Influence of growth striations on paraferroelectric phase transitions: Mechanism of theformation of periodic laminar domains in LiNbO3 and LiTaO3Jun Chen, Qun Zhou, Jinfeng Hong, Wenshang Wang, Naiben Ming, Duan Feng, and Chengao Fang Citation: Journal of Applied Physics 66, 336 (1989); doi: 10.1063/1.343879 View online: http://dx.doi.org/10.1063/1.343879 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/66/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Intrinsic anomalous ferroelectricity in vertically aligned LiNbO3-type ZnSnO3 hybrid nanoparticle-nanowire arrays Appl. Phys. Lett. 105, 212903 (2014); 10.1063/1.4902557 Formation mechanism for ferroelectric domain structures in a LiNbO3 optical superlattice Appl. Phys. Lett. 68, 2642 (1996); 10.1063/1.116267 Study on the formation mechanism of a complex domain structure in LiNbO3 J. Appl. Phys. 77, 4136 (1995); 10.1063/1.359502 Characteristics of periodically domaininverted LiNbO3 and LiTaO3 waveguides for second harmonic generation J. Appl. Phys. 70, 1947 (1991); 10.1063/1.349477 Enhancement of secondharmonic generation in LiNbO3 crystals with periodic laminar ferroelectric domains Appl. Phys. Lett. 37, 607 (1980); 10.1063/1.92035

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Influence of growth striations on paragferroelectric phase transitions: Mechanism of the formation of periodic iaminar domains in UNb03

and UTa03 Jun Chen,a) Qun Zhou, Jin-feng Hong, Wen-shang Wang, Nai-ben Ming, and Duan Feng Laboratory of Solid State Microstructures, Nanjing University, Nanjing, The People's Republic of China

Chen-gao Fang Center of Structure Analysis, University of Science and Technology of China, Hefei, Anhui, The People's Republic a/China

(Received 10 November 1988; accepted for publication 28 February 1989)

The concentration distributions of Y dopings in LiTaO, across the growth striations in as­grown crystals arc obtained by means of x-ray energy dispersive spectrum analysis in a scanning transmission electron microscope at room temperature. The results show that it is the concentration gradient of the dopings which determines the configurations offerroelectric domains. The mechanism for the formation of the periodic laminar ferroelectric domains (PLFDs) is explained theoretically by considering the influence of the distributions of the dopings on the para-ferroelectric phase transitions in LiTa03 and LiMb03 crystals. It is shown that the coupling between the order parameter and the conjugate external field, which results from the ambipolar diffusion or from the strain-induced polarization mechanism associated with the dopings and their compensated point defects, is just the physical reason for the formation of such PLFDs in these two materials.

t INTRODUCTION

The idea of quasiphase matching in second harmonic generation (SHG) was first proposed by Armstrong and Bloembergen in 1962. r It is wen known that there have been various methods used in obtaining quasi phase matching. The method of using a stack of crystal plates has been ap­plied to GaAs, LiMbO:! and quartz, etC. 2

,3 Because only small numbers of the crystal plates can be used, it is not a practical method for SHG, Efforts to find a good method to obtain high-efficiency SHG with quasiphase matching had gone unabated until Feng and Co-workers succeeded in achieving enhanced SHG efficiency based on quasiphase matching in LiNb03 through the use of periodic laminar ferroelectric domains (PLFDs) Levinston, and Loio­cono.4-o

In the early 1960s Nassau,7 and latcr in 1976 Peuzin and Tasson 8 found that there is a one-to-one correspondence be­tween the growth striations and the domain configurations in as-grown doped as wen as undoped LiNb03 crystals. However, they did not report considering the possihility for quasiphase matched SHG in LiNb01 with PLFDs. By con­trolling the parameters of crystal growth, Feng and co­workers were first to succeed in growing yttrium-doped LiNbO, crystals with periods of about the coherence length for SHG, at the beginning of 1980s. The SHG output based on quasiphase matching in such a crystal was approximately proportional to the square of the number of the laminar do­mains, which is in good agreement with the theory proposed by Armstrong and Bloembergen. J Later (in 1986), Feng et al. 'I and Wang et al. 10 achieved for the first time the strong

a, Present address,

SHG output in the Y-doped LiTa03 crystals with I~LFDs.'>·lO LiTa03 cannot be phase matched for SHG be­cause it is a uniaxial crystal with a positive birefriengence index. Recently, Ming et al. showed that LiNb03 as well as LiTaOJ crystals with PLFDs could be widely used in other areas such as piezoelectricity, electromechanical trans­ducers, etc. 1

