RESPONSE TO COMMENTS ON PAPER ON GRAIN BOUNDARY SCATTERING MODEL FOR METALS

The electrical conductivity of a polycrystalline metal film has been studied for a model in which the background scattering and grain boundary scattering are independent. The external surface electron scattering has been analyzed by assuming it to be independent of background scattering and thus the external surface scattering can be conveniently described with the Cottey method.


INTRODUCTION
An attempt has been recently made to give a new model of electric conduction in thick metal films. Some points from this model are discussed by Tellier and Tosser. The proposed model in paper 1 discussing scattering on grain boundary is similar to the Mayadas-Shatzkes model (Section 2). In Section 3 we shall prove, that to describe the external surface scattering in thin films we can use the Cottey method.

GRAIN BOUNDARY ELECTRON SCATTERING
In this Section we shall prove, that our model of electron scattering on grain boundaries is similar to the Mayadas-Shatzkes modeP , 4 and we shall also demonstrate, that the electron transmission coefficient r has the value: 0 < r < 1, similar to the electron reflection coefficient R at a grain boundary, which has also the value: 0 < R < 1. The conductivity for metal films ,4 in the presence of both grain-boundary and background scattering is found from e2 j '*V2 dSF g Vl where it is the background mean free path, D is the average grain diameter, and R is the reflection coefficient kx kvl cos 1 kel cos sin 01 q =cos=cosOsin0 (4) where kF-is the magnitude of the Fermi wavevector.  (9) has been obtained by Mayadas and Shatzkes 4 by considering the total resistivity of a thin metal film in which three types of electron scattering mechanisms are simultaneously present: an isotropic background scattering, scattering due to a distribution of planar potentials grain boundaries, and scattering due to external surfaces. The intrinsic or bulk resistivity is obtained by solving the Boltzmann equation in which both grain-boundary and background scatterings are accounted for. The total resistivity is obtained by imposing boundary conditions due to the external surfaces as in the Fuchs 5,6 theory using the Boltzmann equation.
Assuming that ,:rf -a 0 dz h3 Vxf(V) dvx dvy dv (10) and imposing boundary conditions, due to the external surface as in the Fuchs-Sondheimer theory5,6, we obtain: The first term on the right hand side of Eq. (15) is the grain boundary scattering function (Eq. (9)). The equation (15) is often described as af G(oO Z(k,p, or) O" 0 G() is a grain-boundary function and can be expressed by Eq. (9) or by Eq. (1). Considering that cos sin 0 cos y q, (see Figure 1) and Eq. (2) Let us examine the equations (9) and (17), because these equations refer to electrical conductivity for the film, for which we consider the background scattering and the grain boundary scattering. The model given in paper considers similar problems and as it will be further shown, the effects of electron scattering can also be calculated by means of Matthiessen's rule starting  26)).

EXTERNAL SURFACE ELECTRON SCATTERING
Let us consider both the Fuchs method 5,6 and the Cottey method used to describe electrical conductivity for metallic films in which the background electron scattering and the external surface scattering are examined. The surface reflection parameter p will be discussed in detail. We consider some of the assumptions made by Fuchs 5,6 and the present author, to describe the external size-effect. The Fuchs theory assumes that the relaxation time of bulk material is independent of the film thickness and the PFuchs parameter.
where The p-parameter can also be conveniently described by the Cottey model, P Cottey" If there is an electric field, E, in the x-direction, the In general, the parameterpFuchs Pcottey ifPFuchs is near 1 and p Cottey is near 1. We, however, are concerned with 0 < P Fuchs < 1 and 0 < Pcottey 1. The Fuchs parameter is the probability that an electron will be specularly reflected upon scattering from a film surface. Our parameterp is the probability that an electron will be reflected upon scattering from a film surface. If Pvuchs 0, then all the models agree for diffuse scattering6---for example: This integral has been calculated explicitly for cylindrical grain boundaries. 1 Recently a three-dimensional model of grain boundaries has been proposed by Pichard, Tellier and Tosser. 11 This is an extension of the present grain scattering model since i't is possible that the reflection parameter p, and the transmission coefficient can assume values less than unity. This contrasts with previous comments (see also Refs. 11 and 12).