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Intrinsic LiNbO3 melt species partitioning at the congruent melt composition. III. Choice of the growth parameters for the dynamic congruent-state growth
.........
CIRYSTAL QNOWTH
Journal of Crystal Growth 155 (1995) 229-239
ELSEVIER
Intrinsic
melt species partitioning at the congruent melt composition III. Choice of the growth parameters for the dynamic congruent-state growth LiNbO 3
Satoshi Uda a,,, Kiyoshi Shimamura b, Tsuguo Fukuda b a Advanced Products Division, Mitsubishi Materials Co., Ltd. 1-5-1 Marunouchi, Chiyoda-ku, Tokyo I00, Japan b Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, Miyagi 980-77, Japan
Received 23 March 1995; manuscript received in final form 10 May 1995
Abstract During crystal growth from a stirred melt, partitioning of the intrinsic LiNbO 3 melt species is significantly influenced by the interface electric field caused by the thermoelectric power and the charge separation effects (crystallization EMF) which are a function of growth parameters such as solute boundary layer thickness, ~c, growth velocity, V and temperature gradient, G L. Thus, combinations of Bc(g), V ( g ) and GL(g) were calculated as a function of solidified melt fraction, g, by using a nonlinear programming technique 'polytope method', which leads to the true dynamic congruent-state growth from a finite amount of melt, i.e. a growth with an almost constant composition throughout the crystal and very little amount of uncoupled ionic species contained in the crystal. The micro-pulling-down technique is the most suitable growth method which can meet the calculated growth conditions.
1.
Introduction
We have seen that the interface electrostatic potential, ~b, and the interface electric fields, E L, and, E s, in the liquid and solid, respectively, b e c o m e important variables for the crystallization process of LiNbO 3 [1], i.e. each intrinsic species has a field-modified equilibrium partition coefficient, kE0 [2] at the interface leading to a complex interface boundary condition for the homogeneous crystal. Chemical reactions in the liquid during crystal growth should also be taken into * Corresponding author.
account when determining concentration profiles for the intrinsic chemical species [1]. Thus, the phase-diagram congruent melt composition is no longer valid when considering the real crystal growth from a finite amount of melt. We discussed the requirements for the true dynamic congruent-state growth, i.e. (1) the charge neutrality should hold in the solid and (2) the bulk concentration in the solid should be equal to that in the bulk liquid [1]. The latter is the conventional condition while the first specifically means if there are uncoupled ionic species remaining in the solid, they can be one of the possible sources for point defects which generate a local electric
0022-0248/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0022-0248(95)0023 1-6
S. Uda et al. /Journal o f Crystal Growth 155 (1995) 229-239
230
field or a strain field. They interfere with the laser beam injected into the crystal when the crystal is used as a second harmonic generation device. In this paper we, thus, proposed how to achieve the true congruent-state growth from a finite amount of melt associated with interface electric fields which are a function of the growth velocity, V, temperature gradient in the liquid, GL, and solute boundary layer thickness, 6c. By using a nonlinear programming technique, called 'polytope method' [3-5], we calculated combinations of these growth parameters as a function of the solidified melt fraction, g, which satisfy the requirements for the true dynamic congruentstate growth, i.e. with the solution, one can grow a LiNbO 3 crystal from a finite amount of melt with uniform chemical composition throughout the body, containing a very little amount of uncoupled ionic species.
2. Electric field at the interface
the temperature dependence of the equilibrium ion concentration. D'yakov et al. [6] point out for congruent LiNbO 3 that the potential difference, A~b, between a location at temperature T1 in the solid and a location at temperature T2 in the liquid is given by A4J=as(Tl-Ti)+aL(Ti-r2)+aiV,
(1)
where T~ is the interface temperature, a~ is the crystallization-EMF coefficient while a s and a L are the thermoelectric coefficients for the solid and the liquid, respectively. They [6] find a s = 0.76_+0.02 mV K -1, a t = - 0 . 4 mV K -1 and a i = 1.25 + 0.2 mV s p.m -1. Thus, the interface electric field operating on the liquid, E, is subdivided into two fields: the charge-separation effect-related field, Ec, and the Seebeck effect-related field, Et, i.e. E =E c +E t,
(2)
E~ = aiV/2~ c
(3)
and Fig. 1 illustrates that during crystallization of ionic melts, an electric field will be present in both the liquid and the solid and arises for at least two reasons: (1) the differential partitioning of opposite valence ions to place a net charge of one sign in the crystal and of the opposite sign in the liquid boundary layer (crystallization EMF) and (2) a Seebeck coefficient effect produced by
Solidi
Temperature
\
•
(5)
where Vt{ is the electric field-dependent effective velocity given by Eq. (6). Ti
V~ = V
~o~+))Tk
(4)
where V is the growth velocity, ~c is the thickness of the solute boundary layer and G L is the temperature gradient in the liquid at the interface. The interface electric field significantly influences the partitioning of the intrinsic LiNbO 3 melt species, i.e. for the jth species of the LiNbO 3 melt, the phase diagram equilibrium partition coefficient, k~ should be replaced by the fieldmodified equilibrium partition coefficient, k~0 [2], shown by Eq. (5) when the diffusion in the solid is neglected, k~0 = k oJV / V ~ J,
/ ) O~.......
l ) : : ~ .
E t = --aLGL,
\
Fig. 1. Illustration of the occurrence of an interface electric field during crystallization of LiNbO 3 ionic melt.
DiE ZJLeE kT '
(6)
where D [ is the diffusion constant of the jth species, z[ is the valence of the jth species, e is the electron charge and k is the Boltzmann constant. DJL and z~, used for later calculations, are listed in Table 1. Here, we see the interface
S. Uda et al./Journal of Crystal Growth 155 (1995) 229-239 Table 1 Diffusion constant D and valence z of the intrinsic LiNbO 3 melt species
LiNbO 3 Li20 Li + OLiNb205 Nb2 O2+ 0 2-
231
bulk liquid, C~, "~ is given by Eq. (8) as a function of the solidification parameter, g:
D (cm 2 s - l )
z
~ J ( g ) = C~0) + ACZ(g).
5 . 0 x 10 -6 5.0x 10 -6 3.4 x 10 - s 1.0 x 10- s 5.0x 10 -6 5.0X 10 -6 1.0X10 - s
0 0 1 - 1 0 2 -2
To know C~(g), we set up a modified Scheil equation by taking account of the species conversions [1]. This is shown by the solute conservation equation in the complete mixing approximation, Eq. (9), where a fraction d g of liquid is solidified when the liquid volume is (1 - g ) , i.e.
d g ( C L - kECL) j i =dg(1electric field E in Eq. (6) and then we know that the partitioning is a function of growth parameters, V, 3c and G e. We should also introduce a field-modified effective partition coefficient, k~ [21, kG
k~= k~ ° +(l_kieo)exp[_(V~/DiL)SC ] .
(7)
What we should do is to choose a best combination of the growth parameters as a function of the solidified melt fraction, g, which will yield a suitable interface electric field dominating the partitioning of the intrinsic species and eventually leading to a desirable distribution of the intrinsic species in the solid.
3. Population change of the intrinsic species during crystal growth Fig. 2 illustrates that the amount of Li + ions is equal to the amount of OLi- ions at the very beginning of the crystal growth from a finite amount of the melt. Here, C~0) is the initial concentration of the jth species in the bulk liquid (at x > 6c). However, during crystal growth these ions must change their populations by a concentration change of the jth species in the bulk liquid, AC~, due to accumulation or depletion phenomena in the liquid boundary layer, repulsion or attraction by the interface electric field and chemical conversion reactions (dissociationassociation and ionization-recombination) [7]. This is also the case with the Nb species. Therefore, the concentration of the jth species in the
(8)
j i kE)CL
= (1 - g)(dC~ + dC~on),
(9)
where dC~, is the concentration change of the jth species due to conversion reactions. The solution to Eq. (9) gives the concentration for each of the intrinsic species, C~(g), in the range x > 3c, represented by ( ~ = C~_~o,(1 _ g ) , ~ - l ,
(10)
where k~ is a complex of k~, Cfl~,, and the equilibrium constants, K~ ) (i ='1,2,3), ~or chemical reactions (Appendix A). The detailed procedure for obtaining solution (10) is described in Ref. [1].
