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Materials Letters 59 (

The relation between residual voltage ratio and microstructural

parameters of ZnO varistors

Shengtao Lia, Feng Xiea, Fuyi Liu, Jianying Lia, Mohammad A. Alimb,*

aMulti-disciplinary Materials Research Center, State Key Laboratory of Electrical Insulation and Equipment,

Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of ChinabDepartment of Electrical Engineering, Alabama A and M University, P. O. Box 297, Normal, Alabama 35762, U.S.A.

Received 27 July 2004; received in revised form 17 September 2004; accepted 3 October 2004

Available online 20 October 2004

Abstract

The effect of the sample thickness d on the residual voltage ratio Kr indicates the existence of the dimensional effect in the commercial

ZnO varistors. The empirical relations derived are based on experimental findings for Kr and breakdown electric field strength E1 mA cm�2 and

the microstructural parameter average grain size l. It is observed that Kr decreases with increasing E1 mA cm�2, while it increases with

increasing l. An integrated hybrid parameter r2l, obtained as the product of average grain size l and grain size variance r2, is found to be a

representative relation between electrical properties and microstructure of the ZnO varistors. A microstructural model is used for computer

simulation that also verified the dimensional effect in these varistors. The simulated results are consistent with the experimental results.

D 2004 Elsevier B.V. All rights reserved.

Keywords: Microstructure; Varistor; Residual voltage

1. Introduction

Zinc oxide varistors have been widely used as the

element of lightning arresters for the protection of the

electrical transmission and distribution systems [1] against

lightning and transient over voltages. They are also used for

the protection of low voltage electronic circuits. The highly

nonlinear I–V (current–voltage) behavior of the varistor

allows such a transient protection. During the past three

decades, the overall behavior of this device has been

improved via advanced ceramic processing techniques

including the type and amount of the raw ingredients used

in the recipe.

The residual voltage ratio Kr is defined as the ratio of

the discharge voltage (or electric field) at a specific current

(or current density) impulse and the dc voltage (or electric

field) at 1 mA/cm2 current density for a fixed diameter and

0167-577X/$ - see front matter D 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.matlet.2004.10.008

* Corresponding author. Tel.: +1 2563725562; fax: +1 2563725855.

E-mail addresses: sli@mail.xjtu.edu.cn (S. Li)8 lijy@mail.xjtu.edu.cn

(J. Li)8 mohammad.alim@email.aamu.edu (M.A. Alim).

length (or thickness) of the device. Usually, the discharge

voltage is considered in the high current region of the I–V

behavior. Thus, the low residual voltage ratio provides

high nonlinearity coefficient while the high residual

voltage ratio indicates low nonlinearity coefficient. The

nonlinearity coefficient a is extracted from the I–V

behavior given by the relation [2]:

a ¼ log I2=I1ð Þlog V2=V1ð Þ ¼

�log V1mA cm�2=V0:1mA cm�2ð Þ

��1; ð1Þ

where V1 mA cm�2 (=V2) and V0.1 mA cm�2 (=V1) are the

voltages corresponding to the current density 1 mA cm�2

(=I2) and 0.1 mA cm�2 (=I1), respectively.

In the development of the ZnO varistor-based gapless

lightning arresters, considerable effort was put into low-

ering the residual voltage ratio. It has been reported by

several investigators [3–5] that Al3+ or Ga3+ addition can

significantly decrease ZnO grain resistivity, resulting in

the increase of a and the decrease of residual voltage

ratio. However, excessive Al3+ or Ga3+ addition leads to

the increase of the leakage current, which is detrimental to

2005) 302–307

S. Li et al. / Materials Letters 59 (2005) 302–307 303

the stability of the device in the application field. In

addition, uniform and advanced ceramic fabrication

techniques have been developed improving both energy

handling and surge withstanding capabilities. It was

reported [6] that the nonlinearity of the I–V characteristics

is improved as the reference voltage gradient of the

element increases. Nevertheless, the eventual electrical

properties of the ZnO varistor materials are related to the

resulting microstructure.

