Localized surface plasmons-polaritons represent collective behavior of free electrons confined to metal particles. This effect may be used for enhancing efficiency of solar cells and for other opto-electronic applications. Plasmon resonance strongly affects optical properties of ultra-thin, island-like, metal films. In the present work, the Finite Difference Time Domain (FDTD) method is used to model transmittance spectra of thin gold island films grown on a glass substrate. The FDTD calculations were performed for island structure, corresponding to the Volmer-Weber model of thin film growth. The proposed simulation model is based on fitting of experimental data on nanostructure of ultra-thin gold films, reported in several independent studies, to the FDTD simulation setup. The results of FDTD modeling are then compared to the experimentally measured transmittance spectra of prepared thin gold films and found to be in a good agreement with experimental data.
©2013 Optical Society of America
Thin metal films have attracted large scientific and practical interest due to their specific properties and multiple applications. One of these properties is the rise of collective behavior of free electrons confined to the metal films under electromagnetic field influence. This phenomenon is called as a surface plasmon-polariton (SPP). Surface plasmons (SPs) arise from the interaction of light with free electrons at dielectric/metal interface. Under certain conditions the free electrons on the metal surface oscillate in resonance with the incident light wave, resulting in excitation of SPs, propagating along the interface or confined to the metal particles [1, 2]. It was found that the width of SPP resonance peak and its position are depending on the particle size, shape and environment [3–5]. SPP-based structures are widely used in science and technology, for example as highly-sensitive chemo- and bio-sensors [6, 7]. The enhancement of the electromagnetic field at the metal/dielectric interface  is responsible for surface-enhanced non-linear optical phenomena  such as Raman scattering and second harmonic generation , fluorescence [11, 12], and infrared absorption .
One of the most interesting features of ultra-thin films of noble metals is their discontinuous, island-like structure. Unique optical properties, related to this discontinuous nature, have been intensively studied for several decades, using various experimental techniques and theoretical models. Results of these studies were reported in numerous publications [14–22] and clearly show that transmittance, absorbance and extinction spectra of ultra-thin metal films exhibit resonant, SPP-related peaks (dips) at specific wavelength. The origin and behavior of these SPP is well-known and attributed to resonant oscillation of free electrons confined to nano-scale metal islands.
In the present study we used the Finite Difference Time Domain (FDTD) method  to simulate transmittance spectra of ultra-thin gold films deposited on glass substrate. In order to build a simulation model of island shape evolution, we collected and processed experimental data from independent sources and got empirical parameters for FDTD calculations.
2. Gold islands on glass substrate: geometrical model
According to the well-known Volmer-Weber model of thin film growth , at initial stages of film growth, isolated, 3D metal islands are formed on the substrate surface, instead of continuous metal film. Figure 1 represents the continuous and island thin gold films grown on a glass substrate.
This growth mechanism was experimentally proved in numerous studies [14–22]. In these studies the developing of nano-islands during the gold film deposition was demonstrated by electron microscopy (EM) and atomic force microscopy (AFM) imaging. Also, excitation of localized surface plasmons, associated with gold islands, was clearly indicated by corresponding dips/peaks in measured transmittance, absorbance or extinction spectra. In parallel with experimental studies, intensive theoretical analysis and numerical simulations were undertaken [14–16, 25–29] in order to characterize the optical properties of nano-scale metal islands deposited on various substrates and to fit theoretical results to experimental data. The optical (spectral) response of metal nano-islands was analyzed using the effective-medium model [14–16] or by calculation of polarizability [25–29] of a truncated spheres and oblate spheroids in electrostatic approximation (island size is much smaller than the wavelength of exciting light).
We have carried out a detailed computer simulation of spectral transmittance of ultra-thin, island-like gold films using the 3D FDTD method. Since 1966, when it was first introduced , the FDTD method has become a common and very effective tool for analysis of electromagnetic problems including modeling and design of nano-photonic and plasmonic structures. In order to simulate the localized surface plasmon resonance in thin gold film, we have adapted the Volmer-Weber model of film growth to our nanophotonic simulation tools. As can be seen in Fig. 2 , in our simulation study the shape of isolated, gold nano-island, deposited on flat surface, is modeled by simple spherical cap (segment) with height h, base diameter D and sphere radius R. The center of spherical cap is placed at depth d below the substrate surface. For comparison, a continuous gold film with nominal thickness t is shown in Fig. 2 as well.
The nominal thickness should be calculated according to specifications of deposition technique (see Section 3.2 below). Since our simulation study is aimed on calculation of spectral transmittance of thin, island-like gold films at various nominal thicknesses, we should find suitable empirical fits for geometrical parameters of the gold spherical cap (Fig. 2) as functions of the nominal thickness: D(t), h(t), R(t) and d(t). The data required for these fits were collected from three independent sources ,  and . From the experimental results presented in these sources, the average lateral size D(t), average height h(t) and the density N(t) of gold islands were extracted and summarized in Table 1 .
