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Landauer fields in electron transport and electromigration
Superlattices and Microstructures, Vol. 23, No. 3/4, 1998
Landauer fields in electron transport and electromigration R. S. Sorbello Department of Physics and Laboratory for Surface Studies, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, U.S.A
Landauer’s classic 1957 paper on electron transport in the presence of localized scatterers has provided a powerful approach for analysing the local transport field and the driving force for electromigration in metals. This approach leads to local fields which are associated with residual resistivity dipoles and with carrier density modulation, and these Landauer fields contribute to the electromigration driving force. The nature of these fields and their role in electromigration are critically examined and comparisons are made with the results of more elaborate quantum-mechanical theories. c 1998 Academic Press Limited
Key words: electromigration, residual resistivity dipoles, electron transport.
1. Introduction Landauer’s pioneering work [1] on electron transport in 1957 provided a new and bold approach that has led to a deeper understanding of the electronic conduction process. Prior treatments of electron transport through metals containing localized defects had assumed an infinite system and the existence of a uniform electric field. However, such treatments missed the all-important fact that it is the scattering of the incident electron current by the localized defects themselves that is crucial in establishing the self-consistent electric field. By analysing the local environment of the defect and using appropriate boundary conditions, Landauer was able to show that the scattered electrons form a dipolar charge distribution surrounding each defect, and that this dipole sets up the long-range electric field associated with residual resistivity. Landauer called this localized dipole the ‘residual resistivity dipole’ (RRD). Although Landauer’s approach did not give any new answers for the residual resistivity in bulk systems, it provided a much more physical way of thinking about the electron transport problem. In particular, attention was focused on a region of space surrounding the localized defect, and the scattering of incident electrons by the defect was viewed as a self-consistent process subject to appropriate boundary conditions. The boundary conditions were dictated by the behaviour of electrons far from the defect, for example, whether the electrons are subjected to uniform background scattering or whether the electrons are supplied to and drained from a mesoscopic system by means of reservoirs attached to the ends of the sample. By applying his approach to mesoscopic systems connected to reservoirs, Landauer [2] was able to derive the celebrated ‘Landauer formula,’ which relates the resistance of a mesoscopic sample to the quantum-mechanical transmission coefficient of the system. The Landauer approach for mesoscopic systems, with its subsequent refinement by B¨uttiker [3], has turned out to be indispensable for analysing electron transport on the mesoscopic scale, where multiple scattering of electrons and electron-localization phenomena come into play. The impact of Landauer’s approach to electron transport is not limited to the mesoscopic scale. Indeed, 0749–6036/98/030711 + 08 $25.00/0
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Landauer was able to apply the results of his classic 1957 paper to elucidate the nature of the driving force for electromigration in bulk systems. Electromigration is the migration of impurities in a conductor that is subjected to an electric field and the accompanying electron current [4–7]. The electromigration driving force on an atom arises from the local self-consistent field accompanying electron transport, and this field is the same entity that plays a central role in Landauer’s approach to electron transport. In particular, Landauer’s RRD occupies a conceptually important position in the electromigration phenomenon, since the RRD acts both as a source of the long-range electric field that drives electrons around an impurity and as an important contribution to the force on the impurity itself [7–9]. These ideas are important for a proper understanding of electromigration and were not well understood by the electromigration community before Landauer wrote a series of important papers in the 1970s on electromigration [8–12]. These papers, like his papers on electron transport in bulk and mesoscopic systems, bear the hallmark of Landauer’s style—highly original, intuitive, and packed with physical arguments and clever analogies that make the point persuasively. His dialogues in the 1970s, via these journal articles and open letters to the electromigration community, which included such well-known condensed matter theorists as Peierls, Friedel, Nozi´eres, and Sham, marked what in retrospect might be called the golden age of electromigration theory. In analysing electromigration, it has proven useful to conceptually separate the net driving force on an impurity into two parts [4–7]. One part is called the ‘electron wind force’ and is associated with the rate of momentum transfer from the incident electrons to the impurity during the scattering process. The other part is called the ‘direct force’ and is associated with the electrostatic force exerted by the long-range, macroscopic electric field on the ion core of the impurity. The direct force contribution has always been controversial, because it is not clear how effective the long-range electric field is when it acts on the ion core during the electron transport process. If the screening is negligible, as it is often claimed [13], this force is simply Z eE, where Z is the nominal valence of the impurity, e is the magnitude of the electron charge, and E is the macroscopic electric field. On the other hand, the direct force would vanish if the field were entirely screened out at the ion core, and for interstitial impurities this has been claimed to be the case by Bosvieux and Friedel [14] and their followers [13]. It turns out that the direct force contribution is neither Z eE nor zero, though for weakly scattering impurities (formally small Z ), the Z eE result is the correct one. This can be deduced from Landauer’s analysis [11] outlined below, and follows from more detailed quantum-mechanical treatments as well [13, 15, 16]. In his analysis of the electromigration problem, Landauer focused his attention on two major contributions to the driving force. One of these he associated with the RRD; the other he associated with what he called ‘carrier density modulation’ (CDM). Both the RRD and CDM effects give rise to local microscopic electric fields that act on the ion core of an impurity and tend to drive it through the background lattice. The RRD is associated with the scattering of the incident electrons by the defect and can therefore be regarded as a contribution to the electron wind force. The CDM effect, on the other hand, is associated with extra carriers locally brought in by the impurity, and is more aptly regarded as a contribution to the direct force in electromigration theory. A discussion of these RRD and CDM fields is given in Sections 2 and 3. A re-examination of these fields in the light of more detailed quantum mechanical calculations and a discussion of mesoscopic systems is presented in Section 4.
