effect of antimony oxide stoichiometry on the nonlinearity of zinc oxide varistor ceramic

12
ELSEVIER Materials Chemistryand Physics 49 (. 1997) 258-269 MATERIALS CHEMISTRYAND PHYSICS Effect of antimony oxide stoichiometry on the nonlinearity of zinc oxide varistor ceramics S. Ezhilvalavan, T.R.N. Kutty * Materials Research Centre, IndiaJ~ Institute of ScieTzce, Bangalore 560 012, India Received 2 August 1996; revised 27 January 1997 Abstract The effect of antimony oxide at higher concentrations ( > 2 moI%) and variable valence states of Sb on the nonlinearity of ZnO varistor ceramics has been investigated. Simplified compositions containing 92.5ZNO + 3Bi203 + 2.5Co304 + 2SbaO5 (mol%) show nonIinearity coefficients (c~) up to 65. Ceramic formulations derived from SbaOs bring about higher e~ than those with Sb20~ or Sb204, provided the concentration of SbzO5 is _> 2 tool%. The secondary phase is predominantly antimony spinel, ZnTSb20~2. Formation and involvement of a liquid phase during sintefing is indicated from the microstructure studies. Energy-dispersive X-ray analysis shows that Sb is distributed more in the grain boundaries and within the secondary phase. Admittance spectroscopy, capacitance-voltage analyses, dielectric dispersion and electron paramagnetic resonance show that the observed defect states and the type of traps in ZnO + Bi203 + Co~O4ceramics remain unaltered whereas the trap density increased with the addition of antimony. The method of formulation of the ceramics by way of higher oxygen content of the additives is critical in attaining high nonlinearity. This can be explained on the basis of formation of the depletion layer at the pre- sintering stage itself, because of the surface states arising out of the chemisorbed oxygen from the incipient Iiquid phase. The depletion layer is retained during sintering as a result of the high valence state of cobalt, as evidenced from the electron paramagnetic resonance spectroscopic results. Keywords: Ceramics;Varistors; Antimonyoxide; Zinc oxide 1. Introduction Zinc oxide varistors are voltage-limiting ceramic devices with highly nonlinear resistance used as overvoltage surge protection elements in electrical circuits [ 1 ]. The origin of their non-ohmic behaviour lies in their microstructure, where ZnO grains are three-dimensionally separated from each other by grain boundary layers formed by the reactions of additives with each other and with ZnO [2]. The current- voltage (I-V) characteristics are dependent on the detailed microstructure of ZnO ceramics [ 3]. It is generally accepted that the secondary phases are predominantly 6-Bi203, Zn7Sb2Oi2 (spinel), and Zn2Bi3Sb30~4 (pyrochlore) although the role of individual additives and their reaction products in determining the nonlinear electrical properties is not fully understood [4-6]. In the ZnO-Bi203 system, antimony oxide is usually added so as to react with ZnO and Bi2Q forming a liquid phase * Corresponding author. 0254-0584/97/$17.00 © 1997Elsevier Science S.A. All rights reserved PILS0254-0584(97)01900-7 during the sintering process and playing an important role in improving the varistor characteristics [7]. Microstructure development, formation of secondary phases and the reac- tions between various phases during the sintering of ZnO ceramics containing antimony have been extensively studied by Matsuoka [ 1], Inada [2,4] and Kim et al. [8-11 ]. How- ever, the effect of antimony oxide on the electrical properties has not been discussed and most of the reports centred around the morphology and formation reactions of spinel and pyr- ochlore phases. Furthermore, the influence of antimony on the nonlinear characteristics of ZnO ceramics with concen- trations varying from parts per million (ppm) levels to 5 tool% is not reported. Influence of antimony oxide on the nonlinear resistance of simplified: ZnO-BizO3-Co304-anti- mony oxide compositions has therefore been investigated. Oxides containing different valence states of Sb, prepared from various routes have been used in formulating the ceram- ics. The main emphasis of this paper is on the role played by antimony in improving the nonlinear I-V characteristics of ZnO varistors.

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Page 1: Effect of Antimony Oxide Stoichiometry on the Nonlinearity of Zinc Oxide Varistor Ceramic

E L S E V I E R Materials Chemistry and Physics 49 (. 1997) 258-269

MATERIALS CHEMISTRYAND

PHYSICS

Effect of antimony oxide stoichiometry on the nonlinearity of zinc oxide varistor ceramics

S. Ezhilvalavan, T.R.N. Kutty * Materials Research Centre, IndiaJ~ Institute of ScieTzce, Bangalore 560 012, India

Received 2 August 1996; revised 27 January 1997

Abstract

The effect of antimony oxide at higher concentrations ( > 2 moI%) and variable valence states of Sb on the nonlinearity of ZnO varistor ceramics has been investigated. Simplified compositions containing 92.5ZNO + 3Bi203 + 2.5Co304 + 2SbaO 5 (mol%) show nonIinearity coefficients (c~) up to 65. Ceramic formulations derived from SbaOs bring about higher e~ than those with Sb20~ or Sb204, provided the concentration of SbzO5 is _> 2 tool%. The secondary phase is predominantly antimony spinel, ZnTSb20~2. Formation and involvement of a liquid phase during sintefing is indicated from the microstructure studies. Energy-dispersive X-ray analysis shows that Sb is distributed more in the grain boundaries and within the secondary phase. Admittance spectroscopy, capacitance-voltage analyses, dielectric dispersion and electron paramagnetic resonance show that the observed defect states and the type of traps in ZnO + Bi203 + Co~O4 ceramics remain unaltered whereas the trap density increased with the addition of antimony. The method of formulation of the ceramics by way of higher oxygen content of the additives is critical in attaining high nonlinearity. This can be explained on the basis of formation of the depletion layer at the pre- sintering stage itself, because of the surface states arising out of the chemisorbed oxygen from the incipient Iiquid phase. The depletion layer is retained during sintering as a result of the high valence state of cobalt, as evidenced from the electron paramagnetic resonance spectroscopic results.

