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Second harmonic generation at single crystal surfaces of metals in the vacuum and in a solution
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Surface Science 291 (1993)226-232 North-Holland
surface science
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Second harmonic generation at single crystal surfaces of metals in the vacuum and in a solution Ezequiel Leiva Unidad Docente de Matematicas, Facultad de Ciencas Quimicas, Uniuersidad Nacional de Cordoba, Sucursal 16, C.C. 61, 5016 Cordoba, Argentina
and Wolfgang Schmickler 1 Physics Department, Utah State University, Logan, UT 84322, USA Received 8 June 1992; accepted for publication 2 March 1993
T h e surface electronic properties of a n u m b e r of metals are calculated using jellium with pseudopotentials as a model; particular attention is paid to the optical second harmonic response due to currents induced perpendicular to the metal surface. T h e structure of the surface is shown to have a pronounced effect on the properties investigated. The presence of a solution lowers the absolute magnitude of the second harmonic response.
I. I n t r o d u c t i o n
During the last decade a number of nonlinear optical techniques have been developed for the study of surfaces and interfaces: sum-frequency generation, hyper-Raman scattering, and second harmonic generation (SHG) [1]. Because of its inherent surface sensitivity and relative ease of implementation, SHG is by far the most commonly used of these techniques. It is particularly useful for studies of the metal-solution interface, for which there are very few other in situ methods which give information about its electronic structure. Theoretical work for SHG has been mostly directed to metal surfaces in the UHV. Recently the jellium model [2-4] has met with significant
1 New address: Abteilung Elektrochemie, Universit~it Ulm, Oberer Eselsberg, D-7900 Ulm, Germany.
success in predicting a good value for the nonlinear polarizability of aluminum [5], and in successfully explaining the dependence of the SHG signal on the polarization of the beam and the angle of incidence [5,6]. Furthermore, it offers an explanation for the effect of an adsorbed metal overlayer [7]. On the other hand, there is clear indication that the SHG response of a metal depends upon its surface structure [5,8], an effect which the simple, structureless jellium model cannot reproduce by its very nature. Also, the jellium value for the nonlinear polarizability of silver is significantly higher than both the experimental value for A g ( l l l ) [6] and a theoretical value obtained from electronic structure calculations for charged A g ( l l l ) surfaces [9]. So there is a definite need to introduce structure into the jellium model. There has been comparatively little theoretical work directed at the metal/solution interface. This is regrettable since, in contrast to the situa-
0039-6028/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
E.
227
Leiva, W. Schmickler / SHG at single crystal surfaces of metals
tion in the U H V , high surface charges and electrostatic fields are easily generated at this interface, and their effect on the S H G signal can be studied. The first publications on this subject [10,11] used a variational solution for the electronic density of jellium based on a simple onep a r a m e t e r family of trial functions, which gives reasonable results only for small surface charge densities and a limited range of electronic densities; furthermore these studies introduced adjustable p a r a m e t e r s to describe the effect of the solution on the electronic density of the metal. Some progress was achieved in a recent p a p e r [12], in which a substantially better family of trial functions [13] was used, and in which the solution was characterized by its optical dielectric constant alone, without any arbitrary parameters. While this work can explain a n u m b e r of features of S H G at the m e t a l / s o l u t i o n interface, again it is not able to account for the effect of surface structure. In this p a p e r we report on density functional calculations p e r f o r m e d with a lattice of pseudopotentials, thus introducing crystal structure into the model. We have calculated the nonlinear polarizability of a few metals both in the vacuum and in contact with an aqueous solution for various crystal orientations. Since this is basically an extension of the jellium model, its application is limited to simple sp metals and to the free-electron-like aspects of the noble metals copper and silver. Since we are mainly interested in the effect of surface structure, we have limited our calculations to the quasi-static limit of the S H G response, which is valid if the frequency of the incident laser b e a m is much lower than the bulk plasmon frequency of the metal.
ground charge but by a lattice of pseudopotentials of the Ashcroft type: for r < r c , for r>~rc,
(1)
where rc is the pseudopotential radius, and zi the valence of the ion. To keep the model one-dimensional this potential is averaged parallel to the metal surface; since we are interested in the average nonlinear polarizability of the surface this should not be a serious approximation. By setting up the pseudopotential lattice at different orientations with respect to the metal surface, we can investigate different single crystal surfaces. The electrons are treated as an inhomogeneous electron gas in the local density approximation. Lang and Kohn treated this model by perturbation theory; Monnier and Perdew [15], have found variational solutions with a constant electronic density in the bulk. Recently one of us (E.L.) has calculated exact solutions to this model for thin metal slabs [16] using an expansion technique developed by Mola and Vicente [17]. In this method a metal slab consisting of N layers ( N even) of pseudopotentials is at the center of a one-dimensional box of length D. As required by charge neutrality the metal surface is half a lattice spacing in front of the first plane of metal ions. If we take the origin z = 0 at the center of the box the metal extends in the region - L / 2 z <~L/2, with L - - ( N + 1)d/2, where d is the lattice spacing (see fig. 1). The total extension D
/
2. The model J
We have used a model which was first quantitatively investigated by Lang and Kohn [14]. It differs from the usual jellium model in that the metal ions are not represented as a positive back-
0 zi/r
v(r) =
f
electronic density
~1.88
~=1
AW
pseudopotentials IP'
ID'
' IP ' z~O
IF
IP' z-LI2
Fig. 1. Schematic diagram of the system in the presence of an electrolyte solution; the walls of the enclosing box are not shown.
