Structure and electrical resistivity of liquid silver-germanium alloys

Journal of Non-Crystalline Solids 117/118 (1990) 383-386 North-Holland

STRUCTURE

383

AND ELECTRICAL

RESISTIVITY

SILVER-GERMANIUM

OF LIQUID

ALLOYS

GASSER Jean-Georges, MAYOUFI Moussa, KLEIM Roland

Laboratoire de Physique des

Liquides MEtaUiques ,Universit6 de METZ fie du Saulcy ,57045 METZ C6dex 1 France BELLISSENT-FUNEL Marie-Claire Laboratoire L6on Brillouin C.E.N.-Saclay 91191 Gif-sur Yvette CEdex France ( Laboratoire commun C.E.A.-C.N.R.S.) Abstract

II - Theory Ziman [4] has shown that the electrical resistivity of

The electrical resistivity of the silver-germanium system has already been studied previously. But the different results are not in agreement. We have measured the resistivity of this system very accurately in a large range of temperature over the whole phase diagram. The temperature coefficient of the resistivity (T.C.R.) versus concentration presents a minimum. However it does not

a pure liquid metal can be computed using the expression : 2 kf

3 n

rn2n o I

a(q) v(q) 2 q3 dq

[l]

P- 4 e2 .i,13 kf6 o where a(q) is the static structure factor, v(q) the pseudo (or model) - potential form factor, ~ o the atomic volume and

become negative, at the opposite of the Ag-Sn system.The

kf the Fermi wave vector. Other symbols have their usual

resistivity has been evaluated using the extended

meaning. This formula can be extended to binary alloys

Faber-Ziman formula with phase shifts and hard sphere

[1] by replacing the product a(q) v(q) 2 by a sum of four

structure factors. The Ag-Ge system is one of the rare

terms:

system for which the partial structure factors have been

c 2 a 11 (q)lt 1(q) 12 + (1-c)2a22(q) It2(q)12

measured by neutron scattering using the isotopic

+ c(1-c)al2(q) [tl(q)t2*(q)+ tl* (q) t2(q)]

substitution method. The resistivity is evaluated using

[2]

+c(1-c){ Itl(q)12+lt2(q)12-0.5[tl(q)t2*(q)+ tl*(q ) t2(q) ] }

these partial structure factors and the different contributions are compared to those obtained from the hard sphere

where c is the atomic fraction of constituent 1, and aij(q)

model.

are the Faber-Ziman [2] partial structure factors describing

I - Introduction

different diameters. For noble, transition and rare earth

a mixture of randomly distributed hard spheres with metals we use the scattering approach replacing the model It is largely agreed that electrical transport in simple

potential form factor Iv(q)] 2 by a t matrix [ti(q)] 2 for

metals has been well resolved from a theoretical point of

species i in the alloy, expressed in term of phase shifts.

view. In liquid metallic alloys, the transport properties are

The different contributions have been grouped in 4 terms.

generally interpreted with the Faber- Ziman formalism [1]

We have expressed the FABER-ZIMAN structure factors

in terms of pseudopotentials (or t matrix) and structure

aij(q) in terms of the analytical ASHCROFF-LANGRETH

factors. We recall the basic formula in § 2 together with the

[3] hard sphere partial structure factors. The chosen

parameters used for the practical calculations. We present

parameters are the two hard sphere diameters relative to

the experimental design in §3. In §4 our experimental

each specie. We held them constant with concentration but

results are compare to those deduced from hard sphere and experimental partial structure factors.

obtained from the pure metal data. At each temperature, the

0022 3093/90/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland)

not with temperature. The hard sphere diameters are

384

J.-G. Gasser et al.// Structure and electrical resistivity of liquid silver-germanium alloys

hard sphere diameters are deduced from the experimental

IV - Experimental resistivity and discussion

densities of the pure metals, compiled by Crawley [4] and from the packing fraction given by the Waseda empirical

The phase diagram of the Ag-Ge system (Hansen

law [5a] where the parameters A i and Bi have been taken

[7]) presents a deep eutectic temperature of 65 I°C at 26 at

from Waseda's book [5b]. The packing fraction and the

% Ge. We have published and discussed earlier [8,9,] the electrical resistivity of pure germanium and silver. The

hard sphere diameters are :

electrical resistivity of Ag-Ge alloys has already been "qi (T) = A i exp (-BIT)

[3]

a i 3 (T) = [ 6 rli(T) flo(T)/n]

[4]

studied before. However the different results do not agree. Indeed OZELTON et al. [10] have observed some anomalies in the eutectic region of the phase diagram.

