ELECTRICALCONDUCTION BY PERCOLATION FILMpok andwherewehave implicity assumedthat factor in Eq. (3)is...

Electrocomponent Science and Technology1976, Vol. 3, pp. 77-83

(C)Gordon and Breach Science Publishers Ltd., 1976Printed in Great Britain

ELECTRICAL CONDUCTION BY PERCOLATION IN THICKFILM RESISTORS

F. FORLANI

Magneti Marelli, Divisione Electtronica FIVRE, 27100 PA VIA, ltaly

and

M. PRUDENZIATI

lstituto di Fisica, Universitd di Modena, 41100 MODENA, Italy

(Received May 2, 19 76)

Thick film resistors are widely used in microelectronic devices, however the mechanism of electrical conduction inthese resistors has not yet been fully understood. In particular the anomalous behaviour of the temperaturecoefficient of resistance (T.C.R.) vs. temperature for a purely ohmic resistor has not been explained. The anomalyis that the T.C.R. is negative at low temperatures, is zero around room temperature and becomes positive at highertemperatures.

This paper demonstrates that the electrical conduction mechanism in thick film resistors can be described by theelectron percolation theory already proposed to explain charge transport in amorphous semiconductors. The thickfilm structure consists of conductive grains with a diameter of 0.1 m to 0.3 m separated by dielectric layers.Some of the conductive grains make contact tlarough dielectric layers so thin that electrons are able to tunnelthrough the layers. The critical percolation path is through these grains. Experimental evidence is given whichconfirms that the resistance vs. temperature characteristics satisfactorily fit the conduction equation provided bythe percolation theory.

The T.C.R. anomaly can be explained in the framework of this theory. The decay length of the electron wave-function is shown to be lower than 370 A for a density of conductive grains in the film in the order of101 cm-3. Such a value is consistent with electron tunnelling through layers about 100 A thick.

INTRODUCTION

The use of thick film technology to fabricateresistors in hybrid circuits is well established.Nevertheless until now the physical modelsproposed to describe the electrical conductionmechanism in thick film resistors-9 are un-convincing.

It is required to explain how a purely ohmicresistor can exhibit an anomalous behaviour of thetemperature coefficient of the resistance (T.C.R.)vs. temperature: as shown in Figure 1, where thenormalized conductivity is plotted vs. temperature,the T.C.R. of a thick film resistor is negative at lowtemperatures, it becomes zero around room tempera-ture and turns positive at higher temperatures.

The first attempts to explain this anomaly2,3were based on the idea that conclusions about theconduction mechanisms in transition oxides could beextended to thick film resistors. With this approachthe T.C.R. anomaly found a qualitative interpreta-tion. However the model failed to provide a

77

quantitative fitting to the experimental data onPdO/Ag based resistors. A satisfactory fitting ofthose experimental data was obtained by Kahan4

whose approach consisted in completing the modelelaborated by Brady and based on the conductivitycharacteristics of semiconductive oxide grains. Bradyassumed the oxide grains to be submitted to apressure applied by the glassy matrix and dependentupon the temperature. Kahan also took into accountthe contribution to the current flow due to metallicgrains. One of us6 has already pointed out thatBrady-Kahan model was based on six independentassumptions with no possibility for these to bechecked independently. Moreover it was shown thatEqs. and 2 for conductivity fitted the experimentaldata used by Kahan very well:

o c exp (1)

at low temperatures, and

Eo o T-n exp (2)

kT

78 F. FORLANI AND M. PRUDENZIATI

1,0000_

0,99

0,98_

0,;_

0,9_

o,gs_

o,4_

o,e_

o,e2_

0,91_

o,o_

50

FIGURE 1

’oo zoo :zoo 400 (* K)

Normalized conductivity of a typical thick-film resistor vs. temperature.

at higher temperatures. In Eqs. (1) and (2), To is aconstant with dimensions of temperature and n is apure number of order 1/3.

It was also noted7 that Eqs. (1)and (2)provide agood fit for experimental data concerning Ir-RuO2based resistors.

Hill8 has confirmed the validity of Eq. (1) to fit,at low temperatures, data on Bi2 Ru2 07 basedresistors and concluded that electrical conduction inthick film resistors can be attributed to hopping ofcharge carriers between states localized in a verynarrow band. However, as already reported,9 thevalues of To deducible from the experimental dataare much lower than those expected by charge-carrier hopping mechanisms.

