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Modelling the dielectric function of Au-Ag alloys
Journal of Alloys and Compounds 694 (2017) 857e863
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Modelling the dielectric function of Au-Ag alloys ~ a-Rodríguez Ovidio Pen n Nuclear (UPM), C/Jos Instituto de Fusio e Guti errez Abascal 2, E-28006, Madrid, Spain
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 May 2016 Received in revised form 4 October 2016 Accepted 10 October 2016 Available online 11 October 2016
Two analytical models are proposed to represent the dielectric function of the AgxAu1-x alloys. The intraband effects are represented by the Drude model whereas interband transitions are modelled by means of either Lorentzian-pole pairs or critical-point terms. Quantitative comparisons are presented to show that, for all the alloy compositions, both models describe very well the experimental data reported ~ a-Rodríguez et al. [Opt. Mater. Express 4 (2014) 403e410] in the energy region between 0.65 and by Pen 6.6 eV. Accuracy of the critical-point terms is considerably better but Lorentzian-pole pairs can be easier to integrate in existing FDTD software. © 2016 Elsevier B.V. All rights reserved.
Keywords: Silver-gold alloys Optical constants Refractive index Drude-Lorentz Drude-critical points
1. Introduction A considerable amount of research has been devoted recently to study the optical properties of metallic nanoparticles (NPs). Most of this interest arises from the phenomenon known as localized surface plasmon resonance (LSPR), which makes them useful for several applications such as surface enhanced Raman scattering (SERS) [1,2], all-optical switching devices [3], and biomedical diagnosis [4]. Shape, size and composition are all viable ways for controlling the characteristics of the LSPR and, due to all the possible variations, several models have been used for calculating the optical response of the NPs. Among the most commonly employed are the Mie theory [5], the transition matrix (TM) [6], the discrete dipole approximation (DDA) [7] and the finite-difference time-domain (FDTD) [8,9], all of which have advantages and disadvantages for certain cases. However, calculating the optical response of a given system is a two-part problem: one has to know first the dielectric function (or the complex refractive index) of the constituent materials and only then, the actual calculation can be performed. The values for the dielectric function can be taken either from theoretical models or from experimental results. An example of the first case is the Drude model (DM) [10], which is frequently used to describe the dielectric functions of many free-electron-like metals. Unfortunately the noble metals and their alloys present interband transitions which
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jallcom.2016.10.086 0925-8388/© 2016 Elsevier B.V. All rights reserved.
are not considered in the DM and therefore its results considerably deviate from experimental values in this zone of the spectrum. Experimental results, on the other hand, abound in the literature [11e13] for most useful materials and interpolations of the experimental data can be readily used for most types of calculations. However, this is not always the case because there are some algorithms, like the FDTD method, where analytic representations of the frequency dependent dielectric functions, ε(u), are required. Additionally, an analytical expression can also be useful for understanding situations where the intrinsic parameters of the metal might be modified by external perturbations. In this paper we apply two different analytical models for the calculation of the dielectric function of the AgxAu1-x alloys. The first model includes a Drude term and five Lorentzian-pole pairs (DL model) whereas in the second one the Lorentzian terms are replaced by two critical-point terms (DCP model). Both models describe reasonably well, for all alloy compositions, the experi~ a-Rodríguez et al. [13] in the energy mental data reported by Pen region between 0.65 and 6.6 eV. The accuracy of the DCP model is higher than that of the DL one but the latter is easier to integrate in existing FDTD software. 2. Analytical models 2.1. Drude-Lorentz model Analytical representations of the frequency-dependent dielectric function of metals can be found in various works in the
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858
literature, which is indicative of the usefulness of those expressions. The more common approach consists in separating the intraband effects (i.e., free-electron effects) from interband effects (bound-electron effects) [14e16]. The contributions from free electrons to ε(u) can be described by the Drude model [17]:
u2p εf ðuÞ ¼ ε∞ 2 ; u þ iuGp
absorption are not allowed. According to this model the interband transitions can be represented by means of two critical points of M1 and M2 type [32]:
" εCP1 ðuÞ ¼ A1
n X
(1)
j¼1
εCP2 ðuÞ ¼
Aj
u2j u2 iuGj
;
(2)
where n is the number of oscillators with frequency uj, strength Aj, and lifetime 1/Gj. Now, the final expression to represent the dielectric function is:
εðuÞ ¼ εf ðuÞ þ εb ðuÞ ¼ ε∞
n u2p X Aj þ : u2 j¼1 u2 u2 þ iuGj
(3)
j
Examples of the usage of the DL model abound in the literature, mainly for the noble metals. For instance, See et al. [20], Lee and Gray [21] and Moskovits et al. [22] have modelled the optical properties of silver by using, respectively, two, three and four Lorentzian terms. Likewise, Hao and Nordlander [23] and Rakic et al. [16] have fitted the dielectric functions of both Au and Ag using four and five terms, respectively. Unfortunately none of the above mentioned models is completely satisfactory; probably the best of them is the one by Hao and Nordlander [23] but even they have problems reproducing the data; for example, at the interband threshold of Ag, between 4 and 4.5 eV. Of course one can add even more terms until the ε(u) curves (and indeed any curve) are fitted but then the physical meaning will be lost. For the DL model used in this work we have used five Lorentzian terms (n ¼ 5). 2.2. Drude-critical points model Recently, Etchegoin et al. [24,25] have proposed a different approach: replacing the Lorentzian terms with a family of analytical models, called critical points (CPs), which have long been used for the analysis of interband transitions in semiconductors [26]. The advantage of using the CPs is that, unlike simple Lorentz oscillators, they allow for an easy adjustment of an asymmetric line shape [24,27]. Etchegoin and collaborators, as well as some other authors like Vial and Laroche [28e30] and Lu and Chang [31], have successfully employed this model with two terms to represent the optical properties of various types of noble and transition metals. This model, however, has some problems because the phase parameter can cause the imaginary part of ε for one or both of these modified Lorentzian functions to become negative, which is an unphysical effect for a metal. Rioux and collaborators [32] have developed a model based in the same approach but they circumvent the problems of Etchegoin's model by taking into account the band structure of the metals and, consequently, negative values of
ðu þ iG1 Þ
2 !
