Influence of d electrons on the dispersion relation of Ag surface plasmons for different single-crystal faces

Transcription

1 PHYSICAL REVIEW B, VOLUME 63, Influence of d electron on the diperion relation of Ag urface plamon for different ingle-crytal face Catalina López Batida and Angar Liebch Intitut für Fetkörperforchung, Forchungzentrum Jülich, Jülich, Germany W. Lui Mochán Centro de Ciencia Fíica, Univeridad Nacional Autónoma de México, Apartado Potal 48-3, Cuernavaca, Morelo, Mexico Received 2 November 2000; publihed 2 April 2001 The urface plamon diperion relation of Ag i calculated for different ingle-crytal orientation. To decribe the dynamical repone propertie of both delocalized 5 electron and more tightly bound 4d electron, the jellium model i combined with the o-called dipolium model, in which the occupied Ag d band are repreented in term of polarizable phere located at the ite of a emi-infinite fcc lattice. The nonlocal uceptibility characterizing the electron repone in the urface region i derived uing denity functional theory. The creening of the Coulomb interaction between conduction electron via the lattice of dipole, and of the dipole interaction via the urrounding ea of conduction electron, i treated elf-conitently. Electron energy lo pectra are calculated for all three low-index face. The urface plamon energy i found to increae with parallel wave vector for all cae. The magnitude of the poitive lope depend on the crytal orientation and, for Ag 110, on the propagation direction. Thee reult are in qualitative agreement with electron energy lo meaurement. DOI: /PhyRevB PACS number : Mf, Ci I. INTRODUCTION The pectrocopic characterization of urface i of interet both from a baic theoretical and practical point of view. On the one hand, electronic excitation at urface reflect the microcopic electronic and tructural propertie in the urface region and the repone to external electromagnetic field. 1 3 On the other hand, the enitivity of urface collective excitation to urface condition, e.g., at metalelectrolyte interface, ha recently been utilized with great ucce in method uch a urface plamon pectrocopy, for the purpoe of developing efficient and robut chemical and biological enor. 4 Noble metal are among the mot thoroughly invetigated ytem in baic urface cience. They are alo particularly intereting for practical application. A central quantity to tudy i the variation of the urface plamon and plamon-polariton excitation energy with parallel wave vector q. In the cae of Ag, electron energy lo meaurement 5 7 howed that the diperion relation of urface plamon differ in everal important way from that oberved on imple metal In the nonretarded mall q limit, the frequency trongly diagree with the relation 12 p / 2, where p i the bulk plama frequency. At finite q, the Ag urface plama frequency increae, wherea on imple metal the frequency firt diminihe and only beyond q 0.15 Å 1 increae. Finally, the poitive lope of the diperion differ appreciably for the three low-index crytal face of Ag while lattice effect are negligible on imple metal. Evidently, the preence of the hallow occupied d band ha a profound influence on the overall frequency and momentum diperion of the Ag urface collective excitation. Theoretical tudie including and d band have been carried out only recently for the optical propertie of Ag bulk 13 and thin film. 14 A dynamical treatment of the urface creening repone including the full Ag band tructure i computationally not yet feaible. For thi reaon, variou implified model have been developed in the pat with the aim of decribing ome of the oberved feature of Ag urface plamon In one of thee approache 17,18 the emphai i on the microcopic decription of the nonlocal repone of the electron in the urface region, while the influence of the d band i qualitatively included via a emiinfinite polarizable medium. In thi cheme, the Ag urface plama frequency in the long wavelength limit * 3.7 ev, and the overall poitive lope at finite q can be undertood in imple phyical term. Lattice effect, however, uch a the dependence of the diperion on crytal orientation, are abent in thi model. A complementary approach 15,16 focue on the dynamical repone of d electron, repreenting them via a emi-infinite fcc lattice of polarizable point dipole. A local Drude model wa ued to decribe the influence of the urrounding electron ga. While crytallinity effect are included naturally in thi dipolium cheme, the profile of the conduction electron denity near the urface i approximated and it nonlocal repone propertie are neglected. The aim of the preent work i to combine the attractive feature of thee two approache in order to arrive at a more realitic decription of the Ag urface plamon diperion with parallel wave vector. Specifically, the local denity approximation for the emi-infinite jellium model i ued to obtain the nonlocal repone function for the electron denity. Dynamical creening i treated within a elf-conitent field approach. Previou experience ha hown that thi method yield a nearly quantitative repreentation of the urface plamon diperion of imple metal. 3 The influence of /2001/63 16 / /$ The American Phyical Society

