided JL is equal to J2 by mass conservation under the steady state, the growth rate of the Y123 single crystal is given by [1] 1 D2/3v-l/6to|/2
CL(Tb)
R = 1.6
C~23
-- C i _
_
Ci
(4)
The Y123 single crystals grow with the spiral growth mode as shown in Fig. 4. So the growth rate taking into account the interface kinetics is given in accordance with the BCF theory as [6,9,10] -
R=k
cL(
)
'
(5)
where k is the kinetic coefficient, o- is the supersaturation and CL(Ti) is the equilibrium Y concentration at temperature Ti at the interface between the crystal and the melt. It is known that under the spiral growth mode the terrace width, A, is inversely proportional to the supersaturation in the BCF theory. According to the approximation by Cabrera and Levine [8], the relation between the terrace width and the supersaturation is given as [6,9,10] ya 1 h = 19 kB-~ ~, (6) where y is the step energy per unit length, a is the area per molecule on the surface of the crystal and ka is Boltzmann's constant. In our previous paper [11], we reported that the factor y a / k a T = 0.13 nm satisfied the Eq. (6) based on the BCF theory [6]. Furthermore, the height of a single step is approximately equal to the lattice constant along the c-axis of the Y123 single crystal as shown in Fig. 4. When the advance speed of the step growing with the spiral growth mode is expressed as R c, the growth rate of the Y123 single crystal along the c-axis is given by [6,9,10] R-
Rcd h
,
(7)
Y. Kanamori, Y. Shiohara / Physica C 264 (1996) 305-310
5.0
where R c is expressed by [6,9,10] R,, = h f e x p - w / k B r
1
- - o (19/Xs)(ya/kaT)
"2,
(8)
where d is the lattice constant, f is the frequency of a molecule, W is the dissolution energy of the attached molecule from the surface of the crystal and x s is the mean diffusive distance. From Eqs. (7), (8) and (5), the interface kinetic coefficient, k, is expressed by 19/xs)(Ta/kaT ) "
(9)
Substituting d = 10 - 7 c m ( t h e lattice parameter of the c-axis of the Y123 single crystal), W - - 5 8 kcal/mol (this value was estimated to be several times larger than the value for growth from the vapor), x~ -~ 100a -- 4 × 10 -6 cm (the lattice parameter of the a-axis of the Y123 single crystal), f = k B T / 3 h .~. 1013 s - l (h is Planck's constant and the coefficient, 1/3, indicates that the direction of the oscillation of the solute molecule is only taking into account the direction perpendicular to the surface of the crystal) into Eq. (9) and using 0.13 nm given in our previous paper [11] as T a / k B T , k = 10 - 3 c m / s was calculated as the interface kinetic coefficient. On the other hand, the kinetic coefficient for the supersaturation can be calculated. Since the supersaturation includes two unknown quantities C i and CL(T~), first of all we have to evaluate them. C~ can be obtained by solving Eq. (4) for C~, by using the experimental results, R, CL(Tb) [12] as shown in Table 2, the typical properties of the liquid oxide ( D = 10 - 6 c m 2 / s , tion
v = 10 - 2
~2.5
.N
00
1 k = dfexp-W/kBr(
cm2/s)
309
and the rota-
speed of 120 rpm. For CL(Ti), an Y concentra-
Table 2 Y concentration and growth rate at the temperature at the bottom of the crucible. T b (°C)
CL(Tb) (at.%)
R ( X 10 -6 c m / s )
1002.8 1006.0 1010.0 !013.9
0.605 0.610 0.640 0.650
2.07 1.32 3.51 3.39
T b = temperature at the bottom of the crucible; CL(Tb) = Y concentration at Tb; R = growth rate of Y123.
0.05
0.1
Supersaturation c~ Fig. 6. Supersaturation dependence of the kinetic coefficient.
tion CL(T~) = 0.58 at.% [12] was used assuming that the temperature, T~, at the surface of the melt is equal to T~. T~ is set at 1000°C before contacting the seed crystal with the surface of the melt. In the procedure mentioned above, supersaturation can be calculated by substituting C~ estimated from Eq. (4) and CL(T~) into Eq. (5). Accordingly, the relation between the kinetic coefficient and the supersaturation was obtained as shown in Fig. 6, which shows that the kinetic coefficient is a constant of about 10-3 ( c m / s ) and independent of the supersaturation. Fig. 5 depicts the relation between the experimental results and the calculated ones. We can see the difference between the experimental results and the calculated ones. The order and tendency between experimental results and calculated ones, however, are in good agreement to each other. As mentioned above, it was clarified that Y123 single crystals grown with a spiral growth mode satisfied the BCF theory [6]. According to the calculation the growth rate increases as the supersaturation increases. The high growth rate is one of the advantageous conditions to obtain larger Y 123 single crystals. The spiral growth, however, takes place under low supersaturation, so we cannot grow Y123 single crystals under high supersaturation. After all we have to seize the optimal supersaturation taking account of the spiral growth mode by changing the growth conditions to fabricate larger single crystals with good crystallinity.