Keywords: Ceramics; Varistors; Antimony oxide; Zinc oxide

1. Introduction

Zinc oxide varistors are voltage-limiting ceramic devices with highly nonlinear resistance used as overvoltage surge protection elements in electrical circuits [ 1 ]. The origin of their non-ohmic behaviour lies in their microstructure, where ZnO grains are three-dimensionally separated from each other by grain boundary layers formed by the reactions of additives with each other and with ZnO [2]. The current- voltage (I-V) characteristics are dependent on the detailed microstructure of ZnO ceramics [ 3]. It is generally accepted that the secondary phases are predominantly 6-Bi203, Zn7Sb2Oi2 (spinel), and Zn2Bi3Sb30~4 (pyrochlore) although the role of individual additives and their reaction products in determining the nonlinear electrical properties is not fully understood [4-6] .

In the ZnO-Bi203 system, antimony oxide is usually added so as to react with ZnO and Bi2Q forming a liquid phase

* Corresponding author.

0254-0584/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved PILS0254-0584(97)01900-7

during the sintering process and playing an important role in improving the varistor characteristics [7]. Microstructure development, formation of secondary phases and the reac- tions between various phases during the sintering of ZnO ceramics containing antimony have been extensively studied by Matsuoka [ 1 ], Inada [2,4] and Kim et al. [8-11 ]. How- ever, the effect of antimony oxide on the electrical properties has not been discussed and most of the reports centred around the morphology and formation reactions of spinel and pyr- ochlore phases. Furthermore, the influence of antimony on the nonlinear characteristics of ZnO ceramics with concen- trations varying from parts per million (ppm) levels to 5 tool% is not reported. Influence of antimony oxide on the nonlinear resistance of simplified: ZnO-BizO3-Co304-anti- mony oxide compositions has therefore been investigated. Oxides containing different valence states of Sb, prepared from various routes have been used in formulating the ceram- ics. The main emphasis of this paper is on the role played by antimony in improving the nonlinear I - V characteristics of ZnO varistors.

Page 2: Effect of Antimony Oxide Stoichiometry on the Nonlinearity of Zinc Oxide Varistor Ceramic

S. Ez.hih, alavan, T.R.N. Kutt3' / Materials Cliehiisto, and Physics 49 (1997) 258-269 259

2. Experimental procedures

2.1. Preparation of antimony oxide

Antimony oxide used in formulating ZnO varistors was prepared by the following methods.

(A) Sb203 was prepared by the hydrolysis of antimony trichloride. A known amount of SbC13 (analytical reagent grade) was dissolved in a minimum quantity of 1:1 hydroch- loric acid and the solution was diluted with distilled water. The water was removed once a week and agitated for three months to complete the hydrolysis. The fine powder obtained was dried at 110°C. The last traces of water in the sample were removed by heating at 250°C under reduced pressure. The dried powder was identified by X-ray diffraction as orthorhombic Sb203 (valentinite), existing in a metastable state [ 12-14].

( B ) Sb204 was prepared by heating antimony trioxide in air below 900°C for 3 h. The structure consists of a network of edge sharing S b m Q and SbVQ octahedra [ 15].

(C) Sb2Q was prepared by the slow decomposition of antimony nitrate. The decomposition temperature was main- tained around 440°C [ 16,17].

2.2. Processing of ZnO varistor ceramics

Pre-annealed ZnO powder was mixed with bismuth oxide (3 tool%), cobalt oxide (2.5 mol%) and antimony oxide (up to 5 mol%). Detailed studies on ZnO-Bi203 as the 'base' system have been reported previously, with respect to the valence state of Bi, nonstoichiometry, oxygen contents and phase relations of BizO3 as well as the compositions of the ceramic formulations [18]. The method of preparing the ceramic discs was presented in that article [18]. The discs were prefired at 850°C to burn off the binder and then sintered at selected temperatures in the range of 1100-1200°C for 1 to 3 h and cooled at the rate of 30-75°C h - l to 500°C. The sintered discs were 12 mm in diameter and 1-1.5 mm thick. They were deep black or greenish black depending upon the composition as well as the processing conditions. Ohmic contacts were provided by electroless deposited silver, which was recrystallized at 250°C. For measuring the dielectric properties and for the evaluation of I -V relations at high currents, samples were electroded with fired-on silver paste (650°C for 10 min). Direct soldering of the electrical con- tacts was possible on these electrodes.

2.3. Ceramic characterization

X-ray powder patterns were recorded using the XDS 2000 from Scintag Inc., USA, with Cu Ka radiation. The micros- tructural and grain size distribution studies were carried out using a scanning electron microscope (Cambridge $360) fitted with an energy-dispersive X-ray analyser (LINK AN 10000) for the elemental distribution studies in the ceram-

iCS. Tt'te grain size was measured with the linear intercept

method on the micrographs. Polished samples were lightly etched with dilute acetic acid (1 N) and ultrasonically cleaned for the microstructuraI investigations. Electron par- amagnetic resonance (EPR) spectra were recorded using a Varian X-band spectrometer having a TEol~ cavity in the range of 78-300 K.

The 1-Vcharacteristics were measured in both d.c. and s.c. modes for the lower current regions (below i0 mA) with a home-built voltage source and a curve tracer. For I -V meas- urements at higher current values, a d.c. pulse generator, producing signals of 8 × 20 b~s wave shape, together with an HP54600A oscilloscope was used. The nonlinear coefficient o~ is evaluated in terms of the relation:

d In I In I 2 - - In 11 c ~ = - - (1)

d l n V l n V z - l n V t

where V~ and V2 are the voltages corresponding to I, - 1 mA and 12 = I0 mA, respectively. Capacitance measurements were carried out as a function of frequency (100 Hz- 40 MHz) using an impedance/gain phase analyser (HP4194A) at a signal strength of 0.5 Vrms. The voltage- dependent capacitance at the breakdown region (CS) was measured using an external bias voltage accessory (HP16065A) connected to the impedance bridge where the d.c. current was circulated through the inductor of a tank circuit, the varistor sample and an ammeter. A large blocking capacitor prevented the voltage from damaging the inputs to the bridge. The tank circuit was carefully tuned to the fre- quency of the bridge so that the power supply arm appeared to be nearly an open circuit. A voltmeter measured the d.c. voltage before and after the capacitor measurement. A.c. con- ductance measurements were made at the above frequencies while the sample temperature varied between 77 and 373 K.

3. Results

3.1. Microstrucmre

The sintered density depends on the concentrations of anti- mony oxide as well as the transition metal oxides. The sin- tered density increases up to 2 mol% antimony oxide. At higher concentrations the density as well as the nonlinearity decreased. Therefore the antimony concentration was main- tained around 2 mol%. The density of the ceramics also depends upon the temperature as well as the duration of sin- tering. However, the nonlinear coefficient, a, decreases on enhancing both these processing parameters. Thus sintering temperature is maintained around 1200°C and the duration is <3 h. Under these conditions, the density was around 92- 96%.

