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Interference second harmonic anisotropy at single crystalline silver-electrodes
PERGAMON
Electrochimica Acta 44 (1998) 897±901
Interference second harmonic anisotropy at single crystalline silver-electrodes Bruno Pettinger a, *, Christoph Bilger a, Guillermo Beltramo b, Elisabeth Santos b, Wolfgang Schmickler b a
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4±6, D-14195 Berlin (Dahlem), Germany b Abteilung Elektrochemie, UniversitaÈt Ulm, D-89069 Ulm, Germany Received 5 November 1997
Abstract Using a new approach, the interference SHG anisotropy, it is possible to unambiguously evaluate the nonlinear susceptibility coecients, which describe the isotropic and anisotropic response of a metal electrode. We report on a ®rst application of this method to single crystalline silver electrodes in a 50 mM aqueous solution of NaClO4. The isotropic coecient a shows the strongest dependence on the electrode potential. Theory predicts that the real part ar=Re(a) should have a minimum near the potential of zero charge. A normalized plot of ar vs. the surface charge density s shows a good agreement between experiment and theory in the range ÿ20 mC cmÿ2
1. Introduction As is well known, the properties of the electric double layer at metal electrodes are not only controlled by the electrode potential, the solvent and adsorbed molecules but also by the crystallographic orientation of the substrate. Together they constitute the geometric and electronic structure of the interface. Optical second harmonic generation (SHG) is, due its intrinsic surface sensitivity, particularly attractive for in-situ investigations of the metal/electrolyte interface [1, 2]. Since the non-linear polarization (created by an intense laser pulse) has components parallel and perpendicular to the surface, it probes distinct properties of the interface [3±7]. The parallel components depend on the symmetry of the sample surface, whereas the perpendicular component is sensitive to the interfacial electric ®eld. Keeping the azimuthal
* Corresponding author.
angle ®xed and recording the SHG intensity as a function of the polar angle j, i.e. recording the SHG anisotropy, provides an easy way to monitor the rotational symmetry of the sample and its changes with potential and adsorption [2, 8±16].
2. Theory Second harmonic generation from metal surfaces has been the subject of fairly intensive theoretical research. In the early work, the rotational anisotropy was disregarded, and attention was focused on the rotationally averaged response. For this purpose, jellium proved to be a useful model for a metal surface; within its framework many experimental ®ndings can be explained at least qualitatively (see, e.g. Refs. [17±19]). For a structureless model surface, such as that of jellium, the SHG response is characterized by three coecients aj, bj, and dj ± the index j is usually not used in the literature, but is added here to distinguish them from the
0013-4686/98/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 4 6 8 6 ( 9 8 ) 0 0 1 9 2 - 3
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B. Pettinger et al. / Electrochimica Acta 44 (1998) 897±901
phenomenological coecients introduced below ± which have the following meaning: aj characterizes the SHG response caused by currents driven in the z direction, perpendicular to the metal surface; bj speci®ed the contribution by currents induced parallel to the metal surface, and dj is due to the magnetic dipole moment from the bulk. Within the jellium model, bj= ÿ 1 and dj=1. The only parameter which depends on the electronic structure of the interphase is the coef®cient aj; in the jellium model it depends both on the electronic density of the bulk and on the charge density at the surface. The simple jellium model does not account for the crystal structure. This can be added by incorporating a lattice of pseudopotentials into the electron gas [20]; Leiva and Schmickler [21] used this model to calculate the coecient aj as a function of the surface charge density for the principal single crystal surfaces of several metals in the low frequency region. We will compare their calculations with our experimental data below. The rotational anisotropy of a Ag(111) surface was recently investigated by Petukhov [22]. According to his model the anisotropic SHG response in the low frequency region is caused by the necks of the Fermi surface near the L-points of the Brillouin zone [23]. This implies that the anisotropic amplitude should be practically independent of the surface charge, and hence also of the electric potential.