2. Residual resistivity dipole In discussing the residual resistivity dipole, we use Landauer’s model but follow a slightly different analysis [17, 18] that better displays the quantum-mechanical detail in the region close to the scatterer. We consider a bulk metal sample in the jellium model. Assuming that there is uniform background scattering described by a scattering time τ and that a macroscopic field E is present, the electron distribution is a shifted Fermi sphere,
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with the perturbed part of the distribution given to an excellent approximation by δ f (k) = −eτ vk · E δ(k − F ),
(1)
where vk = h¯ k/m is the velocity of an electron in state k, k = h¯ 2 k 2 /2m is the energy of that state, F is the Fermi energy, and m is the electron mass. The sample resistivity is ρ0 = m/n 0 e2 τ, where n 0 is the electron density. When an impurity is placed in the electron gas, the electron states in a region contained in a sphere that is centered on the impurity and whose radius is much less than a mean free path have the usual asymptotic scattering-state form 1 f (θ 0 ) exp(ikr ) , (2) ψk (r) ∼ 1/2 exp(ik · r) + r where is the sample volume, f (θ 0 ) is the scattering amplitude, and θ 0 is the angle between k and r. The asymptotic form (2) assumes that k F r >> 1, and r << l, where k F is the Fermi wavevector and l is the mean free path. The wavefunctions ψk (r) give rise to an electron density X δ f (k)|ψk (r)|2 , (3) δn w (r) = k
where we have used the symbol δn w to denote an electron density due to the electron wind, or incident electron current. After self-consistent screening, there is an electrostatic potential, δ8, that arises from δn w and the induced screening charge which attempts to locally neutralize δn w . Although one can use more sophisticated screening approximations to find δ8, it is simplest to use the Thomas–Fermi approximation [1, 17], in which case one obtains the analytic form 1 (4) δ8(r) = − (dn/d E)−1 δn w (r), e where dn/d E is the density of states at the Fermi level. Evaluating eqn (4) with the help of eqns (1)–(3), one finds that in the asymptotic region the potential has precisely the form of Landauer’s RRD, namely, p cos θ , (5) r2 where θ is the angle between r and E, and p is the magnitude of the electric dipole moment p, which is given by 3π h¯ S (6) p = − 2 2 J0 . 4k F e δ8(r) = −
In this expression, S is the scattering cross section for transport, and J0 is the uniform charge-current density far from the impurity. S is related to the scattering amplitude by Z (7) S = | f (θ )|2 (1 − cos θ )d, where the integration is over all solid angle. It is straightforward to calculate the force on the impurity from the net momentum transfer per second from the electron system to the impurity. This force is clearly in the direction of the electron particle current, or electron wind. Denoting this wind force as Fw , one finds Fw
= =
− h¯ keF S J0 4k 3F e p. 3π
(8)
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Precisely the same result is obtained if the wind force is evaluated from the local electric field −∇δ8 acting on the bare potential of the scatterer. This was first demonstrated by Bosvieux and Friedel [14] for the special case of weak scattering potentials. It is also true for strong scattering potentials, as can be inferred from the work of Schaich [19], Sham [20] and others [7, 13]. The dipole field of eqn (5) has been referred to as an RRD field, but we have not yet established that it yields the long-range field that is associated with the residual resistivity of a dilute concentration of impurities. To establish this connection, we need to show that a dilute concentration of N randomly located RRDs per unit volume will create an average macroscopic field that gives the potential drop that one ascribes to the presence of defects. (Following Landauer, we take this potential drop to mean the additional voltage measured across the sample due to the presence of defects assuming a measurement under constant current conditions, i.e. fixed J0 .) To determine the long-range macroscopic field due to a dilute concentration of RRDs, we need an expression for δ8 that is valid for distances much greater than the mean free path from each impurity. Unfortunately, the derivation leading to eqn (5) is valid only for a single impurity and only for r << l because it is only in this limit that the wavefunction form (2) holds. To extend these results to the r > l region one needs to solve a transport equation in real space to find the pile-up of electrons as they are scattered by the background surrounding the impurity. In such an analysis, the outgoing wavefunctions (2) act as a particle current source that feeds into the surrounding medium, and a self-consistent solution of the transport equation is required in the presence of the particle current source, background scattering, and the Coulomb interaction between electrons. This is the problem that Landauer [1] solved, and he found that the resulting long-range potential is given precisely by eqn (5). (For an alternative derivation which is more closely related to the present discussion, see [17, 21].) Because eqn (5) does hold in the region r >> l, the extra electric field associated with residual resistivity can be found from the polarization field induced by a dilute concentration of N dipoles per unit volume, with each dipole having dipole moment p given by expression (6). As Landauer [1] showed, the resulting polarization field −4π N p does turn out to equal the additional field associated with defect scattering, namely, J0 ρi , where ρi is the residual resistivity due to scattering by the impurities. This result follows immediately from eqn (6) and the relation ρi = 3π 2 h¯ S N /k 2F e2 , which is valid for a dilute concentration of impurities in an electron gas. Returning to the electromigration driving force, Fw , it is apparent that the strength of the RRD determines the force, and that the force is always in the direction of the electron wind. On the other hand, it is apparent from the fact that p is aligned along the electron particle current, that the charge distribution of the RRD corresponds to a build-up of electrons on the up-wind side of the impurity and a depletion of electrons on the down-wind side. This would, in itself, tend to pull the impurity in the up-wind direction, which seemingly contradicts the statement that the net wind force is in the direction of the electron wind. This contradiction vanishes when it is realized that the force on an impurity depends upon the microscopic electric field at the ion core of the impurity, and for this purpose one has to use the close-in form of the wavefunctions, and not the asymptotic wavefunction forms given by expression (2). Looked at from another viewpoint, the RRD potential δ8 of eqn (5) is proportional to the scattering cross section and therefore gives contributions of order Z 2 and higher. Thus, even if the RRD potential (5) were to persist close to the impurity’s core, it would lead to a wind force contribution of order Z 3 and higher. On the other hand, there are other contributions to the close-in dipole potential that are of order Z , and these lead to a wind force that starts at order Z 2 . These contributions always dominate the RRD contributions. The close-in dipole field that is of order Z is often referred to as the Bosvieux–Friedel dipole because it was first explicitly calculated by these authors [14]. In reality, both the Bosvieux–Friedel dipole and the Landauer RRD are describing the same dipolar charge distribution surrounding the impurity, but they are entities that are dominant in different regions of space. If accurate expressions for the wavefunctions are used in eqn (3), and an accurate screening calculation is performed, one can obtain the actual charge distribution surrounding the impurity. This, in turn, would lead to an accurate determination of the wind force. Such exactness is not needed, however, for determining the asymptotic RRD, which, as seen from eqns (5)–(7),
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depends only on the scattering cross section. Of course, an accurate determination of the scattering cross section can present significant problems of its own, especially if one goes beyond the jellium model and includes multiple scattering with other defects and lattice atoms [13].