2 LÓPEZ BASTIDAS, LIEBSCH, AND MOCHÁN PHYSICAL REVIEW B the d electron i included by repreenting them in term of an fcc lattice of point dipole. The effective atomic polarizability of the d hell i choen to reproduce the meaured bulk dielectric function of Ag. 20 The key feature of our combined jellium-dipolium approach i that the mutual polarization of and d electron denitie i treated elfconitently without further approximation. We may view thi cheme uing two equivalent phyical picture: i The Ag urface i repreented in term of a emi-infinite, nonlocal homogeneou electron ga whoe effective Coulomb interaction i modified a a reult of the dipole lattice. ii The Ag urface i repreented in term of a emi-infinite lattice of polarizable d hell whoe dipole interaction i creened via the urrounding nonlocal ga of electron. An analogou two-component decription wa ued earlier to evaluate the effect of core polarization on the bulk plama frequency of everal metal in the long wavelength limit. 21 The optical repone of noble metal cluter ha recently alo been treated within a imilar model. 22 The macrocopic bulk dielectric function of Ag can be conveniently expreed a 23 ( ) ( ) d ( ) 1, where ( ) 1 2 p / ( i ) i the Drude function characterizing the electron and d ( ) the bound part due to d electron. p 9.2 ev i the bulk plama frequency correponding to the electron denity and a damping parameter. Below the onet of interband tranition at about 3.9 ev, d ( ) i real, with magnitude 5.5. Thu, taking into account the creening via d band, the effective bulk and urface plama frequencie of Ag are p * p / d 3.8 ev and * p / d ev. The anomalou bluehift of the Ag urface plama frequency with increaing q can be undertood in term of the patial variation of the d interaction in the urface region. 17,16 Since the pill out of the electronic denity in the vacuum tem primarily from the electron, the Coulomb interaction due to the outer part of the fluctuating urface plamon charge i not ubject to d creening. Moreover, at finite q d creening inide the metal diminihe becaue of the reduced penetration depth of the dynamical potential. Both mechanim give rie to a bluehift of the urface plama frequency. The calculation how that thi effect i large enough to offet the uual redhift obtained for imple metal, i.e., the overall diperion i poitive in agreement with experimental obervation. An analogou phyical mechanim i believed to be reponible for the anomalou blue hift of the Mie reonance of Ag particle with invere radiu. 22,24 Here we focu on the influence of the crytalline geometry on the diperion of the Ag urface plamon. A mentioned above, thi effect manifet itelf in the different lope of the diperion detected for different crytal orientation and, on the 110 face, for orthogonal parallel wave vector. The latter effect i intimately related to the optical reflectance aniotropy oberved on Ag ,26 In the preent approach, lattice effect are aociated with the dipole-dipole interaction within atomic plane parallel to the urface and between plane. The geometry and denity of dipole within plane, and the interplanar pacing vary for different crytal orientation. Becaue of the limited penetration depth of the dynamical potential at finite q, creening via thee dipole lead to a dependence of the urface plama frequency on crytal orientation. The outline of thi paper i a follow. In Sec. II, a general dicuion of the two-component d electron model for Ag i preented. Section III addree ome detail concerning the choice of the local d electron polarizability. In Sec. IV the reult are dicued and compared with experimental pectra. A ummary i given in Sec. V. Atomic unit are ued unle noted otherwie 1 Hartree 27.2 ev, 1 a Å. II. TWO COMPONENT ELECTRON MODEL Let u conider a metal ubject to a weak perturbating potential ext (r, ). Within a elf-conitent field approach, the induced denity n(r, ) and the total dynamical potential (r, ) are related via the linear repone equation n r, d 3 r 0 r,r, r,, 1 r, ext r, d 3 r K r,r n r,, where 0 (r,r, ) i the nonlocal independent particle uceptibility. In the following, we aume that electronic interaction can be treated within the random phae approximation RPA, i.e., the kernel K(r,r ) i the bare Coulomb interaction 1/ r r. In the time dependent extenion of the LDA, K alo include an exchange-correlation contribution. 27 In a two-component valence electron ytem uch a the d electron denitite of the noble metal, one can ditinguih two type of interaction: i on the one-electron level, there are hybridization effect which modify the orbital energie and wave function; ii in the dynamical repone to the external field, there are mutual polarization which affect the effective potential and the frequency dependence of excitation pectra. Since the frequencie of the Ag collective mode are below the onet of d interband tranition, they lie in a range of weak d hybridization. We therefore neglect the ingle-particle coupling and focu intead on d polarization effect. Accordingly, we eparate the full uceptibility into independent and d contribution: 0 r,r, 0 r,r, d 0 r,r,. The induced denity therefore may be written a n n n d, where 2 3 n,d r, d 3 r 0,d r,r, r,. 4 For thi charge eparation the total dynamical potential take the form ext d, where the potential induced by,d electron are given by

3 INFLUENCE OF d ELECTRONS ON THE DISPERSION... PHYSICAL REVIEW B ,d r, d 3 r K r,r n,d r,. 5 Let u now place the origin of the coordinate ytem at ite i. The induced d electron denity at thi ite i given by To olve thi ytem of repone equation, we now eliminate the induced electron denity n and potential and derive modified external and d electron potential which account for electron creening. Combining Eq. 4, 5, we find n r, d 3 r r,r, ext r, d r,, where i the renormalized electron uceptibility. Omitting momentarily the integration ymbol we can expre in term of 0 chematically via 0 /(1 K 0 ). Thu, Kn K ( ext d ) and (1 K )( ext d ). Thi reformulation of the repone equation how that the total dynamical potential now conit only of external and induced d electron contribution which are, however, creened due to the preence of the nonlocal electron ga. We point out that thi electron creening involve the renormalized uceptibility rather than the bare 0. Thi enure that the creening of the external and d electron potential take place via a fully interacting electron ga rather than a ga of independent electron. For completene we note that elimination of n d and d rather than n and from the initial repone equation lead to an equivalent expreion for the total dynamical potential: (1 K d )( ext ), where d i the renormalized d electron uceptibility d d 0 /(1 K d 0 ). In thi cae, the conit of external and induced electron contribution which are creened due to the preence of interacting d electron. So far, the reorganization of the repone equation i purely formal and depend only on the ditinction of and d contribution to