In the alloy we have taken into account the modification of

However they measured the resistivity by the electrodeless

the mean atomic volume ~ o ( T ) and of the Fermi

rotating field method of ROLL and MOTZ [11]. It has been

wavevector kf :

ADAMS and LEACH [14], GASSER [6]) for different

shown (DAVIES and LEACH [12], MERA et al.[13], kf3 = (3x 2 Z(c)/f2o(T,c))

[5]

where Z(c) and f~o(T,c) are respectively the mean valence

liquid alloys that these anomalies do not appear with an electrode technique. SOIFER et al. [15] results are about

and the mean atomic volume of the alloy obtained by a

30 ~

linear interpolation of the pure metal values.

al. observed negative temperature coefficients while it is

cm above those of UEMURA et al.[16]. SOIFER et

not the case for UEMURA et al.. In order to clarify the HI - Experimental method and arrangement for resistivity

situation, we measured the resistivity of Ag-Ge alloys by

measurements We measured the electrical resistivity of metals and

at. % steps near the eutectic composition. Our experimental

alloys in a quartz cell which has several advantages. The

results are represented on fig. 1 as a function of

first one is the possibility of filling the capillary from the

composition.

10 atomic % steps over the whole phase diagram and by 2

bottom in order to avoid the presence of bubbles. The second one is the possibility of changing the composition

160

of the alloy during the experiment. The measurement cell

'

I

.

Ag-Ge

has been described in earlier work [6]. The geometrical

.

.

.

1000 ° C

constant of the cell was calibrated by measuring the

120

resistivity of triple distilled mercury.The whole arrangement is heated under vacuum until the metals are

:=i.

melted. An absolute pressure of argon of 1 to 3 bars is then applied over the liquid sample and pushes the liquid alloy

~ 80

into the capillary tube. The resistivity is measured by a four probes method. A stable constant current is furnished by a General Resistance DIAL DAS 86 generator. The voltage

N 4o

drop is measured with a 1 I.tV resolution 120 000 points Hewlett Packard 3490 voltmeter. Thermoelectric e.m.f, are eliminated by inverting the current. The accuracy of the electrical resistivity is estimated to 0.4 %, that of the composition of the alloy to 0.3 at % and that of the temperature to. 3 %. The experience is fully described in GASSER's thesis [6].

0

a

0,0

I

+

Ozelton et al

•

Soiferetal

0

This work

I

I

i

0,5

I

l

1,0

COMPOSITION (at% Ge) Figure 1 Experimental resistivity of liquid Ag-Ge alloys

J.-G. Gasser et al./ Structure and electricM resistivity of liquid silver-germanium alloys

385

We did not observe OZELTON's two maxima curve. Our values are about 10% lower than their maxima and 6% higher for germanium rich alloys. SOWER et al. results are more than 20% higher than our values near the eutectic

160

~

Ag-Ge

00 o

composition. We obtain a reasonable agreement with U E M U R A et al measurements which however differ from about 3 %. We have represented on fig. 2 both experimental

and

calculated

resistivity

g

versus

concentration, at 1000 o C. All the curves present a maximum. However the experimental one lies at about 40 at % Ge while the calculated ones give a value of about 20 at % and is less sharp than the calculated values. The

40~

question is to know if the extended F A B E R - Z I M A N

t/o

his work

A WasedalO00C

~

formula fails or if the hard sphere description is inadequate

[] Dreirach 1000C • Waseda A(q)exp

to describe the structure of this alloy. The eutectic Ag-Ge o

alloy is one of the rare systems for which the partial

0,0

0,5

1,0

structure factors have been measured using the neutron COMPOSITION

isotopic substitution method (Bellissent et al [15]). We

(at% Ge)

have calculated the four different contributions of formula

Figure 2 Comparison of our experimental and calculated

2 to the electrical resistivity with both hard sphere and

resistivities with experimental and hard sphere structure

experimental partial structure factors. The results are

factors

summarized on table 1 and are represented on fig 2. The difference in the temperature coefficient is more

Resistivity obtained with hard sphere structure factors

important even with U E M U R A et al values. Our

Phase FIRSTSECONDTHIRD FOUFnHrOTAL shifts of ~ TEt~

experimental temperature coefficient, (together with

WASEDA 17.00 24.24 25.62 62.31 129.17

calculated values) is represented on fig.3 and is compared

DREIRACH22.07 20.30 27.97 83.17 153.52

with experimental WASEDA 6.34 23.72 18.32! 62.31 110.69 structure factors DREIRACH 8,08 19.92 20.03 83.17 131.20

to earlier experimental results. As can be observed, the temperature coefficient presents a minimum at 15 at % G e , but it remains positive.The difference with UEMURA et al. curve is more important on the germanium rich side of the phase diagram. The difference is considerable with

