J. Phys. Chem. Solids Pergamon Press 1971. Vol. 32, pp. 2135-2143. Printed in Great Britain.

VACANCY FLOW E F F E C T ON E L E C T R O M I G R A T I O N IN SILVER

NGUYEN VAN DOAN

Section de Recherches de M~tallurgie Physique, Commissariat a I'Energie Atomique, Centre d'Etudes Nucl~aires de Saclay, France

(Received 8 May 1970; in revised form 28 December 1970)

Abs t rac t -The 'apparent effective valence Z**, deduced from experiments on electromigration using tracer atoms, differs from the true 'effective valence' Z* by the 'vacancy flow term', previously intro- duced by Howard and Lidiard and by Manning in ionic crystals. This corrective term could explain the small values obtained for Z** of the transitional tracers: 54Mn, SaFe, 5aCo and ~Ni in Silver.

An experiment has been car/ied out with saFe in copper to confirm this hypothesis.

1. INTRODUCTION

IN ORDER tO study the electronic structure of ions at the saddle point, i.e. ions halfway between two vacancies during a jump, we have investigated the electromigration of some solutes in Silver[I ,2]: l'~ 1~ 1'4rain, nsSn and '24Sb. In fact, many such experi- ments have been performed in recent years [3-8], but the results vary over a wide range according to the authors and to the experi- mental procedures. For the saddle point in- vestigation, it is necessary to undertake the experiments in the same laboratory. For this reason, to supplement the above mentioned solutes, we have been interested in the behaviour of transition elements.

The choice of the" iron group is governed by the well-known variation of the excess resistivity of ions distr ibutedhomogeneously at the lattice site in a noble metal matrix[9]. This variation is thoroughly explained by the virtual bound states model introduced by Friedel[10-13]. It would be interesting to examine to what extent the model can be used to describe the electronic structure of transi- tional ions at the saddle point.

For the time being we will limit ourselves to the presentation of the preliminary experi- mental results. We will see that they are very unusual. In fact, they could be explained by the presence of the vacancy flow effect. In

2135

Section 2 we show by means of the Thermo- dynamics of Irreversible Processes how this effect is introduced. In Section 3, we point out its importance in the explanation of the small v/~iues obtained for the apparent effective valences.

2. VACANCY FLOW EFFECT

In the previous paper[l] , we have neg- -~, ;k

lected the term (LAB/LB,)(ZA--Z;i'/ZB--Z~) in the expression of the flux of 124Sb in silver. It will be shown that this term can.become impor- tant in some cases, especially for the tracer atoms investigated in the present work.

In a metallic alloy where the diffusion occurs by a vacancy mechanism, the flux J of the components A and B can be written in the following form [ 1 ]:

JA

J~-

LAA ~(~A --~) T

LA, d _ T dx tze

LBA d , _ 7 Ux ~A -ff~)

Lse d _ T dx tze

LABT dx(~8--~L)

LBB d T d~x (~" - ~a)

(I)

(2)

2136 NGUYEN VAN DOAN

where ff~ are the potentials from which the thermodynamic forces are derived:

pL-a = p~a + zA eth

"~s = ~s + zBe4~

I~L = Iz L = -

and E = --(d~b/dx) is the electric field. These fluxes are measured with respect to

the lattice reference, defined by a thin layer of Hf~8~O2 tracer or simply by the initial inter- face of welding (revealed by chemical etching). The phenomenological coefficients LAA, LSB are related respectively to the self- and hetero- diffusion coefficients D~., D~.. The cross-term LAB = Lsa describes the coupling between fluxes of atoms A and B. The coefficients LA~, Ls~ originate from the interaction of atoms A and B with the charge carriers.

Let z~eE and z~:eE be the friction forces due to charge carriers on atoms A and B, averaged over a jump path:

LAe LAAZ~: * = + LABZB (3)

Lse = Ls~z~ ~ + LBBZ~ ~. (4)

It means that this relation is valid only when the fluxes are independent. On the other hand, it is important to note that the correction term Lsa/LBB is identical to the 'vacancy flow effect' introduced by Howard and Lidiard [ 14] in the ionic crystals and calculated by Manning [ 15] in a more complete kinetic analysis.

