Surface-state mediated three-adsorbate interaction

Transcription

1 EUROPHYSICS LETTERS 15 July 2002 Europhys. Lett., 59 (2), pp (2002) Surface-state meiate three-asorbate interaction P. Hylgaar 1 an T. L. Einstein 2 1 Department of Applie Physics, Chalmers University of Technology an Göteborg University - S Gothenburg, Sween 2 Department of Physics, University of Marylan, College Park - MD , USA (receive 31 December 2001; accepte in final form 16 April 2002) PACS Hb Impurity an efect levels; energy states of asorbe species. PACS At Surface states, ban structure, electron ensity of states. PACS Ef Scanning tunneling microscopy (incluing chemistry inuce with STM). Abstract. The interaction energy of three asorbates on a surface consists of the sum of the three pair interactions plus a trio contribution prouce by interference of electrons which propagate the entire perimeter, 123, of the three-asorbate cluster. Here we investigate this triple-asorbate interaction that is meiate by the isotropic Shockley surface-state ban foun on noble-metal (111) surfaces. Our experimentally testable result epens on the s-wave phase shift, δ F 0, characterizing the staning-wave patterns seen in scanning-tunneling microscopy (STM) images. Compare with the asorbate-pair interactions, an in contrast to bulk-meiate interactions, the trio contribution has a slightly weaker amplitue an asymptotically ecays slightly faster, 5/ It also has a istinctive oscillation perio epenent on 123. We finally compare the asymptotic escription with exact moel calculations. Progress in scanning-tunneling microscopy (STM) has mae possible the stuy of physical properties of surface-state electrons in real space, as reveale by staning surface-wave patterns forme in the vicinity of asorbates, efects, an steps [1 5]. Most stuies of these wave patterns have concerne Shockley-type surface states which on a clean surface are characterize by a free-particle like ispersion with effective mass m e, in-surface Fermi wave vector q F an Fermi level ɛ F =( hq F ) 2 /2m e. The corresponing in-surface Fermi wavelength, λ F =2π/q F, significantly excees the bulk-electron counterpart, an the envelope of the response function ecays far more slowly than for the bulk states. Such slow ecay of the response functions opens the possibility for very-long-range interference an mutual efect-interaction effects meiate by the surface-state ban. Exciting examples inclue quantum corrals forme by Fe atoms on Cu(111) [3] an small islans on Ag(111) [4]. Surface waves are strongly scattere from asorbates, an most experiments reveal significant surface-state variation in the local ensity of state (LDOS) characterize by large Fermi-level phase shift, δ F π/2 [2, 5]. The oscillatory nature of the inirect interaction between chemisorbe atoms on metal surfaces [6] has attracte theorical attention for over three ecaes [7]. Usually, the meiation is by bulk electronic states which prouce anisotropic interactions with a rapi 5 ij ecay with the asorbate separation ij [8]. Although qualitative unerstaning of the features of the inirect interaction are known since long [8], quantitative agreement between theory an

2 266 EUROPHYSICS LETTERS experiment prove elusive ue to the complicate nature of the substrate electronic states an the interplay of all occupie energy levels at small ij [6]. While only states near the Fermi level ɛ F are important at asymptotically large ij [9], the rapi ecay reners these interactions unmeasurable. Lau an Kohn [10] recognize that when there are surface states. However, only recently i theorists [11,12] apply these ieas to the surface states on (111) noble metals. These surface states consist of a single, circularly symmetric ban. For two asorbates, i an j, at asymptotic