In the present paper we consider a proposal,already put forward, i.e. the electrical conductionin thick film resistors can be described by electron

percolation, according to the theory initially pro-posed by Miller and Abrahams 0 and used byAmbegaokar, Halperin and Langer 11 to interpretconduction phenomena observed in disorderedsystems.

2 ADAPTATION OF THE PERCOLATIONMODEL FOR THICK FILM RESISTORS

The percolation model assumes that the resistivematerial can be considered equivalent to a resistancenetwork so that its over-all electrical conductivitycan be written as

(3)

where Gc is some characteristic value of the conduc-

CONDUCTION IN THICK FILM RESISTORS 79

tances in the network and r/is some characteristiclength scale for the network. Let Gi be the conduc-tance between any two directly interacting sites and], where more resistors of the assumed network jointogether. Even though the explicit dependence of Gijon the material parameters varies according to themechanism for the electron exchange between sitesand ], it can be asserted that for a network, where thevalues of individual resistances vary over many ordersof magnitude, Gc in Eq. (3) represents the criticalpercolation conductance. In order to have a betterunderstanding of the significance of Gc consider theresistance network as composed of three parts: 11

i) a set of isolated regions of high conductivity,each region consisting of a group of sites linkedtogether by conductances with Gij >> Gc;

ii) a relatively small number of resistors with Gi]of order Gc connecting together a subset of highconductance regions as defined in (i);

iii) the remaining resistors with Gi] " Gc.A thick film resistor can be represented by a resistancenetwork with the characteristics described above. Athick film resistor is obtained through the firing ofspecially prepared screenable compositions attemperatures ranging from 750 C to 1150 C. Suchcompositions are made of three main constituents:a conductive ingredient (generally, one or moreconductive oxides as PdO, RuO2, NbOs, etc.), aglassy content (generally a mixing of PbO, SiO2,B 03, P Os, etc.) and an organic vehicle (butyl-carbitol acetate, ethylcellulose, etc.). This last givesto the composition the thixotropy necessary forprinting on to a ceramic substrate (generally,96% A1203) through a metal screen and is almostcompletely eliminated during the drying of thescreened film before firing. The glassy content, onthe contrary, remains in the film after firing andprovides the adhesion of the film itself to theceramic substrate. Even though after a correct firingthe glass is mainly located at the interface betweenthe film and the ceramic substrate, the as-fired filmconsists of a dense dispersion of small conductivegrains (with diameters of 0.1/am to 0.3/am8) in aglassy matrix.

With this physical structure, a thick film resistorcan be considered as consisting of: (1) a set ofisolated regions with Gi] Gc, corresponding bothto simple conductive grains and to groups of conduc-tive grains in purely ohmic contact; (2) a set ofconductive grains in electrical contact throughcontact resistances having a non-zero value and

consisting of resistors with Gi] of the order of Gc.These form, together with at least a part of the firstset, a critical subnetwork shorting out those resistorswith Gij " Gc which correspond to higher contactresistances between conductive grains.

The model we assume for the conduction in thick-t’rims is, first of all, based on the following considera-tions: the conductances of order Gc determine anddominate the resistance of the network electricallyequivalent to the thick-film resistor. This means thatwhether the conductive grains exhibit metallic orsemiconductive conduction mechanisms becomesunimportant. In order to elucidate Gc, it is assumedthat conduction is due to electron hopping betweenthe percolation sites because of tunnelling effectthrough the potential barriers provided by the glassinto which the conductive grains are embedded.

Using, for Gc, the same expression given byAmbegaokar et al. 11 for the conduction inamorphous materials, even though in this case thepercolation sites are made of conductive grainsinstead of electron traps, we have

e2

Gc=--f Fc (4)where e is the electron charge and Fc is the averagetransition rate from sites to sites along the criticalpercolation path. It is given by

(4t)cOt3) x/4rc 3’0 exp- p- (5)where 3’0 is a constant depending upon the electronphonon coupling strength, the phonon density ofstates and other elastic properties of the material, Vcis a dimensionless constant,/90 is the density ofpercolation sites per unit volune and unit energy anda- is a length whose physical meaning is associatedwith the tunnelling mechanism from one percolationsite to the other, and therefore to the averagediameters of the conductive grains along thepercolation path. In other words, the meaning ofa- is the basic difference between the conductionmodel by electron percolation applied to amorphoussemiconductors and to thick films. While in amor-

-1phous semiconductors we must expect a value of aof the order of a few A, in thick films its value mustbe comparable with the diameter of the conductivegrains, i.e. some 100 A. By assuming the validity ofEq. (5) in the case of thick film resistors, we considerthe charge on a single grain concentrated at the centreof the grain and the separation between percolationsites a bit larger than the grain diameter.