At this stage, in order to avoid confusion with previous notat ions[I ,2] we will define the various valences as following:

za, ZB are valences of the matrix and of the tracer atom;

* z~: are valences relative to the friction ZA, forces;

Z,i* = zA -- z,i, ZB" = zs -- z* are effective va- lences which are to be compared with theo- retical expression [ 16-18];

Z~i**, Z~* are apparent effective valences measured in our experiments and related to the true effective valences by the relations (10) and (I 1). Let us consider two following cases:

( I ) Self-diffusion B = A * Equation (5) becomes:

L .A eE =

For dilute solutions (ns "~ nA), equation (2) reduces to:

Js = - D g . an~ _~ nBD~, Ox kT

where [ 19]:

. 1

X[(ZB--Z~) LB,t. _z j : ) ]eE . +~-~BB (ZA (2')

It follows that, in our tracer diffusion experi- ments, the drift velocity vB resulting from the electric field is given by:

Dg, F Lsa. - - z~ ) ] eE. vB = (z , + (z , (5)

This expression may be compared to the Nernst-Einstein 's relation which states that:

D] . , vB ~ = - ~ - tzn -- z~) eE. (6)

fA being the correlation factor, equal to 0-78 for f.c.c, lattice.

Therefore:

vA. = D~'Z~: eE (7) fAkT

where: D]/fA is the uncorrelated self-diffusion coefficient. Hence, in the case of self-diffusion, the vacancy flow effect compensates exactly the correlation effect. This result is not sur- prising, for in a vacancy mechanism, the flux of solvent atoms is equal and opposite to the

E L E C T R O M I G R A T I O N IN SILVER 2137

flux of vacancies whose jumps are not cor- related [1], [20], [21].

(2) Heterodiffusion B = B* Here:

D~. [ ~, LnA ...l v,. = ZB Z, JeE. (8)

We can therefore write (7) and (8) in the Nernst-Einstein 's form:

DiAZi**eE ( i = A * o r B *) (9) vi -- kT

where:

= z1'/A (lO)

= [ l + L.A Z*] LnB Zj]' J"

( l l )

IfLna/LBn is small (for example 1245b in Ag):

ZB* ~ ZB:.

But it should be mentioned that LBA/LBB could be negative and approach the limiting value -2 ; that is, instead of accelerating the diffusion of the tracer B(LBA/Lss > 0), the vacancy flow effect can in certain cases slow it down considerably (LsA/LBB < 0). This can be seen by using Howard and Lidiard's expres- sions for LBA and Lnn [22]:

LBA __ --2 + 3 ( W3/W, ) LBB I+�89

(12)

where Wi's are standard vacancy jump fre- quencies.

W3 , �9 LBA When: ~ ~ 0, nm LBB ~ --2.

On the other hand, Manning has derived a more general expression[23] for the vacancy

flow effect:

namely:

2(np) -- LnA LBB

OF:

-- 2W, + (W3/W4) ( W o - W4)7(! - -F ) 2 WI + 7FWa

(13)

-2+ 'rl = 2 + 7 F r [

where: r = W3/WI

(14)

~ o = 3 + 7 ( I - - F ) ( W o / W 4 - - 1 ) (15)

7F and ~ are plotted as a function of Wo/W4 [23, 24] in Fig. 1. The Fig. 2 represents (n,) vs. the ratio r for various values of Wo/W4. In particular, for W0 = W4, 2(rip) is approxi- mately given by equation (12).

To determine the true effective valences Z~: from experiments, one therefore need to know these ratios Wo/W4 and W3/WI, which can be deduced from both the isotope effect measure- ment and the solute-enhanced self-diffusion [25]. Unfortunately, there are at present only a few cases where the data are available.