ln 1 u01 2ðu þ iG1 Þ2 ! pffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 u1 1 tanh u1 u01=u þ 1 ðu þ iG1 Þ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u þ iG1 u1 1 u1 u01 tan u þ iG1 u1 ðu þ iG1 Þ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !# u þ iG1 u1 u1 u01 1 tanh ; u þ iG1 u1 ðu þ iG1 Þ2
being i the imaginary number, ε∞ the high-frequency limit dielectric constant, up the plasma frequency, and Gp the plasma (bulk) damping term. Gp is related with the mean free path of electrons in the bulk material, le, by the expression Gp ¼ vF/le [18,19], where vF is the Fermi velocity. There are several ways of representing interband transitions but the most common one consists in using a certain number of Lorentzian terms. We will call this approach the Drude-Lorentz model:
ε b ð uÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1 u01
A2 2ðu þ iG2 Þ
! ðu þ iG2 Þ 2 ln 1 ; 2
(4a)
u02
(4b)
where, for each transition j ¼ 1, 2, Aj is the amplitude, u0j the threshold, uj the gap and Gj the broadening. Finally, the frequencydependent dielectric function in this model is:
u2p þ εCP1 ðuÞ þ εCP2 ðuÞ: u2 þ iuGp
εðuÞ ¼ ε∞
(5)
The form of the dielectric function described by equation (5) conforms to the requirements of the Kramers-Kronig (KK) relations [22].
3. Results and discussion In order to fit the experimental datasets using the DL and DCP models, we have employed the chi-square distribution as the objective function:
c2 ðx1 ; …; xk Þ ¼
" 2 N nci ðx1 ; …; xk Þ nei 1 X n e 2N i¼1 i c # ki ðx1 ; …; xk Þ kei þ ke
(6)
i
where N is the number of points used for the fit, x1, …, xk all the variables of the theoretical model and nci and kci (nei and kei ) the values of the real and imaginary parts of the calculated (experimental) complex refractive index. Then, the optimization process was performed using the limited memory Broyden-FletcherGoldfarb-Shanno (L-BFGS-B) multivariate optimization algorithm [33,34]. Lower and/or upper limits are set for all the variables, in order to ensure that their optimized values have physical meaning (e.g., positive plasma frequency, etc.). The optimization process is finished when one of the following criteria is met:
kGk k εG ;
(7)
2 ðiþ1Þ 2 ðiÞ ðiÞ ðiþ1Þ εF $max c2 c c ; c2 ;1 ;
(8)
ðiþ1Þ ðiÞ xk εX ; xk
(9)
where εG, εF and εX are positive numbers which define a precision of search, k$k means Euclidian norm, Gk, gradient projection onto a
~ a-Rodríguez / Journal of Alloys and Compounds 694 (2017) 857e863 O. Pen
variable xk and the super index represents the iteration number. Additionally, the optimization can be finished after a certain number of iterations. For the fittings presented in this work no limit was set for the number of iterations and the precision of search was taken as: εG ¼ εF ¼ εX ¼ 108. ~ a-Rodríguez et al. [13], The experimental data reported by Pen for different compositions of the silver-gold alloys have been used in this work to obtain the model parameters. Those optical properties were obtained experimentally using spectroscopic ellipsometry measurements on thin films fabricated by electron beam evaporation. This dataset (freely available at zenodo.org, DOI: 10.5281/zenodo.50484) has the advantage of covering a wide spectral range in the UV-visible region, being KK consistent and including the full compositional range. The complex refractive index was used for the calculation of c2 instead of the dielectric ~ a-Rodríguez function because this is the quantity reported by Pen et al. [13] but this selection is not really important, since similar results are obtained by fitting ε(u) instead. It should be noted that Rioux et al. [32] applied their model to the AuxAg1-x alloy but their dataset is restricted to only three alloy compositions (plus the pure metals) and a narrow spectral range (1e4.5 eV). Hence, the
859
~ a-Rodríapplication of their model to the dataset reported by Pen guez et al. [13] should provide a much more realistic representation of the optical properties of the gold-silver alloys. Some representative fits using both models are depicted in Fig. 1 and the model parameters yielding the best fits are shown in Tables 1 and 2 for the DL and DCP models, respectively. It can be seen in Fig. 1 that the fit of the experimental data is rather good with both models. There are, however, some inaccuracies in the fits, particularly in the spectral region close to the onset for interband transitions, where the DL model has some clear oscillations. The DCP model, on the other hand, fits better the data but it also exhibits some deviations in the same region. For a more quantitative assessment of the quality of the fits we also plotted the minimum values of c2 as a function of the silver fraction (Fig. 2). This plots confirms that the DCP model yields considerably better fits. In addition, it can be seen there that for both models the best fits are obtained for the alloys with similar quantities of Ag and Au. Finally, is it worth noting that in spite of the differences between the models and the experimental data, the fits obtained with both models are perfectly capable of reproducing all the important variations of the optical properties for all the alloys.
Fig. 1. Best fits of the complex refractive indices of some AgxAu1-x alloys using the DL and DCP models, for some representative compositions with silver fractions (xAg) of (a) 0.00, (b) 0.15, (c) 0.35, (d) 0.48, (e) 0.74 and (f) 1.00. The insets are a zoom of their respective plots in the onset of interband transitions of the alloys.