the full uceptibility 0 a indicated in Eq. 3. To proceed we now aume that lattice effect on the ground tate electron denity are weak o that 0 can be approximated in term of the nonlocal repone function of a emi-infinite homogeneou electron ga. The local denity approximation LDA i ued for thi jellium ytem 28 to evaluate the wave function and Green function needed for the contruction of 0. We aume furthermore that the d electron are well localized and that overlap between neighboring ite can be neglected. For implicity we repreent thee occupied d hell by dipole located at the ite of a emi-infinite fcc lattice. The noninteracting d electron uceptibility d 0 can then be approximated a a lattice um of ingle-ite contribution 6 n i r, d 3 r i 0 r,r, r,. The total potential (r, ) acting on the d electron hell can be repreented in term of four contribution: the external potential, the potential due to electron, the potential due to d electron at other lattice ite, and the potential induced by the d hell at ite i itelf. The firt three contribution will be referred to a local potential loc and the lat one a i. Thu, loc i. Since the d hell are aumed to be localized, the local potential varie lowly acro the ite and can be expanded a loc r, loc R i, r E loc R i,, where E loc i the local electric field at ite i. To lowet order, the induced d electron denity i therefore determined by the ingle-ite repone equation n i r, d 3 r i 0 r,r, i r, r E loc R i, c r, r E loc R i, The lat identity follow from the aumed pherical ymmetry of the d hell and c(r, ) pecifie the radial dependence of the polarization function. The dipole moment of n i i p i ( ) ( ) E loc (R i, ), where the local polarizability i defined a ( ) 1 3 d 3 rr 2 c(r, ). For implicity we aume thi polarizability to be the ame on all lattice ite. In a more refined treatment, thi retriction can eaily be relaxed. Within the dipole approximation the potential generated by the d electron i d r, i 1 r R i p i. 11 Thi expreion how that the problem of calculating the d electron contribution to the total potential i reduced to finding the dipole moment p i ( ). For point like dipole the above expreion for d i valid throughout pace and the induced d electron denity i given by n d (r, ) i p i ( ) (r R i ). The local field at ite i can be written a for brevity we omit frequency argument of electric field and dipole moment d 0 r,r, i i 0 r R i,r R i,, 7 where R i denote a lattice vector. From Eq. 4 it follow that the induced d electron denity n d i alo given by a um over lattice ite, n d i n i, where n i i the d electron denity induced at ite i. E loc R i r, i r, R r i E ext R i j UJ ij p j. 12 The firt term repreent the external field at ite i creened via the electron

4 LÓPEZ BASTIDAS, LIEBSCH, AND MOCHÁN PHYSICAL REVIEW B E ext R i ext r, r R i ext 1 K ext. Note that the creened external potential ext 13 correpond to the elf-conitent local potential of a emi-infinite electron ga in the abence of d electron. The econd term in Eq. 12 account for the creened dipole-dipole interaction UJ ij UJ ij UJ ij. The bare interaction i given by 1 UJ ij 14 r R j R r. i We ue the convention UJ ii 0 ince the field due to the dipole at ite i i ubtracted in Eq. 12. The creening via electron i decribed by the tenor integration ymbol are again uppreed : UJ ij K 1 r R j. 15 r R i From the definition of p i we now obtain the following elfconitent equation for the dipole moment p i E ext R i j UJ ij p j. 16 Except for the creening of the external field and dipoledipole interaction, thee equation are analogou to thoe of a pure dipole lattice. The derivation dicued o far i rather general and applie to arbitrary three-dimenional ytem. We now conider explicitly a emi-infinite metal expoed to an external potential whoe patial variation i given by ext (r, ) (2 /q) e iq r e qz, where q i a two-dimenional wave vector parallel to the urface and q q. The urface lie in the xy plane and the z axi point toward the vacuum. Becaue of the tranlational ymmetry parallel to the urface it i convenient to perform a two-dimenional Fourier tranform of quantitie aociated with the electron component of the ytem. Since the ground tate electron denity i aumed to be homogeneou in the xy plane, the uceptibility 0 (r,r, ) depend only on the difference (r r ). Thu the Fourier tranform may be written a 0 (z,z,q, ). The creened electron uceptibility alo ha thi form ince the Coulomb interaction doe not modify the ymmetry in the urface plane, i.e., we have (z,z,q, ). Of coure, 0 normal to the urface tranlational ymmetry i broken, o and are nonlocal function of z,z. The dipole lattice introduce periodic modulation in the induced electron denity which are characterized by two-dimenional reciprocal vector g. The overall patial dependence of n i therefore of the form The dipole moment within plane parallel to the urface differ only by a phae factor. Let u define a layer index n and expre an arbitrary lattice vector a R i (P d n,z n ), where P i a two-dimenional intraplanar lattice vector and (d n,z n ) pecifie the origin of the nth plane. Thu, the dipole moment at ite i can be repreented a p i p n e iq P. Within thi repreentation, Eq. 16 for the ite-dependent dipole moment p i can be reformulated in term of an equivalent equation for the planar dipole moment p n, p n E ext z n m with TJ nm dipole interaction between plane i TJ nm p m, 18 TJ nm TJ nm. The tenor decribing the uncreened TJ nm P e iq P r d m P,z m, (d r n,z n ) 19 where the prime implie that the term P 0 mut be excluded from the um in the diagonal element TJ nn. The evaluation of thee tenor element uing the Ewald ummation technique 29 wa dicued previouly. 16 The tenor TJ nm decribing the creening of the dipole interaction between plane via the urrounding electron denity can be derived by a Fourier tranformation of Eq. 15. Thu, the two-dimenional tranform of the Coulomb kernel i given by K(z,z,q g ) 2 e q g z z / q g, and TJ nm 1 e i(q g ) (d n d m ) dz L z A g n,z g dz z,z,q g, L z,z g m. 20 Here, A i the area of the urface unit cell and L g (z,z ) 2 i(q g ), z e q g z z / q g. Equation 18 may readily be olved via matrix inverion. Becaue of the limited penetration depth of the creened external and total dynamical field at finite q, only a finite number of lattice plane need to be taken into account. The number of plane depend on the magnitude of q and on the interplanar ditance for a given crytal orientation. To determine the frequency dependence of the urface excitation pectrum we evaluate firt the Fourier component of the d electron potential Eq. 11, d z,q g, 1 A n e i(q g ) d n p n L g z,z n. 21 n r, g e i(q g ) r n z,q g,. 17 The component of the induced electron denity are then given by

