polarization currents in varistors
TRANSCRIPT
Polarization currents in varistorsF. A. Modine, R. W. Major, S. I. Choi, L. B. Bergman, and M. N. Silver Citation: Journal of Applied Physics 68, 339 (1990); doi: 10.1063/1.347138 View online: http://dx.doi.org/10.1063/1.347138 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/68/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Polarization currents in zinc oxide varistors from 77 to 450 K J. Appl. Phys. 76, 7367 (1994); 10.1063/1.357960 Destruction mechanism of ZnO varistors due to high currents J. Appl. Phys. 56, 2948 (1984); 10.1063/1.333836 Longtime polarization currents in metaloxide varistors J. Appl. Phys. 47, 3177 (1976); 10.1063/1.323113 Highfrequency and highcurrent studies of metal oxide varistors J. Appl. Phys. 47, 3116 (1976); 10.1063/1.323059 HighCurrent Characteristics of Silicon Carbide Varistors J. Appl. Phys. 42, 889 (1971); 10.1063/1.1660124
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Polarization currents in varistors F. A. Modina and R. W. Majora) Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6030
S. I. Choi, L. B. 8ergman. and M. N. Silver University of North Carolina, Chapel Hill, North Carolina 27514
(Received 3 October 1989; accepted for publication 12 March 1990)
The time, voltage, and temperature dependencies of transient polarization currents are reported for two types of varistors (Le., ZnO and a SiC composite). The current transients exhibit a power-law time response to a step change in voltage (i.e., 1 z IoItm, where m is slightly less than unity) that persists over a time scale exceeding 10-8_104 s. The polarization current increa..,es linearly with low applied voltage, but at more than a linear rate for higher voltage. The temperature dependence of the polarization current in medium voltage ZnO varistors is described by an Arrhenius plot with a change of slope near 200 K, which suggests thermal activati.on energies of about 160 and 10 meV. The time dependence of the polarization currents is confirmed and extended to short times by the ac admittance measured as a function of frequency. Transient changes in the ac admittance accompany the transient polarization currents, and exhibit time and temperature dependencies that reveal their close relationship to the polarization currents. By comparing transient admittance data to predictions of the Mott-Schottky theory of a barrier, it is concluded that the theory gives an inadequate account of the ac conductance, even though the voltage dependence of the capacitance is predicted well. Theoretical explanations of the polarization currents that are based upon a distribution of exponential relaxation times are examined. A reasonable account of the polarization current is provided, but the origin of the distribution is uncertain. Possible origins are a distribution of thermal activation energies or electron hopping among randomly distributed donors.
!. INTRODUCTION
Zinc oxide varistors are electrical ceramics that are usually made by sintering a mixture of metal oxide powders. In one of its simpler formulations, the ZnO varistor consists of semiconducting ZnO grains containing dissolved Co, Mn, and Cr with intergranular phases containing Bi20 3 and Sb20 3•
1 The major constituent is ZnO, and the other constituents are present in roughly mole percent amounts. Varistors are insulators at low voltages, but they become conductors at high voltages. Varistors are used extensively for transient overvoltage suppression in electronic circuits2 and electrical power distribution systems.3
The physics of ZnO varistors has been the subject of numerous publications. Overviews of varistor properties and bibliographies are provided by review articies.4--6 Most varistor publications are concerned with electrical properties. Conduction at high applied voltages (i.e., above breakdown Voltage) and short times (e.g., microsecond response times) have received considerable attention. Less attention has been given to the currents produced by applied voltages that are less than the breakdown threshold, which are the primary subject of this article. However, Philipp and Levinson have studied the subject.7 These currents consist primarily of transient polarizati.on currents that are characterized by a power law rather than an exponential time dependence. The currents are large at short times, and they decay slowly. It may take many minutes or
alDepartment of Physics, University of Richmond, Richmond, V A 23173.
even many hours before varistor conduction reaches a steady state. The transients are typical of the non-Debye electrical response that has been observed in materials other than varistors. Review articlesil
,9 treat the general subject and provide many examples. Even though there is general interest in the phenomenon, it is poorly understood and a recognized but only occasionally studied feature of varistor conduction. Nevertheless, polarization currents influence many aspects of varistor conduction, and they are of considerable practical importance, It is necessary to recognize and to account for these currents in order to measure and to interpret varistor electrical properties correctly. The polarization currents contribute a power dissipation and louIe heating that adversely affects a varistor's stability against thermal runaway. Also, the varistor time response provides a strict test of theoretical models. Transient polarization currents cannot be ignored, and any interpretation is sure to be complicated by the non-Debye response characteristics.