4. Requirements for the true dynamic congruent melt composition For true dynamic congruent-state growth from the congruent melt, the following equations are
Interface Solid
Liquid \
~
~
,~Li Li*~ ~ .
f,OLi "~10) (V = G = 0)
-..._/
AC °El (V ~ G , 0)
AC~ (V ~ G , 0 )
I Fig. 2. Illustration of compositional change of ionic species in LiNbO 3 melt during crystal growth.
S. Uda et al. /Journal of Crystal Growth 155 (1995) 229-239
232
required [1] for both the Li-species and the Nbspecies in a certain range of the solidified melt fraction, g * < g < g*: kLi+ggLi _ b. O L i - / - , O L i E ~-"L(~) r~E "-"L(~)
= hl[V(g),GL(g),6c(g)] ~ 0
(lla)
and k N b 2 0 2 + [ g N b 2 O2+
E
~"L(~)
02-
-O 2-
-- kE CL(=)
= h2[V(g),GL(g),6c(g)] -~ O,
(llb)
in order ideally to have charge neutrality in the solid and, by approximating koLimb°3 -- 1 [8], b L i 2 0 / ' ~ L i 2 0 -].- ~1r[~b LEi + ( g" L " Li ( ~+) 4-- b'OLi-/'gOLi-~ '* ~ L(~,) '~E ~-"L(~) ] /ggLi20 1 / / ' g L i + -.1- / g O L i - ) ' ~ - - [ ~L(oo) "[- 2 ~ ~-"L(oo) - - ~" L(=) 1 J
=h3[V(g),GL(g),~c(g)] ~0
(11c)
5. C a l c u l a t i o n of growth parameters
Before we discuss the procedure to calculate a suitable combination of growth parameters, we should note that G and V are normally related to each other for crystal growth via the CZ technique and their relationship is described via heat conservation at the interface,
KsG s = KLG L + p A H V
and
KsG s - p A H V
[,~Nb~O~ ,~Nb~01+) - nt- "--~L(oo) ]
G L =
-- ~ ~L(~)
(lld)
in order ideally to have the concentration in the solid equal to that in the bulk liquid. After the crystal starts growing, there is some time needed for forming a solute diffusion boundary layer and the fraction g * is the point at which we can begin to control ~c- While the fraction g* is the point at which the terminal transient starts. The extent of completion of each condition is represented by the difference, h i (i = 1,2,3 and 4) which is a function of growth parameters. It may not be physically possible that these four equations have zero values at the same time, however, we can numerically simulate crystal growth to find the proper combination of V(g), GL(g) and 6c(g) to yield
foranyg*
(12)
In our previous paper [1], we proposed one way to calculate to meet Eq. (12), wherein CJs(g) = constant i.e., dC~(g) dg
dk~(g) = C~(g) -
-
dg
(14b)
K L
=h4[l/(g),GL(g),6c(g)] ~0,
hl--h2~h3-~h4=O
(14a)
so that
b. N b 2 0 5 ( g N b 2 0 5 4- b Nb2042+ f '-N b 2 0 4 2+
-
and this seems to be effective since assigning a constant value to each of the intrinsic species in the solid is the mathematically easiest way for the case where each of six intrinsic species tries to vary its concentration, interfering with the concentration of other species, to meet the conditions described by Eqs. (11).
dC~(g) + k~(g)--
dg
-0
(13)
where K s and K L are thermal conductivities for the solid and liquid, respectively, A H is the heat of fusion per gram and p is the density. However, we can vary V and G L independently since we can change G s by heating or cooling the growing crystal.
5.1. Polytope method We investigated one best combination of the growth parameters which satisfies the condition of the true dynamic congruent-state growth represented by Eqs. (11) by seeking a minimum value of the objective function, Dyn, which is given by g=gt
Dyn=
~
4
~ (wihi) 2,
(15)
g~g* i = 1
where w i is the weight for h( We used a nonlinear programming 'polytope' method for the calculation. The algorithm was ingeniously introduced by Spendley et al. [3] and modified to significantly improve its speed performance by Nelder and Mead [4] and further elaborated by Parkinson and Hutchinson [5]. It was
S. Uda et al./Journal of Oystal Growth 155 (1995) 229-239
233
Table 2 Population of intrinsic LiNbO 3 melt species at 1270°C as a function of bulk melt composition (mol%) tool% Li ,O
48.35
48.38
48.40
48.45
48.50
48.55
48.60
48.65
LiNbO 3 Li20 Li ÷ OLiNb20 5 Nb2 O2+ O 2-
68.78 2.57 5.67 5.67 5.02 6.14 6.14
68.79 2.58 5.69 5.69 5.00 6.13 6.13
68.80 2.59 5.69 5.69 4.98 6.12 6.12
68.81 2.61 5.71 5.71 4.95 6.10 6.10
68.83 2.62 5.73 5.73 4.92 6.08 6.08
68.84 2.64 5.75 5.75 4.89 6.06 6.06
68.85 2.66 5.77 5.77 4.86 6.04 6.04
68.87 2.68 5.79 5.79 4.83 6.02 6.02
straightforward and easy for computer coding. It differs completely from well-known reducedgradient methods or various Lagrangian methods and the idea is intuitive that one can find the best solution by tracing optimum performing conditions through evaluating the output from a system which is represented as a set of (n + 1) points defining the polytope of n-dimensional space. This procedure continually forms a new polytope, responds to the regional landscape, draws down along inclined planes, adjusts its direction when confronting a deep gap, and moves toward a minimum. This method, therefore, is very favorable when one deals with problems which include several variables interacting with each other and it is applied not only to experimental optimization [9] but is also applied extensively to mathematical optimization problems such as the nonlinear least-squares fitting of data like ours. We adopted here the algorithm modified by Nelder and Mead [4] for our calculation. When the system has n variables, the polytope is made of (n + 1) vertices and deformed into a new polytope by replacing the vertex with the highest evaluation (worst one) by another point. The new polytope formation is based on one of
three operations depending on the order of evaluation that a new point will receive in the system, i.e., (1) reflection, (2) expansion and (3) contraction. The polytope method, like other nonlinear programming methods, requires a good initial setting of each variable that determines the size and orientation of an initial polytope which has an effect on the speed of convergence. In the calculation we made the following assumptions: (1) The initial melt bulk compositions used for calculation ranged from 48.35 to 48.65 mol% Li20. Specifically, 48.38 mol% Li20-is the one which was experimentally obtained as the best congruent melt composition by Bordui et al. [10] who characterized crystal composition through Curie temperature measurement and phasematch temperature measurement for non-critical 1.06 I~m frequency doubling. (2) The melting point for these melts is 1256°C while the temperature of the bulk liquid (x >_6 c) is 1270°C, which requires the temperature gradient in the liquid, GL, to be larger than 14/6c°C e r a - 1.