I

In 1982, Ming et at. measured the Y concentration dis­tributions over the rotational growth striations in LiNb03

by means of x-ray energy dispersive spectrum analysis (EDS) attachment of a scanning electron microscope (SEM), and obtained results showing that the ferroelectric domain structure depends on the Y solute concentration gradient. 12 Results were in agreement with what was ob­tained by Peuzin and Tasson (1975)8 and Tasson et al. ( 1976),13 They also discussed qualitatively, the mechanism ofthe formation of the ferroelectric domain structures in the LiNb03 crystals with nonuniform distributions of solute concentration.

In this paper, the Y solute concentration distributions over the PLFDs in LiTa03 crystals measured with EDS in a scanning transmission electron microscope (STEM) are presented for the first time. Then, the theoretical treatment on the mechanism of the formation of PLFDs in Y -dooed LiNb03 and LiTaO, crystals is elaborated in more detail.

II. EXPERIMENT

Crystals of LiNbO, and LiTa03 doped with 1 and 0.1 wt % yttrium, respectively, were grown by the Czoch­ralski method, lU,12 The as-grown crystals were first etched and examined in a MM-5 optical microscope so as to ascer­tain the period as well as the homogeneity of the PLFDs. Then the crystals with well-spaced PLFDs were cut me-

336 J. Appl. Phys. 66 (1),1 July 1989 0021-8979/89/130336-06$02.40 © 1989 American Institute of Physics 336

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chanicaUy into several plates along the direction perpendic­ular to the laminar domains. The thickness of such plates were further reduced by mechanical grinding and polishing from 500 to 50 11m, and then by ion milling to make it thin enough for TEM and STEM observations on a H-800 elec­tron microscope.

PLFDs in the specimen were first observed using either bright or dark image techniques in TEM mode. While the Y solute concentration distribution over the PLFDs was mea­sured with EDS in a STEM mode, it should be mentioned that great care was taken both i.n protecting the specimen from shift and choosing the suitable detector count number range during the EDS measurements. The x-ray dispersive spectrum analysis was performed by using the manufac­turer's software. The measurement points were uniformly spaced along the line perpendicular to the laminar domain wans. The electron beam used for EDS measurement was about 40-80 urn in diameter and carried about 100 pA. The x-ray number count for each measurement point was con­trolled within the range of about 4000 to 4500. By using the manufacturer's software, the intensity of the Kline of an Y atom and the L line of the Ta atom were converted to chemi­cal compositions. The only varied parameter in the calcula­tion is the film thickness. A wedge-shaped region was select­ed so that the thickness changes almostly linearly along the line on which the measured points were situated. The vari­ation ofthe thickness was determined from the relative con­centrations of the Y solute corresponding to two equivalent measurement points (i.e., two points spaced by an amount equal to the period of the PLFDs) should be nearly equiva­lent in Y concentration. This is based on the postulation that in the crystal the Y concentrations corresponding to such two equivalent points cannot be greatly changed.

III. EXPERIMENTAL RESULTS

Figure 1 (a) is a sketch of the distribution of the relative concentration ofY dopings over the PLFDs in LiTa03 _ Fig­ure 1 (b) is the bright field electron micrograph of the PLFDs selected for EDS measurements. As snmvn in the figure, the domain walls are, on the whole, perpendicular to the crystal growth direction; i.e. [00.1). From Fig. 1 the following main results can be obtained.

(a) The period of the PLFDs is about 4.5 pill which coincides wen with what is deduced from the parameters of the crystal growth. 10

As shown in the figure, there are two kinds of domain walls, one is shaped like a piece of smooth ribbon, marked with DWl and DWl', the other shape is like interlocking sawteeth, denoted by DW2. As shown in Fig. 1 (b), the spac­ing from DWI to DW2, i.e., dw is not equal to that from DW2 to DW 1', i.e., d 21' • The period of the PLFDs equals the sum of both, that is,

(1)

Why d l2 is not equal to d 21' can be accounted for from the foHowing two aspects. It should be stressed that the crystal with PLFDs which corresponds to the growth striati.ons is grown off-axis so as to impose a periodic temperature vari­ation on the growth interface, that is, the solid-liquid inter-

337 J. Appi. Phys., Vol. 66, No.1, 1 July 1989

b

FIG. L The one-to-one correspondence between Y solute distributions (a) and PLFDs (b).

face. Therefore, on the one hand, the time of the former half­cycle of the temperature variation is not equal to that of the latter one. On the other hand, the microscopic growth rates corresponding to th~<;e two half-cycles are not the same.