separation, ij λ F /2, the single-asorbate scattering can then be characterize by the measurable phase shift δ F, leaing to a simple expression for the asorbate-pair interaction energy [11, 12]: near ɛ F, the ecay is much slower, going like 2 ij ( ) 2 E pair ( ij ; δ F ) E asym pair ( 2sin(δF ) sin(2q F ij +2δ F ) ij; δ F )= ɛ F π (q F ij ) 2. (1) Recent STM investigations of Cu on Cu(111) [12, 13] an Co on Cu(111) an Ag(111) [13] have not only reveale that inter-asorbate istances epen on the perio λ F /2=π/q F of the surface-wave oscillations aroun the asorbates, but have also experimentally etermine both the asymptotic ecay an strength of this inirect electronic asorbate-pair interaction. In this letter, we begin the task of extening the analysis to multiasorbate interactions on metal surfaces with isotropic surface-state bans. We investigate the mutual inirect electronic interaction of three asorbates, focusing on the recently investigate [12,13] Cu(111) an Ag(111) surfaces. A simple tight-bining analysis provies the essential formalism [6, 14], which can be aapte to provie a non-perturbative three-asorbate interaction estimate, E triple ( 12, 23, 31 ; δ F ) 3 i>j=1 E pair ( ij ; δ F )+ E trio ( 12, 23, 31 ; δ F ). (2) Such energies can be important in the formation of ilute superlattices [12,13], in etermining the shape of clusters, can be non-negligible ingreients in a lattice-gas parametrization of chemisorbe overlayers [15], an can lea to gross asymmetries in temperature-coverage phase iagrams by breaking the particle-hole symmetry of the lattice-gas Hamiltonian [16]. To leaing orer in the asorbate scattering, the trio contribution E trio ( 12, 23, 31 ; δ F ) comes from electrons which scatter at all three asorbate locations an traverse the perimeter, We expect the asorbates to couple to the same local environment an so assume that ientical phase-shifts, δ F, (evaluate moulo π) escribe them. (Otherwise, 3δ F δ 1F + δ 2F + δ 3F below.) For asymptotic ij we erive the simple analytic result E trio ( 12, 23, 31 ; δ F ) ɛ F sin 3 (δ F ) ( 16 2 π 5/2 ) γ 123 sin(q F δ F 3π/4) (q F 123 ) 5/2, (3) where γ 123 3/2 123 / is a shape-epenent imensionless ratio. For typical threeaatom configurations (i.e., with comparable 12, 23, 31 ), γ [17], an the leaing trio contribution of eq. (3) oscillates istinctly as q F 123 while ecaying barely faster than the asymptotic-pair interaction of eq. (1). However, for an isosceles configuration with one leg, say s, much shorter than the other two legs of length, γ 123 2(2/ s ) 1/2 ; then E trio ecays asymptotically with the ientical envelope 1/ 2 an oscillation perio 2q F of (but ifferent oscillation phase from) eq. (1), an thus effectively screens (or antiscreens ) E pair. The magnitue an slow asymptotic ecay of the preicte surface-state meiate trio interaction (3) can have important effects at finite coverages of the noble-metal surfaces. While the sum of asorbate-pair interaction contributions [11], E pair ( ij ; δ F ) ominates the

3 P. Hylgaar et al.: Surface-state meiate three-asorbate interaction 267 δ F 31 δ F δ F Fig. 1 Schematics of the triple-asorbate interaction. Three asorbates couple at positions i =1, 2, 3, separate by mutual istances ij, to a metal surface which supports a surface-state ban near ɛ F. The scattering at each asorbate causes oscillations (concentric rings) in the surface-state ensity of state. This scattering is characterize by a finite Fermi-level phase shift, δ F π/2, corresponing to significant, slowly ecaying Frieel oscillations inuce in the surface-state electron gas. general triple-asorbate cluster energy (2), the trio interaction E trio ( 12, 23, 31 ; δ F ) will contribute a non-negligible quarter of E triple ( 12, 23, 31 ; δ F ) an ecays only slightly