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Atomic configuration effects in electromigration
I Phys. Chem. Solids. Vol. Printed in Great Britain.
42, pp. 3%316.
1981
W22.-3697/81/@403OMW32.00/0 Pergamon Press Ltd.
ATOMIC CONFIGURATION EFFECTS IN ELECTROMIGRATION RICHARD S.SORBELLO
Department of Physics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201,U.S.A. (Received 12 February 1980; accepted in revised form 21 August 1980) Abstract-We investigate the effects of atomic configuration on the driving force for electromigation. The validity of the pseudo-atom picture is discussed. General pseudopotential expressions are used to calculate the configuration-dependent force for lattice distortion fields around impurities as well as the force appropriate to the interstitialcy-translation-rotation mechanism for fast-diffusion. Lattice-distortion fields typically give rise to a l&30% change in the electron-wind force. Substantial reduction in the wind-forcewas found to occur during the rotation phase of fast diffuser migration, in agreement with a conjecture by Hsieh et al. The effects of grain-boundary structure were considered and found to be small within the pseudoatom picture. Finally, we show how configurational effects may lead to the observed isotope separation in liquid metals (HHffnereffect). 1. INTRODUCTION
The migration of atoms in a metal which is subjected to
an electric field is the phenomenon known as In the early theoretical electromigration [ l-31. treatments[4-6], the driving force for electromigration was separated into two distinct contributions. One contribution is called the electrostatic or direct force and is the force due to the applied electric field acting on the atom or ion of interest. The second contribution is called the electron-wind force and arises from the scattering of electrons by the atom or ion of interest. In the ballistic theories of Fiks[4] and Huntington[5], the electron-wind force is given by the momentum transfer per set from the electron current to the scattering center. For an isolated impurity, the wind-force F is given by [4,5] F=
_npd& %p
(1)
where n is the electron density, nd the density of impurities, p is the resistivity and &, is the impurity resistivity. e = le( is the magnitude of the electron charge and E is the applied electric field. The resistivity is related to the relaxation time r by the expression p = tn/ne2r, where m is the electron mass. Similarly, pd = m/n&d where Td is the relaxation time for impurity scattering. We are here assuming electron conductors; for hole conductors, the minus sign is absent in eqn (1). According to the ballistic models, the total force is obtained by adding the electrostatic force ZeE to the wind force F, where 2 is the valence of the impurity. Denoting the total force as F, we have F==ZeE+F.
(2)
It is customary [ l-31 to express the total force in terms of an effective valence Z* defined such that FT = Z*eE. Clearly from eqn (2), Z* can be written in the form
z*=z-$
where the parameter K measures the wind-force F and is defined according to F= -(K/p)eE. For the particular case of the ballistic expression (l), we have K = n&/n& In more general models or for more complex scattering centers than an isolated impurity, we expect different values of K. In recent years, considerable effort has gone into determining K and establishing whether or not the electrostatic force is simply ZeE as we have asserted in eqn (2). According to Bosvieux and Friedel[6], there is no such electrostatic force on an isolated interstitial, since the impurity acts like a screened neutral object in the external field. The consensus now, however, is that the presence of ZeE is correct. Perhaps, the most convincing analyses center on calculations developing out of the linear-response formalism introduced by Kumar and Sorbello [7] and further developed by Schaich[d], Sham[9] and Rimbey and Sorbello[lO]. Related linearresponse work was performed by othersIll-131 while semi-classical analyses were performed by Das and Peierls[l4], Rohrschach[lS] and Landauer[16]. In these more recent approaches, the a priori separation of the force into wind and electrostatic contributions was not made. The picture that emerges from the most recent work (see especially Ref. [lo]) is the following. First of all, there are corrections to the ZeE term in eqn (2), but for weak-scattering (Ze 1) these are of order 2. (This was also pointed out in Refs. [9 and 171).There is thus no way that the electrostatic term ZeE can be fully screened by these corrections, at least in the small Z limit[9,10,17]. Even when Z is taken to be the actual valence, it is found that the corrections are small compared to ZeE[8,13]. In the strong-coupling regime there are corrections to ZeE typically amounting to a fraction of eE[lO, 121. When there is a virtual-bound-state near the bottom of the conduction band, however, the contribution from this state is similar to that of a bound charge[lO, 121. This correction can then be comparable to ZeE. In the absence of virtual-bound states, it appears that eqns (1) and (2) are rather good approximations for the isolated impurity in the jellium model.
R. s.
310
SORBELLO
Aside from the question of virtual-bound states, there is still some doubt of the validity of eqns (1) and (2) in that there are corrections to F which may be formally comparable to ZeE although they are formally negligible in the usual kJ% 1 limit, where kF is the Fermi wavevector and I is the mean-free-path[lO]. (Typically kJ 1 100 at room temperature.) This difficulty arises because formally ZeE is of order (k&-l times F. For example, if we use the Das-Peierls value of F, we find FlZeE = ZkJ/6. In a more applicable model, we use pseudopotential theory[l8] to calculate pd in eqn (1). We find FlZeE =(ZkJ)a where a is a parameter typically around 0.01-0.05, and gives the factor by which the average squared form-factor over the range 0 < q < 2kF is smaller than its q = 0 value. (This average is weighted by q3.) The smallness of the parameter a accounts for the fact that typically ZeE is observed to be much larger than (k&IF, despite the fact that formally F is of order kJ times ZeE. The smallness of LYalso assuages some of our fears surrounding the validity of eqns (1) and (2) without the inclusion of the higher-order corrections to F. These corrections higher-order in (k&’ are extremely difficult to calculate, but are most probably negligible compared either to ZeE or to F itself. Our conclusion is then that unless there are virtual-boundstates low in the conduction band, eqns (1) and (2) are acceptable. These virtual-bound states wduld change Z in eqn (2) and bring it closer to the value it would have if the virtual-bound-states became true bound states. Since expression (2) appears justifiable within the context of recent work, we are led to calculate the electron-wind force F within the pseudopotential picture. According to this picture[19], the force F is determined by a self-consistent pseudopotential calculation in which a displaced Fermi distribution is associated with the electron current. Although the details of the calculation[l7,19] differ from that of Bosvieux and Friedel[6] the approach is similar. The result is that the ions acquire dipolar-like charge distributions arising from the electron scattering. These ions with their associated electron distributions are the distorted “pseudoatoms” which compose the crystal. By determining the electric fields arising from these pseudoatoms we can compute the wind-force on any ion in the crystal. We present the basic equations of this pseudoatom picture in Section 2. Applications of these equations to specific physical systems are made in Section 3. We there consider the contributions to the wind force arising from the following: (i) lattice distortion around interstitials and vacancies; (ii) the configuration of atoms during fast-diffusion of noble metals in lead; (iii) grainboundary structure; and (iv) isotope-dependent atomic configuration. A brief discussion of these results is given in Section 4. 2.
ELRCTRON-WINDFORCE IN PSEUDOATOMPKTURE
Within a pseudoatom picture of a metal we can consider the ions in the absence of a field to be locally neutral objects resembling atoms [ 181. These objects, called pseudoatoms, interact with the electron gas through a screened pseudopotential denoted by w. The
total pseudopotential W is the sum of the individual w’s over all the ions in the crystal. When a current of electrons is present, the pseudoatom charge distributions distort and the associated electric fields give rise to the electron-wind force. Within pseudopotential theory one can obtain a useful expression for the electron-wind force on the jth pseudoatom. Denoting this force by Fi, one can write[19]
Fi = -
2nkTiqwi(q)wi(q)fk eiq. ‘R’-R~“i’6(~k - Ed+,,) .. (4)
where the sum is over wavevectors k and q and over all the atoms in the crystal. These atoms are in the sum over i and are at positions Ri. Their form-factors are Wi(q). In eqn (4) fk is the perturbed Fermi-distribution in the presence of the field E, and is given by fh = - (ehT/m)k ES(eI,- h2kF2/2m). We have applied eqn (4) to simple situations in our original work[l9]. Recently we have used eqn (4) to treat liquid metals in a phase-shift parameterization scheme. It is easily demonstrated[20] that eqn (4) when applied to liquid metals yields Faber’s formula [21]. Equation (4) can be cast in a useful form by inserting the explicit form of ft and performing the integrations over k and the angle of q. One then finds that F, can be expressed in terms of a potential function CJjas follows F.= I
_?!Q aRj
where
and
for RjZRi
(7a)
=_B(+)r Wi(q)Wi(q)q’dq
for Rj =R;
(7b)
where &, is the atomic volume of the host lattice (both Wj and Wi are normalized to this atomic volume). vd is the electron drift velocity and is related to E by vd = - eET/m. In eqn (7a), j,(x) = X-* sin x -x-l cos x is the spherical Bessel function. vF = IkJm is the Fermi velocity. In applying the above expressions it is useful to make use of the fact that the contribution to the driving force from a perfect lattice of pseudo-atoms is zero. This can
Atomic configuration effects
be seen directly from eqn (4), where if all Ri are at perfect lattice positions then q must be restricted to be a reciprocal lattice vector, qo, say. This gives Fi a spatial dependence of the form sin (4,. RI) which would vanish over the diffusion jump-path of a lattice distance. (For the case of monovalent metals there is no contribution at all from q = q. since the Bragg plane does not intersect the Fermi surface in our nearly-free-electron picture.) Accordingly we restrict the sum over i in eqn (6) to be over aII pseudo-atoms deviating from the fuIIy occupied perfect lattice. This requires treating a vacancy as a missing pseudoatom, i.e. a negative pseudopotential form factor. We therefore can write in place of eqn (6) that
z
Uj = -
"
Uj.