(3) The equilibrium constants K~ ), (Appendix
Table 3 k o of intrinsic LiNbO 3 melt species as a function of bulk melt composition mol% L i 2 0
48.35
48.38
48.40
48.45
48.50
48.55
48.60
48.65
LiNbO 3 Li20 Li + OLi Nb20 5 Nb204z + O2
1.05 1,76 0,70 0.70 1.53 0.47 0.47
1.05 1.75 0.70 0.70 1.54 0.47 0.47
1.05 1.74 0.70 0.70 1.54 0.47 0.47
1.05 1.72 0.70 0.70 1.55 0.47 0.47
1.05 1.70 0.70 0.70 1.56 0.47 0.47
1.05 1.69 0.69 0.69 1.57 0.48 0.48
1.05 1.67 0.69 0.69 1.58 0.48 0.48
1.05 1.65 0.69 0.69 1.60 0.48 0.48
S. Uda et al. /Journal of Crystal Growth 155 (1995) 229-239
234
approximated to be a polynomial function of g,
A) are constant in the temperature range considered. (4) Initial bulk melt composition, C~0 ) and an equilibrium partition coefficient, k 0, of each intrinsic species for these melts at 1270°C are listed in Tables 2 and 3, respectively. C~0 ) is a function of both temperature and the initial bulk melt composition, and is obtained by combining the compositional relationship between intrinsic species with K~ ), while k0 is composition-dependent [8] but assumed to be constant in the temperature range considered. (5) We assume w i to be unity in the calculation. (6) Each of the growth parameters is simply
8c . . . x 100 V ...xl
Li20
Li 20 + Nb205
G L . . . x Joo
x
i.e. ~c = ao + a l g + a 2 g 2 + . . . + a m g m,
(16a)
V = b o + b i g + b2g 2 + . . . + b m g m,
(16b)
G L = CO + c l g + c2 g 2 + . . . + c r a g m.
(16c)
(7) The evaluation is via the sum-of-squares of hi(g) for hundred points in the range of the solidified melt fraction indicator, g = g * to g = g*. As far as the desirable steady-state growth is concerned we should omit the initial uncontrollable region (0 < g < g ") as well as the terminal transient region (g* < g < 1.0). The extent of these regions is dependent on the ratio of the
IOO= 48.38
7.0 Nb~O~ +
20
6.0
0 2
- Li__~'~,...... N b 205
16
~- 5.0 fi~ 4.0
12
8c ~m) m - - m V (mrffh)
3.0
Li20
Gc (°C/cm) i
0.0
0.2
i
0.6
0.8
0.0
(a)
i
i
I
0.4
0.6
0.8
g
(b)
48.40
0.10 ] 0.05
×
0.00
o~ _ _ h --____h
-0.05
m '
0.0
i
0.2
i
0.4 g
0.2
i 2
48.39 48.38
h
Crystal
48.37
h 3 _ _h 4
- - ~ Melt
'
0.4
g
0.6
0.8
(c)
0.0
I
I
I
0.2
0.4
0.6
g
I
0.8
(d)
Fig. 3. Growth conditions for dynamic congruent-state growth as a function of solidified melt fraction g. (a) Choice of growth parameters, 8 o V and GL, (b) population change of intrinsic species in the crystal, (c) variation of h j, h 2, h 3 and h 4, and (d) congruency indicator variation. The initial bulk liquid composition is 48.38 m o l % L i 2 0 .
48.35 1300+369 g 15.0+8.64 g 1160+672 g 0.002391 0.002186 0.000238 0.000278 0.005093
tool% Li20
~c (p,m) V ( m m h -1) GL(oC cm -~) H ~ (tool%) H 2 (mol%) H 3 (mol%) H 4 (mol%) H 1 + H 2 + H 3 + H 4 (mol%)
1720+290 g 13.8+2.74 g 1090+214 g 0.001603 0.000878 0.000430 0.000049 0.002959
48.38 1700+374 g 9.34+3.63 g 746+282 g 0.010142 0.009941 0.001058 0.001356 0.022498
48.40
Table 4 Combination of growth parameters for dynamic congruent-state growth 1690+498 g 8.34+3.99 g 682+304 g 0.016830 0.016003 0.001915 0.001909 0.036657
48.45 1840+452 g 8.34+3.39 g 691 +256 g 0.013670 0.011150 0.002078 0.001234 0.028132
48.50
1690+491 g 8.64+3.81 g 716+298 g 0.014047 0.013607 0.000898 0.001840 0.030392
48.55
1720+251 g 10.6+3.80 g 884+295 g 0.007945 0.004652 0.001868 0.000303 0.014769
48.60
1670+645 g 7.94+8.46 g 684+667 g 0.017662 0.013033 0.002737 0.001011 0.034444
48.65
i
t,,o
t.,n
S. Uda et al. /Journal of Crystal Growth 155 (1995) 229-239
236
growing crystal size to the crucible size and the complexity of the partitioning, in other words, the degree of congruency. For the initial transient region, we assume a very small ratio in size between the growing crystal and the crucible which may lead to reasonably small g* < 0.01. For the congruency, there is no definite reasoning, but we assume that the terminal transient takes place at g t > 0.9. All of these assumptions need to be evaluated after the calculation to see if we were fully justified in using these simplifications. We began with m = 3, i.e. a cubic polynomial form of g for each of the three growth parameters and then decreased the magnitude of m. This is because a higher m easily yields a more complex 3(m + 1)-dimensional space leading to many local minimums so that we should use an m as small as is practical for the obtained solution. However, one should note that the global minimum is not a necessarily right solution if it is far from the real crystal growth conditions. We may find a satisfactory solution from local minimums only when taking account of the feasibility of the real crystal growth.
position are shown in Table 4 where the difference from the ideal condition is described in terms of the integration of ]hil (i = 1 to 4) in the range g = 0.01 to 0.9, i.e. •"0.9
The Hitachi supercomputer S-3800/380 at the Institute for Materials Research, Tohoku University, was used for the optimization. Calculation results for each case of different initial melt com-
LizO
(17)
We finally obtained good results by taking m = 1 in Eqs. (16). One should note that each of the growth parameters 6 o V and G L varies with g in a similar manner for all cases, i.e. positive intersects at g = 0 and positive slopes, which suggests that the optimization calculation worked fine. Although there is a slight difference in h i between the melts, the 48.38 mol% L i 2 0 melt yields the least difference. The calculation results for the 48.38 mol% L i 2 0 melt are illustrated in Fig. 3 as a function of the solidified melt fraction, g: (a) a combination of the growth parameters, (b) the intrinsic LiNbO 3 species distribution in the crystal, (c) h i variation and (d) L i / N b ratio change. One should note that the terminal transient region never takes place at g < 0.9 and we found that g* > 0.96. It should also be noted that the optimization leads to a fairly constant species concentration in the solid as we expected before (Eq. (13)). It is hard to differentiate Li ÷ from O L i - and Nb2 O2÷ from 0 2- in the concentration diagram (Fig. 3b) since h 1 and h 2 are so small. A small value of h i leads to a constant L i / N b ratio (Figs. 3c and 3d). However, the
5.2. Calculation results
Li2 °
i
H i = )0.01[h [dg.
x 100= 48.35
+ Nb205
48.37 0.10 x
0.05
48.36
C ~ ~,~ 48.35
0.00 E
_ _ h __ m Dh
-0.05
m
0.0
0.2
~
~
0.4
0.6 g
j
2 h 3 - _h 4
0.8 (a)
Crystal
48.34
- - - -
0.0
Melt
I
I
I
I
0.2
0.4
0.6
0.8
(b)
g
Fig. 4. Illustration of (a) population change of intrinsic species in the crystal, and (b) congruency indicator wlriation with solidified melt fraction g. The initial bulk liquid composition is 48.35 mol% LizO.
S. Uda et al./Journal of Crystal Growth 155 (1995) 229-239
*b
Q
Melt
o o
(+)
Fig. 5. Illustration of the micro-pulling-down(Ix@D) growth method.
reverse is not true. Fig. 4 illustrates the calculation results in the sense that the charge neutrality holds less when compared with Fig. 3c even if the L i / N b ratio looks extremely constant throughout the crystal (Fig. 4b). This superficial compositional matching yields some uncoupled ionic species leading to ion clusters in the crystal. They may generate local electric fields or stress-strain fields which would cause laser beam scattering when a LiNbO 3 crystal is used as a second harmonic generation device.