faster than the pair interaction [11 13]. Previous attempts to compare theoretical an experimental values of trio energies have met with marginal success an have consiere small separations between asorbates [18, 19]. However, impressive agreement between theory an experiment regaring pair interactions meiate by surface states [12, 13] an preliminary inications of interactions in aition to those between pairs in multiasorbate systems inspire us to revisit the trio interaction at intermeiate an larger separations. The schematic fig. 1 illustrates the experimental situation an ientifies the asorbateasorbate (Frieel) interaction mechanism. Three asorbates are locate above a (noble) metal surface, on which there is a surface-state ban crossing ɛ F. Table I ientifies the key parameters which characterize the Shockley state on the (111) face of Cu, Ag, an Au. Table I also inclues values for the effective mass m eff which relates the Fermi energy to the Fermi wave vector, q F = h 1 2m eff ɛ F. Each asorbate inuces scattering an causes spatial oscillations in the surface-ban ensity of states (DOS). At ɛ F these DOS oscillations suffer a phase shift δ F. The interference between such DOS variations prouces a Frieel-type asorbate-asorbate interaction, which also oscillates with the mutual asorbate separation. Several theoretical stuies have aresse this inirect asorbate interaction mechanism [6,11]. The theoretical expression for the interaction becomes simple only in the asymptotic regime, where exclusively states at the Fermi level matter. While state-of-the-art first-principles calculations [23, 24] can compute the total energy of perioic overlayer structures, reaching the asymptotic regime requires cells bigger than current capabilities. Also, unless the interaction Table I Shockley surface-state parameters an Thomas-Fermi (bulk-screening) lengths of the Cu, Ag, an Au (111) surfaces. The Shockley ban is characterize by the effective electron mass m eff, a Fermi energy ɛ F (measure relative to the bottom of the surface-state ban), an a corresponing in-surface Fermi wave vector q F = h 1 2m eff ɛ F an half wavelength λ F/2 =π/q F. m eff /m e ɛ F (ev) q F (Å 1 ) λ F/2 (Å) k 1 TF (Å) Cu 0.44 a /0.46 b 0.38 a /0.39 b 0.21 a /0.217 b 15.0 a /14.5 b Ag 0.40 a /0.53 b a /0.12 b a /0.129 b 37.9 a /24.4 b Au 0.28 b 0.41 b b 18.2 b Si-Ag (7) c 0.25(5) c 0.010(3) c 31(9) c [k 1 λf at 6 K]e a Ref. [20]. b Ref. [21]. c Ref. [22]. Aapte from ref. [11]. DH e TF Debye-Hückel (DH).

4 268 EUROPHYSICS LETTERS is transmitte by surface states, the interaction will be weak [24]. In contrast, in a scattering theory approach, the interaction energy is calculate by computing the propagators for the clean, high-symmetry surface. Lau an Kohn s lanmark investigation of the asymptotic regime use a perturbative approach [10] but the LCAO tight-bining formalism prouces equivalent results non-perturbatively [6, 9]. Here we use the measure [2, 5, 12, 13] phase shifts (δ F ±π/2,π/3) for a non-perturbative evaluation of the triple-asorbate interaction. Our analysis starts with a Harris functional expression [11, 25] an shoul inclue an electrostatic correction. However, as for the pair interaction [11], screening by the bulk electrons shoul make this correction neglible. Thus, the one-electron contributions etermine the interaction, which, as in the tight-bining approach [6], is given by an energy integral ɛf E triple ( 12, 23, 31 ; δ F )=2 ɛ (ɛ ɛ F ) ρ triple (ɛ; 12, 23, 31 ) (4) of the change ρ triple in the DOS of the three asorbates relative to three isolate asorbates. The factor of two comes from spin egeneracy. Aapting previous formalism [6, 14], we fin ρ triple (ɛ, 12, 23, 31 )= 1 π ɛ Im x x ln[1 i>j K (i,j) pair K trio] x, (5) K (i,j) pair = T a ii(ɛ)g 