310
Y. Kanamori, Y. Shiohara / Physica C 264 (1996) 305-310
5. Conclusions A lot of Y123 single crystals with several sizes were obtained by utilizing the SRL-CP method. In this work, the supersaturation based on the difference of the temperature gradient in the solution was focused on to look for the optimal condition necessary to fabricate larger single crystals of Y 123 with good crystallinity. As a result of the investigation of the relation between the growth rate of the Y123 single crystals and the supersaturation, we found that the growth rate of the Y 123 single crystal is proportional to the squared supersaturation, which satisfies the BCF theory [6]. However, there remain unsettled problems on the SRL-CP method such as the decline of the surface of the melt in the crucible and so forth. In the future the SRL-CP method controlled under the optimal condition will pave the way to constantly fabricate Y123 single crystals with the proper size and good crystallinity.
Acknowledgements This work was performed under the management of the International Superconductivity Technology Center as a part of the R&D of Industrial Science
and Technology Frontier Program supported by the New Energy and Industrial Technology Development Organization.
References [I] Y. Yamada and Y. Shiohara, Physica C 217 (1993) 182. [2] Y. Yamada, K. Ishige, K. Ohtsu and Y. Shiohara, Program and Ext. Abstr. Int. Workshop on Superconductity, co-sponsored by ISTEC and MRS, Hawaii, 1992, p. 268. [3] Y. Yamada, M. Tagami, M. Nakamura, Y. Shiohara and S. Tanaka, Advances in Superconductivity V, 1992 (Springer, Tokyo, 1993) p. 561. [4] Y. Kanamori, K. Ohtsu, S. Koyama, Y. Yamada and Y. Shiohara, Advances in Superconductivity VI, 1993 (Springer, Tokyo, 1994) p. 783. [5] Y. Yamaha, Y. Shiohara and S. Tanaka, Advances in Superconductivity VI, 1993 (Springer, Tokyo, 1994) p. 799. [6] W.K. Burton, N. Cabrera and F.C. Frank, Phil. Trans. R. Soc. A 243 (1951) 299. [7] W.G. Cochran, Proc. Camb. Phil. Soc. 30 (1934) 365. [8] N. Cabrera and M.M. Levine, Phil. Mag. 1 (1956) 450. [9] D. Elwell and H.J. Scheel, Crystal Growth from High-Temperature Solutions (Academic Press, New York, 1975). [10] P. Bennema and G.H. Gilmer, in: Crystal Growth: an Introduction, Kinetics of Crystal Growth, Ed. P. Hartman (NorthHolland, Amsterdam, 1973). [11] Y. Kanamori and Y. Shiohara, to be published. [12] Ch. Krauns, M. Sumida, M. Tagami, Y. Yamada and Y. Shiohara, Z. Phys. B 96 (1994) 207.
Physica C 264 (1996) 305-310
Effect of temperature gradient in the solution on the growth rate of YBa2Cu307_xbulk single crystals Y. Kanamori *, Y. Shiohara Superconductivity Research Laboratory, 1-10-13 Shinonome, Koto-ku, Tokyo 135, Japan Received 8 February 1996; revised 18 March 1996
Abstract
The bulk single crystals of Y123 are required to clarify the superconductivity phenomenon and to develop electronic devices using unique superconductive properties. Only the solute rich liquid crystal pulling method developed by Yamada et al. has succeeded in continuous growth of the YI23 single crystal. In this paper, we investigated the growth rate of Y123 single crystals under different temperature gradients in the solution in order to understand the growth mechanism of Y 123. It was revealed that Y123 single crystals grow with a spiral growth mode, which is in good agreement with the BCF theory. Keywords: YBa2Cu307_x bulk single crystal; Modified top seeded solution growth method; Growth rate
1. Introduction Since high temperature superconductive oxides of L a - B a - C u - O system have been discovered, new materials possessing a higher critical temperature (Tc) and a higher critical current density (Jc) have been studied actively, as well as the elucidation of superconductivity theory and the application to electronic devices. Bulk single crystals with good crystatlinity are required to clarify the superconductivity phenomena. Larger single crystals, which will be utilized as substrates for thin film deposition, are needed to develop electronic devices using unique superconductive properties. In general, following the synthesis of new materials, single crystals are fabricated by the flux method and then the physical properties are studied. Considering the requirement
* Corresponding author. Fax: + 81 3 3536 5714.
