Dielectric relaxation in YTTRIA-doped Ceria solid solutions

Dielectric relaxation in YTTRIA-doped Ceria solid solutions

I. Phys. Chem. Solids Vol. 44, No. 7, pp. 639-646, i983 Printed in Great Britain. 0022-3697183 $3.00+ .W PergamonPressLtd. DIELECTRIC RELAXATION IN ...

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I. Phys. Chem. Solids Vol. 44, No. 7, pp. 639-646, i983 Printed in Great Britain.

0022-3697183 $3.00+ .W PergamonPressLtd.

DIELECTRIC RELAXATION IN YTTRIA-DOPED CERIA SOLID SOLUTIONS DA YIJ WAN@ and A. S. NOWICK Henry Krumb School of Mines, Columbia University, NY 10027,U.S.A. (Receioed 19 July 1982;accepted 19 August 1982) Ahstrnct-The nature of the defects present in Y20,-doped Ce02 is explored by means of dielectric relaxation, employing primarily the thermally-stimulated depolarization current (TSDC) technique. For YIO, concentrations 51 mole.%, two relaxation peaks are observed. The lower-temperature peak shows a dielectric relaxation rate, r-‘, which is i the relaxation rate of the corresponding anelastic relaxation. This proves that the peak is due to nn, (11I) oriented, (Y V,)’ pairs, where V, represents the oxygen vacancy. These pairs are positivelycharged defects which are compensated by an equal number of isolated Y’. The upper peak is a broad peak whose position varies with Yz03 concentration in the same manner as the activation enthalpy for the ionic conductivity.The peak is not due to simple dipoles, but to relaxation of the array of alternately charged (Y Vo)’ and Y’ defects by the redistribution of oxygen-ion vacancies. A simple model, in which these defects form a superlattice containing wrong pairs, explains the essential features of the upper peak

1.

INTRODUCTION

ability. The presence of an electric field or a stress field, however, results in a preferential occupation of these positions. It has been shown [8,9] that the relaxation rate for dielectric relaxation, rdi:,, is given by 7di:, = 2w, while that for anelastic relaxation is given by T::~, = 4w, where w is the specific jump frequency of the V, between any two adjacent nn sites. Thus, a characteristic of this S-position model, first pointed out by Wachtman[9], is that

Oxides of the fluorite structure, when doped with divalent or trivalent cations, are good solid electrolytes at elevated temperatures due to the introduction of oxygen-ion vacancies as compensation for the dopant cations[l,2]. Because of this behavior, such oxides are of interest in applications to high-temperature fuel cells and oxygen sensors[f, 31. Although zirconia (ZrOJ with CaO or Yz03 additions is most widely known, it has been shown[4,5] that doped ceria (CeO,) is a more ideal material for fundamental studies, since it enables low dopant compositions to be studied in order to ascertain the basic defects involved and their interactions. In particular, the ionic conductivity of ceria has been studied in some detail both for divalent and trivalent dopants[S, 61. In most work, Ca” or Y3+ were used, but other trivalent dopants have also been studied[7]. In the case of divalent (say Ca”) doping, one oxygenion vacancy (V,) is introduced for each Ca*’ ion. Further, since a substitutional Ca*’ has an effective charge of -2 and an V, a charge of +2 (Ca& and V& respectively, in Kreger-Vink notation), one anticipates that at low and moderate temperatures the coulomb interaction will produce neutral nearest-neighbor (nn) pairs of the type (Ca,. VJ. Such a pair is illustrated in Fig. 1, which shows one-half the unit cell of the fluorite lattice. The presence of such pairs controls the dependence of conductivity, 0; on temperature, T, by introducing the association enthalpy of the pair into this dependence[S, 61. A more direct demonstration of the presence of such pairs, however, comes from relaxation experiments. Specifically, such a nn pair acts as both an electric and elastic dipole of trigonal symmetry, and gives rise to dielectric and anelastic relaxation[8]. In the absence of an external lieid, the V, occupies any one of eight equivalent positions about the dopant with equal probtPresent address. GTE Laboratories, Waltham, MA 02254, U.S.A. PCS Vol. 44. No. 7-C

