Size and Grain-Boundary Effects in the Electrical Conductivity of Thin Monocrystalline Films

By assuming that the scattering processes from other sources than grain-boundaries can be described by a single relaxation time τ∗ and then by solving a Boltzmann equation in which grain-boundary scattering is accounted for, we have obtained an analytical expression for the thin monocrystalline film conductivity in terms of the reduced thickness k and the grain-boundary reflection coefficient r. Numerical tables are given to show the agreement of the above expression with the Mayadas-Shatzkes expression.


INTRODUCTION
Several investigators 1-8 have reported both experimental results a-s and theoretical expressions [6][7][8] for the thickness dependence of the electrical resistivity of polycrystalline and monocrystalline metallic films on the basis of the Mayadas-Shatzkes model (M-S model). 6'7 However this theory leads to a rather complicated expression which involves the use of a computer to obtain numerical solutions. The purpose of this paper is to derive an analytical expression of the thin monocrystalline film resistivity by assuming that in such films the transition probability of a carrier in state k being scattered to a state k' by all types of scatterers other than the grain-boundaries, can be expressed in term of a relaxation time r*.
Let us recall that in monocrystalline metallic films, carriers suffer scattering from phonons and point defects (background scattering), external surfaces and grain-boundaries and that, in the M-S model the operative grain-boundaries can be represented by a series of randomly spaced partially reflecting planes, perpendicular to the electric field E x whose normal lies in the substrate surface.
Mayadas and Shatzkes have solved the general problem by following the lines of the Fuchs-Sondheimer (F-S) calculation; 9 'a o in particular they have introduced into the Boltzmann equation an effective relaxation time r eff which takes into account the background and grain-boundaries scattering processes occuring simultaneously within the film. 127 Then the total film conductivity o F is expressed as; where o_ is the bulk conductivity (i.e. the conductivity of an infinitely thick monocrystalline film) and where A* is given by r/2 exp -ktH(t,qb) p exp -ktH(t,qb) The parameter c is related to the bulk mean free path o' average grain diameter,a,g and "grainboundary reflection coefficient r by Eq. (5) In monocrystalline films the average grain diameter a_ is found to be equal to the film thickness a; from a simplistic point of view the contributions of grain-boundaries or external surfaces to the total film resistivity become comparable at this point. Consequently, in this paper we propose an analysis which consists of superimposing the grain boundaries effect and the F-S effect.

EXTERNAL SURFACES SCATTERING (F-S EFFECT)
The F-S theory is based upon the assumptions of a free electron, isotropic bulk relaxation time r o and a boundary condition for electronic distribution function which states that a fraction p of electrons is specularly reflected from both surfaces of the film,1 o the remainder being diffusely scattered.
The film conductivity o (without grain boundaries effect) is given by where A(k) is a function of the reduced thickness k which is generally expressed as (7) The A(k) function has been tabulated for different values of the specularity parameter p by several authors. o , 2,3 It is assumed that we may define a total relaxation time r* for the simultaneous background and external surface scattering effects so that the film conductivity (l F can be rewritten in the form Thus from Eq. (6) we derive grain boundaries we follow the lines of the M-S analysis 6 7 and we suppose that the grain-boundaries are represented by N planes whose positions x are distributed according to a Gaussian probability distribution with a standard deviation s. With  1) When the surface scattering is entirely specular (i.e. p 1) Eq. (18)  Recently, Chaudhuri and Pal 22 have analyzed their experimental data in the light of the M-S theory and have obtained a value of the grainboundary reflection coefficient r equal to 0.005. When the reflection coefficient r approaches zero the function m(r) is not easy to evaluate; on the contrary, in the present model, the analytical Eq. 19 allows the calculation of the thin monocrystalline film resistivity, even for very low values of r, without any tabulation.
Hence, we will attempt, in a future paper, to derive from Eq. 19 an analytical expression of the thin monocrystalline film t.c.r which is valid in a large r-range and reduces to the F-S equation when r becomes equal to zero.

CONCLUSION
It is now well established 4 ,1 5-20 that thin metal films thicker than 100 A (i.e k generally greater than 0.5 at room temperature) may be regarded as continuous; furthermore in the range 0.5 _< k _< 10 Eq. 18 deviates by only 3% in the case of diffuse scattering on external surfaces and by only 1% when a fraction p of electrons is specularly scattered from external surfaces (p 0.5). It thus appears that the procedure  proposed for analyzing the monocrystalline film resistivity is suitable and we obtain a simple expression which is easy to evaluate over large ranges of the r and p parameters and allows a direct comparison of the theoretical results with experimental data.