3. Carrier density modulation According to Landauer, there is another contribution to the local transport field that is not contained in the RRD analysis. This contribution arises from the modulation of carrier density in the vicinity of the impurity because of the extra electrons brought in by the impurity. If for example, the bare ion is positively charged (Z > 0), the electron density is locally increased as electrons are locally drawn-in to neutralize the impurity. This increased electron density, in effect, increases the local conductivity. Continuity of the current then implies that the local electric field is diminished in the region of higher electron density. The analogy used by Landauer is that of a local macroscopic inclusion of relatively high conductivity in a conducting material. (For Z < 0, the inclusion would have relatively low conductivity.) Assuming that this CDM effect gives rise to a local perturbation in the transport field which has a dipolar form about each impurity ion, Landauer found that the long-range polarization field associated with these N dipoles per unit volume causes a fractional resistivity change given by [11] 3N Z δρ =− , ρ 3n 0 + βδn
(9)
where n 0 is the unperturbed electron density and δn is the extra density induced in the region of the defect. (For purposes of estimation, δn ≈ Z /d , where d is the volume of the defect.) The parameter β is on the order of unity, and arises from the approximate nature of the calculation. The extra resistivity δρ can be viewed as a contribution to the deviation from Matthiessen’s rule due to modification of the mobile charge density by the defect [11]. Landauer’s derivation of eqn (9) is based on the concept of local momentum generation and the long-range field that is set-up to be consistent with it. Local momentum generation arises from the local, inhomogeneous electric field acting on the local excess charge density δn. Specifically, if the total electric field in the vicinity of the impurity is written as E0 + δE, where E0 is the constant (average) macroscopic field due to all sources far from the impurity in question and δE is the variation in the local field that arises as a consequence of the CDM dipole formation, then Landauer argues that the excess local momentum generated in the vicinity of the impurity is given by [11] Z Z dP (10) = E0 δnd 3r + δEδnd 3r, dt where at this stage it is assumed that δn is a function of position. In the same way that the RRD is related to the local momentum generation associated with the wind force, so too is a CDM dipole related to the momentum generation given by eqn (10). This implies that the CDM dipole moment is given by [11] 1 dP , (11) p= 4π n 0 dt which can be explicitly verified for the RRD case by using eqn (8) together with the relations Fw = dP/dt and n 0 = k 3F /3π 2 . After the simplifying assumption that δn is approximately constant over the atomic volume, Landauer was able to use eqns (10) and (11) to solve for p. His result is p=
Z E0 3 . 4π 3n 0 + βδn
(12)
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Finally, the long-range polarization field due to the CDM dipoles was found from −4π N p, and this extra field was used to obtain the extra resistivity δρ of eqn (9). Now, what is relevant for the electromigration driving force is the close-in detail of the CDM dipolar charge distribution. As such detail falls outside the scope of Landauer’s CDM calculation, he found it necessary to use an alternate, more cunning approach in which the force on an impurity could be expressed in terms of the change in resistivity caused by the impurity. The necessary expression had already been derived by Das and Peierls [22] on the basis of their semiclassical model, and the result was referred to by Landauer [9] as the ‘Das–Peierls electromigration theorem.’ He then showed how their result could be obtained simply from very general arguments based on momentum conservation [9, 23]. The Das–Peierls–Landauer result for the total force on an impurity is ! ρ − ρ 0 n0 E, (13) F = −e ρ N where ρ 0 is the resistivity in the absence of the defect ions. As pointed out by Landauer [9] and Das and Peierls [23], ρ−ρ 0 contains contributions from defect scattering and from the CDM effect. The defect-scattering contribution can be written as ρi = m/ne2 τi , where n is the electron density, and τi is the electron-impurity scattering time, which is related to the scattering cross section (7) by τi = 1/N v F S. The CDM contribution to ρ − ρ 0 is the quantity δρ, which appears in eqn (9). With these substitutions, eqn (13) gives Z eE n 0 ρi E+ . (14) F = −e Nρ 1 + βδn/3n 0 The first term, apart from the presence of n 0 rather than n, turns out to be just the electron wind force, whose magnitude is given by expression (7). (The distinction between n 0 and n is actually irrelevant here because the impurity density is assumed to be very dilute.) Having identified the first term as the wind force, we can therefore regard the second term in eqn (14) as the direct force. Note that this direct force reduces to the unscreened value Z eE in the formal limit of small Z . However, the (βδn/3n 0 ) term implies that the direct force can be appreciably smaller than Z eE. This reduction of the effective electrostatic field acting on the bare ion is a consequence of CDM. A similar reduction of the direct force has been found within a many-body Green function analysis [16].
4. Discussion Landauer’s approach to electron transport has yielded a great deal of insight into the local transport field and into the nature of the driving force for electromigration. Despite the limitations of the jellium model and approximations made by Landauer in obtaining explicit analytical results, his force expression (14) contains the important physics underlying the electromigration force. Specifically, there is a wind force, and there is an additional force associated with the effect of local carrier modulation on the electron transport process. This additional force, which can be considered as part of the direct force, does not arise from a simple static screening response to the applied macroscopic field. Indeed, the assumption of a static screening response would lead to the incorrect result that the impurity feels no direct force because it is a locally neutral object. As Landauer’s derivation indicates, the CDM effect is a more complex quantity having to do with local momentum generation associated with carrier density modulation in the vicinity of the impurity. The CDM contribution to the electromigration driving force, unlike the RRD contribution, cannot be simply subsumed into the scattering cross section in the wind force expression (8), despite some claims to the contrary [13, 20]. In the Green function and density matrix calculations, the CDM effect has been traced to the off-energy-shell