5 INFLUENCE OF d ELECTRONS ON THE DISPERSION... PHYSICAL REVIEW B n z,q g, dz z,z,q g, ext z,q g, d z,q g,. 22 The urface repone function g(q, ) can be obtained from the aymptotic behavior of the g 0 component of the induced and d electron potential far in the vacuum 30 d (2 /q) e qz g(q, ) for z 0. Thu, g q, dz e qz n z,q, 1 A n g q, g d q,. e iq d n e qz n p n iq, q 23 The imaginary part of thi quantity provide the urface lo function which can be compared with energy lo pectra obtained in inelatic electron cattering meaurement. The poition of the maxima of Im g(q, ) plotted a a function of for a given q will be ued to determine the reonance frequency (q ) of the collective urface excitation for a given crytal orientation. We cloe thi ection by pointing out that the computational effort to evaluate the urface excitation pectra within the jellium-dipolium model i not ignificantly greater than for the individual model. The matrix equation for the induced dipole moment p n ha about the ame dimenion a in the pure dipolium cae and the incorporation of electron creening require evaluation of 0 and inverion of (1 K 0 ). The main new tak i that 0 (z,z,q g, ) mut be calculated for all q g rather than only g 0 a for bare jellium. III. LOCAL POLARIZABILITY The key quantity determining the influence of the d electron on the urface lo function derived above i the local polarizability ( ). In principle, thi function could be calculated elf-conitently uing the ingle-ite repone equation 10. The olution of Eq. 16 for a three-dimenional bulk ytem then yield a relationhip between the total potential (r, ) and the applied external potential ext (r, ). Fourier tranforming thee quantitie and taking the long wavelength limit q 0 give the macrocopic dielectric function of the ytem. Thi procedure wa ued previouly 21 to determine the effect of core polarization on the dielectric repone of imple metal. Since our main interet here i not the bulk but the urface, we intead chooe ( ) to reproduce the meaured bulk dielectric function ( ). 20 In the bulk ( ) i given by the modified Clauiu-Mootti CM relation n 3 1, 24 FIG. 1. Re ( ) a a function of the number N G of bulk reciprocal vector. The Lindhard dielectric function i evaluated for an electron ga with r 2.97 a 0 at 3.6 ev. The number denote the cloed hell of G correponding to ome N G. where n i the atomic denity, ( ) the electron Drude dielectric function, and the coefficient f G, 1 1/ L G, 25 G 0 account for electron denity fluctuation induced by the hort-wavelength local field due to the d hell. L (G, ) i the Lindhard dielectric function at the bulk reciprocal lattice vector G. In the cae of point dipole, f (G, ) 1 for all G while, for dipole of finite ize, the magnitude of f diminihe rapidly with growing G. 21 Neglecting the local field induced in the electron denity, i.e., ( ) 0, one recover the uual CM expreion for the polarizability. The derivation of the urface lo function dicued in the previou ection i baed on the aumption of point dipole. The quantity affected by thi aumption i the creening part of the dipole-dipole interaction defined in Eq. 15 and 20. Since for computational reaon only a finite number of urface reciprocal lattice vector g can be conidered in the expanion of TJ, truncation error are unavoidable. To etimate the influence of uch a truncation in the cae of the bulk, it i intructive to tudy the convergence of the coefficient ( ) with the number N G of vector G included in the um in Eq. 25. Thi i illutrated in Fig. 1 which how Re ( ) a a function of N G for a frequency in the range of interet for Ag urface plamon. The low convergence of ( ) i a conequence of the highly localized creening charge induced by the ingular point dipole potential. A mentioned above, taking into account the finite ize of the dipole lead to a ignificantly more rapid convergence a a function of G. 21 On the other hand, truncating the um in Eq. 25 after a few hell lead to a reduction of Re ( ) of the order of 20 to 30 %. Thu, in order to reproduce the meaured dielectric function, the local polarizability mut be adjuted. Figure 2 how the frequency dependence of ( ) near the interband onet for everal value of Re. Since the frequency variation of in the range of interet i negligible compared to the truncation effect, we omit it in the following. The reduction of

6 LÓPEZ BASTIDAS, LIEBSCH, AND MOCHÁN PHYSICAL REVIEW B FIG. 2. Atomic d electron polarizability ( ) per unit volume for different value of Re (n Å 3 ). Solid curve: real part left cale, dahed curve: imaginary part right cale. Re due to truncation of the um over G i een to give an enhancement of Re ( ) of the order of 10%. Since a larger d hell polarizability implie tronger d creening, the frequency of the urface plama ocillation i lowered. In the cae of the Ag low-index crytal face, the different intraplanar geometrie and interplanar pacing give rie to lightly different frequency hift if a finite number of urface reciprocal lattice vector i retained in the calculation of urface lo pectra. One mut therefore be careful when comparing excitation frequencie for different crytal face. Thi problem will be addreed in more detail in the next ection. We briefly point out here that the preent definition of ( ) cannot be ued if the creening of dipole interaction i treated uing a local electron ga. In thi cae the interaction of a dipole with the charge induced by it