3. The principles of the interference technique for SHG Let us consider SHG only in the so-called electric dipole approximation, e.g. we neglect higher multipole contributions (later on we will see that these contributions ± giving rise, for instance, to a four-fold rotational symmetry ± are rather weak). Based on this simpli®cation, the nonlinear polarization is expressed by the dyadic product of a susceptibility tensor of second rank with the ®eld vector of the incident beam: (2) P(2) i (2o, nj)Awijk (nj):Ej(o)Ek(o). The susceptibility tensor exhibits an angular dependence. Thus, the nonlinear polarization, P(2) i , varies if the sample is rotated around its normal. The rotation is described by a tensor transformation, which relates the tensor elements of the sample, w(2) rst given in crystal coordinates, to (2) polar laboratory coordinates: w(2) ijk (nj) = RirRjsRktwrst , with n = 0±3, and Rir being the rotation matrix; j is the polar angle [24]. For a given polarization condition, such as pp, ps, sp or ss polarization, and for a rather low surface symmetry, such as Cs, the SHG anisotropy can be described by a superposition of up to four eective susceptibility terms, auv... duv, having a complex
character in general, Cjauv buv cos j a I SHG uv cuv cos2 j b duv cos3 j gj2 ,
1
where C is an appropriate proportionality constant. Note, that we have introduced here angular shift terms, a, 2b, 3g which are zero for v = p and p/2 for v = s (e.g. in the later case the cos(nj + p/2) 4 sin(nj)). Each of the four terms, {auv, buv, cuv, duv}, carries an index uv which indicates the polarization conditions and determines its composition by standard susceptibility elements (=(1/m)(w(2) rst2. . .)uv, m = 1, 2 or 4); the index uv also indicates which Fresnel factors are implicitly included [15, 24]. The ®rst term, denoted by the complex term auv, is associated with the isotropic susceptibility whereas the other three are connected with the anisotropic susceptibilities, buv, cuv and duv, multiplied by the appropriate trigonometric function. The latter three terms indicate the possible presence and superposition of one-, two- or threefold rotational symmetry elements of the surface. Since we are interested in the angular dependence, we will set C = 1. In addition, we will omit the uv indices of the four terms where appropriate. A general Cs symmetry can lead to a rather complex anisotropy pattern. However, for silver(111) where no surface reconstruction is known, b = c = 0, and the SHG anisotropy exhibits a typical C3v symmetry [2]. Au(111) p electrodes can exhibit surface reconstruction; the 3 23 reconstruction occurs in domains along the three [110] directions and their contributions nearly cancel. Thus, the b and c coecients are still small enough and, so, the coecients a and |d| can be evaluated together with the relative complex phase da±d. A correlation with the metallic charge density, sM, indicates a linear relationship between sM and the Re(a) term for the case of bromide adsorption whereas the strong variation of da±d pointed to a substantial change of the electronic surface levels upon adsorption. Evidently, SHG probes the interfacial symmetries (via the b, c, d terms) as well as the interfacial electric ®eld and electronic resonance conditions (via a and da±d) [12, 15, 16]. Since the SHG anisotropy curve is composed by superposition of a few harmonics (in general n R 6), a Fourier analysis appears to be, at ®rst glance, a straightforward approach to evaluate the coecients, a, b, c and d. Unfortunately, Eq. (1) contains an absolute square over a sum of susceptibility terms having complex character in general. Its Fourier analysis gives rise to mixed terms and, therefore, to a system of equations having multiple solutions. In general, this means distinct sets of coecients which reproduce the same anisotropy curve exist (this holds also if, in addition to the a±d-terms, an e-term, the neglected
B. Pettinger et al. / Electrochimica Acta 44 (1998) 897±901
bulk-contribution, would be signi®cant). This has been shown in Refs. [25, 26] and will not repeated here. Only in favorable cases, an analysis of Eq. (1) is possible. Usually the complete analysis of the SHG anisotropy is dicult, questionable or even impossible [26]. To overcome this problem, we have developed the so-called interference SHG anisotropy (ISHGA), using external SHG from a quartz lamella that interferes with the SHG from the sample [32]. The ISHGA measurement consists of recording three anisotropy curves subsequently, two of them with the quartz lamella present in the beam at two dierent positions, x1 and x2, and one without the quartz lamella. The intensity of the SHG from the quartz lamella is 2 Iquartz SHG =|pÄ| , with pÄ=|pÄ|exp(idp) being also a complex quantity. In the ®rst anisotropy measurement, with the quartz lamella at x1, there is an (unknown) phase relationship between the SHG from the quartz lamella and the SHG from the sample. In the second anisotropy measurement, we have moved the quartz lamella to the location x2 such that x2ÿx1 33.18 cm. In this way the phase relationship between the two SHG wave changes by p/2 compared to its value of the ®rst anisotropy experiment. This is due to the dierences in refractive indices of n(o) and n(2o) causing distinct optical path lengths for the fundamental and the SH wave (see Fig. 1). The root of Eq. (1) can be multiplied by an exponential, exp(id), with an arbitrary phase angle, d, because |exp(id)| = 1. This freedom can be exploited to make p=pÄ exp(ÿidp) a real quantity at x1 and an imaginary quantity at x2, because ip=pÄ exp(ÿidp+ p/2). Calculating the numerical dierence of the ISHG and SHG anisotropy yields two curves which can be easily analyzed due to their simple structure: 3 X DI ISHG x 1 ÿ p2 2 p Re wn n j an , pp
2
3 X DI ISHG x 2 ÿ p2 2 p Im wn n j an , pp
3
n0
899
Fig. 1. Scheme of the interference SHG anisotropy experiment. At the quartz lamella, located at x1 or x2, the ®rst SHG wave is created (dotted line); it travels together with the fundamental beam towards the sample where the second SHG wave is created (dashed line). Upon re¯ection at the sample the ®rst and the second SHG waves propagate on parallel routes and interfere at the detector.