E. Leiva, W. Schmickler / SHG at single crystal surfaces of metals
228
of the box must be so large that it does not affect the electronic density and the quantities derived from it; D - L = 20-30 a.u. #1 proved to be a good choice for our purposes. The electronic density is calculated by expanding the electronic wave functions in terms wave functions for a particle in a box. Typically, for N = 8 layers a basis set of 20-30 functions is enough; results reported here are for 32 basis functions. For further details of the calculation we refer to the original papers [16,17]. The S H G signal is formed by the interference of three contributions [18]: (i) a bulk contribution due the induced magnetic dipole moment, (ii) a surface contribution from the nonlinear polarizability parallel to the surface, and (iii) a surface contribution due to the deformation of the electronic density perpendicular to the surface. We shall focus our attention on the p a r a m e t e r a(to) characterizing the response perpendicular to the surface, which is the most sensitive to the electronic surface structure, and for which values extracted from experimental data are available in the literature. T h e calculations proceeded in the following way: the electronic distribution n(z; q) was calculated both in the absence and in the presence of surfaces charge densities q at both ends. By comparing the electronic density profiles for a charged and an uncharged surface the centroid of the induced charge, which is also the position of the effective image plane Xim, was calculated: o~
z[n(z; q) -n(z;
Xim = f0
0)]
dz-L/2.
(2)
The p a r a m e t e r a characterizing the S H G response due to currents driven perpendicular to the surface is then [2,12]: [
dxim
d2xim
a = 2 n 0 / 2 - - ~ q + q dq------T-
(3)
n o is the bulk electronic density. The usual problems associated with numerical differentiation
#1 Atomic units are usedo unless stated otherwise; 1 a.u. of length equals 0.529 A, 1 a.u. of surface charge density equals 55.56 C m -2.
limit our calculations to the range of small and intermediate excess charge densities q; in contrast to the jellium model, there is no simple form of the pressure theorem (Budd and Vannimenus sum rule) [19] to facilitate the calculations. We performed sample calculations for 6, 8, and 10 layers of metal atoms. In all cases investigated results for 8 and 10 layers agreed within the numerical accuracy, and even those for N = 6 are close. The S H G response is less sensitive to the thickness of the slab than the position of the image plane. This supports the view expressed earlier [2, 20] that the S H G response is limited to a thin surface region. We report results for N = 8. When we performed model calculations for the m e t a l / s o l u t i o n interface, the regions outside the metal (i.e. z>L/2 and z < - L / 2 ) were filled with a background dielectric constant eo~, which screens the electrostatic interactions. For e= we took the optical dielectric constant of the solvent (E~o= 1.88 for water), for the following reasons: (i) The vibrational and librational modes of the solvent, which contribute to the static but not to the optical dielectric constant, are too slow to follow the oscillating electric field of the incoming laser beam~ (ii) the correlation lengths of these modes ( 3 - 6 A) are substantially larger than the distance over which the electrons spill into the solution (0.5-1 ,~). We have not tried to develop a more detailed model for the interaction between jellium and the solvent since too little is known about it. Attempts to do so [10,11] were done at the expense of introducing empirical parameters into the model, which we want to avoid. Since the interactions between jellium and the solution is stronger when high negative charge densities reside on the metal surface and the electrons spill out further, we expect our calculations to be less accurate in this region. When a charged metal surface is immersed into an electrolyte solution the cations and anions rearrange to form a space charge layer compensating the excess charge on the metal. The corresponding Debye length is large compared to the distance over which the electrons spill into the solution, so we need to account only for the electrostatic interaction of these ions with the
E. Leiva, W. Schmickler / SHG at single crystal surfaces of metals
metal electrons. In o t h e r words, the ionic space charge layer in the solution simply serves as a c o u n t e r charge to the electronic charge on the metal. Obviously, this is no longer true w h e n the ions are specifically a d s o r b e d at the interface, so we have excluded this case f r o m o u r considerations.
3. R e s u l t s a n d d i s c u s s i o n
3.1. A l u m i n u m in v a c u u m
Since there are g o o d experimental values for the (111) plane o f a l u m i n u m in the v a c u u m [5], which agree well with the jellium calculations [4], we have investigated the electronic properties o f A I ( l l l ) and Al(100) surfaces - the (110) plane shows a substantial relaxation of the position o f the surface layer and has h e n c e not b e e n included [21]. F o r the p s e u d o p o t e n t i a l radius r c we used the value o f 1.12 a.u. r e c o m m e n d e d by A s h c r o f t and L a n g r e t h [22]. Since there is some interest in the position of the effective image plane [13,23,24], which plays a role b o t h in inverse photoemission and in scanning tunneling spectroscopy, we have plotted this in fig. 2 as a f u n c t i o n o f the surface c h a r g e density q. T h r o u g h o u t the investigated range the image plane of A1(111) is closer to the metal surface than that of Al(100). This is not unexpected,
2.00 ',
.
I,50
[
\
AI(100)\\ \
I \
jellium
o
E x
1.00
A<111) ~ ~ ~ ~ ~ ~ ~ ~l ! 0.50 , -20.0
+ -15.0
~ -10.0
+ -5.0
q
/
+--0.0
a.u.
5.0 *
I 10.0
- - - i 1510 20.0
10 4
Fig. 2. Position of the effective image plane on aluminum as a function of the surface charge density.
229
Table 1 Calculated properties at zero charge; the work functions of single crystal silver contain an extra contribution of 0.55 eV from the d-band, those of copper an extra 0.32 eV; details are found in ref. [25], which also compares the work functions with experimental values Surface AI(lll) AI(100) Alfjellium) Ag(lll) Ag(100) AgOellium) Cu(lll) Cu(100)
• (eV) 4.238 4.227 3.815 4.48 4.50 3.72 4.17 4.22
Xim (vac) 1.10 1.26 1.38 0.41 1.00 1.32 0.76 1.39
- a (vac) 20.6 14.6 27.6 8.1 14.9 13.3 17.3 20.6
x~ (sol) 0.27 0.93 0.94 1.44
- a (sol) 4.8 10.1 16.6 19.6
generally Xim is the smaller the higher the work function of the plane (see table 1), since a high surface barrier results in a low surface polarizability. O u r value o f Xim = 1.26 a.u. for the AI(100) surface agrees very well with the value o f x ~ = 1.1 a.u. obtained by Inglesfield [24] f r o m a truly three-dimensional calculation. F o r comparison, we also show the results of a simple jellium calculation (without pseudopotentials), which gives a lower work function and a c o r r e s p o n d ingly higher Xim. AS observed before [13,23] Xim b e c o m e s larger with increasing negative charge density, since an accumulation of electrons on the surface leads to a higher surface polarizability. F o r large negative surface charges, the electrons can leave the metal t h r o u g h field emission, and Xim b e c o m e s infinite. Note, however, that this increase is m o r e m a r k e d for A I ( l l l ) than for AI(100). This is probably due to the effect of the ion cores: since the (111) plane is m o r e densely packed, the interplanar spacing is larger, and the electrons in the surface region are further r e m o v e d f r o m the strong field of the triply c h a r g e d A1 ions, which makes t h e m m o r e susceptible to the effect o f large electrostatic surface fields. T h e c o r r e s p o n d i n g S H G response a is shown in fig. 3. Over the investigated range the (111) surface shows a greater response l al than the (100) surface, in accord with the greater curvature of the Xim versus q curve in fig. 2. T h e jellium m o d e l gives higher values for l al than
E. Leiva, W. Schmickler / SHG at single crystal surfaces of metals
230
40.0 55.0
A1(111)'\ "\, \""'-~\ \ \\
30.0 25.0 20.0
\.,,, \
"\ \
jell[urn \ x
15.0 10.0 5.0
0.0 I I -15.0 -I0.0 -5.0
I
I
0.0 5.'0 10.0 15'.0 20.0 25.0 q /o.u. "10 4 Fig. 3. Nonlinear optical response a on aluminum in the vacuum.