0 ij(ɛ)t ajj(ɛ)g 0 ji(ɛ), K trio =2T a11(ɛ)g 0 12(ɛ)T a22(ɛ)g 0 23(ɛ)T a33(ɛ)g 0 31(ɛ). (6) The last contribution in eq. (6) contains a factor of 2 because the electrons can traverse the triple-cluster perimeter clockwise or counterclockwise. As note above, eq. (5) shoul be calculate using frozen asorbate-inuce scattering potentials [25]. The screening by bulk electrons justifies an assumption that the asorbate scattering potentials are non-overlapping. We express the formal result, eq. (5), in terms of the one-electron (retare) Green function G 0 (ɛ) for the bare surface an the T -matrices, T aii(ɛ), which characterize the (non-perturbative) scattering at asorbates a i, where i =1, 2, 3. Meiation by a Shockley surface-state ban greatly simplifies the evaluation of ρ triple of eq. (5). First, at large asorbate istances the ominant contribution to ρ triple (ɛ; 12, 23, 31 ) then arises from scattering of surface-state electrons: the amplitue for propagation along the surface in a 2D surface state has a significantly slower ecay with istance than within the bulk electron bans. Secon, the surface-state interaction is then ominate by s-wave scattering contributions because the Fermi wavelength λ F =2π/q F for the Shockley surface states typically is much larger than the bulk Thomas-Fermi screening length, 1/k TF ; cf. table I. At finite asorbate separations, the change in the DOS (5) ue to the inirect interaction between the three asorbates thus becomes ominate by ρ triple (ɛ; 12, 23, 31 ) 1 Im ln 1 K pair ( ij ; δ F ) K trio ( 12, 23, 31 ; δ F ). (7) π ɛ i>j To obtain K pair ( ij ; δ F )fromk (i,j) pair an K trio( 12, 23, 31 ; δ F )fromk trio, replace T aii(ɛ) by t 0 (ɛ; δ F )ang 0 ij (ɛ) byg 0(q ij ), where the Green function g 0 (q) escribes the propagation a istance along the surface at wave vector q = h 1 2m eff ɛ via the isotropic surface ban. Describing asorbate scattering of surface-ban electrons, the new effective T -matrix, t 0 (ɛ; δ F )= (2 h/m eff )sin{δ 0 (ɛ)} exp[iδ 0 (ɛ)], is etermine by the s-wave phase shift δ 0 (ɛ) which, in turn, is specifie by the bounary conition δ 0 (ɛ F )=δ F. The cylinrical Hankel

5 P. Hylgaar et al.: Surface-state meiate three-asorbate interaction 269 function of the first kin, H (1) 0, solves the Helmholtz equation in two imension, so that g 0 (x) =i m eff 2 h H(1) 0 (x) im eff exp[ix iπ/4] for x. (8) h 2πx The expansion emphasize the slow amplitue ecay for propagation in a 2D surface state; when only 3D states contribute, the surface Green function typically ecays as x 2 [6, 9]. Our simple formulas for the asorbate interaction (1) an (3) follow from an asymptotic evaluation of one-electron energy integrals like eq. (4). In the asymptotic region, K trio an K pair become small, allowing expansion of the logarithm in E triple. To orer t 3 0 we get an integral of each of three K pair, corresponing to the asymptotic-constituent-pair interactions E pair of eq. (1), plus the asymptotic limit of the trio contribution E trio ( 12, 23, 31 ; δ F ): 2 ɛf π Im ɛk trio = 4 ɛf π Im ɛ [t 0 (ɛ; δ F )] 3 g 0 (q 12 )g 0 (q 23 )g 0 (q 31 ). (9) 0 0 For parabolic ispersion, the Jacobian for changing the integration variable from ɛ to q is simple. The leaing asymptotic term comes from simply integrating the phase factor, which has the generic form exp[iqr], an evaluating it at the upper limit. The result, (2/π)ImK trio (ɛ F )/qr [26], is just our asymptotic trio-interaction contribution, eq. (3). As escribe in ref. [11] asorbate scattering into the bulk ban can be aresse by allowing the phase shift to become complex: δ 0 (ɛ) =δ 0(ɛ)+iδ 0 (ɛ). Then eq. (3) becomes E trio ɛ F ( r sin 2 {δ 0(ɛ F )} + ) 3/2 ( (1 r) ) 2 sin(q F θ 0 3π/4) γ π 5/2 123, (10) (q F 123 ) 5/2 where r =exp[ 2δ 0 (ɛ F )] is