4. Experimental The silver single crystals were ®rst mechanically polished, then repeatedly etched in NaCN + H2O2 solutions and ®nally exposed to a moderate ¯ame annealing. The ¯ame annealing was repeated prior to each experiment. The SHG setup is essentially the same as described earlier [12, 15, 16]. In short, a Nd:YAG laser provides pulses of about 8 ns width, a power of ca. 100 mJ at 1064 nm and ca. 10 Hz repetition rate. To perform the interference experiment, the cell with a window parallel to the sample surface has to be used. Sample and window are adjusted perpendicular to the rotational axis. The schematic arrangement is shown in Fig. 1 together with the movable quartz lamella. Because of the new con®guration, the angle of incidence at the electrode is ca. 338. Supra pure chemicals and triple distilled water have been used. The electrolyte is generally an aqueous solution of 0.1 M NaClO4.
n0
where p is the SHG intensity of the quartz lamella alone, and wn={a, b, c, d} denotes the complex susceptibility coecients. They can be separated into the complex isotropic susceptibility (n = 0: a) and the complex anisotropic susceptibilities (n = 1, 2, 3: b, c and d). Using this approach we obtain two experimental curves which have to be analyzed according the two expressions given above. They are linear in the coecients Re(wn) and Im(wn). From a Fourier analysis all the coecients can be unambiguously determined along with the oset angles an (the later can describe, in addition to the above given meaning, a possible angular oset of the eective polarization vector associated with the nth-fold rotational symmetry).
5. Results and discussion The analysis of the ISHGA curves yields unambiguous and reliable data of all anisotropy coecients. The behavior of the anisotropic terms (c, b, d) for Ag(111), Ag(110) and Ag(100) will be discussed elsewhere. Here we like to state that, for Ag(111), the three-fold susceptibility, dpp, is rather independent on potential, and the bpp and cpp terms are of the same height as the noise level, as expected for an unreconstructed surface. The same holds also for the hypothetical four-fold term, denoted as epp. Therefore, the potential dependence of the SHG response has to be attributed to the isotropic susceptibility, app. Fig. 1 illustrates this with
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B. Pettinger et al. / Electrochimica Acta 44 (1998) 897±901
Fig. 2. Potential dependence of the a-term. Top panel: |a| vs. E; bottom panel: complex phase da vs. potential. Electrolyte: 0.1 M NaClO4.
The relationship between the charge density and the electrode potential has been determined in independent measurements for Ag(111) and Ag(100) and is used in the following. Fig. 2 shows the comparison of the ISHGA experiment and the theoretical calculation of the a-term, using density functional theory and introducing the crystal structure via pseudopotentials [21] (unfortunately, there are no theoretical data available for Ag(110)). To establish a comparison, only the real part of the experimental a-term (for pp-polarization) could be used: ar=Re(a). In addition, it is given in a normalized fashion, as the ratio of [ar(s) ÿ ar(pzc)]/ ar(pzc) (Fig. 3). The comparison shows a fairly good agreement for low charge densities. For higher charge densities we expect a deviation between theory and experiment, because the model describing the electric double layer is then no longer adequate. This is also evident from the comparison. There are two reasons for these deviations: Firstly, at high negative charge densities the electrons spill out further into the solution. Therefore, the details of the interaction of the electrons with the solvent becomes important. These are not accounted for in the model. Secondly, at high positive charge densities an oxide layer starts to form on the surface, which is also not incorporated in the model.
a plot of three |app| curves vs. potential for Ag(111), Ag(110) and Ag(100). The curves show a minimum around the pzc as proposed by Schmickler et al. [21]. In the framework of the semiempirical model of Sipe et al., the isotropic amplitude is expected to have a dependence on the charge density: At the interface, the static electric ®elds exhibits an extremely large ®eld strength comparable with that of the electromagnetic ®eld of the incident laser beam. This can give rise to an non-linear process of third order to the second order polarizability, . P 2 w 2 :E o E o w 3 ..E o E o EH 2 w 2 w 2 red :E o E o weff :E o E o :
4
Since the static ®eld EH has components only in the direction perpendicular to the surface, the last expression can be reduced to a tensor product of second order, where some tensor elements in w(2) red carry a dependence on the static electric ®eld. This holds in particular for the app term, which is roughly proportional to w(2) e,zzz: sM app a0 a1 EH z ' 00a0 a1 , 5 eH e0 where we omit the pp indices on a0 and a1. At the point of zero charge we have |app(sM=0)|0 |a0|.
pzc Fig. 3. Normalized form of ar=Re(a), (arÿapzc vs. r )/ar charge density sM. Top panel: Ag(100), bottom panel: Ag(111). Electrolyte: 0.1 M NaClO4.