A few recent articles [7–11] indicate that the dimensional

effect, i.e., the change in the breakdown electric field

strength with the thickness d of the ZnO varistor, originates

from the heterogeneity of the ZnO grain size distribution

and the irregularity of the ZnO grain shape. The information

gained from the dimensional effect is utilized in extending

the concept. In this paper, the influence of the sample

thickness on the residual voltage ratio Kr of the ZnO

varistors is reported. The relation between the breakdown

electric field strength E1 mA cm�2 and the residual voltage

ratio Kr incorporating the variation in the microstructural

parameter l is also investigated. A microstructural model is

proposed for computer simulation. Two relations are

simulated: first relation involves E1 mA cm�2, Kr, and d,

and the other involves grain boundary breakdown voltage

and grain resistivity. The present investigation is a part of

the continuing effort such that the total information on the

device reflects segmented protective properties pertaining to

the structural dimensions.

Fig. 1. Thickness dependence of residual voltage ratio Kr of ZnO varistors.

(E1 mA cm�2 is in the range of 205–250 V/mm due to the dimensional effect

of E1 mA cm�2).

2. Experimental

All samples used in this study were procured from a

commercial source and based on the ZnO–Bi2O3 varistor

material system. To the best of the working knowledge, it

is apprehended from the commercial source that this

varistor system includes small amount of additives

containing Bi2O3, Sb2O3, MnCO3, Cr2O3, Co2O3, NiO,

B2O3, and SiO2 to ZnO. Thus, the experimental results

and the microstructure–property relationships reflected in

this paper may be regarded as reasonable representation

of the commercial ZnO–Bi2O3-based varistor materials

having specific recipe for certain uniform fabrication

techniques.

The varistor samples have varied breakdown strength,

diameter, and thickness so as to carry out an extensive

investigation of the microstructure–property relationships.

The breakdown strength (field) ranged from 38 to 280

V/mm. The diameter ranged from 7.4 to 21 mm, while

the thickness ranged from 1 to 4.5 mm. In order to obtain

samples below 1 mm in thickness, the sample was

carefully lapped on both sides to obtain desired thickness

below 1 mm. After polishing, each sample was preferen-

tially etched in 10% concentrated hydrofluoric acid (HF)

solution. These samples were cleaned via ultrasonic

cleaning, using de-ionized water and allowing at least a

few minutes such that the tiny particles are separated from

the surfaces of the varistor samples, yielding clean

surfaces. Then these samples were dried in air. Optical

microscopy was carried out on the surfaces of both sides

of each sample. The longest and the shortest diameters of

more than 150 grains were measured for every sample

and their average value with standard deviation was

determined.

Parallel opposite faces of each sample were electroded

with fired Ag paste (as done for the commercial

samples). Each of the samples was passivated with the

epoxy resin coating to prevent flashover and surface

conduction. Also, this coating protects from moisture

absorption or penetration onto the varistor surfaces. The

varistor voltage V1 mA cm�2 (voltage corresponding to the

current density 1 mA cm�2) and impulse residual voltage

corresponding to 550 Acm�2 for the 8�20 As waveshape

were measured. The breakdown field E1 mA cm�2 is the

electrical field corresponding to a current density of 1

mA cm�2, and obtained from the V1 mA cm�2.

3. Results

3.1. Dimensional effect of residual voltage ratio Kr

The dimensional effect of the breakdown field in ZnO

varistors has already been reported in the literature [7–

11]. The residual voltage ratio of the samples having

thickness ranging between 0.5 and 4.5 mm was meas-

ured. For samples with thickness of 1.0 mm, the

experimental results corresponded to breakdown strength

of 220 V/mm. The relation between the residual voltage

ratio Kr and the reciprocal of thickness (1/d) is shown in

Fig. 1. It is interesting to observe that these two

parameters bear a fairly reasonable straight line relation-

ship indicating decrease of Kr with the increase of the

thickness d. This is a clear indication of the dimensional

Fig. 3. The experimental relation between the ratio Kr and the average grain

size l (E1 mA cm�2 is in the range of 35–280 V/mm).

S. Li et al. / Materials Letters 59 (2005) 302–307304

effect of the residual voltage ratio Kr. The empirical

relation follows as

Kr ¼ a0 þa1

d; ð2Þ

where d is the sample thickness expressed in mm. For

samples with a breakdown field of about 220 V/mm, it

was calculated: a0=1.46 and a1=0.52 mm. Such a

calculation is specific to the varistor material type.

