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Exact and approximate equations for the thickness dependence of resistivity and its temperature coefficient in thin polycrystalline metal films
Thin Solid Films. 18 (1973) 137-144 0 Elsevier Sequoia S.A., Lausanne-Printed
137
in Switzerland
EXACT AND APPROXIMATE EQUATIONS FOR THE THICKNESS DEPENDENCE OF RESISTIVITY AND ITS TEMPERATURE COEFFICIENT IN THIN POLYCRYSTALLINE METAL FILMS
E. E. MOLA AND J. M. HERAS Institute de Incestigaciones Fisicoquimicus Teciriccrs y Aplicadas (Argentina] (Received April 13, 1973; accepted July 5, 1973)
(INIFTA),
Calle 47 y 115. La Plats
Starting from the Mayadas-Shatzkes model for the conductivity in thin polycrystalline metal films, an exact expression for the dependence of the temperature coefficient of resistivity (t.c.r.) on film thickness is derived. In order to allow a quick comparison of experimental data with the model, approximate equations are also derived for the dependence on thickness of both resistivity and its temperature coefficient. These approximations are valid in the range 0.2 5 k, 5 5 (where k, is the ratio between film thickness d and the electron mean free path &,). Numerical tables are also given (i) for the values of the exact function of the t.c.r. and (ii) to show the agreement of the approximate equations with the exact functions.
1. INTRODUCTION
The electrical resistivity of thin evaporated metal films changes when foreign atoms or molecules chemisorb on their surfaces’. In order to interpret this influence on the electrical properties of such films, a study of the scattering mechanism of conduction electrons in very pure films is unavoidable. This has been the object of our study for a number of years2-4. We have found experimentally that the well-known electrical conductivity model of Fuchs-Sondheimer (F-S model) can only be applied to thin evaporated polycrystalline films (thickness < 100 A) which have been thoroughly annealed at temperatures higher than 373 “K 4, 5. This behaviour can be ascribed to a morphological parameter such as the crystallite size, which can be modified by an increasing annealing temperature. The F-S model has nothing to say about this parameter. It is known that the crystallites in evaporated films have a columnar shape, rising from the substrate to the film surface 5-7 . Hence, there is a distribution of grain boundaries perpendicular to the direction of the applied electric field. The potential barrier set up between crystallite boundaries may have a variable reflecting power for the approaching electron. When the crystallite size and film thickness become comparable, the contribution of the crystallite boundaries to electron scattering increases with decreasing film thickness. Consequently,
138
E. E. MOLA,
J. M. HJXAS
as long as the crystallite size is smaller than the electron mean free path (mfp) the F-S model cannot be applicable. Taking these circumstances into account, Mayadas and Shatzkes recently developed a model’ (M-S model) introducing a structural parameter tl, which includes the crystallite size D (or its dependence on film thickness 4 and the probability r for an electron to be reflected by a crystallite boundary perpendicular to the applied electric field. The model can be summarized by the following equations* :
PfIPO= m4 -4-
1
(1)
in which pr is the actual film resistivity, p0 is the resistivity of an infinitely thick monocrystalline jilm (i.e. without size effects),
and f(a)= l--iu+3r2-3!x3
In
(
l+’
(lb)
CI>
Equation (1) was deduced assuming that the crystallite diameter D is strictly equivalent to film thickness d. Hence, the following dependence between cxand d is valid: 1 “k,
r w
l-r
where k, = d/& (I, is the electron mfp within a crystallite) and r is the electron reflection probability parameter with values between 0 and 1. Under the assumption that a is thickness independent with values ~20 (when rx = 0, the M-S model changes into the F-S model) eqn. (1) can be modified to
P& = [l -A/fCW’
(2)
in which the new symbol ps is the resistivity of an infinitely thick polycrystalline .film.
2.
THE M-S MODEL APPLIED
TO THE TEMPERATURE
COEFFICIENT
OF RESISTIVITY
Equations (1) and (2) also allow the derivation of equivalent expressions for the t.c.r.t Assuming that the rigid band model of metals is valid, the relation between the electron mfp 1 and resistivity p is given by the Sommerfeld equation * Details about the possibilities of the model have been given in previous papers’, 9. t We distinguish between the temperature coefficient of resistivity (t.c.r.) and that of resistance
(TCR).