vacancies
t
z
Uji
311
purity as FIN=. We then write (9)
FINT=F&+@'~NT
where FyNTis the force in the absence of distortion and SFlNT is the contribution due to lattice distortion. From eqns (5) and (7) we obtain SFINT within the isotropic continuum model of lattice distortion. In this model, the structure factor describing the displaced host atoms is written as &S(q) where{181 AS(q) = i z exp (- iq . Ri) ‘
(10)
and N is the number of atoms in the crystal. &S(q) has the explicit form [18]
i impurities
where in the sum over impurities we include host atoms not at a regular site as well as foreign atoms. Thus, e.g. the condition Rj = Ri in eqn (7b) does not require i = j; it could also apply to the force contribution on a substitutional impurity due to the host vacancy it necessarily sits in. Assuming one determines the average force on the atom (or atoms) involved in the diffusion jump, the parameter K of eqn (3) is easily found. Writing the average force inatomicunits in the form (vd/+)rwhere I’is an expression in atomic units, we can calculate K using the conversion K = 664r/(k,)4 where K is in PO-cm if kF is in atomic units. In the numerical results to be presented for Al and Pb,one would thenuse K = 871rforAland K = 132Or for Pb.
3.
in electromigration
APPLlCATtONS
The expressions (4)-(8) allow us to determine the force on an impurity for various physical situations. In this section, we consider the application of these expressions to the problems of lattice distortion, fast diffusers, grain boundaries, and isotope effects. (a) Lattice distortion The positions of the host atoms are perturbed in the vicinity of an impurity of a vacancy in a metal. Because of this lattice distortion we may expect additional contributions to the force on any impurity and on the host atoms. The contributions to the force arising-from the dispIaced Mice atoms can be obtained from expression (8) if we formally regard the ith displaced lattice atom as an impurity at Ri and a vacancy at &i, where Ri is the actual position of the ith lattice atom and hi is its unperturbed position in the perfect lattice. According to our discussion in Section 2, there is no contribution arising from host atoms at their perfect lattice sites. The effect of lattice distortion can then be restricted to the contributions from host atoms treated as impurities at Ri in the sums (4) or (8). For the case of an interstitial impurity we can then obtain from eqns (4) or from eqns (S)-(S) an expression for the force on the impurity. For clarity, we drop the subscript i, and denote the force on the interstitial im-
WI) = -g g(q)
(11)
where sti is the local volume change associated with the defect and g(q) is a function calculated by Harrison [ 181. From eqns (5)-(?) or upon evaluating eqn (4) explicitly, we have (12) where here and in what follows we denote the form factor of an impurity by w(q), omitting the subscript. Equation (12) is precisely the ballistic expression (1) for the electron-wind force on an impurity in jellium. This can be seen if one uses the standard pseudopotential expression[lO] for ?d or pd. Similarly we can evaluate eqn (4) using the structure factor (IO) for the Iattice distortion. The resuIt of the integrations over k and the angle of q give NqMdwo(dq3
dq (13)
where the form factor of the host-atom species is denoted by we(q) here and in what follows. If we imagine the distortion cloud to be fixed around the original impurity site and imagine the impurity to move a distance X away from its original site without affecting the distortion cloud, then we can easily extend our anafysis and find the X-dependence of the force. The result can be written FINTQJ = [1+ &UF&
(14)
where the enhancement parameter h(X) is given by =i= @(q)w(q)wo(q)il(qx)q3 dcl (15) w(q)‘q’ dq where jl(qX) = dj,(qX)/d(qX). By definition, FINT(0) is equal to the sum of expressions (12) and (13).
R.S.
312
SORBELLO
We have computed F&T and SFINT for various impurities in both lead and aluminum host metals. The results are given as the first entries in Table 1 for Al and in Table 2 for Pb. Some curves of A(X) for impurities in aluminum are shown in Fig. 1. In obtaining these results we have used both the Ashcroft empty-core pseudopotential[22] and the Heine-Abarenkov model potential[23]. In applying the latter, we have used a simple local re-screening procedure[U] to obtain the
impurity potential in the foreign host lattices. In the case of Cu, Ag and Au impurities we have applied a similar procedure to Moriarty’s form factors[25]. The structure factor for the distortion field was taken to be that of eqn (11). In displaying the calculated results we have set 8MI,, = 1 and udup = 1. The tabulated values of SFINT should therefore be multiplied by (u,,/u~). (6fi/Q) to obtain the actual values. Thus far we have considered only the lattice distortion
Table 1. Calculated electron-wind forces on impurities in Al for different pseudopotentials. The pseudopotentials used are denoted as follows: ASH is the Ashcroft potential (Ref. [22]); M is the Moriarty potential (Ref. [25]); HAA is the Heine-Abarenkov-Animalu potential (Ref. [23]);TCP is the Tomlinson-Carbotte Piercy potential (Ref. [33]). Positive values indicate forces in the direction of electron drift, negative values indicate the opposite direction. FIN7is the force on an interstitial, F soa is the force on a substitutional impurity, FNNis the force on the nearest neighbor up-wind from a vacancy. 6FrNT,Wsue and SF,, are the respective lattice-distortion contributions to these forces. Units are chosen so that the actual forces in atomic units are given by multiplying the listed values of FtNT.Fsua and FNNby u,+/uFand by multiplying the listed values of 6FIW, S&o, and SFNNby (vd/uF)(Sn/Q) Impurity
POINl
C,,(*=)
.0363
-.0541
.0226
-.0035
.0281
-.0029
C"(n)
.3865
-.0975
-.1770
.a042
.0777
-.0092
**(ASS)
.0212
-.0325
.0443
.OlOO
.0242
-.0025
Ag W)
.2996
-SO724
-.08?0
.0175
.0711
-.0072
*“(Aw
.0363
-.0541
.0226
-.0035
.0281
-.0029
Au CH)
.3912
-.0577
-.1472
.0318
.0753
-.0083
)(*
.1263
-.0005
.0104
.0579
.0335
-.0036
n8:BAA)
.1511
-.0699
.0285
.0293
.0622
-.0058
ng(IcP)
.0885
-.0218
.0399
.0451
.0370
-.0037
(ASH)
6P
INT
PO
SUB
6F
SUB
%N
6FNN
Zn(ASB)
.0966
-.0216
.0407
.0460
.0389
-.0041
Zn@AA)
.I473
-.0776
.0175
.0174
.0602
-.0060
Si(*sa)
.3689
-.1505
-.1124
.0273
.1007
-.0105
SicaMa)
.4406
-.1272
-.1373
.0511
.1057
-.0113
Table 2. Calculated electron-wind forces on impurities in Pb for different pseudopotentials. Symbols have same meaning as in Table 1
ImPurit?
'"INT
"UT
GJB
61
SUB
';N
"NN
Cu(ASE)
.0364
-.0263
.orzs
-.0105
.0170
-.0030
CU(")
.2505
-.0350
.0472
-.0402
.0378
-.0051
,S(*Sa)
.0203
-.0300
.0708
-.0071
.0198
-.0028
.0404
-.0051
A*(n)
.1924
-.0446
.0712
-.0366
AuusB)
.0364
-.0263
.0725
-.0104
.0170
-.0030
Au (‘I
.2506
-.0449
.0320
-.0389
.0447
-.0051
,,(ASE)
.0760
-.0710
.0394
-.0019
.0472
-.0047
,,(a")
.1005
-.0766
.0921
-.0159
.0514
-.0060
Zn(ASB)
.0651
-.0675
.0747
-.0064
.0448
-.0050
Zm(BAA)
.0990
-.0623
.1017
-.0190
.0436
-.0056
,,(ASE)
.3688
-.1286
.0136
-.0317
.0765
-.0113
8i(aAA)
.3008
-.1165
.0375
-.0211
.0876
-.Olll
Atomic configuration effects in electromigration
313
where RN,., is the nearest-neighbor distance, and it is assumed that the nearest neighbor is an impurity up-wind from the vacancy. Similarly, the distortion contribution SFNNis given by
I
*kF
0
-0.2..
-a3..