both these single crystal planes; at the point of zero charge we obtain the long wavelength limit of Liebsch and Schaich [4]. The S H G response of single crystal surfaces of aluminum in the vacuum has been investigated by Murphy et al. [5] using a laser b e a m wavelength of 1.06 /xm, which is somewhat above the long wavelength limit for this metal. This results for the A I ( l l l ) plane fit quite well to the theoretical value of a = - 3 6 - 9i obtained by Liebsch and Schaich [4] from the jellium model for that wavelength. The observed intensities for AI(100) correspond to a ~- - 2 2 . While an exact comparison with our calculations is not possible, since we calculated the long wavelength limit only, it is gratifying to note that our model gives the correct order of magnitude for a, and predicts a smaller value of l al for the (100) plane in accord with the experimental data.
work functions of the principal planes of silver. Fig. 4 shows the S H G response a in the vacuum and in an aqueous solution. Over the investigated range it is of the same order of magnitude as that of aluminum, but the curves for the various planes are now ordered in the same way as the image plane positions Xim (table 1): l al is now smallest on the densest plane, which has the highest work function, and highest on the most open plane with the lowest work function. Since the charge on the Ag ions is smaller than that of the A1 ions, it seems to have a smaller effect on I a l (see our remark above). The jellium curve falls in between the others. For the Ag(100) plane at zero charge our value for the position of the effective image plane ( X i m = 1 . 0 0 a . u . ) i s in excellent agreement with the value of Xim = 0.97 a.u. obtained by Aers and Inglesfield [9] from a truly three-dimensional calculation; our value of l a I = 14.9 a.u. is somewhat higher than their value of 8.83 a.u., but considering the fair scatter of the numerical data in the latter work the agreement is still satisfactory. A comparison of their work and ours shows both the strengths and the weaknesses of our approach: we use a simplified one-dimensional model so that we can only obtain average surface properties, but the numerical accuracy is greater in our work, and the computational effort considerably less. The A g ( l l 0 ) plane undergoes a charge-induced surface reconstruction [26], which we can-
400 I 350 30.0
3.2. Silver in vacuum and in a solution
~50
Silver has been a very popular metal for the study of a variety of optical techniques at the m e t a l / s o l u t i o n interface, and is therefore of particular interest. Since silver is an sd-metal, its representation by jellium and similar models is not straightforward. We have used the work of Russier and Badiali [25], which accounts for sd mixing by assigning the Ag ions an effective charge of 1.5, and a pseudopotential radius r c = 0.85 a.u.; in this way they obtain good values for the
', / '
4
" ~ 20.0
2::: I
~:,
Ag(100)vae
"
~,'/
~ ~,°°)~o~\
, 15.0 ~ \ •
-~
\ "•,~
lo.o ,
-~.
5o-i
agl(111)s~1
0"020 0 -- 1 5 0
\
\
~
10 0
I ---V----:---5.0 q*104
00
5.0
/
a.u
~I10.0
150
20.0
Fig. 4. N o n l i n e a r o p t i c a l r e s p o n s e a o n silver in the v a c u u m a n d in a n a q u e o u s solution.
E. Leiva, W. Schmickler / SHG at single crystal surfaces of metals
not treat within our model; so we have not performed calculations for this plane. The presence of a solution lowers the absolute values of a in all cases since the electrostatic field generated by the surface charge is screened in the solution. The ordering of the curves is the same as in the vacuum. It is particularly interesting to compare our values for the (111) plane with experimental data by Guyot-Sionnest et al. [6] in aqueous solution. Since in this environment silver is oxidized at higher electrode potentials we only show experimental points up to q = 10 .3 a.u. (see fig. 5). N e a r the point of zero charge our calculated data agree quite very well with experiment, but at negative charge densities our values for I a l are substantially too high. This is probably due to the absence of short range interactions between the electrons and the solvent in our model discussed above. For illustration we also show the vacuum and the jellium calculation to show the substantial improvement achieved by incorporating the structure and the effect of the solution into our model. 3.3. Copper in vacuum and in a solution
Copper has been used in a n u m b e r of investigations (see e.g. ref. [8]), and its surface electronic properties also seem to be described well by the model of Russier and Badiali [25], which
40.0
", ",
35.0 3o.o
~25.0
\
\
,,,
\
Cu/( 111)va c
" ".
'"" 'x \ \ " /
~ .
~ \ \ "" ,\
lO0)sol
- . q-,.
"deoo I
231
150
4 -TL-~. :_ -
100 5.0
0.0 mo
I
I
5.0
0.0
~ 5.0
q*lO 4 /
I *0.0
*5.0
eo.o
a.u.
Fig. 6. Nonlinear optical response a on copper in the vacuum and in an aqueous solution.
assigns the metal ions an effective charge of 1.5 and a pseudopotential radius of 0.869 a.u. It has a higher electronic density than silver and a similar pseudopotential radius, and hence at the point of zero charge l al is larger, the value for the (100) plane being higher than that for the (111) plane (see fig. 6). The presence of a solution lowers l al for negative charge densities as it does on silver. The C u ( l l 0 ) surface shows lattice relaxation at the surface [27], and was hence not included in our calculations.