This article adds to the knowledge of varistor transient currents as follows: Time- and temperature-dependent measurements are presented in Sec. III to show that lowlevel current transients exhibit a power-law time dependence over a time scale extending from about 10-6 to 105 s. The transient currents reveal a weak temperature dependence that can be characterized by an activation energy 10 $E$ 200 meV. The Arrhenius plot of discharge
339 J. Appl. Phys. 68 (1),1 July 1990 0021-8979/90/130339-08$03.00 CO) 1990 American Institute of Physics 339
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current displays a change of slope that suggests two dif~ ferent processes are present, whereas only one activation energy was previously noted,7 Added perspective is ob~ tained from a brief examination of transient currents in a new composite varistor material that contains no ZnO,
In Sec, IV, an examination is made of the connection between the transient currents and the frequency depen~ dence of the ac admittance that is provided by simple Fou~ rier theory, It is demonstrated that the admittance measurements verify and extend to shorter times the timedomain results of Sec, IlL Voltage-induced transient changes in the ac admittance measured at fixed frequencies are examined in Sec, V, When the ac admittance is measured as a function of applied de voltage, a slow approach to steady state is observed, The time and temperature dependencies of the transient admittance are shown to be similar to that of the transient polarization current. The simple Matt-Schottky description of a varistor barrier is found to give an inconsistent explanation of the transient admittance, A close connection between transient currents and the time derivative of the transient capacitance is demonstrated,
In Sec, VI, the theory of the observed transients is consi.dered, Theories invoking a distribution of time constants are found to give a reasonable description of the phenomena, The time constant distribution can be attributed to deep donors with a nearly uniform distribution of thermal activation energies, or to hopping between occupied and ionized donor sites with a distribution of separations,
U. EXPERIMENTAL PROCEDURES
Most of the experimental measurements were made on commercial low~ or medium-voltage varistors manufactured by the General Electric Co, (V12ZAl and V130L20), The characteristics of these types vary, and to a lesser extent, even varistors of the same type have varying characteristics, In general, the same varistor was used for a complete set of measurements, A few measurements were made on samples of a new varistor material lO that were provided by the Applied Pulse Corp,
The transient polarization currents were measured by essentially the same method used by Philipp and Levinson,7 except that a Kepco model OPS 1000 B programmable power supply replaced the battery, and a Hewlett-Packard mode! 3455 A digital voltmeter and a Hewlett-Packard model 9825 computer replaced the re~ corder. The computer acquired the current measurements on a logarithmic time base, Temperature was controlled with a cold gas-flow system, When very low-level currents were measured, the DVM offset currents were a problem, It was necessary to correct for these offset currents by averaging measurements of positive and negative discharge currents, Measurements at short times (i,e" less than 1 ms) utilized an EH 136A pulse generator and a Tektronix model 7854 oscilloscope,
Admittance measurements were made with a HewlettPackard model 4192 impedance analyzer. The instrument has a built-in dc bias voltage and a computer interface, It
340 J, Appl. Phys., Vol. 68, No.1, 1 July 1990
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FIG, 1. Currents induced in a medium-voltage varistor by voltages that differ by multiples of 2, Solid line: 295 K; dashed line: 77 K.
was interfaced to a Hewlett-Packard model 236 computer, which was used for data acquisition and analysis, For these measurements, the temperature was controlled with a Lake Shore Cryotronics model DTC-500 controller.