as mentioned above, we attempted to fabricate bulk single crystals of YBa2CU3OT_ x (hereby Y123). One of the useful methods for obtaining large scale single crystals under controlled crystal growth is the crystal pulling method. At the Superconductivity Research Laboratory (SRL), the solute rich liquid crystal pulling (SRL-CP) method was developed by Yamada et al. [1]. So far, only the SRL-CP method has succeeded in continuous growth of Y 123 single crystals. In this paper, the growth of Y123 single crystals under different temperature gradients in the solution was investigated by focusing on the growth rate of Y123.
2. Experimental
The characteristics of the SRL-CP method [1-5] is to use the high temperature stable phase, Y211, as
0921-4534/96/$15.00 Copyright © 1996 Published by Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 3 4 ( 9 6 ) 0 0 2 3 3 - X
306
Y. Kanamori, Y. Shiohara / Physica C 264 (1996)305-310
Y
,tal
B
les
s
crucible Fig. 1. Inside of the crucible used in the SRL-CP method.
the solute supply. Fig. 1 schematically shows the inside of the crucible used in the SRL-CP method. Since the crucible in which Y211 is placed on the
~Tb,
...........!.1!..............,..h "~(2)
Tp
............................ (3) ! +Y123 l
i
: i
Fig. 3. Schematic illustration of the crystal pulling apparatus.
bottom is set under the temperature gradient, the solute is transported from the higher temperature region, which is at the bottom of the melt, to the lower temperature region, which is at the surface of the melt, through the melt by natural and forced convections. Fig. 2 briefly shows the solute transport path during the growth in the phase diagram. Since the SRL-CP method is a modified top seeded solution growth (TSSG) method, an appropriate crucible and solution should be selected. High density yttria is used as crucible material. The ratio of the composition of each metal element (cation) is Y : Ba: Cu = 5 : 36 : 59, which amounts to about 300 g. After melting the solution completely, the temperature at the surface of the melt is controlled to be
/
Y211
Y123 )~
~ 3BaO -5CuO CL(Tb) CL(Ts) Composition
arrow (1) Y211 ~
L(solute rich)
arrow (2) L(solute rich) L(supersaturated) arrow (3) L(supersaturated)
> Y123
Fig. 2. Schematic phase diagram of the Y system.
Table 1 Crystal growth conditions Solute Solvent Crucible Rotation speed (rpm) Pulling rate ( m m / h ) Temp. at the surface (Ts) (°C) Temp. at the bottom (T b) (°C) Atmosphere Growth time (h)
Y211 BaCuO 2 + CuO 2 (Ba: Cu = 3 : 5) Y203 60, 120, 180 0.02 ~ 1000 1005-1015 Air ~ 5
Y. Kanamori, Y. Shiohara / Physica C 264 (1996) 305-310
307
about 1000°C. A YBCO thin film deposited on a MgO single crystal is used as a seed crystal, which is pulled with relatively high rotation speed. Fig. 3 shows the schematic illustration of the crystal pulling apparatus. In order to investigate the effect of temperature gradients in the solution on the crystal growth, the growth rates, R, were measured at each condition by dividing the length along the c-axis by the growth time of about 5 h. The growth conditions are shown in Table 1. Furthermore, the growth pattern of the as-grown a-b-plane of the Y123 single crystal was observed by AFM (atomic forced microscopy) to measure the terrace width for estimating the interface supersaturation.
3. Results
Fig. 4. AFM image of the as-grown a - b - p l a n e of the Y123 single crystal.
f
E x p e r i m e n t a l El: to= 60rpm O: Co= 120rpm
--- 5.0 "~
A: co= 180rpm
Calculated
~ 2.5
O t_
~
0 0
0.05 Supersaturation
0.1 G
Fig. 5. Supersaturation dependence of the growth rate of Y 123.