Fig. l (a ) shows the microstructure of ZnO+Bi203+ Co304 ceramics prepared by adding 100 ppm Sb2Q; grain size ranges from 3 to 10 btm. More grain growth is seen with higher SbaO5 (2 mol%), and grain size ranges from 13 to

27 gm (Fig, 1 (b)). The low Sb (ppm) content samples con-

Page 3: Effect of Antimony Oxide Stoichiometry on the Nonlinearity of Zinc Oxide Varistor Ceramic

260 S. Ezhilvalavan, ZR.N. Kutty / Materials Chemistry and Physics 49 (]997) 258-269

Fig. 1. Microstructure of ZnO (92.5-94.5%)+BiaO3 (3%)+Co304 (2.5%) ceramics containing (a) 100 ppm (etched), (b) 2 mol% Sb205 (thermally etched), (c) 2% SbzO5 (fractured surface) (d) 2% Sb205 (as sintered surface) and (e) sample (b) with 0.5% Bi,O3.

rain more angular grains with anhedral morphology whereas higher Sb ( > 2 mol%) samples have grains with rounded corners and distinctly faceted morphology. Phases other than ZnO are distinguishable by the triangular morphology of antimony spinel and also by the darker appearance of pyr- ochlore or Bi203 phases. These secondary phases are seen randomly distributed and are mostly confined to the grain boundary regions and triple junctions. No separate spinel grains were observed in the ceramics containing 100 ppm to 0.2mo1% antimony oxide. Whereas at concentrations > 2 tool%, spinel grains were observed in the form of trian- gular-shaped particles distributed amidst the elongated ZnO grains (Fig. 1 (c)) . Formation and involvement of a liquid phase during the sintering process is indicated from the microstructure by way of the rounded comers of ZnO grains (Fig. 1 (b)) and striations or growth lineage on the surface of the grains (Fig. l (d) )and on the intergranular regions representing the solidification front. Fig. 2 shows the energy- dispersive X-ray (EDAX) analyses for the ceramics,

ZnO + Bi203 + Co304, containing antimony oxide. The emis- sion peak of Sb in the grain interiors, for ceramics containing 100 ppm to 0.2 mol% of Sb, indicates the presence of anti- mony ions in the ZnO grains (Fig. 2(b) ). Whereas at higher concentrations of Sb ( > 2 mol%), EDAX analysis shows more Sb at the grain boundaries (Fig. 2(c)) . Thus EDAX analyses show that antimony is distributed more in the grain boundaries. The emission peaks at 3.604 and 3.843 keV cor- respond to Sb (L~,.m peaks. Cobalt is more uniformly dis- tributed as indicated by the presence of Co (L~ j) = 0.776 keV and Co (K,~L) = 0.692 keV peaks for both grain interiors as well as at the grain boundaries.

3.2. Effect o f Sb/Bi ratio on the microstructure

During sintering, densification, grain growth and chemical reactions occur in presence of a minor liquid phase resulting in the final microstructure of ZnO varistor ceramics. The liquid phase initially consists of BizO 3 which then reacts with

Page 4: Effect of Antimony Oxide Stoichiometry on the Nonlinearity of Zinc Oxide Varistor Ceramic

s. Ezhih, alavan, T.R.N. Kutry / Materials Chemistt 3' and Physics 49 (1997) 258-259 261

*6

tn c-

=

?~ { (d)

{" x :

71 f3 --_ Nit g ~": ~ (c)

3 # <b) ,,,q g.

_j ._1 ...a -.e-

"5 NZ

~ ~ ~ (a)

,A

I I I I I I _ 2 4 6 8 10 12

E(keV) Fig. 2. Energy dispersive X-ray analysis (EDAX) of ZnO (92.5%) + Bi203 (3%) +Co304 (2.5%) ceramics. (a) 0.2% Sb205 (GB), (b) 0.2% Sb205 (GI), (c) 2% Sb205 (GB) and (d) SbzQ (GI) (GI=grain interior, GB = grain boundary).

phases. Fig. 3 indicates that the weight fraction ofpyrochlore is lower than that of spinel in the sintered ceramics. The pyrochlore content decreases as the concentration of anti- mony oxide is increased above 2 mol%. Fig. 3(e) reveals weak reflections corresponding to pyrochlore as well as the spinel, with the latter dominating in relative phase contents. XRD patterns of ceramics are shown in Fig. 3 (a ) - ( e ) with three components (ZnO (95%)+Bi203 ( 3 % ) + S b 2 Q (2%) and ZnO (94.5%) +Bi203 (3%) +C%O4 (2.5%)) as well as four components (ZnO (92.5% ) +Bi203 (3%) + C%O4 (2.5%) + Sb205 (2%)) systems sintered at 1200°C for 2h and either slow cooled at 30°Ch -1 or quenched to room temperature. XRD results indicate that the spinel is more dominant than the pyrochlore phase irrespec- tive of the cooling rate.

Rapid quenching tends to preserve the high temperature phases whereas slow cooled samples tend more to equilib- rium. The dominance of antimony spinel even in the rapidly cooled samples indica[es that this phase is formed at 1200°C and not through the decomposition reaction

Zn2Bi3Sb3014 (Pyrochlore)

-+Zn7Sb2Ol2 (Spinel) +Bi203 (2)

# # (e) 1 ~ ~ ~ I

other additives as well as the ZnO particles. The formation temperature of the liquid phase depends on the ratio SbaOs/ Bi203 as much as on the concentration range of the individual oxide. At lower Bi203 concentrations, the microstructure con- tains angular grains and the grains with rounded corners are minor which shows the least involvement of liquid phase during sintering (Fig. 1 (e)) . In comparison, at low concen- trations of Sb205 (ppm) (Sb/Bi <<< 1), the liquid melt will form above 830°C (melting point of Bi203) and very few grains have rounded corners, Thus, altering the antimony to bismuth ratio cannot eliminate the liquid formation. In the present ceramics, irrespective of the Sb/Bi ratio, the spinel phase (ZnvSb2012) is predominant whereas pyrochlore (Zn2Bi3Sb3012) is minor. When SbzQ is replaced by Sb203 or Sb204, there is no change in the microstructure nor mor- phology. However, the spinel phase is less in comparison with Sb2Os-added ceramics. This shows that Sb (V) stabilizes more of the spinel phases than Sb(III). Since high values of nonlinearity coefficient are observed for the ratio Sb/ Bi = 0.6, the same ratio was used for further studies.