4. Conclusion 400 35.0
-
30.0 4 ~250
vacuum solution "\\//
\\
~
,
\\
"~200ce , experiment~
40
30
jellium
\\\ /
\
eo q*104
\
" '
-m /
a.u.
1
o
1o
eo
Fig. 5. Nonlinear optical response a on A g ( l l l ) comparing our calculated values for the solution, the vacuum and the jellium model in the vacuum with experimental data from ref.
[5].
O u r calculations confirm the important influence of the surface structure on the electronic and nonlinear optical properties of metal surfaces: the S H G response [a[ and the effective position Xtm of the image charge are significantly different on the various single crystal planes investigated. In the few cases where we can compare our results with those obtained from threedimensional band structure calculations the overall agreement is quite good. For silver immersed in an aqueous solution, our values for [a] agree well with experimental data except for the region of high negative charge densities; for aluminum we obtain the right order of magnitude for I a I, and a smaller value for AI(100) than for A I ( l l l ) in agreement with experimental results.
232
E. Leiva, W. Schmickler / SHG at single crystal surfaces of metals
W h i l e we t h i n k that this constitutes a signific a n t a d v a n c e over results o b t a i n e d from the simple j e l l i u m model, a n u m b e r of i m p o r t a n t feat u r e s are still missing from o u r model. (i) O u r p r e s e n t calculations are limited to the long wavel e n g t h limit; following W e b e r a n d Liebsch [2], it should be possible to o b t a i n the f r e q u e n c y d e p e n d e n c e of the S H G r e s p o n s e by applying the time d e p e n d e n t density f u n c t i o n a l formalism. (ii) By s m e a r i n g out the p s e u d o p o t e n t i a l s parallel to the m e t a l surface we have r e d u c e d the m o d e l to a o n e - d i m e n s i o n a l one, which by its very n a t u r e c a n n o t r e p r o d u c e the d e p e n d e n c e of the S H G signal o n the polar angle of incidence; it is therefore desirable to i n t r o d u c e the surface corrugat i o n of the electrons into the model. (iii) Surface states, which have recently b e e n observed o n A g ( l l l ) by S H G [28], c a n n o t be o b t a i n e d from the p r e s e n t model.
Acknowledgements F i n a n c i a l s u p p o r t from the C o n s e j o N a c i o n a l de I n v e s t i g a c i o n e s Cientificas y T e c n i c a s de Arg e n t i n a ( C O N I C E T ) , C o n s e j o de Investigaciones Cientificas y T e c n o l o g i c a s de C o r d o b a ( C O N I C O R ) , F u n d a c i o n A n t o r c h a s , Secretaria de Ciencias y T e c n o l o g i a de la U n i v e r s i d a d de Cordoba, a n d of U t a h State University is gratefully acknowledged.
References [1] For a recent review see: O. Richmond, in: Electrochemical Interfaces, Ed. H.D. Abruna (VCH Publishers, New York, 1991).
[2] M. Weber and A. Liebsch, Phys. Rev. B 35 (1987) 7411; Phys. Rev. B 36 (1987) 6411. [3] A. Liebsch, Phys. Rev. B 40 (1989) 3421. [4] A. Liebsch and W.L. Schaich, Phys. Rev. B 40 (1989) 5401. [5] M. Murphy, M. Yeganeh, K.J. Song and E.W. Plummer, Phys. Rev. Lett. 63 (1989) 318. [6] P. Guyot-Sionnest, A. Tadjeddine and A. Liebsch, Phys. Rev. Lett. 64 (1990) 1678. [7] A. Liebsch, Phys. Rev. B 40 (1989) 3421. [8] See e.g.J.M. Robinson and G.L. Richmond, Chem. Phys. 141 (1990) 175. [9] G.C. Aers and J.E. Inglesfield, Surf. Sci. 217 (1989) 367. [10] P.G. Dzhavakhidze, A.A. Kornyshev, A. Liebsch and M.I. Urbakh, Electrochim. Acta 36 1835 (1991). [11] P.G. Dzhavakhidze, A.A. Kornyshev, A. Liebsch and M.I. Urbakh, Phys. Rev. B, in press. [12] W. Schmickler and M. Urbakh, Phys. Rev. B, in press. [13] W. Schmickler and D. Henderson, Phys. Rev. B 30 (1984) 3081. [14] N.D. Lang and W. Kohn, Phys. Rev. B 1 (1970) 4555; Phys. Rev. B 4 (1971) 1215. [15] R. Monnier and J.P. Perdew, Phys. Rev. B 17 (1978) 2595. [16] E. Leiva, Chem. Phys. Lett. 187 (1991)143. [17] E. Mola and J. Vincente, J. Chem. Phys. 84 (1986) 2876. [18] J. Rudnick and E.A. Stern, Phys. Rev. B 4 (1971) 4274. [19] H.F. Budd and J. Vannimenus, Phys. Rev. Lett. 31 (1973) 1218. [20] K.J. Song, D. Heskett, H.L. Dai, A. Liebsch and E.W. Plummer, Phys. Rev. B 61 (1988) 1380. [21] J.R. Noonan and Davis, Phys. Rev. B 29 (1984) 4349. [22] N.W. Ashcroft and D. Langreth, Phys. Rev. 155 (1967) 682. [23] P. Gies and R.R. Gerhardts, Phys. Rev. B 33 (1986) 6843. [24] J.E. Inglesfield, Surf. Sci. 188 (1987) LT01. [25] V. Russier and J.P. Badiali, Phys. Rev. B 39 (1989) 13193. [26] C.L. Fu and K.M. Ho, Phys. Rev. Lett. 15 (1989) 1617. [27] D.L. Adams, H.B. Nielsen, J.N. Anderson, I. Stensgaard, R. Feidenhansl and J.E. Sorensen, Phys. Rev. Lett. 49 (1982) 669. [28] R.A. Bradley, R. Georgiadis, S.D. Kevan and G.L. Richmond, J. Vac. Sci. Technol., in press.