III. TRANSIENT POLARIZATION CURRENTS
Figure 1 shows charging currents in medium-voltage ZnO varistors, Polarization currents dominate initially, but the polarization currents gradually decay, and steady~state conduction i.s evident at long times, The data confirm the results of Philipp and Levinson 7 and, in addition, illustrate the voltage dependence of the phenomenon, Both the polarization and steady-state currents exhibit a linear response to applied voltages that are much less than the breakdown voltage of about 200 V, but at higher voltage the nonlinearity of varistor conduction becomes evident even in the polarization current. At room temperature and low voltages, about 103 s is required for the polarization currents to become negligible in comparison to the steadystate conduction, The polarization currents are predominant for much longer times at low temperatures, because the thermally activated steady-state conduction is greatly diminished,
Figure 2 shows the discharge currents that follow the removal of an applied voltage, At shorter times than 0,1 s, the varistor capacitance discharges at an exponential rate established by the R C time constant of the circuit, 11 but the current decreases almost inversely with time at greater than 0,1 s, If the steady-state currents are subtracted from the results shown in Fig, 1, a si.milar time dependence is seen, The polarization currents are roughly described by the power law
I =101t"', (1)
where m is nearly unity, However, this law cannot be obeyed at either short or long times because the stored charge is finite, and the integral of the current must, therefore, converge, Indeed, the slopes of the curves in Fig. 2 do
Modine et al. 340
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FIG. 2. Discharge current transients induced in a medium-voltage varifr tor at 295 K by removal of voltages that differ by factors of 2. Inset is an Arrhenius plot of the current measured at 0.1, 1, and 10 s.
increase at times greater than 100 s. Between 0.1 and 100 s the curves are nearly straight, but their slopes increase from about 0.8 to almost 1 as the voltage increases from 5 to 80 V, which indicates that highly charged varistors discharge somewhat faster. A similar increase of m with voltage has been seen in anodic oxide films.9
The discharge currents exhibit a temperature dependence as well as a time. dependence. The inset in Fig. 2 is an Arrhenius Plot of the temperature dependence of the current measured at fixed times after removing the applied voltage. The curves are fits to the data of the function
1=1\ exp( -EllkBT) + [2 exp( -EllkBT) , (2)
where II and h are time dependent. The results suggest two processes with activation energies of about 160 and 10 meV. The higher energy is close to the 160-180 meV observed in deep-level transient spectroscopy and can be interpreted as thennal emission from a deep level. l2
-14 How
ever, 10 meV is too Iowan energy to be interpreted in terms of thermal emission because the current persists too long. Deeper trapping levels are also known in ZnO varistors (e.g., 0.22, 0.32, and 0.36 eV).12-17 No evidence of these levels is seen in Fig. 2, but if present in sufficient numbers, they might be revealed by measurements at higher temperature.
Philipp and Levinson interpret the temperature dependence of the polarization currents in medium-voltage GE varistors in terms of a single 35-meV activation energy.7 However, the carefully measured data of Fig. 2 cannot be so interpreted. We conclude that an Arrhenius plot of the data of Philipp and Levinson are consistent with a change of slope, but there are too few data points to identify two distinct slopes. In fact, if the data points between room and liquid-nitrogen temperature are neglected in the inset of Fig. 2, a single activation energy of 34±3 meV provides a reasonably good fit to the remaining data.
Evidence that a step voltage change induces a powerlaw time response in the varistor current for 10-7
.;;; t
341 J. Appl. Phys., Vol. 68, No. '\,1 July 1990
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FIG. 3. Discharge current transients that follow application of a voltage pulse to a low-voltage varistor at 295 and 420 K. The inset is an Arrhenius plot of the current measured at 3 X 10-4 and 4 X 10-5 S after the pulse .
.;;; 10-3 s is provided by Fig. 3, which shows the current decay in a low-voltage ZnO varistor after a fast voltage pulse is applied. A pulse voltage below breakdown was employed. Initially, the discharge exhibits the exponential time dependence expected of a circuit with an RC time constant of about 10-7 s. Evidence of exponential time dependence can be seen in Fig. 3 for t < 10-6 s. After the initial discharge of the varistor capacitance, the current exhibits a r- m power-law decay with m close to unity. Once again, a weak temperature dependence is found. An Arrhenius plot of the current measured at fixed times is shown in the inset of Fig. 3. To the extent that an activation energy can be identified from the plot, a value of about 38 me V is obtained.
Additional insight and some appreciation for the generality of the polarization transients is provided by a different varistor material. Figures 4 and 5 show transient polarization currents in a recently developed composite
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FIG. 4. Charging current transients induced in a composite varistor material at 295 K by applied voltages that differ by iactors of 2.
Modina et al. 341
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FIG. 5. Discharge current transients in a c{)mposite varistor material at 295 K that follow removal of applied voltages differing by factors of 2.
varistor that contains nickel metal, silicon dioxide, and silicon carbide in a silicon rubber binder. 10 The conduction mechanism at high current is believed to be different from that of ZnO varistors, but the new material exhibits remarkably similar transient polarization currents. Log-log plots of discharge current versus time yield straighter lines than found for ZnO varistors. The slope of these lines increases from about 0.76 to 0.93 as the charging voltage increases from 25 to 400 V.