THIN
F’OLYCRYSTALLINE
METAL
139
FILMS
I= 1.27x lo4 N-2’3 p-r
(3)
in which N is the number of conduction electrons per unit volume. Assuming N to be temperature independent, the logarithmic differentiation of eqn. (3) with respect to temperature gives 1 dl _ 1 dT
ldp_ P dT
(4)
P
which is by definition the t.c.r. Applying these concepts to eqns. (1) and (2), the following expressions for the t.c.r. dependence on thickness in terms of the M-S model can be obtained. (i) Assuming a dependence of o! on thickness (crystallite diameter D equals film thickness d), ELl+ PO
g(x)-A+B+C f(x) - A
(5)
and (ii) assuming u 2 0 and thickness independent,
Ll+
D-A
PO
f(cd- A
(6)
where & is the actually measured t.c.r., PO is the t.c.r. of an infinitely thick monocrystalline film, g(u) = -;
a+6a2++&9a31n 42
(
l+d
>
cc
B = z (1 -P)~ 1 d4 i;;;,$)(r;
r;)
exp {-k,t [l-pew
{--kd
H(t, 4)) W,$))]2
dt
1- exp { - kot Wt, N> ’ 1-pexp{-k,tH(t,4)} and
with H(t,$h)=
l++os~ (1-~)li2
dt
140
E. E. MOLA, .I. M. HERAS
Equations (l), (2), (5) and (6) have been evaluated analytically with a digital computer. The results obtained with eqns. (1) and (2) and some details of the calculus have already been given elsewhere4, 9. Those corresponding to eqns. (5) and (6) are given in Table I for two values of the Fuchs specularity parameter p (the probability that an electron will be specularly reflected upon scattering on a film surface). With the data of Table I, theoretical curves can be drawn for each of the given values of a and Y.These curves allow a comparison between the experimental data for t.c.r. and the Mayadas-Shatzkes model in order to find the structural TABLE
I
VALUES OF THE EXACT /&/&,
ON FILM
FUNCTION
THICKNESS
IN THIN POLYCRYSTALLINE
d,
FOR THE DEPENDENCE
CALCULATED
WITH
OF THE TEMPERATURE
THE MAYADAS-SHATZKES
COEFFICIENT MODEL
OF RESISTIVITY
FOR THE RESISTANCE
FILMS
For more details, see text Equution k,
(51
Equution
(61
r = 0.1
Y = 0.22
r = 0.42
r = 0.62
r=O
r. = 0.5
z = 1.0
r* = 2.0
0.1862 0.2898 0.3623 0.4184 0.4641 0.5026 0.5358 0.5650 0.5908 0.6139 0.7584 0.8285 0.8686 0.8939
0.1219 0.2102 0.2790 0.3351 0.3823 0.4228 0.458 I 0.4893 0.5171 0.5421 0.7010 0.7805 0.8275 0.8582
0.0704 0.1304 0.1824 0.228 1 0.2686 0.3049 0.3376 0.3673 0.3944 0.4192 0.5877 0.6804 0.7390 0.7793
0.0395 0.0757 0.1091 0.1401 0.1688 0.1956 0.2206 0.2440 0.2660 0.2867 0.4418 0.5401 0.6082 0.6583
0.3397 0.4118 0.4668 0.5109 0.5478 0.5796 0.6076 0.6324 0.6546 0.6746 0.8021 0.8637 0.8981 0.9193
0.3899 0.4841 0.5516 0.6043 0.6473 0.6832 0.7136 0.7395 0.7620 0.7815 0.8870 0.9264 0.9457 0.9570
0.4314 0.5396 0.6143 0.6708 0.7152 0.7508 0.7798 0.8038 0.8237 0.8405 0.9221 0.9493 0.9624 0.9702
0.4982
0.2603 0.3962 0.4847 0.5486 0.5974 0.6363 0.6681 0.6948 0.7174 0.7369 0.8453 0.8913 0.9166 0.9324
0.1619 0.2733 0.3560 0.4205 0.4724 0.5154 0.5516 0.5826 0.6094 0.6330 0.7707 0.8334 0.8693 0.8924
0.0845 0.1547 0.2141 0.2651 0.3095 0.3485 0.3832 0.4141 0.4420 0.4672 0.6311 0.7168 0.7697 0.8058
0.0432 0.0825 0.1185 0.1516 0.1821 0.2104 0.2366 0.2610 0.2839 0.3052 0.4628 0.5603 0.6270 0.6756
0.4790 0.5818 0.6471 0.6937 0.7292 0.7572 0.7801 0.7991 0.8152 0.8290 0.9041 0.9349 0.9512 0.9611
0.5574 0.6712 0.7374 0.7816 0.8134 0.8375 0.8563 0.8715 0.8840 0.8944 0.9461 0.9645 0.9736 0.9790
0.6126 0.7277 0.7902 0.8301 0.8578 0.8782 0.8938 0.9061 0.9160 0.9242 0.9625 0.9753 0.9816 0.9853
0.6885 0.7977 0.8515 0.8838 0.9052 0.9203 0.9316 0.9401 0.9469 0.9523 0.9766 0.9845 0.9884 0.9908
p=o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0
p= 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0
0.6234 0.7041 0.7606 0.8017 0.8323 0.8558 0.8740 0.8886 0.9002 0.9521 0.9686 0.9766 0.9814
0.5
THIN POLYCRYSTALLINE
141
METAL FILMS
parameters of the films. The procedure is similar to that explained in a previous paper4 with reference to eqns. (1) and (2). However, this procedure is quite cumbersome. Hence it is desirable to linearize eqns. (l), (2) (5) and (6) in order to simplify the calculus. This will be done in the next section. 3.