For trivalent doping, say with Y3+, the conductivity shows interesting behavior, viz. g rises sharply to a maximum at -3 mole % YzOX, while the activation enthalpy correspondingly decreases. This behavior in the low-concentration range is attributed to the presence of an array or network of (Y V,)’ and Y’ defects. Such defects occur because the cations are essentially immobile below - lOOO”C[l],so that the Y3+ ions may be assumed to be present in an almost random distribution. An isolated Yc, possesses an effective charge of - 1. The mobile V& is then free to associate with a single Y& to produce a bound pair (Y Vo)‘, but at low concentrations, it cannot associate with the two Y3+ ions required to produce a neutral entity. Thus only half of the Y3+ are associated while the other half remain as unassociated Y&, defects in a cubic environment. The evidence from conductivity for this model is indirect; it is thus desirable to better establish its validity. The (Y Vo)’ pairs, even though they are charged, should give rise to dielectric relaxation in the same way as the neutral (CaVJ pairs. Furthermore, an anelastic peak attributed to (Y V,)’ pairs has already been reported by Anderson[lO] who observed a single Debye peak at 5% YzO,, a slightly broadened peak at 1% and a very substantially broadened one at 2.5% YzO, and above. (All compositions quoted herein are in mole %). The present work is a dielectric study, using primarily the thermally

639

DA Yu WANG and A. S. NOWICK

640

n

n

where Z(T) is the depolarization current density, E the polarizing electric field, Nd the number of dipoleslvol of dipole moment Z.L,T, the polarizing temperature, T,, the low temperature from which the depolarization heating is begun, b the heating rate, and 7-l the relaxation rate whose dependence on temperature is given by 7-l = V&exp(- Z-&/U).

(3)

Here H, is the activation enthalpy for relaxation and vb the frequency factor (which also includes the term: exp(AS,/k), in which AS, is the activation entropy). From eqns (2) and (3), Z(T) vs T gives an asymmetric peak whose peak temperature T,,, is given by

0

0

02-ION

cl

VACANCY

HOST

T,,,*= bHq( T,,,)/k.

CATION(4+)

DOPANT CATION (2+ or 3+)

@

Fig. 1. Diagram of half the unit cell of an oxide of the fluorite structure containing a nn MV, pair, where M is a divalent or

trivalent dopant ion.

(4)

Although H, can be determined from the logarithmic slope of the low-temperature side of the Z(T) peak or from the fit to the entire curve, for high precision it is best obtained by measuring peaks under a wide range of heating rates, or by also employing a.c. methods over a range of frequencies. 3. METHODS

stimulated depolarization current (TSDC) (also known as ionic thermocurrent, ITC) technique [ 11,121. In this method, the sample is first polarized by applying a static electric field, E, at a temperature, 7”, at which dipoles can reorient freely. It is then cooled down under the field to a temperature, Ta, where the polarization produced is frozen in. Finally, after the field is removed, the depolarization current Z(T) is observed as the sample is heated at a constant rate through the temperature range where dipoles can reorient. In order to carry out these objectives, it is necessary to eliminate a large relaxation effect due to the presence of a low-conductivity layer along the grain boundaries of most of these materials. Such a layer gives rise to an extra arc in the a.c. complex impedance plot[l3], and to a large TSDC peak of the Maxwell-Wagner (layered dielectric) type[l4]. In Ref. 14 we developed two methods for eliminating this peak: (a) by very rapid quenching following polarization; and (b) by the use of inert dielectric inserts. Both of these methods have been employed in the present work. In so doing, we have observed not one, but two dielectric relaxation processes at low Y20, concentrations. In this paper we consider the detailed experimental results for these peaks in order to arrive at their atomic mechanisms. A preliminary account of part of this work was previously given[ IS]. 2. BASIC TSDC EQUATIONS

For dipolar relaxation in which a single relaxation time is operative, and for cubic crystals, the TSDC obeys the following equation[ll, 121: Z(T) N,/J* -, -=T (T)exp E