~ a-Rodríguez / Journal of Alloys and Compounds 694 (2017) 857e863 O. Pen
860
Table 1 ~ a-Rodríguez and collaborators [13]. Parameters in Eq. (3), obtained from the best fit with the DL model of the data reported by Pen xAg
0.00
0.08
0.15
0.24
0.35
0.48
0.66
0.74
0.85
1.00
ε∞
u5 [eV] G5 [eV]
3.104 8.769 0.089 3.624 2.784 0.414 5.113 3.183 0.600 13.37 3.799 0.980 21.07 4.695 1.630 37.22 6.349 2.751
3.210 8.974 0.090 3.520 2.822 0.445 5.717 3.243 0.652 14.14 3.886 1.037 21.76 4.761 1.669 36.15 6.408 2.800
3.025 8.932 0.117 3.823 2.910 0.512 6.198 3.394 0.723 13.30 4.008 1.054 20.09 4.914 1.694 34.46 6.525 2.653
2.924 8.880 0.158 3.422 2.980 0.531 6.454 3.495 0.753 12.49 4.104 1.061 18.68 5.006 1.686 30.93 6.576 2.534
2.899 8.956 0.160 3.241 3.071 0.570 6.728 3.608 0.793 12.28 4.218 1.089 17.64 5.114 1.685 28.20 6.641 2.432
2.711 8.893 0.183 2.366 3.190 0.527 6.136 3.722 0.758 10.88 4.323 1.046 16.08 5.193 1.608 24.93 6.645 2.257
2.476 8.703 0.169 2.480 3.478 0.707 6.821 4.022 0.867 10.24 4.628 1.157 13.36 5.481 1.651 20.56 6.874 2.149
2.390 8.773 0.149 2.924 3.681 0.739 5.891 4.163 0.798 9.234 4.728 1.054 13.24 5.560 1.501 21.72 6.891 1.626
2.456 8.902 0.115 3.899 4.014 0.757 5.942 4.402 0.734 8.285 4.973 0.956 11.57 5.726 1.352 17.93 6.892 1.674
2.503 9.015 0.079 2.621 4.217 0.286 5.484 4.608 0.532 7.080 5.160 0.700 10.02 5.847 1.001 15.13 6.868 1.222
c2
1.160
0.735
0.686
0.597
0.443
0.428
0.114
0.131
0.301
1.106
up [eV] Gp [eV] A1
u1 [eV] G1 [eV]
A2
u2 [eV] G2 [eV]
A3
u3 [eV] G3 [eV]
A4
u4 [eV] G4 [eV]
A5
Table 2 ~ a-Rodríguez and collaborators [13]. Parameters in Eq. (5), obtained from the best fit with the DCP model of the data reported by Pen xAg
0.00
0.08
0.15
0.24
0.35
0.48
0.66
0.74
0.85
1.00
ε∞
u2 [eV] G2 [eV]
2.568 8.905 0.109 56.39 2.342 2.714 0.125 32.13 3.516 0.264
2.621 9.115 0.108 56.5 2.363 2.77 0.149 33.02 3.564 0.257
2.522 9.062 0.135 52.67 2.428 2.82 0.166 33.78 3.568 0.292
2.429 8.986 0.174 51.38 2.525 2.866 0.195 33.68 3.583 0.289
2.398 9.056 0.175 46.16 2.58 2.951 0.19 34.49 3.626 0.289
2.193 8.97 0.196 41.42 2.735 3.079 0.189 35.33 3.717 0.291
2.047 8.82 0.179 23.73 2.689 3.372 0.164 35.98 3.833 0.292
2.027 8.974 0.159 29.09 2.839 3.77 0.134 31.19 4.09 0.293
2.022 9.116 0.126 33.42 3.086 3.997 0.109 25.15 4.224 0.237
1.909 9.104 0.086 62.13 3.759 4.155 0.045 18.29 4.796 0.217
c2 ( 102)
0.656
0.359
0.413
0.381
0.269
0.269
0.059
0.093
0.158
0.304
up [eV] Gp [eV] A1
u01 [eV] u1 [eV] G1 [eV]
A2
Threshold (u0i), frequency (ui), and broadening (Gi) parameters obtained for all alloy compositions from the DL and DCP models are depicted in Figs. 3 and 4, respectively. In both cases the frequency of the transitions (Figs. 3a and 4a) have a linear behaviour and the plasma frequency, in particular, is almost constant. Variations of the
Fig. 2. Minimum c2 values obtained from the fits of the complex refractive indices of AgxAu1-x alloys using the DL and DCP models, function of the silver fraction. Continuous lines are just a guide to the eye.
damping constants (Figs. 3b and 4b), on the other hand, exhibit a much more interesting behaviour: they have relatively low values for the pure metals and increase for the alloys, with a maximum for the alloys with similar quantities of silver and gold. The exception of this trend is the last transition of the DL model. In all likelihood this term does not represents a physical transition but it is representing some of the higher-energy transitions grouped together, which would explain this different behaviour. A quadratic polynomial fits very well the amplitude (Ai) in the DCP model and all the values of Gi whereas all the other parameters exhibit a linear dependence on the silver fraction. The polynomial parameters yielding the best fits for all the model variables are shown in Table 3. The linear dependence on xAg observed for all the thresholds, u0j, and frequencies, uj, is consistent with the linear variation of the band separation described by Gaudry et al. [35]. The nearlyconstant plasma frequency can be explained by considering that Au and Ag have the same crystal structure and similar lattice constants. Hence, the density of valence electrons for both metals and their alloys is almost identical, resulting in similar plasma frequencies across all compositions. The quadratic dependence of Gp 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 [36]. Moreover, the differences in electronegativity between gold and silver (2.54 and 1.93,
~ a-Rodríguez / Journal of Alloys and Compounds 694 (2017) 857e863 O. Pen
Fig. 3. (a) Frequency (ui), and (b) broadening (Gi) parameters of DL model for the AgxAu1-x alloys as a function of the silver fraction, obtained from the fit of the n and k values. The continuous lines are either linear (a) or quadratic (c) fits of the data.