APPROXIMATE
EQUATIONS OF THE M-S
MODEL
For the study of size effects in thin films, thicknesses ranging from 100 to 1000 A are usually employed and the experimental electron mfp’s are in the range 200-500 A. In consequence, the reduced mfp, k. = d/Z,, takes values 0.2 5 k, _I 5, i.e.close to k, = 1. In this k, interval it is incorrect to employ approximate equations for “ thick films ” (k, > 1) or “ thin films ” (k, < 1) because the mathematical requirements are not fulfilled. Nevertheless, the exact equations of the M-S model can be linearized in the k, range of major interest to a fairly good approximationt. Starting from the assumption that k,,> 1 and taking into account that CI= (l/k&/( 1 -r) we conclude that a--+0. Hence, the limit of eqn. (la) is given by lim A = 3/8k, a-0
and that of eqn. (lb) is given by
Consequently,
E=
eqn. (1) can be rewritten in the form
[l--&(S) (l--p)]‘-i
which can still be approximated by taking into account that k,> 1. Then, expanding in a power series and neglecting higher power terms, we obtain p’,
1+3_ 3r+l l_r > (1-P) gko ( If allowance is made for the fact that the slope of the exact function and its approximation are not the same”, this last equation can be rearranged in the following linearized manner: PO
prd = p. [d+M(P, r)
&,I
(7)
in which M(p, r) is a function of the specularity parameter p and the reflection parameter r. By analogy with eqn. (1) the case of a thickness-independent grain size (constant LX)given by eqn. (2) can be linearized: pf d = pg [d+ W, a) &,I t For a discussion
of the linearized
equations
(8) of the F-S model,
see ref. 10.
142
E. E. MOLA,
J. M. HERAS
where N(p, a) is a function of p and ~1. The equations for the temperature coefficient of resistivity can also be linearized in the same way. Equation (5) changes into
d/Of= (l/PO)v+RP> r)
L31
(9)
and eqn. (6) can be replaced by d/P, = (l/&J V+
Qh 4 44
(10)
In Table II, values of M, N, P and Q are given for different r and CIvalues, assuming thatp = 0. It is evident that the introduction of the parameter r, coupled with the assumption of a thickness-dependent crystallite size, gives ordinate intercepts which are greater than that corresponding to the F-S equation, S(p) = 0.460 (ref. lo), whereas on assuming a thickness-independent crystallite size, the intercepts are always smaller. TABLE VALUES
II OF THE FUNCTIONS
FROM THE MAYADAS-SHATZKES
For more details, Thickness-dependent
M, N, P
AND
Q
OBTAINED
WITH
THE APPROXIMATE
EQUATIONS
DEDUCED
MODEL
see text a
Thickness-independent
r
M ip. ri feqn. (7))
Pip.
0.10 0.22 0.32 0.42 0.52
0.60 0.83 1.08 1.38 1.88
0.58 0.83 1.08 1.34 1.84
r) (eqn. 191 :
0L
Nip,
0.0 0.5 1.o 1.5 2.0
0.460 0.260 0.173 0.135 0.110
c(
cc) ieqn. flyi]
Q ip. ri ieqn. (IO)) 0.460 0.260 0.178 0.145 0.120
From examination of Table II it follows that the structural p& ,ieter tl is needed in order to define the electron scattering mechanism in highly polycrystalline films more rigorously. In order to apply the approximate equations, the experimental data of resistivity pf and thickness d should be plotted in the form p,d versus d. According to eqn. (7), for instance, the ordinate intercept will be M(p, r) I,. Deviations of experimental data from the F-S equation can be explained in terms of the different values that M(p, r) can take. However, I, must be determined previously if an unequivocal interpretation is desired. In order to show the accuracy of the approximations in the range of k, to which they are applicable, in Table III the values of pfk,/po and p‘k,/p, given by the exact equations (1) and (2)land those given by the approximations of eqns. (7) and (8) are compared as functions of ko. Also given are the values of k, /?J,$, according to eqns. (5), (6);.+9) and (10). A value of the specularity parameter p = 0 has been selected because, generally, the experimental data fit this assumption better. The values of r and c(have been chosen at random only to show that
THIN POLYCRYSTALLINE
143
METAL FILMS
the percentage deviations from the exact functions are small. The best agreement is given by the approximate eqns. (7) and (8). TABLE
III
COMPARISON AS GIVEN
OF THE EXACT
AND APPROXIMATE
BY THE EQUATIONS
VALUES
p = 0; r = 0.22 k,
p=O;a=
k,p,lp, Exact
Approx.
