## Abstract

Using high-speed picometrology, the complete cluster-to-film dielectric trajectories of ultra-thin gold films on silica are measured at 488 nm and 532 nm wavelengths for increasing mass-equivalent thickness from 0.2 nm to 10 nm. The trajectories are parametric curves on the complex dielectric plane that consist of three distinct regimes with two turning points. The thinnest regime (0.2 nm – 0.6 nm) exhibits increasing dipole density up to the turning point for the real part of the dielectric function at which the clusters begin to acquire metallic character. The mid-thickness regime (0.6 nm ~2 nm) shows a linear trajectory approaching the turning point for the imaginary part of the dielectric function. The third regime, from 2 nm to 10 nm, clearly displays the Drude circle, with no observable feature at the geometric percolation transition.

©2010 Optical Society of America

## 1. Introduction

The noble metals lie at the interface between optical and electronic physics [1]. As the scale of metal nanophotonic and plasmonic elements shrink, metallic behavior becomes size dependent, and thin layers fragment into clusters that have different optical properties than the bulk [2,3]. Despite the importance of gold as an archetypical metal and its importance for photonic applications, there has been no study revealing the continuous cluster-to-film-to-bulk transition of the complex refractive index $\tilde{n}$ and dielectric constant $\tilde{\epsilon}$, especially when the metal film average thickness is below 2 nm. The ultra-thin regime is inaccessible to current techniques for thin film optical studies. Because $\tilde{n}$ is complex with two variables (real part and imaginary part), a one-parameter measurement cannot acquire the full information needed to calculate $\tilde{n}$. Traditional methods select two or more photonic states of the probe light, such as wavelength, polarization or incidence angle [4] to execute multiple-parameter measurements. For example, spectral reflectometry measures the reflectance change caused by the thin film in a broad wavelength range and uses the Kramers-Kronig relation to find $\tilde{n}$ [5–7]. Ellipsometry measures $\tilde{n}$ by monitoring reflectances at two different polarization states of oblique-incident light [8,9]. However, changing the photonic state of the probe light potentially limits the measurement of $\tilde{n}$ because of optical dispersion or anisotropy which occurs for sub-nanometer metal films. These consequences limit the validity of traditional methods in the ultra-thin regime. For example, ellipsometric measurements become erratic when the gold film thickness is below 6 nm [7,10].

We have developed interferometric picometrology which measures $\tilde{n}$ using a single photonic state (single wavelength, single polarization and normal incidence) [11]. Picometrology monitors the complex change of the normal-incidence reflection coefficient $\tilde{r}$ instead of reflectance *R* to calculate $\tilde{n}$ of the film by analyzing the far-field diffraction of a patterned film in a single measurement which efficiently excludes errors introduced by multiple measurements. The change of $\tilde{r}$ is obtained by monitoring the change of the far-field diffraction pattern as the focal spot scans over the edge of a thin film. The phase change of $\phi (\tilde{r})$ is obtained by monitoring the spatial asymmetry of the diffraction pattern, and the amplitude is obtained by monitoring the overall intensity. The scanning system shown in Fig. 1a
is employed to perform the picometrology measurement. Both the modulus and the phase change of $\tilde{r}$ caused by the thin film are acquired using a split detector that simultaneously monitors intensities and asymmetries of the reflected beam. The split detector is located on the Fourier plane and consists of two semi-circular halves that contribute signals A and B. The intensity (I) signal of the diffraction pattern is acquired as I = A + B, and the spatial asymmetry signal (phase contrast, PC) is acquired as PC = A-B. Typical I and PC channel signals are shown in Fig. 1b. The complex index $\tilde{n}$ is calculated using the picometrology equation [12]

*λ*is the free space wavelength, ${i}^{I}(x)$ and ${i}^{PC}(x)$ are the normalized signals of the I and PC channels, and $A\left[{i}^{I}(x)\right]$ and $A\left[{i}^{PC}(x)\right]$ represent the amplitudes of ${i}^{I}(x)$ and ${i}^{PC}(x)$. The mass-equivalent thickness is defined as $d={\sigma}_{m}/{\rho}_{m}$ where ${\sigma}_{m}$ is the surface mass density measured by a quartz crystal monitor, and ${\rho}_{m}$ is the molecular mass density of gold. The platform consists of a spinner and a linear stage to produce a two-dimensional picometrology scan. We previously applied this system to measure anomalous dispersion of graphene [12].

