1. Introduction

Silver island films grown by plasma or evaporation methods have been extensively studied over the last century due to their interesting optical properties [1]. More recently, interest in semicontinuous and discontinuous silver films has experienced a renaissance due to applications in surface plasmon sensors [2], plasmonics [3] and metamaterials [4]. A critical issue in the development of these rapidly expanding fields is the ability to accurately measure the material optical properties.

Spectroscopic ellipsometry is a key method for determining the electric permittivity, ε(ω)=ε₁+iε₂ which allows direct comparison with theoretically predicted optical structures [5]. However, as it is an indirect measurement technique, spectroscopic ellipsometry requires the application of complex models, such as effective medium theories, to extract physical parameters from the experimental data. Although such models can provide valuable information concerning the dynamics of nucleation, percolation and the early stages of bulk film growth, it is not trivial to incorporate the details of the complex nanoscale interactions, especially around the percolation threshold.

We present here an in situ spectroscopic ellipsometry study of nucleation and percolation in silver films. The data is modelled using the inversion method of Arwin and Aspnes [6] to determine the film thickness and dielectric function. This method uses an iterative process of adjusting the film thickness to remove optical features of the substrate which appear as artefacts in the film dielectric function when the film thickness is incorrect. Once the correct thickness has been found the dielectric function can be uniquely determined. The method is therefore model independent and ideal for studying ultra-thin and discontinuous films which exhibit dielectric properties markedly different from the bulk material. Silver is chosen due to its topical interest for plasmonic applications and its tendency to form relatively large nanoparticles in the early stages of growth.

To our knowledge, there are two other works of note in the literature on this subject. The first in situ spectroscopic ellipsometry on silver films was reported by An and Oh [7] in 1996. They used a simple Lorentz oscillator to fit the data and extract the dielectric functions. In a recent work, De Vries, et al. present a detailed analysis of the growth of silver films nucleated on gold nanoparticles using a Lorentz oscillator and the Maxwell-Garnett effective medium theory (MGT) [8]. They extended the analysis by incorporating terms for anisotropic interactions and non-spherical particles and showed that the fundamental plasmon resonance predicted from the MGT was equivalent to a Lorentz oscillator. However, taking the measurements ex situ and using gold as seed particles introduces significant uncertainties in the results. By using the inversion method, the results presented here provide in situ, model-independent values of the dielectric function of a growing silver film.

By comparing the effective dielectric functions and film thickness determined using a Lorentz oscillator fit with those of the inversion method we assess its accuracy and applicable range. The approach is recommended to assess the accuracy of more complex effective medium theories. This may be especially important when analysing artificially magnetic metamaterials since it is not trivial to disentangle the magnetic and electric contributions from the complex dielectric function [9, 10]. Differences in the plasmon resonances and the film structure are discussed for two different deposition temperatures. Finally, by comparing the ellipsometric data with simultaneous in situ resistivity measurements the percolation threshold is determined using two independent methods.

2. Experimental

The deposition conditions have been described previously [11]. Briefly, depositions were performed in a high-current pulsed filtered cathodic vacuum arc (HCP-FCVA) designed and built in-house [12, 13]. A silver cathode (99.95%) was ablated using 1.2 ms long pulses at 3 Hz repetition rate. The average arc current was 1kA during the pulse and the deposition rate was approximately 0.03nm per pulse. The arc was paused periodically during deposition to allow us to investigate relaxation effects during the film deposition. A series of ten pulses (at 3Hz) was deposited and the film was allowed to relax for 30 seconds, before the next 10 pulses were deposited. The data presented here represents the end of each relaxation pause. The relaxation effects, discussed in detail elsewhere [11], are due to thermodynamic changes in the film nanostructure and affect the values of the thickness by up to 0.3nm at room temperature when the thickness is around 10nm. The dielectric constants change by up to 1.0 at wavelengths close to the resonance.