respectively, in the Pauling scale) result in a considerable transfer of electronic density from Ag to Au (Agdþ-Aud-), with a maximum for the Ag0.5Au0.5 composition [13,37]. This charge transfer will likely stress and/or distort the lattice, destroying the periodicity of the structure. This disorder lowers the mean free path of the conduction electrons for alloys, resulting in a higher value of Gp for alloys compared to pure metals. Finally, the increase in Gj for the alloys compared to pure metals, is similar to the behaviour of Gp. The most likely explanation in this case is that the non-periodic structure broadens every energy state, resulting in a global broadening of the interband electronic transitions. Finally, the fits obtained in this work for the pure metals are compared with the corresponding fits reported by Hao and Nordlander [23], Etchegoin et al. [24,25] and Rakic et al. [16] (Fig. 5). It should be noted that we have fitted the experimental data reported ~ a-Rodríguez at al [13], whereas in all the other cases was by Pen used the dataset of Johnson and Christy [11]. However, this should not be a problem for comparing the models because both experimental datasets are very similar, as can be seen in Fig. 5. All the models give reasonably good fits for gold (Fig. 5a) but the ones reported in this work (particularly the DCP model) are slightly better, mainly in the region around the onset of interband transitions. On the other hand, the models reported by Hao and Nordlander [23] and Rakic et al. [16] (Etchegoin et al. [24,25] only use their model for gold) completely fail to represent the refractive index of silver for energies above 3.8 eV whereas our models give good fits in all the spectral range of the experimental data. Hence, the theoretical models described in this paper not only are one of
861
Fig. 4. (a) Threshold (u0i), frequency (ui), and (b) broadening (Gi) parameters of DCP model for the AgxAu1-x alloys as a function of the silver fraction, obtained from the fit of the n and k values. The continuous lines are either linear (a) or quadratic (c) fits of the data.
Table 3 Parameters of the polynomial fit for the thresholds (u0i), frequency (ui), and broadening (Gi) obtained for the DL and DCP models. Parameter
y ¼ a1 þ a2 xAg þ a3(xAg)2 DL model
ε∞
up [eV] Gp [eV] A1
u01 [eV] u1 [eV] G1 [eV]
A2
u02 [eV] u2 [eV] G2 [eV]
A3
u3 [eV] G3 [eV]
A4
u4 [eV] G4 [eV]
A5
u5 [eV] G5 [eV]
DCP model
a1
a2
a3
a1
a2
a3
2.143 8.877 0.076 3.548 e 2.659 0.354 6.001 e 3.139 0.582 14.21 3.770 0.953 21.77 4.700 1.599 36.99 6.408 2.777
0.819 0.007 0.395 0.783 e 1.442 1.159 0.103 e 1.418 1.005 6.778 1.346 0.859 11.90 1.165 0.834 22.57 0.569 0.606
e e 0.396 e e e 1.057
2.614 8.996 0.096 64.64 2.231 2.561 0.128 30.59 3.364 e 0.256 e e e e e e e e e
0.747 0.033 0.378 106.6 1.107 1.513 0.327 30.64 1.071 e 0.202 e e e e e e e e e
e e 0.392 92.62 e e 0.413 42.07 e e 0.242 e e e e e e e e e
e e 1.012 e e 1.049 e e 1.370 e e 0.928
the few available for gold-silver alloys, they also represent the refractive index of the pure metals more accurately than most of
862
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Acknowledgements The author is grateful with Moncloa Campus of International Excellence (UCM-UPM) for the PICATA postdoctoral fellowship. References
Fig. 5. Comparison between the fits of the complex refractive index of gold (a) and silver (b) obtained in this work for the Drude-Lorentz (DL) and Drude-Critical Points (DCP) models and those reported by Hao and Nordlander (HN) [23], Etchegoin et al. (EA) [24,25] and Rakic et al. (RA) [16]. The models fit the experimental data reported ~ a-Rodríguez et al. (PR, closed symbols) [13] or by Johnson and Christy (JC, either by Pen open symbols) [11]. The insets are a zoom of their respective plots in the onset of interband transitions.
the models available in the literature.
4. Conclusions In this work we have proposed two theoretical models to represent the optical properties of the AgxAu1-x alloys using a Drude term to represent intraband transitions and either five Lorentzian or two critical-point terms to account for the interband transitions. We have shown that both models (especially the DCP) provide excellent fits of the experimental data over a wide spectral range going from 0.65 eV to 6.6 eV. Both models (particularly the DL) can straightforwardly be incorporated into the existing implementations of the FDTD algorithm. Availability of these theoretical models opens the door to better understand the optical properties of plasmonic alloys. Likewise, they will allow researchers in the field to perform a wide number of advanced calculations (mainly using the FDTD method) to predict the optical response of nanostructures for the full compositional range of the aforementioned alloys. This will facilitate the use of alloys in plasmonic applications, a topic that has great potential but so far has not been fully developed.