## 2. Experiments

To study $\tilde{n}$ and $\tilde{\epsilon}$ of gold at arbitrary thicknesses in the range of 0~10 nm, we created gold samples with average gold film thicknesses that increased continuously and linearly from 0 to 10 nm along the y-direction (Fig. 1b) on a single substrate. This was achieved by a metal evaporator modified by attaching a stepper-motor-controlled shutter between the evaporation source and the silicon chip. The shutter moves with constant speed and exposes the chip surface continuously and incrementally during evaporation. The gold deposition rate in these experiments was 0.2 nm/s, and the silicon chip was at room temperature (300 K). The optical study of gold with mass-equivalent thicknesses from 0 to 10 nm is performed on a single chip to eliminate the error due to variations among different chips. Because picometrology is based on Gaussian beam diffraction from the edge of a film, the gold film is deposited in a stripe pattern (Fig. 1b) fabricated using photolithography. The periodic pattern along the x-direction provides multiple edges and improves the detection sensitivity. The substrate is a silicon chip with a 134 nm SiO_{2} layer (measured by a spectroscopic analysis of the reflectance) that is chosen to give nominally equal I and PC signals in Eq. (1). Prior to gold evaporation, the substrate was cleaned by acetone in a sonic bath for 15 min and rinsed with methanol and deionized water and dried with a nitrogen stream.

Two-dimensional picometrology scanning was performed on the gold sample at wavelengths of 488 nm and 532 nm. Images of I and PC scans are shown in Fig. 2b
and 2c. Normalization is performed by dividing the I and PC signal by the intensity of the reflected beam on the adjacent land of the substrate. As a demonstration, in Fig. 3a
, the I and PC signals are extracted for 4 nm gold, and $\tilde{n}$ for gold is calculated at this thickness. Although the approximation in Eq. (1) is suitable to calculate $\tilde{n}$ when the thickness is much less than the probe wavelength, to improve calculation precision, we used a rigorous computational transfer matrix method [13] to extract the precise $\tilde{n}$ from the data. The $\tilde{r}$ of the substrate ($\tilde{r}=-0.38-0.42i$ at 488 nm wavelength and $\tilde{r}=-0.19-0.46i$ at 532 nm wavelength), the film thickness *d*, and the I and PC responses are measured, hence we plot $\tilde{n}=n+ik$ on the complex plane to find a value of $\tilde{n}$ that satisfies both the I and PC responses. For each thickness there is a single curve on which all $\tilde{n}$ give a constant I response (which explains why reflectometry alone cannot determine the $\tilde{n}$). All $\tilde{n}$ satisfying the PC response form a separate curve. The intersection of the I and PC curves uniquely determines the target $\tilde{n}$ (Fig. 3b) at this thickness. By these means, $\tilde{n}$ of gold is calculated for any thickness less than 10 nm. The curves of $\tilde{n}=n+ik$ are shown in Fig. 3d. Gold with effective thickness below 10 nm is usually not a continuous medium but has the topology of heterogeneous clusters with a connectivity that goes through a percolation transition [14].