own field diverge. To exclude thi creened elf-interaction a different definition of the polarizability in term of the remaining creened field mut be found. The relation between ( ) and the macrocopic dielectric function i in thi cae given by 4 3 n 3 26 FIG. 4. Comparion of real part of creened external potential ext (z,q, ) in the abence of d electron with bare external potential ext (z,q, ) for 3.6 ev. Solid curve: q 0.05 Å 1, dahed curve: q 0.15 Å 1. which i alo conitent with the CM formula. A comparion between the polarizability obtained for a local creening model and the preent nonlocal Lindhard creening i not meaningful ince an important phyical difference exit between the definition of the creened field in thee two model. IV. RESULTS AND DISCUSSION A chematic repreentation of the two-component d electron model i given in Fig. 3. The ground tate denity profile i obtained uing the LDA for a emi-infinite electron ga of denity n b 3/(4 r 3 ) with r 2.97 a 0 yielding the Ag Drude plama frequency 9.2 ev. The dipole occupy the ite of a emi-infinite fcc lattice with room temperature lattice contant a 4.09 Å. The ymbol indicate the poition of the atomic plane for the three low-index face. The firt plane i half a lattice pacing away from the jellium edge and the interplanar ditance are 0.5 a, a/ 3, and a/(2 2) for the 100, 111, and 110 face, repectively. The repone FIG. 3. Schematic repreentation of Ag two component d electron model. The ground tate electron denity normalized to n b ; olid curve i calculated within the LDA for a emi-infinite jellium ytem. The poitive background dahed line occupie the halfpace z 0. The ymbol mark the poition of the lattice plane for the three low-index face of the fcc crytal. The 4d hell at the lattice ite are repreented via point dipole. FIG. 5. Real part of xx component of creening contribution TJ nm (q, ) to dipole interaction tenor a a function of layer index m for fixed n 10. The tenor element are multiplied by the atomic volume of Ag 3.7 ev, q 0.15 Å 1, 100 face. Solid curve: N g 21; dahed curve: N g 5, where N g repreent the number of urface reciprocal vector included in the creening calculation. Dotted curve: xx component of uncreened tenor TJ nm (q )

7 INFLUENCE OF d ELECTRONS ON THE DISPERSION... PHYSICAL REVIEW B calculation are carried out at complex frequencie, with the imaginary component accounting for the Drude damping derived from the meaured bulk dielectric function. According to Eq. 18, the effective electric field driving the dipole i determined by the creened external potential ext (z,q, ) which correpond to the elf-conitent potential in the abence of d electron. Figure 4 illutrate thi potential for typical wave vector q. The comparion with the bare external potential ext e qz how that although the creened potential ha a much maller amplitude within the metal, it tail alo decay a e qz toward the interior. Thi penetration depth determine the number of dipole plane for a given crytal face that need to be taken into account to achieve convergence. In the limit of mall q thi depth become very large, indicating that polarization of the entire half-pace contribute to the dielectric repone of the urface. The key quantity pecifying the variation of urface excitation pectra with crytal tructure in the preent model i the creened dipole interaction tenor TJ nm (q, ). Of particular interet i the competition between the bare or direct interaction TJ nm (q ) and the indirect contribution J Tnm (q, ) mediated via the electron denity ee Eq. 19 and 20. Figure 5 how the xx component of TJ nm and TJ nm a a function of m for a fixed n. The magnitude of both element decay to zero a n m increae, implying a limited range of interaction between neighboring plane. Thi i the general behavior of all tenor component at finite value of q. The creening element TJ nm depend on. However, in the narrow frequency range of interet for Ag urface plamon thi variation i very light. The direct and indirect interplanar contribution to TJ nm are een be of imilar magnitude with oppoite ign. Thi tendency i quite general, indicating a ignificant overall change of interaction between d hell a a reult of the urrounding ga of electron. The intraplanar tenor element cannot be compared directly ince the elf-interaction i excluded from the bare interaction while it i preent in the indirect creening element. For thi reaon, TJ nn can be much larger than TJ nn. The interaction tenor have a complicated dependence on wave vector. In general increaing q lead to a horter interaction range and to decaying tenor element for plane n,m which are far apart. On the other hand, the interaction between cloely paced plane can be enhanced for increaing q ince it depend alo on the relative lateral poition of the dipole. Moreover, thee effect vary with crytal orientation becaue of the different intraplanar and interplanar geometrie. For n m the creened tenor TJ nm involve indirect interaction between dipole and the charge induced in other plane. A hown Fig. 5, the um over reciprocal vector g in Eq. 20 converge after a few term becaue of the rapidly decaying Coulomb field. On the other hand, for n m the tenor element include the interaction of a dipole with the charge induced at the ame ite. The Fourier repreentation of thi localized charge require many g. Accordingly, the FIG. 6. Real part of g 0 component of induced electron denity n (z,q, ) for Ag 100, 3.67 ev. Solid curve: q 0.1 Å 1, dahed curve: q 0.2 Å 1. The dot indicate the poition of the lattice plane. um over g converge more lowly than for n m ee Fig. 5. Evidently, the contribution to TJ nn due to large g are generated primarily by thi elf-interaction. They carry no information about neighboring ite within the ame plane. A imilar effect wa dicued in the previou ection when addreing the local polarizability in the bulk cae. For computational reaon it i neceary to limit the calculation of TJ nm to a finite et of reciprocal vector g.a pointed out above, thi amount to an approximation of the elf-field of each dipole. In order to keep the urface repone calculation conitent with the bulk model ued for the definition of ( ), it i therefore important to exclude in the bulk cae the correponding term aociated with the creening contribution due to the dipolar elf-field. Thu, in Eq. 25 the large G term hould be omitted. However, term ariing in the three-dimenional G repreentation cannot be directly compared with the two-dimenional urface analog. It i poible to expre defined by Eq. 25 uing a two-dimenional repreentation of the dipole interaction FIG. 7. Contribution to induced electron denity hown in Fig. 6 for q 0.1 Å 1. Solid curve: Re n 0 (z,q, ) induced by external field; dahed, dotted, dah-dotted curve: Re n n (z,q, ) induced by dipole plane n 1, 5, 10, repectively. The dot indicate the poition of the dipole plane