-0.4-m
Fig. 1. Enhancement parameter A(X) giving the increase in wind-force on a Mg, Zn, or Si interstitial as it moves downstream from the interstitial distortion cloud in an Al lattice. The nearest neighbor distance is RNN.The curves shownare calculatedusing the Heine-Abarenkov-Animalu pseudopotentials for Zn and Si in Ai, and the Tomiinson-Carbo~te-Piercy pseudopotential for Mg in Al. Actual value of A(x) is found by multiplying the displayed values by - 60/f&.
field of an interstitial. We now turn to substitutional impurities and vacancies. In the absence of distortion, the force on a substitutional impurity can be written as F&=-
&,*mkF v.j 2kF 127r3fi2( OF)I 0 w(q)[w(q) -
w&h3 dq
w(q)wo(q)SS(q)il(qR~~)q3 dq
where again AS(q) is the structure-factor correction for distortion around a vacancy. The calculated values for F&, 6FsuB, F& and 8FNN are given in Tables I and 2. In the special, although common, case of vacancy diffusion of the host atoms, w(q) should be replaced by we(q) in eqns (18) and (19). For this case the distortion correction SFNN is about the same order as the correactions for the impurity case given in the tables, i.e. about 10%. (LXI/&- - 1 for vacancies.) As the host atom moves toward the vacancy, however, the force on it increases substantially above F&[19]. We roughly estimated the lattice distortion effect by modelling the defect complex as an atom moving between two vacancies with their individual vacancy distortion fields fixed. The atom was surrounded by a negative vacancydistortion field tied to it so as to allow a smooth transition from one site to the other. In the case of Al we obtained about a 50% increase in the wind-force averaged over the path between sites assuming Sal& = - 1. Our expressions for SF,, and SFsu, show that lattice distortion effects will not change the wind force and the resistivity equally. Consider, for example, the case of interstitials. The defect resistivity in the absence of lattice distortion can be written as [ 181
pd”= end
06) which follows from eqns (5) and (7b) by regarding the substitutional impurity to consist of an impurity and a lattice vacancy occupying the same site. When the appropriate distortion field is put around this site, we obtain the correction SFsue where
(19)
I
*kF
0
W2q3dq
(20)
where C is a constant depending only on host-lattice parameters. (C can be easily obtained by applying eqn (1) with F and pd replaced by FfNT and pd”, respectively, and using eqn (12) for FyNT.) In the presence of distortion, we haveU81 *kF
I
pd = end o SFsuB =+$$
F w(~)wc4q)~S(q)q’dq. ( F )I (17)
Again AS(q) = - (8M&,,)g(q) but now g(q) is the function calculated by Harrison[iB] for a vacancy rather than an interstitial. In the case of vacancy diffusion, the relevant force is that on the nearest-neighbor lattice atom to the vacancy since this is the atom which will jump into the vacancy. Denoting the force on the nearest neighbor to a vacancy as F&, we can apply eqns (5) and (7) and obtain *‘V w(q)w&)iKqRNN)q3 dq (18)
[w(q)+ wo(q)Wd12q3 dq.
(21)
We conclude that the relative change in resistivity due to distortion is not the same as the relative change in wind-force due to distortion, i.e. +~pd” # 8FrN-&F&, where &d = &+- p,+“.However for sufficiently small distortion, we need only retain terms to order H(q) in the integrand of eqn (21). This gives
(22) It should be pointed out that there is no simple connection between 8pd and SF for the more complicated scattering complexes associated with the diffusion of vacancies or substitutional impurities. In general, one
314
R. s.
SORBELLO
must also average the wind-force over the path of the migrating atom, and this further obscures the connection between the measured defect resistivity and the wind force. On the basis of eqn (22) we can appreciate that the negative values of 6Fr, in Tables 1 and 2 reflect the fact that for small outward lattice-displacements (Sf&, positive), the defect resistivity is decreasing. This is the behavior expected from the decreased scattering associated with the decreased “effective charge” of the impurity[lQ (In the continuum lattice theory, “effective charge” decreases for positive &Cl/&, because background charge is displaced away from the impurity[B].) This positive correlation between wind forces and the effective charges does not always hold, however. For polyvalent metals, and large momentum transfers (q t kF), there is an enhancement of scattering due to interference effects between the impurity and the displaced neighborsU81 (S(q) becomes positive[18]). Thus in some cases, 6Fr, will be positive when L#&, is positive. (b) Fast diffusers We can apply our pseudoatom expressions to more complicated diffusion mechanisms such as the interstialcy translation-rotation mechanisms which are believed to occur in the fast diffusion of noble metals in lead [26]. Relatively small electromigration driving forces have been observed for these systems. Hsieh et a/.[261 have suggested that the reduction in driving force might be due to the motion of host partner atoms taking part in the diffusion process. By performing a model calculation, we find that their conjecture appears correct. We wish to investigate the possibility that in the pure rotation phase of the postulated diffusion process [26] the wind force may be significantly diminished by the presence of a displaced host atom. Our pseudoatom model consists of a vacancy at the origin, a noble metal atom at a distance of R, to one side of the origin and a lead atom at a distance R2 to the opposite side of the vacancy. The vacancy is present since the lead atom has been displaced from its original lattice position. This leaves behind a vacancy at the origin, according to the viewpoint which must be used in applying eqn (8). We now compute the net work done by the electron-wind force on the noble metal-vacancy-lead atom complex as the complex is rotated by 90”. For simplicity we keep the system rigid so that the atoms and the vacancy always lie on a straight line, and we set R, = Rz = R. The work SW done by the force on the moving atoms during the rotation then turns out to have the simple form
in Pb. The quantity SW turns out to be negative indicating a contribution opposed to the electron wind force according to the sign convention we have assumed for SW in eqn (23). We have evaluated the corrections arising from SW and compared them with the contribution from F& acting directly on the noble metal atoms. Using the Moriarty form factors for Ag and Au we find that the wind term due to F&)NT is very large, giving a contribution to K equal to 250 for Ag and 330 for Au (all units are @-cm). The SW correction reduces the magnitude of these values by about 15% for R - 0.2RNN and about 25% for R -O.SRNN. Upon using the Ashcroft form factors we find a much smaller contribution due to F&. Now we calculate K = 27 for Ag and 47 for Au, with the SW correction being very large, reducing these values by about 50% for R = 0.05RNN and by almost 100% for R -O.IRNN. (For small R, SW is linear in R.) Although the calculated numbers are markedly different for the two model potentials assumed, it is interesting that in both cases the reduction in the windforce due to SW is significant, tending to substantiate the conjecture of Hsieh et al.[26]. It remains to be seen whether the use of more accurate pseudopotentials for noble metals in lead wiil give numerical results which converge to experimental values. The experimental values obtained by Hsieh et al. [26] are K = 75 * 72 for Ag and K = 1852 42 for Au. Herzig and Strache [26] give the experimental value of K = 120+ 20 for Au and report that the value of K for Ag is considerably smaller. (One may roughly estimate from their tables that K - 40 + 40 for Ag.)