B. Pettinger et al. / Electrochimica Acta 44 (1998) 897±901
6. Conclusion In this note we have reported on our ®rst application of the interference method to measuring the SHG anisotropy of silver electrodes. With this method all relevant susceptibility coecients can be accurately determined. Thus the evaluation of the data is no longer plagued by the occurrence of several solutions for the coecients, which makes the interpretation of the results so dicult in the conventional method. From an electrochemical point of view the most interesting parameter is the coecient a, which characterizes the response of the surface electrons in the direction perpendicular to the surface. This is the only coecient which changes drastically when the electrode potential is varied, or when particles are adsorbed on the surface. Our results for Ag(111) in a dilute solution of NaClO4 agree quite well with the predictions of the calculations by Leiva and Schmickler in the range of low surface charge densities. Acknowledgements The authors thank Professor G. Ertl for continuing support, various discussions and stimulating interest in these studies. This work was partly supported by the Deutsche Forschungsgemeinschaft. References [1] Y.R. Shen, The principles of nonlinear optics, Wiley, New York, 1984. [2] G.L. Richmond, in: H. Gerischer, C.W. Tobias (Eds.), Adv. Electrochem. Sci. Eng., vol. 2, VCH Publishers, Weinheim, 1992. [3] R.A. Bradey, S. Arekat, R. Georgiadis, J.M. Robinson, S.D. Kevan, G.L. Richmond, Chem. Phys. Lett. 168 (1990) 468.
901
[4] J. Bloch, D.J. Bottomley, S. Janz, H.M. van Driel, R.S. Timsit, J. Chem. Phys. 98 (1993) 9167. [5] R.M. Corn, M. Romagnoli, M.D. Levenson, M.R. Philpott, Chem. Phys. Lett. 106 (1984) 30. [6] T. Rasing, T. Stehlin, Y.R. Shen, M.W. Kim, P.J. Valint, J. Chem. Phys. 89 (1988) 3386. [7] R.R. Naujok, D.A. Higgins, D.G. Hanken, R.M. Corn, J. Chem. Soc. Faraday Trans. 91 (1995) 2353. [8] A. Friedrich, B. Pettinger, D.M. Kolb, G. Luepke, R. Marowsky, G. Marowsky, Chem. Phys. Lett. 163 (1989) 123. [9] S. Janz, K. Pedersen, H.M. Van Driel, Phys. Rev. B: Condens. Matter 44 (1991) 3943. [10] H.W.K. Tom, X.D. Zhu, Y.R. Shen, G.A. Somorjai, Surf. Sci. 167 (1986) 167. [11] A. Priem, C.W. van Hasselt, M.A.C. Devillers, T.H. Rasing, Surf. Sci. 352±354 (1996) 612. [12] B. Pettinger, J. Lipkowski, S. Mirwald, A. Friedrich, Surf. Sci. 269±270 (1992) 377. [13] K.A. Friedrich, G.L. Richmond, Ber. Bunsenges. Phys. Chem. 97 (1993) 386. [14] R.M. Corn, D.A. Higgins, Chem. Rev. 94 (1994) 107. [15] B. Pettinger, J. Lipkowski, S. Mirwald, K.A. Friedrich, J. Electroanal. Chem. 329 (1992) 289. [16] B. Pettinger, J. Lipkowski, S. Mirwald, Electrochim. Acta 40 (1995) 133. [17] A. Liebsch, W.L. Schaich, Phys. Rev. B 40 (1989) 5401. [18] P. Guyot-Sionnest, A. Tadjeddine, A. Liebsch, Phys. Rev. Lett. 64 (1990) 522. [19] W. Schmickler, M. Urbakh, Phys. Rev. B 47 (1993) 6644. [20] N.D. Lang, W. Kohn, Phys. Rev. B 1 (1970) 4555. [21] E. Leiva, W. Schmickler, Surf. Sci. 291 (1993) 226. [22] A.V. Petukhov, Surf. Science 347 (1996) 143. [23] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Saunders College, Philadelphia, 1976, p. 290. [24] J.E. Sipe, D.J. Moss, H.M. van Driel, Phys. Rev. B 35 (1987) 1129. [25] D.J. Bottomley, G. LuÈpke, C. Meyer, Y. Makita, Opt. Lett. 20 (1994) 543. [26] B. Pettinger, C. Bilger, to be published. [27] B. Pettinger, C. Bilger, Chem. Phys. Lett. 286 (1998) 355.
