Phil. Mag. B 84, 2543-2557

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Phil. Mag. B 84, 2543-2557

Philosophical Magazine, 21 August 2004Vol. 84, No. 24, 25432557Ab initio determination of the optical properties of bulkAu and free surfaces of AuI. Reichly, A.Vernesy, L.Szunyoghyz, C.Sommers},P. Mohny and P. Weinbergeryy Center for Computational Materials Science, Technical University of Vienna,Getreidemarkt 9/134, A-1060 Vienna, Austriaz Department of Theoretical Physics, Institute of Physics, Budapest University ofTechnology and Economics, Budafoki u´t 8, H-1521 Budapest, Hungary} Laboratoire de Physique des Solides, Universite´ de Paris-Sud,91405 Orsay Cedex, France[Received 5 March 2004 and accepted 5 March 2004]AbstractThe optical properties of bulk Au and a (100) free surface of Au are determinedby solving the Helmholtz–Fresnel equations for a geometry reflectinglayered systems. This approach is based on the use of the microscopical conductivitytensor evaluated fully relativistically and, for later purposes, does includethe option of choosing an arbitrary uniform direction of a possibly presentmagnetization. It will be shown that, while so-called experimental bulk dataagree reasonably well with their theoretically obtained counterparts, in the caseof free surfaces of Au (semi-infinite systems) they not only disagree substantiallyin size between different experiments but also with the theoretical values.The shapes of the curves for the real and the imaginary parts of the diagonalpermittivity tensor elements " xx and " zz , however, are rather similar.} 1. IntroductionThere is no question that optical measurements, and in particular magnetoopticalinvestigations based on the Kerr effect are of great importance not onlyin academic research but also on the industrial level (see for example the reviewarticle by Weller (1995)). A theoretical description of such experiments andtechniques, however, has to deal with two important aspects, namely with a descriptionof the actual optical phenomenon, which of course belongs to classical physics(Mansuripur 2002) and the underlying microscopic origin in terms of a quantummechanicallydefined optical conductivity tensor. This distinction between macroscopicand microscopic quantities becomes especially important in dealing withmultilayer systems such as magnetically coated metals serving as a suitable substrate.Furthermore, since any kind of optical measurement has to be carried out from the‘outside’ of the system to be investigated, that is it refers to a realistic system witha defined surface, a surface near region and a buried interior, theoretical means haveto reflect properly the condition of being adequately realistic.y Author for correspondence. Email: pw@cms.tuwien.ac.at.Philosophical Magazine ISSN 1478–6435 print/ISSN 1478–6443 online # 2004 Taylor & Francis Ltdhttp://www.tandf.co.uk/journalsDOI: 10.1080/14786430410001693445


2544 I. Reichl et al.In the present paper the main emphasis is put on the optical part of determiningpermittivities. For this reason a system is chosen which very often serves as asubstrate for various kinds of magnetic multilayer system, namely pure Au.Although this appears to be an easy example it will be shown that any sort ofcomparison with experimental data has to be made with great care; dependingon the methods applied to produce experimentally suitable samples, substantialdifferences in the optical properties can be found in the literature. In the followinga method is described to deal with all reflections and interferences of a system causedby incident light parallel to the surface normal, but keeping the orientation of apossibly present magnetization general, that is not restricting the magnetization topoint along the surface normal.} 2. Optical properties of a layered systemThe optical properties of a material can be described by the well-knownHelmholtz equationn 2 E eE ðnEEÞn ¼ 0,where e is the permittivity tensor, n the refraction vector and E the electric field.Since in thin-film systems the permittivity varies with the distance to the surface, thisequation has to be reformulated such that it can be expressed in layer-resolvedquantities. In the following, first the solutions of the Fresnel equation are discussedin a single layer and then boundary conditions will be introduced in order to join upthe layer-resolved quantities.2.1. The solution of the Helmholtz equation in one layerConsider a solid magnetized homogeneously in an arbitrary direction, and letthe incident light be parallel to the surface normal (z axis, normal to the x–y plane).The layer-resolved permittivity tensor e p and refraction vector n p are then of theform01 0 1" xx p " xy p " xzpe p B¼@" yx p " yy p " yzp CA, n p ¼B@00CA:" p zx " p zy " p zzn pIt should be noted that, in contrast with the work of Vernes et al. (2002a, b), inthis paper this form of e p corresponds to a general orientation of the magnetizationof the surface.In order to determine n p the Fresnel equationdet n 2 p " p n p, n p, ¼ 0, , 2fx, y, zg,has to be solved. Since in this equation only even powers of n p occur, the correspondingsolutions fn ðkÞp g, k ¼ 1, ..., 4, have the property thatn ð p 1Þ¼ n ðÞp 3 , n ð p 2Þ¼ n ðÞp 4 ,which in turn implies that these four solutions yield only two different systemsof equations and therefore only two linear independent solutions can be obtainedfor the time-independent part of the electric field vector E ðkÞp .