237
151.665 and N A = 6.02 x 1023 into Eq. (18), we obtain exions = 3.26 × 1017 cm -3. One should note that this is the average value and the there are much less uncoupled ions at g > 0.4 (see Fig. 3c). Future work should deal with exions in connection with the generation of charged point defects which interfere with a laser beam. It may be difficult to assign the values of the growth parameters found in Table 4 to the crystal growth via a normal Czochralski method, i.e. all of the three growth parameters, 3c, V and G L have larger values by one order of magnitude than those for the normal CZ method. However, these growth conditions may be possible by using a micro-pulling-down (Ix-PD) method (Fig. 5) which is one of the growth techniques from a finite amount of melt. It has a micro nozzle which gains a large solute diffusion volume and its thickness is adjustable by stirring the melt with a micro paddle inserted to the nozzle. The temperature gradient at the interface in the liquid can be more than 2000°C c m - i [11] and it is controllable by adjusting the power loaded to the afterheater. Since the temperature gradient at the interface is so high, one can easily achieve a growth velocity larger than 10 mm h ~. If the nozzle size is 1 mm thd × 2 mm h and the crucible size is 10 mm ¢bd × 10 mm h, then the volume ratio of nozzle to crucible is 1/500, which may lead us to justify g * < 0.01.
6. Discussion The charge imbalance caused by the quantity difference between Li ÷ and O L i - or Nb2 O2+ and O 2- in the crystal is expressed by h t or h 2, respectively. Thus, we can evaluate the average concentration of uncoupled ions remaining in the unit volume of the grown crystal. For instance, the number of excess Li ÷ or O L i - in 1 cm 3 LiNbO 3 crystal, exions, is obtained by
1.0 H1 Ps exions =
NA,
0.9 100 M W
(18)
where Ps is the density of the crystal, M w is the molecular weight of LiNbO 3 and N A is Avogadro's number. Inserting H ~= 0.001603, Ps = 4.61 g / c m 3, M w for 48.38 mol% L i 2 0 crystal =
7. Summary (1) We have stressed that the interface electric field associated with crystal growth significantly influences on the partitioning of the intrinsic LiNbO 3 melt species and the magnitude of the field is dominated by the growth parameters, i.e. thickness of the solute diffusion layer, 6 c , growth velocity, V and temperature gradient in the liquid, G L. (2) Thus, different growth conditions provide different interface electric fields, which leads us to consider the dynamic congruent-state growth as a function of the growth parameters, which yields a crystal with fairly constant and balanced concentration profiles throughout its volume.
S. Uda et al. /Journal of Crystal Growth 155 (1995) 229-239
238
(3) By using a powerful nonlinear programming 'polytope' method, we calculated the best combinations of growth parameters for the dynamic congruent-state growth from a finite amount of melts with the initial compositions ranging from 48.35 to 48.65 mol% LIE0. A combination of ~c, V and G L with a simple linear change with g leads to a good dynamic congruent-state growth. (4) A micro-pulling-down growth technique may be a good technique which can meet the calculated growth conditions.
k~:Li÷ = k~ i. -
K(r2)(azC[~ °
- mfeon)
(A.4.2)
h / "~Li+ u2v" L~(0 )
kffLi- = kOLi- _
Ktr2)(a2C~o?-AC'con) b g, OLi2~Lw.(0)
(A.4.3) k,Nb205 = kNb205 _ a 3
_ ,~Nb~Os_ AC'
//'r'Nb2°'~+
con ] I "" Dx(O)
t~ 31.., L~(b)
02-
+ CLw.(0))
b f , Nb2O5 3~-" L~(0)
Acknowledgements
(A.4.4)
The authors appreciate the useful discussions with Professor W.A. Tiller at Stanford University who has stressed the importance of the dynamic congruent-state growth with one of the authors several years ago. The kind offer by Prof. Y. Kawazoe to use the supercomputer at the Institute for Materials Research, Tohoku University, is also gratefully acknowledged.
k~Nb~OP =
kENb20~ +
K(3){aT ~ 3""I.~o )/''Nb205 -- A C ' c o n ) _
_
b pNb202 +
3~-" L~(0)
(A.4.5) kkO2- = k o2- _
K(ra)(aaCNLb~?'- AC'con) 02b3CL~(O)
(A.4.6)
Appendix A
where
Precise description of k~ For the following chemical reactions, we define chemical equilibrium constants K~ ) (i =
a2 = kLi2°- (kLi* + k°Li-) + 1,
1,2,3),
bl
2LiNbO 3 ¢* Li20 + Nb205, K(T1) _ .. Li ,O _ Nb,O 5 , . . LiNbO 2 . -- a" L(~')'at L(~') / ~ L(T)
L i 2 0 ~ Li++ O L i - ,
(A.
1)
Ktr2)_ _Li ÷ _OLi-,_Li,O. -- "~ L(T)"t L(T) / ~ L(~') (A.2)
Nb20 s ¢~ NbzO42+ + 0 2 - , K(T3) _--.~L(/~) _ Nb,O 2 +_ 0 2 - ~,_ N b , O , . .'tL(T)/.,tL(~) ,
a3=kNb2°s-(kr~b=°~++k°2-)+
(A.5.1)
1,
(A.5.2)
-- ?/C(1)/-'LiNbO3 4- /'~Li20 4- (~Nb205 - - ~ * ~ T "-~Loe(0) -- "~Lo=(0)- "-'L~(0) ,
(A.5.3)
Li + [-,OLib 2 = K(T2) + CL~(0) -1- ',-'Lo~(0),
(A.5.4)
pNb202+ 02b 3 = K(T3) + "~ L=(0) + CLc~(0),
(A.5.5)
and A C ~ , = [ C~i~)CLW°~o~S{b2b3(2 - k Li2° - k Nb2°s)
(A.3)
where X~r ) is the mole fraction of the jth intrinsic melt species and T = 1270°C. For the intrinsic melt species, k~ is deduced by solving Eq. (9) [1], and we finally obtain
+a2baK(r2) + a3b2rcr3)}] X [ -bjb2b 3 a. h ["Nb2Osk'(2) h rLi20 k'(3)]-1 v 2,,-- L~(0)-x T ]
(A.5.6)
k,Li20 = kLi20 _ a 2
_ (a2CL~°o)-- AC'~o,)(CL~o, + C°,~i0;) References
b2C~dO)
(A.4.1)
Ill W . A . Tiller and S. Uda, J. Crystal G r o w t h 129 (1993) 341.
S. Uda et al. /Journal of Crystal Growth 155 (1995) 229-239 [2] S. Uda and W.A. Tiller, J. Crystal Growth 121 (1992) 93. [3] W. Spendley, G.R. Hext and F.R. Himsworth, Technometrics 4 (1962) 441. [4] J.A. Nelder and R. Mead, Computer J. 7 (1965) 308. [5] J.M. Parkinson and D. Hutchinson, in: Numerical Methods for Non-Linear Optimization, Ed. F.A. Lootsma (Academic Press, New York, 1972) pp. 115-135. [6] V.A. D'yakov, D.P. Shumov, L.N. Rashkovich and A.L. Aleksandrovskii, Bull. Acad. Sci. USSR Phys. Ser. 49 (1986) 117.
239
[7] S. Uda and W.A. Tiller, J. Crystal Growth 121 (1992) 155. [8] W.A. Tiller and S. Uda, J. Crystal Growth 129 (1993) 328. [9] S.N. Deming and S.L. Morgan, Anal. Chem. 45 (1973) 278A. [10] P.F. Bordui, R.G. Norwood, C.D. Bird and G.D. Calvert, J. Crystal Growth 113 (1991) 61. [11] D.H. Yoon, I. Yonenaga, T. Fukuda and N. Ohnishi, J. Crystal Growth 142 (1994) 339.