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parts of response functions [7]. However, such contributions are difficult to calculate, and further work is required before reliable estimates of the CDM effect in electromigration can be made. The identification of the CDM effect as part of the direct force is to a certain extent a matter of interpretation. What is important is the total force on the impurity, and this is given by eqn (14) in Landauer’s analysis. It is, however, reasonable, and in many cases quite useful, to separate the total force into component parts that are associated with a wind force and a direct force if it is found that the total force consists of two separate contributions each of which has the characteristic signature of either the wind force or the direct force. Specifically, these signatures are that the wind force is proportional to E/ρ, while the direct force is proportional to E and independent of ρ. These contributions can be isolated experimentally [4–7], and it thus makes sense to isolate them theoretically as well. The theoretical separation is not conceptually a clean one, however, because corrections to the wind force that are of order 1/ F τ times the leading term in the wind-force would obviously have the signature of a direct force. This has been discussed in detail elsewhere [7, 13]. An important point that is often overlooked in electron transport studies is Landauer’s finding that the long-range RRD fields from a volume distribution of impurities give rise to that part of the macroscopic field E associated with the residual resistivity. This point should be kept in mind when performing calculations that treat E as if it were an externally applied field rather than as a consequence of an internal distribution of RRDs coexisting self-consistently with themselves and with other sources. Ultimately, this has implications for quantum-mechanical treatments of electromigration and electron transport that are based on the assumption of a uniform applied field, as, for example, in the usual Kubo formalism [13]. The assumption of a uniform applied field is really a kind of mean-field approximation, and one should realize this when evaluating correlation functions. The far-away impurities contribute to the field in the vicinity of an ion, but these contributions are subsumed into the macroscopic field in the vicinity of the ion. We conclude that Landauer’s [24] caveat concerning the uncritical application of Kubo formulae is well founded. This is especially important in analysing disordered mesoscopic systems, where the system may be smaller than the inelastic mean free path, and the concept of an applied field requires some care [25, 26]. There are, however, some conceptual simplifications that arise in considering the local field in disordered mesoscopic structures, especially if the dimensions of such structures are much less than an inelastic mean free path. In this regime, electrons supplied from one reservoir travel through the structure, undergo elastic multiple scattering with defects and finally end up in one of the attached reservoirs. What can we say about the RRD, CDM, wind forces and direct forces in this case? Landauer’s basic approach is of course very applicable. As in the analysis for bulk metals, the scattering of the incident electrons results in local charge pile-ups around the various scatterers in the sample, and these lead to a self-consistently screened local potential δ8(r). This, in turn, results in wind forces on each impurity’s ion core. Although the net force on the impurity system acts along the direction of the electron wind, each individual impurity scatterer may be subjected to a force that is quite different from the average wind force [25]. Indeed, the direction of the force on an individual impurity might actually be opposite to the electron wind [25]! Because there is no background scattering in this simple mesoscopic system, the long-range RRD field does not come into play. Rather, one is generally dealing with complicated scattering wavefunctions extending coherently throughout the system, and these can lead to complicated profiles for the electron density and δ8(r). The entire collection of scatterers is acting as one giant impurity that extends over several, or many, electron wavelengths. Neither the direct force nor the CDM effect arises in this case because there is no driving field E0 associated with sources far from this giant impurity, and therefore by eqn (12) there is no CDM dipole. The only force present in this case is the electron wind force, i.e., the force associated with the scattering of the incident electrons. The CDM effect and the direct force would come into play, however, if inelastic background scattering were appreciable. The simple mesoscopic system in which inelastic scattering is negligible provides an extreme case of quantum transport that can be analysed within the framework of the Lang and Kohn [27] density-functional
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formalism. To apply this formalism to the transport problem, one needs to invoke the proper boundary conditions, namely, that: (1) the electron wavefunctions are those appropriate for electrons incident from reservoirs attached to the sample, and (2) the self-consistent electric field vanishes deep within the reservoirs. Calculations of this sort have been made for the purpose of determining I –V curves for an STM geometry [28–30]. It would be interesting to extend such calculations to treat inelastic scattering, and thus apply these calculations to investigate the CDM effect and the direct force. Such a program would require a more detailed analysis of the electron reservoirs, and this brings us back to Landauer.
References [1] }R. Landauer, IBM J. Res. Develop. 1, 223 (1957). This paper has been republished as: R. Landauer, J. Math. Phys. 37, 5259 (1996). See also: R. Landauer, Z. Phys. B21, 247 (1975). [2] }R. Landauer, Philos. Mag. B21, 863, (1970); See also, IBM J. Res. Develop. 32, 306 (1988). [3] }M. B¨uttiker, Phys. Rev. Lett. 57, 1761 (1986). [4] }H. B. Huntington, in Diffusion in Solids, edited by A. S. Nowick and J. J. Burton (New York, Academic, 1975), p. 303. [5] }A. H. Verbruggen, IBM J. Res. Dev. 32, 93 (1988). [6] }P. S. Ho and T. Kwok, Rep. Prog. Phys. 52, 301 (1989). [7] }R.S. Sorbello, in Solid State Physics, Vol. 51, edited by H. Ehrenreich and F. Spaepen, (New York, Academic Press, 1997), p. 159. [8] }R. Landauer and J.W.F. Woo, Phys. Rev. B10, 1266 (1974). [9] }R. Landauer, J. Phys. C8, L389 (1975). [10] }R. Landauer, J. Phys. C8, 761 (1975). [11] }R. Landauer, Phys. Rev. B14, 1474 (1976). [12] }R. Landauer, Phys. Rev. B16, 4698 (1977). [13] }For a detailed reference list, see [7]. [14] }C. Bosvieux and J. Friedel, J. Phys. Chem. Solids 23, 123 (1962). [15] }P. R. Rimbey and R. S. Sorbello, Phys. Rev. B21, 2150 (1980). [16] }R. S. Sorbello, Phys. Rev. B31, 798 (1985). [17] }R. S. Sorbello and C.S. Chu, IBM J. Res. Develop. 32, 58 (1988); [18] }R. S. Sorbello, Phys. Rev. B23, 5119 (1981). [19] }W. L. Schaich, Phys. Rev. B13, 3350 (1976); 3360 (1976). [20] }L. J. Sham, Phys. Rev. B12, 3142 (1975). [21] }C.S. Chu and R. S. Sorbello, , Phys. Rev. B38, 7260 (1988). [22] }A. K. Das and R. Peierls, J. Phys. C8, 3348 (1975). [23] }A similar derivation to Landauer’s was given by L. J. Sham in [20], except that Sham’s derivation does not consider the effect of the dipole polarization field. This is discussed in [7]. [24] }R. Landauer, IBM J. Res. Develop. 32, 306 (1988). [25] }R.S. Sorbello, Phys. Rev. B39, 4984 (1989). [26] }A.D. Stone and A. Szafer, IBM J. Res. Develop. 32, 384 (1988). [27] }N.D. Lang and W. Kohn, Phys. Rev. B1, 4555 (1970). [28] }N.D. Lang, A. Yacoby, and Y. Imry, Phys. Rev. Lett. 63, 1499 (1989); N.D. Lang, Phys. Rev. B45, 13599 (1992); 49, 2067 (1994); 55, 9364 (1997). [29] }K. Hirose and M. Tsukada, Phys. Rev. B51, 5278 (1995). [30] }N. Kobayashi, K. Hirose, and M. Tsukada, Surf. Sci. 345, 299 (1996).