(b) It is the concentration gradient which determines the domain waH orientation and domain structures.

As shown in Fig. 1, there is a one-to-one correspondence between the concentration distribution of the Y solute and the PLFDs. Domain wans are always situated at the place where the gradient of the Y concentration changes its sign from plus to minus or vice versa. The smooth ribbon-shaped walls like DWl or DWl' are at the place where the Y con­centration gradient undergoes a sudden change while those rough sawtooth-like domain walls like DWl are situated in the regions where the gradient changes gradually from a lit­tle bit more than zero, through zero, and then to less than zero. From the figure one can see that the concentrations fluctuate a little near DW2. This phenomenon was not ob­served in LiNb03 crystals with PLFDs. 12 One reason for the fluctuati.on may be that the temperature field within the cru­cible has no radial symmetry. Under the condition of Hnear growth, such a temperature field without radial symmetry (which may originate from the presence of the observation window) may lead to some fluctuations of solute concentra­tions. However, on the other hand, it is also a possible reason for the nonlinear dependencle of temperature fluctuation on crystal growth rate.

IVo THEORY: THE INFLUENCE OF GROWTH STRIATIONS ON PARA~FERROELECTRIC PHASE TRANSITIONS-THE MECHANISM OF THE FORMATION OF PlFDs

We probe the effect of the definite distribution of solute concentrations on the phase transition and further on the domain configurations in the ordered phase only, based on

Chen eta!. 337

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the relationship between the phase transition and the do­main structures. The physical model will be established after analyzing the characteristics of the point defects present in LiNb03 and LiTa03 crystals. And then the experimental results will be explained within the framework of Landau's theory of second-order phase transitions. 14

Many experimental results, such as the investigation of the deviation from chemical compositions in LiNb03,15 the measurements of the electric conductivity,16 the photore­fractive effect,17 the influence of oxygen pressure on the Cu­rie temperature of LiTa03, 18 and more recently the investi­gations on the Mg:LiNb03 crystals,19.20 all indicate that the point defects present in Lil\.'b03 and LiTa03 crystals have several consequences21 such as:

(a) Doping is easy because the lattice is able to accom­modate for large amounts offoreign atoms. Transition metal ions can be incorporated in amounts of up to several moles %, rare earth ions to about 1 mole %0

(b) There are large amounts of point defects such as oxygen and lithium vacancies in the doped and undoped LiNb03 and LiTaO, crystals with various compositions.

(e) Oxygen vacancies can migrate easily even at lower temperatures. The diffusion of oxygen vacancies is isotropic.

(d) The electrons exist as small polarons in the crystals due to the strong coupling between electrons and phonons. So the electron mobility is very low.

Nassau pointed OUf2 that the sites for which the alien atoms substitute are governed by the foHowing factors: ionic radii, charge misfit, the character of the chemical bonds, and the coordination. In the following, we will analyze the sub­stitutions of Y ions, set up the physical model, and then do some calculations.

A. Establishment of the physical mode!

It is well known that the valence of the Y ion is + 3 or + 2 and the radii is about 0.088 nm compared to those ofNb

and Ta ions, + 5 and 0.067, 0.073 nm, respectively. There­fore, ifthere are two ions ofy3 + substitute for both Li + and NbH or Ta5+, respectively, forming the substitution pair, although the misfit charge can be compensated fairlY, the elastic distortion energy related to the pair should be com­paratively high due to the substantial difference of the Y ion radius from those of Li and Nb or Ta. On the other hand, in the crystals there are large amounts of mobile point defects, so the compensation for the misfit charges can be easily made. Thus, we can give the first postulate so as to solve the problem of the space charge related to Y solute concentra­tions. That is, yH may substitute separately for NbS + or Li +. Next, according to the proposal given by Megaw,23 the frame\vork of oxygen octahedra is sufficiently solid. There­fore, the topology of the framework cannot be varied no mat­ter how the Y ions enter the lattice sites of Nb or Li. How­ever, the lattice may be locally distorted to some extent if the substitutions have been made.