3. EXPERIMENTAL RESULTS AND DISCUSSION

The experimental techniques are destribed in [1,2].' A thin layer of radioisotopes is deposited between two cylinders of Ag metal. The isotopes are obtained from the D6part- ment des Radio616ments at Saclay in the form of chlorides. The plating bath consists of a saturated ammonium oxalate solution whose pH is reduced to 4 by addition of oxalic acid [26]. The welding of the two cylinders and of the whole specimen to electrodes is carded out at the same time, in a vacuum of 10 -6 mm Hg, at 750~ during 1 hr[1]. In order to avoid the evaporation of silver, the anneal is made under a purified argon atmosphere for periods

2138 N G U Y E N VAN DOAN

0

- 1

- 2 I/. 0 �9 5 1 . 5

" w~/,w4 b ~ w4/w o

Fig. I. 7F and ~ as a function of Wo/W4 and WgWo.

of 2 to 3 weeks depending on the tracer. After the experiment, the tracer has diffused accord- ing to agaussian distribution with its maximum shifted to the anode from the original welding interface by a quantity Xm. The following tracers have been investigated: ~4Mn, 59Fe, sSCo and 63Ni (Table 1). Typical curves ob- tained are shown in Fig. 3(a, b, c). The Fig. 4 shows the plots of Ln (specific activity) vs. the square of X--Xm (distance from the maxi- mum).

The apparent effective valence is related to xm by:

4pkTxra (16) Z~*-- epoJ

where: poJ = E and p is the slope of the plots in Fig. 4.

The temperature is determined by using values of Q and Do from the literature [26-29]. In general the temperature thus obtained does not differ by more than 10~ from that in- dicated by two insulated thermocouples placed at 2.5 mm from the isotope layer, excepting the case of 5SCo where the calculated tem- perature is 20 to 40~ higher than that of the thermocouples. On the other hand, the recent diffusion data for this tracer [30] are consistent with observed temperatures given in Table 1.

In order to show the importance of the vacancy flow effect, let us consider first the case where it is neglected:

Z~: = z~ :': (17)

and we will see that the results are quite

E L E C T R O M I G R A T I O N IN SILVER 2139

,<Np~

W ~ o o _ _ - - =

W4

. 5 =1

= . 5

I , I , I i I ~ R=W3__ 2 3 4 5

= . 2 WI

_ . 5

=.1

-1. W,=o

Fig. 2. Vacancy flow term (n~) as a function of W31W, for some values of Wo/W4.

anomalous. From (16) and (17), one can cal- culate the sum of the excess resistivities due to the ion at the stable position and at the saddle point, using Bosvieux and Friedel's model [ 18]:

'~-'~B ~ AP (stable) Jr Ap (saddle)

= ( l + f o - - 2 Z ~ ) p o / l O 0 ( ] 8 )

f0 is equal to --2.5 in Silver. The results are given in Table 2. It can be noticed that the two last columns show almost the same values, namely:

EB--~ APLinde (in Ag or Au). (19)

The case of ~aNi is questionable in view of the

Table 1. Electromigration o f transitional tracers in silver

D t (cmZ/sec) po J xm

Tracer (sec) 10 ~ 10 -a (p.l~-cm) (amp/cm z) (p.) Z** ---AZ~*

S4Mn 0.7293 3-17 7.53 4062 70 -9.83 3-2 1-4211 2.95 7.50 5170 200 - 12.21 2-3

59Fe 1.2354 1.212 7-55 4194 250 --52.41 3.8 * 1-3644 1-69 8-82 3904 365 - 5 i .74 4.4

ssCo 1.6344 1-022 7-55 4549 250 -43.32 4.6 1.3948 1.011 7.55 4847 200 --38.53 5.2

~Ni 2.088 0.826 7-52 5468 150 --20.97 9.2

*Experiment performed in copper.

2 1 4 0 N G U Y E N V A N D O A N

O

3

O ,7 2

(n

O I

ggQQ

IQ QQ

5 , 1 0 :

-I- Om O

54Mn in Ag

I i

I I I I 2 3 C M

Fig . 3 (a) .

large experimental error for this run. For the sake of simplification, let us assume that Ap cstable) is comparable to the value of the resistivity given by Linde[9] for correspond- ing substitutional impurities in Au (excepting Mn). This assumption implies further that:

(a) the contribution to ApLinde from lattice relaxation around the impurity atom is neg- lected;

3 ~ 1 0 5 . -

2

D- O <

O LI.

tU 1 O.

o Q

m

I �9 I I 1

(b) Ap (stable) does not include the contribu- tion from the vacancy adjacent to the jumping atom, for it arises only from the momentum transferred to the latter.