The increase of the polarization current with the voltage is not truly linear, because the slope increases with the applied voltage. Instead, there is a different linear relation. To the extent that it can be established by integrating discharge currents from different thicknesses of the composite material when different voltages are applied, the stored charge is proportional to voltage and independent of thickness. Hence, if the same voltage is applied to pieces of the material with different thicknesses, the same amount of charge is stored, but the thinnest piece exhibits the fastest rate of discharge. This result probably applies to ZnO varistors as well, but the thickness independence of the stored charge was not checked.
IV. FREQUENCY RESPONSE
The frequency dependence of the varistor admittance can be used to verify and to extend the time-domain resuIts. Fourier theory relates the time response of a varistor to its frequency response. The admittance as a function of frequency is simply the ratio of the Fourier transforms of current and voltage that are measured as a function of time. Since a step change in the voltage (Le., V = Vo for t> 0) induces a polarization current that obeys a powerlaw time dependence [see Eq. (1)J, the admittance Y( w) is easily calculated:
Y(w) =I(w)IV(w) = UoIVo)r(l-m) Uw)m=KUw)m, (3)
342 J, Appl. Phys., Vol. S8, No.1, i July 1990
FREQUENCY (Hz)
FIG. 6. Conductance and susceptance va frequency for a low-voltage varistor at 295 K.
where j = (- 1) 112, Vo is the magnitude of the voltage step, r 0-m) is the gamma function of argument ( 1-m), and K is a constant. The exponent m is a little less than unity, and since r = exp (j1TmI2) ,
Y=G + jwC=Kwm[cos( 1TmI2) + j sine 1Tm/2)]. (4)
Consequently, the polarization currents imply that the conductance G and the susceptance (vC should exhibit the same power-law frequency dependence and, therefore, have a constant ratio. Figure 6 illustrates that this frequency dependence is measured in ZnO varistors. The slope of the curve is m::::;O.93, which is in good agreement with the time-domain results. The results reveal a constant phase shift S of the current relative to the applied voltage (i.e., a constant dissipation factor and loss tangent):
8=m1T/2=tan -l(wCIG) =tan- 1C liD). (5)
In actuality, the dissipation factor is not quite constant for varistors, but neither does it exhibit the strong frequency dependence (i.e., a decade change per decade of frequency) that is typical of electrical circuits with lumped parameters.
The admittance of varistors has been studied by others,16-19 and it is not a primary subject of this article. However, while the connection between the time response and the frequency response is well known, the implications of this connection for varistors have not been emphasized. Moreover, the frequency response verifies that the r m
power law for the current holds for times less than implied by the time-dependent data. In fact, there is a simple connection between the time and the frequency response.20 A rm power law is implied by the frequency response data for time.<; as short as to- 8 s, and this power law could hold at still shorter times.
V. TRANSIENT ac ADMITTANCE
The application or removal of a dc voltage induces transients in the ac admittance which show behavior similar to that of polarizati.on currents. These admittance transients provide further insight into the nature and origin of
Modine et al. 342
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the polarization currents as wen as other aspects of varistor physics. Figure 7 shows changes in the capacitance and dissipation factor at 10 kHz that follow removal of an applied voltage. The changes in C and D as a function of time and temperature show the slow response and the increased rate of change at higher temperatures that is seen in the polarization currents.
The varistor capacitance measured as a function of voltage yields information on barrier properties. For example, the height and width of the barriers and the spatial and energy distributions of donors and interface traps have been inferred from such measurements. 21
-27 It is surprising
that little if any notice has been made of the slowness with which the measurements approach a steady state. Figure 8 shows hysteresis loops which illustrate the slow changes in the capacitance and dissipation that are induced by the application of a voltage. As with measurements of J- V characteristics at low voltages, many hours are required to measure the steady-state capacitance as a function of voltage.
