NANO LETTERS
Double Effective Medium Model for the Optical Properties of Self-Assembled Gold Nanoparticle Films Cross-Linked with Alkane Dithiols
2004 Vol. 4, No. 2 335-339
S. Schelm,*,† G. B. Smith,† G. Wei,‡ A. Vella,†,‡ L. Wieczorek,‡ K.-H. Mu1 ller,‡ and B. Raguse‡ Department of Applied Physics, UniVersity of Technology Sydney, P.O. Box 123 Broadway, 2007 NSW, Australia, and Telecommunications and Industrial Physics, CSIRO, Sydney, 2070 NSW, Australia Received October 27, 2003; Revised Manuscript Received November 27, 2003
ABSTRACT We present a comprehensive and structurally correct optical model for layers of nanometer sized gold particles, cross-linked with r,ω-alkane dithiol molecules, HS−(CH2)n−SH with n ) 2, 8, 15. The solution requires a two-stage, three-phase effective medium model based on the observed nanostructure of the three constituents, gold, thiol, and voids. Voids and their topology, which were neglected in previous models, prove to be crucial in the explanation of the optical properties.
A unifying first-principles understanding of the diversity of optical data found in a range of self-assembled nanoparticle gold films is found for the first time through simple models that correctly include various key aspects of the final nanostructure. Effective optical functions in amorphous arrays of self-assembled conducting nanoparticles can, in principle, be controlled via interparticle spacing, using different linker molecules1 and nanovoid content. The former is achieved via variation of the length of molecule used to link the particles into an array and the latter via the process used to deposit the layers. Void spatial characteristics, that is, where the voids form in relation to the other two phases, are also critically important in the models, a feature that has not been recognized until now. Apart from optical applications, the electrical properties of these films2,3 and the possibility of using the enhanced off-resonance field concentration within the voids of the films for catalytic purposes or enhanced dye excitation, are making these composites quite interesting for applications and fundamental science. The key optical feature of conducting particles is their surface plasmon (SP) resonance,4,5 which may cause strong spectral features due to an absorption band centered around the surface plasmon peak position, plus a quite inhomogeneous near-field intensity profile near these frequencies.6 The resonant width and peak intensity depends on electron relaxation rates which are modified from bulk * Corresponding author. † University of Technology Sydney. ‡ CSIRO. 10.1021/nl034948d CCC: $27.50 Published on Web 12/20/2003
© 2004 American Chemical Society
values by scattering off the particle surface, internal structure (e.g., grain boundaries in polycrystalline particles),7 and interface effects (e.g., chemical interface damping).8 We focus here on gold nanoparticles, for which selfassembled films have recently received considerable attention (see, e.g., refs 1, 9 and references therein). One interesting point is that it is possible in these systems to achieve moderate and high metallic volume fractions without creating conductive networks, as in gold “blacks”.10 This is accomplished by using the linker molecules as spacers between the individual particles, which can be varied by the use of different molecules. A wide variety of optical responses have been found, but unifying explanations are lacking and only semiempirical models for specific films have been reported so far.1,9 Attempts to apply effective medium approximations (EMA) have been inconclusive because they have been unable to explain the diversity of optical data, though this is not due to problems with EMA in general, as claimed in ref 9, but an incomplete treatment of the structure of these composites. A point of general interest in effective medium theory is also answered, possibly for the first time, in these films. This question is: are multipoles and symmetric and antisymmetric mode splitting important in denser random arrays of metal nanoparticles with fill factors over 40%? The only work in this to date has utilized lattice calculations,11 and based on them we would say, yes. More general arguments, based on the random unit cell concept,12 indicate that as long as we
Figure 1. Example SEM images of films filtrated from 3 mL of solution for two of the studied linkers: (a) C8-dithiol and (b) C15dithiol.
