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A novel approach to analyze the optical second harmonic generation anisotropy at surfaces employing interference techniques. Example: the Au(110) electrode
10 April 1998
Chemical Physics Letters 286 Ž1998. 355–360
A novel approach to analyze the optical second harmonic generation anisotropy at surfaces employing interference techniques. Example: the Au ž110 / electrode Bruno Pettinger, Christoph Bilger Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4–6, D-14195 Berlin (Dahlem), Germany Received 3 November 1997; in final form 15 December 1997
Abstract Different sets of susceptibility coefficients a, b, c and d can reproduce the same second harmonic generation ŽSHG. anisotropy curve, but only one set corresponds to the electronic and geometric structure of the sample surface. A new approach, interference SHG anisotropy, is presented which is generally suitable to determine this physically relevant set of coefficients. It is applied to the study of the SHG anisotropy of the AuŽ110. electrode, which could not be analyzed hitherto. The distinct SHG response upon Bry adsorption indicates corresponding changes of the structural and electronic state of this interface. q 1998 Elsevier Science B.V.
1. Introduction Optical second harmonic generation ŽSHG. at metal surfaces has received much attention due to its intrinsic surface sensitivity gained by the absence of centrosymmetry at interfaces w1,2x. It has proved to be a versatile probe for quite different interfaces such as the solidrvacuum w3,4x, the solidrelectrolyte w5x or the solutionrsolution w6,7x phase boundaries. A rotation of the sample around its normal gives rise to SHG anisotropy observed for crystalline metals w2,8,9x, semiconductors and insulators w10,11x. Its pattern and variation indicate which rotational symmetry elements are present in the sample surface and how they change upon adsorption and – in the case of electrodes – upon variation of the potential w2,12– 16x. Considering SHG in the so-called electric dipole
approximation, we will neglect higher multipole contributions, then the nonlinear polarization is described by the dyadic product of a susceptibility tensor of second rank with the field vector of the Ž . Ž . Ž . incident beam: PiŽ2. Ž2 v . A x iŽ2. j k n w : E j v Ek v . Evidently, the SH response depends on the orientation of the electromagnetic field vector relative to the spatial orientation of the crystalline sample. This can be quantified by a tensor transformation, which relates the tensor elements of the sample, xrŽ2.s t given in crystal coordinates, to laboratory coordinates. The latter then depends on the azimuthal angle w : x iŽ2.j k Ž n w . s R i r R j s R k t xrŽ2.s t , with n s 0–3 and R i r the rotation matrix w17x. For a given polarization condition, such as pp, ps, sp or ss polarization and for a rather low surface symmetry, such as C s , the SHG anisotropy can be described by a superposition of up to four effective
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 Ž 9 7 . 0 1 4 7 1 - 1
B. Pettinger, C. Bilgerr Chemical Physics Letters 286 (1998) 355–360
356
susceptibility terms, j n,u Õ , with a complex character in general: 3
IuSHG Õ sC
Ý j n ,u Õ P f Õ Ž n w .
2
;
u,Õ s p, s
Ž 1.
ns0
where C is an appropriate proportionality constant. Note, f Õsp Ž n w . s cosŽ n w . and f Õss Ž n w . s sin Ž n w .. Each of the four terms j n , u Õ s a u Õ ,bu Õ ,c u Õ ,d u Õ 4 , carries indices n and uÕ. They refer to a specific symmetry element of the sample and to the polarization conditions. Therefore, they determine which group of susceptibility elements Ž j n,u Õ s m1 Ž xrŽ2.s t q . . . . n,u Õ , m s 1,2 or 4. each term represents and which Fresnel factors have to be included w15,17x. The first term Ž n s 0. is associated with the isotropic susceptibility, denoted by the complex term a u Õ , whereas the other three are connected with the anisotropic susceptibilities, bu Õ ,c u Õ and d u Õ , multiplied by the appropriate trigonometric function. The latter three terms indicate the possible presence and superposition of one-, two- or threefold rotational symmetry elements in the surface. Since we are interested in the angular dependence, we will set C s 1. In addition, we will omit the uÕ indices of the four terms where appropriate. In the past, various attempts were made to correlate the SHG anisotropy patterns with superposition of surface rotational symmetries. Based on Eq. Ž1. such an analysis seems to be straightforward Žsee next paragraph.. There is an early study of the symmetry and disordering of a reconstructed SiŽ111.-Ž7 = 7. interface by Tom et al. w10x. In cases, such as silverŽ111., where surface reconstruction is absent Ži.e. b s c s 0., a C3 Õ symmetry is evident w2x. If surface reconstruction is present, as in the case of AuŽ111. electrodes, but the b and c coefficients are still small enough, the coefficients a and < d < can be determined together with the relative complex phase d ay d . Remarkable changes of d ayd upon adsorption of pyridine and pyridine derivatives have been associated with changes in the adsorption geometries of these molecules. A correlation with the metallic charge density sM , indicated a linear relationship between sM and the ReŽ a. term in some cases. Thus SHG is able, not only to probe the interfacial symmetries Žvia the b,c,d terms., but also the interfacial electric field and electronic resonance conditions Žvia a and d ay d . w12,15,16x.