Accurate measurement of film growth is often hampered by access problems due to the vacuum conditions required for deposition. Imaging techniques such as electron and scanning probe microscopy are difficult to perform in situ. Important advances have recently been achieved using synchrotron-based X-ray techniques [14]. One significant drawback of synchrotron based experiments, apart from cost, is that the deposition chamber must be located at, and designed to fit on the beam line. It is more desirable to have a measurement system that can be adapted to any deposition chamber. Ellipsometry has been used for many years for in situ analysis during film deposition because it is non-destructive and capable of real-time measurement [15–19]. In recent years the advent of real-time spectroscopic ellipsometers with computerised data collection and automated controls has made it possible to acquire large data sets at all stages of the growth process. The deposition was monitored in situ using a J.A. Woollam M2000 spectroscopic ellipsometer. The instrument operates in a rotating compensator configuration, with a white light source, refracting prism and CCD detector providing fast data acquisition capabilities. Data was acquired every 7 seconds at 400 wavelengths in the range 370–1000nm at an angle of incidence of 75 degrees from the normal. The data was analysed using the WVASE32 software.

Silicon substrates with a 500nm thermal silicon oxide were used. In situ resistivity was measured between evaporated aluminium contacts defining a 12mm square using a HP 34401A multimeter. For heated substrate depositions the substrate was heated up to 150°C using a quartz lamp. The substrate temperature was measured using a Luxtron fluorescence temperature probe.

 

Fig. 1. ψ and Δ data as a function of time at 3 representative wavelengths. 399 wavelengths were collected in all.

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3. Results and discussion

3.1 Raw results

Figure 1 shows the ellipsometric parameters Ψ and Δ measured at three wavelengths during a room temperature deposition, where tanΨe^iΔ=R _p/R _s and R _p and R _s are the complex reflection coefficients for parallel and perpendicular polarisations, respectively. The instrumental errors are less than 0.1 degree in Ψ and Δ, with 1.6 nm wavelength resolution. In all 399 wavelengths were recorded. Figure 2 shows the in situ resistance data as a function of the number of arc pulses for films deposited at room temperature and 150°C. The data is shown from the time when the resistance first became measurable (<1×108 Ω/□).

3.2 Film growth dynamics

In the initial stages of film formation, silver grows in a Volmer-Weber mode to reduce the surface free energy associated with interface formation [20]. Adatoms migrate on the substrate surface until they nucleate on the surface or form a dimer with a second adatom, whereby their mobility is significantly hindered. Addition of further adatoms results in the formation of silver islands. The density and morphology of these islands is primarily determined by the surface energies of the substrate and island material, the surface temperature and the deposition rate [21]. Further growth occurs with continued deposition until the islands begin to agglomerate and eventually form a percolated network. At this point the optical and electronic properties of the film alter significantly. Further deposition fills in the remaining holes, after which the effects of the substrate on the film growth is negligible and the film grows in a layer-by-layer mode.

Cathodic arcs generate plasmas containing ions with a high directed velocity and comparatively high degree of ionisation and charge state, all of which contribute additional energy to the growing film [22]. The kinetic energy of the arriving atoms may also lead to sub-plantation of the ions into the first few layers of the substrate, although for silver, high mobility and low reactivity generally result in spontaneous demixing. The percolation threshold in arc deposited silver has been observed to occur at a smaller thickness compared to magnetron sputtered films under the same conditions [23]. This is conjectured to be due to the additional energy of the deposited ions.

 

Fig. 2. Resistance as a function of the number of arc pulses for the deposition at room temperature (black squares) and 150°C (red circles).

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In the island growth stage silver films exhibit a strong optical absorption peak associated with plasmonic resonances [1]. For metal nanoparticles, the collective response of the conduction electrons to an electromagnetic perturbation acts to polarise the islands, which induces a restoring force on the electrons. Depending on the island dimensions and the surrounding medium there will be a resonance condition for a given wavelength corresponding to the system’s natural frequency where absorption is significantly enhanced. Silver nanoparticles show a strong resonance in the visible part of the electromagnetic spectrum. As the volume fraction of the metal in the film increases the plasmon resonance red-shifts due to changes in the dielectric environment.