(c) Grain boundaries Grain boundary electromigration is of paramount importance in thin film technology. Here we comment on the application of the pseudoatom theory to grain boundaries. Our conclusion is that the effect of the atomic contiguration of the grain boundaries is of relatively minor importance compared to the wind-force which is directly exerted on the moving atom according to the term i = j in eqn (4). The latter contribution is the force F& given by eqn (12). The reason that the detailed structure of a grain boundary does not contribute appreciably to the electromigration driving force is that the latter is an average over the jump path. In modelling grain boundaries in a nearly-free-electron pseudo-atom picture, the atoms of the boundary are in a regular periodic’ structure. According to eqn (4) these atoms contribute a net force to atom j which is periodic along the grain boundary and has zero average. Since only the average force is relevant to z*, we conclude that there is [w(q) - wdq)lw&MdOs2 dq. no effect due to the periodic grain boundary structure. Of course, there would be higher-order contributions (23) from the grain boundary structure. However, these conHere SW includes all the two-body interference effects tributions, which are of order w3, are formally negligible between the two atoms and the vacancy but not the wind for a nearly-free-electron-metal in the pseudoatom picforce exerted directly on the noble-metal impurity. The ture. In a quantitative evaluation of these corrections, latter is given by F& of eqn (12) for the case of no one should abandon the pseudoatom picture and inlattice distal I. We have calculated 6W for Ag and Au corporate band-structure effects from the outset, i.e. F
Atomic configuration effects in electromigration
work within a Bloch-function representation with the functions matched across the boundary. A more likely effect of grain boundaries on electromigration would arise from local distortion around the diffusion complex within the grain boundary. One might thus expect a significant change in the force on an impurity in the grain boundary as compared to the bulk because of the different local distortion fields. It would then not be surprising to find Z* values differing by 20-50% between bulk and grain boundaries. Larger discrepancies would not appear to be consistent with the pseudoatom picture. Thus far the experimental results do not seem to indicate large discrepancies between Z* values in bulk and grain boundaries[27]. (d) Isotope effects We comment on the role played by atomic cotiguration on isotope effects. The two isotope effects which have been observed are the Haffner effect in liquid metals[28] and the isotope effect for hydrogen electromigration in metals [29-311. The Haffner effect is isotope separation in liquid metal by electromigration. In pure liquid metals containing two isotopes, it has been found that the heavy isotope is pushed to the cathode while the lighter isotope is pushed to the anode. Thus it appears that the heavy isotope feels a smaller wind-force than does the light isotope. Since the windforce depends on charge and not mass, it seems difficult to reconcile theory with experiment. On the other hand, if the configuration of atoms is different for the diffusion mechanisms of each of the two isotopes, an explanation is possible, since as we have seen, atomic configuration effects can be significant. The view that there should be a connection between atomic configuration and isotope species was suggested by Klemm[32] and reviewed by Huntington[28] and Faber[21]. According to these discussions the Haffner effect can be explained if it is assumed that during the migration process the wind force on a jumping atom is on average greater than it is for the atom in its equilibrium position. This larger wind-force on a jumping atom tends to push it toward the anode. Since the lighter atoms have the higher mobility, they respond more readily to the enhanced wind-force. Thus the lighter atoms are pushed more effectively toward the anode than the heavier atoms, in agreement with experiment. The crucial assumption upon which the above argument rests is that the migrating atom feels a greater wind force during the jump process than it feels when it is in the equilibrium condition before the jump. In principle, our expressions in Section 2 can tell us whether this assumption is valid. To apply eqns (S)-(7), we would need to know the actual configuration of all the atoms during the jump process. Since this information is not currently available, we present only a tentative argument indicating how enhanced wind-forces during migration may arise. We expect that during migration an atom will pass quite closely to neighboring atoms as it attempts to push its way past these atoms. Assuming this to be true, we
315
can deduce that there will be an enhancement of the wind force. This enhancement can be derived from the two-body interaction of eqn (7a) applied to the present case where both Wiand wi refer to the pure-metal form factor. Enhancement occurs when the distance between atoms is sufficiently small so that the force is positive (roughly, this requires the distance to be less than about half the average interatomic spacing at equilibrium). Physically, this enhancement arises from one atom coming within the dipolar electron-wind charge[6,19] surrounding another atom. Both atoms then feel enhanced wind forces. We remark that an analogous effect contributes to the wind-force reduction in the fast-diffuser calculation of Section 3(b). There a negative contribution to the wind force arises because the impurity has entered the dipolar charge around the vacancy, and the vacancy dipole is aligned oppositely to the impurity dipole. Isotope effects in hydrogen electromigration[29-311 are also affected by lattice distortion fields. Indeed, in the polaron model of Flynn and Stoneham[31] the local distortion fields must rearrange around interstitial sites in order for the hydrogen to diffuse. The phonon-induced changes in these distortion fields give rise to isotopedependent activation energies and electromigration driving forces[31]. A proper evaluation of the latter would have to include the atomic configuration effects we have been considering. In particular, the force on the hydrogen atom will contain contributions like those we have calculated in eqn (14) for an interstitial as it moves away from its original site. Corrections of the order of 20-30% in the wind force due to distortion might be anticipated. 4. WMMARYANDCONCLUSION In Section 2, we presented general expressions for the force on an atom in a metal in the presence of an electric field. These expressions, which include interference contributions from neighboring atoms, can be applied for any assumed configuration of atoms around the atom in question. Applications to lattice distortion fields, fast diffusers, grain boundaries and isotope effects were considered. The configuration dependence of the force can be understood from the picture of the dynamically screened pseudoatom. Each pseudoatom acquires a dipolar-like charge distribution due to the scattering of electrons by the pseudoatoms. The dipolar-like field surrounding one pseudoatom exerts a force on nearby pseudo-atoms. The effective potential for this 2-body interaction is the uii of expression (7a). This interaction is a non-conservative one in the sense that the net work done for a given rearrangement of two or more atoms depends upon the particular paths assumed during the rearrangment. The effect of atomic configuration can be large. For typical values of the relative volume change sfl/&, say 0.1-0.3, the corrections due to lattice distortion can amount to around l&30% of the wind-force. In the case of other configurations, such as that of the interstitialcyrotation complex postulated for fast diffusers, the corrections may be considerably larger.
316
R. S. SORBELLO
The results displayed in Tables 1 and 2 indicates that the calculations are sensitive to choice of form factor. That is discrepancies of around 25-50% often occur in calculated values obtained with the Ashcroft potential as compared with the Heine-Aberenkov-Animalu potential for the same system. In the case of the noble metals, we have already remarked on the great discrepancy between calculations using Ashcroft and Moriarty pseudo-potentials. To obtain more reliable results one should abandon the local re-screening approach used here to construct the form-factors w(q). Instead one should calculate w(q) in the appropriate host lattice. With the exception of the MgAA system[33] little work has done on accurately determining impurity pseudopotentials. The calculated values in the Tables should therefore be eyed cautiously. Despite the great sensitivity of the fast-diffuser calculations to the choice of pseudopotential, we have been able to show that the wind force can be substantially reduced in the pure rotation stage of the fast-diffusion mechanism. This lends validity to the conjecture that the relatively small wind-force observed for fast diffusers is due to the presence of the displaced host atom in the vicinity of the impurity. We also argued that the regular atomic configuration of a grain boundary does not lead to a strong effect for diffusion along grain boundaries. There may be significant contributions, however, due to local distortions and band-structure effects not included in the pseudoatom picture. Finally, we saw how the Haffner effect can be ascribed to the local configuration of atoms during the diffusion jump. We conclude that the pseudoatom picture can be profitably used to estimate the effects of atomic contiguration on the driving force for electromigration. Aclmowledgemenf-This work was supported by the Air Force Office of Scientific Research under grant AFOSR-763082. REPERENCES 1. Huntington H. B., In Diffusion in Solids: Recent Deuelopmenfs (Editedby A. S. Nowickand J. J. Burton). Academic, New York (1974). 2. Wever H., Electra- und Thermotransport in Metallen. Barth, Leipzig (1973); Pratt J. N. and Sellors R. G. R., Elecin Metals and Alloys. Trans. Tech. SA, Riehen, -trotranspotl . . . ,_^__. Swrtzerland (lY73).
3. Electra- and Thermo-transport in Metals and Alloys (Edited by R. E. Hummel and H. B. Huntington). The Metallurgical Society of AIME, New York (1977). 4. Fiks V. B., Sou. Phys.-Solid State 1, 14 (1959). 5. Huntinaton H. B. and &one A. R.., J. Phvs. , Chem. Solids 20. 76 (1%:). 6. Bosvieux C. and Friedel J., I. Phys. Chem. Solids 23, 123 (1%2). 7. Kumar P. and Sorbello R. S., Thin Solid Films 25.25 (1975). 8. Schaich W. L., Phys. Reu. 813, 3350 (1976); 13, 3360(1976); 19,620 (1979). 9. Sham. L., Phys. Reo. B12, 3142(1975). 10. Rimbey P. R. and Sorbello R. S., Phys. Rev. B (to bek 21,215O(1980). 11. Turban I., Nozieres P. and Gerl M., 1. Phys. (Paris) 37, 159 (1976). 12. Bell B., unpublished.
13. Sorbello RI S. and Dasgupta B., Phys. Rev. B16,5193 (1977), and Phys. Reu. B, 21, 21% (1980). 14. Das A. K. and Peierls R., J. Phys. C 6.2811 (1973). 15. Rorschach H. E.. Ann. Phvs. 98. 70 (1976). 16. Landauer R., L ihys. C E,‘L389(1975);Phys. Reu. B14, 1474 (1976);16.4698 (1977). 17. Sorbello R. S., Ph.D. Thesis, Stanford University (1970) (unpublished). 18. Harrison W. A., Pseudopotentials in the Theory of Metals. Beniamin, New York (1966). 19. So&lo R. S., J. Phys.. Chem. Solids 34,937 (1973). Note the following typographical error: In eqn (4.3) e x eC *j should read x e-i. ‘1. 20. Sorbello R. S., Phys. Status Solidi (b)%, 671 (1978). 21. Faber T. E.. Theorv of Liauid Metals. Cambridge Universitv Press, Cambridge (1972). . 22. Ashcroft N. W., Phys. Lett. 23, 48 (1%6). Values of core radii were obtained from Fukai Y., Phys. Rev. 186, 697 (1%9). 23. Animalu A. 0. E. and Heine V., Phil. Mug. 12, 1249 (1965).
See also Ref. [18). 24. Sorbello R. S., J. Phys. F 4, 1665 (1974). 25. Moriarty J., Phys. Rev. Bl, 1363(1970). 26. Hsieh M. Y., Huntington H. B. and Jeffrey R. N., J. Lattice Defects 7.9 (1977); Herzig Chr. and Stracke E., Phys. Status Solidi (e)27, 75 (1975). 27. See Ho P. S., in Ref. [3]. 28. Huntinzton H. B.. In Encvclooedia of Chemical Technolonv. II, Supplement. Interscience,.New York-(1971). 29. Wipf H., In Hydrogen in Metals II (Edited by G. Alefeld and J. Volkl), p. 273. Springer-Verlag, Berlin (1978). 30. Einziger R. E. and Huntington H. B., J. Phys. Chem. Solids 35, 1563(1974). 31. Flynn C. P. and Stoneham A. M., Phys. Rev. Bl, 3966(1970); J. Phys. F 3, SO5(1973). 32. Klemm A., 2. Naturforsch. A9, 1031(1954). 33. Tomlinson P. G., Carbotte J. P. and Piercy G. R., J. Phys. F 7, 1305 (lY77).
