THREE DIMENSIONAL ANALYTICAL SEPARATION OF GRAIN BOUNDARY AND SURFACE SCATTERINGS IN POLYCRYSTALLINE METAL FILMS IN THE CASE OF NON CUBIC GRAINS

Analytical approximate expressions for the resistivity and its temperature coefficient of thin polycrystalline metal films have been derived by considering separately the contributions of the grain-boundaries perpendicular to the x-, y- and z-axes. Provided that the grain-boundaries act as moderately efficient scatterers reasonable deviations from the three-dimensional model are obtained; an approximate model then seems convenient with which to perform the calculations of the strain coefficients of such fine-grained films.


INTRODUCTION
We have previously shown that to make more tractable the problem of evaluating the strain coefficients of infinitely thick polycrystalline films in terms of the three dimensional grain boundary model 2'3 it was convenient to consider the separate contributions of the three distributions of grain boundaries to the total film resistivity PEp.In a similar manner when the film thickness 'a' becomes sufficiently thin so that the external surface scattering must be taken into account 3 the problem of finding the strain coefficients of such thin polycrystalline films may be solved by calculating as a first step the three following contributions to the resistivity, i.e.: a) the contribution p+/-of the grain-boundaries perpendicular simultaneously to the x-axis and to the applied electric field E x (Figure 1), b) the contribution Pll of the grain-boundaries parallel to the electric field Ex and perpendicular to the y-axis, c) the contribution/9* due to the three following electron mechanisms occurring simultaneously, i.e. the scattering on the grain-boundaries perpendicular to the z-axis, the scattering at external surfaces and tLaboratoire de Chronom6trie et Pi6zoelectricit6.In this communication an attempt is made to derive in this framework approximate expressions for both the total polycrystalline film resistivity and its temperature coefficient (t.c.r.) and then to determine the range of applicability of the relations.
The grain boundary parameter vi (i x, y, z) is related as usual TM to the background mean free path/o, the average grain size ag and the transmission coefficient t of electrons through grain-boundaries, To determine the contribution/9* we suppose that the scattering events occur independently of each other so that the resultant mean free path 1" is expressed as, 3'6 l* lo ls lz the subscripts s and z referring respectively to external surface scattering and to electron scattering at the grain boundaries perpendicular to the z-axis.
The mean free path (m.f.p.) related to scattering at external surfaces 1 s is given as usual a ,7 by (7)   1 where Vz is the z-component of the electron velocity and p, as defined earlier 8 is the fraction of electron specularly scattered at external surfaces.
The m.f.p, lz describing the grain-boundaries scattering is found to be 9 (8)   Introducing the spherical coordinates (q, q,, v) with vz v cos 0 and following the line of previous calculations the contribution Jx* to the total current density Jx becomes for the geometry of the model (Figure 1) sin a 0 dO l+a -x cos0 (9)  with 0/-1 =/g-1 + 0;1 In the preceding equation the external surface parameter is defined as: a'7 p=a.1o -1.ln Integration of Eq. ( 9) leads to 30/(0/ a*/ao G(a) 5 5 + (1 a2) In (1 where Oo is the background conductivity (Oo (8 ne2m2v21o)/3ha).
Assuming that Matthiessen's rule holds an approximate form of the total film resistivity is: One interesting way to analyze the range of validity of the present method before undertaking study of the strain coefficients is to derive an approximate formulation of the film t.c.r, from logarithmic differentiation of Eq. ( 13).
This gives after some mathematical manipulations: Assuming, as usual, 3'1 o,11 that the thermal coefficients of linear expansion of film thickness a and grain size a are negligible in regard with that of the bulk mof.p, lo, it yields: Taking into account the definition of the t.c.r./jl /3 d In p/dT (20)   and assuming that the rigid band model of metal is valid and that the number of conduction electrons per unit volume is temperature independent the bulk t.c.r./3o is given by: o d In po/dT d In lo/dT (21) The resultant t.c.r.yp is then evaluated { Vx f(Vx) + uy g (vy) Fp/3o M (Vx, vy, 0/) F (Vx'-----G 2 (v,"-----+-O2ia),I When electron scattering at external surfaces becomes specular or when the film thickness becomes infinite the film t.c.r.p tends to the approximate grain-boundary t.c.r.g: which constitutes the reduced form of Eq. ( 22) in the limit of very large v.We then represent the grain- boundaries and external surface effects on the polycrystalline film t.c.r, in the standard form 3Fp/g e.g.
It must be pointed out that when the effect of grain-boundary scattering becomes negligible (i.e. when (vi -+ oo) the functions G(vi) and F(vi) tend to unity whereas the functions g(vi) and f(vi) approach zero.As in the limit, of very large v the G (a) and g(a) functions respectively reduce to G ) and g(p) we note that the present approximate method satisfies, as the three dimensional method, the physical requirement which states that when the grain-boundaries do not contribute to the resistivity Eqs. ( 13) and ( 22) simply become the following Cottey's relations: 7 PFp/PO For a comprehensive comparison of the exact and approximate expressions of film conductivity and t.c.r, we give in Tables I, II and III the numerical results related to the conductivity ratio OFp/Oo and to the t.c.r, ratios Fp/0 and [JFp/[Jg respectively, obtained from the exact three dimensional model equations (Eqs.(25) (28) and ( 29)) and from the

