Phys. kondens. Materie 16, 327--333 (1973) 9 by Springer-Verlag 1973
d-Resonance Scattering in the Resistivity of Liquid Cu-La, Cu-Nd and Cu-Gd Alloys H.-J. Gfintherodt and A. Zimmermann Laboraf~rium ffir Festk5rperphysik ETH Ziirich, H6nggerberg, Switzerland Received May 23, 1973 We present measurements of the electrical resistivity of Cu-La, Cu-Nd and Cu-Gd alloys in the liquid state. The experimental results indicate that the electrical resistivity of the RE metals is due to resonance scattering by the 5d and not the 4/states.
Introduction A theory far the electrical resistivity was given by Ziman [1] for liquid normal metals and by Faber-Ziman [2] for liquid alloys of normal metals. This theory was then extended to liquid transition metals [3]. In the formula for the resistivity of a transition metal the normal metals pseudopotential of the Ziman formula must be replaced by the t-matrix of a muffin-tin-potential to take into account the strong scattering of the conduction electrons by the 3d electrons of the transition metal. The theory has been further extended to liquid alloys of transition metals [4]. The experimental results on transition metal alloys with monovalent and polyvalent normal metals [5] could then be explained in a verysatisfactory manner. The theory accounted quantitatively for the concentration dependence and in a qualitative manner also for the temperature dependence of the electrical resistivity of the alloys [6]. The success in understanding the 3 d transition metals encouraged us to apply the theory also to alloys with Rare Earth (RE) metals. We expect interesting effects in the R E alloys due to the 4 / o r the 5d electrons. In this paper we report on measurements of the electrical resistivity of liquid alloys of Cu with the fight R E metals La and Nd and with the heavy R E Gd. They are compared with measurements of the electrical resistivity of Cu-Ce and Cu-Pr alloys C5]. From the measurements we conclude that the electrical resistivity of the light R E metals is mainly due to the strong resonance scattering by the 5d and not the 4/states [7].
Experimental Results The measurements were performed with the four probe method in a special ceramic cell in high vacuum. The samples were prepared from 99.999% pure Cu and 99.9% pure R E metals, and the melt is normally outgased at ~ 1050~ for at least 12 h, The resistivity was measured by cooling the sample to the solid state, remelting it, and heating it up in the liquid state. Fig. 1 shows the electrical resistivity as a function of concentration for the different R E raetais. With increasing R E concentration the electrical resistivity increases, goes through a maximum, and then attains the value of the pure R E
328
H.-J. Gfintherodt and A. Zimmermann
F~'~cm 150
I00
9 0o,o / ~
9oo:c
9 Cu-Nd I000 C 9 Cu-Gd IO00"C
/ 50
i
Cu
20
40
60 at % R.E.
80 La ,Nd = Gd
Fig. 1. Electrical resistivity of liquid Cu-La, Cu-Nd and Cu-Gd alloys as a function of concen~ration
metal. As the measurements show, the absolute inaccuracy does not exceed 5%, in spite of the corrosion of the RE. The resistivity of Cn-Ce and Cu-Pr alloys [5] shows the same concentration dependent behavior. Up to a concentration of 30 at~ R E they follow exactly the curve of Cu-Nd. The resistivities of the pure R E metals in the liquid state are taken from Ref. [5] and given in Table 1. The resistivity of pure Gd in the liquid state has not yet been measured, but seems to be quite high. Table 1 ~ cm La Ce Pr
144 127 140
kF A-1 0.924 0.946 0.949
Nd Gd
Q tL~ cm
kF A-1
150
0.953 0.963
For all the measured Cu-RE systems, the temperature coefficient of the electrical resistivity is quite small and positive. As an example we show in Fig. 2 the temperature dependence of the electrical resistivity for different alloys of the Cu-Nd system in the liquid state. Open circles correspond to measurements by cooling, full points to measurements by heating the sample. The following points should be explained b y a model of the R E metals : I. The concentration dependence of the electrical resistivity of Cu-RE alloys is the same for all the light R E and for Gd in Cu-rich alloys.
d-Resonance Scattering in the Resistivity of Liquid Cu-Alloys
329
1
At%Nd 4.0 160
25 120 ~ ~.