Based on two such postulates we can get the following main characteristics for y3+ substitutions.

(a) y3+ is a charged point defect which can be compen­sated by some mobile point defects such as oxygen or lithium

338 J. Appl. Phys., VoL 66, No.1, 1 July 1989

vacancies. Hence, within a small volume, which can be thought of as both macroscopically small and microscopi­cally large enough, it is electrically neutralized. However, since the radius of y3+ is much larger than those of NbH ,

YaH, and Li''-, then both the mobility and the diffusion coefficient of y3+ substitutions are sufficiently low com­pared with those of the compensated point defects, this may lead to the existence of a space-charge field due to the ambi­polar diffusion24 (provided that the y3 + substitutions have a definite distribution in the crystals). On the other hand, cor­respondingly, an elastic distortion field distribution must be present. The y3 + substitutions and their compensated de­fects can all be viewed as elastic dipoles. 25 Under the action of the inhomogeneous elastic field, such dipoies will be dis­placed different distances; that is, a relative displacement Ax between y3+ and its compensated defects will be produced. Then the strain-induced polarization can be obtained. It can be described with an equivalent electric field:

Psi =q°!:1x= (E-l)Ee'l'

Eeq = qb.x/(£ - 1),

(2)

(3)

q is the charge the substitution ion loads, and E, the dielectric constant, in the first-order approximation, is assumed iso­tropic.

It is inevitable for the electric field to influence the phase transition and the relevant domain structures. We will calcu­late both the space-charge field and the elastic equivalent field.

B. Calculation of the electric field

Let Cy (x) and Cp (x) represent the concentrations of yH and its compensated point defects. They fluctuate peri­odically along the direction of crystal growth. And within the small volume concerned,

(4)

and the charges the Y substituti.on and its compensated de­fect carried are equal but in opposite sign, that is,

qy = -- gp' (5)

Let Esc (x) denote the space-charge field in the crystal concerned, then the current density is the sum of both drift and the diffusion components24

:

Jy = qy [ - DJ' VCy (x) - Cy,u.yESC (x)], (6)

Jp = qp r - D/vCp (x) + Cp,upEsc (x)]. (7)

Here the apparent charge of the Y ion has been supposed negative, D and f.l are diffusion coefficient and drift mobility, respectively, with subscripts h.aving the same meaning as those of the q's. Since there is neither source nor sink in the crystals, thus

(8)

Substituting (6) and (7) into (8), we get

E.c(x) = ± (Dp -Dy·VCy{x}/[Cy{,up +f.ly )]. (9)

If the apparent charge ofy3+ is positive, " - " will be taken. From Ego (9) one can see that if Dy is not equal to Dp'

Esc cannot be zero, otherwise not only space-charge field but also the fluctuation of solute concentrati.on would disappear.

Chen etaf. 338

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In order to get the elastic field resulting from the fluctu­ation of Y concentration, a method similar with that pro~ posed by Cahn26 in dealing with the spinodal decomposi~ tions will be used.

The fluctuation of the Y solute concentration is given by

.6.C(x) = C(x) - Co, (10)

where Co is the mean concentration. The presence of AC(x) will indisputably effect the ion radii. It may be described as

aR(x)IRo = [R(x) - Rol/Ro = TjaC(x), (11)

where 17 is the linear contractabiHty. The linear relation is valid only in the case of dilute solution. Then the stress-free strain is given by

e~ = Tj.6.C(X)Dij' (12)

During steady growth phase, the growth front is the interface between the crystal and the melt, so it is reasonable to suppose that the total normal strains on the interface are zero. If we consider the crystal pulled along the x, direction, then

aC(x) = aC(X3) ==A cos(!3x3 ),

e~ = ATjDij cos(!3x3 ),

(13)

(14)

and the boundary conditions discussed above can be written as

e~l = e~2 = 0,

U;3 = 0,

(15)

(16)

where the subscripts t and e signify the total strain and the elastic strain, respectively. In considering the presence of the stress-free strain, Hook's law in the isotropic media can be formulated as

e~! = [a~l - V(0-;2 + ~3) ]/E + A17 cosC,8x3),

e~2 = (er2 -V(e:r11 +oj3)]IE+Ancos(.Bx3)' (17)

e~3 = [oj3 -v(011 +G"~2)]lE+A1]cos(t5'X3)'

