Abstract

We report a study of the optical and electric properties of Au nanoparticle networks grown on the porous alumina membrane by dry atom sputtering deposition approach. An effective cluster model was developed to evaluate the dielectric function and the electrical conductivities of the nanoparticle networks by taking into account the effects of the Au particle size, the Au volume fraction, and the particle-particle interaction. The calculated transmission spectra from the model were in good agreement with the experimental data. The percolation threshold of the as-fabricated structure was predicted to occur at Au volume fraction of 0.18, consistent with the dc electric resistance measurement.

© 2009 OSA

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    [CrossRef]

2009 (2)

J. Henzie, J. Lee, M. H. Lee, W. Hasan, and T. W. Odom, “Nanofabrication of Plasmonic Structures,” Annu. Rev. Phys. Chem. 60(1), 147–165 (2009).
[CrossRef] [PubMed]

X. Y. Lang, L. Y. Chen, P. F. Guan, T. Fujita, and M. W. Chen, “Geometric effect on surface enhanced Raman scattering of nanoporous gold: Improving Raman scattering by tailoring ligament and nanopore ratios,” Appl. Phys. Lett. 94(21), 213109 (2009).
[CrossRef]

2008 (1)

M. Lai and D. J. Riley, “Templated electrosynthesis of nanomaterials and porous structures,” J. Colloid Interface Sci. 323(2), 203–212 (2008).
[CrossRef] [PubMed]

2007 (1)

T. Rindzevicius, Y. Alaverdyan, B. Sepulveda, T. Pakizeh, M. Kall, R. Hillenbrand, J. Aizpurua, and F. J. Garcia de Abajo, “Nanohole Plasmons in Optically Thin Gold Films,” J. Phys. Chem. C 111(3), 1207–1212 (2007).
[CrossRef]

2006 (1)

I. Willner, R. Baron, and B. Willner, “Growing metal nanoparticles by enzymes,” Adv. Mater. 18(9), 1109–1120 (2006).
[CrossRef]

2004 (1)

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, “Optical transmission properties of a single subwavelength aperture in a real metal,” Opt. Commun. 239(1-3), 61–66 (2004).
[CrossRef]

2003 (3)

H. Xu, “A new method by extending Mie theory to calculate local field in outside/inside of aggregates of arbitrary spheres,” Phys. Lett. A 312(5-6), 411–419 (2003).
[CrossRef]

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107(3), 668–677 (2003).
[CrossRef]

J. J. Mock, D. R. Smith, and S. Schultz, “Local refractive index dependence of plasmon resonance spectra from individual nanoparticles,” Nano Lett. 3(4), 485–491 (2003).
[CrossRef]

2002 (2)

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116(15), 6755–6759 (2002).
[CrossRef]

N. Félidj, J. Aubard, G. Levi, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg, “Enhanced substrate-induced coupling in two-dimensional gold nanoparticle arrays,” Phys. Rev. B 66(24), 245407 (2002).
[CrossRef]

1998 (1)

D. Stroud, “The effective medium approximations: Some recent developments,” Superlattices Microstruct. 23(3-4), 567–573 (1998).
[CrossRef]

1985 (1)

M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57(3), 783–826 (1985).
[CrossRef]

1983 (1)

1980 (3)

J. M. Gérardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell's equations. The long-wavelength limit,” Phys. Rev. B 22(10), 4950–4959 (1980).
[CrossRef]

J. E. Sansonetti and J. K. Furdyna, “Depolarization Effects in Arrays of Spheres,” Phys. Rev. B 22(6), 2866–2874 (1980).
[CrossRef]

P. Sheng, “Pair-cluster theory for the dielectric constant of composite media,” Phys. Rev. B 22(12), 6364–6368 (1980).
[CrossRef]

1978 (2)

C. G. Granqvist and O. Hunderi, “Conductivity of Inhomogeneous Materials - Effective-Medium Theory with Dipole-Dipole Interaction,” Phys. Rev. B 18(4), 1554–1561 (1978).
[CrossRef]

T. Yamaguchi, H. Takahashi, and A. Sudoh, “Optical Behavior of a Metal Island Film,” J. Opt. Soc. Am. 68(8), 1039–1044 (1978).
[CrossRef]

1977 (1)

C. G. Granqvist and O. Hunderi, “Optical-Properties of Ultrafine Gold Particles,” Phys. Rev. B 16(8), 3513–3534 (1977).
[CrossRef]

1976 (1)

P. Clippe, R. Evrard, and A. A. Lucas, “Aggregation Effect on Infrared-Absorption Spectrum of Small Ionic-Crystals,” Phys. Rev. B 14(4), 1715–1721 (1976).
[CrossRef]

1973 (1)