3.2. The effect of microstructural parameters on the residual

voltage ratio Kr

Fig. 2 shows the experimental relation between the

residual voltage ratio Kr and the breakdown strength

E1 mA cm�2 provided the thickness of all samples was 1.0

mm. It is observed that Kr is reasonably inversely propor-

tional to the breakdown field E1 mA cm�2 indicating Kr

decreasing with the increasing breakdown field E1 mA cm�2.

The inset in Fig. 2 provides the fitted relationship via and

empirical equation as

Kr ¼ b0 þb1

E1mAcm�2

: ð3Þ

It can be calculated from the inset of Fig. 2 that b0=1.21

and b1=137 V/mm. Again, such a calculation is specific to

the type of varistor material.

For the samples with thickness of 1.0 mm, the relation

between the parameter Kr and the average grain size l is

shown in Fig. 3, where Kr increases with an increase in the

average grain size l. The data were fitted in the binomial

form as

Kr ¼ c0 þ c1l þ c2l2; ð4Þ

where l is expressed in Am (micrometers). It is determined

from Fig. 3 that c0=1.17, c1=6.9�10�2(Am)�1, and

c2=1.3�10�3(Am)�2. Thus, the experimental results suggest

that for the smaller average grain size l and standard

deviation r, higher breakdown field E1 mA cm�2 and smaller

Fig. 2. Dependence of breakdown strength E1mA on residual voltage ratio

Kr (E1 mA cm�2 is in the range of 35–280 V/mm).

Kr values are obtained. Therefore, the grain size and its

distribution and the grain shape play important roles in

determining the resulting electrical response of the ZnO

varistors.

The dimensional effect [9–11] of the breakdown field

E1 mA cm�2 indicated a critical thickness dc which is

proportional to the product of the average grain size l and

variance r2. In this study, it is observed that the quantity r2lis related to the parameter Kr with the microstructure of the

ZnO varistors. The relation between Kr and the logarithm of

r2l is shown in Fig. 4. The parameter Kr nearly increases

linearly with an increase of ln(r2l). It is worth noting that

the nonlinear characteristics of ZnO varistors is attributed to

the material system and fabrication process in the first

instance, thus, the dimensional effect is strongly dependent

on the material system and fabrication process.

4. Discussion

4.1. Effect of average grain size l on Kr

The excellent nonlinear I–V characteristics of ZnO

varistor originates from the double Schottky barrier often

Fig. 4. The experimental relation between the residual voltage ratio Kr and

ln(r2l) (E1 mA cm�2 is in the range of 35–280 V/mm).

Fig. 5. The microstructural model for computer simulation.

Fig. 6. (a) Simulated and experimental results of thickness dependence

of ratio Kr (E1 mA cm�2 is 220 V/mm). (b) Simulated and experimental

results of grain size dependence of ratio Kr (E1 mA cm�2 is in the range

of 35–280 V/mm). (c) Simulated and experimental results of relation

between ratio Kr and product r2l (E1 mA cm�2 is in the range of 35–

280 V/mm).

S. Li et al. / Materials Letters 59 (2005) 302–307 305

referred to as the back-to-back Schottky barrier formed at

each grain boundary. The I–V response in the ohmic

region is attributed to the thermal emission of the carriers

(electrons) crossing over the electrical barriers (Schottky

barriers), and the nonlinear region is attributed by the hole-

induced breakdown [12]. It is generally accepted that for

the ZnO–Bi2O3-based varistor materials studied here, the

breakdown voltage per grain boundary (GB) is in the

vicinity [13–17] of 3 V/GB. Such information provided

reasonably estimated barrier height [17] as 0.7 eV obtained

by using ac small-signal electrical measurements, and was

identically similar to that obtained from the I–V response

at elevated temperatures. This technique employed the

number of GBs between the two opposite electrodes in an

operative electrical path parallel to others.