eqn. (Ii
eqn. (7)
k,tWBr Exact eqn. (51
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0
0.912 1.021 1.127 1.231 1.333 1.435 1.536 1.636 1.736 1.836 2.826 3.819 4.812 5.810
0.930 1.030 1.130 1.230 1.330 1.430 1.530 1.630 1.730 1.830 2.830 3.830 4.830 5.830
0.820 0.951 1.075 1.193 1.308 1.419 1.528 1.635 1.741 1.845 2.852 3.843 4.832 5.825
4.
CONCLUSIONS
0.1
OF THE FUNCTIONS
k,p,/p,,
kopr/pg
AND
k&,/&
INDICATED
Appro
I.
k&p, E.UKI
eqn. i Y
eqn.
0.930 1.030 1.130
0.238 0.343 0.441 0.538 0.635 0.733 0.831 0.929 1.027 1.126 2.118 3.117 4.116 5.115
1.230 1.330 1.430 1.530 1.630 1.730 1.830 2.830 3.830 4.830 5.830
(2)
1.5 k,Bo/Br Exact
Approx. eqn. (8)
w.
0.235 0.335 0.435 0.535 0.635 0.735 0.835 0.935 1.035 1.135 2.135 3.135 4.135 5.135
0.214 0.342 0.452 0.555 0.654 0.752 0.849 0.946 1.043 1.141 2.126 3.121 4.119 5.117
(6)
Appros. eqn. (IO) 0.245 0.345 0.445 0.545 0.645 0.745 0.845 0.945
1.045 1.145 2.145 3.145 4.145 5.145
The Mayadas-Shatzkes model for the electrical conductivity in thin polycrystalline films explains the discrepancies observed in values of the mean free path of conduction electrons for the same metal quoted in the literature. These discrepancies are probably due to the different experimental conditions and techniques employed in the preparation of the films, causing them to have quite a different structure (morphology). When the major portion of the total resistivity in thin polycrystalline metal films is due to electron scattering at the crystallite boundaries, the FuchsSondheimer model and its approximate equations will no longer be applicable because the model does not include any structural parameters. However, it is applicable to thoroughly annealed and moderately thick3 films, even though, strictly speaking, they maintain their polycrystalline character. From such films an effective mfp can be obtained. Once the magnitude of I, is known, the M-S model allows the calculation of the reflection parameter r, or more sophisticated assumptions about the dependency between crystallite size, film thickness and the potential barrier set up between crystallites4.
144
E. E. MOLA,
J. M. HERAS
However, as the fitting procedure of data and theory is always very cumbersome, we propose approximate linearized equations which give fairly good results. In the range 0.2 5 k, 5 5, the most useful approximate equation (eqn. (7)) deviates by only 1 ‘4. ACKNOWLEDGEMENTS
The authors are indebted to the Director of the Institute, Prof. Dr. H. J. Schumacher, for his encouraging interest and the facilities offered. Also the authors gratefully acknowledge the financial support offered by the Consejo National de Investigaciones Cientificas y Tecnicas de la Republica Argentina (CONICET). REFERENCES I
R. Suhrmann, J. M. Heras, L. Viscid0 de Heras and G. Wedler. Ber. Bunsenge.~. Physsik. Char.. 6X (1964)511;990;72(1968)854. 2 J. M. Heras, Thin Films. 2 (1971) 25; 39. -3 J. Borra_jo and J. M. Heras. Swface Sci.. 25 (1971) 132. SC..34(1973) 561. 4 E. E. Mola. J. Borrajo and J. M. Heras, Surfucc 5 J. M. Heras and E. Toscano. Ber. Bunsenges. Physik. Chem., 75 (1971) 1135. More details to be published. in R. Niedermayer and H. Mayer (eds.), Bus? Problems in 6 C. Kooy and J. M. Nieuwenhuiaen. Thin Film Plqsics. Proc. Intern. SFmp., Cluusthul. GBttingen. 1965. Vandenhoek and Ruprecht, Gottingen. 1966. p. 181. 7 J. W. Geus. in J. R. Anderson (ed.), Chemisorption and Reaction on Metallic Films. Academic Press. London, 1971, p. 400. 8 A. F. Mayadas and M. Shatzkes. Phqa. Rec. B. I (1970) 1382. Twhnol.. in the press. 9 E. E. Mola and J. M. Heras, Electrocomponent 10 J. Borrajo and J. M. Heras. Thin Solid Films, in the press.