Surface Science 291 (1993)226-232 North-Holland
surface science
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Second harmonic generation at single crystal surfaces of metals in the vacuum and in a solution Ezequiel Leiva Unidad Docente de Matematicas, Facultad de Ciencas Quimicas, Uniuersidad Nacional de Cordoba, Sucursal 16, C.C. 61, 5016 Cordoba, Argentina
and Wolfgang Schmickler 1 Physics Department, Utah State University, Logan, UT 84322, USA Received 8 June 1992; accepted for publication 2 March 1993
T h e surface electronic properties of a n u m b e r of metals are calculated using jellium with pseudopotentials as a model; particular attention is paid to the optical second harmonic response due to currents induced perpendicular to the metal surface. T h e structure of the surface is shown to have a pronounced effect on the properties investigated. The presence of a solution lowers the absolute magnitude of the second harmonic response.
I. I n t r o d u c t i o n
During the last decade a number of nonlinear optical techniques have been developed for the study of surfaces and interfaces: sum-frequency generation, hyper-Raman scattering, and second harmonic generation (SHG) [1]. Because of its inherent surface sensitivity and relative ease of implementation, SHG is by far the most commonly used of these techniques. It is particularly useful for studies of the metal-solution interface, for which there are very few other in situ methods which give information about its electronic structure. Theoretical work for SHG has been mostly directed to metal surfaces in the UHV. Recently the jellium model [2-4] has met with significant
1 New address: Abteilung Elektrochemie, Universit~it Ulm, Oberer Eselsberg, D-7900 Ulm, Germany.
success in predicting a good value for the nonlinear polarizability of aluminum [5], and in successfully explaining the dependence of the SHG signal on the polarization of the beam and the angle of incidence [5,6]. Furthermore, it offers an explanation for the effect of an adsorbed metal overlayer [7]. On the other hand, there is clear indication that the SHG response of a metal depends upon its surface structure [5,8], an effect which the simple, structureless jellium model cannot reproduce by its very nature. Also, the jellium value for the nonlinear polarizability of silver is significantly higher than both the experimental value for A g ( l l l ) [6] and a theoretical value obtained from electronic structure calculations for charged A g ( l l l ) surfaces [9]. So there is a definite need to introduce structure into the jellium model. There has been comparatively little theoretical work directed at the metal/solution interface. This is regrettable since, in contrast to the situa-
0039-6028/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
E.
227
Leiva, W. Schmickler / SHG at single crystal surfaces of metals
tion in the U H V , high surface charges and electrostatic fields are easily generated at this interface, and their effect on the S H G signal can be studied. The first publications on this subject [10,11] used a variational solution for the electronic density of jellium based on a simple onep a r a m e t e r family of trial functions, which gives reasonable results only for small surface charge densities and a limited range of electronic densities; furthermore these studies introduced adjustable p a r a m e t e r s to describe the effect of the solution on the electronic density of the metal. Some progress was achieved in a recent p a p e r [12], in which a substantially better family of trial functions [13] was used, and in which the solution was characterized by its optical dielectric constant alone, without any arbitrary parameters. While this work can explain a n u m b e r of features of S H G at the m e t a l / s o l u t i o n interface, again it is not able to account for the effect of surface structure. In this p a p e r we report on density functional calculations p e r f o r m e d with a lattice of pseudopotentials, thus introducing crystal structure into the model. We have calculated the nonlinear polarizability of a few metals both in the vacuum and in contact with an aqueous solution for various crystal orientations. Since this is basically an extension of the jellium model, its application is limited to simple sp metals and to the free-electron-like aspects of the noble metals copper and silver. Since we are mainly interested in the effect of surface structure, we have limited our calculations to the quasi-static limit of the S H G response, which is valid if the frequency of the incident laser b e a m is much lower than the bulk plasmon frequency of the metal.
ground charge but by a lattice of pseudopotentials of the Ashcroft type: for r < r c , for r>~rc,
(1)
where rc is the pseudopotential radius, and zi the valence of the ion. To keep the model one-dimensional this potential is averaged parallel to the metal surface; since we are interested in the average nonlinear polarizability of the surface this should not be a serious approximation. By setting up the pseudopotential lattice at different orientations with respect to the metal surface, we can investigate different single crystal surfaces. The electrons are treated as an inhomogeneous electron gas in the local density approximation. Lang and Kohn treated this model by perturbation theory; Monnier and Perdew [15], have found variational solutions with a constant electronic density in the bulk. Recently one of us (E.L.) has calculated exact solutions to this model for thin metal slabs [16] using an expansion technique developed by Mola and Vicente [17]. In this method a metal slab consisting of N layers ( N even) of pseudopotentials is at the center of a one-dimensional box of length D. As required by charge neutrality the metal surface is half a lattice spacing in front of the first plane of metal ions. If we take the origin z = 0 at the center of the box the metal extends in the region - L / 2 z <~L/2, with L - - ( N + 1)d/2, where d is the lattice spacing (see fig. 1). The total extension D
/
2. The model J
We have used a model which was first quantitatively investigated by Lang and Kohn [14]. It differs from the usual jellium model in that the metal ions are not represented as a positive back-
0 zi/r
v(r) =
f
electronic density
~1.88
~=1
AW
pseudopotentials IP'
ID'
' IP ' z~O
IF
IP' z-LI2
Fig. 1. Schematic diagram of the system in the presence of an electrolyte solution; the walls of the enclosing box are not shown.
E. Leiva, W. Schmickler / SHG at single crystal surfaces of metals
228
of the box must be so large that it does not affect the electronic density and the quantities derived from it; D - L = 20-30 a.u. #1 proved to be a good choice for our purposes. The electronic density is calculated by expanding the electronic wave functions in terms wave functions for a particle in a box. Typically, for N = 8 layers a basis set of 20-30 functions is enough; results reported here are for 32 basis functions. For further details of the calculation we refer to the original papers [16,17]. The S H G signal is formed by the interference of three contributions [18]: (i) a bulk contribution due the induced magnetic dipole moment, (ii) a surface contribution from the nonlinear polarizability parallel to the surface, and (iii) a surface contribution due to the deformation of the electronic density perpendicular to the surface. We shall focus our attention on the p a r a m e t e r a(to) characterizing the response perpendicular to the surface, which is the most sensitive to the electronic surface structure, and for which values extracted from experimental data are available in the literature. T h e calculations proceeded in the following way: the electronic distribution n(z; q) was calculated both in the absence and in the presence of surfaces charge densities q at both ends. By comparing the electronic density profiles for a charged and an uncharged surface the centroid of the induced charge, which is also the position of the effective image plane Xim, was calculated: o~
z[n(z; q) -n(z;
Xim = f0
0)]
dz-L/2.