42, pp. 3%316.
1981
W22.-3697/81/@403OMW32.00/0 Pergamon Press Ltd.
ATOMIC CONFIGURATION EFFECTS IN ELECTROMIGRATION RICHARD S.SORBELLO
Department of Physics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201,U.S.A. (Received 12 February 1980; accepted in revised form 21 August 1980) Abstract-We investigate the effects of atomic configuration on the driving force for electromigation. The validity of the pseudo-atom picture is discussed. General pseudopotential expressions are used to calculate the configuration-dependent force for lattice distortion fields around impurities as well as the force appropriate to the interstitialcy-translation-rotation mechanism for fast-diffusion. Lattice-distortion fields typically give rise to a l&30% change in the electron-wind force. Substantial reduction in the wind-forcewas found to occur during the rotation phase of fast diffuser migration, in agreement with a conjecture by Hsieh et al. The effects of grain-boundary structure were considered and found to be small within the pseudoatom picture. Finally, we show how configurational effects may lead to the observed isotope separation in liquid metals (HHffnereffect). 1. INTRODUCTION
The migration of atoms in a metal which is subjected to
an electric field is the phenomenon known as In the early theoretical electromigration [ l-31. treatments[4-6], the driving force for electromigration was separated into two distinct contributions. One contribution is called the electrostatic or direct force and is the force due to the applied electric field acting on the atom or ion of interest. The second contribution is called the electron-wind force and arises from the scattering of electrons by the atom or ion of interest. In the ballistic theories of Fiks[4] and Huntington[5], the electron-wind force is given by the momentum transfer per set from the electron current to the scattering center. For an isolated impurity, the wind-force F is given by [4,5] F=
_npd& %p
(1)
where n is the electron density, nd the density of impurities, p is the resistivity and &, is the impurity resistivity. e = le( is the magnitude of the electron charge and E is the applied electric field. The resistivity is related to the relaxation time r by the expression p = tn/ne2r, where m is the electron mass. Similarly, pd = m/n&d where Td is the relaxation time for impurity scattering. We are here assuming electron conductors; for hole conductors, the minus sign is absent in eqn (1). According to the ballistic models, the total force is obtained by adding the electrostatic force ZeE to the wind force F, where 2 is the valence of the impurity. Denoting the total force as F, we have F==ZeE+F.
(2)
It is customary [ l-31 to express the total force in terms of an effective valence Z* defined such that FT = Z*eE. Clearly from eqn (2), Z* can be written in the form
z*=z-$
where the parameter K measures the wind-force F and is defined according to F= -(K/p)eE. For the particular case of the ballistic expression (l), we have K = n&/n& In more general models or for more complex scattering centers than an isolated impurity, we expect different values of K. In recent years, considerable effort has gone into determining K and establishing whether or not the electrostatic force is simply ZeE as we have asserted in eqn (2). According to Bosvieux and Friedel[6], there is no such electrostatic force on an isolated interstitial, since the impurity acts like a screened neutral object in the external field. The consensus now, however, is that the presence of ZeE is correct. Perhaps, the most convincing analyses center on calculations developing out of the linear-response formalism introduced by Kumar and Sorbello [7] and further developed by Schaich[d], Sham[9] and Rimbey and Sorbello[lO]. Related linearresponse work was performed by othersIll-131 while semi-classical analyses were performed by Das and Peierls[l4], Rohrschach[lS] and Landauer[16]. In these more recent approaches, the a priori separation of the force into wind and electrostatic contributions was not made. The picture that emerges from the most recent work (see especially Ref. [lo]) is the following. First of all, there are corrections to the ZeE term in eqn (2), but for weak-scattering (Ze 1) these are of order 2. (This was also pointed out in Refs. [9 and 171).There is thus no way that the electrostatic term ZeE can be fully screened by these corrections, at least in the small Z limit[9,10,17]. Even when Z is taken to be the actual valence, it is found that the corrections are small compared to ZeE[8,13]. In the strong-coupling regime there are corrections to ZeE typically amounting to a fraction of eE[lO, 121. When there is a virtual-bound-state near the bottom of the conduction band, however, the contribution from this state is similar to that of a bound charge[lO, 121. This correction can then be comparable to ZeE. In the absence of virtual-bound states, it appears that eqns (1) and (2) are rather good approximations for the isolated impurity in the jellium model.
R. s.
310
SORBELLO
Aside from the question of virtual-bound states, there is still some doubt of the validity of eqns (1) and (2) in that there are corrections to F which may be formally comparable to ZeE although they are formally negligible in the usual kJ% 1 limit, where kF is the Fermi wavevector and I is the mean-free-path[lO]. (Typically kJ 1 100 at room temperature.) This difficulty arises because formally ZeE is of order (k&-l times F. For example, if we use the Das-Peierls value of F, we find FlZeE = ZkJ/6. In a more applicable model, we use pseudopotential theory[l8] to calculate pd in eqn (1). We find FlZeE =(ZkJ)a where a is a parameter typically around 0.01-0.05, and gives the factor by which the average squared form-factor over the range 0 < q < 2kF is smaller than its q = 0 value. (This average is weighted by q3.) The smallness of the parameter a accounts for the fact that typically ZeE is observed to be much larger than (k&IF, despite the fact that formally F is of order kJ times ZeE. The smallness of LYalso assuages some of our fears surrounding the validity of eqns (1) and (2) without the inclusion of the higher-order corrections to F. These corrections higher-order in (k&’ are extremely difficult to calculate, but are most probably negligible compared either to ZeE or to F itself. Our conclusion is then that unless there are virtual-boundstates low in the conduction band, eqns (1) and (2) are acceptable. These virtual-bound states wduld change Z in eqn (2) and bring it closer to the value it would have if the virtual-bound-states became true bound states. Since expression (2) appears justifiable within the context of recent work, we are led to calculate the electron-wind force F within the pseudopotential picture. According to this picture[19], the force F is determined by a self-consistent pseudopotential calculation in which a displaced Fermi distribution is associated with the electron current. Although the details of the calculation[l7,19] differ from that of Bosvieux and Friedel[6] the approach is similar. The result is that the ions acquire dipolar-like charge distributions arising from the electron scattering. These ions with their associated electron distributions are the distorted “pseudoatoms” which compose the crystal. By determining the electric fields arising from these pseudoatoms we can compute the wind-force on any ion in the crystal. We present the basic equations of this pseudoatom picture in Section 2. Applications of these equations to specific physical systems are made in Section 3. We there consider the contributions to the wind force arising from the following: (i) lattice distortion around interstitials and vacancies; (ii) the configuration of atoms during fast-diffusion of noble metals in lead; (iii) grainboundary structure; and (iv) isotope-dependent atomic configuration. A brief discussion of these results is given in Section 4. 2.
ELRCTRON-WINDFORCE IN PSEUDOATOMPKTURE
Within a pseudoatom picture of a metal we can consider the ions in the absence of a field to be locally neutral objects resembling atoms [ 181. These objects, called pseudoatoms, interact with the electron gas through a screened pseudopotential denoted by w. The
total pseudopotential W is the sum of the individual w’s over all the ions in the crystal. When a current of electrons is present, the pseudoatom charge distributions distort and the associated electric fields give rise to the electron-wind force. Within pseudopotential theory one can obtain a useful expression for the electron-wind force on the jth pseudoatom. Denoting this force by Fi, one can write[19]
Fi = -
2nkTiqwi(q)wi(q)fk eiq. ‘R’-R~“i’6(~k - Ed+,,) .. (4)
where the sum is over wavevectors k and q and over all the atoms in the crystal. These atoms are in the sum over i and are at positions Ri. Their form-factors are Wi(q). In eqn (4) fk is the perturbed Fermi-distribution in the presence of the field E, and is given by fh = - (ehT/m)k ES(eI,- h2kF2/2m). We have applied eqn (4) to simple situations in our original work[l9]. Recently we have used eqn (4) to treat liquid metals in a phase-shift parameterization scheme. It is easily demonstrated[20] that eqn (4) when applied to liquid metals yields Faber’s formula [21]. Equation (4) can be cast in a useful form by inserting the explicit form of ft and performing the integrations over k and the angle of q. One then finds that F, can be expressed in terms of a potential function CJjas follows F.= I
_?!Q aRj
where
and
for RjZRi
(7a)
=_B(+)r Wi(q)Wi(q)q’dq
for Rj =R;
(7b)
where &, is the atomic volume of the host lattice (both Wj and Wi are normalized to this atomic volume). vd is the electron drift velocity and is related to E by vd = - eET/m. In eqn (7a), j,(x) = X-* sin x -x-l cos x is the spherical Bessel function. vF = IkJm is the Fermi velocity. In applying the above expressions it is useful to make use of the fact that the contribution to the driving force from a perfect lattice of pseudo-atoms is zero. This can
Atomic configuration effects
be seen directly from eqn (4), where if all Ri are at perfect lattice positions then q must be restricted to be a reciprocal lattice vector, qo, say. This gives Fi a spatial dependence of the form sin (4,. RI) which would vanish over the diffusion jump-path of a lattice distance. (For the case of monovalent metals there is no contribution at all from q = q. since the Bragg plane does not intersect the Fermi surface in our nearly-free-electron picture.) Accordingly we restrict the sum over i in eqn (6) to be over aII pseudo-atoms deviating from the fuIIy occupied perfect lattice. This requires treating a vacancy as a missing pseudoatom, i.e. a negative pseudopotential form factor. We therefore can write in place of eqn (6) that
z
Uj = -
"
Uj.