From equation (19), this would mean that the excess resistivity of the ion at the saddle point would be very small for all these tracers. Physically this is very surprising, for one would expect ApCSaddJe) to be at least of the

e I

59Fe in Ag

+

I I i 2 3

e e

I L C M

Fig . 3 (b) .

E L E C T R O M I G R A T I O N IN S I L V E R 2141

<c

o u.

o3

_3=10 s

- -2 0

e

@

Q

- - 0

@

@

0

0

e

I

I

e

P

=

~

~

o

, g e l ~ g

~

2 5 0 ~

1 2 3

58Co in Ag

+

o�9

1 " " '~1 7 6 1 4 9 1 7 6 4 c~

Fig. 3(c). Typical curves of electromigration in Silver (a) 5*Mn; (b) SaFe; (c) 58Co.

same order of magnitude a s Ap (stable). A tentative explanation to this unexpected

result can be given by taking into account the vacancy flow effect. Z* is no longer equal to Z**, but it is given by equation (11):

Z* = Z** -- 2(np)Z,~. (11')

Therefore if (np) is negative, the absolute value Iz*l would be much greater than ]Z**I. Unfortunately, the data necessary to the (np) calculation for the tracers of interest in Silver, namely the ratios Wo/W4 and W3/WI are not

available. For this reason, we have carried out an experiment with saFe in Copper (see Table 1) where at 980~ we have[31]:

W3=0 .04 and W4 Wx ~ = 0"15

Hence: (np) =- -0 .789 with: Z ~ u = - 1 8 1 3 2 - 3 4 ] and Z*~*=--51.8, the equations (11 ') and (18) give respectively:

Z ~ = --80-2

Er~ = 14,2 p,12-cm/at. %. (in copper)

Table 2. Comparison between resistivities deduced from the present work and from Linde's measurement (/xf/-cm/at. %)

ApLlnde Tracer N ~ T (~ Z~ ~ Z~* XB (Ag) (Au) (Cu)

S4Mn 1 877 --9-8 1-37 1.60 2-41 2-83 2 873 - 1 2 . 2 1-72

saFe 1 880 --52-4 7.80 - - 7-66 9.3 2* 994 --51.8 9-14

58Co 1 882 -43-3 6"43 - - 6-1 6-4 2 880 - 3 8 "5 5 "71

SaNi I 876 - 2 0 . 9 3.04 - - 1.0 l "25

2142 NGUYEN VAN DOAN

I I I

- ~ 54M~

A

> . D

I - - R

O r m

I

I l l

10 a

8 6

4

2

,o = I I 0 -5 1 CM 2 1.5 2 d 0-:~

Fig. 4. Ln (specific activity) versus the square of X--Xm (distance from the maximum).

In the same system, Kuz'menko [35] has given for Z ~ results varying from --28 to --65.7, but they have to be divided by 0-78 to give exact values of Z*~*.

In the same way, if one assumes the same value of <n~> for SgFe in Silver where Z*g = --12"5 [1], one would obtain:

Z*e = --71"2

XFe = 10"8/zf~cm/at. %. (in Silver)

Therefore these values of EFe are more reason- able in view of Linde's data.

In conclusion, the small values of Iz *l in

Table 2 suggest that for all transitional tracers investigated in the present work the vacancy flow term <n~) is negative and plays an impor- tant role in the interpretation of the experi- mental results. Considerations on the relative magnitude of Ap (stable) and Ap (saddle), for e x a m p l e Ap (saddle) ~ Ap (stable), would enable us to predict an upper limit of the ratio Wa/WI for a given Wo/W4. On the other hand, it would be very interesting to take into account this vacancy flow effect for results in Ref. [2]. However , the correction in this case is thought to be less important[36], for the term (np) would be small in view of estimated vacancy jump frequency ratios [25]. This is also, to a

E L E C T R O M I G R A T I O N IN SILVER 2143

certain extent, supported by the reasonable experimental values obtained for the different excess resistivities considered.

Acknowledgements-We are grateful to Dr Y. Adda and G. Brebec of this laboratory for their interest and en- couragement, to Professor H. B. Huntington and Pro- fessor J. Friedel for helpful correspondence and dis- cussions. We are also very pleased to thank Dr. J. R. Manning for his interesting comments.

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