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Computer simulations on the electrical resistivities of metallic superlattices based on simple model
Journal of Magnetism and Magnetic Materials 126 (1993) 475-478 North-Holland
Adm
Computer simulations on the electrical resistivities of metallic superlattices based on simple model Y. Inoue, K. Mae, N. Kyuno, T. Kaneko and R. Y a m a m o t o Institute of Industrial Science, University of Tokyo, Japan We have calculated the resistivity and temperature coefficient resistance (TCR) of multilayers by a theory analogous with the Fuchs-Sondheimer theory. The calculated results of the bilayer thickness dependence of resistivity and TCR of Au/Pd, AI/Ag, Mo/Ta, Pd/Co and Ag/Co are in good agreement with experiments. I. Introduction
described as follows:
A lot of studies of metallic multilayer films have been performed and several novel properties were reported. Most of the novel properties arise from the introducing of interfaces in metallic multilayers. The electrical transport properties of metallic multilayer films are expected to have a particular behavior because of electron scattering at the interfaces. Several resistivity measurements have been performed on metallic multilayers [1-10]. But only a few theoretical calculations on this subject have appeared [11-13] and quantitative comparison of experimental results with the theoretical calculations has not been made. Here we present a simple theoretical model which explains the dependence of resistivity and temperature coefficient resistance (TCR) of multilayers on bilayer thickness. Furthermore, the parameters obtained from the calculations give some general features of electrical conduction in metallic multilayers.
~ri°
2. Calculation model
We have proposed a model [14] analogous with the Fuchs-Sondheimer theory which is applied to calculate the resistivity of thin films. The multilayered films are assumed to have two different kind of metals, alternatively grown on each other with constant layer thickness d 1 and d 2, where A = d I + d 2 is called bilayer thickness. The details of the calculation are presented in our previous paper [14], the conductivities and TCR of each layer are
Correspondence to: Dr. Y. Inoue, Institute of Industrial Science, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106, Japan.
-l+Qi =1+
3-1F'~-
{1-exp(-
T))d ki
(1)
1 OQikl/3°+~Qik2/3° 1 + Qi Ok1 or%
/3i =/30
( i = 1, 2), (2)
where ~ri° and/30 are the conductivity and TCR of the infinitely thick film of metals i (i = 1, 2), respectively. Effects of thickness of the layers of metals i in the multilayer film are described by Oi. The conductivity O"m and resistivity Pm of the whole multilayer can be calculated by
~,.
1
dlO-1 + d2tr 2
Pm
da + d2
,
(3)
and TCR of the multilayer,/3m, is given by
/3m
=
dlO'l/31 + d20r2/32 ]. dw1 + d2°'2 ]
(4)
3. Results and discussions
We have calculated dependence of resistivity and TCR on the bilayer thickness for several kinds of metallic multilayers and have made systematic and quantitative comparisons between the experimental and calculated results. The resistivities, p0 and p2°, the mean free paths, A1 and A2, the temperature coefficients of resistance, /30 and/3 °, of the very thick films of metal 1 and 2 should be known to calculate the resistivity and TCR of the
0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. (North-Holland)
476
E lnoue et al. / Computer simulations on electrical resisticities
Table 1 The parameters used for the calculation
V v [10 n cm/s] m */ m p°/311 [10 3 bt~Qcm/K] p°a [~1) cm A]
AI
Ni
Cu
Nb
Mo
Pd
Ta
Au
1.32 1.13 9.96 545
0.3 5.0 31.0 350
1.13 1.34 6.12 741
0.63 3.22 53.0 412
0.81 1.93 22.6 679
0.35 6.06 38.2 370
0.65 2.93 45.8 460
1.32 t .03 7.65 92(/
multilayer. We also need the values of electron mass m * and the Fermi velocity VF for metal 1 and 2. The parameters used for the calculations are listed in table 1. The values of Fermi velocity were taken from the results of calculation by Papaconstantopoulus [15]. The electron mass was estimated by the nearly free electron model to reproduce the density of state, Fermi velocity, Fermi energy and plasma energy of the metal. The product p°A was calculated from Fermi velocity and the plasma frequency of the bulk metal by O°A = m V v / ne 2 = 4,rrVv /~2 p.
We assumed that the product p°/3° takes the same value as the ideal bulk metal, p°/3° =pbj~b The bulk values p6 and /3 b were obtained from the data collected by Meaden [16]. For simplicity of calculations electrons are assumed to have the same probability for specular transmission from metal 1 to metal 2 and from metal 2 to metal 1. The reflection coefficients for the metal 1 and metal 2 are also assumed to have the same value. The resistivity, p0 and pO, the transmission parameter t and the reflection coefficient r were used as adjustable parameters and determined to fit the experimental data.