vacancies
t
z
Uji
311
purity as FIN=. We then write (9)
FINT=F&+@'~NT
where FyNTis the force in the absence of distortion and SFlNT is the contribution due to lattice distortion. From eqns (5) and (7) we obtain SFINT within the isotropic continuum model of lattice distortion. In this model, the structure factor describing the displaced host atoms is written as &S(q) where{181 AS(q) = i z exp (- iq . Ri) ‘
(10)
and N is the number of atoms in the crystal. &S(q) has the explicit form [18]
i impurities
where in the sum over impurities we include host atoms not at a regular site as well as foreign atoms. Thus, e.g. the condition Rj = Ri in eqn (7b) does not require i = j; it could also apply to the force contribution on a substitutional impurity due to the host vacancy it necessarily sits in. Assuming one determines the average force on the atom (or atoms) involved in the diffusion jump, the parameter K of eqn (3) is easily found. Writing the average force inatomicunits in the form (vd/+)rwhere I’is an expression in atomic units, we can calculate K using the conversion K = 664r/(k,)4 where K is in PO-cm if kF is in atomic units. In the numerical results to be presented for Al and Pb,one would thenuse K = 871rforAland K = 132Or for Pb.
3.
in electromigration
APPLlCATtONS
The expressions (4)-(8) allow us to determine the force on an impurity for various physical situations. In this section, we consider the application of these expressions to the problems of lattice distortion, fast diffusers, grain boundaries, and isotope effects. (a) Lattice distortion The positions of the host atoms are perturbed in the vicinity of an impurity of a vacancy in a metal. Because of this lattice distortion we may expect additional contributions to the force on any impurity and on the host atoms. The contributions to the force arising-from the dispIaced Mice atoms can be obtained from expression (8) if we formally regard the ith displaced lattice atom as an impurity at Ri and a vacancy at &i, where Ri is the actual position of the ith lattice atom and hi is its unperturbed position in the perfect lattice. According to our discussion in Section 2, there is no contribution arising from host atoms at their perfect lattice sites. The effect of lattice distortion can then be restricted to the contributions from host atoms treated as impurities at Ri in the sums (4) or (8). For the case of an interstitial impurity we can then obtain from eqns (4) or from eqns (S)-(S) an expression for the force on the impurity. For clarity, we drop the subscript i, and denote the force on the interstitial im-
WI) = -g g(q)
(11)
where sti is the local volume change associated with the defect and g(q) is a function calculated by Harrison [ 181. From eqns (5)-(?) or upon evaluating eqn (4) explicitly, we have (12) where here and in what follows we denote the form factor of an impurity by w(q), omitting the subscript. Equation (12) is precisely the ballistic expression (1) for the electron-wind force on an impurity in jellium. This can be seen if one uses the standard pseudopotential expression[lO] for ?d or pd. Similarly we can evaluate eqn (4) using the structure factor (IO) for the Iattice distortion. The resuIt of the integrations over k and the angle of q give NqMdwo(dq3
dq (13)
where the form factor of the host-atom species is denoted by we(q) here and in what follows. If we imagine the distortion cloud to be fixed around the original impurity site and imagine the impurity to move a distance X away from its original site without affecting the distortion cloud, then we can easily extend our anafysis and find the X-dependence of the force. The result can be written FINTQJ = [1+ &UF&
(14)
where the enhancement parameter h(X) is given by =i= @(q)w(q)wo(q)il(qx)q3 dcl (15) w(q)‘q’ dq where jl(qX) = dj,(qX)/d(qX). By definition, FINT(0) is equal to the sum of expressions (12) and (13).
R.S.
312
SORBELLO
We have computed F&T and SFINT for various impurities in both lead and aluminum host metals. The results are given as the first entries in Table 1 for Al and in Table 2 for Pb. Some curves of A(X) for impurities in aluminum are shown in Fig. 1. In obtaining these results we have used both the Ashcroft empty-core pseudopotential[22] and the Heine-Abarenkov model potential[23]. In applying the latter, we have used a simple local re-screening procedure[U] to obtain the
impurity potential in the foreign host lattices. In the case of Cu, Ag and Au impurities we have applied a similar procedure to Moriarty’s form factors[25]. The structure factor for the distortion field was taken to be that of eqn (11). In displaying the calculated results we have set 8MI,, = 1 and udup = 1. The tabulated values of SFINT should therefore be multiplied by (u,,/u~). (6fi/Q) to obtain the actual values. Thus far we have considered only the lattice distortion
Table 1. Calculated electron-wind forces on impurities in Al for different pseudopotentials. The pseudopotentials used are denoted as follows: ASH is the Ashcroft potential (Ref. [22]); M is the Moriarty potential (Ref. [25]); HAA is the Heine-Abarenkov-Animalu potential (Ref. [23]);TCP is the Tomlinson-Carbotte Piercy potential (Ref. [33]). Positive values indicate forces in the direction of electron drift, negative values indicate the opposite direction. FIN7is the force on an interstitial, F soa is the force on a substitutional impurity, FNNis the force on the nearest neighbor up-wind from a vacancy. 6FrNT,Wsue and SF,, are the respective lattice-distortion contributions to these forces. Units are chosen so that the actual forces in atomic units are given by multiplying the listed values of FtNT.Fsua and FNNby u,+/uFand by multiplying the listed values of 6FIW, S&o, and SFNNby (vd/uF)(Sn/Q) Impurity
POINl
C,,(*=)
.0363
-.0541
.0226
-.0035
.0281
-.0029
C"(n)
.3865
-.0975
-.1770
.a042
.0777
-.0092
**(ASS)
.0212
-.0325
.0443
.OlOO
.0242
-.0025
Ag W)
.2996
-SO724
-.08?0
.0175
.0711
-.0072
*“(Aw
.0363
-.0541
.0226
-.0035
.0281
-.0029
Au CH)
.3912
-.0577
-.1472
.0318
.0753
-.0083
)(*
.1263
-.0005
.0104
.0579
.0335
-.0036
n8:BAA)
.1511
-.0699
.0285
.0293
.0622
-.0058
ng(IcP)
.0885
-.0218
.0399
.0451
.0370
-.0037
(ASH)
6P
INT
PO
SUB
6F
SUB
%N
6FNN
Zn(ASB)
.0966
-.0216
.0407
.0460
.0389
-.0041
Zn@AA)
.I473
-.0776
.0175
.0174
.0602
-.0060
Si(*sa)
.3689
-.1505
-.1124
.0273
.1007
-.0105
SicaMa)
.4406
-.1272
-.1373
.0511
.1057
-.0113
Table 2. Calculated electron-wind forces on impurities in Pb for different pseudopotentials. Symbols have same meaning as in Table 1
ImPurit?
'"INT
"UT
GJB
61
SUB
';N
"NN
Cu(ASE)
.0364
-.0263
.orzs
-.0105
.0170
-.0030
CU(")
.2505
-.0350
.0472
-.0402
.0378
-.0051
,S(*Sa)
.0203
-.0300
.0708
-.0071
.0198
-.0028
.0404
-.0051
A*(n)
.1924
-.0446
.0712
-.0366
AuusB)
.0364
-.0263
.0725
-.0104
.0170
-.0030
Au (‘I
.2506
-.0449
.0320
-.0389
.0447
-.0051
,,(ASE)
.0760
-.0710
.0394
-.0019
.0472
-.0047
,,(a")
.1005
-.0766
.0921
-.0159
.0514
-.0060
Zn(ASB)
.0651
-.0675
.0747
-.0064
.0448
-.0050
Zm(BAA)
.0990
-.0623
.1017
-.0190
.0436
-.0056
,,(ASE)
.3688
-.1286
.0136
-.0317
.0765
-.0113
8i(aAA)
.3008
-.1165
.0375
-.0211
.0876
-.Olll
Atomic configuration effects in electromigration
313
where RN,., is the nearest-neighbor distance, and it is assumed that the nearest neighbor is an impurity up-wind from the vacancy. Similarly, the distortion contribution SFNNis given by
I
*kF
0
-0.2..
-a3..