Landauer fields in electron transport and electromigration R. S. Sorbello Department of Physics and Laboratory for Surface Studies, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, U.S.A
Landauer’s classic 1957 paper on electron transport in the presence of localized scatterers has provided a powerful approach for analysing the local transport field and the driving force for electromigration in metals. This approach leads to local fields which are associated with residual resistivity dipoles and with carrier density modulation, and these Landauer fields contribute to the electromigration driving force. The nature of these fields and their role in electromigration are critically examined and comparisons are made with the results of more elaborate quantum-mechanical theories. c 1998 Academic Press Limited
Key words: electromigration, residual resistivity dipoles, electron transport.
1. Introduction Landauer’s pioneering work [1] on electron transport in 1957 provided a new and bold approach that has led to a deeper understanding of the electronic conduction process. Prior treatments of electron transport through metals containing localized defects had assumed an infinite system and the existence of a uniform electric field. However, such treatments missed the all-important fact that it is the scattering of the incident electron current by the localized defects themselves that is crucial in establishing the self-consistent electric field. By analysing the local environment of the defect and using appropriate boundary conditions, Landauer was able to show that the scattered electrons form a dipolar charge distribution surrounding each defect, and that this dipole sets up the long-range electric field associated with residual resistivity. Landauer called this localized dipole the ‘residual resistivity dipole’ (RRD). Although Landauer’s approach did not give any new answers for the residual resistivity in bulk systems, it provided a much more physical way of thinking about the electron transport problem. In particular, attention was focused on a region of space surrounding the localized defect, and the scattering of incident electrons by the defect was viewed as a self-consistent process subject to appropriate boundary conditions. The boundary conditions were dictated by the behaviour of electrons far from the defect, for example, whether the electrons are subjected to uniform background scattering or whether the electrons are supplied to and drained from a mesoscopic system by means of reservoirs attached to the ends of the sample. By applying his approach to mesoscopic systems connected to reservoirs, Landauer [2] was able to derive the celebrated ‘Landauer formula,’ which relates the resistance of a mesoscopic sample to the quantum-mechanical transmission coefficient of the system. The Landauer approach for mesoscopic systems, with its subsequent refinement by B¨uttiker [3], has turned out to be indispensable for analysing electron transport on the mesoscopic scale, where multiple scattering of electrons and electron-localization phenomena come into play. The impact of Landauer’s approach to electron transport is not limited to the mesoscopic scale. Indeed, 0749–6036/98/030711 + 08 $25.00/0
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Landauer was able to apply the results of his classic 1957 paper to elucidate the nature of the driving force for electromigration in bulk systems. Electromigration is the migration of impurities in a conductor that is subjected to an electric field and the accompanying electron current [4–7]. The electromigration driving force on an atom arises from the local self-consistent field accompanying electron transport, and this field is the same entity that plays a central role in Landauer’s approach to electron transport. In particular, Landauer’s RRD occupies a conceptually important position in the electromigration phenomenon, since the RRD acts both as a source of the long-range electric field that drives electrons around an impurity and as an important contribution to the force on the impurity itself [7–9]. These ideas are important for a proper understanding of electromigration and were not well understood by the electromigration community before Landauer wrote a series of important papers in the 1970s on electromigration [8–12]. These papers, like his papers on electron transport in bulk and mesoscopic systems, bear the hallmark of Landauer’s style—highly original, intuitive, and packed with physical arguments and clever analogies that make the point persuasively. His dialogues in the 1970s, via these journal articles and open letters to the electromigration community, which included such well-known condensed matter theorists as Peierls, Friedel, Nozi´eres, and Sham, marked what in retrospect might be called the golden age of electromigration theory. In analysing electromigration, it has proven useful to conceptually separate the net driving force on an impurity into two parts [4–7]. One part is called the ‘electron wind force’ and is associated with the rate of momentum transfer from the incident electrons to the impurity during the scattering process. The other part is called the ‘direct force’ and is associated with the electrostatic force exerted by the long-range, macroscopic electric field on the ion core of the impurity. The direct force contribution has always been controversial, because it is not clear how effective the long-range electric field is when it acts on the ion core during the electron transport process. If the screening is negligible, as it is often claimed [13], this force is simply Z eE, where Z is the nominal valence of the impurity, e is the magnitude of the electron charge, and E is the macroscopic electric field. On the other hand, the direct force would vanish if the field were entirely screened out at the ion core, and for interstitial impurities this has been claimed to be the case by Bosvieux and Friedel [14] and their followers [13]. It turns out that the direct force contribution is neither Z eE nor zero, though for weakly scattering impurities (formally small Z ), the Z eE result is the correct one. This can be deduced from Landauer’s analysis [11] outlined below, and follows from more detailed quantum-mechanical treatments as well [13, 15, 16]. In his analysis of the electromigration problem, Landauer focused his attention on two major contributions to the driving force. One of these he associated with the RRD; the other he associated with what he called ‘carrier density modulation’ (CDM). Both the RRD and CDM effects give rise to local microscopic electric fields that act on the ion core of an impurity and tend to drive it through the background lattice. The RRD is associated with the scattering of the incident electrons by the defect and can therefore be regarded as a contribution to the electron wind force. The CDM effect, on the other hand, is associated with extra carriers locally brought in by the impurity, and is more aptly regarded as a contribution to the direct force in electromigration theory. A discussion of these RRD and CDM fields is given in Sections 2 and 3. A re-examination of these fields in the light of more detailed quantum mechanical calculations and a discussion of mesoscopic systems is presented in Section 4.