3kTp

1

I I -b

l-

ToT-‘(T’)dT

I

(2)

The experiments are carried out on sintered polycrystalline pellets, whose preparation has been described [ 13). Both mechanical and chemical preparation methods were used, but almost all of the Y203doped samples were chemically prepared. Samples were disks of - I cm* in area and 1 to 1.5mm in thickness. The TSDC apparatus was described previously[ 141. Either fast quenching or the dielectric-insert method was used, in order to avoid the large grain-boundary peak[l4]. For fast quenching, liquid nitrogen was poured directly into the sample chamber just as a potential lead was being touched to the sample to polarize it (at 20 or 0°C). For dielectric insert, pieces of polyethylene film were placed on both sides of the bare sample, and metal electrodes (a copper disk for the upper electrode and the bottom plate of the sample chamber for the lower electrode) were then placed about this composite dielectric. 4. RESULTS

AND DISCUSSION

TSDC data were obtained for sintered pellets containing various concentrations of yttria. The effect of the manner of conducting the run is illustrated in Fig. 2, which shows the results on the same (1% Y203) sample using three methods. The conventional run in which the sample was cooled relatively slowly following polarization shows the huge grain-boundary peak that we wish to eliminate in the present work. The second run, marked “fast cooled” is carried out by the fast quenching method and reveals the presence of a double peak. It gives the full height of the lower peak but not necessarily the full value of the upper peak, due to the rapid quench. The third run, marked “spacer inserted” employed the dielectric insert method. Here the current level is quite low because of the reduced (and unknown) potential drop across the sample. However, this method more

Dielectric relaxation in yttria-doped ceria solid solutions

641 T ("Cl

-90 I

E=

-70 I

965

-30 1

-50 I

V/cm

0 I

30 50 1 1

cto2:

lo-lo-

1

YZO,

2.5 %

1.0 %

10-l ’ -

0.50%

a 5 5 E 1 ”

0.15 % lo-‘2

-

0.05%

10-13

-

10-14 '

6.0

I

I

50

4.0 lOOO/

I

3.0

5.0 1000

T

Fig. 2. Comparison of TSDC peaks obtained by three methods for a CeO*: 1% Y201 sample. Upper curve is obtained by the conventional method, middle curve by fast cooling and lowest curve by insertion of a dielectric spacer. Polarization in each case was at room temperature.

I 4.0

I

6.0

I 3.0

J

/T

Fig. 3. Thermallystimulateddepolarizationcurves for five dilute Ce02: YZO,solid solutions.

T(T)

-80 I

-60 I

I

-40 I

-20 I

0 I

20 !

1

reliably gives the shape (i.e. relative form) of the entire

TSDC curve. Thus, by using both methods the shape as well as the absolute values of the TSDC curve can best be obtained. Figure 3 shows typical results for compositions ranging from 0.05 to 2.5 mole % Y,O,. Both the rapid quenching and dielectric-insert methods were employed, with polarization temperatures close to 20°C. This figure shows that the double peak is obtained only for compositions below 2.5% Y,O,. The low temperature peak, which we call “peak l”, occurs at a fixed temperature of -64°C for the present heating rate of Z”C/min, while the upper peak, called “peak 2”, moves down in temperature until the two peaks blend into one at the 2.5% YzOs composition. Figure 4 shows results for the higher concentrations, 6, 8 and 10% yttria. Here it can be seen that the peak remains a single broad peak, but that it moves back to higher temperatures with increasing yttria content. In contrast to these results for Y,Os-doped ceiia, we have found only a single TSDC peak for dilute CaOdoped ceria, in agreement with our earlier work[6] and with a.c. measurements on CeOz:CaO and on the analogous system of ThOz : CaO [9,16]. As already mentioned., the peak in that case has been established as due to (Ca VJ dipoles.

10-13

1 t

1

I

I

5

4

3

1000/T

Fig. 4. Thermally stimulated depolarization curves for three CeO*:Y203 solid solutions of higher concentrations.

In the remainder of this section we will examine in detail the characteristics of each of the two peaks occurring in yttria-doped ceria in order to deduce the responsible mechanisms.