Electrochimica Acta 44 (1998) 897±901
Interference second harmonic anisotropy at single crystalline silver-electrodes Bruno Pettinger a, *, Christoph Bilger a, Guillermo Beltramo b, Elisabeth Santos b, Wolfgang Schmickler b a
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4±6, D-14195 Berlin (Dahlem), Germany b Abteilung Elektrochemie, UniversitaÈt Ulm, D-89069 Ulm, Germany Received 5 November 1997
Abstract Using a new approach, the interference SHG anisotropy, it is possible to unambiguously evaluate the nonlinear susceptibility coecients, which describe the isotropic and anisotropic response of a metal electrode. We report on a ®rst application of this method to single crystalline silver electrodes in a 50 mM aqueous solution of NaClO4. The isotropic coecient a shows the strongest dependence on the electrode potential. Theory predicts that the real part ar=Re(a) should have a minimum near the potential of zero charge. A normalized plot of ar vs. the surface charge density s shows a good agreement between experiment and theory in the range ÿ20 mC cmÿ2
1. Introduction As is well known, the properties of the electric double layer at metal electrodes are not only controlled by the electrode potential, the solvent and adsorbed molecules but also by the crystallographic orientation of the substrate. Together they constitute the geometric and electronic structure of the interface. Optical second harmonic generation (SHG) is, due its intrinsic surface sensitivity, particularly attractive for in-situ investigations of the metal/electrolyte interface [1, 2]. Since the non-linear polarization (created by an intense laser pulse) has components parallel and perpendicular to the surface, it probes distinct properties of the interface [3±7]. The parallel components depend on the symmetry of the sample surface, whereas the perpendicular component is sensitive to the interfacial electric ®eld. Keeping the azimuthal
* Corresponding author.
angle ®xed and recording the SHG intensity as a function of the polar angle j, i.e. recording the SHG anisotropy, provides an easy way to monitor the rotational symmetry of the sample and its changes with potential and adsorption [2, 8±16].
2. Theory Second harmonic generation from metal surfaces has been the subject of fairly intensive theoretical research. In the early work, the rotational anisotropy was disregarded, and attention was focused on the rotationally averaged response. For this purpose, jellium proved to be a useful model for a metal surface; within its framework many experimental ®ndings can be explained at least qualitatively (see, e.g. Refs. [17±19]). For a structureless model surface, such as that of jellium, the SHG response is characterized by three coecients aj, bj, and dj ± the index j is usually not used in the literature, but is added here to distinguish them from the
0013-4686/98/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 4 6 8 6 ( 9 8 ) 0 0 1 9 2 - 3
898
B. Pettinger et al. / Electrochimica Acta 44 (1998) 897±901
phenomenological coecients introduced below ± which have the following meaning: aj characterizes the SHG response caused by currents driven in the z direction, perpendicular to the metal surface; bj speci®ed the contribution by currents induced parallel to the metal surface, and dj is due to the magnetic dipole moment from the bulk. Within the jellium model, bj= ÿ 1 and dj=1. The only parameter which depends on the electronic structure of the interphase is the coef®cient aj; in the jellium model it depends both on the electronic density of the bulk and on the charge density at the surface. The simple jellium model does not account for the crystal structure. This can be added by incorporating a lattice of pseudopotentials into the electron gas [20]; Leiva and Schmickler [21] used this model to calculate the coecient aj as a function of the surface charge density for the principal single crystal surfaces of several metals in the low frequency region. We will compare their calculations with our experimental data below. The rotational anisotropy of a Ag(111) surface was recently investigated by Petukhov [22]. According to his model the anisotropic SHG response in the low frequency region is caused by the necks of the Fermi surface near the L-points of the Brillouin zone [23]. This implies that the anisotropic amplitude should be practically independent of the surface charge, and hence also of the electric potential.