8 LÓPEZ BASTIDAS, LIEBSCH, AND MOCHÁN PHYSICAL REVIEW B FIG. 8. Induced electron denity n (z,q, ) for Ag 111, q 0.1 Å 1. a Real part; b imaginary part. The frequency varie between 3.65 and 3.9 ev. The curve are vertically diplaced for clarity. 3 4 lim lim n q 0 m TJ nm q, xx. 27 Unfortunately, the definition of TJ nm given in Eq. 20 cannot be ued in the limit q 0 becaue of the diverging range of the Coulomb interaction. Qualitatively, however, it i clear that truncation of the um over g in thi definition i related to a truncation of the um over G in the definition of, Eq. 25. A illutrated in Fig. 1 and 2, thi implie a reduction of the magnitude of and an enhancement of ( ). In principle, the truncation could affect the parallel and perpendicular tenor component in a different manner, reulting in different enhancement of ( ) and ( ). In the calculation dicued below, we ignore uch aniotropy effect. However, to etimate the overall influence of the truncation of the g um in Eq. 20, we preent reult for typical value of. Fortunately, the uncertainty implied by thi truncation i reduced appreciably by the requirement that in the q 0 limit all three low-index face mut yield the ame urface plama frequency. The induced electron denity n (r, ) defined in Eq. 17 exhibit patial fluctuation due to variou wave vector contribution. The mot relevant component i the g 0 term which i ued to calculate the urface repone function, Eq. 23. Becaue of the form of the external potential the firt term in Eq. 22 i finite only for g 0. Thi induced denity i therefore the ame a for pure jellium in the abence of d electron. We denote thi contribution a n 0 (z,q, ). Becaue of the form of the d electron potential defined in Eq. 21, the econd term in Eq. 22 induced by the dipole can be expreed a a um over lattice plane. Thu, we may write the g 0 contribution a n n 0 n n n. Thi induced denity i hown in Fig. 6 for two value of q. With increaing q the denity decay more rapidly becaue of the horter penetration depth of the perturbating potential. Superimpoed on the ocillation due the dipole plane are Friedel ocillation caued by the Fermi cut off in the um over occupied tate in the uceptibility (z,z,q g, ). To analyze the denity ocillation induced by the d electron, we how in Fig. 7 the contribution to n (z,q, ) due to the external potential and due to everal dipole plane. Apart from weak Friedel ocillation, n 0 (z,q, ) i localized in the region cloe to the urface. Although the contribution from a given dipole plane i localized near that plane note, however, the Friedel ocillation due to the fifth plane extending all the way to the urface, even rather deep plane produce ignificant induced denitie o that the overall penetration depth of the total n (z,q, ) i very much larger than that of n 0 (z,q, ) for the bare jellium urface. Figure 8 how the induced electron denity n (z,q, ) for Ag 111 at everal frequencie. The reonance frequency 3.75 ev i characterized by the ign change of Re n and the FIG. 9. Real part of induced electron denity n (z,q, ) for Ag dahed curve and Ag olid curve. q 0.1 Å 1, 3.67 ev. The dot indicate the poition of the dipole plane. FIG. 10. Surface excitation pectrum for Ag 100. Solid curve: full pectrum Im g(q, ); dahed and dotted curve: and d electron contribution Im g,d (q, ); dah-dotted curve: bare jellium lo function Im g 0 (q, ). q 0.1 Å 1,

9 INFLUENCE OF d ELECTRONS ON THE DISPERSION... PHYSICAL REVIEW B FIG. 11. Surface lo pectra Im g(q, ) for Ag 111 for everal parallel wave vector increae in magnitude of Im n. Again, only the g 0 Fourier component i plotted. A dicued above, the patial ocillation originate from the dipole located at the atomic plane and from the frequency dependent Friedel ocillation. Note alo that the frequency range hown i jut below the range of propagating bulk plamon ince the q 0 value of the creened volume plama frequency of Ag i about 3.8 ev. Thi explain the large penetration depth of the induced charge denity at higher frequencie. The local field at a given ite i enitive to variation in both magnitude and direction of the wave vector. On Ag 110 thi lead to an aniotropy of the dipole moment and the induced electron denity for orthogonal q. Figure 9 illutrate thi effect for the denity n. Although the effect of orthogonal q on the d creening interaction are difficult to trace in detail, it appear that the lower atomic denity in the 001 direction implie a weaker dipole polarization. Thi mechanim reduce the d creening and give rie to a higher urface plama frequency than for the 011 direction. Thi trend i found alo for other value of q. The urface lo function Im g(q, ) can be eparated into and d electron contribution a indicated in Eq. 23. The local electric field polarizing the d hell yield an overall d electron polarization with oppoite ign. Accordingly, a illutrated in Fig. 10, there i a ignificant cancellation of and d contribution to the lo function. Neverthele, the frequency variation i nearly the ame for both term ince the urface plamon i a collective excitation of the combined d electron ytem. In fact, the bare jellium contribution g 0 (q, ) correponding to n 0 (z,q, ) i very mall and nearly independent of. It maximum occur at about 6.5 ev. Thu, the electron contribution induced by the dipole field d i governed by the frequency dependence of the induced dipole p n (q, ). The ame cancellation of and d contribution to the net urface polarization wa found previouly in the polarizable background model. 