-0.4-m
Fig. 1. Enhancement parameter A(X) giving the increase in wind-force on a Mg, Zn, or Si interstitial as it moves downstream from the interstitial distortion cloud in an Al lattice. The nearest neighbor distance is RNN.The curves shownare calculatedusing the Heine-Abarenkov-Animalu pseudopotentials for Zn and Si in Ai, and the Tomiinson-Carbo~te-Piercy pseudopotential for Mg in Al. Actual value of A(x) is found by multiplying the displayed values by - 60/f&.
field of an interstitial. We now turn to substitutional impurities and vacancies. In the absence of distortion, the force on a substitutional impurity can be written as F&=-
&,*mkF v.j 2kF 127r3fi2( OF)I 0 w(q)[w(q) -
w&h3 dq
w(q)wo(q)SS(q)il(qR~~)q3 dq
where again AS(q) is the structure-factor correction for distortion around a vacancy. The calculated values for F&, 6FsuB, F& and 8FNN are given in Tables I and 2. In the special, although common, case of vacancy diffusion of the host atoms, w(q) should be replaced by we(q) in eqns (18) and (19). For this case the distortion correction SFNN is about the same order as the correactions for the impurity case given in the tables, i.e. about 10%. (LXI/&- - 1 for vacancies.) As the host atom moves toward the vacancy, however, the force on it increases substantially above F&[19]. We roughly estimated the lattice distortion effect by modelling the defect complex as an atom moving between two vacancies with their individual vacancy distortion fields fixed. The atom was surrounded by a negative vacancydistortion field tied to it so as to allow a smooth transition from one site to the other. In the case of Al we obtained about a 50% increase in the wind-force averaged over the path between sites assuming Sal& = - 1. Our expressions for SF,, and SFsu, show that lattice distortion effects will not change the wind force and the resistivity equally. Consider, for example, the case of interstitials. The defect resistivity in the absence of lattice distortion can be written as [ 181
pd”= end
06) which follows from eqns (5) and (7b) by regarding the substitutional impurity to consist of an impurity and a lattice vacancy occupying the same site. When the appropriate distortion field is put around this site, we obtain the correction SFsue where
(19)
I
*kF
0
W2q3dq
(20)
where C is a constant depending only on host-lattice parameters. (C can be easily obtained by applying eqn (1) with F and pd replaced by FfNT and pd”, respectively, and using eqn (12) for FyNT.) In the presence of distortion, we haveU81 *kF
I
pd = end o SFsuB =+$$
F w(~)wc4q)~S(q)q’dq. ( F )I (17)
Again AS(q) = - (8M&,,)g(q) but now g(q) is the function calculated by Harrison[iB] for a vacancy rather than an interstitial. In the case of vacancy diffusion, the relevant force is that on the nearest-neighbor lattice atom to the vacancy since this is the atom which will jump into the vacancy. Denoting the force on the nearest neighbor to a vacancy as F&, we can apply eqns (5) and (7) and obtain *‘V w(q)w&)iKqRNN)q3 dq (18)
[w(q)+ wo(q)Wd12q3 dq.
(21)
We conclude that the relative change in resistivity due to distortion is not the same as the relative change in wind-force due to distortion, i.e. +~pd” # 8FrN-&F&, where &d = &+- p,+“.However for sufficiently small distortion, we need only retain terms to order H(q) in the integrand of eqn (21). This gives
(22) It should be pointed out that there is no simple connection between 8pd and SF for the more complicated scattering complexes associated with the diffusion of vacancies or substitutional impurities. In general, one
314
R. s.
SORBELLO
must also average the wind-force over the path of the migrating atom, and this further obscures the connection between the measured defect resistivity and the wind force. On the basis of eqn (22) we can appreciate that the negative values of 6Fr, in Tables 1 and 2 reflect the fact that for small outward lattice-displacements (Sf&, positive), the defect resistivity is decreasing. This is the behavior expected from the decreased scattering associated with the decreased “effective charge” of the impurity[lQ (In the continuum lattice theory, “effective charge” decreases for positive &Cl/&, because background charge is displaced away from the impurity[B].) This positive correlation between wind forces and the effective charges does not always hold, however. For polyvalent metals, and large momentum transfers (q t kF), there is an enhancement of scattering due to interference effects between the impurity and the displaced neighborsU81 (S(q) becomes positive[18]). Thus in some cases, 6Fr, will be positive when L#&, is positive. (b) Fast diffusers We can apply our pseudoatom expressions to more complicated diffusion mechanisms such as the interstialcy translation-rotation mechanisms which are believed to occur in the fast diffusion of noble metals in lead [26]. Relatively small electromigration driving forces have been observed for these systems. Hsieh et a/.[261 have suggested that the reduction in driving force might be due to the motion of host partner atoms taking part in the diffusion process. By performing a model calculation, we find that their conjecture appears correct. We wish to investigate the possibility that in the pure rotation phase of the postulated diffusion process [26] the wind force may be significantly diminished by the presence of a displaced host atom. Our pseudoatom model consists of a vacancy at the origin, a noble metal atom at a distance of R, to one side of the origin and a lead atom at a distance R2 to the opposite side of the vacancy. The vacancy is present since the lead atom has been displaced from its original lattice position. This leaves behind a vacancy at the origin, according to the viewpoint which must be used in applying eqn (8). We now compute the net work done by the electron-wind force on the noble metal-vacancy-lead atom complex as the complex is rotated by 90”. For simplicity we keep the system rigid so that the atoms and the vacancy always lie on a straight line, and we set R, = Rz = R. The work SW done by the force on the moving atoms during the rotation then turns out to have the simple form
in Pb. The quantity SW turns out to be negative indicating a contribution opposed to the electron wind force according to the sign convention we have assumed for SW in eqn (23). We have evaluated the corrections arising from SW and compared them with the contribution from F& acting directly on the noble metal atoms. Using the Moriarty form factors for Ag and Au we find that the wind term due to F&)NT is very large, giving a contribution to K equal to 250 for Ag and 330 for Au (all units are @-cm). The SW correction reduces the magnitude of these values by about 15% for R - 0.2RNN and about 25% for R -O.SRNN. Upon using the Ashcroft form factors we find a much smaller contribution due to F&. Now we calculate K = 27 for Ag and 47 for Au, with the SW correction being very large, reducing these values by about 50% for R = 0.05RNN and by almost 100% for R -O.IRNN. (For small R, SW is linear in R.) Although the calculated numbers are markedly different for the two model potentials assumed, it is interesting that in both cases the reduction in the windforce due to SW is significant, tending to substantiate the conjecture of Hsieh et al.[26]. It remains to be seen whether the use of more accurate pseudopotentials for noble metals in lead wiil give numerical results which converge to experimental values. The experimental values obtained by Hsieh et al. [26] are K = 75 * 72 for Ag and K = 1852 42 for Au. Herzig and Strache [26] give the experimental value of K = 120+ 20 for Au and report that the value of K for Ag is considerably smaller. (One may roughly estimate from their tables that K - 40 + 40 for Ag.)