TABLE
Comparison of the exact and approximate values of the conductivity ratio oFp/tr (as given by respective Eqs. ( 25) and ( 13)).
TABLE II Exact and approximate values (as given by respective Eqs. (28), ( 29), ( 22) and ( 24)) of the t.c.r, ratios #Fp/[3o and Fp/g for u 0.  II clearly shows that the Fp/[3o ratio exhibits larger discrepancies than the t3Fp/3g ratio.As this behaviour is typical of small v Table II represents only the results for v 0.4 but it is interesting to note that for increasing values of v the discrepancy decreases; for example for v 2, and/ 0.4 we obtain deviations of about 17% for the t.c.r, ratio 13Fp/o and of about 11% for the [JFp /[Jg ratio.
Furthermore Table III indicates that the approxi- mate model leads to reasonable deviations (less than about 10%) in a relatively large g range (g > 0.4) provided that the grain parameter o remains larger than 0.4 as previously observed for the conductivity ratio.
As at room temperature the films may be regarded as continuous [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] in this defined/ parameter range the approximate Eqs (13) ( 22) and (24) generally remain satisfying to represent the thickness variations of the t.c.r.approximate equations (Eqs.(13) ( 22) and ( 24)) in which we have assumed Vx vy Vz v. Table shows that the approximate values of the conductivity lead to discrepancies even for large thickness when the v parameter takes small values, whereas for v > 0.4 the approximate values deviate from the exact values, by only 10% until the/ parameter keeps values greater than 0.1.This feature is not surprising because grain-boundary scattering becomes predominant for small v and the validity of Matthiessen's rule is then altered as suggested earlier. 6t since this feature is essentially due to inaccuracies in the grain-boundaries terms of the present model thus we may expect less significant resultant devi- ations between exact and approximate equations by considering Fp/t3g the reduced t.c.r, related to the t.c.r.,/g of an infinitely thick film instead of the 4. CONCLUSION As it must be kept in mind that to perform the calculation of the strain coefficient 1,1 6,1 9,20 we have to undertake a logarithmic differentiation of the approximate expression of the film resistivity it is reasonable to expect, in the light of the preceding discussion, the present approximate model to constitute a convenient form for describing the effects that thickness and grain-boundaries have on the coefficient of resistivity.
The effect of thermal strains 21 on the t.c.r, could also be examined from this three dimensional point of view.
Comparison of experimental data on polycrystalline film strain coefficients of resistivity ")/'fp with the theoretical predictions deduced from the present u=l Exact values model then seems possible provided that the com- parison is carried out in the reduced form 7Pp/Tg where 3'g is the strain coefficient of an infinitely thick film.These relatively long calculations will be reported in a future paper.
125the background scattering due to point defects and phonons.

FIGURE
FIGUREThe geometry of the model.