Optical properties of AgxAu1-x alloys were obtained experimentally using spectroscopic ellipsometry measurements on thin films fabricated by electron beam evaporation. Thin film thicknesses varied between 170 and 330 nm, making size effects negligible. Values of the complex refractive index of the pure metals were in good agreement with literature reports. The optical data set reported in this work can accurately reproduce experimental results. This is very important because there are not reliable and systematic optical constants for the alloys. Moreover, we show that the weighted average of the refractive indices of the pure metals fails to represent those of the alloys, not only in the region near the onset for interband transitions but also in the near-IR region.
© 2014 Optical Society of America
Surface plasmon resonance (SPR), defined as the collective oscillation of conduction electrons, dominates the optical spectra of noble metal nanoparticles (NPs) . This interesting phenomenon makes NPs attractive for many applications such as optoelectronics , medicine , sensing , catalysis  and solar energy conversion . However, a precise control of SPR intensity and position is required in order to optimize the NPs for each particular application. SPR can be controlled either through changes in geometrical factors such as size and shape , or by modifying their composition . The nearly endless number of tuning possibilities makes simulations very important for finding optimal configurations. Hence, the wide availability of good data sets of optical constants is of paramount importance for the design of plasmonic structures and devices.
Optical properties of metals have been measured since the beginning of the twentieth century, when it became apparent that the actual optical constants deviated from Drude’s model . Nowadays, the rise of plasmonics has spawned a renewed interest in the optical properties of the noble metals and their alloys. Unfortunately, good experimental data sets of optical properties are not as easy to find for the alloys as for the pure metals [10,11]. This is a great limitation since the composition is a very important variable for controlling the optical response of nanocomposites. For instance, silver has a very intense SPR but lacks chemical stability whereas for gold the opposite is true. Hence, by alloying both metals it is possible to obtain structures with a fair compromise between chemical stability and good optical response.
Lack of reliable optical data sets is even true for Au-Ag alloys, the two most important metals in plasmonics. In this case one can actually find some experimental reports of the optical properties [12–14] but often they apply only to either a small subset of compositions, a narrow spectral range or, more importantly, cannot reproduce accurately the experimental results, which probably indicates that they were obtained from low-quality samples. This lack of data considerably limits the possibility of performing optical calculations to predict the most interesting range of concentration for any given application. Hence, the researchers in the field have had to resort to solutions as inaccurate as using the weighted average of the pure metals [8,15]. This approach is clearly incorrect because for the alloy films the energy of the transitions changes with the composition, which makes it impossible to study their optical properties through a simple effective medium analysis but, surprisingly, it often gives better results than the reported optical properties. It is clear, then, that there exists a great need for rigorous and systematic experimental data sets of optical constants for noble metal alloys.
In this work, we have systematically determined the optical constants for AgxAu1-x alloy thin films, with compositions ranging from pure gold (x = 0) to pure silver (x = 1). Thin films were fabricated by electron-beam deposition, and their compositions and thicknesses were measured using Rutherford backscattering (RBS). Refractive indices were determined by means of spectroscopic ellipsometry (SE). Our experimental results were compared with values available in the literature. Finally, we performed Mie calculations [16,17] with the obtained optical data sets. Excellent agreement was obtained between our calculations and experimental reports of AgxAu1-x alloy nanoparticles. We are confident that the experimental data sets presented in this paper will be a reference for future works on Ag-Au alloys, in particular, for accurate simulations of plasmonic structures.
2.1. Specimen preparation and characterization
AgxAu1-x alloy thin films were co-deposited on silicon substrates by an electron beam evaporation method at room temperature under high vacuum (base pressure: 6.7x10−6 Pa). The purity of Au and Ag source material was more than 99.9%. The deposition rate for Ag and Au was in the range of 0.5 and 4.5 nm/s, respectively. The thickness of the films was determined by the combination of Rutherford backscattering spectrometry (RBS) measurement with 7.5 MeV 4He ions, and RBS analysis program package of RUMP . A suite of thin films were prepared with varying Ag:Au ratios. Thin film thickness was limited to 170–330 nm range.