Applying the boundary conditions to these equations, fol­lowed by some algebraic manipulations one obtains

e~,3 = [2Av17 cos(/3x3 }}le 1 - v) + ATl cos(!3x3 )

= A17(1 + v)/(1 - v)cos(/3x3 ). (18)

Since the elastic strain equals the total strain minus the stress~free strain, one can further get

e~3 = e~3 - ArJ cos ({3x3) = A'I]2v cos(/Jx})/( 1 - v),

(19)

Because the y3+ ion is effectively compensated electri­caHy, the concentration fluctuation associated with the com­pensated point defects must, as a whole, undergo the same variation as the Y substitutions. And we also have the elastic field similar to (19) but corresponding to the compensated defects. Therefore, the total elastic field should involve these two parts, however, the formulation is also like (19) as ex­pected, but

l] = Tjy + 'l'}p. (20)

339 J, Appl. Phys., Vol. 66, NO.1, 1 July 1989

It is dear that if rl y is not equal to - 11 P' the total elastic field cannot be zero. Fortunately in Y -doped LiNb03 and LiTaOJ

crystals, yH has a much larger radius than those of Nb5 +-,

Ta5+ as wen as Li+ and 0 2- • It is not necessary to worry

about the disappearance of the total elastic field. If the difference between the permanent polarizations

P and P' associated previously with two different kinds of elastic dipoles, which characterize the above-mentioned two kinds of point defects, is simply taken into account, the rel­evant displacement between these two dipoles under the ac­tion of the inhomogeneous elastic field is then given by25

t:..x = K'j)P-ij - P~ )Veij (x), (21) ij

where K = Kp = Ky is a constant. Strictly speaking Kp is not equal to K y, but the difference is very small. The equiva­lent field can be represented as

Eeq = qy t1x/(€ - 1)

= KI(Pt - ppVeij(x)'qyl(E -- 1), (22) ij

or

E K "'(P" PP)( l)Jeij(X) eq = qy £.. 'ij - ij € - -- .

ij aXj

(23)

In the isotropic approximation,

Pt = PPoij' p·u = P Y8 jj • (24)

In consideration of Eq. (20)-( 24), the equivalent field can be derived as follows if k is the basic vector along the X3 direction.

Eeq = - 2K71v/3qyA sin(Bx3 ) (pY - PP)okf

[(1- V)(E - 1)]. (25)

Substitute (10) and (13) into (9) and we can also get the space-charge field as Esc = ± (Dp - Dy)A ( - (3) sin (/3x3 ) "k I[ Cv (f-lp + liy)] >

(26) From (9), (26) and (23), (25) one can conclude that both space-charge field and the equivalent electric field are always parallel or anti parallel to the direction of the concentration gradient If the gradient reverses its direction, the total field will reverse its direction as well. If the concentration fluctu­ation is quite low, the Y concentration in (26) can be seen as a constant, then the total electric field can be represented as

E t = Esc + EC<J = (C1 + C2 )VC(x), (27)

where C 1 and Cz are the constants associated with the me­chanical and electrical parameters of the materials.

c. The influence of the electric field on the phase transitions and the relevant domain structures

It is known that the electric field is the conjugate field for the para-ferroelectric phase transitions. Since the field is inhomogeneous, the term related to the gradient of the order parameter should be added into the expansion series of the functional of the free energy. In LiNb03 and LiTa03 crys­tals the order parameter, that is, the spontaneous polariza­tion is along the positive direction of the X3 axis. That is, P s = P.., k, the conjugate field is the only Xl component of the total electric field. The other two components cannot

Chen etal. 339

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effect the phase transition. Therefore, we may account only for the X3 component of the field in the free-energy func­tional, It is given as

f( aP;3 bP;3 ) G=Go + -2-+-4--E3P,3 +!hVPS3·VPd dV.