CIRYSTAL QNOWTH
Journal of Crystal Growth 155 (1995) 229-239
ELSEVIER
Intrinsic
melt species partitioning at the congruent melt composition III. Choice of the growth parameters for the dynamic congruent-state growth LiNbO 3
Satoshi Uda a,,, Kiyoshi Shimamura b, Tsuguo Fukuda b a Advanced Products Division, Mitsubishi Materials Co., Ltd. 1-5-1 Marunouchi, Chiyoda-ku, Tokyo I00, Japan b Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, Miyagi 980-77, Japan
Received 23 March 1995; manuscript received in final form 10 May 1995
Abstract During crystal growth from a stirred melt, partitioning of the intrinsic LiNbO 3 melt species is significantly influenced by the interface electric field caused by the thermoelectric power and the charge separation effects (crystallization EMF) which are a function of growth parameters such as solute boundary layer thickness, ~c, growth velocity, V and temperature gradient, G L. Thus, combinations of Bc(g), V ( g ) and GL(g) were calculated as a function of solidified melt fraction, g, by using a nonlinear programming technique 'polytope method', which leads to the true dynamic congruent-state growth from a finite amount of melt, i.e. a growth with an almost constant composition throughout the crystal and very little amount of uncoupled ionic species contained in the crystal. The micro-pulling-down technique is the most suitable growth method which can meet the calculated growth conditions.
1.
Introduction
We have seen that the interface electrostatic potential, ~b, and the interface electric fields, E L, and, E s, in the liquid and solid, respectively, b e c o m e important variables for the crystallization process of LiNbO 3 [1], i.e. each intrinsic species has a field-modified equilibrium partition coefficient, kE0 [2] at the interface leading to a complex interface boundary condition for the homogeneous crystal. Chemical reactions in the liquid during crystal growth should also be taken into * Corresponding author.
account when determining concentration profiles for the intrinsic chemical species [1]. Thus, the phase-diagram congruent melt composition is no longer valid when considering the real crystal growth from a finite amount of melt. We discussed the requirements for the true dynamic congruent-state growth, i.e. (1) the charge neutrality should hold in the solid and (2) the bulk concentration in the solid should be equal to that in the bulk liquid [1]. The latter is the conventional condition while the first specifically means if there are uncoupled ionic species remaining in the solid, they can be one of the possible sources for point defects which generate a local electric
0022-0248/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0022-0248(95)0023 1-6
S. Uda et al. /Journal o f Crystal Growth 155 (1995) 229-239
230
field or a strain field. They interfere with the laser beam injected into the crystal when the crystal is used as a second harmonic generation device. In this paper we, thus, proposed how to achieve the true congruent-state growth from a finite amount of melt associated with interface electric fields which are a function of the growth velocity, V, temperature gradient in the liquid, GL, and solute boundary layer thickness, 6c. By using a nonlinear programming technique, called 'polytope method' [3-5], we calculated combinations of these growth parameters as a function of the solidified melt fraction, g, which satisfy the requirements for the true dynamic congruentstate growth, i.e. with the solution, one can grow a LiNbO 3 crystal from a finite amount of melt with uniform chemical composition throughout the body, containing a very little amount of uncoupled ionic species.
2. Electric field at the interface
the temperature dependence of the equilibrium ion concentration. D'yakov et al. [6] point out for congruent LiNbO 3 that the potential difference, A~b, between a location at temperature T1 in the solid and a location at temperature T2 in the liquid is given by A4J=as(Tl-Ti)+aL(Ti-r2)+aiV,
(1)
where T~ is the interface temperature, a~ is the crystallization-EMF coefficient while a s and a L are the thermoelectric coefficients for the solid and the liquid, respectively. They [6] find a s = 0.76_+0.02 mV K -1, a t = - 0 . 4 mV K -1 and a i = 1.25 + 0.2 mV s p.m -1. Thus, the interface electric field operating on the liquid, E, is subdivided into two fields: the charge-separation effect-related field, Ec, and the Seebeck effect-related field, Et, i.e. E =E c +E t,
(2)
E~ = aiV/2~ c
(3)
and Fig. 1 illustrates that during crystallization of ionic melts, an electric field will be present in both the liquid and the solid and arises for at least two reasons: (1) the differential partitioning of opposite valence ions to place a net charge of one sign in the crystal and of the opposite sign in the liquid boundary layer (crystallization EMF) and (2) a Seebeck coefficient effect produced by
Solidi
Temperature
\
•
(5)
where Vt{ is the electric field-dependent effective velocity given by Eq. (6). Ti
V~ = V
~o~+))Tk
(4)
where V is the growth velocity, ~c is the thickness of the solute boundary layer and G L is the temperature gradient in the liquid at the interface. The interface electric field significantly influences the partitioning of the intrinsic LiNbO 3 melt species, i.e. for the jth species of the LiNbO 3 melt, the phase diagram equilibrium partition coefficient, k~ should be replaced by the fieldmodified equilibrium partition coefficient, k~0 [2], shown by Eq. (5) when the diffusion in the solid is neglected, k~0 = k oJV / V ~ J,
/ ) O~.......
l ) : : ~ .
E t = --aLGL,
\
Fig. 1. Illustration of the occurrence of an interface electric field during crystallization of LiNbO 3 ionic melt.
DiE ZJLeE kT '
(6)
where D [ is the diffusion constant of the jth species, z[ is the valence of the jth species, e is the electron charge and k is the Boltzmann constant. DJL and z~, used for later calculations, are listed in Table 1. Here, we see the interface
S. Uda et al./Journal of Crystal Growth 155 (1995) 229-239 Table 1 Diffusion constant D and valence z of the intrinsic LiNbO 3 melt species
LiNbO 3 Li20 Li + OLiNb205 Nb2 O2+ 0 2-
231
bulk liquid, C~, "~ is given by Eq. (8) as a function of the solidification parameter, g:
D (cm 2 s - l )
z
~ J ( g ) = C~0) + ACZ(g).
5 . 0 x 10 -6 5.0x 10 -6 3.4 x 10 - s 1.0 x 10- s 5.0x 10 -6 5.0X 10 -6 1.0X10 - s
0 0 1 - 1 0 2 -2
To know C~(g), we set up a modified Scheil equation by taking account of the species conversions [1]. This is shown by the solute conservation equation in the complete mixing approximation, Eq. (9), where a fraction d g of liquid is solidified when the liquid volume is (1 - g ) , i.e.
d g ( C L - kECL) j i =dg(1electric field E in Eq. (6) and then we know that the partitioning is a function of growth parameters, V, 3c and G e. We should also introduce a field-modified effective partition coefficient, k~ [21, kG
k~= k~ ° +(l_kieo)exp[_(V~/DiL)SC ] .
(7)
What we should do is to choose a best combination of the growth parameters as a function of the solidified melt fraction, g, which will yield a suitable interface electric field dominating the partitioning of the intrinsic species and eventually leading to a desirable distribution of the intrinsic species in the solid.
3. Population change of the intrinsic species during crystal growth Fig. 2 illustrates that the amount of Li + ions is equal to the amount of OLi- ions at the very beginning of the crystal growth from a finite amount of the melt. Here, C~0) is the initial concentration of the jth species in the bulk liquid (at x > 6c). However, during crystal growth these ions must change their populations by a concentration change of the jth species in the bulk liquid, AC~, due to accumulation or depletion phenomena in the liquid boundary layer, repulsion or attraction by the interface electric field and chemical conversion reactions (dissociationassociation and ionization-recombination) [7]. This is also the case with the Nb species. Therefore, the concentration of the jth species in the
(8)
j i kE)CL
= (1 - g)(dC~ + dC~on),
(9)
where dC~, is the concentration change of the jth species due to conversion reactions. The solution to Eq. (9) gives the concentration for each of the intrinsic species, C~(g), in the range x > 3c, represented by ( ~ = C~_~o,(1 _ g ) , ~ - l ,
(10)
where k~ is a complex of k~, Cfl~,, and the equilibrium constants, K~ ) (i ='1,2,3), ~or chemical reactions (Appendix A). The detailed procedure for obtaining solution (10) is described in Ref. [1].