3.3 Ellipsometric analysis

Arwin and Aspnes [6] showed that if the substrate onto which the film is deposited exhibits an absorption feature in the range of measurement, the thickness and dielectric function of the deposited film can be determined by fitting the dielectric function using an assumed thickness value. If the assumed thickness is incorrect the substrate feature will show up in the dielectric function of the film. By minimising or eliminating the substrate feature the correct film thickness can be established and the complex dielectric function determined. This approach has been successfully applied in a number of studies [12, 15, 24]. Here we utilise the interference fringes from the 500nm silicon oxide layer as the substrate feature.

Since the above method is time consuming, a common method of extracting the dielectric functions, ε(ω)=ε₁+iε₂ from the ellipsometric measurement of a metallic film is to fit a linear sum of oscillators based on the Drude-Lorentz theory. The physical origins of these oscillators may be attributed to free (Drude) and bound (Lorentz) electrons [1]. For measurements of silver in the visible region, the contribution from interband transitions can be well approximated to a constant value ε _∞, since the first interband transition occurs in the UV [25]. The Drude-Lorentz formulism then simplifies to

where ω_p is the bulk plasma frequency and Γ is the free electron relaxation frequency. For composite films where the constituent materials are much smaller than the interacting wavelength, optical modelling is simplified by utilising effective medium approximations (EMAs) such as the Maxwell-Garnett theory (MGT) [26]. EMAs determine the effective dielectric functions of a composite medium as a mathematical combination of the measured dielectric functions of the constituent materials. Wormeester, et al. recently showed that for a collection of spherical particles with the dielectric function of Eq. (1) and a refractive index of the embedding medium equal to unity, the effective dielectric function, ε_eff, as determined by the MGT is described by a Lorentzian of the form [27]

where

for a surface area coverage, Q. Since ω_p and ε _∞ are considered to be constant in silver at 9.0 and 4.0 eV respectively [28] the amplitude and resonance frequency are functions of Q, and the broadening depends on Γ. Therefore, as the film grows, percolates and tends toward a continuous slab the resonance frequency red shifts to zero as Q approaches unity. In the limiting case that Q=1, (1) reduces to the Drude equation.

The Lorentzian description for a thin island film from the MGT was originally developed by Yamaguchi [29]. Since the MGT was originally formulated for low concentrations of small spherical particles in a 3-dimensional matrix its application to thin island films is debateable. However it is clear from previous reports that the Lorentzian lineshape is a good approximation to the plasmonic resonance observed in thin island films up to relatively large values of Q, when the interactions between contiguous particles become significant. The basic Lorentzian form has been applied to describe the plasmon resonance in all reported ellipsometric investigations of growing silver films [7, 8, 30]. In other spectroscopic investigations of thin films the Thin Island Film theory of Bedeaux and Vlieger [31] is often used, which can also be described by a Lorentzian form [27]. It is therefore of great interest to determine the applicable range of the Lorentzian description of the plasmonic resonance as a function of surface coverage. It is interesting to note that the Bruggmann EMA [32] does not well describe the thin island film case except when Q is small [33] when it converges with the MGT.

Doremus [34] showed that by applying this method to the thin film case, Q, as determined from electron micrographs, can be accurately calculated by the position of the resonance maxima determined from reflection and transmission measurements. The method was extensively verified for 0.19>Q>0.63 for gold, silver and copper island films, and we have confirmed it using ellipsometry (ref). Interestingly, although the particle is in contact with the substrate on one side only, Doremus argues that the embedding dielectric function should be that of the substrate. It should be noted that his analysis does not account for changes in the free electron relaxation frequency due to increased surface scattering.

The methods described above were applied using the software package WVASE32. Variables are determined by performing an iterative fit which reduces the mean squared error (MSE). For the inversion method the thickness is fixed and the dielectric functions are determined for each wavelength. The mean squared error (MSE) is the sum of the squares of the differences between the measured and calculated data for each (Δ, Ψ) pair. The MSE is related to the statistical indicator chi-squared, χ², by

where N is the number of (Δ, Ψ) pairs and M is the number of variable parameters in the model. A regression algorithm is then used to minimize the MSE by adjusting the variable parameters. The MSE gives a relative indication of the fit. As a rule, MSE values below 10 are in most cases considered acceptable.