The trajectory of the dielectric constant ${\tilde{\epsilon}}_{g}=\epsilon \text{\'}+i\epsilon "={\tilde{n}}^{2}$ across the complex plane, parameterized by the film thickness, is shown in Fig. 4 for the wavelengths 488 nm and 532 nm for which the collective plasmon contributions to the Drude response are small [7]. Three distinct regimes are observed in the ${\tilde{\epsilon}}_{g}$ trajectories as film thickness decreases from 10 nm to 0.2 nm. In the first regime (gold thickness decreases from 10 nm to 2 nm), ${\tilde{\epsilon}}_{g}$ evolves in a circular trajectory. In the second regime (gold thickness from 2 nm to 1 nm), ${\tilde{\epsilon}}_{g}$ evolves in a linear trajectory. In the third regime (gold thickness from 1 nm to 0.2 nm), ${\tilde{\epsilon}}_{g}$ evolves with the imaginary part approaching zero as the discontinuous film fragments completely into finely scattered clusters.

## 3. Discussion

To interpret the ${\tilde{\epsilon}}_{g}$ trajectory of ultra-thin gold, the microscopic topology of the film was analyzed by a Field-Emission scanning electron microscopy (SEM, Hitachi S-4800 Field Emission SEM). For instance, a gold film with a mass-equivalent thickness of a nanometer is a collection of isolated gold clusters. The size, shape (aspect ratio) and surface coverage of the gold clusters determines the effective dielectric constant. SEM imaging was performed to measure the surface coverage, aspect ratio and size of the gold clusters at thicknesses of 1, 2, 5 and 10 nm. The data and topology analysis are shown in Fig. 5 , enabling us to interpret the evolution of ${\tilde{\epsilon}}_{g}$.

In the thinnest regime (mass-equivalent thickness 0.2 – 0.6 nm), the dipole density on the surface increases causing an increase of the real part of the dielectric constant as the gold clusters grow in lateral size up to 10 nm (4 nm in height) up to a surface coverage of 15%. Approximately at this size, the clusters begin to take on metallic character, and the dielectric trajectory reaches the turning point in the real part of the dielectric constant as the imaginary part increases rapidly. In the mid-thickness regime (mass-equivalent thickness 0.6 to 2 nm) the coverage increases from 15% up to 40%, as the lateral size of the clusters increases to 16 nm (6 nm height). The trajectory is approximately linear in this regime, with the real part decreasing and the imaginary part increasing.

In the thickest regime of the trajectory of ${\tilde{\epsilon}}_{g}$ (Drude circle regime where gold thickness increases from 2 nm to 10 nm), the aspect ratio of gold clusters changes little as the coverage varies slowly (coverage above 50%), but the average lateral size of the gold clusters changes from 16 nm to 42 nm (6 nm to 16 nm in height). The cluster size is the major factor determining the effective ${\tilde{\epsilon}}_{g}$ of the gold film in this regime causing ${\tilde{\epsilon}}_{g}$ to evolve in a circular trajectory in Fig. 4 as the film thickness increases. This phenomenon is caused by gradual deconfinement of the free electrons in gold clusters caused by the increasing cluster size. According to the Drude theory [15,16], the ${\tilde{\epsilon}}_{g}$ of nanoscale gold structures is described as

*ω*is the probe light angular frequency, ${\epsilon}_{bound}$ is the dielectric constant contributed by bound electrons, and $\mathrm{\Gamma}\left(d\right)$ is the collision frequency of free electrons. $\mathrm{\Gamma}\left(\infty \right)$ is a constant (bulk value), but when the cluster size is on the order of the electron mean free path (25 nm in Au [17]), $\mathrm{\Gamma}\left(d\right)$ increases due to surface collisions. In Ref [18], the approximate behavior was parametrized by $\mathrm{\Gamma}=\mathrm{\Gamma}(bulk)+2\pi {v}_{F}/s$ where ${v}_{F}$ is the Fermi velocity of free electrons and

*s*is the radius of a metal nano-particle. Therefore, ${\tilde{\epsilon}}_{g}$ evolves in a circular trajectory, called the Drude Circle, as $\mathrm{\Gamma}\left(d\right)$ varies and