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Huffman, Absorption and Scattering of Light by Small Particles, Wiley-Interscience, 1998. [18] U. Kreibig, C.v. Fragstein, The limitation of electron mean free path in small silver particles, Z. Phys. 224 (1969) 307e323, http://dx.doi.org/10.1007/ BF01393059. € vel, S. Fritz, A. Hilger, U. Kreibig, M. Vollmer, Width of cluster plasmon [19] H. Ho resonances: bulk dielectric functions and chemical interface damping, Phys. Rev. B 48 (1993) 18178e18188, http://dx.doi.org/10.1103/ PhysRevB.48.18178. [20] K.C. See, J.B. Spicer, J. Brupbacher, D. Zhang, T.G. Vargo, Modeling interband transitions in silver nanoparticle-fluoropolymer composites, J. Phys. Chem. B 109 (2005) 2693e2698, http://dx.doi.org/10.1021/jp046687h. [21] T.-W. Lee, S. Gray, Subwavelength light bending by metal slit structures, Opt. Express 13 (2005) 9652e9659, http://dx.doi.org/10.1364/OPEX.13.009652. [22] M. Moskovits, I. Srnov a-Sloufova, B. Vlckova, Bimetallic Ag-Au nanoparticles: extracting meaningful optical constants from the surface-plasmon extinction spectrum, J. Chem. Phys. 116 (2002) 10435e10446, http://dx.doi.org/10.1063/ 1.1449943. [23] F. Hao, P. Nordlander, Efficient dielectric function for FDTD simulation of the optical properties of silver and gold nanoparticles, Chem. Phys. Lett. 446 (2007) 115e118, http://dx.doi.org/10.1016/j.cplett.2007.08.027. [24] P.G. Etchegoin, E.C. Le Ru, M. Meyer, An analytic model for the optical properties of gold, J. Chem. Phys. 125 (2006) 164705, http://dx.doi.org/10.1063/ 1.2360270. [25] P.G. Etchegoin, E.C. Le Ru, M. Meyer, Erratum: “An analytic model for the optical properties of gold” [J. Chem. Phys. 125, 164705 (2006)], J. Chem. Phys. 127 (2007) 189901, http://dx.doi.org/10.1063/1.2802403. [26] P. Etchegoin, J. Kircher, M. Cardona, Elasto-optical constants of Si, Phys. Rev. B 47 (1993) 10292e10303, http://dx.doi.org/10.1103/PhysRevB.47.10292. [27] M. Campoy-Quiles, G. Heliotis, R. Xia, M. Ariu, M. Pintani, P. Etchegoin, D.D.C. Bradley, Ellipsometric characterization of the optical constants of polyfluorene gain media, Adv. Funct. Mater. 15 (2005) 925e933, http:// dx.doi.org/10.1002/adfm.200400121. [28] A. Vial, T. Laroche, Description of dispersion properties of metals by means of
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Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom
Modelling the dielectric function of Au-Ag alloys ~ a-Rodríguez Ovidio Pen n Nuclear (UPM), C/Jos Instituto de Fusio e Guti errez Abascal 2, E-28006, Madrid, Spain
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 May 2016 Received in revised form 4 October 2016 Accepted 10 October 2016 Available online 11 October 2016
Two analytical models are proposed to represent the dielectric function of the AgxAu1-x alloys. The intraband effects are represented by the Drude model whereas interband transitions are modelled by means of either Lorentzian-pole pairs or critical-point terms. Quantitative comparisons are presented to show that, for all the alloy compositions, both models describe very well the experimental data reported ~ a-Rodríguez et al. [Opt. Mater. Express 4 (2014) 403e410] in the energy region between 0.65 and by Pen 6.6 eV. Accuracy of the critical-point terms is considerably better but Lorentzian-pole pairs can be easier to integrate in existing FDTD software. © 2016 Elsevier B.V. All rights reserved.
Keywords: Silver-gold alloys Optical constants Refractive index Drude-Lorentz Drude-critical points
1. Introduction A considerable amount of research has been devoted recently to study the optical properties of metallic nanoparticles (NPs). Most of this interest arises from the phenomenon known as localized surface plasmon resonance (LSPR), which makes them useful for several applications such as surface enhanced Raman scattering (SERS) [1,2], all-optical switching devices [3], and biomedical diagnosis [4]. Shape, size and composition are all viable ways for controlling the characteristics of the LSPR and, due to all the possible variations, several models have been used for calculating the optical response of the NPs. Among the most commonly employed are the Mie theory [5], the transition matrix (TM) [6], the discrete dipole approximation (DDA) [7] and the finite-difference time-domain (FDTD) [8,9], all of which have advantages and disadvantages for certain cases. However, calculating the optical response of a given system is a two-part problem: one has to know first the dielectric function (or the complex refractive index) of the constituent materials and only then, the actual calculation can be performed. The values for the dielectric function can be taken either from theoretical models or from experimental results. An example of the first case is the Drude model (DM) [10], which is frequently used to describe the dielectric functions of many free-electron-like metals. Unfortunately the noble metals and their alloys present interband transitions which
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jallcom.2016.10.086 0925-8388/© 2016 Elsevier B.V. All rights reserved.
are not considered in the DM and therefore its results considerably deviate from experimental values in this zone of the spectrum. Experimental results, on the other hand, abound in the literature [11e13] for most useful materials and interpolations of the experimental data can be readily used for most types of calculations. However, this is not always the case because there are some algorithms, like the FDTD method, where analytic representations of the frequency dependent dielectric functions, ε(u), are required. Additionally, an analytical expression can also be useful for understanding situations where the intrinsic parameters of the metal might be modified by external perturbations. In this paper we apply two different analytical models for the calculation of the dielectric function of the AgxAu1-x alloys. The first model includes a Drude term and five Lorentzian-pole pairs (DL model) whereas in the second one the Lorentzian terms are replaced by two critical-point terms (DCP model). Both models describe reasonably well, for all alloy compositions, the experi~ a-Rodríguez et al. [13] in the energy mental data reported by Pen region between 0.65 and 6.6 eV. The accuracy of the DCP model is higher than that of the DL one but the latter is easier to integrate in existing FDTD software. 2. Analytical models 2.1. Drude-Lorentz model Analytical representations of the frequency-dependent dielectric function of metals can be found in various works in the
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858
literature, which is indicative of the usefulness of those expressions. The more common approach consists in separating the intraband effects (i.e., free-electron effects) from interband effects (bound-electron effects) [14e16]. The contributions from free electrons to ε(u) can be described by the Drude model [17]:
u2p εf ðuÞ ¼ ε∞ 2 ; u þ iuGp
absorption are not allowed. According to this model the interband transitions can be represented by means of two critical points of M1 and M2 type [32]:
" εCP1 ðuÞ ¼ A1
n X
(1)
j¼1
εCP2 ðuÞ ¼
Aj
u2j u2 iuGj
;
(2)
where n is the number of oscillators with frequency uj, strength Aj, and lifetime 1/Gj. Now, the final expression to represent the dielectric function is:
εðuÞ ¼ εf ðuÞ þ εb ðuÞ ¼ ε∞
n u2p X Aj þ : u2 j¼1 u2 u2 þ iuGj
(3)
j
Examples of the usage of the DL model abound in the literature, mainly for the noble metals. For instance, See et al. [20], Lee and Gray [21] and Moskovits et al. [22] have modelled the optical properties of silver by using, respectively, two, three and four Lorentzian terms. Likewise, Hao and Nordlander [23] and Rakic et al. [16] have fitted the dielectric functions of both Au and Ag using four and five terms, respectively. Unfortunately none of the above mentioned models is completely satisfactory; probably the best of them is the one by Hao and Nordlander [23] but even they have problems reproducing the data; for example, at the interband threshold of Ag, between 4 and 4.5 eV. Of course one can add even more terms until the ε(u) curves (and indeed any curve) are fitted but then the physical meaning will be lost. For the DL model used in this work we have used five Lorentzian terms (n ¼ 5). 2.2. Drude-critical points model Recently, Etchegoin et al. [24,25] have proposed a different approach: replacing the Lorentzian terms with a family of analytical models, called critical points (CPs), which have long been used for the analysis of interband transitions in semiconductors [26]. The advantage of using the CPs is that, unlike simple Lorentz oscillators, they allow for an easy adjustment of an asymmetric line shape [24,27]. Etchegoin and collaborators, as well as some other authors like Vial and Laroche [28e30] and Lu and Chang [31], have successfully employed this model with two terms to represent the optical properties of various types of noble and transition metals. This model, however, has some problems because the phase parameter can cause the imaginary part of ε for one or both of these modified Lorentzian functions to become negative, which is an unphysical effect for a metal. Rioux and collaborators [32] have developed a model based in the same approach but they circumvent the problems of Etchegoin's model by taking into account the band structure of the metals and, consequently, negative values of
ðu þ iG1 Þ
2 !