The components of each wave can be expressed in terms of one component,for example0E ðkÞ 1p, xE ðkÞp ¼ B Ep, ðkÞyðEp, ðkÞxÞ C@ A , k ¼ 1, 3:Ep, ðkÞzðEp, ðkÞxÞDescribing multiple reflections of light at normal incidence, it is therefore sufficientto take only the x and the y components into account. The monochromatic,homogeneous and harmonic plane waves are then given byE ðkÞp ðz, tÞ ¼E ðkÞp exp ið ~q p z ~!tÞwhere q 0 , given by , ~qp ¼ q 0 np ðkÞ , ~! ¼ ! i, ð1Þq 0 ¼ ! c ,is the propagation constant in vacuum, ! is the photon frequency and is the lifetimebroadening parameter (Szunyogh and Weinberger 1999, Vernes et al. 2002a, c). Forlater purposes it is necessary to identify the incident and the reflected waves at eachinterface. It is convenient to define waves with n ð1Þp and n ð2Þp (Im ðnp ðkÞ Þ < 0, k ¼ 1, 2)as incident waves and waves with n ð3Þp and n ð4Þp (Im ðnpðkÞ Þ > 0, k ¼ 3, 4) as reflectedwaves.The total incident wave (indicated by the superscript inc) is a linear combinationof Epð1Þ and Ep ð2Þ , which by introducing a 2 2 matrix can be written as a function ofthe x component of E ð1Þp and the y component of E ð2Þp :!!Ep, incx Ep, ð1Þx¼ A p , ð2ÞwhereOptical properties of bulk Au and free surfaces of Au 25450E incp, y1A p ¼B@ Ep, ð1ÞyðE ð1Þp, xÞEp, ðÞ 1xEp, ð2ÞyEp, ð2ÞxðEp, ð2Þ 1yÞEp, ð2ÞyCA :1It is easy to find the matrix A 0 p which connects the reflected wave to E ð3Þp and E ð4ÞThe waves EpðkÞ and E ðkþ2Þp , k ¼ 1, 2, differ only by a phase factor such thatE ð3Þp, xE ð3Þp, y¼ Eð1Þ p, xEp, ð1Þy,E ð4Þp, xE ð4Þp, y¼ Eð2Þ p, xEp, ð2Þywhich in turn implies that A 0 p A p .Defining now a general 2 2 reflection matrix R p which transforms the x and ycomponents of the incident waves, E ð1Þ and E ð2Þ , into their reflected counterparts(indicated by the superscript ref), E ð3Þ and E ð4Þ , the total reflected wave is given byE refp, xE refp, y!¼ A p R pE ð1Þp, xE ð2Þp, y,p .!: ð3Þ


2546 I. Reichl et al.As for each solution n ðkÞp (the superscript ðkÞ numbers the solutions) the magneticfield vector HpðkÞ is given byH ðkÞp¼ n ðkÞpand the above 2 2 matrix formalism yields0 1 0Hp, incx@ A ¼ N p@0@H incp, yHp, refxH refp, y E ðkÞp , k ¼ 1, ...,4,Ep, ð1ÞxE ð2Þp, y1A, ð4Þ1 0 1Ep, ð1ÞxA ¼ N p R p@ A, ð5ÞE ð2Þp, ywhere N p is given by0N p ¼B@n ð1Þp E ð1Þp, yðE ð1Þp, xÞE ð1Þp, xn ð1Þpn ð2Þpn ð2Þp E ð2Þp, xðE ð2ÞE ð2Þp, yp, yÞ1CA :ð6Þ2.2. Boundary conditionsBetween different layers the electric and the magnetic fields have to be matchedat each boundary z p (figure 1). The wave with the subscript z þ p is supposed to be thesolution in the pth layer at the lower boundary z þ p ; the wave with the subscript z p isthe solution in the ð p 1Þth layer at the upper boundary z p .Atz þ p the sum of thep+1pN-1N-2N-3p-1NVacuumzN+1zNzN -1zN -2zN -3zp+ 2zp+1zpzp -143210Bulkz1z2z5z4z3zp+1 -zp=dpFigure 1.Layers are numbered from 1 to N; for the bulk regime the index 0 applies. Eachlayer p has a lower boundary z p and an upper boundary z pþ1 .