*ω*is held constant. This is a direct consequence of the equation

The radius of this circle on the complex plane is ${\omega}_{p}^{2}/2{\omega}^{2}$ centered at ${\epsilon}_{bound}-\frac{{\omega}_{p}^{2}}{2{\omega}^{2}}$. The Drude circle describes the parametric trajectory of ${\tilde{\epsilon}}_{g}$ for metal clusters as *Γ* varies. By fitting the data in the first regime on Fig. 4, the radii are found to be 3.8 at 488 nm and 5.2 at 532 nm, giving a plasmon energy of 7.3 eV and ${\epsilon}_{bound}\approx 6+1.6i$. The plasmon frequency of bulk Au is between 7.5 – 8.5 eV in [19–20] [21]. Therefore, the Drude equation is consistent with our experimental results. The turning point in the imaginary part of the dielectric constant (the apex of the trajectory) occurs when the free electron scattering rate equals the optical angular frequency $\mathrm{\Gamma}=\omega $. To study Drude behavior, previous approaches tuned the frequency *ω* while treating *Γ* as a constant. Here, we observe the Drude circle for the first time by using the graded-thickness gold sample combined with picometrology in a single photonic state, allowing $\mathrm{\Gamma}\left(d\right)$ to vary parametrically and to observe ${\tilde{\epsilon}}_{g}$ at different gold thicknesses, while keeping *ω* constant (constant radius on the complex plane).

Our experiment tracks the metal-to-insulator (I/M) transition of gold film from an optical perspective. The geometric percolation threshold is reached when a single giant cluster spans the probe scale, which occurs around a mass-equivalent thickness of 7 nm in our samples. Therefore, although the geometric percolation thickness is well within the Drude circle, there is no measurable feature or deviation from the circular Drude trajectory in that thickness range. For optical frequencies, continuous percolation across long-range gold clusters has little or no influence on the optical properties of a gold film. There is evidence [22] in the microwave region (4.9 GHz, 60 mm wavelength) that the conductivity of a gold film decreases abruptly below a thickness of 8 nm, which is consistent with the percolation threshold. Higher-frequency infrared light ($\lambda =24\text{}\mu m$) displayed a transmission anomaly at the thinner value of 2 nm [3], which is where, in our results, the Drude circle commences (metallic optical behavior) as a continuation of the mid-thickness regime of the dielectric trajectory.

## 4. Conclusion

In conclusion, we have measured the dielectric trajectory on the complex plane of thermally-evaporated gold on silica at the wavelengths 532 nm and 488 nm with mass-equivalent gold film thickness as the parametric variable. Picometrology is able to extract the full complex dielectric trajectory using a single-wavelength measurement without polarization control and at normal incidence. We extracted ${\tilde{\epsilon}}_{g}$ of gold films with arbitrary thickness in the range of 0.2 nm to 10 nm, and demonstrated that the ${\tilde{\epsilon}}_{g}$ trajectory traces out the Drude circle for mass-equivalent thicknesses greater than 2 nm.

## Acknowledgement

This project is sponsored by a grant from the 21st Century Fund of the Indiana Economic Development Corporation. We sincerely thank Dr. Vladimir Shalaev for fruitful discussions. We also thank Jacob Millspaw and Chris Petrovitch for help processing the gold samples.

## References and links

**1. **E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science **311**(5758), 189–193 (2006). [CrossRef] [PubMed]

**2. **R. B. Laibowitz and Y. Gefen, “Dynamic Scaling near the Percolation-Threshold in Thin Au Films,” Phys. Rev. Lett. **53**, 380–383 (1984). [CrossRef]

**3. **J. J. Tu, C. C. Homes, and M. Strongin, “Optical properties of ultrathin films: evidence for a dielectric anomaly at the insulator-to-metal transition,” Phys. Rev. Lett. **90**(1), 017402 (2003). [CrossRef] [PubMed]

**4. **A. Gray, M. Balooch, S. Allegret, S. De Gendt, and W. E. Wang, “Optical detection and characterization of graphene by broadband spectrophotometry,” J. Appl. Phys. **104**, 053109 (2008). [CrossRef]

**5. **P. Grosse and V. Offermann, “Analysis of Reflectance Data Using the Kramers-Kronig Relations,” Appl. Phys. A52, 138–144 (1991). [CrossRef]