2. The interference SHG anisotropy According to Eq. Ž1. a SHG anisotropy curve is composed of superpositions of a few harmonics Žin general n F 6.. At first glance, Fourier analysis seems to be a suitable approach to evaluate the coefficients a, b, c and d. Unfortunately, Eq. Ž1. contains an absolute square over a sum of susceptibility terms with in general a complex character. This gives rise to mixed terms and therefore to a system of equations with multiple solutions and, thus in general, yielding several, distinct sets of coefficients. They reproduce the same anisotropy curve Žthis holds also if, in addition to the a–d-terms, an e-term, the neglected bulk-contribution, is significant.. The mathematical proof will be presented in Ref. w19x. Here, Table 1 illustrates this by four evaluated sets of coefficients with the left column corresponding to the initial set Žfor a Ž111. surface, therefore only a and d are non-zero. and with the last row giving the standard deviation s, which is lower the more accurate the computation. Thus, among the four sets of coefficients only one describes the physical situation of the sample, but which one that is, is unknown in advance. This result forced us to modify the SHG anisotropy experiment and the corresponding analysis, in order to extract all the coefficients unambiguously. The new approach we will present can be extended easily to the Žneglected. contributions of higher multipoles as well as to other optical spectroscopies. Note, the coefficients, j n,u Õ , may contain surface and bulk contributions. Their separation is possible, using the Table 1 Set of coefficients Ž0–3. reproducing the same anisotropy; in this simulation set 0 is initially given but represents also one of the four solutions. The last row shows the standard deviation s between the initial and the computed curves A
Initial
1
2
3
ar ai br bi cr ci dr s
20 25 0.1 0 0 0 25 y
25.11800 y8.51452 2.47351 y22.55490 0 11.34150 25 0.000119
4.48527 y9.71725 y9.56020 35.30280 0 21.97950 25 0.00188
11.50330 3.44441 y22.10950 12.74880 0 33.32380 25 0.000101
B. Pettinger, C. Bilgerr Chemical Physics Letters 286 (1998) 355–360
approach of Bottomley et al. w18x. However, to begin with a technique, such as our approach, is needed to uniquely evaluate these coefficients. The interference between two SHG sources is a well known method of extracting phase information Žsee for instance, Chang et al., Ref. w20x; Shen, Ref. w1x, pp. 98–102; Schwarzberg et al., Ref. w21x.. We describe here the use of this technique in a novel way: let us consider the interference of a constant SHG signal, created by a quartz lamella at two different location x 1 or x 2 , with the variable SHG response of the sample during an azimuthal angle scan and considering pp polarization. The SHG anisotropy and the interference SHG ŽISHG. anisotropy Žsee Fig. 1. are described by Eqs. Ž2. and Ž3.: 2
3
IpSHG p s
Ý j n , p p cos Ž n w .
Ž 2.
ns0 3
IpISHG p
Ž x . s pŽ x . q
Ý j n , p p cos Ž n w .
2
Ž 3.
ns0
The SH field of the quartz lamella enters the latter Eq. Ž3. via the term pŽ x .. In the above equations the phase is arbitrary. Therefore, we define the relative phase relationship of the two SH waves such that pi s 0 and pr s < pŽ x 1 .< if the quartz lamella is located at x 1. Moving the quartz lamella to a specific location x 2 , this phase relationship changes by pr2; thus, pr s 0 and pi s < pŽ x 2 .< Žnote that < pŽ x 1 .< s < pŽ x 2 .<.. Hence, our approach requires the recording of three anisotropy curves, one without the quartz lamella and two with the quartz lamella at locations x 1 and x 2 . The next step is to compute the difference between Eq. Ž3. and Ž2. and to subtract the SHG intensity of the quartz lamella alone, < pŽ x .< 2 : Ž k . y IpSHG < <2 D Ip p Ž x . s I pISHG p p y p . We obtain in this way two anisotropy difference curves:
357
trigonometric terms. Thus, a Fourier analysis yields the full set of these coefficients unambiguously. In addition, but not shown in this Letter, this analysis can also provide angular shift terms a n , hitherto inaccessible Ži.e. w ™ w q a n .. These terms have a non-trivial physical meaning which will be discussed in Ref. w19x.