The residual voltage ratio Kr is estimated as

Kr ¼V1mAcm�2 þ Vg

V1mAcm�2

¼ 1þ Vg

VmAcm�2

¼ 1þ iqlVGB

; ð5Þ

where Vg is the electric field corresponding to the ZnO grain

resistance (rZnO). q is the ZnO grain resistivity, i is the peak

impulse current density, and VGB is the average breakdown

voltage of the grain boundary. Since Kr is affected by both

the grain resistivity q and the average grain size l, an

empirical equation may be described as

DKr ¼iqVGB

Dl þ ilVGB

Dq: ð6Þ

Integrating with respect of l, it is obtained as

Kr ¼Z

iqVGB

dl þZ il

dqdl

VGB

dl: ð7Þ

Therefore, the following equation can be obtained,

Kr ¼ c0 þiqVGB

l þidqdl

2VGB

l2: ð8Þ

It is observed that the format of Eq. (8) is consistent

with that of Eq. (4). In this context, it may be noted that the

grain resistivity q is estimated accounting for the average

surface area of a ZnO grain assuming an average grain size

(grain diameter) obtained from 150 lumped ZnO grains.

This approach provided reasonable value of the grain

resistivity.

S. Li et al. / Materials Letters 59 (2005) 302–307306

4.2. Computer simulation of effect of microstructural

parameters on Kr

The proposed microstructural model of the ZnO varistor

considered for the computer simulation is shown in Fig. 5

depicting the diameter D and length (thickness) d. The

ratio Kr is found to be dependent on the sample thickness

for various diameter devices, average grain size, and r2lwhich are shown in Fig. 6. The simulated result for the 5-

mm diameter device is shown in Fig. 6a. The other

simulated results are depicted in are shown in Fig. 6b and

c. The grain size distribution in the entire ceramic body is

considered to obey normal distribution, and then the

probability density function f(l) is given by [9–11,18,19],

f lð Þ ¼ 1ffiffiffiffiffiffiffiffi2pr

p exp

�� l � lð Þ2

2r2

�; ð9Þ

where l is the grain size variable, l is the mathematical

expectation of the variable l, r is the standard deviation,

whose estimated values are obtained through sample

analysis. Assuming the grain shape is a square with a side

length of l, the number of grains N is estimated by [10]

N ¼ pD2

4l2: ð10Þ

Furthermore, assumption of the N axial grains and the

variable l for the grain size obey normal distribution. Because

the average grain number of the vertical axial grains in series

is d/l, and the variance of the variable l is r2/d/l,incorporation of the variance in the appropriate form to the

Eq. (9) provides a new arrangement of the probability

function as

f ð l Þ ¼ 1ffiffiffiffiffiffi2p

p rffiffiffiffiffiffiffiffid=l

p exp

�� ð l

P� lÞ2

2r2

d=l

�: ð11Þ

The number of grains in the column Dn(l) in the range

from l to l+Dlis then given by,

Dnð lPÞ ¼ Nf ð lPÞDlP: ð12Þ

Assuming the currents flowing through the vertical axis

do not interfere with each other, implying maintaining

parallel paths between the opposite electrodes and the

applied voltage V for the sample, satisfying the condition

VN(d/l) VGB will cause breakdown in the device. The

relation between the voltage V and the current Ii(l) flowing

through the grain column is given by

V ¼ d

lP VGB þ Iið l

PÞ d

Dnð lPÞl2q; ð13Þ

where VGB represents the breakdown voltage of a single

grain boundary electrical barrier. Considering another

situation satisfying the condition Vb(d/l)VGB the current

flowing through the grains of the vertical axis is zero as

the vertical axis grains cannot be segmented. The analysis

for the foregoing arguments is identical considering back-

to-back Schottky barrier across each grain boundary. Thus,

at the applied voltage V, the total current I is the sum of

the current flowing through N grains in that column is

estimated as

I ¼X

Iið lPÞ: ð14Þ

As the applied voltage V is increased step wise, the I–V

response and the residual voltage ratio Kr are obtained. The

simulated results are consistent with the experimental

results.

5. Conclusions

The dimensional effect of the residual voltage ratio Kr

exists in ZnO varistors implying Kr decreases with an

increase of the sample thickness d. It is empirically

obtained as

Kr ¼ a0 þa1

d;

where a0=1.46 and a1=0.52 mm are recipe and processing

specific for the device. Concurrently, it is observed that the

empirical relation between Kr and the average grain size lis given by

Kr ¼ c0 þ c1l þ c2l2;

where c0=1.17, c1=6.9�10�2(Am)�1 and c2=1.3�10�3

(Am)�2. The values for these parameters are also recipe

and processing specific for the device. However, the

general trend of these two equations for the device is

universal and independent of the constituents or raw

material types of the device. The quantity r2l is a

representation of the microstructural feature of the ZnO

varistors that relates electrical behavior via the residual

voltage ratio Kr. A microstructural model has been used in

the computer simulation yielding consistency with the

experimental results.