can keep the particles apart and randomly distributed, only dipoles play a role up to quite high fill factors, as the randomness cancels higher order modes. This in turn means Maxwell-Garnett models should remain valid. Short linker molecules enable exploration of this high fill factor domain. The nanoparticle films used in this study are produced by filtration of a partially aggregated solution of gold nanoparticles of 6 nm diameter and R,ω-alkane dithiols through a nanoporous substrate, which was polycarbonate for ellipsometry measurements and alumina for cross-section analysis. Full details of the preparation of the solution and the films are given elsewhere.13 The volume of solution filtered dictates the final film thickness, and as the area for the filtration was the same for all films we take the thickness in this paper as proportional to the filtrated volume(Vsol). Further, the amount of deposited mass controls the optical properties and the structure of the films within two regimes: (1) the region with incomplete substrate coverage (up to a solution volume of 2-3 mL) and (2) the region with completely covered substrate surface. Unless otherwise specified, the following discussion deals with the continuous region. Example nanostructures for C8-dithiol and C15-dithiol films can be seen in Figure 1. The solid matter is predominantly linker molecules with the nanoparticle gold inclusions showing up as the bright features. The gold particles seem to be well separated with a similar average spacing. Voids with interesting topography can be seen, as well. They appear to be due to the preformation of chains or clusters of goldthiol material and their subsequent linkage in the continuous films. As can be seen on these images there is a difference in the structure for the different linker lengths. In the case of the n ) 8 dithiol, a lot of the voids seem to be enclosed by planar rings of dithiol-gold wires, whereas in the case of the C15-dithiol the solid matter seems to be organized in almost round, aggregated globules. We note at this point that the build up of material in our samples is occurring under 336
Figure 2. Effective n and k values for the whole films (including substrate) for two volumes and the three linker lengths studies from ellipsometry and as calculated from our model (the inset shows bulk values from Ref 17): C2-dithiol (a) 1.0 mL, (b) 7.5 mL; C8dithiol (c) 1.0 mL, (d) 7.5 mL; C15-dithiol (e) 1.0 mL, (f) 7.5 mL.
pressure as the solution is pulled through the porous substrate, as opposed to the unforced layer-by-layer (LBL) method,1,9 which will affect the formation and volume fraction of the voids. Reflectance ellipsometry data were acquired at 70° incidence angle using a rotating analyzer instrument from 300 to 800 nm. Experimental n and k values were extracted from these data, and some examples can be seen in Figure 2, along with the results of our model, which will be explained below. Thicknesses were known approximately from atomic force microscopy,13 but accurate FEG-SEM cross sections were used to measure thicknesses. In common with many vacuum deposited thin films, a thickness threshold needs to be reached before a constant optical response is found. This is usually associated with full coverage of the substrate, but since voids are intrinsic to these films it can take between 30 to 50 nm before this asymptotic behavior sets in. Initial impressions on visible observation of these films is that they are not too different from uniform thin gold films. However, the physical reasons for this appearance are quite different than bulklike behavior, which is clear from the comparison of n and k values for our films with bulk values (inset of Figure 2b). Standard effective medium models such as basic MaxwellGarnett (MG) and Bruggeman (BR) theory do not apply to these films, though resonance peak positions are reasonably well predicted by MG models, provided the gold is embedded in a dense linker medium. An MG model is expected for well-separated inclusions and possibly a dense film of Nano Lett., Vol. 4, No. 2, 2004
separated, randomly arranged particles.12,15 It is well known that the microstructure of composites is playing a crucial role in the applicability of EMAs. Hence the void topology has to be considered for the modeling of the optical response. Considering our observations from the SEM analysis, especially the separation of the gold particles, an MG approach seemed to be justified for the gold-thiol system (the isolated particle case holds also for the LBL technique1). The appropriate choice for the treatment of the voids is more subtle. For instance, voids could be distributed uniformly among the linker system, or the linker system could be dense and the voids separated into a distinct region. These two cases will give quite distinct responses. The model we now present can include results from other deposition methods as well, if the specifics of these methods are properly accounted for. Earlier results,1,9 for example, show a significant blue shift with respect to our results, which can be explained assuming a less dense linker medium with a reduced refractive index. This can be explained by the use of the LBL method as it is a two-dimensional growth process with a part of the particle surface hidden in the layer underneath. In our films the linking takes place in solution, leaving the complete surface free to interact with the dithiol molecules. We developed a model which builds on basic MG theory according to observed topology and resonance positions, but also includes a phase-separated, well-distributed void system. The model is constructed in two steps. The starting composite material is gold spheres embedded in a dithiol linker medium. It has an effective dielectric function AuL given by the MG relation for spherical inclusions of material Au in host material L: AuL - L Au - L ) fAuL AuL + 2L Au + 2L