(28)

Minimizing the functional by the method of variation, one can get the famous Lagrange-Euler equation:

(a + bP;3 + hV'2)PS3 = E,

that is,

ha 2

(a + bP~J + -a 2 )P,} = E 3 •

X3

(29)

It is a nonlinear inhomogeneous differential equation. It is, in general, impossible to get an analytical solution. However, in the crystals ofY-doped LiNb03 or LiTa03, the concen­tration ofy3+ is varied slowly with the coordinate, and the total electric field is, of course, a slowly varied function. So it is reasonable to assume the field is constant within a sman region where it does not reverse its direction. Thus, one can simplify Eq. (29) in such a region by canceling out the gradi­ent term, and the functional can be written as

G = Go + aP;J2 +- bP;3 - EJPsJ • (30)

It is clear that the free energy is minimized only when the field has the same direction as P,. If the field does not change its direction, P s will remain unchanged. P s changes its direction only within the region where the field reverses its direction no matter how slowly or suddenly. As we know, there must be a domain waH in the region where the P s

changes its direction. Neglecting the effect of the field we have

( ha2) a + bP ~3 + ax; P,) = o. (31)

One special solution is

Ps3 = P~3 tanh(x31/ihla), (32)

where P~3 is the spontaneous polarization in the region far from the domain wall. Since there is no effect of field, both positive and negative kinks are solutions of Eq. (31). Thus, in the region where the Y concentration fluctuates slowly a little bit, there results, on the one hand, both positive and negative kinks or domain wans, and on the other hand, the walls are less influenced by the electric field, thus, the walls in such areas are correspondingly shaped like the so-called interlocking teeth of a saw. However, as the field is strong enough, the direction of the P s depends on it. It is obvious that the walls must be localized within a comparatively nar­row region as the electric field changes sharply in direction. The walls in such region should be smooth enough. The schematic diagrams showing the correspondence among the Y concentration distributions, the total electric field, and the domain structures are included in Fig. 2. In summary, it is the total electtic field corresponding to the Y solute concen­tration distribution which governs the domain structures in the ordered phase, i.e., ferroelectric phase, of LiNb03 and LiTa03 crystals.

340 J. Appl. Phys., Vol. 66, No.1, 1 July 1989

c

FIG. 2. Schematic diagrams showing the correspondence among the Y sol­ute distributions (a), the total electric field (b), and the domain configura­tions (c).

v. DiSCUSSION

In fact, there are PLFDs in the Y-doped LiNb03 and LiTa03 crystals grown along the X3 axis as well as along the Xl axis. As a matter offact, the walls ofPLFDs shown in Fig. 1 are charged walls, because the spontaneous polarization is almost perpendicular to the domain walls. One of the possi­ble di.stributions of the electric field is shown in Fig. 2. It shows that the field is parallel to the crystal growth axis. However, during the crystal growth, it is hard to keep the growth interface strictly parallel to the growth axis aU the time. Furthermore, the growth interface cannot be an ideal fiat plane, thus, even for the crystal grown along the c axis, the gradient ofY concentrations and subsequently, the total electric field cannot be rigorous along the x~ axis. This is also the case for the crystals grown along the XI axis. That is to say, there is a nonzero E3 component of the total electric field present in crystals grown even along the X 1 axis. It is just this nonzero field component E3 which couples with the param­eter of the phase transition. As a result, it governs the do­main structures, although it is weak enough, because the phase transition in LiNbOJ and LiTaOJ is the strong second­order one. It is well known that 14 for a second-order transi­tion, whether the conjugate external field is strong or not is of only relative importance. That is, even for the weak field, however, the influence on phase transition can become strong enough provided the temperature is approaching the Curie point Tc closely enough. It is the reason why PLFDs appear in crystal grown along the x 1 axis.

Since the ribbon-like domain walls in Fig. 1 are charged, they will bend themselves to reduce the static electrical energy.

From Eqs. (25) and (26), one can conclude that the foHowing conditions will be in favor of forming a strong elec­tric field: (a) The apparent charge of the doped ions should not be so large or so small, (b) there should be a pronounced difference of the diffusion coefficients of the two kinds of defects, and (c) there should also be pronounced differences of the linear contractibility as well as the permanent elastic dipole moments.

Chen etal. 340

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Therefore, from the point view of the production of PLFDs, the doping ion should be large enough compared with Nb, Ta, or Li ions, but on the other hand, in view of crystal growth, it should have a large effective segregation coefficient so as to make more doped ions on the definite lattice sites. However, the most important thing is that the concentration fluctuation must be sufficiently high; but this will be limited or governed by the growth parameters or technology, such as the designing of the temperature field, the exact control of the rotation, the pulling rate, and so on, in order to achieve pronounced striations ill the doped crys­tals.

ACKNOWLEDGMENTS

Thanks are due to Professor Li Qi and Dr. Liu Zhi-guo for valuable discussions. J. Chen is very grateful to Dr. Lee Chow for all his help and many useful discussions.

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