137
in Switzerland
EXACT AND APPROXIMATE EQUATIONS FOR THE THICKNESS DEPENDENCE OF RESISTIVITY AND ITS TEMPERATURE COEFFICIENT IN THIN POLYCRYSTALLINE METAL FILMS
E. E. MOLA AND J. M. HERAS Institute de Incestigaciones Fisicoquimicus Teciriccrs y Aplicadas (Argentina] (Received April 13, 1973; accepted July 5, 1973)
(INIFTA),
Calle 47 y 115. La Plats
Starting from the Mayadas-Shatzkes model for the conductivity in thin polycrystalline metal films, an exact expression for the dependence of the temperature coefficient of resistivity (t.c.r.) on film thickness is derived. In order to allow a quick comparison of experimental data with the model, approximate equations are also derived for the dependence on thickness of both resistivity and its temperature coefficient. These approximations are valid in the range 0.2 5 k, 5 5 (where k, is the ratio between film thickness d and the electron mean free path &,). Numerical tables are also given (i) for the values of the exact function of the t.c.r. and (ii) to show the agreement of the approximate equations with the exact functions.
1. INTRODUCTION
The electrical resistivity of thin evaporated metal films changes when foreign atoms or molecules chemisorb on their surfaces’. In order to interpret this influence on the electrical properties of such films, a study of the scattering mechanism of conduction electrons in very pure films is unavoidable. This has been the object of our study for a number of years2-4. We have found experimentally that the well-known electrical conductivity model of Fuchs-Sondheimer (F-S model) can only be applied to thin evaporated polycrystalline films (thickness < 100 A) which have been thoroughly annealed at temperatures higher than 373 “K 4, 5. This behaviour can be ascribed to a morphological parameter such as the crystallite size, which can be modified by an increasing annealing temperature. The F-S model has nothing to say about this parameter. It is known that the crystallites in evaporated films have a columnar shape, rising from the substrate to the film surface 5-7 . Hence, there is a distribution of grain boundaries perpendicular to the direction of the applied electric field. The potential barrier set up between crystallite boundaries may have a variable reflecting power for the approaching electron. When the crystallite size and film thickness become comparable, the contribution of the crystallite boundaries to electron scattering increases with decreasing film thickness. Consequently,
138
E. E. MOLA,
J. M. HJXAS
as long as the crystallite size is smaller than the electron mean free path (mfp) the F-S model cannot be applicable. Taking these circumstances into account, Mayadas and Shatzkes recently developed a model’ (M-S model) introducing a structural parameter tl, which includes the crystallite size D (or its dependence on film thickness 4 and the probability r for an electron to be reflected by a crystallite boundary perpendicular to the applied electric field. The model can be summarized by the following equations* :
PfIPO= m4 -4-
1
(1)
in which pr is the actual film resistivity, p0 is the resistivity of an infinitely thick monocrystalline jilm (i.e. without size effects),
and f(a)= l--iu+3r2-3!x3
In
(
l+’
(lb)
CI>
Equation (1) was deduced assuming that the crystallite diameter D is strictly equivalent to film thickness d. Hence, the following dependence between cxand d is valid: 1 “k,
r w
l-r
where k, = d/& (I, is the electron mfp within a crystallite) and r is the electron reflection probability parameter with values between 0 and 1. Under the assumption that a is thickness independent with values ~20 (when rx = 0, the M-S model changes into the F-S model) eqn. (1) can be modified to
P& = [l -A/fCW’
(2)
in which the new symbol ps is the resistivity of an infinitely thick polycrystalline .film.
2.
THE M-S MODEL APPLIED
TO THE TEMPERATURE
COEFFICIENT
OF RESISTIVITY
Equations (1) and (2) also allow the derivation of equivalent expressions for the t.c.r.t Assuming that the rigid band model of metals is valid, the relation between the electron mfp 1 and resistivity p is given by the Sommerfeld equation * Details about the possibilities of the model have been given in previous papers’, 9. t We distinguish between the temperature coefficient of resistivity (t.c.r.) and that of resistance
(TCR).