4. Requirements for the true dynamic congruent melt composition For true dynamic congruent-state growth from the congruent melt, the following equations are
Interface Solid
Liquid \
~
~
,~Li Li*~ ~ .
f,OLi "~10) (V = G = 0)
-..._/
AC °El (V ~ G , 0)
AC~ (V ~ G , 0 )
I Fig. 2. Illustration of compositional change of ionic species in LiNbO 3 melt during crystal growth.
S. Uda et al. /Journal of Crystal Growth 155 (1995) 229-239
232
required [1] for both the Li-species and the Nbspecies in a certain range of the solidified melt fraction, g * < g < g*: kLi+ggLi _ b. O L i - / - , O L i E ~-"L(~) r~E "-"L(~)
= hl[V(g),GL(g),6c(g)] ~ 0
(lla)
and k N b 2 0 2 + [ g N b 2 O2+
E
~"L(~)
02-
-O 2-
-- kE CL(=)
= h2[V(g),GL(g),6c(g)] -~ O,
(llb)
in order ideally to have charge neutrality in the solid and, by approximating koLimb°3 -- 1 [8], b L i 2 0 / ' ~ L i 2 0 -].- ~1r[~b LEi + ( g" L " Li ( ~+) 4-- b'OLi-/'gOLi-~ '* ~ L(~,) '~E ~-"L(~) ] /ggLi20 1 / / ' g L i + -.1- / g O L i - ) ' ~ - - [ ~L(oo) "[- 2 ~ ~-"L(oo) - - ~" L(=) 1 J
=h3[V(g),GL(g),~c(g)] ~0
(11c)
5. C a l c u l a t i o n of growth parameters
Before we discuss the procedure to calculate a suitable combination of growth parameters, we should note that G and V are normally related to each other for crystal growth via the CZ technique and their relationship is described via heat conservation at the interface,
KsG s = KLG L + p A H V
and
KsG s - p A H V
[,~Nb~O~ ,~Nb~01+) - nt- "--~L(oo) ]
G L =
-- ~ ~L(~)
(lld)
in order ideally to have the concentration in the solid equal to that in the bulk liquid. After the crystal starts growing, there is some time needed for forming a solute diffusion boundary layer and the fraction g * is the point at which we can begin to control ~c- While the fraction g* is the point at which the terminal transient starts. The extent of completion of each condition is represented by the difference, h i (i = 1,2,3 and 4) which is a function of growth parameters. It may not be physically possible that these four equations have zero values at the same time, however, we can numerically simulate crystal growth to find the proper combination of V(g), GL(g) and 6c(g) to yield
foranyg*
(12)
In our previous paper [1], we proposed one way to calculate to meet Eq. (12), wherein CJs(g) = constant i.e., dC~(g) dg
dk~(g) = C~(g) -
-
dg
(14b)
K L
=h4[l/(g),GL(g),6c(g)] ~0,
hl--h2~h3-~h4=O
(14a)
so that
b. N b 2 0 5 ( g N b 2 0 5 4- b Nb2042+ f '-N b 2 0 4 2+
-
and this seems to be effective since assigning a constant value to each of the intrinsic species in the solid is the mathematically easiest way for the case where each of six intrinsic species tries to vary its concentration, interfering with the concentration of other species, to meet the conditions described by Eqs. (11).
dC~(g) + k~(g)--
dg
-0
(13)
where K s and K L are thermal conductivities for the solid and liquid, respectively, A H is the heat of fusion per gram and p is the density. However, we can vary V and G L independently since we can change G s by heating or cooling the growing crystal.
5.1. Polytope method We investigated one best combination of the growth parameters which satisfies the condition of the true dynamic congruent-state growth represented by Eqs. (11) by seeking a minimum value of the objective function, Dyn, which is given by g=gt
Dyn=
~
4
~ (wihi) 2,
(15)
g~g* i = 1
where w i is the weight for h( We used a nonlinear programming 'polytope' method for the calculation. The algorithm was ingeniously introduced by Spendley et al. [3] and modified to significantly improve its speed performance by Nelder and Mead [4] and further elaborated by Parkinson and Hutchinson [5]. It was
S. Uda et al./Journal of Oystal Growth 155 (1995) 229-239
233
Table 2 Population of intrinsic LiNbO 3 melt species at 1270°C as a function of bulk melt composition (mol%) tool% Li ,O
48.35
48.38
48.40
48.45
48.50
48.55
48.60
48.65
LiNbO 3 Li20 Li ÷ OLiNb20 5 Nb2 O2+ O 2-
68.78 2.57 5.67 5.67 5.02 6.14 6.14
68.79 2.58 5.69 5.69 5.00 6.13 6.13
68.80 2.59 5.69 5.69 4.98 6.12 6.12
68.81 2.61 5.71 5.71 4.95 6.10 6.10
68.83 2.62 5.73 5.73 4.92 6.08 6.08
68.84 2.64 5.75 5.75 4.89 6.06 6.06
68.85 2.66 5.77 5.77 4.86 6.04 6.04
68.87 2.68 5.79 5.79 4.83 6.02 6.02
straightforward and easy for computer coding. It differs completely from well-known reducedgradient methods or various Lagrangian methods and the idea is intuitive that one can find the best solution by tracing optimum performing conditions through evaluating the output from a system which is represented as a set of (n + 1) points defining the polytope of n-dimensional space. This procedure continually forms a new polytope, responds to the regional landscape, draws down along inclined planes, adjusts its direction when confronting a deep gap, and moves toward a minimum. This method, therefore, is very favorable when one deals with problems which include several variables interacting with each other and it is applied not only to experimental optimization [9] but is also applied extensively to mathematical optimization problems such as the nonlinear least-squares fitting of data like ours. We adopted here the algorithm modified by Nelder and Mead [4] for our calculation. When the system has n variables, the polytope is made of (n + 1) vertices and deformed into a new polytope by replacing the vertex with the highest evaluation (worst one) by another point. The new polytope formation is based on one of
three operations depending on the order of evaluation that a new point will receive in the system, i.e., (1) reflection, (2) expansion and (3) contraction. The polytope method, like other nonlinear programming methods, requires a good initial setting of each variable that determines the size and orientation of an initial polytope which has an effect on the speed of convergence. In the calculation we made the following assumptions: (1) The initial melt bulk compositions used for calculation ranged from 48.35 to 48.65 mol% Li20. Specifically, 48.38 mol% Li20-is the one which was experimentally obtained as the best congruent melt composition by Bordui et al. [10] who characterized crystal composition through Curie temperature measurement and phasematch temperature measurement for non-critical 1.06 I~m frequency doubling. (2) The melting point for these melts is 1256°C while the temperature of the bulk liquid (x >_6 c) is 1270°C, which requires the temperature gradient in the liquid, GL, to be larger than 14/6c°C e r a - 1.