T. Yamaguch, S. Yoshida, and A. Kinbara, “Anomalous Optical-Absorption of Aggregated Silver Films,” Thin Solid Films 18(1), 63–70 (1973).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical-Constants of Noble-Metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

1970 (1)

U. Kreibig, “Kramers Kronig Analysis of Optical Properties of Small Silver Particles,” Z. Phys. 234(4), 307–318 (1970).
[CrossRef]

1969 (1)

U. Kreibig and C. V. Fragstein, “Limitation of Electron Mean Free Path in Small Silver Particles,” Z. Phys. 224(4), 307–323 (1969).
[CrossRef]

1935 (1)

D. A. G. Bruggeman, “Calculation of various physics constants in heterogenous substances I Dielectricity constants and conductivity of mixed bodies from isotropic substances,” Ann. Phys. Berlin 24, 636–664 (1935).
[CrossRef]

1904 (1)

J. C. M. Garnett, “Colours in Metal Glasses and in Metallic Films,” Philos. Trans. R. Soc. Lond. A 203(1), 385–420 (1904).
[CrossRef]

Aizpurua, J.

T. Rindzevicius, Y. Alaverdyan, B. Sepulveda, T. Pakizeh, M. Kall, R. Hillenbrand, J. Aizpurua, and F. J. Garcia de Abajo, “Nanohole Plasmons in Optically Thin Gold Films,” J. Phys. Chem. C 111(3), 1207–1212 (2007).
[CrossRef]

Alaverdyan, Y.

T. Rindzevicius, Y. Alaverdyan, B. Sepulveda, T. Pakizeh, M. Kall, R. Hillenbrand, J. Aizpurua, and F. J. Garcia de Abajo, “Nanohole Plasmons in Optically Thin Gold Films,” J. Phys. Chem. C 111(3), 1207–1212 (2007).
[CrossRef]

Aubard, J.

N. Félidj, J. Aubard, G. Levi, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg, “Enhanced substrate-induced coupling in two-dimensional gold nanoparticle arrays,” Phys. Rev. B 66(24), 245407 (2002).
[CrossRef]

Ausloos, M.

J. M. Gérardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell's equations. The long-wavelength limit,” Phys. Rev. B 22(10), 4950–4959 (1980).
[CrossRef]

Aussenegg, F. R.

N. Félidj, J. Aubard, G. Levi, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg, “Enhanced substrate-induced coupling in two-dimensional gold nanoparticle arrays,” Phys. Rev. B 66(24), 245407 (2002).
[CrossRef]

Barbic, M.

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116(15), 6755–6759 (2002).
[CrossRef]

Baron, R.

I. Willner, R. Baron, and B. Willner, “Growing metal nanoparticles by enzymes,” Adv. Mater. 18(9), 1109–1120 (2006).
[CrossRef]

Bruggeman, D. A. G.

D. A. G. Bruggeman, “Calculation of various physics constants in heterogenous substances I Dielectricity constants and conductivity of mixed bodies from isotropic substances,” Ann. Phys. Berlin 24, 636–664 (1935).
[CrossRef]

Chen, L. Y.

X. Y. Lang, L. Y. Chen, P. F. Guan, T. Fujita, and M. W. Chen, “Geometric effect on surface enhanced Raman scattering of nanoporous gold: Improving Raman scattering by tailoring ligament and nanopore ratios,” Appl. Phys. Lett. 94(21), 213109 (2009).
[CrossRef]

Chen, M. W.

X. Y. Lang, L. Y. Chen, P. F. Guan, T. Fujita, and M. W. Chen, “Geometric effect on surface enhanced Raman scattering of nanoporous gold: Improving Raman scattering by tailoring ligament and nanopore ratios,” Appl. Phys. Lett. 94(21), 213109 (2009).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical-Constants of Noble-Metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Clippe, P.

P. Clippe, R. Evrard, and A. A. Lucas, “Aggregation Effect on Infrared-Absorption Spectrum of Small Ionic-Crystals,” Phys. Rev. B 14(4), 1715–1721 (1976).
[CrossRef]

Coronado, E.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107(3), 668–677 (2003).
[CrossRef]

Degiron, A.

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, “Optical transmission properties of a single subwavelength aperture in a real metal,” Opt. Commun. 239(1-3), 61–66 (2004).
[CrossRef]

Ebbesen, T. W.

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, “Optical transmission properties of a single subwavelength aperture in a real metal,” Opt. Commun. 239(1-3), 61–66 (2004).
[CrossRef]

Evrard, R.

P. Clippe, R. Evrard, and A. A. Lucas, “Aggregation Effect on Infrared-Absorption Spectrum of Small Ionic-Crystals,” Phys. Rev. B 14(4), 1715–1721 (1976).
[CrossRef]

Félidj, N.

