Numerical Evaluation of the Resistivity of Polycrystalline Metal Films with the Mayadas-Shatzkes Model

In the past years, it has been a common practice to analyze electrical resistivity data of polycrystalline metal films by means of the FuchsoSondheimer model (F-S model) for the size effects. However, the granular structures of evaporated films impose a restriction about which the F-S model says nothing. In order to take this into account, Mayadas and Shatzkes developed the model (M-S model) given briefly later on. The purpose of this communication is to present numerical tables. Model possibilities and applications have been discussed elsewhere


INTRODUCTION
In the past years, it has been a common practice to analyze electrical resistivity data of polycrystalline metal films by means of the FuchsoSondheimer model 2 (F-S model) for the size effects. However, the granular structures of evaporated films impose a restriction about which the F-S model says nothing. In order to take this into account, Mayadas  For the evaluation of the overall film resistivity pf, M-S assumed that crystallite boundaries can be represented by a Gaussian distribution of partially electron reflecting planes randomly spaced with an average distance a and a standard deviation s.
Based on the available experimental data, which shows that thin evaporated films have a column-like structure with crystallites towering from the substrate to the surface, they assumed that (i)only those planes perpendicular to the applied electric field cause the extra electron scattering; (ii)the average interplanar distance a is equivalent to the average 77 crystallite diameter D as obtained by electron microscopy. In order to fit the model to experimental data, the electron reflecting power of the potential barrier between crystallites is interpreted by a phenomenological parameter r.
The model can be summarized by the two following equations, rearranged according to the heading assumptions to make the mathematical problem and numerical evaluation mo re tractable.  Then, the ratio between film resistivity pf and the resistivity of a single crystal f'flm infinitely thick, Po, is:

Crystallite
Po in which A, f(a) and t have the same meaning as in eq. (1). When ko > 10 or r -+ 0, eq. (2) reduces to the F-S equation.

NUMERICAL TABLES
Unfortunately Eq. (1) and (2) cannot be evaluated analytically. Hence, in order to interpret quantitatively the experimental results it was necessary to elaborate numerical tables. These have not been given in the literature yet and were obtained with a IBM-360/50 computer with a program written in PLI. Excessively high computer time is necessary to solve the integrals by conventional methods This problem was overcome by generalizing Gauss' equation 6 to any number of variables. This can be done changing the variables in such a way as to transform the original interval for each variable into another one, ranging from 0 to 1, as required by the Gauss' formula. The integral is then evaluated over n 2 points, each one of them given by the roots of the Legendre polynomials of order n with their respective weight. 7 The numerical evaluation was carried over n z 100 points to obtain an acceptable approximation. Nevertheless, 25 points are enough to maintain error below 1% for the integrals, but their place in the function magnifies the overall error.
In Table the values of pf/pg vs. ko calculated with Eq. (1) are given under the assumption e a thickness independent constant. Only two values of the specularity parameter p are taken into account, because in general the experimental data fit the theoretical values better by postulating p 0. Table II shows the dependence of Pf]Oo on ko calculated with Eq. (2) for the same p-values as in Table I, but with a f(d), (i.e. D d and r thickness independent constant). There are more curves given with different r, than with different a because actually the available data fit the former more closely. In Table the column corresponding to a 0 is actually the numerical data of the F-S equation. A column with r 0 in Table II would be identical. These values are given in order to estimate the error of the method. For this purpose they should be compared with the data obtained by Soffer s with Simpson's rule using 213 subdivisions. The error is less than 0.1% up to k0 =0.05. However, the first value for ko 0.01 has an error of 6%.
The evaluation method is thus justified because the computation time was reduced to 30 seconds per column.