80
_= o-o-
9
--
10(30
IlO0~
15
8
40 700
800
900
T
Fig. 2. Temperature dependence of the electrical resistivity of differenb Cu-Nd alloys in the liquid state
2. The resistivities of the different pure light R E do not vary much from one to another compared to the difference in the resistivities of the liquid transition metals [5]. 3. The temperature coefficient of Cu-RE alloys up to about 60 a t % R E is positive for all the R E investigated so far. Discussion
a) Model In this section we deduce the model for the resistivity behavior of R E metals in the liquid state. We star~ with the resonance scattering formula which has been applied successfully to liquid transition metals, to evaluate the resistivity behavior of Cu-RE alloys in the dilute case. We show that the resistivity cannot be due to 4/resonance scattering, but is due to 5d resonance scattering and that reasonable values for the numbers of d and s electrons are obtained. The model also explains the data from concentrated alloys and may therefore give a good suggestion of a way to understand the resistivity behavior of pure R E metals in the liquid state. The resistivity of liquid transition metal alloys is given by i -- e9'~v~
a
~
4 2~-F
I T a l l o y l 9'
o
where [ ~l~lloy 12 ~- C1 [~1[ "2(1 - - ~1 "-}- C1 a l l (q)) + c~ ] t2 [2 (1 - - C~. + c2 a2. (q))
+ cic2(t*~t~ + 6t*)(a~2(q) -- 1) 23 Phys. kondens. Materie, Vol. 16
330
H.-J. Gfintherodt and A. Zimmermann
with --
2z~h3
1
t ( k , k') -= m ( 2 m E) 1/2 ~c2 ~z (2/-4- 1) sin2~2Z(E) exp i ~7~(E) P~ (cos ~),
where kr is the radius of the Fermi sphere on which the scattering takes place, ~2 is the atomic volume, vr is the Fermi velocity, tl and t2 are the t-matrices of the muffin-tin-potentials and a n , a12 and a22 are the partial structure factors of components 1 and 2 [6]. I f the contribution to the t-matrix is dominated by a single phase shift, the resistivity of the pure component can be approximated by the backward scattering contribution 6 ~3 h a
Q ~ e2 k ~ s
( 2 / + 1) sin 2 ~z(EF) a(2kr)
where m is the electron mass and EF the Fermi energy, or
Q ,~ const.. /'2 2
2 -4- (Eres -- Er) 2
where F is the width of the resonance and Eres is the center of the resonance scattering level. As we only consider Cu-RE alloys, all and tl are the same for all the alloys. We also expect that a12 and a22 do nolb vary much from La to Nd and Gd. The same increase in the electrical resistivity of the Cu-RE alloys on the Cu-rich side then definitely means that the t-matrix for the different R E metals must be equal, i.e. the phase shifts at the Fermi level of the different R E metals or the energy difference between the resonance energy and the Fermi level must be equal. This can only be fulfilled if the resistivity is not due to resonance scattering by the 4/electrons. Recent XPS measurements on R E [8] can be interpreted as showing a rather high density of states at the Fermi level due to the 5 d electrons and a peak with increasing intensity from Ce to Nd due to the 4/electrons. The important point is that the energy difference between the 4] level and the Fermi energy increases from Ce to Nd. The same energy variation of the 4/ level is calculated by Herbst et al. [9]. I f the resistivity were due to resonance scattering b y the 4f electrons it would decrease from Ce to Nd in contradiction to the measurements. So we suggest that the resistivity of the R E metals is due to the strong scattering by the 5d electrons. From our measurements, we deduce that the energy difference Es~ -- EF is equal for the R E measured so far. Some idea of the number of 5d electrons can be gathered from a virtual bound state (VBS) analysis. I f the resistivity is dominated by a single phase shift, the resistivity increase in the dilute case [10] is given by 20z~/~ A~o -sin2 W (EF) nA e2 kr where kr and nx are the Fermi wave number and the number of conduction electrons of the matrix. With the measured resistivity increase, the d phase shift of the impurity at the Fermi energy can be determined, and the number of 5d
d-Resonance Scattering in the Resistivity of Liquid Cu-Alloys
331
electrons at the impurity site can be calculated from 10 Z~ = --
~ (Er).