THIN
F’OLYCRYSTALLINE
METAL
139
FILMS
I= 1.27x lo4 N-2’3 p-r
(3)
in which N is the number of conduction electrons per unit volume. Assuming N to be temperature independent, the logarithmic differentiation of eqn. (3) with respect to temperature gives 1 dl _ 1 dT
ldp_ P dT
(4)
P
which is by definition the t.c.r. Applying these concepts to eqns. (1) and (2), the following expressions for the t.c.r. dependence on thickness in terms of the M-S model can be obtained. (i) Assuming a dependence of o! on thickness (crystallite diameter D equals film thickness d), ELl+ PO
g(x)-A+B+C f(x) - A
(5)
and (ii) assuming u 2 0 and thickness independent,
Ll+
D-A
PO
f(cd- A
(6)
where & is the actually measured t.c.r., PO is the t.c.r. of an infinitely thick monocrystalline film, g(u) = -;
a+6a2++&9a31n 42
(
l+d
>
cc
B = z (1 -P)~ 1 d4 i;;;,$)(r;
r;)
exp {-k,t [l-pew
{--kd
H(t, 4)) W,$))]2
dt
1- exp { - kot Wt, N> ’ 1-pexp{-k,tH(t,4)} and
with H(t,$h)=
l++os~ (1-~)li2
dt
140
E. E. MOLA, .I. M. HERAS
Equations (l), (2), (5) and (6) have been evaluated analytically with a digital computer. The results obtained with eqns. (1) and (2) and some details of the calculus have already been given elsewhere4, 9. Those corresponding to eqns. (5) and (6) are given in Table I for two values of the Fuchs specularity parameter p (the probability that an electron will be specularly reflected upon scattering on a film surface). With the data of Table I, theoretical curves can be drawn for each of the given values of a and Y.These curves allow a comparison between the experimental data for t.c.r. and the Mayadas-Shatzkes model in order to find the structural TABLE
I
VALUES OF THE EXACT /&/&,
ON FILM
FUNCTION
THICKNESS
IN THIN POLYCRYSTALLINE
d,
FOR THE DEPENDENCE
CALCULATED
WITH
OF THE TEMPERATURE
THE MAYADAS-SHATZKES
COEFFICIENT MODEL
OF RESISTIVITY
FOR THE RESISTANCE
FILMS
For more details, see text Equution k,
(51
Equution
(61
r = 0.1
Y = 0.22
r = 0.42
r = 0.62
r=O
r. = 0.5
z = 1.0
r* = 2.0
0.1862 0.2898 0.3623 0.4184 0.4641 0.5026 0.5358 0.5650 0.5908 0.6139 0.7584 0.8285 0.8686 0.8939
0.1219 0.2102 0.2790 0.3351 0.3823 0.4228 0.458 I 0.4893 0.5171 0.5421 0.7010 0.7805 0.8275 0.8582
0.0704 0.1304 0.1824 0.228 1 0.2686 0.3049 0.3376 0.3673 0.3944 0.4192 0.5877 0.6804 0.7390 0.7793
0.0395 0.0757 0.1091 0.1401 0.1688 0.1956 0.2206 0.2440 0.2660 0.2867 0.4418 0.5401 0.6082 0.6583
0.3397 0.4118 0.4668 0.5109 0.5478 0.5796 0.6076 0.6324 0.6546 0.6746 0.8021 0.8637 0.8981 0.9193
0.3899 0.4841 0.5516 0.6043 0.6473 0.6832 0.7136 0.7395 0.7620 0.7815 0.8870 0.9264 0.9457 0.9570
0.4314 0.5396 0.6143 0.6708 0.7152 0.7508 0.7798 0.8038 0.8237 0.8405 0.9221 0.9493 0.9624 0.9702
0.4982
0.2603 0.3962 0.4847 0.5486 0.5974 0.6363 0.6681 0.6948 0.7174 0.7369 0.8453 0.8913 0.9166 0.9324
0.1619 0.2733 0.3560 0.4205 0.4724 0.5154 0.5516 0.5826 0.6094 0.6330 0.7707 0.8334 0.8693 0.8924
0.0845 0.1547 0.2141 0.2651 0.3095 0.3485 0.3832 0.4141 0.4420 0.4672 0.6311 0.7168 0.7697 0.8058
0.0432 0.0825 0.1185 0.1516 0.1821 0.2104 0.2366 0.2610 0.2839 0.3052 0.4628 0.5603 0.6270 0.6756
0.4790 0.5818 0.6471 0.6937 0.7292 0.7572 0.7801 0.7991 0.8152 0.8290 0.9041 0.9349 0.9512 0.9611
0.5574 0.6712 0.7374 0.7816 0.8134 0.8375 0.8563 0.8715 0.8840 0.8944 0.9461 0.9645 0.9736 0.9790
0.6126 0.7277 0.7902 0.8301 0.8578 0.8782 0.8938 0.9061 0.9160 0.9242 0.9625 0.9753 0.9816 0.9853
0.6885 0.7977 0.8515 0.8838 0.9052 0.9203 0.9316 0.9401 0.9469 0.9523 0.9766 0.9845 0.9884 0.9908
p=o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0
p= 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0
0.6234 0.7041 0.7606 0.8017 0.8323 0.8558 0.8740 0.8886 0.9002 0.9521 0.9686 0.9766 0.9814
0.5
THIN POLYCRYSTALLINE
141
METAL FILMS
parameters of the films. The procedure is similar to that explained in a previous paper4 with reference to eqns. (1) and (2). However, this procedure is quite cumbersome. Hence it is desirable to linearize eqns. (l), (2) (5) and (6) in order to simplify the calculus. This will be done in the next section. 3.