3.4 Dielectric functions

The complex dielectric functions of the growing films as a function of the number of arc pulses determined using the inversion method are shown in Figs. 3(a), 3(c) and Figs. 4(a), 4(c) for the room temperature and 150°C depositions respectively. There is some noise in the very early stages of the deposition, especially in the UV region, associated with a strong influence of the substrate in the case of very thin films when applying the inversion method. The uncertainty in determining the effective thickness is +/- 0.1nm. The dielectric functions extracted from Δ and Ψ shift by less than +/- 0.1 for this thickness change, except at wavelengths where the interference fringes in the substrate occur (around 400 and 600 nm). At these wavelengths the values become increasingly noisy for thinner films [12].

Considering first the imaginary part of the dielectric functions we can observe the plasmon resonance associated with metallic nanoparticles growing from around 50 arc pulses with centre wavelength of around 400nm (3.0eV). As the deposition progresses the resonance increases in amplitude and the centre energy red-shifts, in accordance with a growing particle radius and increasing interaction between adjacent particles [1]. For the room temperature deposition, the width of the resonance is comparatively broad due to greater scattering of the free electrons, indicating that the particles are comparatively small. As the deposition progresses the broadening increases and the centre energy moves out of the measurement range.

 

Fig. 3. Real (a,b) and imaginary (c,d) parts of the dielectric functions of the growing silver film for the room temperature deposition determined using (a,c) the inversion method and (b,d) the Lorentz oscillator method. e1 = 0 is shown as a black line.

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In comparison, the width of the plasmon resonance for the 150°C deposition is comparatively narrow, indicating an increased particle size due to increased adatom mobility on the substrate surface with elevated temperature. The resonance remains comparatively narrow and red-shifts at a much slower rate than the film deposited at room temperature. As the silver film thickness increases the inversion method becomes increasingly difficult to implement and less accurate due to the reduction in intensity of the substrate features in the ellipsometric data. Although the data was available for thicker films the uncertainties increased after 600 pulses to the point where we could no longer confidently determine the film thickness using the inversion method.

As well as using the inversion method to obtain the film dielectric functions we fit the ellipsometric data using the Lorentz oscillator method fitting for the thickness, Γ and Q. The real and imaginary parts of the dielectric function for the room temperature and 150°C deposition are shown in Figs. 3(b), 3(d) and Figs. 4(b), 4(d) respectively. The general features of the dielectric functions determined using the Lorentzian fit are reproduced: namely, the position of the plasmon resonance, and the point where ε₁ becomes negative in the infra-red. This suggests that the Lorentz oscillator method provides a good qualitative approximation for the position of the resonance up until the percolation threshold. However the shape of the resonance is only well described by a Lorentzian in the early stages of deposition at room temperature (below approximately 75 pulses, or 5nm effective thickness). As percolation is approached, dipole-dipole and multipole interactions increase which distort the shape of the resonance. At higher temperatures the increased particle size renders the Lorentz oscillator method less useful (discussed below).

 

Fig. 4. Real (a,b) and imaginary (c,d) parts of the dielectric functions for the 150°C deposition determined using (a,c) the inversion method and (b,d) the Lorentz oscillator method. ε₁=0 is shown as a black line.

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3.5 Film thickness

In many instances, ellipsometry is employed as a method to determine the film thickness with little regard for the optical properties. Figure 5 compares the film thickness for the room temperature and 150°C deposition determined using the two methods above. In the early stages of deposition, up until 90 pulses for the room temperature (film thickness 5.6nm) and 140 pulses for the 150°C (film thickness 11.5nm), the two methods are in good agreement. For subsequent deposition the thickness values diverge significantly. We are confident that the inversion method results are more reliable for the thickness determination since firstly, the deposition rates are more consistent for the inversion method and secondly, for larger particle sizes and larger fill factors the Lorentz oscillator method will be increasingly compromised by the exclusion of dipole interactions from the model. We have cross-checked the inversion method with transmission electron microscopy and found excellent agreement [12].

For the room temperature case the inversion method shows that the film growth rate is initially high, before levelling out and eventually reducing toward the dashed line which indicates the constant deposition rate, as determined by depositing a thick film under identical conditions (0.0325 nm/pulse, time averaged growth rate of 0.0975nm/s). This effect is well documented [15, 16, 24] and arises from the Volmer-Weber growth of metal islands, followed by percolation, infilling and eventually continuous growth. The results suggest that the film has percolated but is not yet a continuous film with some void content still present. In this case the percolation threshold occurs after around 160 pulses (discussed below).