2548 I. Reichl et al.the corresponding reflectivity matrix R 0 can be assumed to be zero and thusD 0 ¼ N 0 A0 1 , ð14ÞR 1 ¼ðN 0 A0 1 A 1 þ N 1 Þ 1 ðN 1 N 0 A0 1 A 1 Þ: ð15Þ2.3.2. Last stepTaking into account that there are only two solutions in the vacuum, namelyone incident and one reflected wave, the reflectivity matrix at the vacuum interface(usually termed the surface reflectivity matrix) is given byR surf ¼ r !xx r xy:r yx r yyFurthermore, the material properties in vacuum are obtained when setting n ð1Þpto unity, that isn ð2Þp N vac ¼ 0 11 0:At z þ Nþ1 the total fields are therefore given by½E vac Š z þ ¼ð1 þ R surfÞE ðincÞvac ,Nþ1½H vac Š z þ ¼ N vacð1 RNþ1 surf ÞH ðincÞvac ,while at z Nþ1 they are of the form ( p ¼ N)!0E vac, x¼ A N C 12N þ C 34 E ð1ÞN R N@E vac, y z Nþ1!0H vac, x¼ N N C 12N C 34 N R N@H vac, y z Nþ1N, xE ð2ÞN, yE ð1ÞN, xE ð2ÞN, yDemanding continuity at the surface for both fields leads to1A;1A:andR surf ¼ ðN vac þ D N Þ 1 ðN vac D N Þ; ð16Þ 1AD N ¼ N N C ð1;2ÞN C ð3;4ÞN R N C ð1;2ÞN þ C ð3;4ÞN R 1N N : ð17ÞEquations (16) and (17) are the corresponding equations for the vacuum-surfaceinterface which are needed to determine R surf .} 3. AB INITIO determination of the permittivity tensor3.1. Recursive determination of e pIt was shown by Vernes et al. (2002a) that the (macroscopic) permittivitytensor e pq can be related to the (microscopic) conductivity tensor r pq in terms ofthe following mapping:" pqij ð!Þ ¼ pqij þ 4piij ð!Þ, p, q ¼ 1, ..., N, pqij ¼ ij pq ,e! pq


where N is the total number of (atomic) layers and i, j 2fx, y, zg. In order to obtaina layer-resolved reduced permittivity e p , the implicit equatione p ð!ÞE p ¼ XNq¼1e pq ð!ÞE q , p ¼ 1, ..., N, ð18Þhas to be solved such that, when for example " yy p ¼ " xx p and " yx p ¼ " xy,p! !! !" xx p " xyp Ep, x" yx p " yyp ¼ XN " pqxx " pqxy Eq, xE p, y " pqq¼1 yx " pq , p ¼ 1, ..., N: ð19Þyy E q, yShifting the index p in equation (10) to p þ 1,!!E pþ1, xð1, 2Þ ð3, 4Þ Eð1Þp, x¼ A p Cpþ CpR p , ð20ÞE pþ1, y E ð2Þz p, ypþ1the pseudovector ðE ð1Þp, x, E ð2Þp, yÞ can be expressed in terms of equation (20) and insertedinto equation (7) to yield!E p, xE p, yz þ p ¼ A p 1 þ R p Cð1, 2Þ ð3, 4Þ 1A1p þ CpR p p!E pþ1, xE pþ1, yðE p Þ z þpdescribes a wave at the boundary between layer p 1 and layer p. At halfdistancebetween z p and z pþ1 this wave accumulates a phase shift which can beðk, kþ1Þtaken into account using a factor ðCpÞ 1=2 , k ¼ 1, 3, such that each partialwave is multiplied automatically by the correct phase factor,!E p, xð1, 2Þ 1=2þ !1=2Rp E ¼ A p CpCð3, 4Þp, ð1Þxp: ð22ÞE p, yz þ p þd p =2E ð2Þp, yBy replacing ðEp, ð1Þx, Ep, ð2ÞyÞ in equation (22) with equation (20) and recursively insertingequation (21) into equation (22) ðE p Þ z þp þd p =2 can finally be written as a functionof ðE N Þ zN:Optical properties of bulk Au and free surfaces of Au 2549!E p, xE p, y¼ YNW kz þ p þd p =2 k¼p!E N, xE N, y¼ YN pW kþpz k¼0N!E N, xE N, ywhere the matrices W kþp are defined in the following way: 1=2 1=2Rp h ið1, 2ÞW pþk ¼ A p Cpþ Cð3, 4Þð1, 2Þ ð3, 4Þ 1A1p Cpþ CpR p p ; k ¼ 0; W pþk ¼ A pþk 1 þ R pþk Cð1, 2Þ 4Þ 1Apþkþ Cð3,pþk R 1pþk pþk; k > 0:The reduced permittivity e p is obtained by inserting the expansion in equation (23)into equation (18):z N,z pþ1:ð21Þð23Þe p ð!Þ ¼ XNq¼1e pq ð!ÞW pq ,! !W pq ¼ YN qN Y p1W kþq W kþp :k¼0k¼0ð24Þ


