Zeitschrift for P h y s i k B
Z. Physik B 33, 25-29 (1979)
© by Springer-Verlag 1979
The Electrical Resistivity of Liquid
_xGax-Alloys
Ch. Holzhey, H. Coufal, G. Schubert, J. Brunnhuber and S. Sotier Physik Department El3, Technische Universitiit Miinchen, Garching, Germany Received May 23, Revised Version October 31, 1978 The electrical resistivity of liquid copper-gallium-alloys has been determined by an ACas well as by a DC-method. The resistivity has its maximum at a gallium-concentration of 35 %, and the temperature coefficient of the resistivity becomes minimal at this concentration. Within the Faber-Ziman-theory this behaviour can be qualitatively understood.
I. Introduction The properties of non-crystalline conductors are of increasing interest. The periodicity or long-range order of the atomic arrangement which characterizes the crystalline state has been replaced by short-range order which can be described using correlation functions. Typical representatives of these non-crystalline materials are amorphous semi-conductors [1], metallic glasses [2], liquid metals and alloys [3] and the socalled liquid semi-conductors [4]. The electronic properties of liquid metals and alloys can be qualitatively described by simple models based on the work of Faber and Ziman [5 I. They calculated the electrical resistivity of these metals using pseudo-potentials and interference functions. Extensive experiments on the electrical resistivity of binary alloys were performed by Busch and Giintherodt [6]. We measured the resistivity of liquid Cu-Ga-alloys [7] and made calculations based on Ziman's model. The Knight shift [8] and the susceptibility [9] of these alloys were also measured.
II. Experiment The resistivity of Cu-Ga-alloys was determined from 300 K to 1,450 K with an accuracy of 1 ~ using ACand DC-methods. A vacuum high temperature furnace (Fig. l a) was
heatyelement !ll
L J J' ...... ~dSo I ....
feedpipe and curmnl e ectmde
14rad.o~j_l_j_sama~i;ioS~ielrts I
storage tank and potential electrodes current electrode
coup~ sample channel b Fig. i. a Experimental set-up and b cross-section of the sample container (made of stenan), scale in centimeters
used for these experiments as described by Th6rmer et al. [7]. The temperature gradient along the ceramic sample container (Fig. lb) inside the furnace was less than
0340-224X/79/0033/0025/$01.00
26
Ch. Holzhey et al.: Electrical Resistivity of Liquid Cul_~Gax-Alloys
0.1K/mm. Samples are molten using 6N pure gallium (Alusuisse) and etched pieces of 5 N pure copper wire (Demetron). Before starting the measurements the liquid alloys were held for 15h at 1,450 K in small crucibles and then filled in the sample container after cooling down at room temperature. Two electrodes (tungsten wire), in the storage tank and in the feed pipe, were used for the current. Small holes drilled perpendicular to the axis of the specimen channel, contain the two electrodes for measuring the potential difference. The temperature and the temperature gradient in the specimen were measured with an accuracy of 1% by two Pt-Pt90Rhlo thermocouples located in the specimen container. After the last measurement with each alloy, the specimen container was broken and the solid sample was checked for small gas inclusions and analysed for its chemical composition. The advantage of the standard four-point-DC-method [7] for resistance measurements is its high accuracy, while the advantage of the AC-method is the suppression of thermoelectric effects. Therefore, we used both methods. The AC-bridge was designed on the basis of a proposal by Logan [10]. The geometry of the sample container was calibrated with mercury 1-11] at room temperature and with gallium [12, 13-1 in the temperature range of the measurements. Chemical analysis after the measurements showed that the concentrations agreed within 1% with those calculated from the weight of the ingredients before alloying. In our sample we found silicon impurities with a concentration of <0.05 %.
6C
ol
Z,C theory
- - experiment
-3C
experimental, error
2C-
1C-
O
I 20
,
I t,0
,
I 60
i
I 80
I
I 100
at % Ga Fig. 3. Comparison of experimental data of the resistivity with numerical calculations for various Cu-Ga-alloys at 1,350 K
I \ 20-
E 1o C~
Results
The temperature dependence of the resistivities p(T)c is shown in Fig. 2 and the resistivity as a function of
2E GO
6c
i
I 1o0
at °/o ~-a
_g -10
at*/,Cu
BO
- - -
theory
-20
t
§ sc Fig. 4. Comparison of experimental data of the temperature dependence of the resistivity with numerical calculations for various Cu-Ga-alloys at their melting point
.,=
j-I
600
I
I
800
I
1
I
1000
I
1200
I
I
I/.00
TIK
Fig. 2. Temperature dependence of the resistivity of various CuGa-alloys. At the melting point of each alloy the copper concentration is indicated
chemical composition, p(C)T=l,350K, in Fig. 3. The resistivity shows a maximum at Cu65Ga35. The temperature coefficient (@/c?T)r m of the resistivity at the melting point (in the liquid state) is negative for the alloys with 15 to 60 at. % Ga (Fig. 4). Cu-Ga-resistivity data of Th6rmer et al. I-7] are in fair agreement with our results.
