COMMENT ON THE DEPENDENCE OF AND CURRENT NOISE ON GRAIN SIZE IN THICK FILM RESISTORS ( TFR ’ s )

The heterogeneous structure of TFR's results in high resistivities and high current noise. Accepting models of 
conduction in TFR's, according to which the resistivity is determined by a resistance independent of bulk-resistivity 
of a metallic-like component, it will be shown, that R□ and Ceff*(describing current noise behaviour) 
increase with d and d3, respectively, when d is the grain size. On the other hand, both quantities depend on the 
volume fraction of the metallic component in the same manner. This leads to the conclusion, that a general 
dependence in the form Ceff*= f(R□) cannot exist.


INTRODUCTION
2 of a TFR is given by For a simple bar-like geometry the current noise voltage uto (1)   with U is the applied dc-voltage, V is the volume of TFIL Ce*ff is a constant, which describes the current noise behaviour of the heterogeneous material+.In previous papers it has been tried i) to find a general dependnce Ceff : R (2)   relating the increase of Ce*ff to that of the sheet resistivity Rt3. 1, ii) to connect the value of a with the conduction mechanism in TFR's (constriction or interface dominated resistanceS).
In t has been shown, that it is not possible to decide between the cases' constriction or interlace dominated resistance considering the value of a.
The aim of the following comment is to discuss (within the chain-like model considered in3) the dependence of Rr and Ce*ff on the volume fraction x and the grain size d of the metallic component used to prepare the TFR.It is found that RD and C*eff depend on both quantities in such a way, that a general relation C'err f(Rr) cannot exist.The correct relation is given and experiments, which support the conclusions, are discussed.
+Compared with the definition in Ceff'-Cef V is valid.Thus Cef is independent of volume and a material m' 1' k' holds for C*eff.Here m' denotes the total number of microscopic resistors in one chain and 1' and k' the number of chains, contributing to conduction (V volume ofTFR).If the thickness of a possible glassy interlayer is small compared with grain size d the (L B, D length, width, thickness of TFR, respectively).
Eq. ( 4) follows, if the volume fraction x of the metallic like compound is considered, where the right hand side (x Xo) has such a form that the experimentally found Rt-x-dependence is described correctly see (6).Physical arguments explaining the x-dependence in (4) are the following: x o denotes threshold concentration for onset of conduction; that means if the volume fraction of the conducting material is smaller than x o the conductivity of TFR material vanishes.The observed values s > are explained by the fact that islands, chains of finite length do not contribute to conduction.The factor 3 at the left, in a crude approximation takes into account that the network of chains is three-dimensional, but the resistance is determined by chains parallel to the current only.By means of (4) from (3) follows.This expression has to be compared with that for Rr R 3___r d (6) D (x Xo) (r is the magnitude of elementary resistor), obtained by considerations according to and Eq. ( 4).The Eq.'s ( 5) and (6) may be summarized, which yields the desired relation between Ceff and Rr: Obviously this expression remains valid, if the x dependence in (4) has another mathematical form.Thus it has been shown, that a general relation C'elf oc R does not exist, because both quantities depend on x and d in quite a different manner.An expression of the form (2) makes senses only if the conditions are defined under which R[] is controlled.In two limiting cases the following behaviour is expected Ce*ff c R if R is controlled by x, d const.
Ceff oc Ra if R= is controlled by d, x const.The assumption of the existence of a resistance r which is independent of the composition is supported by the results of the investigatiorfs. 3,4The limiting case (I) is verified by the results ot.The independence of r on grain size is suggested by the data given in for the Bi2Ru20 system.
Here it is interesting to note, that in the cermet system W-A1203 the resistivity is also increased, if the diameter of the W particles is increased by annealing?The magnitude of microresistor r, determined by tunnelling processes in that case, is not changed by annealing the cermet structure, as follows from TCR measurements.
Thus, for further verification of the described model of TFR's, it would be highly interesting to prepare TFR's containing the same volume fraction of conducting material with different grain size and to investigate their noise behaviour.In this way the scarce published data concerning the Rrq-d-dependence would also be completed.For this aim commercialkly available inks are not suitable, because they cannot be characterized sufficiently well as regards grain size, glass, composistion, TCR modifier etc., and not only x and d are changed to control Rrq.Furthermore one must be sure that the microscopic features of the inhomogeneous structures for different d remain comparable, e.g. the x-dependence in Eq. ( 4) is independent of d.This may easily be verified by investigations of x and d dependence of Rrq.The presented conclusions are valid, ifr is constant regardless ofthe origin ofthe resistivity.Therefore, as already has been pointed out in from a relation C*eff oc Rr, concerning the conduction mechanism no conclusions can be drawn.
The grain size dependence of Ceff has already been discussed in2, but that result is correct only in the case of constriction resistance (cf.also).
Summarizing, within the model discussed the noise coefficient Cff should obey the relation Eq. (7).A power dependence C'elf R would follow only under very special conditions, e.g. the "limiting cases" (I) and (II), but it is impossible to define a a which depends on x and d.If it is found experimentally that for one series of inks C*eff R[] does not hold, it has to be strongly suggested, that not only the volume fraction has been changed to develop the series.
, F. MOLLER AND H. HEMSCHIK 2. THEORETICAL CONSIDERATIONS In the chain-like model for TFR's proposed in Ce Rr AND CURRENT NOISE ON GRAIN SIZE IN TFR's DISCUSSION