3. Experimental The experimental realization of the method is sketched in Fig. 1. It shows a quartz lamella which can be placed into the fundamental beam Ž1064 nm, 100 mJ, 8 ns pulse width and 10 Hz repetition rate. at two different positions, either at x 1 or x 2 , or not at all. The SH wave generated in the quartz lamella propagates with the fundamental beam towards the sample, but with distinct optical path lengths compared to the fundamental beam because of distinct dispersion in air ŽShen, Ref. w1x, pp. 98–102.. Consequently, for a suitable length difference, x 2 y x 1 Ž( 3.18 cm., the relative phase shift amounts to the required pr2 shift. The parallel configuration of window and sample surface shown in Fig. 1 is essential for the interference, mainly compensating the differences in the angles of refraction and emission due to differences in e Ž2 v . and e Ž v . of the electrolyte. The fundamental beam creates the second SHG-beam upon reflection at the sample sur-
3
D I p p Ž pr . s 2
Ý
pr Re Ž j n , p p . cos Ž n w .
Ž 4.
pi Im Ž j n , p p . cos Ž n w .
Ž 5.
ns0 3
D I p p Ž pi . s 2
Ý ns0
thereby separating the contributions of the real and imaginary parts of the coefficients, respectively. Their angular dependence is a linear superposition of
Fig. 1. Interference SHG anisotropy experiment. The first SHG wave is created by the quartz lamella Ždotted line. located at x 1 or x 2 ; it propagates together with the fundamental beam towards the sample where the second SHG wave is created Ždashed line.. Upon reflection at the sample both SHG waves travel on parallel routes and interfere at the detector.
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B. Pettinger, C. Bilgerr Chemical Physics Letters 286 (1998) 355–360
face. The detector then records the SHGŽ-interference. intensity during the angular scan. Each series of measurements started with flame treatment to clean and anneal the surface. The electrode potential was controlled ŽU s y0.6 V. as the electrolyte Ž0.1 M NaClO4 . filled the cell. The measurements consisted of a series of cycles composed of Ži. a potential step of 100 mV from the negative startpoint towards the positive end; Žii. a subsequent waiting time of 1 min to allow a sort of stationary state of the sample surface to be reached; Žiii. the three azimuthal scans. These cycles were repeated until the positive potential limit was reached. Finally, the potential was set back to the negative startpoint. The same sequence was repeated by adding a sufficient amount of bromide ions to the solution.
Fig. 2. Variation of the isotropic susceptibility a p p , with potential. Top panel: < a p p < vs. U; bottom panel: complex phase d a vs. U Žgiven vs. SCE.. Both panels: solid line and v for 0.1 M NaClO4 ; curve with ' for 0.1 M NaClO4 q 0.001 M NaBr, respectively.
4. Results Contrary to the case of AgŽ110. w2x, the SHG anisotropy at the AuŽ110. electrodes does not show the expected twofold rotational symmetry. It exhibits a rather complex rotational anisotropy with a strong threefold component varying substantially with electrode potential and upon adsorption of ions or organic molecules w15x. Up to now, the complexity of this pattern prohibited the analysis of the SHG data in terms of the a, b, c, d coefficients and their relative complex phases. Only the absolute quantity 1 < < 2 could be determined with confidence via 2 d Fourier analysis. It was considered as a measure of the AuŽ110. microfaceting w16x, also observed by Gao et al. in STM investigations w22x. Employing the interference SHG anisotropy approach, a complete analysis of the anisotropy pattern will be illustrated for the case of an AuŽ110. electrode inserted into two different electrolytes, either into pure 0.1 M NaClO4 or into 0.1 M NaClO4 q 0.001 M NaBr. In order to associate the potential dependence with essentially a single element of the susceptibility . tensor Ž x iŽ2. jk , we used either the pp or ps polarization condition Žsee Table 2 in Ref. w15x.. For pp polarization essentially a p p ŽU . A x z z z ŽU ., while for ps polarization a p s should be zero Žand is indeed weak., but both c p s A x x x z ŽU . and d p s A x x x x ŽU . are comparatively large. Figs. 2–4 show the potential variation of the < a <, < c < and < d < terms in the top panels
Fig. 3. The anisotropic susceptibility c p s , vs. potential. Top panel: < c p s < vs. U; bottom panel: dc vs. U Žgiven vs. SCE.. Curves Ž v, '. for the same electrolytes as in Fig. 2.
Fig. 4. The anisotropic susceptibility d p s , vs. potential. Top panel: < d p s < vs. U; bottom panel: d d vs. U Žgiven vs. SCE.. Curves Ž v, '. for the same electrolytes as in Fig. 2.