ln 1 u01 2ðu þ iG1 Þ2 ! pffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 u1 1 tanh u1 u01=u þ 1 ðu þ iG1 Þ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u þ iG1 u1 1 u1 u01 tan u þ iG1 u1 ðu þ iG1 Þ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !# u þ iG1 u1 u1 u01 1 tanh ; u þ iG1 u1 ðu þ iG1 Þ2
being i the imaginary number, ε∞ the high-frequency limit dielectric constant, up the plasma frequency, and Gp the plasma (bulk) damping term. Gp is related with the mean free path of electrons in the bulk material, le, by the expression Gp ¼ vF/le [18,19], where vF is the Fermi velocity. There are several ways of representing interband transitions but the most common one consists in using a certain number of Lorentzian terms. We will call this approach the Drude-Lorentz model:
ε b ð uÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1 u01
A2 2ðu þ iG2 Þ
! ðu þ iG2 Þ 2 ln 1 ; 2
(4a)
u02
(4b)
where, for each transition j ¼ 1, 2, Aj is the amplitude, u0j the threshold, uj the gap and Gj the broadening. Finally, the frequencydependent dielectric function in this model is:
u2p þ εCP1 ðuÞ þ εCP2 ðuÞ: u2 þ iuGp
εðuÞ ¼ ε∞
(5)
The form of the dielectric function described by equation (5) conforms to the requirements of the Kramers-Kronig (KK) relations [22].
3. Results and discussion In order to fit the experimental datasets using the DL and DCP models, we have employed the chi-square distribution as the objective function:
c2 ðx1 ; …; xk Þ ¼
" 2 N nci ðx1 ; …; xk Þ nei 1 X n e 2N i¼1 i c # ki ðx1 ; …; xk Þ kei þ ke
(6)
i
where N is the number of points used for the fit, x1, …, xk all the variables of the theoretical model and nci and kci (nei and kei ) the values of the real and imaginary parts of the calculated (experimental) complex refractive index. Then, the optimization process was performed using the limited memory Broyden-FletcherGoldfarb-Shanno (L-BFGS-B) multivariate optimization algorithm [33,34]. Lower and/or upper limits are set for all the variables, in order to ensure that their optimized values have physical meaning (e.g., positive plasma frequency, etc.). The optimization process is finished when one of the following criteria is met:
kGk k εG ;
(7)
2 ðiþ1Þ 2 ðiÞ ðiÞ ðiþ1Þ εF $max c2 c c ; c2 ;1 ;
(8)
ðiþ1Þ ðiÞ xk εX ; xk
(9)
where εG, εF and εX are positive numbers which define a precision of search, k$k means Euclidian norm, Gk, gradient projection onto a
~ a-Rodríguez / Journal of Alloys and Compounds 694 (2017) 857e863 O. Pen
variable xk and the super index represents the iteration number. Additionally, the optimization can be finished after a certain number of iterations. For the fittings presented in this work no limit was set for the number of iterations and the precision of search was taken as: εG ¼ εF ¼ εX ¼ 108. ~ a-Rodríguez et al. [13], The experimental data reported by Pen for different compositions of the silver-gold alloys have been used in this work to obtain the model parameters. Those optical properties were obtained experimentally using spectroscopic ellipsometry measurements on thin films fabricated by electron beam evaporation. This dataset (freely available at zenodo.org, DOI: 10.5281/zenodo.50484) has the advantage of covering a wide spectral range in the UV-visible region, being KK consistent and including the full compositional range. The complex refractive index was used for the calculation of c2 instead of the dielectric ~ a-Rodríguez function because this is the quantity reported by Pen et al. [13] but this selection is not really important, since similar results are obtained by fitting ε(u) instead. It should be noted that Rioux et al. [32] applied their model to the AuxAg1-x alloy but their dataset is restricted to only three alloy compositions (plus the pure metals) and a narrow spectral range (1e4.5 eV). Hence, the
859
~ a-Rodríapplication of their model to the dataset reported by Pen guez et al. [13] should provide a much more realistic representation of the optical properties of the gold-silver alloys. Some representative fits using both models are depicted in Fig. 1 and the model parameters yielding the best fits are shown in Tables 1 and 2 for the DL and DCP models, respectively. It can be seen in Fig. 1 that the fit of the experimental data is rather good with both models. There are, however, some inaccuracies in the fits, particularly in the spectral region close to the onset for interband transitions, where the DL model has some clear oscillations. The DCP model, on the other hand, fits better the data but it also exhibits some deviations in the same region. For a more quantitative assessment of the quality of the fits we also plotted the minimum values of c2 as a function of the silver fraction (Fig. 2). This plots confirms that the DCP model yields considerably better fits. In addition, it can be seen there that for both models the best fits are obtained for the alloys with similar quantities of Ag and Au. Finally, is it worth noting that in spite of the differences between the models and the experimental data, the fits obtained with both models are perfectly capable of reproducing all the important variations of the optical properties for all the alloys.