DAYu WANG and A. S. NOWICK

642

(a) The low-temperature peak (peak 1) At compositions of 0.5% YzO, or below, peak 1 can be described by a process with essentially a single relaxation time. Figure 5 shows this peak for 0.5% YZ03, obtained separately from peak 2 by using a T, of - 25°C. The dots represent the fit to eqns (2) and (3) with H, = 0.64 eV and vb = 0.5 x lOI set-‘. The results show that a single relaxation time provides a satisfactory fit.

CeOz: 0.5%

. . y4, -48

-56

In Fig. 6 we compare the dependence of the peak height, I,.,, for peak 1 on yttria concentration, and compare it with similar results for CaO-doped ceria. The solid line, drawn with 45” slope on this log-log plot, represents a linear composition dependence. The yttria results show a definite departure from the linear relation, particularly at 1%. The agreement in magnitude for the two dopants, however, suggests that the same mechanism, viz. vacancy-impurity dipole reorientation, may be operating in both cases. A relatively unique characteristic of the nn impurityvacancy pair relaxation is the occurrence of a factor of 2 in the ratio of anelastic to dielectric relaxation rafes (eqn 1). The anelastic measurements[lO] showed a single Debye peak for 0.5% Y203 at 77°C for a frequency of 7750Hz. In order to compare results at the same temperature, we have carried out a.c. dielectric measurements at four different frequencies (between 900 Hz and 50 kHz) on the 0.5% Y,O, sample. These results are plotted, as logarithm of the frequency vs reciprocal peak temperature, in Fig. 7. An additional point was obtained from the TSDC data, since T(T,) is given by eqn (4), and the corresponding frequency is f = 1/2~r~. Substituting T, = 209 K and the approximate value of H, = 0.65 eV yields f= 9 x 10m4set-‘. A plot such as Fig. 7 which combines quasi-static (TSDC) and a.c. results, enables one to obtain 7(T) over many decades, and therefore, precise values of H, and v: in eqn (3). Thus, the best straight line drawn in Fig. 7 yields H, = 0.678 eV and f. = 2.2 x 10” sec~‘.t This frequency factor corresponds to vb = 27rjo = 1.4 x 10”’set-‘, which is quite reasonable since vb includes the entropy of activation, AS, which is usually positive[ 171. Substituting the temperature of 350K, at which Anderson’s anelastic peak was observed, into the Arrhenius equation for f with the above parameters, we obtain f = 4.0 x lo3 Hz. This is the frequency for which

YzO3

I

I

-64

-72

\ .

i

-80

T PC)

Fig. 5. The lower TSDC peak for CeOZ:0.5%YZO,after polarizing at - 25°Cduring continuous cooling from room temperature. The solid curve is experimental and the dots are calculated, with H, = 0.64eV and v; = 0.5 x lOI set-‘. tThe fit to the data of Fig. 5 using these parameters in eqns (2) and (3) is even better than the fit shown in that figure.

10-l’

E = 365

V/cm

I

I

IO

0.1 MOLE

%

DOPANT

Fig. 6. Peak height I,,,,, as a function of dopant concentration for Ce02:YzOj and Ce02:Ca0 solid solutions. The solid line represents a direct proportionality between I,,,, and concentration.

Dielectric relaxation in yttria-doped ceria solid solutions

linear at low concentrations and agrees with the magnitude of the peak height for calcia-doped samples. The totality of this evidence leaves little doubt that peak 1 is due to nn (Y V,)’ pairs corresponding to the 8-position model shown in Fig. 1. The existence of such pairs in the dilute range (i.e. below 1% YZ03), previously deduced from conductivity studies[5], may now be regarded as more directly confirmed by these relaxation studies. It is interesting to consider to what limits of concentration this simple (Y V,,)’ pair model can be expected to apply. Clearly, as the yttria concentration increases, higher defect clusters, involving more than one Y3+ ion, will be formed. A simple approach is to define an isolated defect by requiring the absence of a second Y 3+ ion in at least the 18 nn and nnn cationic positions about a given Y”. Letting cy be the fraction of Y ions on cation sites, i.e.