The values of average diameter and height presented in Table 1 are based on two different techniques: analysis of AFM images [20, 22] and analysis of EM images . As can be seen from Table 1, combination of data from several sources [16, 20, 22] covers the entire thickness range of interest (1.1 - 15 nm). These data were fitted to suitable functions. The choice of fitting functions for geometrical parameters of gold islands D(t) and h(t) is based on following consideration: As it was reported in experimental studies [16, 20–22, 30], the plasmon resonance wavelength exhibit red shift with increase of nominal thickness of deposited films. According to theoretical analysis [4, 30] and computer simulations  this red shift is attributed to increase of aspect ratio of oblate metal nanoparticles, e. g. the plasmon resonance shifts to the red for more oblate particles. In our case of gold spherical caps the aspect ratio is defined as follows:
In order to find empirical relations between the nominal thickness t and the geometrical parameters of gold islands D(t) and h(t), we have tried different fitting functions (linear, power, second-order polynomial and exponential) and combinations of them. It was found that expected behavior of aspect ratio (1) (increase with nominal thickness) can be obtained with the second-order polynomial fit for D(t) and linear fit for h(t).
Figure 3 and Fig. 4 represent the dependences (presented by symbols) D(t) and h(t), plotted using the data from Table 1, and corresponding fitting curves (dashed curves). As shown in Fig. 4, the linear fit for h(t) was calculated as average of the linear fits for AFM [20, 22] and EM data . The found empirical relations between nominal thickness and geometrical parameters of gold spherical caps (islands) are given by
The islands' aspect ratio, corresponding to the empirical relations (2) and (3), is plotted in Fig. 5 . Using the geometry, presented in Fig. 1, we get the radius of spherical cap and the depth of its center below the substrate surface, both versus the nominal thickness:
Figure 6 represents the evolution of cap's shape calculated with (4) and (5) for various nominal thicknesses.
As shown in Fig. 7 , the density of gold islands can be fitted fairly well by power functionEqs. (2)- (6) in the next section in order to build simulation setup for various nominal thicknesses.
3. Simulation and experimental details
3.1 FDTD simulation setup
To simulate transmittance spectra of gold (Au) nano-caps deposited on glass substrate we used commercially available FDTD Solutions software from Lumerical Solutions, Inc . The simulation was based on the setup shown in Fig. 8 . In this setup, identical spherical gold caps are randomly distributed on the top planar surface of glass (SiO2) substrate. The geometrical parameters of the caps and their surface density were calculated using the empirical expressions (2) - (6).
The optical properties of materials, used for simulation, were taken from  (SiO2 substrate) and  (Au thin films). As shown in Fig. 8, the simulation area has square cross-section in x-y plane. The gold nano-caps are illuminated from the top by the normally-incident (along z-axis) plane wave which is linearly-polarized in x-z plane. The transmitted optical field is recorded by planar monitor (virtual 2D detector), placed inside the SiO2 substrate, parallel to its top surface. The simulation area is bounded in x- and y-directions by parallel planes in which periodical boundary conditions are defined. To prevent any reflections, a Perfect Matching Layers (PMLs) are applied as bottom and top boundaries of the simulation area. The transmittance spectra of the simulated structure were obtained by broadband pulse excitation and calculation of Fourier transform of time-dependent, transmitted field, recorded by the monitor.
3.2 Practical implementation
The island gold films were prepared using a laboratory thermal evaporation setup working at residual pressure of (2.5 - 4) × 10−6 Pa. The deposition setup was equipped with the two-stage vacuum system based on the diffusion pump. The films were deposited on the glass substrates cleaned in the ultrasound bath in isopropyl alcohol and drained by a compressed air flux. During the deposition, all substrates were kept at room temperature.
Gold films were evaporated at residual pressure from tungsten baskets. Deposition was provided up to full evaporation of material from the cell. Portions of the material for evaporation were prepared with help of the microbalance ViBRA. Value of this mass was found by calculation from the defined (nominal) film thickness using follows relation written for the point evaporation source :
4. Results and discussion
4.1 Simulated and measured transmittance spectra
Figure 9 represents evolution of transmittance spectra of ultra-thin, island-like gold films calculated by FDTD technique. The plasmon resonance-related dips are clearly observed in the spectral curves corresponding to nominal thicknesses in the range 1 - 10 nm. The position of plasmon resonance dip monotonically shifts toward the longer wavelengths with increase of nominal thickness, according to the red shift, reported in experimental [16, 20–22, 30] and theoretical [4, 30] studies.
