THE ELECTRICAL RESISTIVITY OF MONOCRYSTALLINE FILMS DEDUCED FROM DERIVATION OF THE MAYADAS-SHATZKES EQUATIONS

Mayadas and Shatzkes have proposed a theoretical calculation of the electrical resistivity of thin metallic films; they have considered the simultaneous action of isotropic electron scattering, surface scattering and grain boundary scattering. Mola and Heras have tabulated the exact values of the electrical resistivity of monocrystalline and polycrystalline metal films and given linearized expressionsa which are valid in some special cases (p 0, 0.1 < k < 5, 0.1 < r < 0.52). In this paper we propose an approximate expression of the resistivity of monocrystalline films which is valid in larger ranges.


INTRODUCTION
Mayadas and Shatzkes have proposed a theoretical calculation of the electrical resistivity of thin metallic films; they have considered the simultaneous action of isotropic electron scattering, surface scattering and grain boundary scattering.Mola and Heras 2 have tabulated the exact values of the electrical resistivity of monocrystalline and polycrystalline metal films and given linearized expressions a which are valid in some special cases (p 0, 0.1 < k < 5, 0.1 < r < 0.52).
In this paper we propose an approximate expression of the resistivity of monocrystalline films which is valid in larger ranges.

APPROXIMATE EXPRESSION OF MONOCRYSTALLINE FILM RESISTIVITY
The resistivity Of of a monocrystalline film is a given by Of/Oo D(a) A -1 (1) in which Po is the resistivity of an infinitely thick monocrystalline film, i.e. without Fuchs-Sondheimer 4 size effects, where and where 1)---+ 3 3 a g +- (3)   The variable p is the fraction of electrons specularly scattered at the external surfaces and k is the atio of film thicess, d, to electron me free path, l. k d/lo (4) where r is the electron reflection coefficient defined by the relation r(1 r) -(V.a).(hke/2m) - (6)   in which V is the heist of the potential well at grain- boundary and a is the potential well width, is the modified Planck's constant, kf the magnitude of the Fermi wave vector and m the electron effective mass.
If *oe is essentially temperature independent, then (pff)(,Oo3o) -Since this assumption is not completely valid, we introduce the function m(r), measuring the deviation from the Mathiessen's rule, in the expression of k,Of/,Oo related to a given value of p. Hence" It may also be expressed as"

FIGURE
Exact values of ko Pf/Po versus r/(1 r) for an electronic specular reflection coefficient, p, of 0.    We thus obtain approximate expressions of kof/o 0 if the exact value of [kpf/po k=ko is known, slight deviations from the exact values (tabulated by Mola and al ) are obtained with p independent values of m(r).
Furthermore, we may observe that the exact values 2 of kpf/po for k 0.1 are linear with r/1 r when p 0 or p 0.5 (Figure 1, 2).Hence: (kPf/Po)k-o.a 0.1 + [0.37(1-p) + 0.06(I-p) 2 + 1.32 This expression is valid with a deviation less than 2% from the exact values in the 0 to 0.5 p-range and in the 0.22 to 0.62 r-range (Table I) Making use of Mathiessen's rule Wissman6 had r FIGURE 3 Variations in re(r) (a measure of deviation from Mathiessen's rule) with r (from Eq. 25).
In the ranges considered, this equation leads to most significant deviations (Table II).

TABLE IV
r 0.62