Thus, from Eqs. (3), (4) and (5), the conductivityof a thick-film resistor, is given by

o oc exp (6)

where

e27o/3=

k(7)

4Pc3To (8)pok

and where we have implicity assumed that factorin Eq. (3) is a relatively slowly varying function of the

parameters of the network and that the dominantvariation of o is contained in Go.

It can be seen that the resistance, R, of a thick-film resistor and associated T.C.R. is given by

R : - expT.C.R. 1--R dT 7’ (9)(10)and T.C.R. becomes zero for a temperature such thatTmin 25---- (11)

-" 100 K./O.oO

o’

-go;’c}

FIGURE 2 Matching between theory and experiments for thick-film resistors with the same aspect ratio and withdifferent sheet resistivities.


This temperature corresponds to the miniznum valueacquired by the normalized resistance, R/Rmin, vs.temperature as shown in Figure 3.

3 EXPERIMENTAL EVIDENCES

A set of measurements was performed on Bi2Ru207based thick film resistors. The connections to theseresistors were provided by screen and fired PtAu-based conduction films.

All the resistors had a constant width of 2 mm,they were printed at the same nominal thickness andfired under the same temperature profile.

TABLEComparison to check the consistency of percolation theory

applied to thick-films

TminSheet resistivity (= To/256) (Tmin)experimental(a/sq) (c) (c)

1,000 59.4 6010,000 67.9 70

100,000 71.2 75

In Figure 2 In R/T is plotted vs. (l/T) 1/4 fordifferent sheet resisfivifies. In Figure 3 the plot ofR/Rmin vs. T is shown.

In Table the values of Tmin calculated using Eq.(11) from the values of To obtained from the slopesof the straight lines in Figure 2 and the experimentallyderived values of Zmin are compared; the agreementis good.

4 DISCUSSION

The behaviour of T.C.R. with temperatureexpressed by Eq. (10) indicates a negative infinityin the temperature coefficient as the absolutetemperature tends to zero and a null temperaturecoefficient for the absolute temperature going toinfinity. This implies that the T.C.R. must exhibita maximum for some temperature larger thanTo/256. An evaluation of the derivative of Eq. (10)indicates that such a maximum is to be found at atemperature of To/(3.2)4. In practice, it means that aresistor having a null temperature coefficient at roomtemperature (300K), and consequently a value ofTo 76800K (see Eq. (11)), should exhibit amaximum T.C.R. at 459 C, i.e. at such a high

%0000

1-7 .50 25 O" 25" 50" 75* 100 12

FIGURE 3 Plot of normalized resistance vs. temperature.


temperature that it cannot be checked by experi-ments. The existence of a maximum in the T.C.R. ata temperature higher than T0/256 explains the lowergradient observed in the range of the high tempera-tures. On the other hand, the derivative indicates thatfbr T < T0/(3.2)4 we have d(T.C.R.)/dT > 0, asconfirmed by experimental evidence.

For a value of To of 76800K we obtain from Eq.(8)

a3 6.62(eV) /904pc

and for Pc 4

a3 4.138 x 10-1po

(12)

(13)We can evaluate Oo from the experimental measure-ments on the diameter of the conductive grains in thefilms reported by Hill8 for Bi2Ru207 based resistors.The reported diameters of 0.1/am to 0.3/m corre-spond to a grain density, n, of 1014 cm-3 to101 cm-3. As already pointed out, 11 the graindensity is related to the sites per unit volume andper unit energy, Oo, by

n 2poEmax (14)

where Emax is the maximum energy distance fromthe Fermi energy, within which the electron densityof states has to be constant. A more recent analysisperformed by Pollak et al. has demonstrated thatsites with an energy distance far away from the Fermilevel contribute in a negligible way to the chargetransport at the percolation path. In other wordsEq. (14) must be written as follows

n < 2PoEmax (15)