(2)
The p a r a m e t e r a characterizing the S H G response due to currents driven perpendicular to the surface is then [2,12]: [
dxim
d2xim
a = 2 n 0 / 2 - - ~ q + q dq------T-
(3)
n o is the bulk electronic density. The usual problems associated with numerical differentiation
#1 Atomic units are usedo unless stated otherwise; 1 a.u. of length equals 0.529 A, 1 a.u. of surface charge density equals 55.56 C m -2.
limit our calculations to the range of small and intermediate excess charge densities q; in contrast to the jellium model, there is no simple form of the pressure theorem (Budd and Vannimenus sum rule) [19] to facilitate the calculations. We performed sample calculations for 6, 8, and 10 layers of metal atoms. In all cases investigated results for 8 and 10 layers agreed within the numerical accuracy, and even those for N = 6 are close. The S H G response is less sensitive to the thickness of the slab than the position of the image plane. This supports the view expressed earlier [2, 20] that the S H G response is limited to a thin surface region. We report results for N = 8. When we performed model calculations for the m e t a l / s o l u t i o n interface, the regions outside the metal (i.e. z>L/2 and z < - L / 2 ) were filled with a background dielectric constant eo~, which screens the electrostatic interactions. For e= we took the optical dielectric constant of the solvent (E~o= 1.88 for water), for the following reasons: (i) The vibrational and librational modes of the solvent, which contribute to the static but not to the optical dielectric constant, are too slow to follow the oscillating electric field of the incoming laser beam~ (ii) the correlation lengths of these modes ( 3 - 6 A) are substantially larger than the distance over which the electrons spill into the solution (0.5-1 ,~). We have not tried to develop a more detailed model for the interaction between jellium and the solvent since too little is known about it. Attempts to do so [10,11] were done at the expense of introducing empirical parameters into the model, which we want to avoid. Since the interactions between jellium and the solution is stronger when high negative charge densities reside on the metal surface and the electrons spill out further, we expect our calculations to be less accurate in this region. When a charged metal surface is immersed into an electrolyte solution the cations and anions rearrange to form a space charge layer compensating the excess charge on the metal. The corresponding Debye length is large compared to the distance over which the electrons spill into the solution, so we need to account only for the electrostatic interaction of these ions with the
E. Leiva, W. Schmickler / SHG at single crystal surfaces of metals
metal electrons. In o t h e r words, the ionic space charge layer in the solution simply serves as a c o u n t e r charge to the electronic charge on the metal. Obviously, this is no longer true w h e n the ions are specifically a d s o r b e d at the interface, so we have excluded this case f r o m o u r considerations.
3. R e s u l t s a n d d i s c u s s i o n
3.1. A l u m i n u m in v a c u u m
Since there are g o o d experimental values for the (111) plane o f a l u m i n u m in the v a c u u m [5], which agree well with the jellium calculations [4], we have investigated the electronic properties o f A I ( l l l ) and Al(100) surfaces - the (110) plane shows a substantial relaxation of the position o f the surface layer and has h e n c e not b e e n included [21]. F o r the p s e u d o p o t e n t i a l radius r c we used the value o f 1.12 a.u. r e c o m m e n d e d by A s h c r o f t and L a n g r e t h [22]. Since there is some interest in the position of the effective image plane [13,23,24], which plays a role b o t h in inverse photoemission and in scanning tunneling spectroscopy, we have plotted this in fig. 2 as a f u n c t i o n o f the surface c h a r g e density q. T h r o u g h o u t the investigated range the image plane of A1(111) is closer to the metal surface than that of Al(100). This is not unexpected,
2.00 ',
.
I,50
[
\
AI(100)\\ \
I \
jellium
o
E x
1.00
A<111) ~ ~ ~ ~ ~ ~ ~ ~l ! 0.50 , -20.0
+ -15.0
~ -10.0
+ -5.0
q
/
+--0.0
a.u.
5.0 *
I 10.0
- - - i 1510 20.0
10 4
Fig. 2. Position of the effective image plane on aluminum as a function of the surface charge density.
229
Table 1 Calculated properties at zero charge; the work functions of single crystal silver contain an extra contribution of 0.55 eV from the d-band, those of copper an extra 0.32 eV; details are found in ref. [25], which also compares the work functions with experimental values Surface AI(lll) AI(100) Alfjellium) Ag(lll) Ag(100) AgOellium) Cu(lll) Cu(100)
• (eV) 4.238 4.227 3.815 4.48 4.50 3.72 4.17 4.22
Xim (vac) 1.10 1.26 1.38 0.41 1.00 1.32 0.76 1.39
- a (vac) 20.6 14.6 27.6 8.1 14.9 13.3 17.3 20.6
x~ (sol) 0.27 0.93 0.94 1.44
- a (sol) 4.8 10.1 16.6 19.6
generally Xim is the smaller the higher the work function of the plane (see table 1), since a high surface barrier results in a low surface polarizability. O u r value o f Xim = 1.26 a.u. for the AI(100) surface agrees very well with the value o f x ~ = 1.1 a.u. obtained by Inglesfield [24] f r o m a truly three-dimensional calculation. F o r comparison, we also show the results of a simple jellium calculation (without pseudopotentials), which gives a lower work function and a c o r r e s p o n d ingly higher Xim. AS observed before [13,23] Xim b e c o m e s larger with increasing negative charge density, since an accumulation of electrons on the surface leads to a higher surface polarizability. F o r large negative surface charges, the electrons can leave the metal t h r o u g h field emission, and Xim b e c o m e s infinite. Note, however, that this increase is m o r e m a r k e d for A I ( l l l ) than for AI(100). This is probably due to the effect of the ion cores: since the (111) plane is m o r e densely packed, the interplanar spacing is larger, and the electrons in the surface region are further r e m o v e d f r o m the strong field of the triply c h a r g e d A1 ions, which makes t h e m m o r e susceptible to the effect o f large electrostatic surface fields. T h e c o r r e s p o n d i n g S H G response a is shown in fig. 3. Over the investigated range the (111) surface shows a greater response l al than the (100) surface, in accord with the greater curvature of the Xim versus q curve in fig. 2. T h e jellium m o d e l gives higher values for l al than
E. Leiva, W. Schmickler / SHG at single crystal surfaces of metals
230
40.0 55.0
A1(111)'\ "\, \""'-~\ \ \\
30.0 25.0 20.0
\.,,, \
"\ \
jell[urn \ x
15.0 10.0 5.0
0.0 I I -15.0 -I0.0 -5.0
I
I
0.0 5.'0 10.0 15'.0 20.0 25.0 q /o.u. "10 4 Fig. 3. Nonlinear optical response a on aluminum in the vacuum.