3.3. X-ray p o w d e r diffraction

X-ray powder diffraction (XRD) patterns of the antimony oxide added ZnO ceramics show 6-Bi203, antimony spinel (,ZnTSb20 ~2) and 9"£och~,ore (Zn2Bi3Sb30 ~4) as the minor

(d)

_ . _ 3 _ j ~ . . J W,~J L

< IITII (c)

I i , l l l , , I r l r i T T P ~ l r T l l r r ~ l 20 30 40 50 60 70

2e (deg)

Fig. 3. X-ray diffraction pattern of ZnO ceramics sintered at 1200°C for 2 h and cooled or quenched to room temperature: (a) 95ZnO + 3Bi203 + 2Sb205 (quenched), (b) 95ZnO+3BizQ+2SbaOs (30°Ch-I), (e) 97ZnO+ 0.5Bi2Q + 2.5CO304 (30~C h- i), (d) 92.5ZNO + 3Bi203 + 2.5Co304 + 2Sb=Q (quenched) and (e) 92.5ZnO+3Bi2Q+2.5C%O4+2Sb,O5 (30°Ch-t).

Page 5: Effect of Antimony Oxide Stoichiometry on the Nonlinearity of Zinc Oxide Varistor Ceramic

262 S. Ezhilvalavan, ZR.N. Kuuy / Mawrials Chemistry and Physic's 49 (]997) 258-269

on cooling. The content of 8-Bi203 is lower in Sb2Os added samples when compared with the ZnO-BizO3-Co304 ceram- ics. In the three-component system (ZnO + Bi:O3 + Sb205), the cooling rate alters the relative intensities of different reflections from spinel which can be explained in terms of the morphological differences in crystallites which lead to orientation effects in X-ray diffraction patterns. Hence (333) reflection of spinel (Sp) dominates for the slow cooled sam- ple whereas the intensity of Sp(440) reflection is increased on quenching (Fig. 3(a) and (b)) . The concentration of pyrochlore increases on slow cooling in the three-component system. For the ceramics containing cobalt oxide (four-com- ponent system), the intensity of the spinel reflections are not altered in both slow cooling as well as under fast quenching. However, spinel is the dominant phase in the presence of cobalt oxide whereas pyrochlore is the minor phase. XRD patterns of the ZnO + Bi20~ + Co304 system indicate &bis- muth oxide as the only secondary phase in minor concentra- tion as shown in Fig. 3(c).

3.4. Effect of antimony o.ride on the I -V curves

3.4.1. ZnO-Bi2Os-antimony oxide Ceramics prepared using 95% ZnO + 3% Bi203 + 2% anti-

mony oxide (from various routes of preparation) sintered 2 h at 1200°C and cooled at 75°Ch -1 show nonlinear (I-V) characteristics (Fig. 4). The leakage current region (at the lower field strengths) depends upon the antimony oxide pre- pared by different methods, with the lowest values for the ceramics formulated using Sb205. The nonlinear coefficient, oe, varies from 6 to 14, with the highest values for samples formulated with Sb205. The turn-on point is not sharp for the

ceramics prepared using Sb2Q or Sb~O4. These results clearly show the importance of the pre-history of antimony oxide used in formulating the ceramics. Since SbzO5 has maximum oxygen content, presence of higher valence anti- mony together with the extra oxygen seems to enhance the nonlinearity. However by itself, antimony oxide is unable to induce high oe values.

3.4.2. ZnO-Bi g-~-Co304--antimony oxide Detailed studies on ZnO-Bi203-Co304 system have been

reported by us, with respect to valence states of cobalt, oxygen contents, preparative routes and compositions of the ceramic formulations [18,19]. It has been demonstrated that high values of c~ can be obtained for ceramics containing 3 tool% Bi203 and 2.5 mol% Co304. Fig. 5 shows the nonlinear I -V characteristics of ZnO ceramics formulated from 92.5ZNO + 3Bi203 + 2.5Co304 (tool%) with 2 tool% anti- mony oxide from various routes of preparation (Sb203,

SbzO4 and Sb205) and processed under the same conditions. The highest nonlinearity was observed for ceramics formu- lated from Sb2Q, with the I -V curves showing a sharp turn- on and oe = 25-65. The I -V characteristics of the ceramics prepared from Sb,O3 (or Sb204) are inferior, with no sharp turn-on points and lower values of o~ (Fig, 5(a) and (b)) . For further investigations, Sb205 was used in the ceramic formulations. The concentration of SbzOs used has consid- erable influence on the l-Vcurves. At a given temperature of sintering (1200°C), duration of sintering (2 h) and cooling rate ( 75°C h - ~ ), the effect of SbaOs concentration on the I -V curves is evident from Fig. 6. With Sb2Q present at ppm levels ( 100 ppm), the leakage currents are low, diminishing o~ values and less sharp in the turn-on behaviour. The same characteristics persist when the concentration of Sb,_O5 is around 0.2 to 0.5 tool%. Whereas with SbaO5 levels at > 1 mol%, the value of a increases reaching a maximum

1

0

v

~- 3

!-0 u

- J - 5

-6

Fig. 4. I-V characteristics of ZnO (95%) + Bi203 (3%) ceramics sintered at 1200°C for 2 h containing 2 tool% (a)SbzO3, (b) SbaO4 and (c) Sb2Os.

(d)/(¢)

" e4 2

///7 g

' ..// / i s -; . . . . y -y / / y . ,

1,5 7D 2.5 3.0 3.5 4.0 4.5 Log field strength(V/cm)

Log field strength (V/cm) Fig. 5. l-V characteristics ofZnO(92.5%) + BizO3 (3%) +Co304 (2.5%) ceramics sintered at 1200°C for 2 h containing 2 moi% (a) Sb203, (b) Sb204 and (c) ShoOs.