2.2. Spectroscopic ellipsometry
The optical properties of the AgxAu1-x thin films were studied by means of spectroscopic ellipsometry (SE). Ellipsometric measurements were carried out at room temperature with an angle of incidence (with respect to the normal of the sample) of 75°, using a spectroscopic ellipsometer of the rotating compensator type (M-2000FI by J.A. Woollam Inc.), in the photon energy range 0.73 - 5.8 eV. Ellipsometry measures the complex reflectance ratio ρ between the Fresnel reflection coefficients (rp and rs) for light polarized parallel and perpendicular to the plane of incidence, respectively. That ratio is usually written as ρ = rp/rs = tan Ψ·exp(i·Δ). In the GES5E rotating polarizer ellipsometer the magnitudes obtained at each energy point are tan Ψ and cos Δ. The film's optical properties can be obtained from SE data with a model-based technique by minimizing the difference between the measured spectrum of the ellipsometric parameters and the values calculated from the model (i.e., the reduced χ2 function ). Additional details can be found elsewhere .
A three-phase multilayer model (Ambient/AgxAu1-x + Voids/AgxAu1-x, schematically represented in the inset of Fig. 1(a)) was used to analyze the ellipsometric spectra of the AgxAu1-x thin films. In this model, the second and third layers represent the surface roughness, which was assumed to be a physical mixture of voids and AgxAu1-x, and the dense region of the alloy thin film, respectively. The silicon substrate was not considered in this model because the films were thick enough to be optically opaque. No evidence of anisotropy was detected and therefore the optical properties of the alloy films were taken to be isotropic. The Bruggeman effective medium approximation (EMA)  was used to calculate the refractive index of the layer roughness representing the surface (nEMA) from the optical properties of the constituent elements. Moreover, we used a value of 1/3 for the depolarization factor (G), corresponding to an isotropic medium with spherical inclusions. After some trials, we found that the best fits were obtained by fixing the thickness of the roughness layer to 1.5 nm and its composition to equal fractions of vacuum and alloy. The complex dielectric function (εa = (na)2 = ε1 + i ε2) of the alloy film was represented by means of cubic B-splines. The main advantage of this method over analytical representations of the dielectric function is that it favors convergence of the fitting procedures . Moreover, we have used in this work a set of B-splines that is Kramers-Kronig (K-K) compliant , which ensures that the parameterized dielectric function has a physical meaning.
2.3. Basis splines
B-splines (i.e., Basis-splines) are a recursive set of polynomial splines. Their basis function is defined as:23]. Note that the basis function is zero outside the interval [ti, ti + k + 1]. Then, the total spline curve S(x), representing the dielectric function of the material, is given by:Eq. (3) is bounded by the control points. Then, if we parameterize a dielectric function with B-splines, enforcing ε2(ω) ≥ 0 is simply a matter of limiting the B-spline coefficients, ci ≥ 0. Moreover, it has been shown that B-splines can be made K-K compliant  and expressions to calculate ε1 from the values of ε2 exist. This reduces the number of fitting parameters by half and, consequently, also reduces the probability of correlation between them. Kramers-Kronig consistency requires that ε2 goes smoothly to zero; this can be enforced by choosing knots at appropriate locations outside the measured range. The number of coefficients used to accurately describe the dielectric function should be kept as low as possible, while still following the shape of the function. Although additional coefficients can provide a better description, too many points can produce an unrealistic result in which only the noise is better described. It also increases the possibility of correlation between the coefficients.