B. Pettinger, C. Bilgerr Chemical Physics Letters 286 (1998) 355–360
and the corresponding changes of the complex phase d a , dc and d d with the potential Žwhich are defined in the form a s < a < expw i d a x etc. in the bottom panels, respectively. Each panel presents two curves for each of the above mentioned electrolytes. Note that in earlier publications only the internal relative phase, d s d a y d d , was given w15x. Here, the phase has to be understood as relative to the constant phase of the quartz lamella at x 1. In the case of pure 0.1 M NaClO4 solution, the < a < term rises monotonically. Obviously, it exhibits no minimum at the point of zero charge Žpzc. at y0.015 V for AuŽ110. w23x. Contrary to that, in the cases of AgŽ111., AgŽ110. and AgŽ100. such a minimum of the < a < term has been proposed theoretically w24x and has recently been observed in ISHG experiments w5,25x, substantiating its earlier observation on polycrystalline Ag w5x. For bromide ion adsorption at AuŽ110. Fig. 2 shows another, but consistent picture: the rise of < a < is steeper at the beginning than seen for the base electrolyte, the < a < term reaches a maximum at potentials which correspond roughly to the maximum of the CV curve associated with the adsorption of this anion. The decrease of < a < at more positive potential is not due to changes in the amount of adsorption; it must be caused by a change of the SHG resonance and of partial charge transfer. The complex phases in Fig. 2 show essentially the same trend, apart from the fact that there is no clear decrease at more positive potentials. Fig. 3 reproduces the potential dependence of the c term. Its rise points to a growth of the twofold rotational symmetry elements in the surface. In the case of pure sodium perchlorate solution, the phase dc is constant and < c < grows monotonically, while for bromide adsorption the < c < quantity varies in a rather complex fashion with potential. A more stringent picture is evident from the behavior of the complex phase dc : it shows a little variation in the cathodic potential regime and a significant drop by about 708 for Bry adsorption. The d term variation in Fig. 4 illustrates the change of the threefold symmetry elements in the surface. For < d < it confirms the AuŽ110. microfaceting reported earlier w16x. The complex phase d d , varies in a similar way to dc indicating changes in the optical resonance of the states which contribute to the anisotropic SHG response.
359
5. Conclusion In this Letter we have presented a novel experimental approach to determine the complex susceptibility coefficients a, b, c and d, constituting the SHG anisotropy and providing structural and electronic information about the interface. It is also applicable to cases which were impossible to analyze up to now, such as the microfaceted AuŽ110.. However, a full understanding of our observations, which requires a theoretical treatment of these processes, is emerging for AgŽhkl. w24x, but is still missing for AuŽhkl. interfaces. 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 ŽPe 282r4-2.. References w1x Y.R. Shen, The principles of nonlinear optics, Wiley, New York, 1984, Ch. 25. w2x G.L. Richmond, in: H. Gerischer, C.W. Tobias ŽEds.., Adv. Electrochem. Sci. Eng., vol. 2, VCH Publishers, Weinheim, 1992. w3x R.A. Bradley, S. Arekat, R. Georgiadis, J.M. Robinson, S.D. Kevan, G.L. Richmond, Chem. Phys. Lett. 168 Ž1990. 468. w4x J. Bloch, D.J. Bottomley, S. Janz, H.M. van Driel, R.S. Timsit, J. Chem. Phys. 98 Ž1993. 9167. w5x R.M. Corn, M. Romagnoli, M.D. Levenson, M.R. Philpott, Chem. Phys. Lett. 106 Ž1984. 30. w6x T. Rasing, T. Stehlin, Y.R. Shen, M.W. Kim, P.J. Valint, J. Chem. Phys. 89 Ž1988. 3386. w7x R.R. Naujok, D.A. Higgins, D.G. Hanken, R.M. Corn, J. Chem. Soc. Faraday Trans. 91 Ž1995. 2353. w8x A. Friedrich, B. Pettinger, D.M. Kolb, G. Luepke, R. Steinhoff, G. Marowsky, Chem. Phys. Lett. 163 Ž1989. 123. w9x S. Janz, K. Pedersen, H.M. Van Driel, Phys. Rev. B: Condens. Matter 44 Ž1991. 3943. w10x H.W.K. Tom, X.D. Zhu, Y.R. Shen, G.A. Somorjai, Surf. Sci. 167 Ž1986. 167. w11x A. Priem, C.W. van Hasselt, M.A.C. Devillers, T.H. Rasing, Surf. Sci. 352–354 Ž1996. 612. w12x B. Pettinger, J. Lipkowski, S. Mirwald, A. Friedrich, Surf. Sci. 269–270 Ž1992. 377. w13x K.A. Friedrich, G.L. Richmond, Ber. Bunsenges. Phys. Chem. 97 Ž1993. 386. w14x R.M. Corn, D.A. Higgins, Chem. Rev. Washington D.C. 94 Ž1994. 107.