3. The principles of the interference technique for SHG Let us consider SHG only in the so-called electric dipole approximation, e.g. we neglect higher multipole contributions (later on we will see that these contributions ± giving rise, for instance, to a four-fold rotational symmetry ± are rather weak). Based on this simpli®cation, the nonlinear polarization is expressed by the dyadic product of a susceptibility tensor of second rank with the ®eld vector of the incident beam: (2) P(2) i (2o, nj)Awijk (nj):Ej(o)Ek(o). The susceptibility tensor exhibits an angular dependence. Thus, the nonlinear polarization, P(2) i , varies if the sample is rotated around its normal. The rotation is described by a tensor transformation, which relates the tensor elements of the sample, w(2) rst given in crystal coordinates, to (2) polar laboratory coordinates: w(2) ijk (nj) = RirRjsRktwrst , with n = 0±3, and Rir being the rotation matrix; j is the polar angle [24]. For a given polarization condition, such as pp, ps, sp or ss polarization, and for a rather low surface symmetry, such as Cs, the SHG anisotropy can be described by a superposition of up to four eective susceptibility terms, auv... duv, having a complex
character in general, Cjauv buv cos j a I SHG uv cuv cos2 j b duv cos3 j gj2 ,
1
where C is an appropriate proportionality constant. Note, that we have introduced here angular shift terms, a, 2b, 3g which are zero for v = p and p/2 for v = s (e.g. in the later case the cos(nj + p/2) 4 sin(nj)). Each of the four terms, {auv, buv, cuv, duv}, carries an index uv which indicates the polarization conditions and determines its composition by standard susceptibility elements (=(1/m)(w(2) rst2. . .)uv, m = 1, 2 or 4); the index uv also indicates which Fresnel factors are implicitly included [15, 24]. The ®rst term, denoted by the complex term auv, is associated with the isotropic susceptibility whereas the other three are connected with the anisotropic susceptibilities, buv, cuv and duv, multiplied by the appropriate trigonometric function. The latter three terms indicate the possible presence and superposition of one-, two- or threefold rotational symmetry elements of the surface. Since we are interested in the angular dependence, we will set C = 1. In addition, we will omit the uv indices of the four terms where appropriate. A general Cs symmetry can lead to a rather complex anisotropy pattern. However, for silver(111) where no surface reconstruction is known, b = c = 0, and the SHG anisotropy exhibits a typical C3v symmetry [2]. Au(111) p electrodes can exhibit surface reconstruction; the 3 23 reconstruction occurs in domains along the three [110] directions and their contributions nearly cancel. Thus, the b and c coecients are still small enough and, so, the coecients a and |d| can be evaluated together with the relative complex phase da±d. A correlation with the metallic charge density, sM, indicates a linear relationship between sM and the Re(a) term for the case of bromide adsorption whereas the strong variation of da±d pointed to a substantial change of the electronic surface levels upon adsorption. Evidently, SHG probes the interfacial symmetries (via the b, c, d terms) as well as the interfacial electric ®eld and electronic resonance conditions (via a and da±d) [12, 15, 16]. Since the SHG anisotropy curve is composed by superposition of a few harmonics (in general n R 6), a Fourier analysis appears to be, at ®rst glance, a straightforward approach to evaluate the coecients, a, b, c and d. Unfortunately, Eq. (1) contains an absolute square over a sum of susceptibility terms having complex character in general. Its Fourier analysis gives rise to mixed terms and, therefore, to a system of equations having multiple solutions. In general, this means distinct sets of coecients which reproduce the same anisotropy curve exist (this holds also if, in addition to the a±d-terms, an e-term, the neglected
B. Pettinger et al. / Electrochimica Acta 44 (1998) 897±901
bulk-contribution, would be signi®cant). This has been shown in Refs. [25, 26] and will not repeated here. Only in favorable cases, an analysis of Eq. (1) is possible. Usually the complete analysis of the SHG anisotropy is dicult, questionable or even impossible [26]. To overcome this problem, we have developed the so-called interference SHG anisotropy (ISHGA), using external SHG from a quartz lamella that interferes with the SHG from the sample [32]. The ISHGA measurement consists of recording three anisotropy curves subsequently, two of them with the quartz lamella present in the beam at two dierent positions, x1 and x2, and one without the quartz lamella. The intensity of the SHG from the quartz lamella is 2 Iquartz SHG =|pÄ| , with pÄ=|pÄ|exp(idp) being also a complex quantity. In the ®rst anisotropy measurement, with the quartz lamella at x1, there is an (unknown) phase relationship between the SHG from the quartz lamella and the SHG from the sample. In the second anisotropy measurement, we have moved the quartz lamella to the location x2 such that x2ÿx1 33.18 cm. In this way the phase relationship between the two SHG wave changes by p/2 compared to its value of the ®rst anisotropy experiment. This is due to the dierences in refractive indices of n(o) and n(2o) causing distinct optical path lengths for the fundamental and the SH wave (see Fig. 1). The root of Eq. (1) can be multiplied by an exponential, exp(id), with an arbitrary phase angle, d, because |exp(id)| = 1. This freedom can be exploited to make p=pÄ exp(ÿidp) a real quantity at x1 and an imaginary quantity at x2, because ip=pÄ exp(ÿidp+ p/2). Calculating the numerical dierence of the ISHG and SHG anisotropy yields two curves which can be easily analyzed due to their simple structure: 3 X DI ISHG x 1 ÿ p2 2 p Re wn n j an , pp
2
3 X DI ISHG x 2 ÿ p2 2 p Im wn n j an , pp
3
n0
899
Fig. 1. Scheme of the interference SHG anisotropy experiment. At the quartz lamella, located at x1 or x2, the ®rst SHG wave is created (dotted line); it travels together with the fundamental beam towards the sample where the second SHG wave is created (dashed line). Upon re¯ection at the sample the ®rst and the second SHG waves propagate on parallel routes and interfere at the detector.