both these single crystal planes; at the point of zero charge we obtain the long wavelength limit of Liebsch and Schaich [4]. The S H G response of single crystal surfaces of aluminum in the vacuum has been investigated by Murphy et al. [5] using a laser b e a m wavelength of 1.06 /xm, which is somewhat above the long wavelength limit for this metal. This results for the A I ( l l l ) plane fit quite well to the theoretical value of a = - 3 6 - 9i obtained by Liebsch and Schaich [4] from the jellium model for that wavelength. The observed intensities for AI(100) correspond to a ~- - 2 2 . While an exact comparison with our calculations is not possible, since we calculated the long wavelength limit only, it is gratifying to note that our model gives the correct order of magnitude for a, and predicts a smaller value of l al for the (100) plane in accord with the experimental data.
work functions of the principal planes of silver. Fig. 4 shows the S H G response a in the vacuum and in an aqueous solution. Over the investigated range it is of the same order of magnitude as that of aluminum, but the curves for the various planes are now ordered in the same way as the image plane positions Xim (table 1): l al is now smallest on the densest plane, which has the highest work function, and highest on the most open plane with the lowest work function. Since the charge on the Ag ions is smaller than that of the A1 ions, it seems to have a smaller effect on I a l (see our remark above). The jellium curve falls in between the others. For the Ag(100) plane at zero charge our value for the position of the effective image plane ( X i m = 1 . 0 0 a . u . ) i s in excellent agreement with the value of Xim = 0.97 a.u. obtained by Aers and Inglesfield [9] from a truly three-dimensional calculation; our value of l a I = 14.9 a.u. is somewhat higher than their value of 8.83 a.u., but considering the fair scatter of the numerical data in the latter work the agreement is still satisfactory. A comparison of their work and ours shows both the strengths and the weaknesses of our approach: we use a simplified one-dimensional model so that we can only obtain average surface properties, but the numerical accuracy is greater in our work, and the computational effort considerably less. The A g ( l l 0 ) plane undergoes a charge-induced surface reconstruction [26], which we can-
400 I 350 30.0
3.2. Silver in vacuum and in a solution
~50
Silver has been a very popular metal for the study of a variety of optical techniques at the m e t a l / s o l u t i o n interface, and is therefore of particular interest. Since silver is an sd-metal, its representation by jellium and similar models is not straightforward. We have used the work of Russier and Badiali [25], which accounts for sd mixing by assigning the Ag ions an effective charge of 1.5, and a pseudopotential radius r c = 0.85 a.u.; in this way they obtain good values for the
', / '
4
" ~ 20.0
2::: I
~:,
Ag(100)vae
"
~,'/
~ ~,°°)~o~\
, 15.0 ~ \ •
-~
\ "•,~
lo.o ,
-~.
5o-i
agl(111)s~1
0"020 0 -- 1 5 0
\
\
~
10 0
I ---V----:---5.0 q*104
00
5.0
/
a.u
~I10.0
150
20.0
Fig. 4. N o n l i n e a r o p t i c a l r e s p o n s e a o n silver in the v a c u u m a n d in a n a q u e o u s solution.
E. Leiva, W. Schmickler / SHG at single crystal surfaces of metals
not treat within our model; so we have not performed calculations for this plane. The presence of a solution lowers the absolute values of a in all cases since the electrostatic field generated by the surface charge is screened in the solution. The ordering of the curves is the same as in the vacuum. It is particularly interesting to compare our values for the (111) plane with experimental data by Guyot-Sionnest et al. [6] in aqueous solution. Since in this environment silver is oxidized at higher electrode potentials we only show experimental points up to q = 10 .3 a.u. (see fig. 5). N e a r the point of zero charge our calculated data agree quite very well with experiment, but at negative charge densities our values for I a l are substantially too high. This is probably due to the absence of short range interactions between the electrons and the solvent in our model discussed above. For illustration we also show the vacuum and the jellium calculation to show the substantial improvement achieved by incorporating the structure and the effect of the solution into our model. 3.3. Copper in vacuum and in a solution
Copper has been used in a n u m b e r of investigations (see e.g. ref. [8]), and its surface electronic properties also seem to be described well by the model of Russier and Badiali [25], which
40.0
", ",
35.0 3o.o
~25.0
\
\
,,,
\
Cu/( 111)va c
" ".
'"" 'x \ \ " /
~ .
~ \ \ "" ,\
lO0)sol
- . q-,.
"deoo I
231
150
4 -TL-~. :_ -
100 5.0
0.0 mo
I
I
5.0
0.0
~ 5.0
q*lO 4 /
I *0.0
*5.0
eo.o
a.u.
Fig. 6. Nonlinear optical response a on copper in the vacuum and in an aqueous solution.
assigns the metal ions an effective charge of 1.5 and a pseudopotential radius of 0.869 a.u. It has a higher electronic density than silver and a similar pseudopotential radius, and hence at the point of zero charge l al is larger, the value for the (100) plane being higher than that for the (111) plane (see fig. 6). The presence of a solution lowers l al for negative charge densities as it does on silver. The C u ( l l 0 ) surface shows lattice relaxation at the surface [27], and was hence not included in our calculations.