Fig. 1. Best fits of the complex refractive indices of some AgxAu1-x alloys using the DL and DCP models, for some representative compositions with silver fractions (xAg) of (a) 0.00, (b) 0.15, (c) 0.35, (d) 0.48, (e) 0.74 and (f) 1.00. The insets are a zoom of their respective plots in the onset of interband transitions of the alloys.
~ a-Rodríguez / Journal of Alloys and Compounds 694 (2017) 857e863 O. Pen
860
Table 1 ~ a-Rodríguez and collaborators [13]. Parameters in Eq. (3), obtained from the best fit with the DL model of the data reported by Pen xAg
0.00
0.08
0.15
0.24
0.35
0.48
0.66
0.74
0.85
1.00
ε∞
u5 [eV] G5 [eV]
3.104 8.769 0.089 3.624 2.784 0.414 5.113 3.183 0.600 13.37 3.799 0.980 21.07 4.695 1.630 37.22 6.349 2.751
3.210 8.974 0.090 3.520 2.822 0.445 5.717 3.243 0.652 14.14 3.886 1.037 21.76 4.761 1.669 36.15 6.408 2.800
3.025 8.932 0.117 3.823 2.910 0.512 6.198 3.394 0.723 13.30 4.008 1.054 20.09 4.914 1.694 34.46 6.525 2.653
2.924 8.880 0.158 3.422 2.980 0.531 6.454 3.495 0.753 12.49 4.104 1.061 18.68 5.006 1.686 30.93 6.576 2.534
2.899 8.956 0.160 3.241 3.071 0.570 6.728 3.608 0.793 12.28 4.218 1.089 17.64 5.114 1.685 28.20 6.641 2.432
2.711 8.893 0.183 2.366 3.190 0.527 6.136 3.722 0.758 10.88 4.323 1.046 16.08 5.193 1.608 24.93 6.645 2.257
2.476 8.703 0.169 2.480 3.478 0.707 6.821 4.022 0.867 10.24 4.628 1.157 13.36 5.481 1.651 20.56 6.874 2.149
2.390 8.773 0.149 2.924 3.681 0.739 5.891 4.163 0.798 9.234 4.728 1.054 13.24 5.560 1.501 21.72 6.891 1.626
2.456 8.902 0.115 3.899 4.014 0.757 5.942 4.402 0.734 8.285 4.973 0.956 11.57 5.726 1.352 17.93 6.892 1.674
2.503 9.015 0.079 2.621 4.217 0.286 5.484 4.608 0.532 7.080 5.160 0.700 10.02 5.847 1.001 15.13 6.868 1.222
c2
1.160
0.735
0.686
0.597
0.443
0.428
0.114
0.131
0.301
1.106
up [eV] Gp [eV] A1
u1 [eV] G1 [eV]
A2
u2 [eV] G2 [eV]
A3
u3 [eV] G3 [eV]
A4
u4 [eV] G4 [eV]
A5
Table 2 ~ a-Rodríguez and collaborators [13]. Parameters in Eq. (5), obtained from the best fit with the DCP model of the data reported by Pen xAg
0.00
0.08
0.15
0.24
0.35
0.48
0.66
0.74
0.85
1.00
ε∞
u2 [eV] G2 [eV]
2.568 8.905 0.109 56.39 2.342 2.714 0.125 32.13 3.516 0.264
2.621 9.115 0.108 56.5 2.363 2.77 0.149 33.02 3.564 0.257
2.522 9.062 0.135 52.67 2.428 2.82 0.166 33.78 3.568 0.292
2.429 8.986 0.174 51.38 2.525 2.866 0.195 33.68 3.583 0.289
2.398 9.056 0.175 46.16 2.58 2.951 0.19 34.49 3.626 0.289
2.193 8.97 0.196 41.42 2.735 3.079 0.189 35.33 3.717 0.291
2.047 8.82 0.179 23.73 2.689 3.372 0.164 35.98 3.833 0.292
2.027 8.974 0.159 29.09 2.839 3.77 0.134 31.19 4.09 0.293
2.022 9.116 0.126 33.42 3.086 3.997 0.109 25.15 4.224 0.237
1.909 9.104 0.086 62.13 3.759 4.155 0.045 18.29 4.796 0.217
c2 ( 102)
0.656
0.359
0.413
0.381
0.269
0.269
0.059
0.093
0.158
0.304
up [eV] Gp [eV] A1
u01 [eV] u1 [eV] G1 [eV]
A2
Threshold (u0i), frequency (ui), and broadening (Gi) parameters obtained for all alloy compositions from the DL and DCP models are depicted in Figs. 3 and 4, respectively. In both cases the frequency of the transitions (Figs. 3a and 4a) have a linear behaviour and the plasma frequency, in particular, is almost constant. Variations of the
Fig. 2. Minimum c2 values obtained from the fits of the complex refractive indices of AgxAu1-x alloys using the DL and DCP models, function of the silver fraction. Continuous lines are just a guide to the eye.