18 Figure 11 how the urface lo function for Ag 111 for everal wave vector. With increaing q the reonance aociated with the urface plamon i een to hift to higher frequency. The width of the reonance alo increae, in particular, on the high-energy ide due to the onet of interband tranition near 3.9 ev. Part of the width i due to bulk damping obtained from the Drude contribution of the meaured dielectric function. The value of i choen uch that the urface plama frequency extrapolate in the long wavelength limit to the value meaured optically. The repone equation in the preent model can be olved for q 0.05 Å 1. At maller value, convergence become difficult becaue of the rapidly increaing penetration depth of the induced potential. Although there i ome uncertainty with repect to the precie form of the diperion curve at mall q, 18 the reonance maxima in Fig. 11 extrapolate in the q 0 limit to about 3.7 ev in agreement with experiment. To illutrate the enitivity of the Ag excitation pectra to the d electron polarizability ( ) we how in Fig. 12 the diperion of the urface plama frequency of Ag 111 and Ag 100 for everal value of. A dicued in the previou ection, for computational reaon the urface repone calculation require a truncation of the um over lateral reciprocal lattice vector g. Since thi amount to an approximate treatment of the induced dipolar elf-field, a conitent bulk decription of thi field require an analogou truncation of the um over bulk reciprocal lattice vector G involved in the local d hell polarizability defined in Eq. 24, 25. A can be een, a reduction of, i.e., a larger value of ( ) implie a lowering of the excitation frequency due to effectively larger d creening. In addition, the diperion with q become lightly teeper ince the reduced penetration depth of the dynamical potential at larger q amount to a more rapid reduction of d creening and a tronger blue hift of the plama frequency. The comparion of the urface plamon diperion for Ag 100 and Ag 111 indicate a lightly larger poitive lope for the 100 face than for Ag 111 in agreement with FIG. 12. Surface plamon diperion (q) for a Ag 111 and b Ag 100 for everal value of. A reduced implie a larger d electron polarizability and tronger d creening, giving rie to lower urface plamon frequencie and a teeper diperion with q

10 LÓPEZ BASTIDAS, LIEBSCH, AND MOCHÁN PHYSICAL REVIEW B FIG. 13. Surface plamon diperion (q) for low-index face of Ag. a Experimental data Ref. 5 7 ; b Calculated diperion for 2.3. experimental obervation ee below. The frequency difference are, however, rather mall and comparable to the change introduced by the adjutment of ( ) due to the truncation of the reciprocal lattice um. To enure that term of imilar ize are included for all low-index face we have truncated the um over g at imilar abolute value g. The number of g term therefore depend on the crytal face. Specifically, we have ued 21, 19, and 31 urface reciprocal vector on the 100, 111, and 110 face, repectively. To compenate for the omitted part of elf-field the polarizability ( ) i renormalized by reducing the parameter a dicued above. In Fig. 13 the calculated urface plamon diperion for the Ag low-index face are compared to the experimental reult. 5 7 We chooe 2.3 for all face in order to approximately reproduce the range of oberved frequencie. A can be een, the trend obtained for the different crytal orientation i conitent with the data: i the diperion for Ag 100 lie above that of the 111 face and ii the lope for Ag i larger than for Ag On Ag 110 the long wavelength limit of the urface plama frequency i the ame for both propagation direction regardle of the value of. The diperion for the two propagation direction illutrate the effect of the intraplanar geometry on the effective local field. Except, of coure, for the lattice effect, the diperion obtained within the preent jellium-dipolium model agree well with thoe of the iotropic model employing a emi-infinite polarizable medium. 17 The main difference with repect to the data i that the overall variation of the poitive lope with crytal orientation i maller than found experimentally. Taking into account the finite ize of the dipole repreenting the d electron preumably lead to even weaker lattice effect becaue of the moother patial variation of the induced d electron potential. Thu, part of the oberved dependence of the Ag urface plamon diperion on crytal tructure appear to be aociated with the true and d electron band tructure not captured in the preent jellium-dipolium approach. In view of the mall difference between the calculated diperion we have invetigated in more detail the influence of the value of. For the purpoe of comparing the diperion for Ag 100 and Ag 111 let u approximate the reult hown in Fig. 12 by linear fit of the form (q ) aq b, where the contant depend on. Figure 14 how the relationhip between the lope a and intercept b for variou value of. Auming the linear diperion to be valid down to the q 0 limit, the diperion for both crytal face are conitent if they yield the ame q 0 urface plama frequency. Thu lope a for identical value of b hould be compared. Although the abolute frequencie for the two face then become more imilar than the one hown in Fig. 13, the reult plotted in Fig. 14 demontrate that regardle of the choice of, the lope on Ag 100 i about 20% larger than on Ag 111. Thi ugget that depite the truncation error introduced in the olution of the repone equation, we can be ure that the preent model i able to make reliable prediction concerning the face dependence of the Ag urface plamon diperion. We note, however, that the aumption of a linear extrapolation of the diperion to the long wavelength limit might not be accurate. 18 In fact, becaue of quadratic term, the true q 0 frequency i mot likely omewhat higher than the intercept b hown in Fig. 14. Thu, the linear fit exaggerate the difference between the long wavelength limit for the two face. The repone calculation dicued above were carried out within the RPA. In the cae of imple metal, exchangecorrelation contribution to the dynamical potential weaken the bare Coulomb interaction between conduction electron, giving rie to a light lowering of the urface plama frequency at finite q. 3 A imilar redhift wa found in the twocomponent d electron model where the d band were repreented in term of a emi-infinite polarizable medium. 18 In the preent jellium-dipolium ytem, thi frequency lowering FIG. 14. Parameter a and b obtained from a linear approximation of the urface plamon diperion, (q) aq b for different value of. Filled dot: Ag 100 ; empty dot: Ag 111. For each face, the ymbol correpond to from top 2.15,2.20,...,