Page 6: Effect of Antimony Oxide Stoichiometry on the Nonlinearity of Zinc Oxide Varistor Ceramic

S. Ezhih,alavan, T.R.N. Kq~tty / Materials C t i e m i ~ T a h y s i c s 49 (79977258-2d9 263

4

2

i l "~ o ~ ~

e=_ a ~o

_J

-5

-6 - ( { : ~

-5 2.0 2.5 3.0 3.5 4.0

Log field strength(V/cm)

Fig. 6. I-V characteristics of ZnO (89.5-94.5%) +Bi203 (3%) +Co304 (2.5%) ceramics sintered at 1200°C for 2 h containing Sb205. (a) I00 ppm, (b) 0.2%, (c) 1%, (d) 2% and (e) 5%.

around 2 mol% and the turn-on point is sharp. Breakdown voltage (VB) values also increases with concentration of Sb205.

The high current measurements under the pulsed d.c. mode are shown in Fig. 6 for the ceramics containing SbzQ. The up-turn region is observed above 8× 102A cm -2 for the ceramics containing 2.0 mol% SbaO> The current values at the up-turn are proportional to the grain interior resistance. The point of up-turn is at lower currents for ceramics con- taining 0.2 mol% SbaOs, whereas the up-turn current has increased marginally with SbeO5 concentration. This indi- cates that the grain interior resistance is more for the ceramics containing low concentrations of antimony. Thus the power handling capability of the varistors can be increased by the addition of Sb205. The energy absorption by the ceramics under the pulse d.c. mode of 8 × 20 >s increases from 200 to 500 J cm- 3 with Sb205 addition. The I-V characteristics are influenced by the formation of a liquid phase during sintering. Ceramics formulated with 0.2 mol% SbeO5 are flee of sec- ondary phases which is reflected in the I-V characteristics with the diminishing c~. In contrast, for the ceramics contain- ing increased Sb205 concentration (2 tool%), fornaation of the secondary phases such as antimony spinel, antimony pyrochlore and &BiaO3 plays a crucial role in improving the nonlinearity. Higher oxygen content of the additives used in the formulation and the slow cooling of the ceramics from the sintering temperature are the important factors in enhanc- ing the nonlinearity. I-V characteristics of the ceramic quenched to room temperature is shown in Fig. 5 (d). The I-V curve is less sharp and the value of c~ is = 16. The preservation of nonlinearity in the quenched sample indicates that the antimony ions can be stabilized in the higher oxidation state ( Sb s + ) which is supported by the presence of more antimony spinel in the quenched sample. Under fast quenching, the pre- breakdown region of the I-V curve is not affected whereas

the up-turn region is modified and extended to higher current regions, indicating the decrease in the grain interior resistance.

3.5. Capacitance-voltage (C-V) analyses

On the basis of a symmetrical double Schottky barrier model for the grain boundary region, parameters such as the barrier height, donor density, interface state density and depletion layer width have been determined from the capac- itance-voltage relations, as reported earlier [ 19-21 ].

( 1 /C- 1/2Co)2= 2( (~b -~- V ) / q ~ . N d (3)

where

1/2Co = ( 2 ~ b / q ~ V d ) 1/2 (4 )

q is the electron charge, e the dielectric constant of ZnO, N~ the donor density, q5 u the barrier height, C the capacitance per grain boundary junction and V the applied voltage per grain boundary. From the slope as well as the intercept of the straight line plots of ( l / C ) 2 versus V, &t, and Nd can be evaluated. Using these values the depletion layer width (W) can be obtained from the relation

W= ( 2eeO&b/ q2Nd)1/2 (5)

whea'e e0 is the permittivity of vacuum. The density of states at the interface (N~), between the ZnO grain and the grain boundary region can be estimated from

N~= ( 2Ndeeo~/ q)1/2 (6)

The capacitance measurements are carried out at 1 kt-Iz with the variable applied bias in the pre-breakdown region of the I-V curves for the ceramics sintered at identical conditions (Table 1 ).

Although the grain boundary parameters are dependent on the pre-treatment of ZnO, the type of additives, the processing conditions such as sintering temperature, sintering duration and cooling rates, marginal differences are observed in the grain boundary parameters with the addition of Sb when compared with the ZnO + Bi203 + Co304 ceramics. The var- iation in donor density (Nd) with the concentration of SbaO5 is low. The values of barrier height (qSb), density of states at the interface (N~) and the depletion width (W) are compa- rable with the values reported for the three-component system [191.

Capacitance at a given frequency varies considerably with the applied voltage depending upon the charge trapping in the depletion regions and the magnitude of potential barriers at the grain boundary. As the electric field is increased, the low frequency capacitance passes through a minimum. On further increase of voltage into the breakdown region, the capacitance increases sharply (Fig. 7). The magnitude of the capacitance increases from nano- to microfarads at low fre- quencies ( < 300 Hz). Further increase of the voltage beyond the breakdown region causes capacitance to decrease abruptly, pass through zero and attain negative values

Page 7: Effect of Antimony Oxide Stoichiometry on the Nonlinearity of Zinc Oxide Varistor Ceramic

264 S. Ezhilvalm,an, T.R.N. Kut 0'/Materials Chemistry and Physics 49 (1997) 258-269

Table 1 Grain boundary properties of the sintered ceramics

Composition "94.5ZnO + 3Bi,_O3 + 2.5Co304 N d ( X 10 ;'~ cm -3) ~ (eV) N, ( X 10 ~3 cm -a) W ( X 10 -~ cm) (tool%) + Sb205 (tool%)

100 ppm 1.13 1.42 3.23 8.98 0.2 1.35 1.23 1.45 3.38 0.5 1.33 0.77 0.98 2.33 1.0 1.30 0.93 1.06 2.59 2.0 1.39 I. 14 1.42 2.23 5.0 1.40 1.17 1.51 3.49

These ceramics were formulated with ZnO (89.5-94.5) + 3Bi203 + 2.5Co304 + 2Sb,Os (moI%) concentration varying from 100 ppm to 5 mol% and sintered at I200°C for 2 h and cooled at 75°C h- ~,

[22,23]. At the breakdown region, the increase in capaci- tance is greater at low frequencies ( < 100 Hz), whereas the capacitance around the breakdown region decreases with fre- quency and remains flat at higher frequencies ( > 1 MHz). Capacitance as a function of applied voltage for ceramics containing 2 tool% Sb2Os is shown in Fig. 7 (inset). The experiment carded out at different frequencies from 100 Hz to 1 kHz reveals that the increase in capacitance is sharp around the breakdown region. Varistors with sharp turn-on and high oe exhibit larger magnitude for the capacitance jump, indicating that the C - V characteristics depends on the nonli- nearity of the ceramics [ 19].