In Table 2 we give the increase of the resistivity, extrapolated to 100% RE, and the calculated phase shifts. They correspond to about 1.9 5d electrons per R E atom. Table 2
Alloy
]CFmatrix
~Amatrix
A~ ~cm
72
Za
Cu-Oe, Cu-Pr, Cu-Nd Cu-La, Cu-Gd
1.36 1.36
1 1
615 580
0.606 0.587
1.93 1.86
As the R E metals are tripositive ions, we can make a reasonable estimate of
kr, the radius of the sphere on which the scattering takes place. In the resonance scattering formula the Fermi wave number used to calculate the resistivity is determined by the free electron result
k~-
3 ~2
nA
/2
where/2 is the atomic volume and nA is the number of conduction electrons before the potential is switched on. The three valence electrons can be specified as about 2 5d electrons and about one free conduction electron. With nA = 1 we determine very small values of kr for the R E metals (listed in Table 1). The corresponding Fermi energy is about 3.5 eV. This Fermi energy is in excellent agreement with band structure calculations of Gd [11]. The value~,~ of k~ and of 72 found from the resistivity measurements are also in good agreement with parameters used to calculate the electron phonon mass enhancement :in co-Ca [12]. They list a value of 0.686 or 0.531 for the d phase shift corresponding to 2.18 or 1.69 d electrons and a value of 1.18 A -1 or 1.07/~-1 for the Fermi wave number according to two different potentials. Additionally their potentials show rather high s and p phase shifts. But when we calculate the resistivity increase with their s, p and d phase shifts, a resistivity increase results which is too high compared to the experimental value. The values of the Fermi wave number deduced from the dilute alloys allow one to explain the experimental fact that no negative temperature coefficients of the electrical resistivity are observed for Cu-RE alloys up to 60 a t % RE. In the theory of Faber-Ziman for normal metal alloys [13] and in the extended theory for transition metal alloys [4], we expect negative temperature coefficients if Kp, the value of t]he first maximum of the structure factor, equals 2 kF of the alloy. As the Kp value is 2 A -1 for La [14] and also for Ce [15] and 2kF is about the same for the light RE, no negative temperature coefficients of the electrical resistivity of liquid Cu-RE alloys at least up to 50 a t % R E are expected, in agreement with the experimental results. 23 *
332
H.-J. Giintherod$ and A. Zimmermann
We therefore propose the following model for the electrical resistivity behavior of pure light R E metals: liquid R E metals consist of statistically distributed ions which are described by a muffin-tin potential and three valence electrons. From the three valence electrons about two electrons screen the ion in a 5d resonant state. The one free conduction electron is resonantly scattered by these 5d states. The model, considering only 5d resonance scattering, does not explain the high resistivity of pure Gd in the liquid state. I t is, however, interesting to note that in the ease of dilute Cu-I~E alloys no difference in the behavior of the lighter R E metals and Gd is observed. The high resistivity of pure liquid Gd is due to an additional contribution to the electrical resistivity, important only for Gd. This additional contribution results mainly from spin disorder scattering [16]. I t is not possible to calculate exactly a spin disorder contribution to the electrical resistivity of R E metals as there are too many unknown parameters. But an estimate indicates that the contribution for the fighter R E metals is about 10 times smaller than the contribution for Gd. Compared to the contribution of d resonance scattering, the contribution of the spin disorder scattering to the electrical resistivity of the other R E metals discussed above is small. As the spin disorder resistivity is linear with R E metal concentration, the resistivity increase due to spin disorder scattering in dilute Cu-RE alloys is small compared to the overall resistivity increase.