Then, the maximum energy distance from the Fermilevel can be expressed by1

70 (16)Emax =kTln Pcand from Eq. (5)

Emax kT 3/4 To1/4 (17)At room temperature for the resistor under consider-ation (with T.C.R. 0 at 300K) we obtain Emax0.103 eV. Then for a density of conductive grains of1014 cm -3 to 10 is cm-3:

Po > 4.85n cm -3 eV -1 (18)where n is 1014 cm -3 to 10 is cm -3 and from Eqs.(12) and (13)

a -1 < (370 to 790) A (19)

Such a length is quite consistent with the discussedmeaning of a- and with a diameter of the conductive-grains on the order of 1000 A to 3000 A.

Finally, it should be noted that the three straightlines of Figure 2 are almost parallel. The values of Toare only weakly dependent upon the sheet resistivityof the film (see Table I). The approach discussed hereand based on a conduction mechanism by electronpercolation does not allow us to determine theproportionality factor in Eq. (9).

5 CONCLUSIONS

The electron percolation theory as proposed byAmbegaokar et al. 11 provides a very simple andcredible interpretation of the conduction mechanismin a thick-film resistor. The apparently anomalousT.C.R. vs. temperature behaviour finds an immediateexplanation in the frame of this theory. It has beenchecked that the matching between theory andexperiments holds also for resistor compositionsother than BiRuzO7 based ones as the extendedseries of tests was limited to Bi2Ru=O7 based resistorsdue to their stability and reproducibility.

It is not surprising that a theory dealing withdisordered systems can be applied to thick-filmstructures. Moreover, further investigations, stillunderway, indicate that the assumption of a sitedensity, Po, constant through the film and express-ing an average order in the disordered configuration,is only a first order approximation.

At least in the range of high sheet resistivity, theT.C.R.’s are dependent upon resistor aspect ratios andsuch a dependence might find an explanation in theframe of electron percolation theory by introducingsome unhomogeneity in the site density. However,more work is necessary to confirm these preliminaryobservations.

ACKNOWLEDGEMENT

The authors wish to thank A. Cattaneo for the preparation ofthe films and for the measurements of the conductioncharacteristics.

REFERENCES

1. C. A. Harper, Handbook of Thick-Film Hybrid Micro-electronics, McGraw-Hill, N.Y. (1974).

2. E. H. Melan and A. H. Mones, "The glaze resistor: itsstructure and reliability", Proc. IEEE Electron. Compon.Confer., Washington D.C. (1964).


3. E. H. Melan, "Stability of Palladium Oxide resistiveglaze films", Microelectronics & Reliability, 6, 53 (1967).

4. G. J. Kahan, "Equivalent circuit for conductivity-temperature characteristics of PdO/Ag--Pd glazeresistors",IBMJ. Res. & Dev., 15,313 (1971).

5. L. J. Brady, "Mechanism of conduction in thick-filmcermet resistors", Proc. 1EEE Electron. Compon.Confer. lashington D.C. (1967).

6. F. l:orlani, "Meccanismi di conduzione in film resistividepositati per serigrafia",Alta Frequenza, 41,921(1972).

7. F. Forlani and M. Prudenziati, "Meccanismi di con-duzione in film resistivi depositati per serigrafia",Comm. at 59th National Confer. Italian Phys. Soc.,Florence (1973).

8. R. M. Hill, "Conduction theory applied to thick filmresistors", lERE Conf. Proc., 31, 251 (1975).

9. F. Forlani, "Relative merits of thin film and thick-filmtechnology", Comm. at 3rd Intern. Confer. on ThinFilms, Budapest (1975); Thin Solid Films, (to bepublished).

10. A. Miller and E. Abrahams, "Impurity conduction atlow concentrations",Phys. Rev., 120, 745 (1960).

11. V. Ambegaokar, B. I. Halperin and J. S. Langer,"Hopping conductivity in disordered systems", Phys.Rev., B4, 2612 (1971).

12. M. Pollak, M. L. Knotek, H. Kurtzman and H. Glick,"D.c. conductivity of amorphous germanium and thestructure of the pseudogap", Phys. Rev. Letts., 30, 856(1973).

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