4. Experimental The silver single crystals were ®rst mechanically polished, then repeatedly etched in NaCN + H2O2 solutions and ®nally exposed to a moderate ¯ame annealing. The ¯ame annealing was repeated prior to each experiment. The SHG setup is essentially the same as described earlier [12, 15, 16]. In short, a Nd:YAG laser provides pulses of about 8 ns width, a power of ca. 100 mJ at 1064 nm and ca. 10 Hz repetition rate. To perform the interference experiment, the cell with a window parallel to the sample surface has to be used. Sample and window are adjusted perpendicular to the rotational axis. The schematic arrangement is shown in Fig. 1 together with the movable quartz lamella. Because of the new con®guration, the angle of incidence at the electrode is ca. 338. Supra pure chemicals and triple distilled water have been used. The electrolyte is generally an aqueous solution of 0.1 M NaClO4.
n0
where p is the SHG intensity of the quartz lamella alone, and wn={a, b, c, d} denotes the complex susceptibility coecients. They can be separated into the complex isotropic susceptibility (n = 0: a) and the complex anisotropic susceptibilities (n = 1, 2, 3: b, c and d). Using this approach we obtain two experimental curves which have to be analyzed according the two expressions given above. They are linear in the coecients Re(wn) and Im(wn). From a Fourier analysis all the coecients can be unambiguously determined along with the oset angles an (the later can describe, in addition to the above given meaning, a possible angular oset of the eective polarization vector associated with the nth-fold rotational symmetry).
5. Results and discussion The analysis of the ISHGA curves yields unambiguous and reliable data of all anisotropy coecients. The behavior of the anisotropic terms (c, b, d) for Ag(111), Ag(110) and Ag(100) will be discussed elsewhere. Here we like to state that, for Ag(111), the three-fold susceptibility, dpp, is rather independent on potential, and the bpp and cpp terms are of the same height as the noise level, as expected for an unreconstructed surface. The same holds also for the hypothetical four-fold term, denoted as epp. Therefore, the potential dependence of the SHG response has to be attributed to the isotropic susceptibility, app. Fig. 1 illustrates this with
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Fig. 2. Potential dependence of the a-term. Top panel: |a| vs. E; bottom panel: complex phase da vs. potential. Electrolyte: 0.1 M NaClO4.
The relationship between the charge density and the electrode potential has been determined in independent measurements for Ag(111) and Ag(100) and is used in the following. Fig. 2 shows the comparison of the ISHGA experiment and the theoretical calculation of the a-term, using density functional theory and introducing the crystal structure via pseudopotentials [21] (unfortunately, there are no theoretical data available for Ag(110)). To establish a comparison, only the real part of the experimental a-term (for pp-polarization) could be used: ar=Re(a). In addition, it is given in a normalized fashion, as the ratio of [ar(s) ÿ ar(pzc)]/ ar(pzc) (Fig. 3). The comparison shows a fairly good agreement for low charge densities. For higher charge densities we expect a deviation between theory and experiment, because the model describing the electric double layer is then no longer adequate. This is also evident from the comparison. There are two reasons for these deviations: Firstly, at high negative charge densities the electrons spill out further into the solution. Therefore, the details of the interaction of the electrons with the solvent becomes important. These are not accounted for in the model. Secondly, at high positive charge densities an oxide layer starts to form on the surface, which is also not incorporated in the model.
a plot of three |app| curves vs. potential for Ag(111), Ag(110) and Ag(100). The curves show a minimum around the pzc as proposed by Schmickler et al. [21]. In the framework of the semiempirical model of Sipe et al., the isotropic amplitude is expected to have a dependence on the charge density: At the interface, the static electric ®elds exhibits an extremely large ®eld strength comparable with that of the electromagnetic ®eld of the incident laser beam. This can give rise to an non-linear process of third order to the second order polarizability, . P 2 w 2 :E o E o w 3 ..E o E o EH 2 w 2 w 2 red :E o E o weff :E o E o :
4
Since the static ®eld EH has components only in the direction perpendicular to the surface, the last expression can be reduced to a tensor product of second order, where some tensor elements in w(2) red carry a dependence on the static electric ®eld. This holds in particular for the app term, which is roughly proportional to w(2) e,zzz: sM app a0 a1 EH z ' 00a0 a1 , 5 eH e0 where we omit the pp indices on a0 and a1. At the point of zero charge we have |app(sM=0)|0 |a0|.
pzc Fig. 3. Normalized form of ar=Re(a), (arÿapzc vs. r )/ar charge density sM. Top panel: Ag(100), bottom panel: Ag(111). Electrolyte: 0.1 M NaClO4.
