ELECTRICAL CONDUCTION BY PERCOLATION IN THICK FILM RESISTORS

Thick film resistors are widely used in microelectronic devices, however the mechanism of electrical conduction in these resistors has not yet been fully understood. In particular the anomalous behaviour of the temperature coefficient of resistance (T.C.R.) vs. temperature for a purely ohmic resistor has not been explained. The anomaly is that the T.C.R. is negative at low temperatures, is zero around room temperature and becomes positive at higher temperatures. This paper demonstrates that the electrical conduction mechanism in thick film resistors can be described by the electron percolation theory already proposed to explain charge transport in amorphous semiconductors. The thick film 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 tunnel through the layers. The critical percolation path is through these grains. Experimental evidence is given which confirms that the resistance vs. temperature characteristics satisfactorily fit the conduction equation provided by the percolation theory. The T.C.R. anomaly can be explained in the framework of this theory. The decay length of the electron wavefunction is shown to be lower than 370 A for a density of conductive grains in the film in the order of 101 cm-3. Such a value is consistent with electron tunnelling through layers about 100 A thick.


INTRODUCTION
The use of thick film technology to fabricate resistors in hybrid circuits is well established.Nevertheless until now the physical models proposed to describe the electrical conduction mechanism in thick film resistors -9 are un- convincing.
It is required to explain how a purely ohmic resistor can exhibit an anomalous behaviour of the temperature coefficient of the resistance (T.C.R.) vs. temperature: as shown in Figure 1, where the normalized conductivity is plotted vs. temperature, the T.C.R. of a thick film resistor is negative at low temperatures, it becomes zero around room temperature and turns positive at higher temperatures.
The first attempts to explain this anomaly 2,3 were based on the idea that conclusions about the conduction mechanisms in transition oxides could be extended to thick film resistors.With this approach the T.C.R. anomaly found a qualitative interpreta- tion.However the model failed to provide a quantitative fitting to the experimental data on PdO/Ag based resistors.A satisfactory fitting of those experimental data was obtained by Kahan 4 whose approach consisted in completing the model elaborated by Brady and based on the conductivity characteristics of semiconductive oxide grains.Brady assumed the oxide grains to be submitted to a pressure applied by the glassy matrix and dependent upon the temperature.Kahan also took into account the contribution to the current flow due to metallic grains.One of us 6 has already pointed out that Brady-Kahan model was based on six independent assumptions with no possibility for these to be checked independently.Moreover it was shown that Eqs. and 2 for conductivity fitted the experimental data used by Kahan  Normalized conductivity of a typical thick-film resistor vs. temperature.
at higher temperatures.In Eqs. ( 1) and (2), To is a constant with dimensions of temperature and n is a pure number of order 1/3.It was also noted 7 that Eqs. ( 1)and (2)provide a good fit for experimental data concerning Ir-RuO2 based resistors.
Hill 8 has confirmed the validity of Eq. (1) to fit, at low temperatures, data on Bi2 Ru2 07 based resistors and concluded that electrical conduction in thick film resistors can be attributed to hopping of charge carriers between states localized in a very narrow band.However, as already reported, 9 the values of To deducible from the experimental data are much lower than those expected by chargecarrier hopping mechanisms.
In the present paper we consider a proposal, already put forward, i.e. the electrical conduction in thick film resistors can be described by electron percolation, according to the theory initially pro- posed by Miller and Abrahams 0 and used by Ambegaokar, Halperin and Langer 11 to interpret conduction phenomena observed in disordered systems.