As can be seen in Fig. 9, the plasmon resonance completely dismisses from the films, thicker than 10 nm. It should be noted that FDTD-simulated transmittance spectra, shown in Fig. 9, are in a good agreement with experimental extinction and transmittance spectra presented in [20, 22].
Figure 10 shows the dependence of plasmon resonance wavelength on the nominal thickness. The data, presented by squares in Fig. 10, where obtained from FDTD-calculated transmittance spectra, shown in Fig. 9. Slightly non-linear dependence of resonant wavelength on nominal thickness in the range 1-10 nm can be fitted by exponential function.
Figure 11 represents measured transmittance spectra of thin gold films on the glass substrates, prepared as specified above, in Section 3.1. All these films have a transmittance maximum at λ ≈500 nm, however at longer wavelength they behave differently. One can see that thinner films have a transmittance minimum which depends on the film’s thickness. The position of this minimum shifts toward the longer wavelengths with increase of film thickness and disappears at the transition to a continuous film. Thus, there are two different types of anomalies of transmittance spectra inherent in the gold films. The first is a maximum, observed at fixed wavelength, corresponding to the nature of the metal. The second is a minimum, associated with discontinuous structure of the film, and observed at specific, thickness-dependent wavelengths. Obviously these two phenomena have different explanations.
In the metals having fully filled d-shells, only interband transitions of electrons are enabled. Therefore, the maxima of transmission spectra for such metals are conditioned by existence of bulk absorption modes according to their dispersion equation 36]. This explains the natural yellowish color of the gold.
The nature of the minima, shown in Fig. 11, can be explained by excitation of localized plasmons-polaritons with the intensity and at the frequency depending on the size and shape of the islands forming the film. In accordance with the size of these nano-particles, the plasmon resonance is shifted to longer wavelengths with increase of particles’ size. So, we can see the same red shift of the SPP spectral location, depended on the film thickness, similar to the red shift, in the FDTD-simulated spectra, presented in Fig. 9. Also, it should be noted that the plasmon resonance exists only on very thin island films. In the coalescing films the localized plasmons-polaritons disappear.
The difference between the simulated (see Fig. 9) and the experimental (see Fig. 11) transmittance spectra may be explained by assumption used in our simulation model. We assumed that all the gold islands have identical sizes (see part 3.1). Most likely, the random distribution of real islands sizes around their average value leads to smoothing of experimental transmittance spectra. The transmittance edge at 300 nm, observed in the experimental spectra, is due to absorption in the glass substrate. This transmittance edge is not presented in the simulated spectra because of short distance between the top surface of glass substrate and the 2D monitor, recording the transmitted light.
4.2 FDTD simulation of electrical field intensity distributions
Figure 12 represents the two-dimensional distributions of electrical field intensity on the top surface of glass substrate (z = 0) calculated using FDTD method.
These distributions were recorded at resonant wavelengths, corresponding to squares in Fig. 10, for nominal thicknesses 1 - 10 nm, and at wavelengths, predicted by dashed curve in Fig. 10, for nominal thicknesses 15 and 20 nm. As can be seen from these simulated distributions, at nominal thicknesses 1 - 7 nm, isolated islands are present on the substrate without significant overlapping (coalescence). Strong field enhancement due to excitation of localized surface plasmons is seen around the isolated islands. This image of plasmon-enhanced field correlates with relatively narrow plasmon dips in the transmittance spectra shown in Fig. 9. At nominal thicknesses 8 - 10 nm, significant overlapping of gold islands can be seen in Fig. 11. This corresponds with the broadening of plasmon resonance dips in Fig. 9. In thicker films (nominal thicknesses 15 and 20 nm), the islands fully overlaps forming quasi-continuous gold films in which plasmon resonance is totally suppressed and corresponding resonant dips dismiss from the transmittance spectra.
In this work, we developed and experimentally verified a simulation model of thin gold island films growth and their optical properties. The proposed model is based on fitting the experimental data on size, shape and surface density of gold nano-islands to suitable functions of nominal thickness of ultra-thin film. The 3D Finite Difference Time Domain (FDTD) method was used to calculate the transmittance spectra of modeled gold island films. The transmittance spectra, calculated using the proposed model, exhibit the well-known, thickness-dependent red shift of the plasmon resonance associated with discontinuous structure of thin gold films. The model also demonstrates that localized surface plasmons-polaritons are excited only in island gold films. With beginning of coalescence process in the island films, the plasmon-polariton resonance disappears. The proposed method can be applied to the investigation and characterization of other metal islands films, for example silver or copper thin films.
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