Adequate Use of the Cottey Model for the Description of Conduction in Polycrystalline Films

those parallel to the electric field, which produce only specular reflection, and those perpendicular whose effects are represented by a grain-boundary relaxation time [2F(I kx I)] -1 (Refer to Mayadas and Shatzkes3, Eqs. 6b and 7b) introduced in the Boltzmann equation. The positions of the planes are defined by a Gaussian law [Mayadas and Shatzkes Eq. 3] whose standard deviation is s. In the limit s 0 no grain boundary resistivity is observed. This means that a periodic array of planes provides no resistance. However the average separation of the planes (which is identified with the average grain diameter D) could be used if the travelling distance of electrons satisfies the condition >> D (2)

], the current density J is written as J= e fVx f(O, q)dap (1) where e is the electron charge, Vx the component of electron velocity o in the direction of the applied electric field and fl stands for the deviation from the equilibrium function of Fermi distribution.
Eq. (1) has been given by COTTEY 2 (p.301) and is valid in spherical polar coordinates (r, 0, ) with: Z r cos 0 x r sin 0 cos as shown in Figure 1.
Let us remember that in the Mayadas-Shatzkes conduction model a the grain boundaries are represented by two types of randomly spaced planes, (Received January 1, 1979) those parallel to the electric field, which produce only specular reflection, and those perpendicular whose effects are represented by a grain-boundary relaxation time [2F(I kx I)] -1 (Refer to Mayadas and Shatzkes3, Eqs.6b and 7b) introduced in the Boltzmann equation.
The positions of the planes are defined by a Gaussian law [Mayadas and Shatzkes 3 Eq.3] whose standard deviation is s.In the limit s 0 no grain boundary resistivity is observed.This means that a periodic array of planes provides no resistance.However the average separation of the planes (which is identified with the average grain diameter D) could be used if the travelling distance of electrons satisfies the condition >> D (2) The transmission of electrons across grain bound- aries can be conveniently described with the COTTEY model, using the same assumptions as COTTEY 2 specially if the effect of boundaries on electrons can be described by an exponential function P of the travelling distance.This feature requires that the effect of any boundary could be expressed by a given differential variation of P and that the electron path between two boundaries could be considered as a differential variation of I.In the new proposed model the first condition must be written in the form: r (3) Furthermore, under this assumption, the term b .grad f can be neglected in the Boltzmann equa- tion [Sondheimer4, Eq. 5].
The second condition must be expressed as: >> D' for 0 .-(4) 2 where D' is the boundary spacing measured in the vertical plane (z 0 p) containing b (Figure 2).
If Eq. 4 is satisfied, the probability of electrons where Xo is the mean free path associated with the influence of grain boundaries on the conduction electrons.
The travelling distance is calculated in the (z O p) plane (Figure 3).nD' Isin0 FIGURE 3 The geometry of the Warkusz modeP.
Assuming that grain boundary and background scattering are independent, the mean free path X(0, , r) for both scatterings is4'.
1 - (11)   x(0, , r) Xo z where o is the bulk mean free path.where og is the grain boundary conductivity, i.e. the conductivity of an infinitely thick polycrystalline film, and oo the bulk conductivity.Eq. 12 is not easily integrated in these coordinates but more convenient coordinates could be used.
As it has been shown above Eq.12 is valid if r takes values near 1.
Finally it appears that the new model recently proposed is of slight physical interest and somewhat questionable whereas the alternative expression for Mayadas-Shatzkes equation is somewhat complicated.Further studies would be necessary.

FIGURE 1
FIGURE 1 Definition of spherical polar coordinates.

FIGURE 2
FIGURE 2The geometry of the grain-boundary model.travelling through n grain boundaries P is defined by1,2 not agree with the expression previously proposed[Warkusz Eq. 4], since the author did calculate the travelling distance from the relationvalid for cylindrical grain boundaries.
SHORT COMMUNICATION Adequate Use of the Cottey Model for the Description of Conduction in Polycrystalline Films C. R. TELLIER and A. J. TOSSER Universite de Nancy 1, Laboratoire d'Electronique, C.O. 140-54037 Nancy Cedex France An attempt has been recently made to give a new model of electrical conduction in polycrystalline metal films.Some points are discussed in this paper.In the theoretical calculations related to a new grain boundary scattering model for conduction in metals