ADAPTATION OF THE PERCOLATION MODEL FOR THICK FILM RESISTORS
The percolation model assumes that the resistive material can be considered equivalent to a resistance network so that its over-all electrical conductivity can be written as (3)   where G c is some characteristic value of the conduc- tances in the network and r/is some characteristic length scale for the network.Let Gi be the conduc- tance between any two directly interacting sites and ], where more resistors of the assumed network join together.Even though the explicit dependence of Gij on the material parameters varies according to the mechanism for the electron exchange between sites and ], it can be asserted that for a network, where the values of individual resistances vary over many orders of magnitude, G c in Eq. ( 3) represents the critical percolation conductance.In order to have a better understanding of the significance of Gc consider the resistance network as composed of three parts: 11 i) a set of isolated regions of high conductivity, each region consisting of a group of sites linked together by conductances with Gij >> Gc; ii) a relatively small number of resistors with Gi] of order Gc connecting together a subset of high conductance regions as defined in (i); iii) the remaining resistors with Gi] " Gc.
A thick film resistor can be represented by a resistance network with the characteristics described above.A thick film resistor is obtained through the firing of specially prepared screenable compositions at temperatures ranging from 750 C to 1150 C. Such compositions are made of three main constituents: a conductive ingredient (generally, one or more conductive oxides as PdO, RuO2, NbOs, etc.), a glassy content (generally a mixing of PbO, SiO2, B 03, P Os, etc.) and an organic vehicle (butyl- carbitol acetate, ethylcellulose, etc.).This last gives to the composition the thixotropy necessary for printing on to a ceramic substrate (generally, 96% A1203) through a metal screen and is almost completely eliminated during the drying of the screened film before firing.The glassy content, on the contrary, remains in the film after firing and provides the adhesion of the film itself to the ceramic substrate.Even though after a correct firing the glass is mainly located at the interface between the film and the ceramic substrate, the as-fired film consists of a dense dispersion of small conductive grains (with diameters of 0.1/am to 0.3/am8) in a glassy matrix.
With this physical structure, a thick film resistor can be considered as consisting of: (1) a set of isolated regions with Gi] Gc, corresponding both to simple conductive grains and to groups of conduc- tive grains in purely ohmic contact; (2) a set of conductive grains in electrical contact through contact 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 first set, a critical subnetwork shorting out those resistors with Gij " Gc which correspond to higher contact resistances 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 and dominate the resistance of the network electrically equivalent to the thick-film resistor.This means that whether the conductive grains exhibit metallic or semiconductive conduction mechanisms becomes unimportant.In order to elucidate Gc, it is assumed that conduction is due to electron hopping between the percolation sites because of tunnelling effect through the potential barriers provided by the glass into which the conductive grains are embedded.
Using, for Gc, the same expression given by Ambegaokar et al. 11 for the conduction in amorphous materials, even though in this case the percolation sites are made of conductive grains instead of electron traps, we have e 2 Gc=--f Fc (4)   where e is the electron charge and Fc is the average transition rate from sites to sites along the critical percolation path.It is given by (4t)cOt3) x/4 r c 3'0 exp-p- (5)   where 3'0 is a constant depending upon the electron phonon coupling strength, the phonon density of states and other elastic properties of the material, Vc is a dimensionless constant,/90 is the density of percolation sites per unit volune and unit energy and ais a length whose physical meaning is associated with the tunnelling mechanism from one percolation site to the other, and therefore to the average diameters of the conductive grains along the percolation path.In other words, the meaning of ais the basic difference between the conduction model by electron percolation applied to amorphous semiconductors and to thick films.While in amor- -1 phous semiconductors we must expect a value of a of the order of a few A, in thick films its value must be comparable with the diameter of the conductive grains, i.e. some 100 A. By assuming the validity of Eq. (5) in the case of thick film resistors, we consider the charge on a single grain concentrated at the centre of the grain and the separation between percolation sites a bit larger than the grain diameter.3), ( 4) and ( 5), the conductivity of a thick-film resistor, is given by o oc exp (6)   where e27o /3= k (7)   4Pc 3 To (8)   pok and where we have implicity assumed that factor in Eq. ( 3) is a relatively slowly varying function of the parameters of the network and that the dominant variation 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 :exp T.C.R.This temperature corresponds to the miniznum value acquired by the normalized resistance, R/Rmin, vs. temperature as shown in Figure 3.