2. Residual resistivity dipole In discussing the residual resistivity dipole, we use Landauer’s model but follow a slightly different analysis [17, 18] that better displays the quantum-mechanical detail in the region close to the scatterer. We consider a bulk metal sample in the jellium model. Assuming that there is uniform background scattering described by a scattering time τ and that a macroscopic field E is present, the electron distribution is a shifted Fermi sphere,
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with the perturbed part of the distribution given to an excellent approximation by δ f (k) = −eτ vk · E δ(k − F ),
(1)
where vk = h¯ k/m is the velocity of an electron in state k, k = h¯ 2 k 2 /2m is the energy of that state, F is the Fermi energy, and m is the electron mass. The sample resistivity is ρ0 = m/n 0 e2 τ, where n 0 is the electron density. When an impurity is placed in the electron gas, the electron states in a region contained in a sphere that is centered on the impurity and whose radius is much less than a mean free path have the usual asymptotic scattering-state form 1 f (θ 0 ) exp(ikr ) , (2) ψk (r) ∼ 1/2 exp(ik · r) + r where is the sample volume, f (θ 0 ) is the scattering amplitude, and θ 0 is the angle between k and r. The asymptotic form (2) assumes that k F r >> 1, and r << l, where k F is the Fermi wavevector and l is the mean free path. The wavefunctions ψk (r) give rise to an electron density X δ f (k)|ψk (r)|2 , (3) δn w (r) = k
where we have used the symbol δn w to denote an electron density due to the electron wind, or incident electron current. After self-consistent screening, there is an electrostatic potential, δ8, that arises from δn w and the induced screening charge which attempts to locally neutralize δn w . Although one can use more sophisticated screening approximations to find δ8, it is simplest to use the Thomas–Fermi approximation [1, 17], in which case one obtains the analytic form 1 (4) δ8(r) = − (dn/d E)−1 δn w (r), e where dn/d E is the density of states at the Fermi level. Evaluating eqn (4) with the help of eqns (1)–(3), one finds that in the asymptotic region the potential has precisely the form of Landauer’s RRD, namely, p cos θ , (5) r2 where θ is the angle between r and E, and p is the magnitude of the electric dipole moment p, which is given by 3π h¯ S (6) p = − 2 2 J0 . 4k F e δ8(r) = −
In this expression, S is the scattering cross section for transport, and J0 is the uniform charge-current density far from the impurity. S is related to the scattering amplitude by Z (7) S = | f (θ )|2 (1 − cos θ )d, where the integration is over all solid angle. It is straightforward to calculate the force on the impurity from the net momentum transfer per second from the electron system to the impurity. This force is clearly in the direction of the electron particle current, or electron wind. Denoting this wind force as Fw , one finds Fw
= =
− h¯ keF S J0 4k 3F e p. 3π
(8)
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Precisely the same result is obtained if the wind force is evaluated from the local electric field −∇δ8 acting on the bare potential of the scatterer. This was first demonstrated by Bosvieux and Friedel [14] for the special case of weak scattering potentials. It is also true for strong scattering potentials, as can be inferred from the work of Schaich [19], Sham [20] and others [7, 13]. The dipole field of eqn (5) has been referred to as an RRD field, but we have not yet established that it yields the long-range field that is associated with the residual resistivity of a dilute concentration of impurities. To establish this connection, we need to show that a dilute concentration of N randomly located RRDs per unit volume will create an average macroscopic field that gives the potential drop that one ascribes to the presence of defects. (Following Landauer, we take this potential drop to mean the additional voltage measured across the sample due to the presence of defects assuming a measurement under constant current conditions, i.e. fixed J0 .) To determine the long-range macroscopic field due to a dilute concentration of RRDs, we need an expression for δ8 that is valid for distances much greater than the mean free path from each impurity. Unfortunately, the derivation leading to eqn (5) is valid only for a single impurity and only for r << l because it is only in this limit that the wavefunction form (2) holds. To extend these results to the r > l region one needs to solve a transport equation in real space to find the pile-up of electrons as they are scattered by the background surrounding the impurity. In such an analysis, the outgoing wavefunctions (2) act as a particle current source that feeds into the surrounding medium, and a self-consistent solution of the transport equation is required in the presence of the particle current source, background scattering, and the Coulomb interaction between electrons. This is the problem that Landauer [1] solved, and he found that the resulting long-range potential is given precisely by eqn (5). (For an alternative derivation which is more closely related to the present discussion, see [17, 21].) Because eqn (5) does hold in the region r >> l, the extra electric field associated with residual resistivity can be found from the polarization field induced by a dilute concentration of N dipoles per unit volume, with each dipole having dipole moment p given by expression (6). As Landauer [1] showed, the resulting polarization field −4π N p does turn out to equal the additional field associated with defect scattering, namely, J0 ρi , where ρi is the residual resistivity due to scattering by the impurities. This result follows immediately from eqn (6) and the relation ρi = 3π 2 h¯ S N /k 2F e2 , which is valid for a dilute concentration of impurities in an electron gas. Returning to the electromigration driving force, Fw , it is apparent that the strength of the RRD determines the force, and that the force is always in the direction of the electron wind. On the other hand, it is apparent from the fact that p is aligned along the electron particle current, that the charge distribution of the RRD corresponds to a build-up of electrons on the up-wind side of the impurity and a depletion of electrons on the down-wind side. This would, in itself, tend to pull the impurity in the up-wind direction, which seemingly contradicts the statement that the net wind force is in the direction of the electron wind. This contradiction vanishes when it is realized that the force on an impurity depends upon the microscopic electric field at the ion core of the impurity, and for this purpose one has to use the close-in form of the wavefunctions, and not the asymptotic wavefunction forms given by expression (2). Looked at from another viewpoint, the RRD potential δ8 of eqn (5) is proportional to the scattering cross section and therefore gives contributions of order Z 2 and higher. Thus, even if the RRD potential (5) were to persist close to the impurity’s core, it would lead to a wind force contribution of order Z 3 and higher. On the other hand, there are other contributions to the close-in dipole potential that are of order Z , and these lead to a wind force that starts at order Z 2 . These contributions always dominate the RRD contributions. The close-in dipole field that is of order Z is often referred to as the Bosvieux–Friedel dipole because it was first explicitly calculated by these authors [14]. In reality, both the Bosvieux–Friedel dipole and the Landauer RRD are describing the same dipolar charge distribution surrounding the impurity, but they are entities that are dominant in different regions of space. If accurate expressions for the wavefunctions are used in eqn (3), and an accurate screening calculation is performed, one can obtain the actual charge distribution surrounding the impurity. This, in turn, would lead to an accurate determination of the wind force. Such exactness is not needed, however, for determining the asymptotic RRD, which, as seen from eqns (5)–(7),
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depends only on the scattering cross section. Of course, an accurate determination of the scattering cross section can present significant problems of its own, especially if one goes beyond the jellium model and includes multiple scattering with other defects and lattice atoms [13].