The parameters thus obtained are listed in table 2 with related values. The calculated results of the bilayer thickness dependence of resistivity and T C R reproduce well the experimental ones for two kinds of A u / P d [1,2], A I / A g [3] and M o / T a [4]. In fig. 1 we show the experimental and calculated results of inplane resistivity and T C R as functions of inverse bilayer thickness I / A for A u / P d [1]. The transmission parameter t is 0.94 and the reflection coefficient r is 0.05. Very few electrons are scattered diffusively at the interfaces in A u / P d . We have successfully fitted the experimental data of the A u / P d multilayers deposited by de Vries et al. [2]. The parameters obtained from the fitting differ from those of the A u / P d multilayer by Carcia et al. [1]. The difference of the parameters would be due to the difference in the deposition methods. Carcia et al. synthesized A u / P d by rf sputtering and de Vries et al. used a method of vacuum evaporation. The bilayer thickness dependence of resistivity and T C R of M o / C u were fitted to the experimental results [5]. Both t and r were zero in the best fitting. In other words all the conduction electrons are scattered diffu-
Au/Pd 50
o•
20
(Exp.) I pp (Cal.)
40 15 ~" 3O
\
7~ \.O.
lO~
20 5
10
• - - ¢'-
0 0
10
20
30 40 50 A z (x103~. q)
60
70
TCR (Exp.)) I TCR (Cal.) i
I
80
Fig. 1. The bilayer thickness dependence of resistivity and TCR of A u / P d multilayer. The solid circles and the solid triangles are the observed values replotted from It]. The open circles and the open triangles are the best fit to the observed data.
477
Y. Inoue et al. / Computer simulations on electrical resistit,ities
Table 2 The transmission probability, the reflection coefficient and the mean free path provide the best fit to experimental results Ref.
t
r
p0
[1]
0.94
0.05
Au/Pd
[2]
0.20
0.43
AI/Ag
[3]
1.0
0.0
Mo/Ta
[4]
0.64
0.36
Mo/Cu
[5]
0.0
0.0
Nb/Cu
[6]
0.0
0.0
Mo/Ni
[7]
0.0
0.0
Nb/AI
[8]
0.0
0.0
Cu/Ta
[9]
0.0
0.0
Au Pd Au Pd Al Ag Mo Ta Cu Mo Nb Cu Mo Ni Nb AI Cu Ta
I
A
[~ Au/Pd
Nb/Cu 150
cm]
1.37 174 2.94 17.8 90.7 5.78 16.7 116 8.97 64.0 144 10.5 24.4 140 144 11.0 4.1 461
Experimental ] Calculated I
o•
[A.] 671 2.13 313 20.8 6.0 159 40.6 3.98 82.6 10.6 2.85 70.8 27.8 2.5 2.85 49.7 183 2.85
100 =I.
50
0 .
Experimental ] Calculated I
o 20
10 o~ c) ,0
0 sively at the interfaces. The multilayers such as N b / C u [6] which has the fcc(1 1 1) and bcc(1 1 0) planes parallel to the surface had the same tendency as M o / C u . The multilayer films which have Ni layer as a constituent metal layer showed relatively high resistivities and T C R decreased drastically as A decreased. Fig. 2 shows the experimental and calculated dependence of T C R of M o / N i [7] on 1 / A . The present model could not explain the sudden decrease of T C R in these multilayers. Fig. 3 describes the dependence of resistivity and T C R of N b / C u synthesized by W e r n e r et al. [6] on l / A . The slope of the increase of resistivity with 1 / A Mo/Ni 30 • o
Experimental Calculated
2O 10 ,.#
oo
,,
.2oo
.............
0
0
20
40
O'-~
60
.....
80
O,
O
O
cl.
10
. . . .
0
I
10
. . . .
I
20
. . . .
I
. . . .