(3) The equilibrium constants K~ ), (Appendix
Table 3 k o of intrinsic LiNbO 3 melt species as a function of bulk melt composition mol% L i 2 0
48.35
48.38
48.40
48.45
48.50
48.55
48.60
48.65
LiNbO 3 Li20 Li + OLi Nb20 5 Nb204z + O2
1.05 1,76 0,70 0.70 1.53 0.47 0.47
1.05 1.75 0.70 0.70 1.54 0.47 0.47
1.05 1.74 0.70 0.70 1.54 0.47 0.47
1.05 1.72 0.70 0.70 1.55 0.47 0.47
1.05 1.70 0.70 0.70 1.56 0.47 0.47
1.05 1.69 0.69 0.69 1.57 0.48 0.48
1.05 1.67 0.69 0.69 1.58 0.48 0.48
1.05 1.65 0.69 0.69 1.60 0.48 0.48
S. Uda et al. /Journal of Crystal Growth 155 (1995) 229-239
234
approximated to be a polynomial function of g,
A) are constant in the temperature range considered. (4) Initial bulk melt composition, C~0 ) and an equilibrium partition coefficient, k 0, of each intrinsic species for these melts at 1270°C are listed in Tables 2 and 3, respectively. C~0 ) is a function of both temperature and the initial bulk melt composition, and is obtained by combining the compositional relationship between intrinsic species with K~ ), while k0 is composition-dependent [8] but assumed to be constant in the temperature range considered. (5) We assume w i to be unity in the calculation. (6) Each of the growth parameters is simply
8c . . . x 100 V ...xl
Li20
Li 20 + Nb205
G L . . . x Joo
x
i.e. ~c = ao + a l g + a 2 g 2 + . . . + a m g m,
(16a)
V = b o + b i g + b2g 2 + . . . + b m g m,
(16b)
G L = CO + c l g + c2 g 2 + . . . + c r a g m.
(16c)
(7) The evaluation is via the sum-of-squares of hi(g) for hundred points in the range of the solidified melt fraction indicator, g = g * to g = g*. As far as the desirable steady-state growth is concerned we should omit the initial uncontrollable region (0 < g < g ") as well as the terminal transient region (g* < g < 1.0). The extent of these regions is dependent on the ratio of the
IOO= 48.38
7.0 Nb~O~ +
20
6.0
0 2
- Li__~'~,...... N b 205
16
~- 5.0 fi~ 4.0
12
8c ~m) m - - m V (mrffh)
3.0
Li20
Gc (°C/cm) i
0.0
0.2
i
0.6
0.8
0.0
(a)
i
i
I
0.4
0.6
0.8
g
(b)
48.40
0.10 ] 0.05
×
0.00
o~ _ _ h --____h
-0.05
m '
0.0
i
0.2
i
0.4 g
0.2
i 2
48.39 48.38
h
Crystal
48.37
h 3 _ _h 4
- - ~ Melt
'
0.4
g
0.6
0.8
(c)
0.0
I
I
I
0.2
0.4
0.6
g
I
0.8
(d)
Fig. 3. Growth conditions for dynamic congruent-state growth as a function of solidified melt fraction g. (a) Choice of growth parameters, 8 o V and GL, (b) population change of intrinsic species in the crystal, (c) variation of h j, h 2, h 3 and h 4, and (d) congruency indicator variation. The initial bulk liquid composition is 48.38 m o l % L i 2 0 .
48.35 1300+369 g 15.0+8.64 g 1160+672 g 0.002391 0.002186 0.000238 0.000278 0.005093
tool% Li20
~c (p,m) V ( m m h -1) GL(oC cm -~) H ~ (tool%) H 2 (mol%) H 3 (mol%) H 4 (mol%) H 1 + H 2 + H 3 + H 4 (mol%)
1720+290 g 13.8+2.74 g 1090+214 g 0.001603 0.000878 0.000430 0.000049 0.002959
48.38 1700+374 g 9.34+3.63 g 746+282 g 0.010142 0.009941 0.001058 0.001356 0.022498
48.40
Table 4 Combination of growth parameters for dynamic congruent-state growth 1690+498 g 8.34+3.99 g 682+304 g 0.016830 0.016003 0.001915 0.001909 0.036657
48.45 1840+452 g 8.34+3.39 g 691 +256 g 0.013670 0.011150 0.002078 0.001234 0.028132
48.50
1690+491 g 8.64+3.81 g 716+298 g 0.014047 0.013607 0.000898 0.001840 0.030392
48.55
1720+251 g 10.6+3.80 g 884+295 g 0.007945 0.004652 0.001868 0.000303 0.014769
48.60
1670+645 g 7.94+8.46 g 684+667 g 0.017662 0.013033 0.002737 0.001011 0.034444
48.65
i
t,,o
t.,n
S. Uda et al. /Journal of Crystal Growth 155 (1995) 229-239
236
growing crystal size to the crucible size and the complexity of the partitioning, in other words, the degree of congruency. For the initial transient region, we assume a very small ratio in size between the growing crystal and the crucible which may lead to reasonably small g* < 0.01. For the congruency, there is no definite reasoning, but we assume that the terminal transient takes place at g t > 0.9. All of these assumptions need to be evaluated after the calculation to see if we were fully justified in using these simplifications. We began with m = 3, i.e. a cubic polynomial form of g for each of the three growth parameters and then decreased the magnitude of m. This is because a higher m easily yields a more complex 3(m + 1)-dimensional space leading to many local minimums so that we should use an m as small as is practical for the obtained solution. However, one should note that the global minimum is not a necessarily right solution if it is far from the real crystal growth conditions. We may find a satisfactory solution from local minimums only when taking account of the feasibility of the real crystal growth.
position are shown in Table 4 where the difference from the ideal condition is described in terms of the integration of ]hil (i = 1 to 4) in the range g = 0.01 to 0.9, i.e. •"0.9
The Hitachi supercomputer S-3800/380 at the Institute for Materials Research, Tohoku University, was used for the optimization. Calculation results for each case of different initial melt com-
LizO
(17)
We finally obtained good results by taking m = 1 in Eqs. (16). One should note that each of the growth parameters 6 o V and G L varies with g in a similar manner for all cases, i.e. positive intersects at g = 0 and positive slopes, which suggests that the optimization calculation worked fine. Although there is a slight difference in h i between the melts, the 48.38 mol% L i 2 0 melt yields the least difference. The calculation results for the 48.38 mol% L i 2 0 melt are illustrated in Fig. 3 as a function of the solidified melt fraction, g: (a) a combination of the growth parameters, (b) the intrinsic LiNbO 3 species distribution in the crystal, (c) h i variation and (d) L i / N b ratio change. One should note that the terminal transient region never takes place at g < 0.9 and we found that g* > 0.96. It should also be noted that the optimization leads to a fairly constant species concentration in the solid as we expected before (Eq. (13)). It is hard to differentiate Li ÷ from O L i - and Nb2 O2÷ from 0 2- in the concentration diagram (Fig. 3b) since h 1 and h 2 are so small. A small value of h i leads to a constant L i / N b ratio (Figs. 3c and 3d). However, the
5.2. Calculation results
Li2 °
i
H i = )0.01[h [dg.
x 100= 48.35
+ Nb205
48.37 0.10 x
0.05
48.36
C ~ ~,~ 48.35
0.00 E
_ _ h __ m Dh
-0.05
m
0.0
0.2
~
~
0.4
0.6 g
j
2 h 3 - _h 4
0.8 (a)
Crystal
48.34
- - - -
0.0
Melt
I
I
I
I
0.2
0.4
0.6
0.8
(b)
g
Fig. 4. Illustration of (a) population change of intrinsic species in the crystal, and (b) congruency indicator wlriation with solidified melt fraction g. The initial bulk liquid composition is 48.35 mol% LizO.
S. Uda et al./Journal of Crystal Growth 155 (1995) 229-239
*b
Q
Melt
o o
(+)
Fig. 5. Illustration of the micro-pulling-down(Ix@D) growth method.
reverse is not true. Fig. 4 illustrates the calculation results in the sense that the charge neutrality holds less when compared with Fig. 3c even if the L i / N b ratio looks extremely constant throughout the crystal (Fig. 4b). This superficial compositional matching yields some uncoupled ionic species leading to ion clusters in the crystal. They may generate local electric fields or stress-strain fields which would cause laser beam scattering when a LiNbO 3 crystal is used as a second harmonic generation device.