**6. **D. A. Crandles, F. Eftekhari, R. Faust, G. S. Rao, M. Reedyk, and F. S. Razavi, “Kramers-Kronig-constrained variational dielectric fitting and the reflectance of a thin film on a substrate,” Appl. Opt. **47**(23), 4205–4211 (2008). [CrossRef] [PubMed]

**7. **M. Hovel, B. Gompf, and M. Dressel, “Dielectric properties of ultrathin metal films around the percolation threshold,” Phys. Rev. B **81**, 035402 (2010). [CrossRef]

**8. **F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. Steinberg, “Measurememt of Thickness and Refractive Index of Very Thin Films and Optical Properties of Surfaces by Ellipsometry,” J. Res. Natl. Bur. Stand. A: Phys. Chem.**67**, 363–377 (1963).

**9. **R. J. Archer, “Determination of Properties of Films on Silicon by Method of Ellipsometry,” J. .Opt. Soc. Amer.52, 970– 977 (1962). [CrossRef]

**10. **M. Yamamoto and T. Namioka, “In situ ellipsometric study of optical properties of ultrathin films,” Appl. Opt. **31**(10), 1612–1621 (1992). [CrossRef] [PubMed]

**11. **X. Wang, M. Zhao, and D. D. Nolte, “Common-path interferometric detection of protein monolayer on the BioCD,” Appl. Opt. **46**(32), 7836–7849 (2007). [CrossRef] [PubMed]

**12. **X. F. Wang, Y. P. Chen, and D. D. Nolte, “Strong anomalous optical dispersion of graphene: complex refractive index measured by Picometrology,” Opt. Express **16**(26), 22105–22112 (2008). [CrossRef] [PubMed]

**13. **O. S. Heavens, *Optical Properties of Thin Solid Films* (Academic Press Inc., 1955), pp. 66–80.

**14. **V. M. Shalaev, *Nonlinear Optics of Random Media: Fractal Composites and Metal-Dielectric Films*, Springer Tracts in Modern Physics (Berlin, 2000).

**15. **H. Y. Li, S. M. Zhou, J. Li, Y. L. Chen, S. Y. Wang, Z. C. Shen, L. Y. Chen, H. Liu, and X. X. Zhang, “Analysis of the drude model in metallic films,” Appl. Opt. **40**(34), 6307–6311 (2001). [CrossRef]

**16. **F. Abeles, *Optical Properties of Solids* (North-Holland, 1972), p. 103.

**17. **M. Walther, D. G. Cooke, C. Sherstan, M. Hajar, M. R. Freeman, and F. A. Hegmann, “Terahertz conductivity of thin gold films at the metal-insulator percolation transition,” Phys. Rev. B **76**, 125408 (2007). [CrossRef]

**18. **U. Kreibig and C. Vonfrags, “Limitation of Electron Mean Free Path in Small Silver Particles,” Zeitschrift Fur Physik224, 307–323 (1969). [CrossRef]

**19. **M. G. Blaber, M. D. Arnold, and M. J. Ford, “Search for the Ideal Plasmonic Nanoshell: The Effects of Surface Scattering and Alternatives to Gold and Silver,” J. Phys. Chem. C **113**, 3041–3045 (2009). [CrossRef]

**20. **C. Noguez and C. E. Roman-Velazquez, “Dispersive force between dissimilar materials: Geometrical effects,” Phys. Rev. B **70**, 195412 (2004). [CrossRef]

**21. **M. Kreiter, S. Mittler, W. Knoll, and J. R. Sambles, “Surface plasmon-related resonances on deep and asymmetric gold gratings,” Phys. Rev. B **65**, 125415 (2002). [CrossRef]

**22. **T. Zychowicz, J. Krupka, and J. Mazierska, “Measurements of Conductivity of Thin Gold Films at Microwave Frequencies Employing Resonant Techniques,” Proceedings of Asia-Pacific Microwave Conference (2006).