The thickness values determined using the Lorentz oscillator method diverge significantly from those determined using the inversion method for the film deposited at 150°C. Percolation for this film occurs much later, at around 950 pulses (discussed below). The effect is due to the larger sizes of the particles grown at higher temperature. For particles larger than the quasistatic limit the influence of multipole oscillations are increasingly important and the lorentzian lineshape is significantly distorted [35]. Due to their larger plasmon oscillations the dipole-dipole (and multipole) interactions between larger particles are also enhanced. These effects both act to reduce the applicability of the Lorentz oscillator method to the analysis of films deposited at high substrate temperatures.

 

Fig. 5. Film thickness as a function of the number of plasma pulses determined from the ellipsometric data using the inversion method (black squares) and the Lorentzian fit (red circles). The upper curves correspond to the film deposited at 150°C and the lower curves at room temperature. Blue line.

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3.6 Surface area coverage and MSE

In Fig. 6(a), the MSE [Eq. (6)] is observed to rise quickly for the room temperature deposition after around 70 pulses from reasonable values (below 10). It reaches a maximum of around 38 at precisely the point of the percolation threshold (160 pulses). After that it reduces to values of around 10 again as the film becomes continuous and the Drude free-electron model becomes increasingly applicable. The Q values here are not expected to be quantitatively accurate since the model is highly simplistic, however they do show the general trend from an island film to a continuous film. After 250 pulses the model predicts that the film is continuous, however as discussed above the thickness curves suggest that the film is still not continuous after 300 pulses. We conclude that Eq. 2 overestimates the area coverage for large Q values, which is in agreement with previous observations [8].

The 150°C deposition exhibits much lower Q values since the plasmon resonance red-shifts significantly slower at higher temperatures (Fig. 4). Since the total amount of silver deposited is more or less the same in both cases it is the nanostructure of the film which influences the resonance. At room temperature the island density is greater and the interaction between islands increases more quickly, acting to red shift the resonance earlier. The 150°C deposition exhibits a much larger MSE after a few tens of pulses, reducing somewhat around 200 pulses. Unlike the room temperature case, this maximum is far from the percolation threshold at around 950 pulses. As the deposition continues above 300 pulses the MSE once more rises (not shown) and increases with the film thickness. The larger MSE values show the limited applicability of the Lorentz oscillator method to films with larger particle sizes such as those grown at higher temperatures.

 

Fig. 6. Parameters from the Lorentz oscillator method fits to the room temperature (black squares) and 150°C (red circles) data. (a) MSE, (b) surface area coverage, Q.

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Various methods of extending the MGT to account for particle-particle interactions, particle-substrate interactions and particle shape have been proposed. It is not trivial to quantify the relative contributions of these effects. A simple approach is to include a depolarisation factor in the MGT [27] which does not distinguish between these effects. We found that the inclusion of a depolarisation factor did not significantly improve the results presented above and therefore was not investigated further. The effect of the interface and particle shape should result in anisotropy with the optical axis in the direction normal to the surface, however we could not detect this anisotropy in ex-situ variable angle ellipsometric and polarimetry measurements. The Bruggeman formula [32] was also applied to fit the data, however the results showed no improvement, with many parameters giving non-physical results.

3.7 Percolation threshold

Determination of the percolation threshold in ultra-thin films is important for a broad range of applications. For example, low-emissivity silver window coatings should be as thin as possible to permit maximum visible transmittance whilst being optimally absorbing in the infra-red i.e. metallic [36]. Similar requirements hold for charge-conducting films used for electron microscopy and ion implantation [37]. Recent theoretical predictions show that silver films with negative refractive indices and high transparency can be used as an important element in negative index metamaterials [9]. All of these conditions are optimised at or near the percolation threshold. It is therefore valuable to have a good method to determine the percolation threshold of thin films, preferably non-destructively, non-invasively and in situ so that the deposition may be stopped at the optimum thickness. Spectroscopic ellipsometry is ideal for this purpose.

