ELECTRON TUNNELING AND HOPPING POSSIBILITES IN RuO 2 THICK FILMS

It is proposed in this paper that the temperature coefficient of resistivity (TCR) in thick film resistors arises from (i) the usual particle-to-particle conduction, (ii) electron tunneling, and (iii) the phononassisted hopping. Equations for activation energies are derived for the temperature minimum of the resistance with and without hopping. New equations for TCR are suggested. Some extensive calculations of TCR and activation energy have been made for RuO thick film resistors, the results of which agree well with available experimental measurements.


INTRODUCTION
It is generally believed 1,2 that the conduction mechanism in thick film resistors is of two types: one is direct particle-to-particle contact and the other is quantum-mechanical tunneling. Fine metallic particles uniformly dispersed in a glassy medium, fired at a suitably high temperature and slowly cooled to room temperature can provide 3 a suitable situation for the above two mechanisms to occur at the same time. For the tunneling mechanism, however, there are two possibilities: (i) resonance-type tunneling which is today the most widely accepted 4 process for thick film phenomena and (ii) phonon-assisted hopping or electron hopping, first proposed by Mort in his theory of amorphous semiconductors.
The theory of electronic conduction in thick film resistors using resonance-type tunneling has been developed by Pike 6 and Pike and Seager. 7 This theory has been recently applied a-l to many thick film resistors, but there are some instances 4' where deviations from this theory could be observed, especially when the percentage of the metallic particles in the glassy medium is rather low. The main objective of this paper is to emphasize that when the phonon-assisted hopping 9 is appropriately included along with the resonance-type tunneling, not only the earlier theory 6'7 can be improved, but also the physics of the situation can be more correctly described. We shall first develop here an equation for the temperature coefficient of resistivity (TCR) with hopping and then obtain equations for activation energy involving temperature minimum of resistance.
We shall then make computer calculations of various TCR and activation energy curves and compare them directly with experimentally measured values of RuO2 thick films.

THEORETICAL CONSIDERATIONS
In Figure 1 we show a typical electron path for particle-to-particle conduction, but for the tunneling mechanism we consider in Figure 2, an energy band diagram for the electron passing through the nonparticle or glassy region. a) Metallic Conduction and Resonance Tunneling If we consider only particle-to-particle conduction and simple resonance-type tunneling the total resistance of the film can be easily written. According to Pike and Seager 7 the METAL GLASS FIGURE The particle to particle conduction in a thick film resistor, TUNNELING BARRIER FIGURE 2 The tunneling mechanism of an electron in a thick film resistor. The resonance-type tunneling occurs when the width of the barrier is thin, such as the one in extreme left barrier. However, the phonon asisted hopping is possible when the width is wide enough, allowing many electron steps before tunneling out the barrier. resistance in a series combination is given by In the present paper, a is different from a' and defined by a R b (Rb + Rm)-I (10) In the Pike-Seager 7 work Eq. (6) was approximated (see their Eq. (10)) and the third term in the denominator of Eq. (9) (compare with their Eq. (12)) was neglected to a first approximation. Furthermore, no distinction was made between a' and a.
b) Metallic Conduction, Resonance Tunneling and Hopping Mott s has shown that for electrons in a higlfly disordered system the de conductivity (tr) has its contribution from thermally activated hopping. If we do not consider a very high temperature regime, then two possibilities exist: the weak field hopping and the strong field hopping.
For weak field hopping (i.e., when eRF , kT) the conductivity is given by I cr 2e2R2vpN(EF) exp 2aR - (11) where W(hopping energy) P N(EF) foraRo >> and 4rR a N(EF) R(hopping distance) where To and C are explicitly def'med in Reference 5.
There is a considerable discussion in the literature about the value of B which supposedly varies from 1.78 to 2.48, but apparently not much is said about A. We will refer to these constants as Mott's A and B constants. In two dimension, however, Eq. (17) takes the form (which may be valid in some thin film studies) For strong field hopping (i.e., when eRF >> kT) the electron hops from higher to lower energy state without thermal activation. In this case the conductivity expression is shown to be where A' is another constant different from Mott's A and B coefficients, We now denote the phonon-assisted (or hopping) resistance where and S respectively represent the length and cross-sectional area of the film. Therefore the total effective resistance should read as R Rm + RB (26) Two different situations of practical interest can now be discussed.
(i) Parallel connection mode: If the barrier resistance Rb and the hopping resistance Rp are considered in the parallel mode, the total effective barrier resistance will be The range of the parameters a, , and E studied in this investigation is given in Table I.  (8). For the sake of comparisons we have also used the Pike-Seager equations. All these results are plotted in Figures 3 and 4. Some small difference was observed between the two sets of 3' in Figure 5, but the general trends of the curves were the same. The activation energy E from the Pike-Seager equation turned out to be 10% at low temperature, but 70% at high temperature larger than those calculated from present equations. In the absence of hopping a most likely value of a -0.7 has been suggested by Pike and Seager. 7 With this value of a and finding out Tm experimentally, it is possible to obtain the activation energy E of the film (the energy range 1.0 to 2.6 x 10 -a eV was investigated for this purpose).  The experimental plots of a for 8% RuO films as measured by Schaffer and Sergent. 12 The dimension of the films were (1) 20 mil. 20 mil., (2) 40 mil. 40 mil., (3) 60 mil. 60 mil., (4) 80 mil. 80 mil. Group and Group II represent particle size of 3.1 and 2.2 t respectively.

