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.

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2010 (1)

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

2009 (1)

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]

2008 (3)

2007 (2)

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]

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]

2006 (1)

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

2004 (1)

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

2003 (1)

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]

2002 (1)

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]

2001 (1)

1992 (1)

1984 (1)

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

Allegret, S.

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]

Arnold, M. D.

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]

Balooch, M.

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]

Blaber, M. G.

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]

Chen, L. Y.

Chen, Y. L.

Chen, Y. P.

Cooke, D. G.

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]

Crandles, D. A.

De Gendt, S.

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]

Dressel, M.

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

Eftekhari, F.

Faust, R.

Ford, M. J.

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]

Freeman, M. R.

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]

Gefen, Y.

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

Gompf, B.

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

Gray, A.

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]

Hajar, M.

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]

Hegmann, F. A.

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]

Homes, C. C.

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]

Hovel, M.

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

Knoll, W.

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]

Kreiter, M.

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]

Laibowitz, R. B.

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

Li, H. Y.

Li, J.

Liu, H.

Mittler, S.

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]

Namioka, T.

Noguez, C.

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

Nolte, D. D.

Ozbay, E.

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

Rao, G. S.

Razavi, F. S.

Reedyk, M.

Roman-Velazquez, C. E.

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

Sambles, J. R.

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]

Shen, Z. C.

Sherstan, C.

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]

Strongin, M.

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]

Tu, J. J.

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]

Walther, M.

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]

Wang, S. Y.

Wang, W. E.

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]

Wang, X.

Wang, X. F.

Yamamoto, M.

Zhang, X. X.

Zhao, M.

Zhou, S. M.

Appl. Opt. (4)

J. Appl. Phys. (1)

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]

J. Phys. Chem. C (1)

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]

Opt. Express (1)

Phys. Rev. B (4)

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]

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

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

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]

Phys. Rev. Lett. (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]

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]

Science (1)

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

Other (8)

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).

P. Grosse and V. Offermann, “Analysis of Reflectance Data Using the Kramers-Kronig Relations,” Appl. Phys. A 52, 138–144 (1991).
[CrossRef]

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).

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

U. Kreibig and C. Vonfrags, “Limitation of Electron Mean Free Path in Small Silver Particles,” Zeitschrift Fur Physik 224, 307–323 (1969).
[CrossRef]

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

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

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

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Figures (5)

Fig. 1
Fig. 1

Schematics of picometrology and the gold sample. (a) Picometrology measures the complex refractive index of ultra-thin films at single wavelengths at normal incidence. The reflected Gaussian beam forms an asymmetric diffraction pattern on the Fourier plane when scanning across film edges. By analyzing the intensity variation (I signal) and spatial asymmetry (PC signal) of the diffraction pattern with a split detector, full information is acquired of the complex refractive index calculation. (b) The gold sample is fabricated for the optical study for thicknesses in the range of 0~10 nm on a single chip. The gold effective thickness (measured mass per area divided by molecular mass volume density) increases from 0 to 10 nm along the –Y direction, and the gold film has a stripe pattern along the X direction to provide multiple film edges for the picometrology analysis.

Fig. 2
Fig. 2

Gold film pattern and I, PC response images. (a) Stripe-patterned gold film with thickness varying continuously from 0 to 10 nm on thermal oxide on silicon (134 nm SiO2). The sample is prepared using a thermal metal evaporator in which the evaporation time is controlled by a stepper motor. The image is captured by microscope under white light. The color of the gold shows a rich transition within 10 nm (Real color). (b) and (c) show the images of the sample in the I and PC channels scanned by the picometrology system at 532 nm (Pseudo color).

Fig. 3
Fig. 3

Data processing to calculate refractive indices of ultra-thin gold. (a) One example of I and PC signals from 4 nm thick gold (one track taken from images Fig. 1b and 1c). The normalized amplitudes of both channels are acquired, and the refractive index of gold at this thickness can be calculated by Eq. (1). (b) The I response is satisfied by multiple n ˜ which form a curve on the complex plane, and similarly for the PC response. The intersection point uniquely determines n ˜ of the gold film. (c). Normalized amplitudes of I and PC responses of gold from 0 to 10 nm at 488 nm and 532 nm wavelengths. (d) The n ˜ is calculated for gold films with 0~10 nm thicknesses at both 488 nm and 532 nm wavelengths.

Fig. 4
Fig. 4

The dielectric trajectories of gold film as the film thickness changes from 10 to 0.2 nm. The dielectric trajectory has three distinct regimes that originate from three types of gold topology: 1) Sparse regime (0.2 nm to 1 nm). In this regime, coverage of gold clusters goes from zero to 20%, and the imaginary part of ε ˜ g vanishes at ultra-low coverage. 2) Intermediate regime (1 nm to 2 nm) connecting the first and third regime. 3) Drude circle regime (2 nm to 10nm). In this regime, the effective dielectric constant of gold film ε ˜ g evolves along a circular trajectory (Drude circle). This pattern is predicted by the Drude equation as the size of the metal clusters decreases below the electron mean free path.

Fig. 5
Fig. 5

SEM images of the gold samples and gold topology analysis. (a), (b), (c), (d) show the images of gold clusters at different mass-equivalent film thicknesses. (e) and (f) show Gold topology in terms of coverage, cluster lateral size and film thicknesses. The cluster height was calculated by dividing the average gold thickness by the coverage, and the cluster lateral size was calculated by auto-correlation analysis of the SEM images. Cluster coverage rapidly and almost linearly increases from 0 to 37% as the average thickness of gold film changes from 0 to 2 nm. The coverage increases at a much slower rate when film thickness grows beyond 3 nm. As a result, coverage increases from 54% to 60% as the gold thickness changes from 5 to 10 nm.

Equations (3)

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( n ˜ 2 1 ) ( 1 + r ˜ ) 2 r ˜ 2 π d λ = 3.565 A [ i P C ( x ) ] + A [ i I ( x ) ] j ,
ε g = ε b o u n d ω p 2 ω 2 + i Γ ( d ) ω ,
| ε ˜ g ( ε b o u n d ω p 2 2 ω 2 ) | = | ( 1 + i Γ / ω 1 + i Γ / ω ) ω p 2 2 ω 2 | = ω p 2 2 ω 2 ,

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