(1)
where L and Au are the dielectric functions of the linker and the gold nanoparticles, respectively, and fAuL is the volume fraction of gold in linker medium (not in the total film). Modifications to Au relative to bulk gold are taken into account and explained below. The final medium has effective dielectric function * and is a composite of voids with the dithiol-gold medium of eq 1. There is a dilemma here on the void topology that determines which generic effective medium model applies.12,15 Some films have voids enclosed entirely by the linker/gold medium (see Figure 1) and thus an MG model is appropriate. Most, however, seem to have approximately similar topology for both components with considerable percolation in both phases. In the latter topological case, a BR model is more suitable. We have thus used both models for the final step and assessed which gives the best overall fit to the optical data in each film. In some cases the difference is not large, with a slightly better fit for BR, whereas in most cases the fit with BR is better than that with MG by a factor of 2 in the mean square error. Only in the three thickest films for the n ) 15 films is the fit with the MG model slightly better, but not enough to force the Nano Lett., Vol. 4, No. 2, 2004
Table 1. Thicknesses and Void Fractions for the Films Produced with the Three Different Cross Linkersa C2-dithiol filtrated volume thickness (mL) (nm) fvoids 0.1 0.2 0.5 1.0 2.0 3.0 4.0 5.0 7.5 10.0
4 8 19 38 75 113 150 188 281 376
0.765 0.724 0.653 0.633 0.625 0.633 0.614
C8-dithiol
C15-dithiol
thickness (nm)
fvoids
thickness (nm)
fvoids
5 10 24 48 96b 144 193 241 360b 481
0.715 0.701 0.613 0.584 0.506 0.491 0.504 0.503 0.505 0.514
6 11 27 55 109b 164 219 273 410b 547
0.862 0.854 0.674 0.611 0.386 0.422 0.442 0.456 0.405 0.441
a The values f AuL derived from the k peak position are 0.74, 0.58, and 0.45 for C2-dithiol, C8-dithiol, and C15-dithiol, respectively. b Actually measured cross-sections.
use of MG, and the BR model was used for all studied films. The BR case yields the following equation for the final dielectric function of the whole film *; again we assume spherical inclusions: AuL - * 1 - * + fvoids )0 (1 - fvoids) AuL + 2* 1 + 2*
(2)
where fvoids is the void volume fraction and AuL as defined in eq 1. We also considered the possibility that the bonding of sulfur to the gold created a unique intermediate layer of distinct refractive index and possibly extra absorption peaks. This approach has been claimed as a necessary mechanism to better fit optical data of self-assembled layers of thiol linker on planar gold films.16 Optical data found in ref 16 were used for a model including a thiol-gold interface layer, but these models were qualitatively quite unacceptable in terms of our data. Thus we conclude that such an interface layer plays no significant role in our films, especially since its effect would be stronger in a nanoparticle than on a planar surface. Another effect such a layer could have is to reduce the electron relaxation rate via chemical interface damping.8 The final model uses a known layer thickness d, determined from SEM cross sections (see Table 1 for an overview), on a substrate of 10 µm thick polycarbonate in the Woollam Software WVase32. The only parameter that was calculated to fit our experimental data was fvoids. The remaining parameter fAuL was resolved from the absorption peak position in k, which is very sensitive to fAuL (maximum error of fAuL is 0.01) and found to be 0.74, 0.58, and 0.45 for C2-dithiol, C8-dithiol, and C15-dithiol, respectively. The thicknesses of films for which no cross sections were analyzed were extrapolated assuming d ∝ Vsol. Another important parameter is the electron relaxation time. It is well established that below a particle size which is comparable to the electron mean free path of the material, the electron relaxation time is reduced. In this case, the Drude part of the dielectric function of metal nanoparticles is usually 337
corrected by the following formula:7 (R) ) bulk + ω2p
(
1 1 ω2 + iωγ∞ ω2 + iωγ(R)
)
(3)
where bulk is the experimental bulk dielectric function of gold,17 ωp the bulk plasma frequency, γ∞ the inverse of the bulk relaxation time, and γ(R) ) γ∞ + AVF/R a sizedependent broadening parameter (where R is the radius of the particle, VF the Fermi velocity, and A a parameter which describes additional broadening mechanisms, e.g., defects, grain boundaries, chemical interface effects).7,8 We tested our model first with the physical size of the gold particles (R ) 3 nm). The positions of the spectral features were well reproduced but they were too strong and too narrow. A reduction of R by a factor of 2 yielded a far better match. This value is not only reasonable (considering additional surface scattering, grain boundary effects from polycrystallinity, and chemical effects) but also it agrees well with the results from Baum et al.18 The actual