The transient ac admittance can be used to examine the simple Mott-Schottky description of a varistor in which the conductance G=mCD and the capacitance Care determined by the height Vb and width w, respectively, of a parabolic barrier4,25-28:
(6)
and
(7)
343 J. Appl. Phys., Vol. 68, No.1, 1 July i 990
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where e is the electron charge, k8 is the Boltzmann constant, Jo = AT2
, A is the Richardson constant, E is the dielectric function, and a unit area is assumed. A uniform donor density N d is assumed, and the height and width of the barrier are uniquely determined by the charge Q per unit area in deep acceptors at the grain-boundary interface:
(8)
and
w = QI eN d= woQI QQ, (9)
where the subscript 0 denotes an equilibrium value in the absence of an applied potential.
Within the Mott-Schottky model, the transient admittance is a result of the interface charge changing with time in response to the applied voltage. The interface charge increases with voltage, out responds slowly because it is deeply trapped. 26
,27 (The influence of minority carriers generated by impact ionization26-29 can be neglected because only lOW voltages are considered.) The model can be checked without detailed knowledge of the interface charge by eliminating Q from Eqs, (6)-( 9) to obtain a relationship between C and G:
1n(GoIG) = (V ooIkBT) [(ColC)2-1]. (lOa)
However, because the changes in C and G are small, a simpler first-order approximation can be used:
(lOb)
Modine €It al. 343
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FIG. 9. Fractional change in conductance vs fractional change in capacitance following the removal of an applied potential of 10 V from a low-voltage varistor. Measurements made at 295 K and several frequencies are shown. The inset shows measurements made at 10 kHz and three different temperatures.
The result implies that V bO can be easily determined from the transient admittance.
The transient admittance induced by removing a potential is plotted according to Eq. (lOb) in Fig. 9 for several frequencies. The results qualitatively conform to the equation, although a frequency dispersion that is not predicted by the theory is evident. The slopes of the curves imply V bO = 30 ± 12 meV, with the highest frequency of 107 Hz corresponding to about 30 me V and the lowest frequency of 102 corresponding to about 18 me V. The 104-Hz data are shown for three temperatures in the inset of the figure, and values of V bO = 28 ± 3 me V are implied. Clearly, a discrepancy exists because values of V bO between 0.5 and 1.5 eV are found from I-V characteristics measured as a function of temperature30 or from capacitance measured as a function ofvoltage.21
-25 Thus it is concluded
that the simple Mott-Schottky description gives an inadequate account of the ac admittance of a varistor barrier. Since the Mott-Schottky theory gives a good account of the voltage dependence of the capacitance,2i-27 the problem must be with the conductance. Apparently, the ac conduction is not thermally activated over the barrier, but is contributed by a separate path. Hence Fig, 9 shows only that the voltage-induced changes in C and G are proportional and weakly dependent upon frequency and temperature.
A connection between the transient capaci.tance and the polarization currents is provided by the equation
de (don €) I = Va dt = VoC dt
dOn w)) dt '
(11)
where Vo is the applied potential, and € and ware the dielectric function and width, respectively, of the depletion region. This equation implies that either the dielectric function or the width of the depletion layer at the grain
344 J. Appl. Phys., Vol. 66, No.1, 1 July 1990
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boundary can change. A change in the width can result from a change in the charge trapped either at the interface or in deep donors.