References

[1] T.K. Gupta, Application of zinc oxide varistors, Journal of the

American Ceramic Society 73 (7) (1990) 1817–1840.

[2] M. Matsuoka, Non-ohmic properties of zinc oxide ceramics, Japanese

Journal of Applied Physics 10 (1971) 736–746.

[3] W.G. Carlson, T.K. Gupta, Improved varistor nonlinearity via

donor impurity-doping, Journal of Applied Physics 53 (8) (1982)

5746–5753.

[4] T. Miyoshi, K. Maeda, K. Takahashi, T. Yamazaki, Effect of dopants

on the characteristics of ZnO varistors, in: L.M. Levinson (Ed.),

Advances in Ceramics: Grain-Boundary Phenomena in Electronic

Ceramics, vol. 1, American Ceramic Society, Columbus, OH, 1981,

pp. 309–315.

S. Li et al. / Materials Letters 59 (2005) 302–307 307

[5] J. Fan, R. Freer, The roles played by Ag Al dopants in controlling the

electrical properties of ZnO varistors, Journal of Applied Physics 77#9

(1995) 4800–4975.

[6] T. Imai, T. Udagawa, H. Ando, Y. Tanno, Y. Kayano, M. Kan,

Development of high gradient zinc oxide nonlinear resistors and their

application to surge arresters, IEEE Transactions on Power Delivery

13#4 (1998) 1182–1187.

[7] S. Li, F. Liu, G. Jia, Y. Ohki, Structural origin of dimensional effect on

dielectric breakdown strength of ZnO varistors, Transactions of the

IEE of Japan 116-A#12 (1996) 1146–1147.

[8] S. Li, F. Liu, G. Jia, The relation between the dimensional effect of

breakdown strength and microstructure in ZnO varistor ceramics,

Journal of Xi’an Jiaotong University 30#11 (1996) 58–62.

[9] S. Li, F. Liu, G. Jia, A study on dimensional effect of ZnO varistors by

means of statistics, Chinese Journal of Inorganic Materials 12#4

(1997) 525–530 (in Chinese).

[10] S.T. Li, J.Y. Li, F.Y. Liu, M.A. Alim, G. Chen, The dimensional effect

of breakdown field in ZnO varistors, Journal of Physics. D, Applied

Physics 35 (2002) 1884–1888.

[11] S. Li, J. Li, M.A. Alim, Structural origin of dimensional effect in ZnO

varistors, Journal of Electroceramics 11 (2003) 119–124.

[12] G.E. Pike, S.R. Kurtz, P.L. Gourley, H.R. Philipp, L.M. Levinson,

Electroluminescence in ZnO varistors: evidence for hole contributions

to the breakdown mechanism, Journal of Applied Physics 57 (12)

(1985) 5512–5518.

[13] M. Bartkowiak, G.D. Mahan, F.A. Modine, M.A. Alim, Influence of

ohmic grain boundaries in ZnO varistors, Journal of Applied Physics

79 (1) (1996) 273–281.

[14] M. Bartkowiak, G.D. Mahan, F.A. Modine, M.A. Alim, R. Lauf, A.

McMillan, Voronoi network model of ZnO varistors with different

types of grain boundaries, Journal of Applied Physics 80 (11) (1996)

6516–6522.

[15] M. Tao, Bui. Ai, O. Dorlanne, A. Loubiere, Different dsingle grain

junctionsT with a ZnO varistor, Journal of Applied Physics 61 (4)

(1987) 1562–1567.

[16] E. Olsson, G.L. Dunlop, Characterization of individual interfacial

barriers in a ZnO varistor materials, Journal of Applied Physics 66 (8)

(1989) 3666–3675.

[17] M.A. Alim, M.A. Seitz, R.W. Hirthe, Complex plane analysis of

trapping phenomena in zinc oxide based varistor grain boundaries,

Journal of Applied Physics 63 (7) (1988) 2337–2345.

[18] J. Holman, Experimental Methods for Engineers, 4th edition,

McGraw-Hill, New York, 1984.

[19] G.L. Tuve, L.C. Domholdt, Engineering Experimentation, McGraw-

Hill, New York, 1966.