3. Results and discussion
Thin film thickness and composition was determined from the RBS spectra (not shown), using RUMP spectroscopy analysis; calculated values are listed in Table 1.As can be seen, the film thickness is similar for all samples and silver fraction (x) varies almost linearly from 0 (pure gold) to 1 (pure silver). Moreover, scanning electron microscope (SEM) micrographs of the thin film alloys (not shown) revealed that thin film specimens have a uniform surface and cross section morphology.
Refractive index values of AgxAu1-x alloys were determined from the fit of the ellipsometric spectra. It should be noted that for each concentration we fabricated five different samples and in all cases the ellipsometric spectra for the same concentration did not vary over 5%. This result shows that the reproducibility of the combined fabrication and measurement techniques used in this work is excellent. A typical fit, corresponding to the Ag0.48Au0.52 alloy, is depicted in Fig. 1(a); as can be seen, the quality of the fits is also very good. Reassuringly, the calculated complex refractive index values for the pure metals closely match the values reported in the literature. For instance, a comparison of our data with those reported by Johnson and Christy  (JC) is shown in Fig. 1(b). Both, the position of the onset for interband transitions and the actual values of n and k, are virtually identical within the limit of the experimental errors. The only differences arise for silver, where our values of n for energies below 1.5 eV are slightly larger than those by JC. We speculate that this discrepancy can come from differences in the samples or because they used a direct point-by-point calculation that does not enforce K-K compliance.
The onset for interband transition of experimentally determined n and k values (Fig. 2), varies smoothly as a function of the composition between the values of the pure metals, which is in line with previously reported results [8,24,25]. However, this linear behavior is not present in the near-IR region. It is apparent in Fig. 2(a) that the values of n for the alloys in this energy zone are higher than those of both pure metals. This is a very reproducible effect and it is more pronounced for the compositions with similar content of silver and gold. A similar trend has been reported recently by Nishijima and Akiyama  but in their case it was present in the values of both, n (increase) and k (decrease).
This unexpected behavior is very important because, as we discussed in the introduction, the optical properties of the alloys are often calculated as the weighted average of those of the constituents. In order to further analyze this problem, experimental n and k values for the Ag0.48Au0.52 alloy are shown in Fig. 3, together with those calculated as the arithmetical average of Au and Ag. It is clear in Fig. 3 that the averaged optical constant is not even close to the actual values in the region < 4 eV because (i) it is not able to reproduce the variations of the onset of interband transitions in the alloys that shifts roughly linearly as a function of the composition and (ii) it cannot account for the anomalous increase of n beyond the values of the pure metals.
In order to better analyze this effect, we fitted n and k values using the Drude model , ε(ω) = ε∞ - ωp2/(ω2 + i ω Γ); where ε∞ is the high-frequency limit dielectric constant, ωp is the Drude plasma frequency and Γ = h/τ is the damping constant (h is the Plank constant and τ is relaxation constant). Fits were performed in the energy region below interband transitions; i.e., ~3.1 eV for silver, ~2.1 eV for gold and for the alloys varying linearly between those values as a function of the composition. The same analysis was applied to the data reported by JC . Calculated values of ε∞ and Γ are shown in Fig. 4, for all alloy compositions. The high-frequency dielectric constant (Fig. 4(a)) varies almost linearly with Ag fraction, whereas the calculated plasma frequency (not shown, the best-fit equation is ωp(xAg) = 8.96 + 0.02*xAg) is almost constant regardless of the alloy composition, which is not surprising since the plasma frequency of silver and gold are very similar. Variations of the damping constant (Fig. 4(b)), on the other hand, exhibit a much more interesting behavior: it has relatively low values for the pure metals and increases for the alloys, with a maximum around the Ag0.5Au0.5 composition. A quadratic polynomial fits very well the values of Γ.