Experimental

resislivity

98.8

Table 1 Different contributions to the resistivity of Ag-Ge

SOIFER et al values which could not be drawn on our figure (they can be more than ten times greater in positive or negative values). A minimum of the temperature coefficient (which can be negative) in metallic alloys

With both sets of phase shifts, the electrical

generally occurs when the limit of integration 2kf in

resistivity is lowered from about 20 I.tf2 c m . We obtain a

Faber-Ziman formula [1] lies near the main peak of the

value of 131 g O cm with DREIRACH et al. phase shifts

structure factors. This is the case when the mean valency is

and a value of 110 g ~ cm with W A S E D A [5c] phase

about 1.7 i.e. for alloys of polyvalent with monovalent

shifts which has to be compared to an experimental value

(noble, alkaline, or transition metals). This classical

of 99 p.f~.cm. A better agreement between experimental

explanation of temperature coefficient minimum can be

and calculated resistivities is obtained when using

used for Ag-Ge systems. We must return to the Ziman

experimental structure factors.

explanation of this effect. When the temperature is raised,

386

J.-G. Gasser et al./Structure and electrical resistivity of liquid silver-germanium alloys

30 1)

Ag-Ge

20

FABER T.E. and ZIMAN J.M. ; Phil. Mag. 11 (1965) 153

2)

FABER T.E.; Introduction to the theory of liquid metals ; (1972) Cambridge at the University press

~ ~ ~Z

3)

'

Rev. 156 (1967) 685

0 1 ~

r.=

D Uemura

~ . -10 ~O

-20 0,0

ASHCROFT N.W. and LANGRETH D.C. ; Phys.

4)

CRAWLEY A.F., Int. Met. Rev. 19 (1974) 32

5)

WASEDA Y. ; The structure of non-crystalline materials (1980) Mac Graw Hill. int. book company;

" dRo/dT Waseda

0,2

0,4

0,6

0,8

a) p 59

1 ,0

COMPOSITION (at% Ge)

6)

Universit6 de Metz (France) 7)

HANSEN M. ; Constitution of binary alloys. Mc Graw Hill book company New-York (1958)

Figure 3 Experimental and calculated temperature coefficients of the resistivity

b) table 3-1 p 54 c) table p 207

GASSER J.G. (1982) Th~se de de doctorat d'Etat ;

8)

GASSER J.G., MAYOUFI M. and BELLISSENT M . C . J . Phys. Condens. Matter 1 (1989) 2409

the disorder increases and the structure factor tends to

9)

KEFIF B. and GASSER J.G. Phys. Chem. Liq. 18

10)

OZELTON,

(1988) 327

unity. Schematically, the temperature coefficient is governed by 4 contributions. A contribution to a positive

M.W.,WILSON J.R. and PRAT

J.N.Rev. Int. Hautes Temp. et Refract. 4(1967) 109

coefficient by the low q region where a(q) is lower than 1 ; a negative contribution in the peak region, a positive one

11)

ROLL A. and MOTZ H. ; Z. Metallk. 48 (1957) 272

near the first minimum of the structure factor, and again a

12)

DAVIES H.A. and LEACH J.S.L. Phys Chem. Liq. 2(1970) 1

negative one because the limit of integration 2kf decreases with temperature. Of course, the position of the node of the form factor (or the t matrix) and the factor q3 modulate the importance of these contributions. The temperature coefficient curves (fig. 3) presents a sharper and negative minimum. However the general shape is about the same. It is not possible to have a quantitative estimation of the influence of the true stucture factors (which have not be measured at two temperatures) on the electrical resistivity temperature coefficient. We hope that temperature dependent measurements of partial structure factors will bring an answer to that question.

11)

MERA Y. KITA Y. and ADACHI A.Tech. Rep. Osaka. Univ.22 (1972) 447

12)

ADAMS P.D. and LEACH J.S.L. Phys Rev. 156

13)

SOIFER L.M., IZMAILOV V.A. AND KASHIN

14)

UEMURA O. and IKEDA S. ;Trans. Jap. Inst. Met.;

(1967) 178 V.I. Teplofizika Vysokikh Temp.12 (1974) 669 14 (1973) 351. 15)

BELLISSENT-FUNEL

M.C.,

DESRE

P.J.,

BELLISSENT R. and TOURAND G. J. PHYS. F 7 (1977) 2485