APPROXIMATE
EQUATIONS OF THE M-S
MODEL
For the study of size effects in thin films, thicknesses ranging from 100 to 1000 A are usually employed and the experimental electron mfp’s are in the range 200-500 A. In consequence, the reduced mfp, k. = d/Z,, takes values 0.2 5 k, _I 5, i.e.close to k, = 1. In this k, interval it is incorrect to employ approximate equations for “ thick films ” (k, > 1) or “ thin films ” (k, < 1) because the mathematical requirements are not fulfilled. Nevertheless, the exact equations of the M-S model can be linearized in the k, range of major interest to a fairly good approximationt. Starting from the assumption that k,,> 1 and taking into account that CI= (l/k&/( 1 -r) we conclude that a--+0. Hence, the limit of eqn. (la) is given by lim A = 3/8k, a-0
and that of eqn. (lb) is given by
Consequently,
E=
eqn. (1) can be rewritten in the form
[l--&(S) (l--p)]‘-i
which can still be approximated by taking into account that k,> 1. Then, expanding in a power series and neglecting higher power terms, we obtain p’,
1+3_ 3r+l l_r > (1-P) gko ( If allowance is made for the fact that the slope of the exact function and its approximation are not the same”, this last equation can be rearranged in the following linearized manner: PO
prd = p. [d+M(P, r)
&,I
(7)
in which M(p, r) is a function of the specularity parameter p and the reflection parameter r. By analogy with eqn. (1) the case of a thickness-independent grain size (constant LX)given by eqn. (2) can be linearized: pf d = pg [d+ W, a) &,I t For a discussion
of the linearized
equations
(8) of the F-S model,
see ref. 10.
142
E. E. MOLA,
J. M. HERAS
where N(p, a) is a function of p and ~1. The equations for the temperature coefficient of resistivity can also be linearized in the same way. Equation (5) changes into
d/Of= (l/PO)v+RP> r)
L31
(9)
and eqn. (6) can be replaced by d/P, = (l/&J V+
Qh 4 44
(10)
In Table II, values of M, N, P and Q are given for different r and CIvalues, assuming thatp = 0. It is evident that the introduction of the parameter r, coupled with the assumption of a thickness-dependent crystallite size, gives ordinate intercepts which are greater than that corresponding to the F-S equation, S(p) = 0.460 (ref. lo), whereas on assuming a thickness-independent crystallite size, the intercepts are always smaller. TABLE VALUES
II OF THE FUNCTIONS
FROM THE MAYADAS-SHATZKES
For more details, Thickness-dependent
M, N, P
AND
Q
OBTAINED
WITH
THE APPROXIMATE
EQUATIONS
DEDUCED
MODEL
see text a
Thickness-independent
r
M ip. ri feqn. (7))
Pip.
0.10 0.22 0.32 0.42 0.52
0.60 0.83 1.08 1.38 1.88
0.58 0.83 1.08 1.34 1.84
r) (eqn. 191 :
0L
Nip,
0.0 0.5 1.o 1.5 2.0
0.460 0.260 0.173 0.135 0.110
c(
cc) ieqn. flyi]
Q ip. ri ieqn. (IO)) 0.460 0.260 0.178 0.145 0.120
From examination of Table II it follows that the structural p& ,ieter tl is needed in order to define the electron scattering mechanism in highly polycrystalline films more rigorously. In order to apply the approximate equations, the experimental data of resistivity pf and thickness d should be plotted in the form p,d versus d. According to eqn. (7), for instance, the ordinate intercept will be M(p, r) I,. Deviations of experimental data from the F-S equation can be explained in terms of the different values that M(p, r) can take. However, I, must be determined previously if an unequivocal interpretation is desired. In order to show the accuracy of the approximations in the range of k, to which they are applicable, in Table III the values of pfk,/po and p‘k,/p, given by the exact equations (1) and (2)land those given by the approximations of eqns. (7) and (8) are compared as functions of ko. Also given are the values of k, /?J,$, according to eqns. (5), (6);.+9) and (10). A value of the specularity parameter p = 0 has been selected because, generally, the experimental data fit this assumption better. The values of r and c(have been chosen at random only to show that
THIN POLYCRYSTALLINE
143
METAL FILMS
the percentage deviations from the exact functions are small. The best agreement is given by the approximate eqns. (7) and (8). TABLE
III
COMPARISON AS GIVEN
OF THE EXACT
AND APPROXIMATE
BY THE EQUATIONS
VALUES
p = 0; r = 0.22 k,
p=O;a=
k,p,lp, Exact
Approx.