L~_,
30 40 A I (xi0 3/kl)
~
,
I
50
,
,
~
,
60
Fig. 3. The bilayer thickness dependence of resistivity and TCR of Nb/Cu synthesized by Werner et al. [6]. was so large that the theoretical calculation could not reproduce the experimental results. The steep increase of resistivity similar to N b / C u was observed in N b / A I [8], C u / T a [9] and other N b / C u prepared by Lowe et al. [10]. The T C R of N b / C u at the shortest A had a negative value, which was also observed in N b / A I [7] and M o / N i [7]. This negative T C R could not be explained by the present model. The negative T C R could be explained as follows; the grain size of multilayer becomes small as A decreases. Therefore, the residual resistivity largely increases by the grain boundary scattering. The large resistivity reduces the positive phonon contribution and enhances the negative temperature dependence described by the D e b y e - W a l l e r factor [17]. 4. Conclusions
l O0
A I (xlO 3~1)
Fig. 2. Dependence of TCR of Mo/Ni [7] on the inverse bilayer thickness.
We have calculated the resistivity and T C R of multilayers by the model analogous with the F u c h s Sondheimer theory and compared the calculated dependence of resistivity and T C R on bilayer thickness with the experimental results. The results of the calculations for two kinds of A u / P d multilayer, A I / A g ,
478
Y. lnoue et al. / Computer simulations on electrical resistit'ities
M o / T a a n d M o / C u have r e p r o d u c e d well the bilayer thickness d e p e n d e n c e . In o t h e r words, the bilayer thickness d e p e n d e n c e of these multilayers is governed by the interface scattering. All the conduction electrons were scattered diffusely at the interfaces in the multilayers which have fcc(1 1 1) and bcc(1 1 0) planes parallel to the surface. T C R of the multilayer films constituted by Ni a n d a n o t h e r m e t a l d e c r e a s e d drastically as the bilayer thickness decreased. T h e calculation could not fit this s u d d e n decrease of TCR. The slope of increase of resistivity of N b / C u a n d C u / T a with the inverse bilayer thickness was so large that the calculation could not r e p r o d u c e the e x p e r i m e n t a l results. Negative T C R was observed in these multilayers at very short bilayer thickness. T h e large increase of resistivity and negative T C R could be due to the scattering at the grain boundaries. Acknowledgements." T h e a u t h o r s arc grateful to Dr. Hideyuki Sato (Tokyo M e t r o p o l i t a n University) for useful discussions a n d e n c o u r a g e m e n t . This work was s u p p o r t e d by a G r a n t - i n Aid for Scientific R e s e a r c h from the Ministry of Education, Science and Culture, Japan.
References [1] P.F. Carcia and A. Suna, J. Appl. Phys. 54 (1983) 2000. [2] J.W.C. de Vries and F.J.A. den Broeder, J. Phys. F 18 (1988) 2653.
[3] H. Sato, I. Sakamoto, K. Yonemitsu, W.P. Pratt, Jr, W. Abudul-Razzaq, J. Slaughter and P.A. Schroeder, in: Multilayers, Mat. Res. Soc. (1988). [4] J.L. Makous, C.M. Falco, R. Vaglio and A. Cucolo, Jpn. J. Appl. Phys. 26 (1987) 1467. [5] T. Kaneko, T. Sasaki, M. Sakuda, R.. Yamamoto, T. Nakamura, H. Yamamoto and S. Tanaka, J. Phys. F 18 (1988) 2053. [6] T.R. Werner, 1. Banerjee, Q.S. Yang, C.M. Falco and I.K. Schuller, Phys. Rev. B 26 (1982) 2224. [7] M.R. Khan, C.S.L. Chun, G.P. Felcher. M. Grimsditch, A. Kunny, C.M. Falco and I.K. Schuller, Phys. Rev. B 27 (1983) 7186. [8] M. Gurvitch, Phys. Rev. B 34 (1986) 540. [9] G. Reiss, K. Kapfberger, G. Meier, J. Vancea and H. Hoffmann, J. Phys.: Condens. Matter I (1989) 1275. [10] W.P. ix)we, T.W. Barbee, Jr., T.ft. Geballe and D.B. McWhan, Phys. Rev. B 26 (1981) 6193. [11] R. Dimmich, J. Phys. F 15 (1985) 2477. [12] C.-X. Chert, J. Phys.: Condens. Matter I (1989) 3919. [13] R.E. Camley and J.E. Barnfis, Phys. Rev. Lett. 63 (1989) 664. [14] T. Kaneko, Y. Inoue and R. Yamamoto, J. Phys. ('ondens. Matter 4 (1992). [15] D.A. Papaconstantopoulos, The Band Structure of Elemental Solids (Plenum Press, New York, 1986). [16] G.T. Meaden, Electrical Resistance of Metals (Plenum Press, New York, 1965). [17] P.L. Rossiter, The Electrical Resistivity of Metals and Alloys (Cambridge University, Press, Cambridge, 1987).