2550 I. Reichl et al.VacuumpRecursionIteration: new εBulkε and nfrom a symmetrizedbulk calculationFigure 2. Recursive algorithm. Firstly, the bulk properties eð!Þ and nð!Þ serving as thestarting values for a layer recursion have to be known from a separate bulk calculation.Secondly, the 2 2 matrix technique starts at the boundary between the bulk andthe first layer considered. Subsequently all reflections, transmissions and interferencesat all boundaries between different layers are taken into account. Thirdly, the obtainedlayer-resolved permittivities e p ð!Þ serve as input for the next iteration. This recursiveapproach is repeated until the desired accuracy for the permittivities e p ð!Þ is obtained.This system of equations for e p ð!Þ (see equation (18)) has to be solved iteratively(figure 2), starting withW ð0Þp ¼ 1, ð25Þimplying thate p ð!Þ ð0Þ ¼ XNq¼1e pq ð!Þ:


Optical properties of bulk Au and free surfaces of Au 2551As a criterion for the accuracy of this iterative procedure the below inequality canbe usede p ð!Þ ðnÞ e p ð!Þ ðnþ1Þ < e p ð!Þ threshold :ð26Þ3.2. Bulk systemsAs described above, the recursive algorithm starts at the interface betweenthe substrate and first layer and ends at the interface between the last layer andthe vacuum. In order to obtain the necessary bulk quantities A 0 and N 0 as startingvalues, the bulk properties e bulk and n bulk , have to be investigated. By definition ina bulk system (‘infinite system’, with three-dimensional periodicity) all physicalproperties have to be the same in all unit cells. For a simple lattice (one atom perunit cell) this implies that all physical quantities have to be the same in all (atomic)layers. As in the above algorithm the total number (N) of atomic layers necessarilyhas to be finite, in the case of a bulk system a kind of symmetric extension of thelayered system such that jp qj 4 N 1 with respect to a particular p can be appliedby which the corresponding " pij , i, j 2fx, y, zg, are obtained ase p ¼q¼pX0þ XNq¼1Nþ1W N;N jp qj N;N jp qjepþNW p;q e p;q þ X 1W 1;1þjp qj e 1;1þjp qj ; ð27Þnamely as the original sum plus the extensions to the left and to the right.q¼N} 4. ResultsFigure 3 shows schematically the types of calculation that have to be carried out.As a first step the effective scattering potentials for bulk Au and a free surface ofAu along the (100) surface normal were calculated using the fully relativistic screenedKorringa–Kohn–Rostoker method (Szunyogh et al. 1994, 1995, U´ jfalussy et al.1995) and the local density functional parametrization by Vosko et al. (1980).Figure 3.Computational scheme for an ab initio calculation of optical properties.