the intra-surface-ban reflection an θ 0 (etermine by tan(θ 0 )= (1/r)csc{2δ 0(ɛ F )} cot{2δ 0(ɛ F )}) is the phase shift of staning waves aroun asorbates. Thus, for maximal limit of black scattering (r 0), E trio is reuce by a factor of 1/8. Figure 2 shows our results for the three-asorbate interaction energy meiate by the surface-state ban assuming the Fermi-level phase shift δ F = π/2. The top (bottom) panel compares estimates for the combine triple-asorbate cluster (for the trio-interaction) energy. For specificity, we assume an equilateral-triangle configuration. The soli curves show our analytical asymptotic evaluation obtaine using eqs. (1) an (3). The ashe curves show an E3δ trio, which we obtain by numerically evaluating the asorbate-inuce change in DOS, eq. (7), for g 0 (x) H (1) 0 (x) with the aitional assumption of a zero-range (δ-function) asorbate potential [11]. This moel assumption of δ-function scattering provies a semiquantitative approximation for escribing efects in metals [27]. For a two-imensional free-electron like ban, the s-wave phase shift [27] takes the simple analytic form given in ref. [11], δ 0 (ɛ) = arccot[π 1 ln(ɛ/ɛ F )+cot(δ F )]; thus, δ 0 (ɛ) is separate interaction estimates, E 3δ triple uniquely specifie by the experimentally observe δ F. As for the asymptotic-pair interaction [6, 8 11], the asymptotic trio interaction is etermine by the behavior of the integran aroun ɛ F an hence eventually approaches the analytical trio result of eq. (3), as illustrate in the insert panel. For non-asymptotic, the numerical E 3δ () results are still vali, although they may become outweighe by bulk-state meiate inirect interactions. The comparison between E triple,trio () an Etriple,trio 3δ () (soli an ashe curves) in fig. 2 ocuments that our asymptotic evaluations of both the triple-asorbate energy an of the trio contributions become aequate for q F 123 > 6π, corresponing to ij >λ F. In contrast, for the simpler two-asorbate interaction problem, the asymptotic-pair interaction result (1)

6 270 EUROPHYSICS LETTERS E triple /ε F δ F = π/2 12 = 23 = 31 = 123 /3 E trio /ε F = ( E triple E pairs ) / ε F E trio /ε F q F 123 /π q F 123 /π = 123 /(λ F /2) Fig. 2 Three-aatom inirect interaction meiate by a surface-state ban. The figure compares estimates both for the full interaction energy E triple of three aatoms (top panel) an for the triointeraction energy (bottom panel), i.e. theenergybywhich E triple iffers from the sum of the three constituent-pair interactions. Our non-perturbative results use the phase-shift δ F = π/2, consistent with experimentally observe staning wave patterns of S, Cu, an Co on Cu(111). The soli curves show our asymptotic results, eqs. (1) an (3). The ashe curves show a numerical etermination, Etriple(), 3δ which arises when we base the single-asorbate scattering approximation, eq. (7), on a zero-range (δ-function) approximation for the asorbate potential an obtain the s-wave phase shift δ 0(ɛ) = arccot[π 1 ln(ɛ/ɛ F)+cot(δ F)]. The insert etails the long-range asymptotic variation. becomes accurate alreay at 12 >λ F /2 [11]. This ifference in the onset of valiity of the two asymptotic formulas (3) an (1) reflects the intrinsic complexity of the multiterm logarithmic integran which efines E triple an hence E trio in eqs. (2)-(6). There is, in general, no simple way to express the trio-interaction result without an explicit subtraction; unlike for the pair interaction problem [6,11], there is no concise expression for a ρ trio that can be inserte into eq. (4) to obtain the trio interaction. It is the single-integral approximative result (9) itself which only becomes applicable at q F 123 > 6π. Isotropic, free-electron-like, partially fille surface states are also foun on semiconuctor