Great Britain STUDY OF THICK FILM RESISTORS BY 1 / f NOISE MEASUREMENTS

Knowledge on the conduction mechanism of thick film resistors are limited and often contradictory.


INTRODUCTION
Numerous efforts have been made to understand the conduction and noise mechanisms of thick film resistors but until now a model with- out contradictions is not available 1-6.There are two extreme views: a) Conduction and noise are determined by the contacts between conducting grains enbedded into the glass [e.g., ref .3]; according to this model the most determining factor is the small effective area of contact b) Conduction and noise are determined by the dopants (or impurities) within the glass [e.g., ref. 4].Some models tried to include both mechanisms [e.g., refs.2, 5].It is important to note that for a satisfactory description not only a) and b) should be taken into consideration but also the results of generalized percolation theory.Percolation effects may increase the fluctuations of resistance and under proper conditions they may even influence the shape of the spectrum.Special attention should be devoted to the fractal dimension character of the system topology and to the effect of the diffusion controlling the conductivity including possible anomalous diffusion7.
In this paper we try to collect information about the conductivity mechanism of thick film resistors by temperature dependent 1/f noise measurements.

THEORY
In [ref.8] a method is proposed for the temperature dependence carrier concentration n(T) and mobility #(T): n(T0) [ a---0) I_a-, - (1)   and /(To)- where L(T) is the (temperature dependent) lattice scattering mobility, an(T) the noise (Hooge) factor lattice scattering, SR(f, T) the measured resistance noise spectrum and S6(f, T) is related to the measured conductance noise spectrum, S6(f, T)/Ra(T), where R(T) is the sample resistance.The first factors in equations ( 1) and ( 2) are usually unknown.However, the temperature dependence of often can be estimated and that of ct is usually weak 9 or saturating1.Additionally, it should be noted that the temperature dependence of these terms is reduced by the powers 1/3 and 1/4, respectively.
We note that Eqs (1) and (2) are valid only for the cases where Ohm's law and the Hooge-Vandamme formula given by the equation; hold for the sample material, where 1 and Vef is the effective noise volume of the sample9.
In thick film resistors both Ohm's law and Hooge's classical 1/f Eqs.(1) and (2) were obtained from Eq. (3) and Ohm's law by elementary calculations8.In thick film resistors both Ohm's law and Hooge's classical 1/f noise formula are valid .However, the validity of the Hooge-Vandamme formula (containing also a reduction factor of the squared ratio of actual and lattice scattering mobilities) is still open.Nevertheless, as it is mentioned in [ref . 11], for the square relationship between noise and mobility (that is, for the validity of Eq. ( 3)) it is enough to postulate a scattering mechanism which is responsible for mobility fluctuation leading to 1/f noise but is not dominant in determining the mean resistivity.Eq. ( 3) has turned out to hold, even in bismuth films made by several deposition methods and in wires 12 '13.At present, at very limited knowledge on the conduction mech- anism of thick film resistors, a natural question is whether the temperature dependence of carrier concentration and mobility given by Eqs.(1 and 2), are consistent with physical expectations.The samples were made 6n 96% alumina substrates with the usual firing temperature of 850C.The lower resistance values (several ten kiloohms) are of a 10 kf2/square ink, the higher values of 100 kf2/square.The length/width radio is always > 1.The ink brand is RX 8000 Remex.
A slight temperature dependence of the shape of SR(f, T) was observed which turned out to be sample dependent.Therefore in Eqs. ( 1) and (2) the cumulated spectrum between 200 Hz and 2 kHz was applied to reduce the errors coming from the above effect and from stochastic nature of the measurements.
In Eqs. ( 1) and (2) the lattice scattering mobility was taken into account with the proportionality of PL (T) : T -3/2 15.This is valid for acoustical phonon scattering at ordinary temperatures.

RESULTS AND CONCLUSIONS
The relative temperature dependence of the carrier mobility given by Eq. ( 2) can be seen in Fig.  between Tand T-3/2.The temperature dependence of the concen- tration given by Eq. ( 1) is not drawn since our samples have a practi- cally constant resistance and hence n(T) is the reciprocal of tr(T).Thus, it can be concluded that the results given by Eqs. ( 1) and (2) are plausible: the dependence of mobility on T -1 T -3/2 and the slightly rising concentration with temperture resembles the behaviour of degenerated semiconductors.Regarding the models of the con- duction mechanism in thick film resistors the following conclusion can be drawn provided Eq. ( 3) is valid for this system.The temperature dependence of mobility refers to the presence of scattering mechan- isms so the hopping models (with increasing mobility vs. tempera- ture) can be excluded.It is interesting to note that a system obeying the "barrier dominated model ''5 would have the same properties as described above.
noise measurements have been done .onfive different samples, provided by Microelectronics Company (MEV), Budapest, Hungary in the temperature range of 298-373 K.The experimental setup reported in [ref.14] was applied.

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