Because of experimental difficulties, for instance, particles floating on the surface of the melt, which prevent the continuous growth of Y123 single crystals, we could obtain Y 123 single crystals only under limited conditions. Y 123 single crystals fabricated by the SRL-CP method have an a-b-plane of about 5 mm × 5 ram. First of all, to investigate the growth pattern of Y123 single crystals, observation of the as-grown a-b-plane of Y123 single crystals by AFM was carried out. As shown in the top of Fig. 4, the spiral growth has a squared symmetry and the terrace widths are almost the same. It is noted that the white inclusions appearing on the surface originated from the wetted solution onto the as-grown single crystal which was solidified during taking the single crystal out of the fumace. Furthermore, whether the step height of the spiral is equal to the lattice parameter of the c-axis of Y123 was observed in detail to calculate the growth rate of the Y123 single crystal based on the BCF theory [6]. As shown in the bottom of Fig. 4, the spiral has a step height of 1.178 nm, which is almost the same as the lattice constant along the c-axis of the Y123 single crystal. Secondly, the growth rate of the Y123 single crystal was measured by dividing the length along the c-axis by the growth time. Provided that under low supersaturation single crystals grow with a spiral mode, which have to possess a step height equal to
Y. Kanamori, Y. Shiohara / Physica C 264 (1996)305-310
308
one unit cell, the BCF theory [6] predicts the growth rate of a single crystal is in proportion to the squared supersaturation. Though there are to some extent errors, the calculated growth rate, which represents a parabolic law as depicted in Fig. 5, is in good agreement with experiment.
4. Discussion In the SRL-CP method, as mentioned above, the supersaturation in the solution necessary to grow Y 123 single crystals is given from the high temperature stable phase Y211 placed on the bottom of the crucible and transported toward the surface of the melt by both natural convection caused by the temperature gradient in the solution and forced convection caused by the rotation of the seed crystal (or the growing Y123 single crystal). The Y concentration in the melt is assumed to be uniform except in the thin solute layer adjacent to the interface between the crystal and the melt. At the surface of the melt, a thin diffusion boundary layer exists in front of the interface due to the high speed rotation of the growing Y123 single crystal. The thickness of the layer, 6, is approximated by the Cochran's analysis [7], as follows:
6 = 1.6DI/3pl/6to - I/2 where D is the diffusivity, v the kinematic viscosity and w the angular rotation speed of the crystal, respectively. Since the interface supersaturation as a driving force is necessary to grow facetted crystals, the Y concentration at the interface must be higher than the equilibrium concentration. In the boundary layer, the transport of the solute is limited by diffusion. The diffusion flux at the interface of the growing Y123 single crystal for the steady state is given by the Fick's first law, as follows:
CL(Tb)C j, = D
6
'
(2)
where CL(Tb) is the Y concentration in the melt given by the equilibrium liquidus composition at the bottom of the crucible which is equal to the liquid concentration at a distance 6 from the crystal/solution interface due to the assumption of homogeneous Y concentration, C~ is the Y concentration at the
interface. On the other hand, the solute flux necessary to grow Y123 at growth rate R in the steady state is given as the product of the growth rate and the difference of the Y concentration between the solid phase and the liquid phase, as follows: j2 = R ( C ~ 2 3 - C i )
(3)
,
where C~23 is the Y concentration of solid Y123. Provided JL is equal to J2 by mass conservation under the steady state, the growth rate of the Y123 single crystal is given by [1] 1 D2/3v-l/6to|/2
CL(Tb)
R = 1.6
C~23
-- C i _
_
Ci
(4)
The Y123 single crystals grow with the spiral growth mode as shown in Fig. 4. So the growth rate taking into account the interface kinetics is given in accordance with the BCF theory as [6,9,10] -
R=k
cL(
)
'
(5)
where k is the kinetic coefficient, o- is the supersaturation and CL(Ti) is the equilibrium Y concentration at temperature Ti at the interface between the crystal and the melt. It is known that under the spiral growth mode the terrace width, A, is inversely proportional to the supersaturation in the BCF theory. According to the approximation by Cabrera and Levine [8], the relation between the terrace width and the supersaturation is given as [6,9,10] ya 1 h = 19 kB-~ ~, (6) where y is the step energy per unit length, a is the area per molecule on the surface of the crystal and ka is Boltzmann's constant. In our previous paper [11], we reported that the factor y a / k a T = 0.13 nm satisfied the Eq. (6) based on the BCF theory [6]. Furthermore, the height of a single step is approximately equal to the lattice constant along the c-axis of the Y123 single crystal as shown in Fig. 4. When the advance speed of the step growing with the spiral growth mode is expressed as R c, the growth rate of the Y123 single crystal along the c-axis is given by [6,9,10] R-