2552 I. Reichl et al.Next, the (complex layer-resolved) conductivity matrices r ð!Þ are evaluated usingthe Kubo–Greenwood equation fully relativistically (Vernes et al. 2002c). Only then,as a third step, can the above-described calculation of the optical properties of Au beperformed. In using this scheme the permittivities eð!Þ for bulk Au and a free surfaceof Au along (100) were calculated for photon energies p! varying from 1:5 to 4 eV.Let D " ðNÞ be the difference between the in-plane and out-of-plane componentsof the permittivity:D " ðNÞ ¼j" xx ðNÞ" zz ðNÞj,where N is the total number of layers considered. Since in a bulk cubic systemcharacterized by three-dimensional periodicity these two components have to bethe same, that islim ½D "ðNÞŠ ¼ 0,N!1D " ðNÞ can serve as a numerical criterion for the calculated bulk permittivities.In figure 4, D " ðNÞ is shown with respect to N together with the relative differences2D " ðNÞ=j" xx ðNÞþ" zz ðNÞj, D " ðNÞ=j" xx ðNÞj and D " ðNÞ=j" zz ðNÞj.For bulk Au the present theoretical results compare reasonably well withthe experimental data obtained by Hagemann et al. (1975) and Weaver et al.(1981). The principal shapes of the curves look alike; for small photon energies,eð!Þ assumes quite large values (figure 5) owing to the finite lifetime broadening (see equation (1)).Figure 4. Convergence of the Au calculation to ‘bulk’ with respect to the number of Aulayers sandwiched between two semi-infinite systems of Au, where the layer quantitiesare bulk like if the difference between the out-of-plane (" zz ) and the in-plane components(" xx and " yy ) of the permittivity vanishes: (.), Re ðD " Þ (ordinate is the left y axis);(), Im ðD " Þ (ordinate is the left y axis); (^), j2D " j=j" xx þ " zz j; (i), jD " j=j" xx j;(œ) jD " j=j" zz j. For the relative differences the ordinate to the right applies.


Optical properties of bulk Au and free surfaces of Au 2553Figure 5. Permittivity for bulk Au: (——), experimental Re ð" xx Þ values (Weaver et al. 1981);(- - - -) experimental values Im ð" xx Þ (Weaver et al. 1981); (), experimental Re ð" xx Þvalues (Hagemann et al. 1975); (— —), experimental Im ð" xx Þ values (Hagemann et al.1975); (.), theoretical Re ð" xx Þ values; (g), theoretical Im ð" xx Þ values; (), theoreticalRe ð" zz Þ values; (œ), theoretical Im ð" zz Þ values.Using now the results of the bulk calculation as initial values in equations (14)and (15), eð!Þ was investigated for an Au(100) surface. It turns out that the order ofmagnitude of the theoretical permittivity differs considerably from the experimentaldata obtained by Bader et al. (2000) and by Truong et al. (2003), which in turndiffer considerably from each other. In order to take into account these differences infigures 6–9, showing the real and imaginary parts of " xx and " zz , the experimentaldata are shifted and therefore different ordinates apply.It should be mentioned that Au surfaces usually suffer somewhat from surfacereconstruction. In the experiments of Bader et al. (2000) the surface was producedby depositing Au nanoparticles in a solvent on to a glass plate and evaporatingthe solvent afterwards. In the experiments of Truong et al. (2003), Au was evaporatedonto glass substrates and annealed for 2 h. Thus, depending on the method ofpreparation, fundamental differences in the surface structure of Au can exist. Quiteclearly the present calculations refer to a perfect single-crystal surface. Besidesthe fact that the experimental ambiguities affect the absolute size of the permittivity,all surface investigations (both experimental work (Bader et al. 2000, Truonget al. 2003) and the present theoretical study) detect one common feature, namelyan anomalous adsorption at the surface, an effect which is not present for bulk Au.It is indeed reassuring that in all three studies the photon energies at which the peakin the permittivity occurs have rather similar values.


2554 I. Reichl et al.Figure 6. Reð" xx Þ for Au(100): (.), calculated values; (œ), (i), (^), experimental data forthe 3 nm, 7:5 nm and 10 nm Au films respectively used by Truong et al. (2003); (r),experimental data of Bader et al. (2000). The left ordinate applies to the experimentaldata, and the right to the calculated values.Figure 7. Im ð" xx Þ for Au(100): (.), calculated values; (œ), (i), (^), experimental data forthe 3 nm, 7:5 nm and 10 nm Au films respectively used by Truong et al. (2003), (r),experimental data of Bader et al. (2000). The outer right ordinate applies to thecalculated values, the inner right ordinate to the experimental data of Bader etal. (2000), and the left to all other values.


Optical properties of bulk Au and free surfaces of Au 2555Figure 8. Reð" zz Þ for Au(100): (.), calculated values; (œ), (i), (^), experimental data forthe 3 nm, 7:5 nm and 10 nm Au films respectively used by Truong et al. (2003); (r),experimental data of Bader et al. (2000). The left ordinate applies to the experimentaldata, and the right to the calculated values.Figure 9. Im ð" zz Þ for Au(100): (.), calculated values; (œ), (i), (^), experimental data forthe 3 nm, 7:5 nm and 10 nm Au films respectively used by Truong et al. (2003); (r),experimental data of Bader et al. (2000). The right ordinate applies to the calculatedvalues, and the left to the experimental data.