Adm
Computer simulations on the electrical resistivities of metallic superlattices based on simple model Y. Inoue, K. Mae, N. Kyuno, T. Kaneko and R. Y a m a m o t o Institute of Industrial Science, University of Tokyo, Japan We have calculated the resistivity and temperature coefficient resistance (TCR) of multilayers by a theory analogous with the Fuchs-Sondheimer theory. The calculated results of the bilayer thickness dependence of resistivity and TCR of Au/Pd, AI/Ag, Mo/Ta, Pd/Co and Ag/Co are in good agreement with experiments. I. Introduction
described as follows:
A lot of studies of metallic multilayer films have been performed and several novel properties were reported. Most of the novel properties arise from the introducing of interfaces in metallic multilayers. The electrical transport properties of metallic multilayer films are expected to have a particular behavior because of electron scattering at the interfaces. Several resistivity measurements have been performed on metallic multilayers [1-10]. But only a few theoretical calculations on this subject have appeared [11-13] and quantitative comparison of experimental results with the theoretical calculations has not been made. Here we present a simple theoretical model which explains the dependence of resistivity and temperature coefficient resistance (TCR) of multilayers on bilayer thickness. Furthermore, the parameters obtained from the calculations give some general features of electrical conduction in metallic multilayers.
~ri°
2. Calculation model
We have proposed a model [14] analogous with the Fuchs-Sondheimer theory which is applied to calculate the resistivity of thin films. The multilayered films are assumed to have two different kind of metals, alternatively grown on each other with constant layer thickness d 1 and d 2, where A = d I + d 2 is called bilayer thickness. The details of the calculation are presented in our previous paper [14], the conductivities and TCR of each layer are
Correspondence to: Dr. Y. Inoue, Institute of Industrial Science, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106, Japan.
-l+Qi =1+
3-1F'~-
{1-exp(-
T))d ki
(1)
1 OQikl/3°+~Qik2/3° 1 + Qi Ok1 or%
/3i =/30
( i = 1, 2), (2)
where ~ri° and/30 are the conductivity and TCR of the infinitely thick film of metals i (i = 1, 2), respectively. Effects of thickness of the layers of metals i in the multilayer film are described by Oi. The conductivity O"m and resistivity Pm of the whole multilayer can be calculated by
~,.
1
dlO-1 + d2tr 2
Pm
da + d2
,
(3)
and TCR of the multilayer,/3m, is given by
/3m
=
dlO'l/31 + d20r2/32 ]. dw1 + d2°'2 ]
(4)
3. Results and discussions
We have calculated dependence of resistivity and TCR on the bilayer thickness for several kinds of metallic multilayers and have made systematic and quantitative comparisons between the experimental and calculated results. The resistivities, p0 and p2°, the mean free paths, A1 and A2, the temperature coefficients of resistance, /30 and/3 °, of the very thick films of metal 1 and 2 should be known to calculate the resistivity and TCR of the
0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. (North-Holland)
476
E lnoue et al. / Computer simulations on electrical resisticities
Table 1 The parameters used for the calculation
V v [10 n cm/s] m */ m p°/311 [10 3 bt~Qcm/K] p°a [~1) cm A]
AI
Ni
Cu
Nb
Mo
Pd
Ta
Au
1.32 1.13 9.96 545
0.3 5.0 31.0 350
1.13 1.34 6.12 741
0.63 3.22 53.0 412
0.81 1.93 22.6 679
0.35 6.06 38.2 370
0.65 2.93 45.8 460
1.32 t .03 7.65 92(/
multilayer. We also need the values of electron mass m * and the Fermi velocity VF for metal 1 and 2. The parameters used for the calculations are listed in table 1. The values of Fermi velocity were taken from the results of calculation by Papaconstantopoulus [15]. The electron mass was estimated by the nearly free electron model to reproduce the density of state, Fermi velocity, Fermi energy and plasma energy of the metal. The product p°A was calculated from Fermi velocity and the plasma frequency of the bulk metal by O°A = m V v / ne 2 = 4,rrVv /~2 p.
We assumed that the product p°/3° takes the same value as the ideal bulk metal, p°/3° =pbj~b The bulk values p6 and /3 b were obtained from the data collected by Meaden [16]. For simplicity of calculations electrons are assumed to have the same probability for specular transmission from metal 1 to metal 2 and from metal 2 to metal 1. The reflection coefficients for the metal 1 and metal 2 are also assumed to have the same value. The resistivity, p0 and pO, the transmission parameter t and the reflection coefficient r were used as adjustable parameters and determined to fit the experimental data.
