EMPIRICAL RESULTS ESTABLISHING THE THERMAL INDEPENDENCE OF THE GRAIN-BOUNDARY REFLECTION COEFFICIENT

Starting from the effective Fuchs–Sondheimer model of electronic conduction in thin polycrystalline films and 
implementing the asymptotic expressions of conductivity it is shown that the hypothesis of temperature independence 
of the grain boundary reflection coefficient agrees with previously published results.

M. S. function.f(a)which characterizes the effect of grain-boundary scattering.An effective mean free path had been defined for this purpose.
Theoretical expressions of the temperature coefficient of resistivity can be deduced from the expressions of the conductivity, ,4 ,s assuming that thermal variations in film thickness, grain size and reflection coefficient r are negligible.
Numerical results ,s related to theoretical formulae have shown that M. S. expressions and modified F. S. expressions coincide within an accuracy of 5%.
Hence the polycrystalline (monocrystalline) film can be considered, from a theoretical point of view, as built in an imaginary isotropic material whose bulk imaginary resistivity is equal to that of an infinitely thick polycrystalline (monocrystalline) film and in which electrons are scattered at the external surfaces.The conduction properties are then described by the following equations for polycrystalline films.

, ,g[-A(g)]
() where cry is the film conductivity and/3/its t.c.r., ag is the grain-boundary conductivity and/3g its t.c.r., a k kg lo (c 0 (a) (3 where a is the film thickness, lo the electronic mean free path in the bulk material and f(a) is the M. S. function defined by: J"(a) 2 a + 3a 2 3a 3 In +--a (4) lo r a (5) ag r where ag is the average grain diameter and r the electronic reflection coefficient on grain-boundary, and: A (kg) -; (1 19) -exp{-kgt) dt (6)  p exp(-kgt) B(kg) -(1 p): t: t4 exp(-kgt) (1 -p exp(-kg-T dt (7)  where p is the fraction of electrons specularly scattered at the external surfaces.
In the F. S. model the corresponding equations are" 1,3,6 of oo [1 A(k) (S) For k > 1, approximate forms of Eqs. ( 8) and (9) 3,7,8 are" Of "" ao 1- /f '" /0 8 k These expressions are valid.for,k > 1, with an accuracy of 5 %.8It follows, from Eqs. (10) and ( 11)" It may be noted that 13o and/3f (Eqs. ( 9 and ( 11) were calculated 6 by logarithmic differentiation of the conductivity equation with the assumption that thermal variation in geometrical parameters is negligible in relation to thermal variation in the mean free path.
Vahes of conductivity and t.c.r.(they are marked with an "exp" index in the equations) of sputtered A1 and Zn polycrystalline films have been obtained experimentally, 8-x and it has been experimentally observed 6,9 that: (Pfexp /fexp )" (Pgexp gexp ) -1 (13)   in a large range of values of kg (Figure 1).
Since Eqs. ( 12) and ( 13) are identical experi- mental evidence is given for the validity of the assumption related to the negligible thermal variations in conduction parameters except in mean free path.Hence, thermal variations in reflection coefficient r can also be neglected in relation to variations in mean free path in the range kg > 1, for the calculation of a; and 2. SOME CONSEQUENCES Consequently the t.c.r, of unsupported films can be easily deduced from the expression of film con- ductivity in the imaginary F. S. model 1.1 For monocrystalline films, Eqs. ( 1), ( 2), ( 14) and ( 15) are still valid if the g index is related to infinitely thick monocrystalline film.Eq. ( 5) is also valid but the grain diameter is now equal to the film thickness and an asymptotic expression of Eq. ( 4) is then: Eq. ( 21) agrees with the theoretical expression previously proposed by Thieme and Kirstein 13 Eq. ( 12).It must be noted that more sophisticated approximate expressions must be used 4 if a more extended range of validity is required.All these consequences agree with previously published results and sustain the validity of the thermal independence of the grain-boundary reflection coefficient.
However, a more detailed analysis is necessary in the case of films firmly attached to a substrate since variations in the t.c.r, are due to thermal strains when there is a mismatch between thermal expansion coefficients of film and substrate.This point would be emphasized in a further paper.
The relation between the t.c.r.3g of an infinitely thick film and the bulk t.c.r, is known: 4

CONCLUSION
The approximate thermal independence of reflection coefficient on grain-boundary is empirically estab- lished.It gives further physical validity to previous theoretical calculations related to conduction properties such as electric thermopower 16 or thermal variations in the strain coefficient of resistivity 17 or in the Hall coefficient. 18() /g +o.
The validity of Eq. ( 15) has been previously shown by experiments related to polycrystalline films. 9 Eq. (20) gives the literal form of a new approximate expression of the t.c.r, of monocrystalline films, whose theoretical validity has been recently established.l 2 Eq. ( 14) is also valid for monocrystalline films and can be transformed in the same way as Eq. ( 15).