b) Conclusion Previous work on R E metals has concentrated largely on the anomalous RE. In particular, the unexpected properties of pure Ce and Ce alloys have received considerable attention. The basic idea used to characterise Ce is the presence of a narrow 4] level close to the Fermi level which produces a large resonant scattering effect [17, 18]. This model describes successfully the main properties of Ce, namely the phase diagram and the main properties of Ce in the ~ and the ? phase. However difficulties have been encountered in trying to explain the pressure dependence of the resistivity in ~-Ce [19] and susceptibility measurements of magnetic impurities in ~-Ce [20] in terms of this model. We believe that the particular strength of our investigation lies in the systematic study of different light R E alloys. The measurements on liquid Cu-La, Cu-Nd and Cu-Gd, by comparison with the measurements on Cu-Ce and Cu-Pr alloys, do not show any systematic variation of the resistivity increase on the Cu-rieh side with the number of 4] electrons. We therefore attribute the resistivity behavior of the dilute alloys to a virtual bound d state. This model also explains the experimental data of concentrated Cu-RE alloys and suggests that the resistivity of pure light R E in the liquid state is mainly due to 5d resonance scattering. Until now, the idea of 5 d states playing an important role in R E has been limited mainly to investigations of heavy R E metals [21, 22, 23]. As a consequence, we suggest that it may be possible to reinterpret the above-mentioned properties of Ce in terms of a model which includes also 5d resonant scattering. Further experiments will be needed to improve our knowledge of the l and d electrons in R E metals. Measurements of liquid alloys of heavy R E will be very interesting. Especially, concentrated alloys of R E in monovalent normal metals and of alloys of R E with polyvalent metals will be very helpful to see if negative temperature coefficients in the liquid state of R E alloys can be observed.
d-Resonance Scattering in the Resistivity of Liquid Cu-Alloys
333
We would like to thank Prof. Busch for his stimulating encouragement and his continuous support of the work. We are also indepted to Prof. Baltensperger and Prof. Friedel, to Dr. Y. Baer, Dr. A. ten Bosch, Dr. H. U. Kfinzi, Dr. It. A. Meier and Mr. L. Schlapbach for helpful discussion and to Dr. D. Pierce for carefully reading the manuscript.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
Ziman, J. M.: Phil. Mag. 6, 1013 (1961) Faber, T. E., Ziman, J. M.: Phil. Mag. 11, 153 (1965) Evans, R., Greenwood, D. A., Lloyd, P. : Phys. Letters A 35, 57 (1971) Evans, R., Giintherodt, H.-J., Kiinzi, H. U., Zimmcrmann, A. : Phys. Letters A 38, 151 (1972) Giintherod~, H.-J., Kiinzi, H. U. : Phys. kondens. Materie 16, 117 (1973) Dreirach, 0., Evans, R., Giintherodt, H.-J., Kiinzi, H . U . : J. Phys. F: Metal Phys. 2, 709 (1972) Giintherodt, H.-J., Stoll, W., Zimmermann, A.: Helv. phys. Aeta (1973) Baer, Y., Busch, G.: Phys. Rev. Letters 31, 35 (1973) Herbst, J. F., Lowy, D. N., Watson, R. E.: Phys. Rev. B 6, 1913 (1972) Friedel, J.: Can. J. Phys. 84, 1190 (1956) Dimmock, J. O.: Solid State Physics 26, 103 (1971) Mukhopadhyay, G., Gyorffy, B. L. : In press Busch, G., Giintherodt, H.-J. : Phys. kondens. Materie 6, 325 (1967) Breuil, M., Tourand, G.: Phys. Letters A 29, 506 (1969) Ruppersberg, private communication Taylor, K. N. R., Darby, M. I. : Physics of RE solids, Chapman and Hall 1972 Coqblin, B., Blandin, A.: Adv. Phys. 17, 281 (1968) Coqblin, B.: J. Phys. 82, C 1 599 (1971) Nicolas-Francillon, M., Jerome, D.: Solid State Commun. 12, 523 (1973) MacPherson, M. R., Everett, G. E., Wohlleben, D., Maple, M. B. : Phys. Rev. Letters 26, 20 (1971) Bekker, F. F. : Phys. Letters A 41, 301 (1972) Eagles, D. M. : Solid State Commun. 12, 291 (1973) Williams, G., ttirst, L. L.: Phys. Rev. 185, 407 (1969) A. Zimmermann Laboratorium fiir FestkSrperphysik Eidgen. Technische ttochschule HSnggerberg CH-8049 Zfirich, Switzerland