Above the breakdown point, the low frequency capacitance increases by > 103 times and becomes negative. These enhanced dielectric resonance features are indicative of the oscillatory charge redistribution between the trap states at the interface under the applied bias V> V~,. The charge redistri- bution can be enhanced by the multivalence states of cobalt. The general response of a charge trapping grain boundary to an applied d.c. plus harmonic voltage of frequency ~o is ana- lysed by Pike et al. [ 22,23 ]. Accordingly the quadrature part of the current crossing the boundary yields the anomalous

&

(O

6 ~ v 0 !2o

40

20

-a0

0

601 1 30 o

/,:o:I

50 100 150 Vo[tog¢ (V)

l O O H z

200

Fig. 7, Capacitance as a function of applied voltage in the breakdown region for ZnO (92.5%)+Bi203 (3%)+Co304 (2.5%) ceramics containing 2 mol% Sb2Os (Inset: pre-breakdown region at higher frequencies.)

capacitance. Detailed calculations by Pike et al. [ 22,23 ] show that when majority carriers alone are important in the barrier charge trapping, the anomalous capacitance always augments the normal capacitance (CD > 0). However, when minority carriers are also involved, it is possible to have the sign of the effect reversed (CD<0) , and thus have the apparent capacitance become negative [22,23]. Notice that the anom- alous component, of either sign, is not capacitance as nor- mally conceptualized. It is not a reversible storage of energy (as is C~), but rather a self-induced modulation of over- barrier, Joule heating current. Thus the C - V studies show that there is a direct correlation between the magnitude of capac- itance jump and the nonlinearity of the ZnO varistors.

3.6. Dielectric dispersion o f varistor ceramics

The frequency dependence of dielectric constant for the nonlinear ceramics of different compositions is shown in Fig. 8. The dielectric constant, Grr decreases with frequency. The effective dielectric constant ranges from I000 to 4000 for the present ceramics, as compared with the bulk dielectric constant of < 10 for bulk ZnO. These high dielectric con- stants indicate the formation of a microstructure of semicon- ducting ZnO grains surrounded by insulating grain boundaries and they also suggest the existence of surface states in the grain boundaries. Grr increases with the concen-

5o00 I ?, 400o ! 3o00, ,~_.~ 2000 { ~

lOOppm_ | ;'=, 1 0 0 0 = =z . . . . .

5.0

02 3 4 5 6 Log F (Hz)

Fig. 8. The frequency dependence of dielectric constant for the ZnO (89.5- 94,5%) +BiaQ (3%) +C%O4 (2.5%) ceramics containing various con- centrations of Sb205 ( I00 ppm to 5 tool%).

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S. EzhiIvalavan, T.R.N. Kutty /Materials Chemistry and PhySics 49 (1997) 258-269 265

A 4

%3

3 2

%

1.8 2.0 2,2 2.4 2,6 CpX~O'gF

/ uo / e.i I

r 4.0 4 . 5

C~xl0-SF 5.0

Fig. 9. Plot of a.c. conductance vs. paraIIeI capacitance of ZnO 94.5%) + Bi203 (3%) + Co~04 (2.5%) ceramics containing 2 mol.% Sb205 (Inset: 5 mol% Sb205).

tration of SbeQ up to 0.2 mol% and a reversal is noticed at higher concentrations. The eefr of the ceramics containing antimony oxide is greater than that of the ZnO + Bi203 + C o 3 0 4 ceramics for identical processing conditions. This indicates that addition of Sb increases the insulating region near the grain boundary by segregation as well as by forming secondary phases which does not occur in the three-compo- nent system.

The type of relaxation present in these ceramics can be determined by drawing either the Cole-Cole plot or the com- plex capacitance Cp versus Gp/co curve, obtained from a.c. conductance measurements as a function of frequency, where Cp is the parallel capacitance Gp the a.c. conductivity and co is the frequency. The complex capacitance behaviour (Fig. 9) clearly shows the presence of multiple relaxations in the present ceramics. Two relaxations are observed at 7= 2.5 × 10 -4 S and r - - 7.7 × t0 -6 S in ceramics containing 2 mol% Sb205. Although the relaxation frequencies of the present ceramics are not changed much by the SbzO 5 addition, there is a large variation in the area under the curve which decreases with increase in Sb205 concentration above 2 mol%, particularly for the relaxation around 7.7 × 10 -6 s (at the high frequency region). Therefore multiple trapping relaxations occur in the present ceramics which may be explained on the basis of Debye-type processes caused due to trapping or detrapping of electrons at the inter-band states located at the interface [24-26]. Thus the addition of Sb has not changed the type of relaxations. However the decrease in the area under the curve for higher concentration of Sb can be explained from the fact that the particular relaxing species (at the high frequency region of the Cole-Cole plot) are more at the depletion regions than at the grain boundary owing to the formation of liquid phase.

3.7. Admittance spectroscopy

In polycrystalline materials, such as ZnO varistors, inter- face states at the grain boundaries provide sites for electron trapping thereby generating double Schottky barriers. The baad bending associated with these barriers causes the bulk

defect levels, whether intrinsic or extrinsic in nature, to cross the Fermi levet in the depletion regions. If the traps are con- sidered above the Fermi level, electrons are excited from the defects to the conduction band and below the Fermi level electrons reside in the defect levels. The bulk traps can strongly affect the dielectric properties as well as the voltage- or temperature-dependence of the potential barrier height, (fib. Therefore, characterization of the interface states as well as the bulk traps is necessary for understanding the varistor behaviour [ 27,28].

Admittance spectroscopy has been used to characterize the deep bulk as well as the interface traps through their density, energy position (activation energy) and capture cross-sec- tion. The details of measurements are reported in our earlier pubiication [19]. A.c. conduc{ance measurements were made at frequencies between 10 kHz to 1 MHz at tempera- tures from 77 to 373 K. By applying a small a.c. signal to the polycrystalline specimen, a conductance resonance occurs when the emission rate (en) of the electron in a particular energy level equals the angular frequency (co) of the a.c. signal. The characteristic emission rate associated with the deep levels is given by

en = r n i ( SnV~hNc) exp( Et/Kt) (7)

where Sn is the capture cross-section, Vt~ is the thermal veloc- ity and Ne is the effective density of states for the conduction band [27,28]. The trap depth (Et), the capture cross-section (Sn) and thermal emission constants ( r~= 1/%) can be obtained by repeated measurements of a.c. conductance over a given temperature range at different frequencies.