surfaces with partial overlayers of metals, e.g., Si(111)- 3 3-Ag [22]. The relevant parameters are inclue in table I. STM measurements at 6 K reveal a remarkable orere asorbate phase with interatomic spacings much smaller than the asymptotic limit. The absence of bulk screening, however, makes it unclear if the concepts evelope here an/or in preceing works an resting on a Harris functional analysis can be irectly applie. In summary, we have presente results for the triple-asorbate interaction energy meiate by an isotropic Shockley surface-state ban, as foun on noble-metal (111) surfaces. We erive

7 P. Hylgaar et al.: Surface-state meiate three-asorbate interaction 271 a general formalism for this energy expresse in terms of experimentally accessible parameters. We also provie an explicit numerical evaluation for the equilateral-triangle configuration on Cu(111). While the sum of pair interaction contributions ominates the triple-asorbate interaction energy, our work inicates that the aitional trio contribution accounts for about a quarter of the interaction an exhibits only a marginally slower asymptotic ecay. We also erive a simple analytic expression for the asymptotic limit of the trio interaction. It epens essentially only on the perimeter of the three-asorbate triangle an so has its own characteristic oscillation wavelength an ecay envelope. We assess its range of valiity. The trio an analogous quarto an higher-orer interaction contributions can play an important role for larger clusters an affect the total interaction energy of relatively ense (orere) overlayers for which asorbate interactions are meiate primarily by surface states. PH gratefully acknowleges financial support from the Wilhelm an Martina Lungrens Founation an from the Sweish Founation for Strategic Research (SSF) through Materials Consortium ATOMICS. TLE s work was supporte by NSF Grant EEC , with supplemental support from University of Marylan NSF-MRSEC, Grant REFERENCES [1] Davis L. C. et al., Phys. Rev. B, 43 (1991) [2] Crommie M. F. et al., Nature, 363 (1993) 524. [3] Crommie M. F. et al., Science, 262 (1993) 218. [4] Hasegawa Y. an Avouris Ph., Phys. Rev. Lett., 71 (1993) [5] E. Wahlström et al., Appl. Phys. A, 66 (1998) S1107. [6] See, e.g., Einstein T. L., Hanbook of Surface Science, eitebyunertl W. N., Vol. 1 (Elsevier Science B. V., Amsteram) 1996, Chapt. 11. [7] Grimley T. B., Proc. Phys. Soc. Lonon, 90 (1967) 751. [8] Einstein T. L. an Schrieffer J. R., Phys. Rev. B, 7 (1973) [9] Einstein T. L., Surf. Sci., 75 (1978) 161L. [10] Lau K. H. an Kohn W., Surf. Sci., 75 (1978) 69. [11] Hylgaar P. an Persson M., J. Phys. Conens. Matter, 12 (2000) L13. [12] Repp J. et al., Phys. Rev. Lett., 85 (2000) [13] Knorr K. et al., Phys. Rev. B, 65 (2002) [14] Einstein T. L., Surf. Sci., 84 (1979) L497. [15] L. Österlun et al., Phys. Rev. Lett., 83 (1999) [16] For a general review, see Einstein T. L., Langmuir, 7 (1991) [17] For isosceles triangles, we fin that γ 123 ranges from 3 3/2 for equilateral to 3 1/4 (2 + 3) 3/2 for obtuse (apex angle ϕ = 120 )to2 5/2 for egenerate/linear (ϕ =180 ). [18] Fink H. W. an Ehrlich G., Phys. Rev. Lett., 52 (1984) [19] Koh S. J. an Ehrlich G., Phys. Rev. B, 60 (1999) 5981 an references therein. [20] Jeanupeux O. et al., Phys. Rev. B, 59 (1999) [21] Kevan S. D. an Gaylor R. H., Phys. Rev. B, 36 (1988) [22] Tong X. et al., Phys. Rev. B, 57 (1998) [23] Fichthorn K. A. an Scheffler M., Phys. Rev. Lett., 84 (2000) [24] Bogicevic A. et al., Phys. Rev. Lett., 85 (2000) [25] Harris J., Phys. Rev. B, 31 (1985) [26] Here R is 123, plusq-erivatives of phase factors that are negligible to leaing orer in [27] Doniach S. an Sonheimer E. H., Green s Functions for Soli State Physicists (W. A. Benjamin, Reaing, Mass.) 1974, pp