eqn. (Ii
eqn. (7)
k,tWBr Exact eqn. (51
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0
0.912 1.021 1.127 1.231 1.333 1.435 1.536 1.636 1.736 1.836 2.826 3.819 4.812 5.810
0.930 1.030 1.130 1.230 1.330 1.430 1.530 1.630 1.730 1.830 2.830 3.830 4.830 5.830
0.820 0.951 1.075 1.193 1.308 1.419 1.528 1.635 1.741 1.845 2.852 3.843 4.832 5.825
4.
CONCLUSIONS
0.1
OF THE FUNCTIONS
k,p,/p,,
kopr/pg
AND
k&,/&
INDICATED
Appro
I.
k&p, E.UKI
eqn. i Y
eqn.
0.930 1.030 1.130
0.238 0.343 0.441 0.538 0.635 0.733 0.831 0.929 1.027 1.126 2.118 3.117 4.116 5.115
1.230 1.330 1.430 1.530 1.630 1.730 1.830 2.830 3.830 4.830 5.830
(2)
1.5 k,Bo/Br Exact
Approx. eqn. (8)
w.
0.235 0.335 0.435 0.535 0.635 0.735 0.835 0.935 1.035 1.135 2.135 3.135 4.135 5.135
0.214 0.342 0.452 0.555 0.654 0.752 0.849 0.946 1.043 1.141 2.126 3.121 4.119 5.117
(6)
Appros. eqn. (IO) 0.245 0.345 0.445 0.545 0.645 0.745 0.845 0.945
1.045 1.145 2.145 3.145 4.145 5.145
The Mayadas-Shatzkes model for the electrical conductivity in thin polycrystalline films explains the discrepancies observed in values of the mean free path of conduction electrons for the same metal quoted in the literature. These discrepancies are probably due to the different experimental conditions and techniques employed in the preparation of the films, causing them to have quite a different structure (morphology). When the major portion of the total resistivity in thin polycrystalline metal films is due to electron scattering at the crystallite boundaries, the FuchsSondheimer model and its approximate equations will no longer be applicable because the model does not include any structural parameters. However, it is applicable to thoroughly annealed and moderately thick3 films, even though, strictly speaking, they maintain their polycrystalline character. From such films an effective mfp can be obtained. Once the magnitude of I, is known, the M-S model allows the calculation of the reflection parameter r, or more sophisticated assumptions about the dependency between crystallite size, film thickness and the potential barrier set up between crystallites4.
144
E. E. MOLA,
J. M. HERAS
However, as the fitting procedure of data and theory is always very cumbersome, we propose approximate linearized equations which give fairly good results. In the range 0.2 5 k, 5 5, the most useful approximate equation (eqn. (7)) deviates by only 1 ‘4. ACKNOWLEDGEMENTS
The authors are indebted to the Director of the Institute, Prof. Dr. H. J. Schumacher, for his encouraging interest and the facilities offered. Also the authors gratefully acknowledge the financial support offered by the Consejo National de Investigaciones Cientificas y Tecnicas de la Republica Argentina (CONICET). REFERENCES I
R. Suhrmann, J. M. Heras, L. Viscid0 de Heras and G. Wedler. Ber. Bunsenge.~. Physsik. Char.. 6X (1964)511;990;72(1968)854. 2 J. M. Heras, Thin Films. 2 (1971) 25; 39. -3 J. Borra_jo and J. M. Heras. Swface Sci.. 25 (1971) 132. SC..34(1973) 561. 4 E. E. Mola. J. Borrajo and J. M. Heras, Surfucc 5 J. M. Heras and E. Toscano. Ber. Bunsenges. Physik. Chem., 75 (1971) 1135. More details to be published. in R. Niedermayer and H. Mayer (eds.), Bus? Problems in 6 C. Kooy and J. M. Nieuwenhuiaen. Thin Film Plqsics. Proc. Intern. SFmp., Cluusthul. GBttingen. 1965. Vandenhoek and Ruprecht, Gottingen. 1966. p. 181. 7 J. W. Geus. in J. R. Anderson (ed.), Chemisorption and Reaction on Metallic Films. Academic Press. London, 1971, p. 400. 8 A. F. Mayadas and M. Shatzkes. Phqa. Rec. B. I (1970) 1382. Twhnol.. in the press. 9 E. E. Mola and J. M. Heras, Electrocomponent 10 J. Borrajo and J. M. Heras. Thin Solid Films, in the press.