The parameters thus obtained are listed in table 2 with related values. The calculated results of the bilayer thickness dependence of resistivity and T C R reproduce well the experimental ones for two kinds of A u / P d [1,2], A I / A g [3] and M o / T a [4]. In fig. 1 we show the experimental and calculated results of inplane resistivity and T C R as functions of inverse bilayer thickness I / A for A u / P d [1]. The transmission parameter t is 0.94 and the reflection coefficient r is 0.05. Very few electrons are scattered diffusively at the interfaces in A u / P d . We have successfully fitted the experimental data of the A u / P d multilayers deposited by de Vries et al. [2]. The parameters obtained from the fitting differ from those of the A u / P d multilayer by Carcia et al. [1]. The difference of the parameters would be due to the difference in the deposition methods. Carcia et al. synthesized A u / P d by rf sputtering and de Vries et al. used a method of vacuum evaporation. The bilayer thickness dependence of resistivity and T C R of M o / C u were fitted to the experimental results [5]. Both t and r were zero in the best fitting. In other words all the conduction electrons are scattered diffu-
Au/Pd 50
o•
20
(Exp.) I pp (Cal.)
40 15 ~" 3O
\
7~ \.O.
lO~
20 5
10
• - - ¢'-
0 0
10
20
30 40 50 A z (x103~. q)
60
70
TCR (Exp.)) I TCR (Cal.) i
I
80
Fig. 1. The bilayer thickness dependence of resistivity and TCR of A u / P d multilayer. The solid circles and the solid triangles are the observed values replotted from It]. The open circles and the open triangles are the best fit to the observed data.
477
Y. Inoue et al. / Computer simulations on electrical resistit,ities
Table 2 The transmission probability, the reflection coefficient and the mean free path provide the best fit to experimental results Ref.
t
r
p0
[1]
0.94
0.05
Au/Pd
[2]
0.20
0.43
AI/Ag
[3]
1.0
0.0
Mo/Ta
[4]
0.64
0.36
Mo/Cu
[5]
0.0
0.0
Nb/Cu
[6]
0.0
0.0
Mo/Ni
[7]
0.0
0.0
Nb/AI
[8]
0.0
0.0
Cu/Ta
[9]
0.0
0.0
Au Pd Au Pd Al Ag Mo Ta Cu Mo Nb Cu Mo Ni Nb AI Cu Ta
I
A
[~ Au/Pd
Nb/Cu 150
cm]
1.37 174 2.94 17.8 90.7 5.78 16.7 116 8.97 64.0 144 10.5 24.4 140 144 11.0 4.1 461
Experimental ] Calculated I
o•
[A.] 671 2.13 313 20.8 6.0 159 40.6 3.98 82.6 10.6 2.85 70.8 27.8 2.5 2.85 49.7 183 2.85
100 =I.
50
0 .
Experimental ] Calculated I
o 20
10 o~ c) ,0
0 sively at the interfaces. The multilayers such as N b / C u [6] which has the fcc(1 1 1) and bcc(1 1 0) planes parallel to the surface had the same tendency as M o / C u . The multilayer films which have Ni layer as a constituent metal layer showed relatively high resistivities and T C R decreased drastically as A decreased. Fig. 2 shows the experimental and calculated dependence of T C R of M o / N i [7] on 1 / A . The present model could not explain the sudden decrease of T C R in these multilayers. Fig. 3 describes the dependence of resistivity and T C R of N b / C u synthesized by W e r n e r et al. [6] on l / A . The slope of the increase of resistivity with 1 / A Mo/Ni 30 • o
Experimental Calculated
2O 10 ,.#
oo
,,
.2oo
.............
0
0
20
40
O'-~
60
.....
80
O,
O
O
cl.
10
. . . .
0
I
10
. . . .
I
20
. . . .
I
. . . .