4. Conclusion 400 35.0
-
30.0 4 ~250
vacuum solution "\\//
\\
~
,
\\
"~200ce , experiment~
40
30
jellium
\\\ /
\
eo q*104
\
" '
-m /
a.u.
1
o
1o
eo
Fig. 5. Nonlinear optical response a on A g ( l l l ) comparing our calculated values for the solution, the vacuum and the jellium model in the vacuum with experimental data from ref.
[5].
O u r calculations confirm the important influence of the surface structure on the electronic and nonlinear optical properties of metal surfaces: the S H G response [a[ and the effective position Xtm of the image charge are significantly different on the various single crystal planes investigated. In the few cases where we can compare our results with those obtained from threedimensional band structure calculations the overall agreement is quite good. For silver immersed in an aqueous solution, our values for [a] agree well with experimental data except for the region of high negative charge densities; for aluminum we obtain the right order of magnitude for I a I, and a smaller value for AI(100) than for A I ( l l l ) in agreement with experimental results.
232
E. Leiva, W. Schmickler / SHG at single crystal surfaces of metals
W h i l e we t h i n k that this constitutes a signific a n t a d v a n c e over results o b t a i n e d from the simple j e l l i u m model, a n u m b e r of i m p o r t a n t feat u r e s are still missing from o u r model. (i) O u r p r e s e n t calculations are limited to the long wavel e n g t h limit; following W e b e r a n d Liebsch [2], it should be possible to o b t a i n the f r e q u e n c y d e p e n d e n c e of the S H G r e s p o n s e by applying the time d e p e n d e n t density f u n c t i o n a l formalism. (ii) By s m e a r i n g out the p s e u d o p o t e n t i a l s parallel to the m e t a l surface we have r e d u c e d the m o d e l to a o n e - d i m e n s i o n a l one, which by its very n a t u r e c a n n o t r e p r o d u c e the d e p e n d e n c e of the S H G signal o n the polar angle of incidence; it is therefore desirable to i n t r o d u c e the surface corrugat i o n of the electrons into the model. (iii) Surface states, which have recently b e e n observed o n A g ( l l l ) by S H G [28], c a n n o t be o b t a i n e d from the p r e s e n t model.
Acknowledgements F i n a n c i a l s u p p o r t from the C o n s e j o N a c i o n a l de I n v e s t i g a c i o n e s Cientificas y T e c n i c a s de Arg e n t i n a ( C O N I C E T ) , C o n s e j o de Investigaciones Cientificas y T e c n o l o g i c a s de C o r d o b a ( C O N I C O R ) , F u n d a c i o n A n t o r c h a s , Secretaria de Ciencias y T e c n o l o g i a de la U n i v e r s i d a d de Cordoba, a n d of U t a h State University is gratefully acknowledged.
References [1] For a recent review see: O. Richmond, in: Electrochemical Interfaces, Ed. H.D. Abruna (VCH Publishers, New York, 1991).
[2] M. Weber and A. Liebsch, Phys. Rev. B 35 (1987) 7411; Phys. Rev. B 36 (1987) 6411. [3] A. Liebsch, Phys. Rev. B 40 (1989) 3421. [4] A. Liebsch and W.L. Schaich, Phys. Rev. B 40 (1989) 5401. [5] M. Murphy, M. Yeganeh, K.J. Song and E.W. Plummer, Phys. Rev. Lett. 63 (1989) 318. [6] P. Guyot-Sionnest, A. Tadjeddine and A. Liebsch, Phys. Rev. Lett. 64 (1990) 1678. [7] A. Liebsch, Phys. Rev. B 40 (1989) 3421. [8] See e.g.J.M. Robinson and G.L. Richmond, Chem. Phys. 141 (1990) 175. [9] G.C. Aers and J.E. Inglesfield, Surf. Sci. 217 (1989) 367. [10] P.G. Dzhavakhidze, A.A. Kornyshev, A. Liebsch and M.I. Urbakh, Electrochim. Acta 36 1835 (1991). [11] P.G. Dzhavakhidze, A.A. Kornyshev, A. Liebsch and M.I. Urbakh, Phys. Rev. B, in press. [12] W. Schmickler and M. Urbakh, Phys. Rev. B, in press. [13] W. Schmickler and D. Henderson, Phys. Rev. B 30 (1984) 3081. [14] N.D. Lang and W. Kohn, Phys. Rev. B 1 (1970) 4555; Phys. Rev. B 4 (1971) 1215. [15] R. Monnier and J.P. Perdew, Phys. Rev. B 17 (1978) 2595. [16] E. Leiva, Chem. Phys. Lett. 187 (1991)143. [17] E. Mola and J. Vincente, J. Chem. Phys. 84 (1986) 2876. [18] J. Rudnick and E.A. Stern, Phys. Rev. B 4 (1971) 4274. [19] H.F. Budd and J. Vannimenus, Phys. Rev. Lett. 31 (1973) 1218. [20] K.J. Song, D. Heskett, H.L. Dai, A. Liebsch and E.W. Plummer, Phys. Rev. B 61 (1988) 1380. [21] J.R. Noonan and Davis, Phys. Rev. B 29 (1984) 4349. [22] N.W. Ashcroft and D. Langreth, Phys. Rev. 155 (1967) 682. [23] P. Gies and R.R. Gerhardts, Phys. Rev. B 33 (1986) 6843. [24] J.E. Inglesfield, Surf. Sci. 188 (1987) LT01. [25] V. Russier and J.P. Badiali, Phys. Rev. B 39 (1989) 13193. [26] C.L. Fu and K.M. Ho, Phys. Rev. Lett. 15 (1989) 1617. [27] D.L. Adams, H.B. Nielsen, J.N. Anderson, I. Stensgaard, R. Feidenhansl and J.E. Sorensen, Phys. Rev. Lett. 49 (1982) 669. [28] R.A. Bradley, R. Georgiadis, S.D. Kevan and G.L. Richmond, J. Vac. Sci. Technol., in press.