H :Q6?0eV f = 2 2 r10'3sec-' 102 -z I

I

I

I

I

I

30

3.5

4.0

4.5

I

‘o-4 2.5

643

I

5.0

CY =

NJ&,

(5)

1000/T

Fig. 7. Plot of log frequency vs reciprocalpeak temperatureof peak I in Ce02:0.5% Y203. The four high-temperaturepoints are from a.c. dielectric loss measurements; the low temperature point is from TSDC.

the dielectric peak is located at 350 K. Comparing this with Anderson’s frequency of 7.75 x 10’ Hz we obtain

which agrees with the expected value of 2.0 to well within experimental error. The relevant evidence concerning peak 1 may be summarized as follows: (a) It occurs at a fixed temperature for various yttria concentrations below 1%. (b) It can be described in terms of a single relaxation time, as can the corresponding anelastic peak. (c) The ratio of relaxation rates is: T&$/T& = 2. (d) The concentration dependence of peak height is

where NY and N,, are the numbers per unit volume of Y and Ce ions, respectively, the probability of such an isolated Y ion is clearly (1 - cy)“. Values for this quantity for the samples of interest are given in the third column of Table 1. They show that for 1% Y203, 70% of the Y ions are so isolated, while by 2.5% only 40% of Y ions are isolated. This result serves as an explanation both for the departure from the linear relation in Fig. 6, and for the fact that the anelastic peak begins to broaden at 1% Y203 and is very much broadened at 2.5% YZ03[10].It may also help to explain why, in the present study (Fig. 3), peak 1 has lost its identity and blended with peak 2 at the composition of 2.5% YzO,. Actually, the existence of such a simple defect for compositions up to 0.5 mole % of solute is unusual. For example, in the much studied case of trivalent-cation doped CaF,, the complexity of defects is considerable even at much lower concentrations[l8]. It is possible that the greater simplicity in the case of ceria is due, at least in part, to its large dielectric constant (-25) which serves to weaken electrical interaction among the defects.

Table I. Calculated quantities and Perptfor various Ce02Y20, solid solutions I

"ZQ3

CY

wy)18

expc-hv/kT

)

max

(10-12amp/cm2)

B CFilC

6

expt

0.05

.OOl

.98

6.30

.053

.1064

8.9

135

'10

0.15

.003

.95

4.36

.073

.0443

5.3

27

22

0.5

.Ol

.83

2.92

.091

.0212

3.8

5.8

5

1

.02

.70

2.32

.096

.0168

3.8

2.9

5

2.5

.049

.40

1.72

.091

.0209

6.4

6

.113

.12

1.30

.109

.0098

4.0

a

.148

.06

1.19

.122

.0055

2.4

10

.182

.03

1.11

.130

.0039

1.9

DAYu WANG and A. S. NOWICK

644

(b) The high-temperature peak (peak 2) It remains to consider the origin of peak 2, which has no analog in the anelastic behavior, and is also absent in the case of calcia-doped ceria. There is no way for this large peak to be due to the reorientation of still another type of dipole, since essentially all of the oxygen-ion vacancies are accounted for by peak 1, while the remaining Y’+ are present as Y’, which are on cubic sites and should not give rise to relaxation effects. Peak 2 is considerably broader than a single relaxation time. Also, for a given composition, experiments that we have conducted with successively higher polarization temperatures show increasing peak temperatures, indicative of a distribution of relaxation times. An interesting feature is that the peak temperature, T,, first decreases as the % Y,O, increases (from - l.8”C for 0.05% Y,O, to -36.7”C for 2.5% Y203) then increases at the higher concentrations (up to +3O”C for 15% Y203). This is the same pattern (but opposite in direction) to that followed by the conductivity of these solid solutions[S]. It is therefore suggestive of the possibility that the mean relaxation rate, TV’, is controlled by long-range vacancy migration just as is the conductivity. It is well known that a relaxation mechanism which involves a redistribution of charge takes place with a relaxation rate 7-l proportional to the conductivity, u, given by [ 191: 7-1 = OIEOQ

(6)

where l0 is the permittivity of free space and 4 the dielectric constant. Since D = (A/T) exp(-HdkT)

(7)

where A is the pre-exponential constant and H, the activation enthalpy, we can obtain an implicit expression for T,,, by differentiating eqn (2) and inserting eqns (6) and (7). The result is HJkT,,, = (A/b~)exp(-HJkT,,,)+

I.