237
151.665 and N A = 6.02 x 1023 into Eq. (18), we obtain exions = 3.26 × 1017 cm -3. One should note that this is the average value and the there are much less uncoupled ions at g > 0.4 (see Fig. 3c). Future work should deal with exions in connection with the generation of charged point defects which interfere with a laser beam. It may be difficult to assign the values of the growth parameters found in Table 4 to the crystal growth via a normal Czochralski method, i.e. all of the three growth parameters, 3c, V and G L have larger values by one order of magnitude than those for the normal CZ method. However, these growth conditions may be possible by using a micro-pulling-down (Ix-PD) method (Fig. 5) which is one of the growth techniques from a finite amount of melt. It has a micro nozzle which gains a large solute diffusion volume and its thickness is adjustable by stirring the melt with a micro paddle inserted to the nozzle. The temperature gradient at the interface in the liquid can be more than 2000°C c m - i [11] and it is controllable by adjusting the power loaded to the afterheater. Since the temperature gradient at the interface is so high, one can easily achieve a growth velocity larger than 10 mm h ~. If the nozzle size is 1 mm thd × 2 mm h and the crucible size is 10 mm ¢bd × 10 mm h, then the volume ratio of nozzle to crucible is 1/500, which may lead us to justify g * < 0.01.
6. Discussion The charge imbalance caused by the quantity difference between Li ÷ and O L i - or Nb2 O2+ and O 2- in the crystal is expressed by h t or h 2, respectively. Thus, we can evaluate the average concentration of uncoupled ions remaining in the unit volume of the grown crystal. For instance, the number of excess Li ÷ or O L i - in 1 cm 3 LiNbO 3 crystal, exions, is obtained by
1.0 H1 Ps exions =
NA,
0.9 100 M W
(18)
where Ps is the density of the crystal, M w is the molecular weight of LiNbO 3 and N A is Avogadro's number. Inserting H ~= 0.001603, Ps = 4.61 g / c m 3, M w for 48.38 mol% L i 2 0 crystal =
7. Summary (1) We have stressed that the interface electric field associated with crystal growth significantly influences on the partitioning of the intrinsic LiNbO 3 melt species and the magnitude of the field is dominated by the growth parameters, i.e. thickness of the solute diffusion layer, 6 c , growth velocity, V and temperature gradient in the liquid, G L. (2) Thus, different growth conditions provide different interface electric fields, which leads us to consider the dynamic congruent-state growth as a function of the growth parameters, which yields a crystal with fairly constant and balanced concentration profiles throughout its volume.
S. Uda et al. /Journal of Crystal Growth 155 (1995) 229-239
238
(3) By using a powerful nonlinear programming 'polytope' method, we calculated the best combinations of growth parameters for the dynamic congruent-state growth from a finite amount of melts with the initial compositions ranging from 48.35 to 48.65 mol% LIE0. A combination of ~c, V and G L with a simple linear change with g leads to a good dynamic congruent-state growth. (4) A micro-pulling-down growth technique may be a good technique which can meet the calculated growth conditions.
k~:Li÷ = k~ i. -
K(r2)(azC[~ °
- mfeon)
(A.4.2)
h / "~Li+ u2v" L~(0 )
kffLi- = kOLi- _
Ktr2)(a2C~o?-AC'con) b g, OLi2~Lw.(0)
(A.4.3) k,Nb205 = kNb205 _ a 3
_ ,~Nb~Os_ AC'
//'r'Nb2°'~+
con ] I "" Dx(O)
t~ 31.., L~(b)
02-
+ CLw.(0))
b f , Nb2O5 3~-" L~(0)
Acknowledgements
(A.4.4)
The authors appreciate the useful discussions with Professor W.A. Tiller at Stanford University who has stressed the importance of the dynamic congruent-state growth with one of the authors several years ago. The kind offer by Prof. Y. Kawazoe to use the supercomputer at the Institute for Materials Research, Tohoku University, is also gratefully acknowledged.
k~Nb~OP =
kENb20~ +
K(3){aT ~ 3""I.~o )/''Nb205 -- A C ' c o n ) _
_
b pNb202 +
3~-" L~(0)
(A.4.5) kkO2- = k o2- _
K(ra)(aaCNLb~?'- AC'con) 02b3CL~(O)
(A.4.6)
Appendix A
where
Precise description of k~ For the following chemical reactions, we define chemical equilibrium constants K~ ) (i =
a2 = kLi2°- (kLi* + k°Li-) + 1,
1,2,3),
bl
2LiNbO 3 ¢* Li20 + Nb205, K(T1) _ .. Li ,O _ Nb,O 5 , . . LiNbO 2 . -- a" L(~')'at L(~') / ~ L(T)
L i 2 0 ~ Li++ O L i - ,
(A.
1)
Ktr2)_ _Li ÷ _OLi-,_Li,O. -- "~ L(T)"t L(T) / ~ L(~') (A.2)
Nb20 s ¢~ NbzO42+ + 0 2 - , K(T3) _--.~L(/~) _ Nb,O 2 +_ 0 2 - ~,_ N b , O , . .'tL(T)/.,tL(~) ,
a3=kNb2°s-(kr~b=°~++k°2-)+
(A.5.1)
1,
(A.5.2)
-- ?/C(1)/-'LiNbO3 4- /'~Li20 4- (~Nb205 - - ~ * ~ T "-~Loe(0) -- "~Lo=(0)- "-'L~(0) ,
(A.5.3)
Li + [-,OLib 2 = K(T2) + CL~(0) -1- ',-'Lo~(0),
(A.5.4)
pNb202+ 02b 3 = K(T3) + "~ L=(0) + CLc~(0),
(A.5.5)
and A C ~ , = [ C~i~)CLW°~o~S{b2b3(2 - k Li2° - k Nb2°s)
(A.3)
where X~r ) is the mole fraction of the jth intrinsic melt species and T = 1270°C. For the intrinsic melt species, k~ is deduced by solving Eq. (9) [1], and we finally obtain
+a2baK(r2) + a3b2rcr3)}] X [ -bjb2b 3 a. h ["Nb2Osk'(2) h rLi20 k'(3)]-1 v 2,,-- L~(0)-x T ]
(A.5.6)
k,Li20 = kLi20 _ a 2
_ (a2CL~°o)-- AC'~o,)(CL~o, + C°,~i0;) References
b2C~dO)
(A.4.1)
Ill W . A . Tiller and S. Uda, J. Crystal G r o w t h 129 (1993) 341.
S. Uda et al. /Journal of Crystal Growth 155 (1995) 229-239 [2] S. Uda and W.A. Tiller, J. Crystal Growth 121 (1992) 93. [3] W. Spendley, G.R. Hext and F.R. Himsworth, Technometrics 4 (1962) 441. [4] J.A. Nelder and R. Mead, Computer J. 7 (1965) 308. [5] J.M. Parkinson and D. Hutchinson, in: Numerical Methods for Non-Linear Optimization, Ed. F.A. Lootsma (Academic Press, New York, 1972) pp. 115-135. [6] V.A. D'yakov, D.P. Shumov, L.N. Rashkovich and A.L. Aleksandrovskii, Bull. Acad. Sci. USSR Phys. Ser. 49 (1986) 117.
239
[7] S. Uda and W.A. Tiller, J. Crystal Growth 121 (1992) 155. [8] W.A. Tiller and S. Uda, J. Crystal Growth 129 (1993) 328. [9] S.N. Deming and S.L. Morgan, Anal. Chem. 45 (1973) 278A. [10] P.F. Bordui, R.G. Norwood, C.D. Bird and G.D. Calvert, J. Crystal Growth 113 (1991) 61. [11] D.H. Yoon, I. Yonenaga, T. Fukuda and N. Ohnishi, J. Crystal Growth 142 (1994) 339.