Figure 10 shows a plot of dC/dt versus time together with the discharge current transient that is measured when the same applied voltage is removed. Both curves exhibit a slope that is near to - 1 and increasing in magnitude with time. The measured current is about an order of magnitude larger than calculated from Eq. (11). However, the capacitance exhibits dispersion, and its low-frequency value is appropriate for Eq. (11), whereas a smaller 1O-kHz value was used for Fig, 10. An Arrhenius plot of dC/dt is shown in the inset to the figure. The plot suggests an activation energy of about 180 meV. The result is dose to the 160-meV value found from current measurements shown in Fig, 2, and it agrees with the 160-180 meV that is observed in deep-level transient spectroscopy. 12-14
VI. THEORETICAL INTERPRETATION
A non-Debye electrical response is seen in many materials, and theoretical interpretations have been developed.8
,9 The phenomena are usually attributed to a distribution of Debye (i.e., exponential) relaxations with different time constants. The relaxations can be associated with either the reorientation of dipoles or the dynamics of charge trapping.8 Application of a voltage to a dielectric with a distribution of Debye relaxations induces a current that is a superposition of exponential decays:
dE I=CgVo dt
I (exp( -th)) G(T) dT,
r (12)
where Cg is the capacitance in the absence of a dielectric, €s and € 00 are the static and high-frequency dielectric constants, and G( .. ) is a distribution of relaxation times. [In
Modineetal. 344
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Eq. (12), any change in the barrier width can be considered to be a change in the effective dielectric constant.] It is well known that a suitable distribution of relaxation times is obtained if the time constant is thermally activated:
(13)
and a uniform distribution of thermal activation energies is assumed:
dE _lknT G(T)=N(E) -d =(Emax-Emin) -, (14)
T 1"
where Emu - Emin is the width of the distribution. From Eqs. (12) and (14),
1 = CgVoknT[ (es-€", )/(Em,ax -Emin)
X [exp( -t17 max) -exp( - t/r min} )It, (15a)
(15b)
Thus a weak temperature dependence and a roughly correct time dependence for the current are obtained. A uniform distribution of activation energies that is 0.7 eV wide leads to 12 orders of magnitude difference in rmin and r max' which would be the minimum required to explain varistor transients. Presumably, a departure of the measurements from Eq. (15) (e.g., a recognizable activation energy) can be explained by a distribution of activation energies that is not completely uniform. Trapping levels have been observed in ZnO varistors at approximately 0,17,0.22,0.32,0.36,0.5, and 0.7 eV. l2
-17 When interac
tions between the traps are taken into account, the result may resemble a continuous distribution.
Although it remains unproven, the existence of a wide, approximately uniform distribution of thermal activation energies is often an ad hoc assumption. Clearly, the essential requirement is a distribution of relaxation times that varies almost inversely with r, and there are other explanations for this distribution that rely on charge hopping or tunneling rather than thermal activation. For example, electron hopping among the donors surrounding deep acceptors has been proposed as an explanation of non-Debye phenomena in silicon.3l It 1s easy to extend the proposal to varistors because varistor barriers are just deep acceptors that are localized in a planar geometry and compensated by ionized donors. The oppositely charged donor and acceptor produce a dipole moment that can reorient or change magnitude when an electron hops from a neutral to an ionized donor. Pollak and GebaHe31 calculated the distribution hopping times for random donor separations, and good agreement was obtained with the noninteger power law [i.e., m=0.8 and G( r) - r- m
] exhibited by their measurements on silicon. Unfortunately, the calculation predicts a response that varies inversely with temperature, which is not observed for either silicon or varistors. Moreover, the theory may not apply at very long times.
345 J. Appl. Phys., Vol. 68, No.1, 1 July 1990
VII. CONCLUDING REMARKS
This paper has described many of the characteristics of polarization currents in varistors along with the close connection of these currents to time- and frequency-dependent variations in the ac admittance. These phenomena (i.e., non-Debye electrical properties) are not restricted to varistors, and this paper describes just one more example of this poorly understood but common property of solids. Varistors may not provide the best means to a better theoretical understanding of non-Debye phenomena, but they surely add some useful insights. Because nearly all the potential drop is at the grain-boundary barriers in a varistor, the non-Debye phenomena must originate in this region which has a roughly constant potential gradient. Moreover, the thinness of the depletion region gives rise to a capacitance that is sensitive to the barrier charge and easy to measure.
It is emphasized again that the polarization currents in varistors are the major source of ac power dissipation in a varistor under quiescent operating conditions, and they substantially reduce a varistors stability against thermal runaway. Moreover, polarization currents must be taken into careful account when varistors are characterized. Accurate current versus voltage or capacitance versus voltage can only be measured when sufficient time is allowed to stabilize the polarization currents. This is particularly crucial when measurements are made at low voltages andlor low temperatures. If this is not done, erroneous values for barrier heights and donor densities may be deduced from the measurements. Furthermore, polarization current transients are not generally small or slowly varying. Given their nearly inverse time dependence, the slowly changing current of 10-1 A measured at 1 s in a medium-voltage varistor was about 10 A and changing rapidly at 10-8 s. Also, a faster than linear increase in the polarization CUf
rent with voltage was noted. This increase presumably originates with a nonlinear change in the barrier height with voltage.
The ac admittance provides a valuable complementary insight into the non-Debye phenomena. The connection between the transient time response of the current and the admittance is provided by well-known linear response theory, but it has received little mention in the varistor literature, where admittance studies are more numerous than the studies of polarization currents, The additional studies of the time-dependent change in the admittance that accompanies the charging and discharging currents provides a direct connection of the polarization currents with changing barrier properties that is original with this work. Moreover, with our equipment, it is somewhat easier to measure the transient capacitance than it is to measure low-level polarization currents.