(c) Grain boundaries Grain boundary electromigration is of paramount importance in thin film technology. Here we comment on the application of the pseudoatom theory to grain boundaries. Our conclusion is that the effect of the atomic contiguration of the grain boundaries is of relatively minor importance compared to the wind-force which is directly exerted on the moving atom according to the term i = j in eqn (4). The latter contribution is the force F& given by eqn (12). The reason that the detailed structure of a grain boundary does not contribute appreciably to the electromigration driving force is that the latter is an average over the jump path. In modelling grain boundaries in a nearly-free-electron pseudo-atom picture, the atoms of the boundary are in a regular periodic’ structure. According to eqn (4) these atoms contribute a net force to atom j which is periodic along the grain boundary and has zero average. Since only the average force is relevant to z*, we conclude that there is [w(q) - wdq)lw&MdOs2 dq. no effect due to the periodic grain boundary structure. Of course, there would be higher-order contributions (23) from the grain boundary structure. However, these conHere SW includes all the two-body interference effects tributions, which are of order w3, are formally negligible between the two atoms and the vacancy but not the wind for a nearly-free-electron-metal in the pseudoatom picforce exerted directly on the noble-metal impurity. The ture. In a quantitative evaluation of these corrections, latter is given by F& of eqn (12) for the case of no one should abandon the pseudoatom picture and inlattice distal I. We have calculated 6W for Ag and Au corporate band-structure effects from the outset, i.e. F
Atomic configuration effects in electromigration
work within a Bloch-function representation with the functions matched across the boundary. A more likely effect of grain boundaries on electromigration would arise from local distortion around the diffusion complex within the grain boundary. One might thus expect a significant change in the force on an impurity in the grain boundary as compared to the bulk because of the different local distortion fields. It would then not be surprising to find Z* values differing by 20-50% between bulk and grain boundaries. Larger discrepancies would not appear to be consistent with the pseudoatom picture. Thus far the experimental results do not seem to indicate large discrepancies between Z* values in bulk and grain boundaries[27]. (d) Isotope effects We comment on the role played by atomic cotiguration on isotope effects. The two isotope effects which have been observed are the Haffner effect in liquid metals[28] and the isotope effect for hydrogen electromigration in metals [29-311. The Haffner effect is isotope separation in liquid metal by electromigration. In pure liquid metals containing two isotopes, it has been found that the heavy isotope is pushed to the cathode while the lighter isotope is pushed to the anode. Thus it appears that the heavy isotope feels a smaller wind-force than does the light isotope. Since the windforce depends on charge and not mass, it seems difficult to reconcile theory with experiment. On the other hand, if the configuration of atoms is different for the diffusion mechanisms of each of the two isotopes, an explanation is possible, since as we have seen, atomic configuration effects can be significant. The view that there should be a connection between atomic configuration and isotope species was suggested by Klemm[32] and reviewed by Huntington[28] and Faber[21]. According to these discussions the Haffner effect can be explained if it is assumed that during the migration process the wind force on a jumping atom is on average greater than it is for the atom in its equilibrium position. This larger wind-force on a jumping atom tends to push it toward the anode. Since the lighter atoms have the higher mobility, they respond more readily to the enhanced wind-force. Thus the lighter atoms are pushed more effectively toward the anode than the heavier atoms, in agreement with experiment. The crucial assumption upon which the above argument rests is that the migrating atom feels a greater wind force during the jump process than it feels when it is in the equilibrium condition before the jump. In principle, our expressions in Section 2 can tell us whether this assumption is valid. To apply eqns (S)-(7), we would need to know the actual configuration of all the atoms during the jump process. Since this information is not currently available, we present only a tentative argument indicating how enhanced wind-forces during migration may arise. We expect that during migration an atom will pass quite closely to neighboring atoms as it attempts to push its way past these atoms. Assuming this to be true, we
315
can deduce that there will be an enhancement of the wind force. This enhancement can be derived from the two-body interaction of eqn (7a) applied to the present case where both Wiand wi refer to the pure-metal form factor. Enhancement occurs when the distance between atoms is sufficiently small so that the force is positive (roughly, this requires the distance to be less than about half the average interatomic spacing at equilibrium). Physically, this enhancement arises from one atom coming within the dipolar electron-wind charge[6,19] surrounding another atom. Both atoms then feel enhanced wind forces. We remark that an analogous effect contributes to the wind-force reduction in the fast-diffuser calculation of Section 3(b). There a negative contribution to the wind force arises because the impurity has entered the dipolar charge around the vacancy, and the vacancy dipole is aligned oppositely to the impurity dipole. Isotope effects in hydrogen electromigration[29-311 are also affected by lattice distortion fields. Indeed, in the polaron model of Flynn and Stoneham[31] the local distortion fields must rearrange around interstitial sites in order for the hydrogen to diffuse. The phonon-induced changes in these distortion fields give rise to isotopedependent activation energies and electromigration driving forces[31]. A proper evaluation of the latter would have to include the atomic configuration effects we have been considering. In particular, the force on the hydrogen atom will contain contributions like those we have calculated in eqn (14) for an interstitial as it moves away from its original site. Corrections of the order of 20-30% in the wind force due to distortion might be anticipated. 4. WMMARYANDCONCLUSION In Section 2, we presented general expressions for the force on an atom in a metal in the presence of an electric field. These expressions, which include interference contributions from neighboring atoms, can be applied for any assumed configuration of atoms around the atom in question. Applications to lattice distortion fields, fast diffusers, grain boundaries and isotope effects were considered. The configuration dependence of the force can be understood from the picture of the dynamically screened pseudoatom. Each pseudoatom acquires a dipolar-like charge distribution due to the scattering of electrons by the pseudoatoms. The dipolar-like field surrounding one pseudoatom exerts a force on nearby pseudo-atoms. The effective potential for this 2-body interaction is the uii of expression (7a). This interaction is a non-conservative one in the sense that the net work done for a given rearrangement of two or more atoms depends upon the particular paths assumed during the rearrangment. The effect of atomic configuration can be large. For typical values of the relative volume change sfl/&, say 0.1-0.3, the corrections due to lattice distortion can amount to around l&30% of the wind-force. In the case of other configurations, such as that of the interstitialcyrotation complex postulated for fast diffusers, the corrections may be considerably larger.
316
R. S. SORBELLO
The results displayed in Tables 1 and 2 indicates that the calculations are sensitive to choice of form factor. That is discrepancies of around 25-50% often occur in calculated values obtained with the Ashcroft potential as compared with the Heine-Aberenkov-Animalu potential for the same system. In the case of the noble metals, we have already remarked on the great discrepancy between calculations using Ashcroft and Moriarty pseudo-potentials. To obtain more reliable results one should abandon the local re-screening approach used here to construct the form-factors w(q). Instead one should calculate w(q) in the appropriate host lattice. With the exception of the MgAA system[33] little work has done on accurately determining impurity pseudopotentials. The calculated values in the Tables should therefore be eyed cautiously. Despite the great sensitivity of the fast-diffuser calculations to the choice of pseudopotential, we have been able to show that the wind force can be substantially reduced in the pure rotation stage of the fast-diffusion mechanism. This lends validity to the conjecture that the relatively small wind-force observed for fast diffusers is due to the presence of the displaced host atom in the vicinity of the impurity. We also argued that the regular atomic configuration of a grain boundary does not lead to a strong effect for diffusion along grain boundaries. There may be significant contributions, however, due to local distortions and band-structure effects not included in the pseudoatom picture. Finally, we saw how the Haffner effect can be ascribed to the local configuration of atoms during the diffusion jump. We conclude that the pseudoatom picture can be profitably used to estimate the effects of atomic contiguration on the driving force for electromigration. Aclmowledgemenf-This work was supported by the Air Force Office of Scientific Research under grant AFOSR-763082. REPERENCES 1. Huntington H. B., In Diffusion in Solids: Recent Deuelopmenfs (Editedby A. S. Nowickand J. J. Burton). Academic, New York (1974). 2. Wever H., Electra- und Thermotransport in Metallen. Barth, Leipzig (1973); Pratt J. N. and Sellors R. G. R., Elecin Metals and Alloys. Trans. Tech. SA, Riehen, -trotranspotl . . . ,_^__. Swrtzerland (lY73).
3. Electra- and Thermo-transport in Metals and Alloys (Edited by R. E. Hummel and H. B. Huntington). The Metallurgical Society of AIME, New York (1977). 4. Fiks V. B., Sou. Phys.-Solid State 1, 14 (1959). 5. Huntinaton H. B. and &one A. R.., J. Phvs. , Chem. Solids 20. 76 (1%:). 6. Bosvieux C. and Friedel J., I. Phys. Chem. Solids 23, 123 (1%2). 7. Kumar P. and Sorbello R. S., Thin Solid Films 25.25 (1975). 8. Schaich W. L., Phys. Reu. 813, 3350 (1976); 13, 3360(1976); 19,620 (1979). 9. Sham. L., Phys. Reo. B12, 3142(1975). 10. Rimbey P. R. and Sorbello R. S., Phys. Rev. B (to bek 21,215O(1980). 11. Turban I., Nozieres P. and Gerl M., 1. Phys. (Paris) 37, 159 (1976). 12. Bell B., unpublished.
13. Sorbello RI S. and Dasgupta B., Phys. Rev. B16,5193 (1977), and Phys. Reu. B, 21, 21% (1980). 14. Das A. K. and Peierls R., J. Phys. C 6.2811 (1973). 15. Rorschach H. E.. Ann. Phvs. 98. 70 (1976). 16. Landauer R., L ihys. C E,‘L389(1975);Phys. Reu. B14, 1474 (1976);16.4698 (1977). 17. Sorbello R. S., Ph.D. Thesis, Stanford University (1970) (unpublished). 18. Harrison W. A., Pseudopotentials in the Theory of Metals. Beniamin, New York (1966). 19. So&lo R. S., J. Phys.. Chem. Solids 34,937 (1973). Note the following typographical error: In eqn (4.3) e x eC *j should read x e-i. ‘1. 20. Sorbello R. S., Phys. Status Solidi (b)%, 671 (1978). 21. Faber T. E.. Theorv of Liauid Metals. Cambridge Universitv Press, Cambridge (1972). . 22. Ashcroft N. W., Phys. Lett. 23, 48 (1%6). Values of core radii were obtained from Fukai Y., Phys. Rev. 186, 697 (1%9). 23. Animalu A. 0. E. and Heine V., Phil. Mug. 12, 1249 (1965).
See also Ref. [18). 24. Sorbello R. S., J. Phys. F 4, 1665 (1974). 25. Moriarty J., Phys. Rev. Bl, 1363(1970). 26. Hsieh M. Y., Huntington H. B. and Jeffrey R. N., J. Lattice Defects 7.9 (1977); Herzig Chr. and Stracke E., Phys. Status Solidi (e)27, 75 (1975). 27. See Ho P. S., in Ref. [3]. 28. Huntinzton H. B.. In Encvclooedia of Chemical Technolonv. II, Supplement. Interscience,.New York-(1971). 29. Wipf H., In Hydrogen in Metals II (Edited by G. Alefeld and J. Volkl), p. 273. Springer-Verlag, Berlin (1978). 30. Einziger R. E. and Huntington H. B., J. Phys. Chem. Solids 35, 1563(1974). 31. Flynn C. P. and Stoneham A. M., Phys. Rev. Bl, 3966(1970); J. Phys. F 3, SO5(1973). 32. Klemm A., 2. Naturforsch. A9, 1031(1954). 33. Tomlinson P. G., Carbotte J. P. and Piercy G. R., J. Phys. F 7, 1305 (lY77).