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w15x B. Pettinger, J. Lipkowski, S. Mirwald, K.A. Friedrich, J. Electroanal. Chem. 329 Ž1992. 289. w16x B. Pettinger, J. Lipkowski, S. Mirwald, Electrochim. Acta 40 Ž1995. 133. w17x J.E. Sipe, D.J. Moss, H.M. van Driel, Phys. Rev. B 35 Ž1987. 1129. w18x D.J. Bottomley, G. Lupke, C. Meyer, Y. Makita, Optics Lett. ¨ 20 Ž1994. 543. w19x B. Pettinger, C. Bilger, to be published. w20x R.K. Chang, J. Ducuing, N. Bloembergen, Phys. Rev. Lett. 15 Ž1965. 6.
w21x E. Schwarzberg, G. Berkovic, G. Marowsky, Appl. Phys. A 59 Ž1994. 631. w22x X. Gao, A. Hamelin, M.J. Weaver, Phys. Rev. B 44 Ž1991. 10983. w23x L. Stolberg, J. Lipkowski, D.E. Irish, J. Electroanal. Chem. 296 Ž1990. 171. w24x W. Schmickler, M. Urbakh, Phys. Rev. B. 47 Ž1993. 6644. w25x C. Bilger, B. Pettinger, G. Beltramo, E. Santos, W. Schmickler, Interference Second Harmonic Anisotropy at Single Crystalline Silver-Electrodes, to be published.
Chemical Physics Letters 286 Ž1998. 355–360
A novel approach to analyze the optical second harmonic generation anisotropy at surfaces employing interference techniques. Example: the Au ž110 / electrode Bruno Pettinger, Christoph Bilger Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4–6, D-14195 Berlin (Dahlem), Germany Received 3 November 1997; in final form 15 December 1997
Abstract Different sets of susceptibility coefficients a, b, c and d can reproduce the same second harmonic generation ŽSHG. anisotropy curve, but only one set corresponds to the electronic and geometric structure of the sample surface. A new approach, interference SHG anisotropy, is presented which is generally suitable to determine this physically relevant set of coefficients. It is applied to the study of the SHG anisotropy of the AuŽ110. electrode, which could not be analyzed hitherto. The distinct SHG response upon Bry adsorption indicates corresponding changes of the structural and electronic state of this interface. q 1998 Elsevier Science B.V.
1. Introduction Optical second harmonic generation ŽSHG. at metal surfaces has received much attention due to its intrinsic surface sensitivity gained by the absence of centrosymmetry at interfaces w1,2x. It has proved to be a versatile probe for quite different interfaces such as the solidrvacuum w3,4x, the solidrelectrolyte w5x or the solutionrsolution w6,7x phase boundaries. A rotation of the sample around its normal gives rise to SHG anisotropy observed for crystalline metals w2,8,9x, semiconductors and insulators w10,11x. Its pattern and variation indicate which rotational symmetry elements are present in the sample surface and how they change upon adsorption and – in the case of electrodes – upon variation of the potential w2,12– 16x. Considering SHG in the so-called electric dipole
approximation, we will neglect higher multipole contributions, then the nonlinear polarization is described by the dyadic product of a susceptibility tensor of second rank with the field vector of the Ž . Ž . Ž . incident beam: PiŽ2. Ž2 v . A x iŽ2. j k n w : E j v Ek v . Evidently, the SH response depends on the orientation of the electromagnetic field vector relative to the spatial orientation of the crystalline sample. This can be quantified by a tensor transformation, which relates the tensor elements of the sample, xrŽ2.s t given in crystal coordinates, to laboratory coordinates. The latter then depends on the azimuthal angle w : x iŽ2.j k Ž n w . s R i r R j s R k t xrŽ2.s t , with n s 0–3 and R i r the rotation matrix w17x. For a given polarization condition, such as pp, ps, sp or ss polarization and for a rather low surface symmetry, such as C s , the SHG anisotropy can be described by a superposition of up to four effective
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 Ž 9 7 . 0 1 4 7 1 - 1
B. Pettinger, C. Bilgerr Chemical Physics Letters 286 (1998) 355–360
356
susceptibility terms, j n,u Õ , with a complex character in general: 3
IuSHG Õ sC
Ý j n ,u Õ P f Õ Ž n w .
2
;
u,Õ s p, s
Ž 1.
ns0
where C is an appropriate proportionality constant. Note, f Õsp Ž n w . s cosŽ n w . and f Õss Ž n w . s sin Ž n w .. Each of the four terms j n , u Õ s a u Õ ,bu Õ ,c u Õ ,d u Õ 4 , carries indices n and uÕ. They refer to a specific symmetry element of the sample and to the polarization conditions. Therefore, they determine which group of susceptibility elements Ž j n,u Õ s m1 Ž xrŽ2.s t q . . . . n,u Õ , m s 1,2 or 4. each term represents and which Fresnel factors have to be included w15,17x. The first term Ž n s 0. is associated with the isotropic susceptibility, denoted by the complex term a u Õ , whereas the other three are connected with the anisotropic susceptibilities, bu Õ ,c u Õ and d u Õ , multiplied by the appropriate trigonometric function. The latter three terms indicate the possible presence and superposition of one-, two- or threefold rotational symmetry elements in the surface. Since we are interested in the angular dependence, we will set C s 1. In addition, we will omit the uÕ indices of the four terms where appropriate. In the past, various attempts were made to correlate the SHG anisotropy patterns with superposition of surface rotational symmetries. Based on Eq. Ž1. such an analysis seems to be straightforward Žsee next paragraph.. There is an early study of the symmetry and disordering of a reconstructed SiŽ111.-Ž7 = 7. interface by Tom et al. w10x. In cases, such as silverŽ111., where surface reconstruction is absent Ži.e. b s c s 0., a C3 Õ symmetry is evident w2x. If surface reconstruction is present, as in the case of AuŽ111. electrodes, but the b and c coefficients are still small enough, the coefficients a and < d < can be determined together with the relative complex phase d ay d . Remarkable changes of d ayd upon adsorption of pyridine and pyridine derivatives have been associated with changes in the adsorption geometries of these molecules. A correlation with the metallic charge density sM , indicated a linear relationship between sM and the ReŽ a. term in some cases. Thus SHG is able, not only to probe the interfacial symmetries Žvia the b,c,d terms., but also the interfacial electric field and electronic resonance conditions Žvia a and d ay d . w12,15,16x.