EXPERIMENTAL EVIDENCES
A set of measurements was performed on Bi2Ru207 based thick film resistors.The connections to these resistors were provided by screen and fired PtAubased conduction films.
All the resistors had a constant width of 2 mm, they were printed at the same nominal thickness and fired under the same temperature profile.In Figure 2 In R/T is plotted vs. (l/T) 1/4 for different sheet resisfivifies.In Figure 3 the plot of R/Rmin vs. T is shown.
In Table the values of Tmin calculated using Eq. ( 11) from the values of To obtained from the slopes of the straight lines in Figure 2 and the experimentally derived values of Zmin are compared; the agreement is good. 4

DISCUSSION
The behaviour of T.C.R. with temperature expressed by Eq. ( 10) indicates a negative infinity in the temperature coefficient as the absolute temperature tends to zero and a null temperature coefficient for the absolute temperature going to infinity.This implies that the T.C.R. must exhibit a maximum for some temperature larger than To/256.An evaluation of the derivative of Eq. (10)   indicates that such a maximum is to be found at a temperature of To/(3.2)4.In practice, it means that a resistor having a null temperature coefficient at room temperature (300K), and consequently a value of To 76800K (see Eq. ( 11)), should exhibit a maximum T.C.R. at 459 C, i.e. at such a high temperature that it cannot be checked by experi- ments.The existence of a maximum in the T.C.R. at a temperature higher than T0/256 explains the lower gradient observed in the range of the high tempera- tures.On the other hand, the derivative indicates that fbr T < T0/(3.2) 4 we have d(T.C.R.)/dT > 0, as confirmed by experimental evidence.
For a value of To of 76800K we obtain from Eq.We can evaluate Oo from the experimental measure- ments on the diameter of the conductive grains in the films reported by Hill 8 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 to 101 cm-3.As already pointed out, 11 the grain density is related to the sites per unit volume and per unit energy, Oo, by n 2poEma x (14 where Emax is the maximum energy distance from the Fermi energy, within which the electron density of states has to be constant.A more recent analysis performed by Pollak et al. has demonstrated that sites with an energy distance far away from the Fermi level contribute in a negligible way to the charge transport at the percolation path.In other words Eq. ( 14) must be written as follows n < 2PoEmax Then, the maximum energy distance from the Fermi level can be expressed by 1 70 (16) Emax =kTln Pc and from Eq. ( 5) At room temperature for the resistor under consider- ation (with T.C.R. 0 at 300K) we obtain Emax 0.103 eV.Then for a density of conductive grains of 1014 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 discussed meaning of aand with a diameter of the conductive- grains on the order of 1000 A to 3000 A.
Finally, it should be noted that the three straight lines of Figure 2 are almost parallel.The values of To are only weakly dependent upon the sheet resistivity of the film (see Table I).The approach discussed here and based on a conduction mechanism by electron percolation does not allow us to determine the proportionality factor in Eq. ( 9). 5

CONCLUSIONS
The electron percolation theory as proposed by Ambegaokar et al. 11 provides a very simple and credible interpretation of the conduction mechanism in a thick-film resistor.The apparently anomalous T.C.R. vs. temperature behaviour finds an immediate explanation in the frame of this theory.It has been checked that the matching between theory and experiments holds also for resistor compositions other than Bi RuzO7 based ones as the extended series of tests was limited to Bi2Ru=O7 based resistors due to their stability and reproducibility.
It is not surprising that a theory dealing with disordered systems can be applied to thick-film structures.Moreover, further investigations, still underway, indicate that the assumption of a site density, 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, the T.C.R.'s are dependent upon resistor aspect ratios and such a dependence might find an explanation in the frame of electron percolation theory by introducing some unhomogeneity in the site density.However, more work is necessary to confirm these preliminary observations.

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