11 INFLUENCE OF d ELECTRONS ON THE DISPERSION... PHYSICAL REVIEW B could, in principle, depend on the crytal face. In view of the overall mallne of thi effect, however, we expect the equence of poitive lope of the diperion for the Ag low index face to be unaffected by exchange-correlation contribution. V. CONCLUSION A model for the dynamical urface repone of the Ag 5,4d valence electron ha been developed in which the mutual polarization of thee denitie i treated elfconitently. Since the conideration of the full band tructure i not yet feaible, the delocalized electron denity i treated within the emi-infinite jellium model uing the local denity approximation and the more localized d electron are repreented by a emi-infinite fcc lattice of point dipole. Thi combined jellium-dipolium improve previou cheme which focu on either or d electron component but treat the other component more approximately. In particular, both the nonlocal dynamical urface repone of the electron and the crytalline geometry aociated with the d electron are taken into account. In addition, the repone to finite lateral wave vector i treated in a nonperturbative manner. Thu, the full range of momenta tudied in electron energy lo meaurement in acceible. We have applied our cheme to evaluate energy lo pectra for the low-index face of Ag with the aim of undertanding the dependence of the urface plamon diperion relation on crytal orientation. The diperion with wave vector i poitive on all three low-index face, and the magnitude of the lope varie in a characteritic manner with crytal orientation and propagation direction. Thee obervation are in agreement with experiment. The main difference i that the overall variation of the theoretical diperion with crytal orientation i maller than oberved in the data. Since the Ag urface collective mode lie jut below the onet of interband tranition, thi dicrepancy might be related to genuine band tructure effect ignored in the preent approach. The quetion concerning the large optical aniotropy pectra that ha been obtained experimentally for the Ag 110 face alo arie. It would be intereting to ee if the preent model yield a large enough aniotropy for the urface plamonpolariton propagation in the q 0 limit for the different propagation direction on thi face. In future work it would be deirable to allow for the finite extent of the d electron denitie induced at the lattice ite. Thi improvement of the point dipole model would eliminate inaccuracie in the preent cheme originating from the approximate treatment of the rapidly varying induced dipolar elf-field. In addition, it would be intereting to conider a d hell polarizability in the urface region that differ from the bulk value. Alo, a more realitic decription of the electron repone could be achieved by incorporating a onedimenional peudopotential in the direction normal to the urface. ACKNOWLEDGMENTS C.L.B. and W.L.M. acknowledge partial upport from DGAPA-UNAM through Grant No. IN and from the Deutcher Akademicher Autauchdient DAAD C.L.B.. 1 H. Raether, Surface Plamon Springer, Berlin, P. J. Feibelman, Phy. Solid State 12, A. Liebch, Electronic Excitation at Metal Surface Plenum, New York, B. Rothenhäuler and W. Knoll, Nature London 332, ; G.Flätgen et al., Science 269, ; C. A. Keller, K. Glamäter, V. P. Zhdanov, and B. Kaemo, Phy. Rev. Lett. 84, R. Contini and J. M. Layet, Solid State Commun. 11, S. Suto, K. Tuei, E. W. Plummer, and E. Burtein, Phy. Rev. Lett. 63, ; G. Lee, P. T. Sprunger, E. W. Plummer, and S. Suto, ibid. 67, ; Surf. Sci. 286, L M. Rocca and U. Valbua, Phy. Rev. Lett. 64, ; M. Rocca, M. Lazzarino, and U. Valbua, ibid. 67, ; 69, ; M. Rocca, Surf. Sci. Rep. 22, ; F.Moreco, M. Rocca, V. Zielaek, T. Hildebrandt, and M. Henzler, Surf. Sci. 388, ; 388, K. D. Tuei, E. W. Plummer, and P. J. Feibelman, Phy. Rev. Lett. 63, K. D. Tuei, E. W. Plummer, A. Liebch, K. Kempa, and P. Bakhi, Phy. Rev. Lett. 64, ; K. D. Tuei, E. W. Plummer, A. Liebch, E. Pehlke, K. Kempa, and P. Bakhi, Surf. Sci. 247, P. T. Sprunger, G. M. Waton, and E. W. Plummer, Surf. Sci. 269Õ270, H. Ihida and A. Liebch, Phy. Rev. B 54, R. H. Ritchie, Phy. Rev. 106, M. A. Cazalilla, J. S. Dolado, A. Rubio, and P. Echenique, Phy. Rev. B 61, P. Monachei unpublihed. 15 J. Tarriba and W. L. Mochán, Phy. Rev. B 46, C. López-Batida and W. L. Mochán, Phy. Rev. B 60, A. Liebch, Phy. Rev. Lett. 71, ; Phy. Rev. B 48, A. Liebch and W. L. Schaich, Phy. Rev. B 52, P. J. Feibelman, Surf. Sci. 282, P. Johnon and R. W. Chrity, Phy. Rev. B 6, K. Sturm, Solid State Commun. 48, ; K. Sturm, E. Zaremba, and K. Nuroh, Phy. Rev. B 42, L. Serra and A. Rubio, Phy. Rev. Lett. 78, H. Ehrenreich and H. R. Philipp, Phy. Rev. 128, J. Tiggebäumker, L. Köller, K. H. Meiwe-Broer, and A. Liebch, Phy. Rev. A 48, Y. Borenztein, W. L. Mochán, J. Tarriba, R. G. Barrera, and A

12 LÓPEZ BASTIDAS, LIEBSCH, AND MOCHÁN PHYSICAL REVIEW B Tadjeddine, Phy. Rev. Lett. 71, V. Fernandez, D. Pahlke, N. Eer, K. Stahrenberg, O. Hunderi, A. Bradhaw, and W. Richter, Surf. Sci. 377, A. Zangwill and P. Soven, Phy. Rev. A 21, N. D. Lang and W. Kohn, Phy. Rev. B 1, M. Born and K. Huang, Dynamical Theory of Crytal Lattice Oxford Univerity Pre, Oxford, 1988, Appendix of Chap. II. 30 B. Peron and E. Zaremba, Phy. Rev. B 31,