A reasonable description of the polarization currents in varistors is provided by theories invoking a distribution of exponential relaxation times. However, the origin of such a distribution is unknown. The observed change of slope near 200 K of the Arrhenius plot of discharge current may reveal a change in the discharge mechanism.
Modlne et al. 345
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ACKNOWLEDGMENTS
This research was sponsored by the U.S. Department of Energy, Office of Energy Storage and Distribution under Contract DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc., and by the Army Research Office under Contract No. DAAL 03-88-K-0028.
1M. Matsuoka, Jpn. J. App!. Phys. 10,736 (1971). 2S. A. Jerabek and S. J. Cardella, Eds., Transient Voltage Suppression
(General Electric, Liverpool, NY, 1986). 3E. C. Sakshang, J. S. Kresge, and S. A. Miske, IEEE Trans. Power Appar. Syst. PAS·96, 647 (1977).
41. D. Levine, Crit. Rev. Solid State Sci. 5, 597 (1975). 5 L. M. Levinson and H. R. Philipp, Am. Ceram. Soc. Bull. 65, 639
(1986). 6R, Einzinger, Annn. Rev. Mater. Sci. 17, 299 (1987). 7H. R.. Philipp and L. M. Levinson, J. Appl. Phys. 47, 3177 (1976). 8 A. K. Jonscher, Dielectric Relaxation in Solids (Chelsea Dielectrics, London, 1983).
9D. M. Smyth, in Oxide and Oxide Films, edited by J. W. Diggle (De-kker, New York, 1973).
JoF. A. Modine and H. M. Hyatt, 1. Appl. Phys. 64, 4229 (1988). II F. A. Modine and R. B. Wheeler, J. App!. Phys. 61, 3093 (1987). 12J. F. Cordaro, Y. Shim, and J. E. May, J. AppL Phys. 60, 4186 (1986). 13 A. Nitayama, H. Saki, and T. Ikoma, Jpn. J. App!. Phys. 19, L743
346 J. Appl. Phys., Vol. 68. No.1, 1 July 1990
(1980). 14M. Rossinelli, G. Blatter, and F. Greater, Electrical Ceramics, edited by
B. C. R Steele (British Ceramic Society, Stoke-on-Trent, England, 1985).
15N. Shohata, T. Matsamuro, and T. Ohno, Jpn. J. Appl. Phys. 19, 1793 ( 1980).
I"L. M. Levinson and H. R. Philipp, 1. Appl. Phys. 47,1117 (1976). 17M. A. Alim, M. A. Seitz, and R. W. Hirthe, J. Appl. Phys. 63, 2337
(1987). 18M. A. Alim, M. A. Seitz, and R. W. Hirthe, J. Am. Ceram. Soc. 71,
C-52 (1988). !9M. A. Alim and M. A. Seitz, 1. Am. Ceram. Soc. 71, C-246 (1988). lOB, V. Hamon, Proc. lEE (London) 99, 151 (1952). 21p. R. Emtage, J. App!. Phys. 48, 4372 (1977). 22W. G. Morris, J. Vac. Sci. Technol. 13,926 (1976). BK, Mukae, K. Tsuda, and I. Nagasawa, J. App!. Phys. 50, 4475 (1979). 24p. L. Hower and T. K. Gupta, J. App!. Phys. 50, 4847 (1979). 25 A. M. Prudenziati, A. Masoero, and A. M. Rletta, J. App!. PhYI!. 58,
345 (1985). 2bG. E. Pike, Phys. Rev . .B 30,795 (1984). 21 G. E. Pike, Mater. Res. Soc. Symp. Proc. 5, 369 (1982). 28 G. E. Pike, S. R. Kurtz, P. L. Gourley, H. R. Philipp, and L. M.
Levinson, J. App!. Phys. 57, 5512 (1985). 29 0. D. Mahan, L. M. Levinson, and H. R. Philipp, J. App/. Phys. 50,
2799 (1979). 30H. R. Philipp and L. M. Levinson, I. Appl. Phys. SO, 383 (1979). 31M. Pollak and T. H. Geba\le, Phys. Rev. 122, 1742 (1961).
Modina et al. 346
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