N. Félidj, J. Aubard, G. Levi, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg, “Enhanced substrate-induced coupling in two-dimensional gold nanoparticle arrays,” Phys. Rev. B 66(24), 245407 (2002).
[CrossRef]

Fragstein, C. V.

U. Kreibig and C. V. Fragstein, “Limitation of Electron Mean Free Path in Small Silver Particles,” Z. Phys. 224(4), 307–323 (1969).
[CrossRef]

Fujita, T.

X. Y. Lang, L. Y. Chen, P. F. Guan, T. Fujita, and M. W. Chen, “Geometric effect on surface enhanced Raman scattering of nanoporous gold: Improving Raman scattering by tailoring ligament and nanopore ratios,” Appl. Phys. Lett. 94(21), 213109 (2009).
[CrossRef]

Furdyna, J. K.

J. E. Sansonetti and J. K. Furdyna, “Depolarization Effects in Arrays of Spheres,” Phys. Rev. B 22(6), 2866–2874 (1980).
[CrossRef]

Garcia de Abajo, F. J.

T. Rindzevicius, Y. Alaverdyan, B. Sepulveda, T. Pakizeh, M. Kall, R. Hillenbrand, J. Aizpurua, and F. J. Garcia de Abajo, “Nanohole Plasmons in Optically Thin Gold Films,” J. Phys. Chem. C 111(3), 1207–1212 (2007).
[CrossRef]

Garnett, J. C. M.

J. C. M. Garnett, “Colours in Metal Glasses and in Metallic Films,” Philos. Trans. R. Soc. Lond. A 203(1), 385–420 (1904).
[CrossRef]

Gérardy, J. M.

J. M. Gérardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell's equations. The long-wavelength limit,” Phys. Rev. B 22(10), 4950–4959 (1980).
[CrossRef]

Granqvist, C. G.

C. G. Granqvist and O. Hunderi, “Conductivity of Inhomogeneous Materials - Effective-Medium Theory with Dipole-Dipole Interaction,” Phys. Rev. B 18(4), 1554–1561 (1978).
[CrossRef]

C. G. Granqvist and O. Hunderi, “Optical-Properties of Ultrafine Gold Particles,” Phys. Rev. B 16(8), 3513–3534 (1977).
[CrossRef]

Guan, P. F.

X. Y. Lang, L. Y. Chen, P. F. Guan, T. Fujita, and M. W. Chen, “Geometric effect on surface enhanced Raman scattering of nanoporous gold: Improving Raman scattering by tailoring ligament and nanopore ratios,” Appl. Phys. Lett. 94(21), 213109 (2009).
[CrossRef]

Hasan, W.

J. Henzie, J. Lee, M. H. Lee, W. Hasan, and T. W. Odom, “Nanofabrication of Plasmonic Structures,” Annu. Rev. Phys. Chem. 60(1), 147–165 (2009).
[CrossRef] [PubMed]

Henzie, J.

J. Henzie, J. Lee, M. H. Lee, W. Hasan, and T. W. Odom, “Nanofabrication of Plasmonic Structures,” Annu. Rev. Phys. Chem. 60(1), 147–165 (2009).
[CrossRef] [PubMed]

Hillenbrand, R.

T. Rindzevicius, Y. Alaverdyan, B. Sepulveda, T. Pakizeh, M. Kall, R. Hillenbrand, J. Aizpurua, and F. J. Garcia de Abajo, “Nanohole Plasmons in Optically Thin Gold Films,” J. Phys. Chem. C 111(3), 1207–1212 (2007).
[CrossRef]

Hunderi, O.

C. G. Granqvist and O. Hunderi, “Conductivity of Inhomogeneous Materials - Effective-Medium Theory with Dipole-Dipole Interaction,” Phys. Rev. B 18(4), 1554–1561 (1978).
[CrossRef]

C. G. Granqvist and O. Hunderi, “Optical-Properties of Ultrafine Gold Particles,” Phys. Rev. B 16(8), 3513–3534 (1977).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical-Constants of Noble-Metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Kall, M.

T. Rindzevicius, Y. Alaverdyan, B. Sepulveda, T. Pakizeh, M. Kall, R. Hillenbrand, J. Aizpurua, and F. J. Garcia de Abajo, “Nanohole Plasmons in Optically Thin Gold Films,” J. Phys. Chem. C 111(3), 1207–1212 (2007).
[CrossRef]

Kelly, K. L.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107(3), 668–677 (2003).
[CrossRef]

Kinbara, A.

T. Yamaguch, S. Yoshida, and A. Kinbara, “Anomalous Optical-Absorption of Aggregated Silver Films,” Thin Solid Films 18(1), 63–70 (1973).
[CrossRef]

Kreibig, U.