The trap depth (&) and the capture cross-section (S~) can be determined by plotting ln(co/T~) as a function of 1/T where Tp is the temperature at which the a.c. resonance max- imum occurs for each frequency and co is the angular fre- quency of the a.c. signal. The plot showed a straight line relation by least-squares fit, the slope of which correspond to the trap depth from the conduction band edge and the inter- cept is related to the capture cross-section. Table 2 shows the summary of results obtained from the admittance spectra. The traps may be associated with either native defects or the dopants in ZnO. The observed trap depths have not varied with the concentration of Sb205 and the trap depth at 0.42 eV

Table 2 Summary of results for traps obtained from the admittance spectra

ConcentratiOn E~ (eV) Sn r (b~s) at of Sb205 ( × 10-Is crn2) 300 K (mol%)

0

0.2

2.0

0.30 2.2 0.5 0.45 5.4 0.2 0.20 1.8 0.5 0.34 2.3 0.4 0.45 2.8 013 0.20 1.2 1.0 0.36 1.3 0.8 0.45 1.8 0.5

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266 S. Ezhilvalavan, ZR.N. Kutty /Materials Chemistry and Physics 49 (1997) 258-269

gain= 1.25x102 ~ tg=1 .96 / " l O O p p m Sb205

2'50x102 0.2'% "

10 xt02 2"1, "

5x103 5'/,, ,

330~3 G )'H

Fig. 10. X-band EPR spectra (T= 25"C) of ZnO (89.5-94.5%) + Bi203 (3%) + Co304 (2.5) cermNcs containing Sb,O~ ( 100 ppm to 5 mol%).

may be associated with the transition metal ions. The density of traps have increased marginally with antimony concentra- tion remaining of the same order as Sb free samples.

3.8. Electron paramagnet ic resonance

EPR has been used to study the role of the transition metal valence state and to find the effect of antimony oxide on the native defects in ZnO. EPR is a sensitive technique for stud- ying the oxidation state of paramagnetic ions as well as the charged native defect centres such as ionized vacancies or interstitials. The detailed EPR and optical reflectance spectral studies on the stability of multivalence state of cobalt have been reported [ 19,20]. Accordingly, the prevalence of higher concentration of Co(III) at the grain boundary regions has been indicated, as compared with those in grain interiors where both Co(II) + Co(III) are present. Charge compen- sation of Co(III) by zinc vacancies renders the grain bound- ary regions more insulating. The optical reflectance spectral studies demonstrated that the black colour of the ceramics can be accounted for in terms of the simultaneous absorption in the blue-violet as well as the red region. The black colour is common in all commercial varistor samples and has not been mentioned in the literature. The black colour of varistors is a direct evidence for the multivalence of cobalt since Co(II)-doped ZnO ceramics are brilliant green under reflected light (Rinnmann's green).

EPR spectra of ZnO ceramics formulated with Sb2Os hav- ing varied concentrations up to 5 mol% are shown in Fig. 10. The g = 1.96 signal arises from the singly ionized native donor centre Vo [20,29]. Fig. 11 shows the integrated inten- sity of the signal g = 1.96 for the ZnO ceramics formulated up to 5 mol% Sb2Os which decreases with the concentration of Sb2Q. The electron paramagnedc resonance spectra of ZnO ceramics formulated with SbzOs (or Sb204) processed under identical conditions show similar change in intensity of the signal g = 1.96 with increased concentration. High valence (Sb 5 + ) and higher oxygen content of SbaOs should stabilize high valent transition metals (Mn or Co). EPR spec-

5 0 0

4OO

T~ 300

o~ 200

"5

~00

0 r o 1

I t I

2 3 4 5

% S b z O s

Fig. i t. Relative intensity of g = 1.96 signal as a function of antimony concentration for the ceramics ZnO (89,5-94,5%) + Bi,O3 (3%) + C%O4 (2,5%) containing SbzOs ( 100 ppm to 5 mol%).

tra were also recorded by intentionally adding 10 ppm of Mn as the EPR probe. The dominance of Mn(II) (d 5) in the sintered ceramics formulated from 2 mol% Sb, O~ is evident from the EPR spectra (Fig. 12). The spectra show all the five FS lines of Mn(II) with the hyperflne splitting parameter A = 70 X 10 - 4 c m - t. T h u s the EPR studies clearly show that stability ofMn(II ) is unaffected by SbzO5 whereas the higher valence state of cobalt ( Co (III)) is preserved and hence the present ceramics show higher nonlinearity. This is indicative of the fact that Sb has a very minor effect in the grain interiors whereas it makes major changes at the grain boundary regions by way of keeping cobalt in the higher oxidation state, Co(III) .

4. Discussion

The above results show the importance of formulating the ceramics, particularly with respect to the oxygen content in the additive, for attaining high nonlinear coefficients. The relevance of the pre-history of ZnO powder with respect to the concentration of donor type defects is known [20,29,30]. Composition of the formulating phases by way of high oxy-

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S. EzJ'~i[va{avan, T.R.N. Ktttty / Materials Chemistry and Physics 49 (1997) 258-269 257

Gain =8x102

[ I I [ I I

- 5 / 2 ,e-~ -3/2

[ I I

s g

P P

l V 3/2 ~ 5/2

-3/2 ,,e--> -1/2 r q l t

-1/2 ~ 1/2

M n2+(g = 2'0 014) P I I F 1/2 ~ 3/2

> H 1 3400G 125G

Fig. 12, X-band EPR spectra (T= 25°C) of the ZnO (92.5%) + Bi203 (3,0%) + Co304 (2.5%) + Sb2Q (2,0%) + 10 ppm Mn.

gen contents in the transition metal additives has also been demonstrated [ 19,31,32]. Highlights of the present obser- vations are as follows. ( l ) The concentration range of anti- mony oxide need to be around 2 tool% for high oe values ( > 5 0 ) . (2) The oxygen content of antimony oxide as an additive in the formulation has to be high for larger oe. (3) The secondary phases consist predominantly of antimony spinel with minor contents of pyochlore and &Bi203. (4) The nature and distribution (between grain boundary layers and grain interiors) of trap states arising out of the dopants have not changed with the addition of antimony ions.