2556 I. Reichl et al.It should be noted that, independent of the film thickness, always only twodimensionalrotational symmetry applies. Therefore all investigations of the permittivityshow different results for the " xx and the " zz components. The calculated valuesof Re ð" xx Þ for Au(100), (see figure 6) have a maximum at about 1:9 eV, and a kindof ondulation at approximately 2:4 eV, which also can be seen in the experimentalvalues at 2:4 eV for the thinnest film and at 2:2 eV for the thickest film. The calculatedvalues of Re ð" zz Þ for Au(100), appear to be in reasonably good qualitativeagreement with the measured data (Truong et al. 2003), although " zz ð!Þ seems todepend strongly on the film thickness. For instance, the experimental curves for 3 nmand 7:5 nm film thicknesses change sign at photon energies above 2:6 eV and 3:4eVrespectively while the calculated curve and the curve measured for 10 nm do notchange sign below 3:5 eV.} 5. ConclusionIt was shown in this paper that, by mapping the quantum-mechanically evaluatedconductivity tensor on to the macroscopic permittivity tensor and by applyingan appropriate scheme for taking into account all interferences and reflections,optical properties can be studied on a truly ab initio level. The present study alsoshowed that, whenever realistic systems, that is systems with a surface, have to bedealt with, a comparison with experimental data has to be carried out with extremecare since, not only can the applied preparation technique be decisive, but also thethickness of the prepared films enters the experimental observations. It also has to benoted that the term ‘bulk’ in many cases can be quite misleading, not only becauseany kind of measurement is carried out ‘from the outside’, that is for an at best semiinfinitesystem, but also because ‘bulk’ very often only refers to a fictitious quantity,obtained by extrapolating certain thickness parameters to infinity. In the presentstudy the permittivity of ‘bulk’ Au is needed as starting value for an a iterativeprocedure aiming at an evaluation of the surface reflectivity. Keeping in mindthat, as mentioned in the introduction, optical measurements are of crucial importancefor the characterization of magnetic properties of multilayered systems, heterostructures,etc., in terms of the Kerr effect a rigorous study of the optical propertiesof a typical substrate is very necessary.AcknowledgementsFinancial support from the Austrian Science Foundation (project W004), theAustrian Science Ministry (GZ 45.531) and the Wissenschaftlich-TechnischesAbkommen (WTA) Austria–Hungary (A-3/03) is gratefully acknowledged. Wealso wish to thank the computing centre IDRIS at Orsay as part of the calculationswas performed using their facilities.ReferencesBader, G., Hache¤, A., and Truong, V.-V., 2000, Thin Solid Films, 375, 73.Hagemann, H. J., Gudat, W., and Kunz, C., 1975, J. opt. Soc. Am., 65, 742.Mansuripur, M., 2002, Classical Optics and its Applications (Cambridge University Press).Szunyogh, L., U´ jfalussy, B., Weinberger, P., and Kollar, J., 1994, Phys. Rev. B, 49,2721.Szunyogh, L., U´ jfalussy, B., and Weinberger, P., 1995, Phys. Rev. B, 51, 9552.Szunyogh, L., and Weinberger, P., 1999, J. Phys. condens. Matter, 11, 10 451.Truong, V.-V., Belley, R., Bader, G., and Hache¤, A., 2003, Appl. Surf. Sci., 212–213, 140.


Optical properties of bulk Au and free surfaces of Au 2557Vernes, A., Szunyogh, L., and Weinberger, P., 2002a, Phys. Rev. B, 65, 144 448; 2002b,ibid., 66, 214 404; 2002c, Phase Transitions, 75, 167.Vosko, S. H., Wilk, L., and Nusair, M., 1980, Can. J. Phys., 58, 1200.U´ jfalussy, B., Szunyogh, L., and Weinberger, P., 1995, Phys. Rev. B, 51, 12 836.Weaver, J. H., Krafka, C., Lynch, D. W., and Koch, E. E., 1981, Optical Properties ofMetals (Fach-Informationszentrum Karlsruhe).Weller, D., 1995, Spin–orbit-influenced Spectroscopies of Magnetic Solids, edited by H. Ebertand G. Schu¨tz (Berlin: Springer), p. 1.

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