(8)

Using our earlier results for A and e[S] and the present results for T,,,, we have used eqn (8) to calculate values of H,; these calculated values may then be compared with the experimental values obtained from conductivity measurements. Such a comparison is shown in Fig. 8. (Since H, is not a monotonic function of YzO, content, the % Yz03 is marked next to each point of the graph). The straight line drawn is for H,(calc) = H,,(expt). The results show very good agreement, consistent with an uncertainty in pre-exponential factors by a factor -2. The conclusion is that eqn (6) indeed appears to be valid. In order to interpret the magnitudes of peak 2, a more specific model is required. For the model we turn to the unique aspect of dilute trivalent (as vs divalent) doping, viz. the creation of an array of (Y Vo)’ and Y’ defects of opposite charges that are present in equal numbers. We seek the mechanism of peak 2 in the relaxation of this array, which can only occur by the migration of oxygen vacancies from one Y’+ to another to produce a redistribution of charge. (Recall that the Yzi ions are completely immobile at these temperatures.) The actual situation is an essentially random arrangement of the Y” ions with a strong preference for those that are paired, as (Y V,)‘, to surround themselves by Y’ defects in order to minimize the Coulomb energy. To treat this situation, we shall make a considerable simplification, and assume that the (Y V,)’ and Y’ defects are arranged in a regular

1.1 Hc7 (CALC) (eV)

1.0

0.8

Fig. 8. Comparison of H, from conductivity experiments with values calculated from TSDC peak temperatures using eqn (8). The line drawn is for H,(calc) = H,(expt).

Dielectricrelaxationin yttria-dopedceria solid “superlattice” of the NaCl type with a half-cell spacing, R,. This spacing is readily calculated from the Y3+ concentration, as R”/Cl = (4cv)P

(9)

where a is the lattice parameter of the fluorite unit cell of CeO?. If this superlattice were perfect, it could not give rise to dielectric relaxation; however, to minimize the free energy of the system, we must allow for the occasional presence of wrong pairs. Such pairs, as illustrated in Fig. 9, constitute large dipoles which can be reoriented in the presence of an electric field through oxygen-vacancy migration. The Coulomb energy of formation of such a wrong pair is easily shown to be Aw = 4eZ(aM- l)/yeRRo

(10)

where e is the electronic charge, aM is the Madelung constant which for this “lattice” is 1.748, and the factor y = 47re,,is required for SI units. Values of Aw for each composition, calculated by employing eqns (5) and (9) and the measured dielectric constants l[5], are listed in column 5 of Table 1. The equilibrium number of such wrong pairs per unit volume N,, at the polarization temperature, T,,, is given by N,, = 3Nv exp(-Aw/kT,).

(11)

The relaxation magnitude (the total polarization per unit electric field) is the quantity a&, Sx being the relaxation of the dielectric susceptibility. This magnitude is given by PI: edx =

Nw,p?vpDkT,

(12)

where F,,.~, the effective dipole moment of a wrong pair, is pwp= 2eRo

+

(13)

- + -/y;8yvi

7

Tb +

Fig. 9. Model

of an array

I-7 \

+

of (Y V,)’ and Y’ defects wrong pair.

showing

a

solutions

645

(see Appendix A). Combining these equations allows for the calculation of eOSx.This quantity is equal to the total charge released per unit area, Q, divided by the polarizing field E. The quantity Q is the area under TSDC peak 2, but experimental values are awkward to determine because of the overlap with peak 1 and because the tails of the peak are difficult to measure. Instead, we take Q as equal to peak height x width, or Q = I,,,,, . AZYb.