Our results can be understood if we consider that the electrical resistivity is usually larger for the alloys, due to the introduction of additional impurities that produce an increase in the number of different scattering mechanisms; this is the so-called Matthiessen's rule . However, the rise of the damping constant produced by this effect is expected to be linear whereas in our case it clearly follows a quadratic curve. The additional contribution might be explained from the differences in electronegativity between gold and silver (2.54 and 1.93, respectively, in the Pauling scale). Hence, a considerable transfer of electronic density from Ag to Au (Agδ+-Auδ-) is expected for the alloys, with a maximum for the Ag0.5Au0.5 composition. This charge transfer will likely stress and/or distort the lattice, producing the additional damping observed in our samples.
It should be recalled that Nishijima and Akiyama  (NA) have reported previously a similar effect for the AgxAu1-x alloy but there are important differences between their results and ours. Firstly, they observed deviations of both n and k in the near-IR region whereas in our case the extinction coefficient follows the expected trend. Secondly, in the Drude analysis, they found larger changes in the plasma frequency than in the damping constant, but we only see meaningful changes in the latter. It is not clear for us the reason for the discrepancies, but one can speculate that they are coming either from differences in the samples or in the analysis. In the former case, we have fabricated thick (170-330 nm) and wide-area films (~6x6 mm2), but their alloy films are very thin (20-30 nm); hence, probably their samples are affected by some size effects that are not likely present in ours. As for the analysis, it would be important to know if they enforced K-K compliancy or not. In case they did not, then their optical constants can be somewhat modified from the real values. More importantly, the large variations in ωp observed by NA are very hard to explain, even resorting to the differences in electronegativity, because Ag and Au have a very similar plasma frequency. More likely is that the charge transfer can affect the collision frequency, due to the stress introduced in the lattice.
Finally, we have compared the calculated extinction spectra for spherical nanoparticles (using Mie theory [16,17]) with the experimental results reported by Kuladeep et al.  (see Fig. 5). It can be seen in Fig. 5(a) that with our optical constants the SPR position varies linearly with the composition between the values of both pure metals. In no way do we see the back and forth oscillations of the SPR position reported by NA. Moreover, if we compare the predicted peak position with those obtained experimentally by Kuladeep et al., they follow exactly the same trend. This gives us confidence that the optical calculations carried out with our optical constants are reliable. It is remarkable the spectral tunability of the SPR position provided by the composition alone (from 410 nm to 530 nm approximately). It is not hard to foresee that if they are combined with additional shape variations (e.g., nanoshells, nanorods, etc.), alloy nanostructures have a great potential for sensing applications.
In summary, we have carried out a systematic determination of the optical constants for the AgxAu1-x alloys in the full composition range. It has been shown that the weighted average of the refractive indices of the pure metals fails to represent those of the alloys, not only in the region near the onset for interband transitions but also (and somewhat surprisingly) in the near-IR region. The latter effect can be explained partly by the reduction of the relaxation time in the alloys by the addition of impurities and electron scattering and partly by the additional effect of the electronegativity difference between both metals that produces a charge transfer from silver to gold and stresses the alloy lattice, further increasing the electron scattering and, consequently, the damping constant. Finally, we have shown that the optical data set reported in this work can accurately reproduce experimental results. Hence, we consider that it will be very useful for the scientific community working in plasmonics, in order to predict the optical response of alloy nanostructures, saving time and experimental resources.
The authors thank E. Bringa for fruitful discussions. J.K. Baldwin, C. Sheehan and the Center for Integrated Nanotechnologies (CINT) are acknowledged for synthesizing the samples and SEM characterization. E.G. Fu, Y.Q. Wang and the Ion Beam Materials Laboratory (IBML) team are also acknowledged, for their help performing ion irradiations and RBS analysis. OPR is grateful with Moncloa Campus of International Excellence (UCM-UPM) for the PICATA postdoctoral fellowship. This work was partially funded by the Los Alamos Laboratory Directed Research and Development (LDRD) Program and by the Spanish ministry MINECO, projects AIC-A-2011-0718 and MAT-2012-38541.