Concentrations of Bi203 and Sb205 in the present ceramics are higher than those previously reported [ 1,2,4,5]. The results show that the concentration of antimony oxide is more crucial in enhancing the nonlinear coefficients. Ceramics for- mulated with antimony oxide in low concentrations ( <0.2 mol%) show that the I -V curves are less nonlinear with low values of o~. Whereas at concentrations ~- 2 tool%, the nonlinearity is maximum. The reason is that at low con- centrations, Sb ions go into the lattice increasing the grain interior resistance. Whereas, at higher concentrations, they are distributed more at the grain boundary regions and within the secondary phases. This is indicated by EDAX analyses showing the presence of Sb in the grain interior for low Sb contents. For high Sb samples, the Sb peak intensities are greater at the grain boundary regions (Fig. 2). Addition of Sb involves formation of a liquid phase and 2 mol% is suf- ficient to give liquid interaction with ZnO and other additives resulting in the recrystallization of ZnO which helps in the densification, grain growth and chemical reactions to form the varistor microstructure, including secondary phases such as the antimony spinel.

The liquid formation temperature depends on the ratio Sb2Os/Bi203 as much as on the concentration of the individ- ual additive [33]. In the present ceramics, although the Sb/ Bi ratio is <0.6, the spinel phase is predominant than the pyrochlore irrespective of the cooling rate (Fig. 3). Rapidly quenched samples preserve the high temperature phases. In

comparison, slowly cooled samples tend to attain equilibrium conditions at lower temperatures. The dominance of anti- mony spinel even in the fast cooled samples indicate that this phase is formed at 1200°C and not through the decomposition reactions as reported in the literature [ 1,2,8,33 ]. In the pres- ent ceramics, the pre-treatment of ZnO powder, defect states formed therein, concentration levels of additives and the pre- cessing conditions are crucial in stabiiizing the spinel phases, yet preserving the nonlinearity.

Addition of antimony oxide does not modify the nature and distribution of trap states, yet the performance of various antimony oxides differs, e.g. that of Sb203 in comparison with Sb2Q. These differences can either be due to higher oxygen contents in the individual additives or due to the stability of higher valence states of specific ions after sinter- ing. The results show that high nonlinearity coefficients can be achieved by using Sb205 and consequently higher oxygen content than that permitted for trivalent antimony in the for- mulations while all the other parameters, compositional and processing, have been kept constant. The results show that the oxygen stoichiometry of antimony oxide used in the for- mulation has to be high, although the respective additive phases do not survive as separate phases after sintering.

The results showed that high nonlinearity coefficients could be achieved with the additives used in the formulation having higher oxygen stoichiometry, although the respective additive oxides did not survive as separate phases after sin- tering. Accordingly, higher oxygen containing additives in situ produce oxygen in the minor liquid phase as the sintering temperature is approached. This could be explained on the basis of the varistor-forming mechanism reported in our ear- lier publications [ 18,21,30 ].

Since the presence of cobalt(III) is essential for maintain- ing high a, addition of antimony ions helps in preserving the higher valence state of cobalt(III) through the formation of the liquid phase. The molten phase formed is rich in Sb 5+, i.e. oxygen excess conditions prevail initially itself and the liquid phase interaction with the other additives of higher

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268 S. Ezhitvalavan, ZR.N. Kutty / Materials Chemistry and Physics 49 (1997) 258-269

oxygen content helps to preserve Sb 5 + state. The liquid phase interaction is more dominant when the ceramics are slow cooled from the sintering temperature and larger Po2 in the surrounding atmosphere during sintering. Preservation of nonlinearity in the quenched samples indicates that antimony ions can be stabilized in the higher oxidation state which is supported by the presence of more antimony spinel phase. However, the I - V curves are inferior with a diminishing o~. The grain interior resistance of such quenched sample has decreased marginally and modify the up-turn region which extends into higher current regions. Whereas the pre-break- down is not affected under quenching.

The difference between Sb and Co is that the former is substituted in ZnO in limited concentrations whereas Co is completely miscible. Addition of antimony ion stabilizes cobalt as Co(III ) which can be compensated through the formation of Vz,. This happens preferentially in the grain boundary regions rather than within the grain interiors because of the better access to oxygen from the surroundings through grain boundary diffusion. Higher concentration of gzn arising out of the charge compensation of Co(III ) , at the grain boundary regions than in the grain interiors enhances the resistivity of the grain boundaries. The influence of oxy- gen present at the grain to grain interface on the height of the potential barrier was demonstrated from X-ray photoelectron spectroscopy studies wherein excess oxygen is proposed to be due to the presence of Bi203 at the grain boundaries of ZnO+Bi203 ceramics [34]. However addition of Bi203 alone does not yield high nonlinear coefficients for ZnO ceramics, whereas the presence of antimony oxide together with Bi203 and Co304 helps in attaining high o~ values.

The interface states at the grain boundaries have been deter- mined using different methods such as admittance spectros- copy, isothermal capacitance transient spectroscopy, deep level transient spectroscopy and V - I deconvolution [27,35- 37] and most of the measurements are carried out for com- mercial varistors, whose methods of formulation as well as the processing details remain proprietary. In polycrystalline ZnO ceramics, the reported energy levels of trap depths are around 0.2, 0.33, 0.45 and 0.7 eV. In addition, model systems by way of thin films or single crystal junctions of ZnO have been prepared for trap depth studies [38]. Trap depths of individual transition metal dopant has been evaluated in sput- tered thin films of ZnO; e.g. Mn =0 .4 eV and Co =0.65 eV [38]. In the present set of ceramics, the observed energy levels of trap depths are 0.2, 0.36 and 0.45 eV for ZnO- Bi203-Co304-Sb205 ceramics. Thus the addition of anti- mony ions have not changed the nature of traps.

5. Conclusions

Simplified ceramic compositions can exhibit large nonlin- ear coefficients if they are doped with antimony oxide. The method of formulating the ZnO ceramics greatly influences their nonlinear electrical conductivity. A suitable combina-

tion of donor concentration in ZnO powders, oxygen excess conditions of the various additives and the controlled sinter- ing parameters, particularly the cooling rates is essential in attaining high nonlinear coefficients. The retention of excess oxygen at the grain boundaries may also be influenced by the combinations of additives. Microstructure development, for-

mation of secondary phases and the reactions between various phases during sintering of ZnO ceramics depends on the method of formulation, ratio as well as the concentration of individual additives and the processing conditions. The oxy- gen excess conditions in the additives can he maintained by the presence of electropositive elements with larger ionic radii. In such cases faster cooling rates after sintering can be adapted, without diminishing the nonlinear coefficients.

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