3. Carrier density modulation According to Landauer, there is another contribution to the local transport field that is not contained in the RRD analysis. This contribution arises from the modulation of carrier density in the vicinity of the impurity because of the extra electrons brought in by the impurity. If for example, the bare ion is positively charged (Z > 0), the electron density is locally increased as electrons are locally drawn-in to neutralize the impurity. This increased electron density, in effect, increases the local conductivity. Continuity of the current then implies that the local electric field is diminished in the region of higher electron density. The analogy used by Landauer is that of a local macroscopic inclusion of relatively high conductivity in a conducting material. (For Z < 0, the inclusion would have relatively low conductivity.) Assuming that this CDM effect gives rise to a local perturbation in the transport field which has a dipolar form about each impurity ion, Landauer found that the long-range polarization field associated with these N dipoles per unit volume causes a fractional resistivity change given by [11] 3N Z δρ =− , ρ 3n 0 + βδn
(9)
where n 0 is the unperturbed electron density and δn is the extra density induced in the region of the defect. (For purposes of estimation, δn ≈ Z /d , where d is the volume of the defect.) The parameter β is on the order of unity, and arises from the approximate nature of the calculation. The extra resistivity δρ can be viewed as a contribution to the deviation from Matthiessen’s rule due to modification of the mobile charge density by the defect [11]. Landauer’s derivation of eqn (9) is based on the concept of local momentum generation and the long-range field that is set-up to be consistent with it. Local momentum generation arises from the local, inhomogeneous electric field acting on the local excess charge density δn. Specifically, if the total electric field in the vicinity of the impurity is written as E0 + δE, where E0 is the constant (average) macroscopic field due to all sources far from the impurity in question and δE is the variation in the local field that arises as a consequence of the CDM dipole formation, then Landauer argues that the excess local momentum generated in the vicinity of the impurity is given by [11] Z Z dP (10) = E0 δnd 3r + δEδnd 3r, dt where at this stage it is assumed that δn is a function of position. In the same way that the RRD is related to the local momentum generation associated with the wind force, so too is a CDM dipole related to the momentum generation given by eqn (10). This implies that the CDM dipole moment is given by [11] 1 dP , (11) p= 4π n 0 dt which can be explicitly verified for the RRD case by using eqn (8) together with the relations Fw = dP/dt and n 0 = k 3F /3π 2 . After the simplifying assumption that δn is approximately constant over the atomic volume, Landauer was able to use eqns (10) and (11) to solve for p. His result is p=
Z E0 3 . 4π 3n 0 + βδn
(12)
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Finally, the long-range polarization field due to the CDM dipoles was found from −4π N p, and this extra field was used to obtain the extra resistivity δρ of eqn (9). Now, what is relevant for the electromigration driving force is the close-in detail of the CDM dipolar charge distribution. As such detail falls outside the scope of Landauer’s CDM calculation, he found it necessary to use an alternate, more cunning approach in which the force on an impurity could be expressed in terms of the change in resistivity caused by the impurity. The necessary expression had already been derived by Das and Peierls [22] on the basis of their semiclassical model, and the result was referred to by Landauer [9] as the ‘Das–Peierls electromigration theorem.’ He then showed how their result could be obtained simply from very general arguments based on momentum conservation [9, 23]. The Das–Peierls–Landauer result for the total force on an impurity is ! ρ − ρ 0 n0 E, (13) F = −e ρ N where ρ 0 is the resistivity in the absence of the defect ions. As pointed out by Landauer [9] and Das and Peierls [23], ρ−ρ 0 contains contributions from defect scattering and from the CDM effect. The defect-scattering contribution can be written as ρi = m/ne2 τi , where n is the electron density, and τi is the electron-impurity scattering time, which is related to the scattering cross section (7) by τi = 1/N v F S. The CDM contribution to ρ − ρ 0 is the quantity δρ, which appears in eqn (9). With these substitutions, eqn (13) gives Z eE n 0 ρi E+ . (14) F = −e Nρ 1 + βδn/3n 0 The first term, apart from the presence of n 0 rather than n, turns out to be just the electron wind force, whose magnitude is given by expression (7). (The distinction between n 0 and n is actually irrelevant here because the impurity density is assumed to be very dilute.) Having identified the first term as the wind force, we can therefore regard the second term in eqn (14) as the direct force. Note that this direct force reduces to the unscreened value Z eE in the formal limit of small Z . However, the (βδn/3n 0 ) term implies that the direct force can be appreciably smaller than Z eE. This reduction of the effective electrostatic field acting on the bare ion is a consequence of CDM. A similar reduction of the direct force has been found within a many-body Green function analysis [16].
4. Discussion Landauer’s approach to electron transport has yielded a great deal of insight into the local transport field and into the nature of the driving force for electromigration. Despite the limitations of the jellium model and approximations made by Landauer in obtaining explicit analytical results, his force expression (14) contains the important physics underlying the electromigration force. Specifically, there is a wind force, and there is an additional force associated with the effect of local carrier modulation on the electron transport process. This additional force, which can be considered as part of the direct force, does not arise from a simple static screening response to the applied macroscopic field. Indeed, the assumption of a static screening response would lead to the incorrect result that the impurity feels no direct force because it is a locally neutral object. As Landauer’s derivation indicates, the CDM effect is a more complex quantity having to do with local momentum generation associated with carrier density modulation in the vicinity of the impurity. The CDM contribution to the electromigration driving force, unlike the RRD contribution, cannot be simply subsumed into the scattering cross section in the wind force expression (8), despite some claims to the contrary [13, 20]. In the Green function and density matrix calculations, the CDM effect has been traced to the off-energy-shell