References and links
2. P. Mazzoldi, G. W. Arnold, G. Battaglin, F. Gonella, and R. F. Haglund Jr., “Metal nanocluster formation by ion implantation in silicate glasses: Nonlinear optical applications,” J. Nonlinear Opt. Phys. Mater. 5(02), 285–330 (1996). [CrossRef]
3. L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U.S.A. 100(23), 13549–13554 (2003). [CrossRef] [PubMed]
5. F. Tihay, G. Pourroy, M. Richard-Plouet, A. C. Roger, and A. Kiennemann, “Effect of Fischer-Tropsch synthesis on the microstructure of Fe-Co-based metal/spinel composite materials,” Appl. Catal. Gen. 206(1), 29–42 (2001). [CrossRef]
7. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107(3), 668–677 (2003). [CrossRef]
8. S. Link, Z. L. Wang, and M. A. El-Sayed, “Alloy formation of gold-silver nanoparticles and the dependence of the plasmon absorption on their composition,” J. Phys. Chem. B 103(18), 3529–3533 (1999). [CrossRef]
9. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998).
10. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
11. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1997).
12. H. Fukutani, “Optical constants of silver-gold alloys,” J. Phys. Soc. Jpn. 30(2), 399–403 (1971). [CrossRef]
13. K. Ripken, “Die optischen konstanten von Au, Ag und ihren legierungen im energiebereich 2, 4 bis 4, 4 eV,” Z. Für Phys. Hadrons Nucl. 250(3), 228–234 (1972). [CrossRef]
14. J. Rivory, “Comparative study of the electronic structure of noble-metal-noble-metal alloys by optical spectroscopy,” Phys. Rev. B 15(6), 3119–3135 (1977). [CrossRef]
15. P. Mulvaney, “Surface plasmon spectroscopy of nanosized metal particles,” Langmuir 12(3), 788–800 (1996). [CrossRef]
16. G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 330(3), 377–445 (1908). [CrossRef]
17. O. Peña and U. Pal, “Scattering of electromagnetic radiation by a multilayered sphere,” Comput. Phys. Commun. 180(11), 2348–2354 (2009). [CrossRef]
18. M. Thompson, “RUMP: Rutherford Backscattering Spectroscopy analysis package,” http://www.genplot.com/RUMP/index.htm.
19. G. E. Jellison Jr., “Data analysis for spectroscopic ellipsometry,” Thin Solid Films 234(1-2), 416–422 (1993). [CrossRef]
20. O. Peña-Rodríguez, C. F. Sánchez-Valdés, M. Garriga, M. I. Alonso, X. Obradors, and T. Puig, “Optical properties of Ceria-Zirconia epitaxial films grown from chemical solutions,” Mater. Chem. Phys. 138(2-3), 462–467 (2013). [CrossRef]
21. D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20(8), 3292–3302 (1979). [CrossRef]
22. M. Garriga, M. I. Alonso, and C. Domínguez, “Ellipsometry on very thick multilayer structures,” Phys. Status Solidi B 215, 247–251 (1999). [CrossRef]
23. B. Johs and J. S. Hale, “Dielectric function representation by B-splines,” Phys. Status Solidi A 205(4), 715–719 (2008). [CrossRef]
24. R. Kuladeep, L. Jyothi, K. S. Alee, K. L. N. Deepak, and D. N. Rao, “Laser-assisted synthesis of Au-Ag alloy nanoparticles with tunable surface plasmon resonance frequency,” Opt. Mater. Express 2(2), 161–172 (2012). [CrossRef]
25. Y. Nishijima and S. Akiyama, “Unusual optical properties of the Au/Ag alloy at the matching mole fraction,” Opt. Mater. Express 2(9), 1226–1235 (2012). [CrossRef]
26. H.-L. Engquist and G. Grimvall, “Electrical transport and deviations from Matthiessen’s rule in alloys,” Phys. Rev. B 21(6), 2072–2077 (1980). [CrossRef]