Ch. Holzhey et al. : Electrical Resistivity of Liquid Cu 1 ~Ga~-Alloys
27
a(K)
IlL Numerical Calculation of the Resistivities Using the Ziman Theory For a pure liquid metal, the Ziman theory gives the resistivity:
/~\ -279 K
,I
3rgm 2 N p - ha eZ /2k2 (}u(K)l 2 a(K))
' L678K
(1)
/1}I1)x~1350 K
where k e is the Fermi radius in the free electron theory, N/f2 = n is the number of atoms in the volume £2 (number density), a(K) is the interference function (the scattering is asof the metal with K = k - k ' sumed to be elastic and the electron states are plane waves Ik) and ]k')),
.....
u(K) is the pseudo-potential due to a single ion, ( ) is the average of some function of K over the range between zero and 2 ke, so that
0
theory experiment
~
!~ x/'~^i
.
o
I
2
I
We use the hard sphere model of Percus-Yewick to calculate the interference function. For pure metals according to Ashcroft and Leckner [14], we have
a ( K , ~ ) = { 1 - n c(K ~)} -~
(2)
where a is the hard sphere diameter and c(K a) the direct correlation function in m o m e n t u m space, a is related to t/, the packing density parameter according to the equation ~/=(~/6)na 3. For most of the metals at the melting point, ~/has a value close to 0.45. To have the results for a temperature of 1,350 K, we made a correction assuming that t/is proportional to
T - o.3-~ 1-15]. For liquid copper the pseudo-potential form factor calculated by Moriarty [16] at the melting point could be used, For liquid gallium we used the form factor of the potential model of Heine and Abarenkov obtained by Animalu and Heine (Harrison 1-17]). Knowing that R ( K = 0 ) ~__ - s 2k e 2, we calculated the values of this form factor at the temperature of 1,350 K. For pure gallium at the melting point, the calculation of the interference function by Eq. (2) gives
Copper at melting point Gallium at melting point Gallium at 1,350K " From [23]
;2o
(ilk) 1
(z~x)3
1.307 1.65 1.61
13.268 18.994 21.262
4 2)k FGo
good agreement with the experimental results of Cusack et al. [18] (see Fig. 5). We found a resistivity p =25.5 gf2cm (the experiment gives 0=25.8 ~tf~cm). k F was calculated with Z = 3 . For 1,350 K the new parameters ~/, n and a are given in Table 1. The resistivity has here the value 0=45.96 gf~cm (the experiment gives p = 46.11 gf~ cm). For pure copper, if we compare the calculated interference function with the experimental results of North and Wagner [19], we find the best agreement by taking ~/=0.48 (see Fig. 6). The number density and hard sphere diameter are given in Table 1. The Fermi radius ke is calculated with Z = 1. Using for the effective mass m*=0.81 m (Dreirach [20]) we found for the resistivity p =24.6 laf~ cm (the experiment gives p =21.6 g ~ cm). Assuming that the N F E model and the Born approximation are valid for alloys, we can use the Faber-Ziman theory which yields: P-
37zm 2 N h3e 2 a k ~ ( C ' u 2 a " + c 2 u 2 a 2 2 + c ' c 2
[u2(1-a11)+u2(1-a22)-2UlUe(1-a12)]).
Table 1 Ke
3
Fig. 5. Interference function of pure gallium. Experimental data are from [18]. The maximum integration limit 2kF.Gais indicated
1
( f ( K ) ) = ~ f ( K ) 4(K/2 ke) a d(K/2 kv). o
Metal
~,.,\'o ^ "e
r/ 0.48 0.45 0.227
n
cr
p gf~ cm
p p~ cm
(ilk)- 3
(~k)
Theory
experiment
0.07537 0.0526 0.0470
2.299 2.54 2.633
24.6 25.5 45.96
21.6a 25.8 46.1
(3)
28
l
Ch. Holzhey et al.: Electrical Resistivity of Liquid Cu t_~Gax-Alloys
a(K)
atomic volume/cm 3
,i
I
t
theory -- experiment
/
x \
13
×
11 lC
i
i
L
i
Cu
20
40
60
80
/x
0
2 k F A ~
21.262
- --
0'
2kFA/~-1
x--x
4x~ 1
•/ ~x j I 2
/
X
1~
3"2
2.2
:~,0 I00 "loGa
KpCu,, I [
3 I
Fig.7. Concentration dependence of the atomic volume f20 and the Fermi-vector of Cu-Ga-alloys, The first maxima of the interference function of pure gallium at Kpaa and of pure copper at Kpcu are indicated
2kFc u 2kFo a Fig. 6. Interference function of pure copper. Experimental data are from [19]. The minimum of the integration limit 2 kvc~, as well as the maximum at 2 kFa~, are indicated cl, c2: concentrations of the components 1 and 2, at a, a22, a12: partial interference functions, ua, u2: pseudopotentials of the metals 1 and 2 as defined above. Thermodynamics shows that for the atomic volume of the alloy, a linear combination of the atomic volumes of the constituents is to be expected if the free energy obeys the so-called "ideal law of mixing" (if we have an ideal r a n d o m distribution). We assume that it is correct for the system C u - G a and can, therefore, write ~o = c~ ~o~ + c2 f2o2- ke is calculated with the average valence of the alloy Z a = c l Z t + c 2 Z 2. The change of k v and Q0 an alloying is shown in Fig. 7. There are no experimental partial interference functions for this system. Therefore, we assume (see Dreirach et al. [20] who refer to the experimental results of North and Wagner [19] for Cu-Sn and of Halder and Wagner [21] for Ag-Sn) that for metal alloys of the noble-polyvalent group, the partial interference functions az~ and a22 are very similar to the interference functions of the corresponding pure metals. Therefore, we calculate ata and a22 with Eq.(2) defining the new parameters th and t/2 as follows (Ashcroft and Langreth [-22]): if c~ is the hard-sphere ratio c~ = a j a 2 (0< c~< 1), and Z the concentration of larger spheres Z--n2/nA and if I?a is set to the total packing fraction for the mixture, tlA=~h +172, we can obtain
t]l and ~/2 in function of c~, Z and ~a. The interference function ale is intermediate between a~l and a22 and is obtained with Eq. (2) using the parameters hA, rlA and aA=(6~7+1] 1/3.