 

Fig. 7. Resistivity as a function of thickness for the room temperature deposition.

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The real part of the dielectric function, ε₁, gives an indication of the conductivity of the film. The decrease from positive to negative values of ε₁ for low photon energies indicates the percolation threshold, where the film transits from a dielectric to a conductive material [15,24]. For the room temperature deposition this occurs abruptly just after 160 pulses [Fig. 3(a)], whilst for the 150°C deposition ε₁ remains positive for much longer [Fig. 4(a)]. We determined with much greater uncertainty than the room temperature case the point where ε₁ becomes negative at the highest wavelength at around 950 pulses for the 150°C deposition. As pointed out by de Vries, et al. in their recent work [8], the method of determining the percolation threshold by noting where ε₁ becomes negative is dependent on the range of the optical measurement; however the transition here is quite abrupt for the room temperature deposition. It is of interest to note that the zero value of ε₁ occurs at the peak of the resonance in ε₂ as may be expected from the Kramer-Kronig relation. The percolation threshold shall be discussed further below.

By using the thickness values determined above we can convert the resistance values of Fig. 2 into resistivity as a function of film thickness, shown in Fig. 7 for the room temperature film. The film becomes conductive within the limit of our experimental range (10 MΩ) at a film thickness of 7.4nm (140 pulses). As further material is deposited the resistivity approaches a constant level as the film approaches the bulk and the contribution to the resistance from surface scattering diminishes. The literature value for the resistivity of bulk silver is 1.6×10^-8 Ω.m [38], which is an order of magnitude below the lowest value we measure. We expect the film is not yet in the bulk stage and that surface scattering is still significant for a film of thickness less than 12nm considering the mean free path of conduction electrons in the bulk is of the order of 40 nm [38].

Although the resistivity is first measurable at a film thickness of 7.4 nm, it reduces significantly between 160 and 170 pulses (7.6 – 8.0 nm) when the rate begins to slow. This is in agreement with the point where ε₁ becomes negative in the infrared from Fig. 3(a). It is also in excellent agreement with the results of our previous study of FCVA deposited silver on glass substrates [23], which exhibited a percolation threshold at a film thickness slightly under 8 nm. It is of interest to note that when the deposition is stopped before percolation we observed the resistivity of the film increases with time and when the deposition is stopped after percolation it decreases with time, as has been reported previously [23].

For the 150°C film, due to difficulties with the inversion method, the thickness values were not determined for the region where the resistance was measurable and a resistivity plot is not available. The raw resistance data in Fig. 2 shows an abrupt jump between 920 and 930 pulses. This compares well with the ellipsometry data showing ε₁ going negative in the infrared after around 950 pulses (not shown). It should be noted that the ε₁ transition is not as abrupt at 150°C as for room temperature. The correlation between the two methods is however significant and supports the use of spectroscopic ellipsometry to experimentally determine the percolation threshold in thin metallic films.

4. Conclusion

Using the model independent inversion method the dielectric functions of a growing silver film were determined and compared to the dielectric functions determined by fitting the data with a Lorentz oscillator. Both methods provide thickness values which compare well for room temperature films below 5.6 nm and 150°C films below 11.5 nm. Above these values the thickness values diverge significantly. The inversion method is valid beyond the percolation threshold however as the film thickness increases the errors associated with this method increase due to the increasing opacity of the film. The Lorentz oscillator gives comparable results for the main features of the dielectric functions although the magnitude and broadening of the plasmon resonance are not correctly determined for higher fill factors and higher temperatures due to larger particle sizes. For silver films at room temperature the percolation threshold is well predicted by ellipsometry when compared with resistance measurements.

The Lorentz oscillator method is easier to implement than the inversion method and it is recommended for real-time analysis of films in the low coverage phase where the particle sizes satisfy the quasistatic approximation (<20nm). It can also be used for a simple determination of the percolation threshold and it approximates the value of the plasmon resonance frequency below the percolation threshold. It does not predict the amplitude and broadening of the resonance and the inversion method should be used when accurate knowledge of these values is required. An algorithm based approach to the inversion method would be extremely valuable for real time applications that require accurate measurement of the dielectric functions of metallic films throughout the deposition.