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parts of response functions [7]. However, such contributions are difficult to calculate, and further work is required before reliable estimates of the CDM effect in electromigration can be made. The identification of the CDM effect as part of the direct force is to a certain extent a matter of interpretation. What is important is the total force on the impurity, and this is given by eqn (14) in Landauer’s analysis. It is, however, reasonable, and in many cases quite useful, to separate the total force into component parts that are associated with a wind force and a direct force if it is found that the total force consists of two separate contributions each of which has the characteristic signature of either the wind force or the direct force. Specifically, these signatures are that the wind force is proportional to E/ρ, while the direct force is proportional to E and independent of ρ. These contributions can be isolated experimentally [4–7], and it thus makes sense to isolate them theoretically as well. The theoretical separation is not conceptually a clean one, however, because corrections to the wind force that are of order 1/ F τ times the leading term in the wind-force would obviously have the signature of a direct force. This has been discussed in detail elsewhere [7, 13]. An important point that is often overlooked in electron transport studies is Landauer’s finding that the long-range RRD fields from a volume distribution of impurities give rise to that part of the macroscopic field E associated with the residual resistivity. This point should be kept in mind when performing calculations that treat E as if it were an externally applied field rather than as a consequence of an internal distribution of RRDs coexisting self-consistently with themselves and with other sources. Ultimately, this has implications for quantum-mechanical treatments of electromigration and electron transport that are based on the assumption of a uniform applied field, as, for example, in the usual Kubo formalism [13]. The assumption of a uniform applied field is really a kind of mean-field approximation, and one should realize this when evaluating correlation functions. The far-away impurities contribute to the field in the vicinity of an ion, but these contributions are subsumed into the macroscopic field in the vicinity of the ion. We conclude that Landauer’s [24] caveat concerning the uncritical application of Kubo formulae is well founded. This is especially important in analysing disordered mesoscopic systems, where the system may be smaller than the inelastic mean free path, and the concept of an applied field requires some care [25, 26]. There are, however, some conceptual simplifications that arise in considering the local field in disordered mesoscopic structures, especially if the dimensions of such structures are much less than an inelastic mean free path. In this regime, electrons supplied from one reservoir travel through the structure, undergo elastic multiple scattering with defects and finally end up in one of the attached reservoirs. What can we say about the RRD, CDM, wind forces and direct forces in this case? Landauer’s basic approach is of course very applicable. As in the analysis for bulk metals, the scattering of the incident electrons results in local charge pile-ups around the various scatterers in the sample, and these lead to a self-consistently screened local potential δ8(r). This, in turn, results in wind forces on each impurity’s ion core. Although the net force on the impurity system acts along the direction of the electron wind, each individual impurity scatterer may be subjected to a force that is quite different from the average wind force [25]. Indeed, the direction of the force on an individual impurity might actually be opposite to the electron wind [25]! Because there is no background scattering in this simple mesoscopic system, the long-range RRD field does not come into play. Rather, one is generally dealing with complicated scattering wavefunctions extending coherently throughout the system, and these can lead to complicated profiles for the electron density and δ8(r). The entire collection of scatterers is acting as one giant impurity that extends over several, or many, electron wavelengths. Neither the direct force nor the CDM effect arises in this case because there is no driving field E0 associated with sources far from this giant impurity, and therefore by eqn (12) there is no CDM dipole. The only force present in this case is the electron wind force, i.e., the force associated with the scattering of the incident electrons. The CDM effect and the direct force would come into play, however, if inelastic background scattering were appreciable. The simple mesoscopic system in which inelastic scattering is negligible provides an extreme case of quantum transport that can be analysed within the framework of the Lang and Kohn [27] density-functional
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formalism. To apply this formalism to the transport problem, one needs to invoke the proper boundary conditions, namely, that: (1) the electron wavefunctions are those appropriate for electrons incident from reservoirs attached to the sample, and (2) the self-consistent electric field vanishes deep within the reservoirs. Calculations of this sort have been made for the purpose of determining I –V curves for an STM geometry [28–30]. It would be interesting to extend such calculations to treat inelastic scattering, and thus apply these calculations to investigate the CDM effect and the direct force. Such a program would require a more detailed analysis of the electron reservoirs, and this brings us back to Landauer.
References [1] }R. Landauer, IBM J. Res. Develop. 1, 223 (1957). This paper has been republished as: R. Landauer, J. Math. Phys. 37, 5259 (1996). See also: R. Landauer, Z. Phys. B21, 247 (1975). [2] }R. Landauer, Philos. Mag. B21, 863, (1970); See also, IBM J. Res. Develop. 32, 306 (1988). [3] }M. B¨uttiker, Phys. Rev. Lett. 57, 1761 (1986). [4] }H. B. Huntington, in Diffusion in Solids, edited by A. S. Nowick and J. J. Burton (New York, Academic, 1975), p. 303. [5] }A. H. Verbruggen, IBM J. Res. Dev. 32, 93 (1988). [6] }P. S. Ho and T. Kwok, Rep. Prog. Phys. 52, 301 (1989). [7] }R.S. Sorbello, in Solid State Physics, Vol. 51, edited by H. Ehrenreich and F. Spaepen, (New York, Academic Press, 1997), p. 159. [8] }R. Landauer and J.W.F. Woo, Phys. Rev. B10, 1266 (1974). [9] }R. Landauer, J. Phys. C8, L389 (1975). [10] }R. Landauer, J. Phys. C8, 761 (1975). [11] }R. Landauer, Phys. Rev. B14, 1474 (1976). [12] }R. Landauer, Phys. Rev. B16, 4698 (1977). [13] }For a detailed reference list, see [7]. [14] }C. Bosvieux and J. Friedel, J. Phys. Chem. Solids 23, 123 (1962). [15] }P. R. Rimbey and R. S. Sorbello, Phys. Rev. B21, 2150 (1980). [16] }R. S. Sorbello, Phys. Rev. B31, 798 (1985). [17] }R. S. Sorbello and C.S. Chu, IBM J. Res. Develop. 32, 58 (1988); [18] }R. S. Sorbello, Phys. Rev. B23, 5119 (1981). [19] }W. L. Schaich, Phys. Rev. B13, 3350 (1976); 3360 (1976). [20] }L. J. Sham, Phys. Rev. B12, 3142 (1975). [21] }C.S. Chu and R. S. Sorbello, , Phys. Rev. B38, 7260 (1988). [22] }A. K. Das and R. Peierls, J. Phys. C8, 3348 (1975). [23] }A similar derivation to Landauer’s was given by L. J. Sham in [20], except that Sham’s derivation does not consider the effect of the dipole polarization field. This is discussed in [7]. [24] }R. Landauer, IBM J. Res. Develop. 32, 306 (1988). [25] }R.S. Sorbello, Phys. Rev. B39, 4984 (1989). [26] }A.D. Stone and A. Szafer, IBM J. Res. Develop. 32, 384 (1988). [27] }N.D. Lang and W. Kohn, Phys. Rev. B1, 4555 (1970). [28] }N.D. Lang, A. Yacoby, and Y. Imry, Phys. Rev. Lett. 63, 1499 (1989); N.D. Lang, Phys. Rev. B45, 13599 (1992); 49, 2067 (1994); 55, 9364 (1997). [29] }K. Hirose and M. Tsukada, Phys. Rev. B51, 5278 (1995). [30] }N. Kobayashi, K. Hirose, and M. Tsukada, Surf. Sci. 345, 299 (1996).