Rcd h
,
(7)
Y. Kanamori, Y. Shiohara / Physica C 264 (1996) 305-310
5.0
where R c is expressed by [6,9,10] R,, = h f e x p - w / k B r
1
- - o (19/Xs)(ya/kaT)
"2,
(8)
where d is the lattice constant, f is the frequency of a molecule, W is the dissolution energy of the attached molecule from the surface of the crystal and x s is the mean diffusive distance. From Eqs. (7), (8) and (5), the interface kinetic coefficient, k, is expressed by 19/xs)(Ta/kaT ) "
(9)
Substituting d = 10 - 7 c m ( t h e lattice parameter of the c-axis of the Y123 single crystal), W - - 5 8 kcal/mol (this value was estimated to be several times larger than the value for growth from the vapor), x~ -~ 100a -- 4 × 10 -6 cm (the lattice parameter of the a-axis of the Y123 single crystal), f = k B T / 3 h .~. 1013 s - l (h is Planck's constant and the coefficient, 1/3, indicates that the direction of the oscillation of the solute molecule is only taking into account the direction perpendicular to the surface of the crystal) into Eq. (9) and using 0.13 nm given in our previous paper [11] as T a / k B T , k = 10 - 3 c m / s was calculated as the interface kinetic coefficient. On the other hand, the kinetic coefficient for the supersaturation can be calculated. Since the supersaturation includes two unknown quantities C i and CL(T~), first of all we have to evaluate them. C~ can be obtained by solving Eq. (4) for C~, by using the experimental results, R, CL(Tb) [12] as shown in Table 2, the typical properties of the liquid oxide ( D = 10 - 6 c m 2 / s , tion
v = 10 - 2
~2.5
.N
00
1 k = dfexp-W/kBr(
cm2/s)
309
and the rota-
speed of 120 rpm. For CL(Ti), an Y concentra-
Table 2 Y concentration and growth rate at the temperature at the bottom of the crucible. T b (°C)
CL(Tb) (at.%)
R ( X 10 -6 c m / s )
1002.8 1006.0 1010.0 !013.9
0.605 0.610 0.640 0.650
2.07 1.32 3.51 3.39
T b = temperature at the bottom of the crucible; CL(Tb) = Y concentration at Tb; R = growth rate of Y123.
0.05
0.1
Supersaturation c~ Fig. 6. Supersaturation dependence of the kinetic coefficient.
tion CL(T~) = 0.58 at.% [12] was used assuming that the temperature, T~, at the surface of the melt is equal to T~. T~ is set at 1000°C before contacting the seed crystal with the surface of the melt. In the procedure mentioned above, supersaturation can be calculated by substituting C~ estimated from Eq. (4) and CL(T~) into Eq. (5). Accordingly, the relation between the kinetic coefficient and the supersaturation was obtained as shown in Fig. 6, which shows that the kinetic coefficient is a constant of about 10-3 ( c m / s ) and independent of the supersaturation. Fig. 5 depicts the relation between the experimental results and the calculated ones. We can see the difference between the experimental results and the calculated ones. The order and tendency between experimental results and calculated ones, however, are in good agreement to each other. As mentioned above, it was clarified that Y123 single crystals grown with a spiral growth mode satisfied the BCF theory [6]. According to the calculation the growth rate increases as the supersaturation increases. The high growth rate is one of the advantageous conditions to obtain larger Y 123 single crystals. The spiral growth, however, takes place under low supersaturation, so we cannot grow Y123 single crystals under high supersaturation. After all we have to seize the optimal supersaturation taking account of the spiral growth mode by changing the growth conditions to fabricate larger single crystals with good crystallinity.
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5. Conclusions A lot of Y123 single crystals with several sizes were obtained by utilizing the SRL-CP method. In this work, the supersaturation based on the difference of the temperature gradient in the solution was focused on to look for the optimal condition necessary to fabricate larger single crystals of Y 123 with good crystallinity. As a result of the investigation of the relation between the growth rate of the Y123 single crystals and the supersaturation, we found that the growth rate of the Y 123 single crystal is proportional to the squared supersaturation, which satisfies the BCF theory [6]. However, there remain unsettled problems on the SRL-CP method such as the decline of the surface of the melt in the crucible and so forth. In the future the SRL-CP method controlled under the optimal condition will pave the way to constantly fabricate Y123 single crystals with the proper size and good crystallinity.
Acknowledgements This work was performed under the management of the International Superconductivity Technology Center as a part of the R&D of Industrial Science
and Technology Frontier Program supported by the New Energy and Industrial Technology Development Organization.
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