(14)

AT being the effective peak width. For peak 2 we estimate AT - 30°C and since b = $‘C/sec, we obtain Al76 = 900 sec. Using E = 9.65 x lo2 V/cm as employed in Figs. 3 and 4 we calculate values of I,,,,, as listed in Table 1, column 7: Comparison with Figs. 3 and 4 shows that the calculated order of magnitude of Zma.is correct and that the concentration dependence of the calculated values shows correct features, specifically, a decrease in I,,,,, from 0.15 to 0.5% Y,O,, then a leveling off, followed by an increase to a maximum at 2.5%. As pointed out in Section 4(a), however, models based on just (Y Vo)’ and Y’ defects cannot be valid for concentrations at or above 2.5%; therefore, the agreement at high concentrations must be, in part, fortuitous. Another approach is to calculate the ratio, ~3,of the magnitude of peak 2 to that of peak 1. Experimentally this ratio is easier to deal with, since it avoids certain problems such as knowledge of the absolute electrode area and electric field. Since the number of nn Y Vo dipoles is NJ2 and the dipole strength is 2e(ad/3/4) (see Appendix A for an explanation), comparison with eqns (ll)-(13) gives p = 32(Ro/a)* exp(-Aw/kT).

(15)

Values of fl, calculated for compositions 5 1% Y,O,, are listed in Table 1 as PO,,. Experimental values are obtained by averaging results of many such runs carried out. The most reliable values of p are obtained from runs done with the dielectric-insert method, since in the quenching method peak 2 is lessened. Furthermore, in taking the ratio, knowledge of the electric field across the sample is not required. Therefore, results for 0.15-l% Y,O, are obtained as an average of all runs using the dielectric insert. The scatter in values of ~3from such runs is *30%. Also, as before, we have used Z,,,,,AT instead of the area, and taken ATJAT, - 2.0 for the ratio of the width of peak 2 to that of peak 1. These values are listed in the last column of Table 1, as &,,. Unfortunately, runs on the 0.05% sample were not as extensive as the others and were done only by the quenching technique. Therefore, the tabulated value given is only a lower limit. Comparison of /?_,, with BeXP,gives reasonably good agreement in the magnitudes as well as showing the decreasing trend with increasing composition. It is doubtful, however, that a value as high as ~3= 135, predicted for the 0.05% sample, would be observed experimentally.

DA Yu WANG and A. S. NOWICK

646

In summary, the correlation of the relaxation time of peak 2 with electrical conductivity results, as shown in Fig. 8, constitutes strong evidence that peak 2 is due to the relaxation of the array of alternately charged (Y VO)’ and Y’ defects through the migration of VO;vacancies. It is further concluded, from both the absolute calculations of peak 2 (I,,,,,) and the relative peak heights (~3)of peak 2 to peak I, that the wrong-pair model gives the essential features of the behavior, although the model is surely too crude to be completely quantitatively valid. This mode1 of an alternating array of charged defects is of interest because it provides a novel source of dielectric relaxation not previously reported for ionic crystals. Acknowledgement-This work was supported by the US. Department of Energy under contract DE-AS 02-78ER 04693. APPENDIX

A

Dipole strengfh

The effective dipole strength of the (Y V,)’ pair can be obtained by recognizing that this defect is both a monopole and dipole. The long-range potential about such a defect, considered as a charge of t 2e separated from a charge of - e by a distance d, is readily shown to be equal to the field of a monopole of charge t e and to that of a dipole made up of opposite charges of ?ie. Since polarization (i.e. dipole rotation) occurs only by motion of the t2e charge (i.e. the Vi), it involvesa contribution both from the monopoleand the dipole.The dipole strength can be calculated most simply by taking the reference point at the Y’+ ion and noting that unit polarization can be regarded as migration of a t2e charge over a separation distance d in the field direction. Ac_cordingly,the effective dipole moment is 2ed and since d = &3/4 for this defect, the quoted result is obtained. In the case of the wrong pair, it is appropriate to take the perfect superlattice as a reference state. Thus, a plus charge on a site that should be minus is an effective charge of t2 and similarly for the minus charge on a wrong site. The wrong pair, therefore, has an effective dipole moment of 2eRo.

REFERENCES

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