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6. Conclusion In this note we have reported on our ®rst application of the interference method to measuring the SHG anisotropy of silver electrodes. With this method all relevant susceptibility coecients can be accurately determined. Thus the evaluation of the data is no longer plagued by the occurrence of several solutions for the coecients, which makes the interpretation of the results so dicult in the conventional method. From an electrochemical point of view the most interesting parameter is the coecient a, which characterizes the response of the surface electrons in the direction perpendicular to the surface. This is the only coecient which changes drastically when the electrode potential is varied, or when particles are adsorbed on the surface. Our results for Ag(111) in a dilute solution of NaClO4 agree quite well with the predictions of the calculations by Leiva and Schmickler in the range of low surface charge densities. Acknowledgements The authors thank Professor G. Ertl for continuing support, various discussions and stimulating interest in these studies. This work was partly supported by the Deutsche Forschungsgemeinschaft. References [1] Y.R. Shen, The principles of nonlinear optics, Wiley, New York, 1984. [2] G.L. Richmond, in: H. Gerischer, C.W. Tobias (Eds.), Adv. Electrochem. Sci. Eng., vol. 2, VCH Publishers, Weinheim, 1992. [3] R.A. Bradey, S. Arekat, R. Georgiadis, J.M. Robinson, S.D. Kevan, G.L. Richmond, Chem. Phys. Lett. 168 (1990) 468.
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[4] J. Bloch, D.J. Bottomley, S. Janz, H.M. van Driel, R.S. Timsit, J. Chem. Phys. 98 (1993) 9167. [5] R.M. Corn, M. Romagnoli, M.D. Levenson, M.R. Philpott, Chem. Phys. Lett. 106 (1984) 30. [6] T. Rasing, T. Stehlin, Y.R. Shen, M.W. Kim, P.J. Valint, J. Chem. Phys. 89 (1988) 3386. [7] R.R. Naujok, D.A. Higgins, D.G. Hanken, R.M. Corn, J. Chem. Soc. Faraday Trans. 91 (1995) 2353. [8] A. Friedrich, B. Pettinger, D.M. Kolb, G. Luepke, R. Marowsky, G. Marowsky, Chem. Phys. Lett. 163 (1989) 123. [9] S. Janz, K. Pedersen, H.M. Van Driel, Phys. Rev. B: Condens. Matter 44 (1991) 3943. [10] H.W.K. Tom, X.D. Zhu, Y.R. Shen, G.A. Somorjai, Surf. Sci. 167 (1986) 167. [11] A. Priem, C.W. van Hasselt, M.A.C. Devillers, T.H. Rasing, Surf. Sci. 352±354 (1996) 612. [12] B. Pettinger, J. Lipkowski, S. Mirwald, A. Friedrich, Surf. Sci. 269±270 (1992) 377. [13] K.A. Friedrich, G.L. Richmond, Ber. Bunsenges. Phys. Chem. 97 (1993) 386. [14] R.M. Corn, D.A. Higgins, Chem. Rev. 94 (1994) 107. [15] B. Pettinger, J. Lipkowski, S. Mirwald, K.A. Friedrich, J. Electroanal. Chem. 329 (1992) 289. [16] B. Pettinger, J. Lipkowski, S. Mirwald, Electrochim. Acta 40 (1995) 133. [17] A. Liebsch, W.L. Schaich, Phys. Rev. B 40 (1989) 5401. [18] P. Guyot-Sionnest, A. Tadjeddine, A. Liebsch, Phys. Rev. Lett. 64 (1990) 522. [19] W. Schmickler, M. Urbakh, Phys. Rev. B 47 (1993) 6644. [20] N.D. Lang, W. Kohn, Phys. Rev. B 1 (1970) 4555. [21] E. Leiva, W. Schmickler, Surf. Sci. 291 (1993) 226. [22] A.V. Petukhov, Surf. Science 347 (1996) 143. [23] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Saunders College, Philadelphia, 1976, p. 290. [24] J.E. Sipe, D.J. Moss, H.M. van Driel, Phys. Rev. B 35 (1987) 1129. [25] D.J. Bottomley, G. LuÈpke, C. Meyer, Y. Makita, Opt. Lett. 20 (1994) 543. [26] B. Pettinger, C. Bilger, to be published. [27] B. Pettinger, C. Bilger, Chem. Phys. Lett. 286 (1998) 355.