Vz n~/ We had to make a correction taking into account the charge transfer between the constituents so that (North and Wagner [19]) ui(K)=u i.... ( K ) . Z j Z A , where uioalo(K) is the pseudopotential calculated for the pure metal i, Z i is the valence of this pure metal. The results of the resistivity calculation compared with the experimental data are shown in Fig. 3. The temperature coefficients (Fig. 4) are calculated from a 10 K interval directly above the melting point of each alloy.
IV, Conclusion Liquid copper-gallium alloys are typical representatives of I b - I I I - g r o u p alloys. Qualitatively their resistivity can be understood by the Faber-ZimanModel which gives reasonable agreement between experimental and numerical results. Thus the maxim u m of the resistivity at a concentration range near 35% and the negative temperature coefficient for about the same concentration range can be explained by this model. We would like to thank Prof. Dr. E. Liischer, in whose Institute this work was done, for many helpful discussions. We are also greatly indebted to Alusuisse for giving us the gallium of the probes and to Mr. Langanke for making chemical analysis of the samples.
Ch. Holzhey et al.: Electrical Resistivity of Liquid Cu 1_~Gax-Alloys
References 1. Spear, W.E., LeComber, P.G.: Phil. Mag. 33, 935 (1976) 2. Cargill, G.S.: Solid State Physics 30, 227 (Academic Press 1975) 3. Shimoji, M.: Liquid Metals, London: Academic Press 1977 4. Cutler, M.: Liquid Semiconductors, London: Academic Press 1977 5. Faber. T.E., Ziman, J.M.: Phil. Mag. 8, 153 (1965) 6. Busch, G., Gtintherodt, H.J.: Solid State Phys. 29, 235 (1974) 7. Th6rmer, K., Coufal, H., Fritsch, G., Diletti, H.: Phys. Lett. 56A, 489 (1976) 8. Zollner, R., Sotier, S., Holzhey, Ch., Ltischer, E.: Z. Naturforsch. 30a, 1250 (1975) 9. Nemura, O., Takeuchi, S.: Trans. Jap. Insl. Met. 37, 252 (1973) 10. Logan, M.A.: Bell Syst. Techn. I. May (1961) 11. Th6rmer, K.: Diplomarbeit Physikdepartment El3, TU Miinchen (1976) 12. Pokorny, M., ~.str6m, H.U.: J. Phys. F. Metal Phys. 6, 559 (1976) 13. Schulz, L.G., Spiegler, P.: Trans. AIME, 215, 87 (1959) 14. Ashcroft, N.W., Lekner, J.: Phys. Rev. 145, 83 (1966) 15. Faber, T.E.: An Introduction to the Theory of Liquid Metals. Cambridge: University Press 1972
29 16. Moriarty, J.A.: Phys. Rew B6, 1239 (1972) 17. Harrison, W.A.: Pseudopotentials in the Theory of Metals, New York: Benjamin 1966 18. Cusack, N,E., Kendall, P.W., Marzwaha, A.S.: Phil. Mag. 7, 1745 (1962) 19. North, D.M., Wagner, C.N.J.: Phys. Chem. Liquids 2, 87 (1970 20. Dreirach, O., Evans, R., Giintherodt, H.-J., Ktinzi, H.-U.: J. Phys. F. Metal Phys. 2, 709 (1972) 21. Halder, N.C., Wagner, C.N.J.: J. Chem. Phys. 47, 4385 (1967) 22. Ashcroft, N.W., Langreth, D.: Phys. Rev. 156, 685 (1967) 23. Cusack, N.E.: Rept. Progr. Phys. 26, 361 (1963)
Ch. Holzhey H. Coufal G. Schubert J. Brunnhuber S. Sotier Physik-Department E 13 Technische Universit~it Mtinchen James-Franck-Strasse D-8046 Garching bei Miinchen Federal Republic of Germany