2. The interference SHG anisotropy According to Eq. Ž1. a SHG anisotropy curve is composed of superpositions of a few harmonics Žin general n F 6.. At first glance, Fourier analysis seems to be a suitable approach to evaluate the coefficients a, b, c and d. Unfortunately, Eq. Ž1. contains an absolute square over a sum of susceptibility terms with in general a complex character. This gives rise to mixed terms and therefore to a system of equations with multiple solutions and, thus in general, yielding several, distinct sets of coefficients. They reproduce the same anisotropy curve Žthis holds also if, in addition to the a–d-terms, an e-term, the neglected bulk-contribution, is significant.. The mathematical proof will be presented in Ref. w19x. Here, Table 1 illustrates this by four evaluated sets of coefficients with the left column corresponding to the initial set Žfor a Ž111. surface, therefore only a and d are non-zero. and with the last row giving the standard deviation s, which is lower the more accurate the computation. Thus, among the four sets of coefficients only one describes the physical situation of the sample, but which one that is, is unknown in advance. This result forced us to modify the SHG anisotropy experiment and the corresponding analysis, in order to extract all the coefficients unambiguously. The new approach we will present can be extended easily to the Žneglected. contributions of higher multipoles as well as to other optical spectroscopies. Note, the coefficients, j n,u Õ , may contain surface and bulk contributions. Their separation is possible, using the Table 1 Set of coefficients Ž0–3. reproducing the same anisotropy; in this simulation set 0 is initially given but represents also one of the four solutions. The last row shows the standard deviation s between the initial and the computed curves A
Initial
1
2
3
ar ai br bi cr ci dr s
20 25 0.1 0 0 0 25 y
25.11800 y8.51452 2.47351 y22.55490 0 11.34150 25 0.000119
4.48527 y9.71725 y9.56020 35.30280 0 21.97950 25 0.00188
11.50330 3.44441 y22.10950 12.74880 0 33.32380 25 0.000101
B. Pettinger, C. Bilgerr Chemical Physics Letters 286 (1998) 355–360
approach of Bottomley et al. w18x. However, to begin with a technique, such as our approach, is needed to uniquely evaluate these coefficients. The interference between two SHG sources is a well known method of extracting phase information Žsee for instance, Chang et al., Ref. w20x; Shen, Ref. w1x, pp. 98–102; Schwarzberg et al., Ref. w21x.. We describe here the use of this technique in a novel way: let us consider the interference of a constant SHG signal, created by a quartz lamella at two different location x 1 or x 2 , with the variable SHG response of the sample during an azimuthal angle scan and considering pp polarization. The SHG anisotropy and the interference SHG ŽISHG. anisotropy Žsee Fig. 1. are described by Eqs. Ž2. and Ž3.: 2
3
IpSHG p s
Ý j n , p p cos Ž n w .
Ž 2.
ns0 3
IpISHG p
Ž x . s pŽ x . q
Ý j n , p p cos Ž n w .
2
Ž 3.
ns0
The SH field of the quartz lamella enters the latter Eq. Ž3. via the term pŽ x .. In the above equations the phase is arbitrary. Therefore, we define the relative phase relationship of the two SH waves such that pi s 0 and pr s < pŽ x 1 .< if the quartz lamella is located at x 1. Moving the quartz lamella to a specific location x 2 , this phase relationship changes by pr2; thus, pr s 0 and pi s < pŽ x 2 .< Žnote that < pŽ x 1 .< s < pŽ x 2 .<.. Hence, our approach requires the recording of three anisotropy curves, one without the quartz lamella and two with the quartz lamella at locations x 1 and x 2 . The next step is to compute the difference between Eq. Ž3. and Ž2. and to subtract the SHG intensity of the quartz lamella alone, < pŽ x .< 2 : Ž k . y IpSHG < <2 D Ip p Ž x . s I pISHG p p y p . We obtain in this way two anisotropy difference curves:
357
trigonometric terms. Thus, a Fourier analysis yields the full set of these coefficients unambiguously. In addition, but not shown in this Letter, this analysis can also provide angular shift terms a n , hitherto inaccessible Ži.e. w ™ w q a n .. These terms have a non-trivial physical meaning which will be discussed in Ref. w19x.