Comparison with Experimental Results
We now examine the above results of TCR in the light of several recent experimental measurements, ix-is In Figures 9 and 10 are shown some results for RuO2 thick films as measured by Schaffer and Sergent 12 and Chen and Smith. 13 The later experiment of Chen et al. is has the same result as their earlier work x3 for resistance and TCR but the range of temperature has been extended. While experimental details of these measurements can be found in these referenced papers, the highlights of these measurements may be summarized here. These films were made of a resistive ink which consists of RuO2 conductor particles, a glassy binder and an organic liquid. To prepare a resistor, the ink was first screen printed on ceramic substrates and then fired on a belt furnace usually around 800C. By this process, the organic binder was evaporated but the RuO particles were fused into the substrate through the glassy background. We do not have actual experimental data for the activation energies and therefore no direct comparison can be made with the calculated plots of  The experimental resistance and TCR data for RuO films as reported by Chen and Smith. 13 The dimensions of the films were 100 mil. X 200 mil. and 0.5 mil. thick. recent experimental work of Chen et al. is indicates a value for Tm 280K, where resistance is minimum for their RuO2 film. If we take this value of Tm to calculate the activation energies we f'md, for ct 0.6, E 3.0 x 10 -a eV from Eq. (5) and E 4.2 x 10 -a eV from Pike-Seager equation. It would seem quite appropriate to quote the values which Pike and Seager 7 found in their work with RuO2 films determined from their measured values of Tin. These values range from 1.5 to 3.8 x 10 -a eV, which are in good agreement within our calculated numbers as observed from Figures 4 and 6. When hopping is present energy curves show much less dependence on temperature. One way to test whether or not hopping plays any meaningful part in the electronic conduction in thick film resistors is to measure E vs. temperature curves. A steeper slope of these curves, i.e., b-) would indicate a possibility of only a resonance-type tunneling. The negative T energy curves (curve numbered 5) in Figures 7(a) and 7(b) indicate a situation for a particular combination of the data t and #, which may not exist in an actual thick film resistor. That is to say, it is not possible to have a thick f'flm with arbitrary values of a and Of the two sets of experimental 12'13 data, only in Figure 8 is the percentage of RuO2 known 12 (8%). Presumably in Fig. 9 the percentage of RuO2 should be about 20% by volume, which is calculated from the data of Chen et al. ts of the resistance equal to 10 kI2/sq., and the conductivity percentage plot of Angus and Gainsbury (see Ref. 2 Vol. II). These two sets of plots seem to agree with our calculated plots in Figure 3(a) or 3(c) obtained without hopping. The general trend including the magnitude agree within 5% for a 0.6. We, however, cannot find any physical reason for the presence of small humps observed around -50C in Gr. and around 0C in Gr. II films in Figure 8. We believe these could very well arise from the experimental error 2 in the measurement or some artifact in plotting the data.
To find results to be compared with the plots of Figure 5, one must experiment with very small-percentage (less than 2%) of RuO2 in the f'rims. Our own experimentation with small-percentage RuO2 thick f'flms has been in the limited range of 25 to 200C. In this range as is evident from Figure 10, the TCR has a slowly increasing trend, but stays negative all the way. We indeed observed such results (-942 to-250 ppm/C). Detailed experimental work for the whole range of temperature is currently in progress for RuO2 thick films. Further measurement on activation energy and barrier height will supplement this work. These results will be reported separately. 4. CONCLUSION This paper calculates, for the first time, the TCR and the activation energy of RuO2 thick tVflm resistors with hopping. A comparison is then made with three sets of recent experi-mental measurements. The results of the present investigation along with the experimental data, indicate that the hopping is important when the percentage of RuO2 in the glassy matrix is rather low in the 2 5% range. When the percentage of RuO2 is high the two well known mechanisms, i.e., particle-to-particle conduction and electron tunneling are sufficient to explain the observed TCR in those films. The present suggestion with regard to the TCR equation and activation energy takes one more step beyond the well known Pike-Seager theory, and helps us to understand the available experimental results better.