damping constants (Figs. 3b and 4b), on the other hand, exhibit a much more interesting behaviour: they have relatively low values for the pure metals and increase for the alloys, with a maximum for the alloys with similar quantities of silver and gold. The exception of this trend is the last transition of the DL model. In all likelihood this term does not represents a physical transition but it is representing some of the higher-energy transitions grouped together, which would explain this different behaviour. A quadratic polynomial fits very well the amplitude (Ai) in the DCP model and all the values of Gi whereas all the other parameters exhibit a linear dependence on the silver fraction. The polynomial parameters yielding the best fits for all the model variables are shown in Table 3. The linear dependence on xAg observed for all the thresholds, u0j, and frequencies, uj, is consistent with the linear variation of the band separation described by Gaudry et al. [35]. The nearlyconstant plasma frequency can be explained by considering that Au and Ag have the same crystal structure and similar lattice constants. Hence, the density of valence electrons for both metals and their alloys is almost identical, resulting in similar plasma frequencies across all compositions. The quadratic dependence of Gp 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 [36]. Moreover, the differences in electronegativity between gold and silver (2.54 and 1.93,
~ a-Rodríguez / Journal of Alloys and Compounds 694 (2017) 857e863 O. Pen
Fig. 3. (a) Frequency (ui), and (b) broadening (Gi) parameters of DL model for the AgxAu1-x alloys as a function of the silver fraction, obtained from the fit of the n and k values. The continuous lines are either linear (a) or quadratic (c) fits of the data.
respectively, in the Pauling scale) result in a considerable transfer of electronic density from Ag to Au (Agdþ-Aud-), with a maximum for the Ag0.5Au0.5 composition [13,37]. This charge transfer will likely stress and/or distort the lattice, destroying the periodicity of the structure. This disorder lowers the mean free path of the conduction electrons for alloys, resulting in a higher value of Gp for alloys compared to pure metals. Finally, the increase in Gj for the alloys compared to pure metals, is similar to the behaviour of Gp. The most likely explanation in this case is that the non-periodic structure broadens every energy state, resulting in a global broadening of the interband electronic transitions. Finally, the fits obtained in this work for the pure metals are compared with the corresponding fits reported by Hao and Nordlander [23], Etchegoin et al. [24,25] and Rakic et al. [16] (Fig. 5). It should be noted that we have fitted the experimental data reported ~ a-Rodríguez at al [13], whereas in all the other cases was by Pen used the dataset of Johnson and Christy [11]. However, this should not be a problem for comparing the models because both experimental datasets are very similar, as can be seen in Fig. 5. All the models give reasonably good fits for gold (Fig. 5a) but the ones reported in this work (particularly the DCP model) are slightly better, mainly in the region around the onset of interband transitions. On the other hand, the models reported by Hao and Nordlander [23] and Rakic et al. [16] (Etchegoin et al. [24,25] only use their model for gold) completely fail to represent the refractive index of silver for energies above 3.8 eV whereas our models give good fits in all the spectral range of the experimental data. Hence, the theoretical models described in this paper not only are one of
861
Fig. 4. (a) Threshold (u0i), frequency (ui), and (b) broadening (Gi) parameters of DCP model for the AgxAu1-x alloys as a function of the silver fraction, obtained from the fit of the n and k values. The continuous lines are either linear (a) or quadratic (c) fits of the data.
Table 3 Parameters of the polynomial fit for the thresholds (u0i), frequency (ui), and broadening (Gi) obtained for the DL and DCP models. Parameter
y ¼ a1 þ a2 xAg þ a3(xAg)2 DL model
ε∞
up [eV] Gp [eV] A1
u01 [eV] u1 [eV] G1 [eV]
A2
u02 [eV] u2 [eV] G2 [eV]
A3
u3 [eV] G3 [eV]
A4
u4 [eV] G4 [eV]
A5
u5 [eV] G5 [eV]
DCP model
a1
a2
a3
a1
a2
a3
2.143 8.877 0.076 3.548 e 2.659 0.354 6.001 e 3.139 0.582 14.21 3.770 0.953 21.77 4.700 1.599 36.99 6.408 2.777
0.819 0.007 0.395 0.783 e 1.442 1.159 0.103 e 1.418 1.005 6.778 1.346 0.859 11.90 1.165 0.834 22.57 0.569 0.606
e e 0.396 e e e 1.057
2.614 8.996 0.096 64.64 2.231 2.561 0.128 30.59 3.364 e 0.256 e e e e e e e e e
0.747 0.033 0.378 106.6 1.107 1.513 0.327 30.64 1.071 e 0.202 e e e e e e e e e
e e 0.392 92.62 e e 0.413 42.07 e e 0.242 e e e e e e e e e
e e 1.012 e e 1.049 e e 1.370 e e 0.928
the few available for gold-silver alloys, they also represent the refractive index of the pure metals more accurately than most of
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Acknowledgements The author is grateful with Moncloa Campus of International Excellence (UCM-UPM) for the PICATA postdoctoral fellowship. References
Fig. 5. Comparison between the fits of the complex refractive index of gold (a) and silver (b) obtained in this work for the Drude-Lorentz (DL) and Drude-Critical Points (DCP) models and those reported by Hao and Nordlander (HN) [23], Etchegoin et al. (EA) [24,25] and Rakic et al. (RA) [16]. The models fit the experimental data reported ~ a-Rodríguez et al. (PR, closed symbols) [13] or by Johnson and Christy (JC, either by Pen open symbols) [11]. The insets are a zoom of their respective plots in the onset of interband transitions.
the models available in the literature.
4. Conclusions In this work we have proposed two theoretical models to represent the optical properties of the AgxAu1-x alloys using a Drude term to represent intraband transitions and either five Lorentzian or two critical-point terms to account for the interband transitions. We have shown that both models (especially the DCP) provide excellent fits of the experimental data over a wide spectral range going from 0.65 eV to 6.6 eV. Both models (particularly the DL) can straightforwardly be incorporated into the existing implementations of the FDTD algorithm. Availability of these theoretical models opens the door to better understand the optical properties of plasmonic alloys. Likewise, they will allow researchers in the field to perform a wide number of advanced calculations (mainly using the FDTD method) to predict the optical response of nanostructures for the full compositional range of the aforementioned alloys. This will facilitate the use of alloys in plasmonic applications, a topic that has great potential but so far has not been fully developed.
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