contributions to this reduced relaxation time are difficult to determine, but we think chemical effects (especially since the bonding between thiols and gold is quite strong) and internal defects are the most likely ones. Figure 2 shows experimental results from ellipsometry and the results from our model for all linkers studied. The graphs represent the two cases in the evolution of these films mentioned earlier: incomplete (1 mL films) and complete substrate coverage (7.5 mL films). Because the microstructure of the films in the incomplete region is evolving, the optical properties in the complete case are better described by our model, which shows a good fit for the thicker films. The difference in the microstructure in the incomplete films might warrant a change to the second step in our model, for example, by treating the gold-thiol matter in a void medium within the MG scheme. Another reason for the discrepancy in the thinner films could be scattering between the goldthiol islands, as this is not considered in EMA. The fact that the properties of the thin film are, at least in general, reproduced by our model indicates that the matter is deposited in such a way that the additional voids in the thinner films do not alter the principal features of the goldthiol component. This threshold effect is also evident in the value of fvoids in Table 1. The values for fvoids reach an asymptotic limit for films created from more than 2 mL of solution. The slight change for the thickest films is possibly due to another change in the microstructure, which expresses itself in a decreased quality of the fits for these thickest films. This is especially true for the longest linker length studied (C15dithiol), in which case the MG EMA starts to yield slightly better results. A comparison with the SEM images for this linker in Figure 1 offers the following explanation: It seems that the thiol-clusters, which are more compactly packed in the C15-dithiol films, might actually separate the voids and hence negate the percolation in the void phase. The BR scheme would not be valid anymore in this case and one would have to use MG for the voids instead, which is exactly 338
what is observed. A more comprehensive interpretation of the evolution of the nanostructure in these films will be presented in a forthcoming paper.14 Table 1 summarizes the thicknesses and void fractions for each linker length studied. It is clear that MG theory works here for AuL, even at gold fill factors in linker as high as 74%, because particles are kept separate with a similar nearest neighbor spacing, and are also randomly arranged with respect to each other. Self-assembled gold nanoparticles linked by dithiol molecules and deposited after partial preformation have a unique set of spectral responses and associated complex refractive index. These can be modeled with a hierarchial effective medium approach which takes into account the arrangement of the void phase, which forms in this deposition process separately from the gold-linker phase. If the voids are considered properly, films from all deposition techniques can be modeled. The gold-linker phase determines the central spectral features in terms of feature shapes and resonant frequencies. The resonance frequency only varies when the length of the linker molecules changes, i.e., the average separation between the gold particles (expressed in fAuL). The magnitudes of each of n, k, however, depend noticeably on the void content. An asymptotic complex refractive index is found beyond a threshold thickness, which enables the use of standard thin film optical design methods with the additional freedom to tune the optical properties of these kind of films. This work also confirms experimentally, possibly for the first time, that simple MG models can apply for quite high fill factors of the particle phase, provided they stay apart and are randomly arranged. The latter ensures that multipoles do not arise. Closely spaced nontouching arrays have also been achieved recently19 by using gold nanoparticles coated with a dielectric shell. In that case at volume fractions > 40% multipole effects are present and the MG model fails at those volume fractions. This is due to their comparative long-range order, in contrast to the random arrangement shown here. Complete models for random dense arrays of spheres have not been formulated, but simple symmetry arguments and our data and models indicate that multipoles may not be found until almost touching densities, or in more regular structures. Acknowledgment. S.S. acknowledges the support from BASF AG, Germany, for support of his PhD. We also thank Richard Wuhrer for help with the SEM characterization and Roger Netterfield for help with the ellipsometry measurements. References (1) Brust, M.; Bethell, D.; Kiely, C. J.; Schiffrin, D. J. Langmuir 1998, 14, 5425. (2) Mu¨ller, K.-H.; Herrmann, J.; Raguse, B.; Baxter, G.; Reda, T. Phys. ReV. B 2002, 66, 075417. (3) Trudeau, P. E.; Orozco, A.; Kwan, E.; Dhirani, A. A. J. Chem. Phys. 2002, 117, 3978. (4) Smith, G. B. In Introduction to complex mediums for optics and electromagnetics; Lakhatkia, A., Weiglhofer, W. S., Eds.; SPIE Press: Bellingham, WA, 2003; Chapter 18. (5) Krenn, J. R.; Ditlbacher, H.; Schider, G.; Hohenau, A.; Leitner, A.; Aussenegg, F. R. J. Microsc.-Oxford 2003, 209, 167. Nano Lett., Vol. 4, No. 2, 2004
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