L~_,
30 40 A I (xi0 3/kl)
~
,
I
50
,
,
~
,
60
Fig. 3. The bilayer thickness dependence of resistivity and TCR of Nb/Cu synthesized by Werner et al. [6]. was so large that the theoretical calculation could not reproduce the experimental results. The steep increase of resistivity similar to N b / C u was observed in N b / A I [8], C u / T a [9] and other N b / C u prepared by Lowe et al. [10]. The T C R of N b / C u at the shortest A had a negative value, which was also observed in N b / A I [7] and M o / N i [7]. This negative T C R could not be explained by the present model. The negative T C R could be explained as follows; the grain size of multilayer becomes small as A decreases. Therefore, the residual resistivity largely increases by the grain boundary scattering. The large resistivity reduces the positive phonon contribution and enhances the negative temperature dependence described by the D e b y e - W a l l e r factor [17]. 4. Conclusions
l O0
A I (xlO 3~1)
Fig. 2. Dependence of TCR of Mo/Ni [7] on the inverse bilayer thickness.
We have calculated the resistivity and T C R of multilayers by the model analogous with the F u c h s Sondheimer theory and compared the calculated dependence of resistivity and T C R on bilayer thickness with the experimental results. The results of the calculations for two kinds of A u / P d multilayer, A I / A g ,
478
Y. lnoue et al. / Computer simulations on electrical resistit'ities
M o / T a a n d M o / C u have r e p r o d u c e d well the bilayer thickness d e p e n d e n c e . In o t h e r words, the bilayer thickness d e p e n d e n c e of these multilayers is governed by the interface scattering. All the conduction electrons were scattered diffusely at the interfaces in the multilayers which have fcc(1 1 1) and bcc(1 1 0) planes parallel to the surface. T C R of the multilayer films constituted by Ni a n d a n o t h e r m e t a l d e c r e a s e d drastically as the bilayer thickness decreased. T h e calculation could not fit this s u d d e n decrease of TCR. The slope of increase of resistivity of N b / C u a n d C u / T a with the inverse bilayer thickness was so large that the calculation could not r e p r o d u c e the e x p e r i m e n t a l results. Negative T C R was observed in these multilayers at very short bilayer thickness. T h e large increase of resistivity and negative T C R could be due to the scattering at the grain boundaries. Acknowledgements." T h e a u t h o r s arc grateful to Dr. Hideyuki Sato (Tokyo M e t r o p o l i t a n University) for useful discussions a n d e n c o u r a g e m e n t . This work was s u p p o r t e d by a G r a n t - i n Aid for Scientific R e s e a r c h from the Ministry of Education, Science and Culture, Japan.
References [1] P.F. Carcia and A. Suna, J. Appl. Phys. 54 (1983) 2000. [2] J.W.C. de Vries and F.J.A. den Broeder, J. Phys. F 18 (1988) 2653.
[3] H. Sato, I. Sakamoto, K. Yonemitsu, W.P. Pratt, Jr, W. Abudul-Razzaq, J. Slaughter and P.A. Schroeder, in: Multilayers, Mat. Res. Soc. (1988). [4] J.L. Makous, C.M. Falco, R. Vaglio and A. Cucolo, Jpn. J. Appl. Phys. 26 (1987) 1467. [5] T. Kaneko, T. Sasaki, M. Sakuda, R.. Yamamoto, T. Nakamura, H. Yamamoto and S. Tanaka, J. Phys. F 18 (1988) 2053. [6] T.R. Werner, 1. Banerjee, Q.S. Yang, C.M. Falco and I.K. Schuller, Phys. Rev. B 26 (1982) 2224. [7] M.R. Khan, C.S.L. Chun, G.P. Felcher. M. Grimsditch, A. Kunny, C.M. Falco and I.K. Schuller, Phys. Rev. B 27 (1983) 7186. [8] M. Gurvitch, Phys. Rev. B 34 (1986) 540. [9] G. Reiss, K. Kapfberger, G. Meier, J. Vancea and H. Hoffmann, J. Phys.: Condens. Matter I (1989) 1275. [10] W.P. ix)we, T.W. Barbee, Jr., T.ft. Geballe and D.B. McWhan, Phys. Rev. B 26 (1981) 6193. [11] R. Dimmich, J. Phys. F 15 (1985) 2477. [12] C.-X. Chert, J. Phys.: Condens. Matter I (1989) 3919. [13] R.E. Camley and J.E. Barnfis, Phys. Rev. Lett. 63 (1989) 664. [14] T. Kaneko, Y. Inoue and R. Yamamoto, J. Phys. ('ondens. Matter 4 (1992). [15] D.A. Papaconstantopoulos, The Band Structure of Elemental Solids (Plenum Press, New York, 1986). [16] G.T. Meaden, Electrical Resistance of Metals (Plenum Press, New York, 1965). [17] P.L. Rossiter, The Electrical Resistivity of Metals and Alloys (Cambridge University, Press, Cambridge, 1987).