3. Experimental The experimental realization of the method is sketched in Fig. 1. It shows a quartz lamella which can be placed into the fundamental beam Ž1064 nm, 100 mJ, 8 ns pulse width and 10 Hz repetition rate. at two different positions, either at x 1 or x 2 , or not at all. The SH wave generated in the quartz lamella propagates with the fundamental beam towards the sample, but with distinct optical path lengths compared to the fundamental beam because of distinct dispersion in air ŽShen, Ref. w1x, pp. 98–102.. Consequently, for a suitable length difference, x 2 y x 1 Ž( 3.18 cm., the relative phase shift amounts to the required pr2 shift. The parallel configuration of window and sample surface shown in Fig. 1 is essential for the interference, mainly compensating the differences in the angles of refraction and emission due to differences in e Ž2 v . and e Ž v . of the electrolyte. The fundamental beam creates the second SHG-beam upon reflection at the sample sur-
3
D I p p Ž pr . s 2
Ý
pr Re Ž j n , p p . cos Ž n w .
Ž 4.
pi Im Ž j n , p p . cos Ž n w .
Ž 5.
ns0 3
D I p p Ž pi . s 2
Ý ns0
thereby separating the contributions of the real and imaginary parts of the coefficients, respectively. Their angular dependence is a linear superposition of
Fig. 1. Interference SHG anisotropy experiment. The first SHG wave is created by the quartz lamella Ždotted line. located at x 1 or x 2 ; it propagates together with the fundamental beam towards the sample where the second SHG wave is created Ždashed line.. Upon reflection at the sample both SHG waves travel on parallel routes and interfere at the detector.
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face. The detector then records the SHGŽ-interference. intensity during the angular scan. Each series of measurements started with flame treatment to clean and anneal the surface. The electrode potential was controlled ŽU s y0.6 V. as the electrolyte Ž0.1 M NaClO4 . filled the cell. The measurements consisted of a series of cycles composed of Ži. a potential step of 100 mV from the negative startpoint towards the positive end; Žii. a subsequent waiting time of 1 min to allow a sort of stationary state of the sample surface to be reached; Žiii. the three azimuthal scans. These cycles were repeated until the positive potential limit was reached. Finally, the potential was set back to the negative startpoint. The same sequence was repeated by adding a sufficient amount of bromide ions to the solution.
Fig. 2. Variation of the isotropic susceptibility a p p , with potential. Top panel: < a p p < vs. U; bottom panel: complex phase d a vs. U Žgiven vs. SCE.. Both panels: solid line and v for 0.1 M NaClO4 ; curve with ' for 0.1 M NaClO4 q 0.001 M NaBr, respectively.
4. Results Contrary to the case of AgŽ110. w2x, the SHG anisotropy at the AuŽ110. electrodes does not show the expected twofold rotational symmetry. It exhibits a rather complex rotational anisotropy with a strong threefold component varying substantially with electrode potential and upon adsorption of ions or organic molecules w15x. Up to now, the complexity of this pattern prohibited the analysis of the SHG data in terms of the a, b, c, d coefficients and their relative complex phases. Only the absolute quantity 1 < < 2 could be determined with confidence via 2 d Fourier analysis. It was considered as a measure of the AuŽ110. microfaceting w16x, also observed by Gao et al. in STM investigations w22x. Employing the interference SHG anisotropy approach, a complete analysis of the anisotropy pattern will be illustrated for the case of an AuŽ110. electrode inserted into two different electrolytes, either into pure 0.1 M NaClO4 or into 0.1 M NaClO4 q 0.001 M NaBr. In order to associate the potential dependence with essentially a single element of the susceptibility . tensor Ž x iŽ2. jk , we used either the pp or ps polarization condition Žsee Table 2 in Ref. w15x.. For pp polarization essentially a p p ŽU . A x z z z ŽU ., while for ps polarization a p s should be zero Žand is indeed weak., but both c p s A x x x z ŽU . and d p s A x x x x ŽU . are comparatively large. Figs. 2–4 show the potential variation of the < a <, < c < and < d < terms in the top panels
Fig. 3. The anisotropic susceptibility c p s , vs. potential. Top panel: < c p s < vs. U; bottom panel: dc vs. U Žgiven vs. SCE.. Curves Ž v, '. for the same electrolytes as in Fig. 2.
Fig. 4. The anisotropic susceptibility d p s , vs. potential. Top panel: < d p s < vs. U; bottom panel: d d vs. U Žgiven vs. SCE.. Curves Ž v, '. for the same electrolytes as in Fig. 2.
B. Pettinger, C. Bilgerr Chemical Physics Letters 286 (1998) 355–360
and the corresponding changes of the complex phase d a , dc and d d with the potential Žwhich are defined in the form a s < a < expw i d a x etc. in the bottom panels, respectively. Each panel presents two curves for each of the above mentioned electrolytes. Note that in earlier publications only the internal relative phase, d s d a y d d , was given w15x. Here, the phase has to be understood as relative to the constant phase of the quartz lamella at x 1. In the case of pure 0.1 M NaClO4 solution, the < a < term rises monotonically. Obviously, it exhibits no minimum at the point of zero charge Žpzc. at y0.015 V for AuŽ110. w23x. Contrary to that, in the cases of AgŽ111., AgŽ110. and AgŽ100. such a minimum of the < a < term has been proposed theoretically w24x and has recently been observed in ISHG experiments w5,25x, substantiating its earlier observation on polycrystalline Ag w5x. For bromide ion adsorption at AuŽ110. Fig. 2 shows another, but consistent picture: the rise of < a < is steeper at the beginning than seen for the base electrolyte, the < a < term reaches a maximum at potentials which correspond roughly to the maximum of the CV curve associated with the adsorption of this anion. The decrease of < a < at more positive potential is not due to changes in the amount of adsorption; it must be caused by a change of the SHG resonance and of partial charge transfer. The complex phases in Fig. 2 show essentially the same trend, apart from the fact that there is no clear decrease at more positive potentials. Fig. 3 reproduces the potential dependence of the c term. Its rise points to a growth of the twofold rotational symmetry elements in the surface. In the case of pure sodium perchlorate solution, the phase dc is constant and < c < grows monotonically, while for bromide adsorption the < c < quantity varies in a rather complex fashion with potential. A more stringent picture is evident from the behavior of the complex phase dc : it shows a little variation in the cathodic potential regime and a significant drop by about 708 for Bry adsorption. The d term variation in Fig. 4 illustrates the change of the threefold symmetry elements in the surface. For < d < it confirms the AuŽ110. microfaceting reported earlier w16x. The complex phase d d , varies in a similar way to dc indicating changes in the optical resonance of the states which contribute to the anisotropic SHG response.
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5. Conclusion In this Letter we have presented a novel experimental approach to determine the complex susceptibility coefficients a, b, c and d, constituting the SHG anisotropy and providing structural and electronic information about the interface. It is also applicable to cases which were impossible to analyze up to now, such as the microfaceted AuŽ110.. However, a full understanding of our observations, which requires a theoretical treatment of these processes, is emerging for AgŽhkl. w24x, but is still missing for AuŽhkl. interfaces. 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 ŽPe 282r4-2.. References w1x Y.R. Shen, The principles of nonlinear optics, Wiley, New York, 1984, Ch. 25. w2x G.L. Richmond, in: H. Gerischer, C.W. Tobias ŽEds.., Adv. Electrochem. Sci. Eng., vol. 2, VCH Publishers, Weinheim, 1992. w3x R.A. Bradley, S. Arekat, R. Georgiadis, J.M. Robinson, S.D. Kevan, G.L. Richmond, Chem. Phys. Lett. 168 Ž1990. 468. w4x J. Bloch, D.J. Bottomley, S. Janz, H.M. van Driel, R.S. Timsit, J. Chem. Phys. 98 Ž1993. 9167. w5x R.M. Corn, M. Romagnoli, M.D. Levenson, M.R. Philpott, Chem. Phys. Lett. 106 Ž1984. 30. w6x T. Rasing, T. Stehlin, Y.R. Shen, M.W. Kim, P.J. Valint, J. Chem. Phys. 89 Ž1988. 3386. w7x R.R. Naujok, D.A. Higgins, D.G. Hanken, R.M. Corn, J. Chem. Soc. Faraday Trans. 91 Ž1995. 2353. w8x A. Friedrich, B. Pettinger, D.M. Kolb, G. Luepke, R. Steinhoff, G. Marowsky, Chem. Phys. Lett. 163 Ž1989. 123. w9x S. Janz, K. Pedersen, H.M. Van Driel, Phys. Rev. B: Condens. Matter 44 Ž1991. 3943. w10x H.W.K. Tom, X.D. Zhu, Y.R. Shen, G.A. Somorjai, Surf. Sci. 167 Ž1986. 167. w11x A. Priem, C.W. van Hasselt, M.A.C. Devillers, T.H. Rasing, Surf. Sci. 352–354 Ž1996. 612. w12x B. Pettinger, J. Lipkowski, S. Mirwald, A. Friedrich, Surf. Sci. 269–270 Ž1992. 377. w13x K.A. Friedrich, G.L. Richmond, Ber. Bunsenges. Phys. Chem. 97 Ž1993. 386. w14x R.M. Corn, D.A. Higgins, Chem. Rev. Washington D.C. 94 Ž1994. 107.
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