U. Kreibig, “Kramers Kronig Analysis of Optical Properties of Small Silver Particles,” Z. Phys. 234(4), 307–318 (1970).
[CrossRef]

U. Kreibig and C. V. Fragstein, “Limitation of Electron Mean Free Path in Small Silver Particles,” Z. Phys. 224(4), 307–323 (1969).
[CrossRef]

Krenn, J. R.

N. Félidj, J. Aubard, G. Levi, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg, “Enhanced substrate-induced coupling in two-dimensional gold nanoparticle arrays,” Phys. Rev. B 66(24), 245407 (2002).
[CrossRef]

Lai, M.

M. Lai and D. J. Riley, “Templated electrosynthesis of nanomaterials and porous structures,” J. Colloid Interface Sci. 323(2), 203–212 (2008).
[CrossRef] [PubMed]

Lang, X. Y.

X. Y. Lang, L. Y. Chen, P. F. Guan, T. Fujita, and M. W. Chen, “Geometric effect on surface enhanced Raman scattering of nanoporous gold: Improving Raman scattering by tailoring ligament and nanopore ratios,” Appl. Phys. Lett. 94(21), 213109 (2009).
[CrossRef]

Lee, J.

J. Henzie, J. Lee, M. H. Lee, W. Hasan, and T. W. Odom, “Nanofabrication of Plasmonic Structures,” Annu. Rev. Phys. Chem. 60(1), 147–165 (2009).
[CrossRef] [PubMed]

Lee, M. H.

J. Henzie, J. Lee, M. H. Lee, W. Hasan, and T. W. Odom, “Nanofabrication of Plasmonic Structures,” Annu. Rev. Phys. Chem. 60(1), 147–165 (2009).
[CrossRef] [PubMed]

Leitner, A.

N. Félidj, J. Aubard, G. Levi, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg, “Enhanced substrate-induced coupling in two-dimensional gold nanoparticle arrays,” Phys. Rev. B 66(24), 245407 (2002).
[CrossRef]

Levi, G.

N. Félidj, J. Aubard, G. Levi, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg, “Enhanced substrate-induced coupling in two-dimensional gold nanoparticle arrays,” Phys. Rev. B 66(24), 245407 (2002).
[CrossRef]

Lezec, H. J.

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, “Optical transmission properties of a single subwavelength aperture in a real metal,” Opt. Commun. 239(1-3), 61–66 (2004).
[CrossRef]

Lucas, A. A.

P. Clippe, R. Evrard, and A. A. Lucas, “Aggregation Effect on Infrared-Absorption Spectrum of Small Ionic-Crystals,” Phys. Rev. B 14(4), 1715–1721 (1976).
[CrossRef]

Meier, M.

Mock, J. J.

J. J. Mock, D. R. Smith, and S. Schultz, “Local refractive index dependence of plasmon resonance spectra from individual nanoparticles,” Nano Lett. 3(4), 485–491 (2003).
[CrossRef]

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116(15), 6755–6759 (2002).
[CrossRef]

Moskovits, M.

M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57(3), 783–826 (1985).
[CrossRef]

Odom, T. W.

J. Henzie, J. Lee, M. H. Lee, W. Hasan, and T. W. Odom, “Nanofabrication of Plasmonic Structures,” Annu. Rev. Phys. Chem. 60(1), 147–165 (2009).
[CrossRef] [PubMed]

Pakizeh, T.

T. Rindzevicius, Y. Alaverdyan, B. Sepulveda, T. Pakizeh, M. Kall, R. Hillenbrand, J. Aizpurua, and F. J. Garcia de Abajo, “Nanohole Plasmons in Optically Thin Gold Films,” J. Phys. Chem. C 111(3), 1207–1212 (2007).
[CrossRef]

Riley, D. J.

M. Lai and D. J. Riley, “Templated electrosynthesis of nanomaterials and porous structures,” J. Colloid Interface Sci. 323(2), 203–212 (2008).
[CrossRef] [PubMed]

Rindzevicius, T.

T. Rindzevicius, Y. Alaverdyan, B. Sepulveda, T. Pakizeh, M. Kall, R. Hillenbrand, J. Aizpurua, and F. J. Garcia de Abajo, “Nanohole Plasmons in Optically Thin Gold Films,” J. Phys. Chem. C 111(3), 1207–1212 (2007).
[CrossRef]

Sansonetti, J. E.

J. E. Sansonetti and J. K. Furdyna, “Depolarization Effects in Arrays of Spheres,” Phys. Rev. B 22(6), 2866–2874 (1980).
[CrossRef]

Schatz, G. C.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107(3), 668–677 (2003).
[CrossRef]

Schider, G.

N. Félidj, J. Aubard, G. Levi, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg, “Enhanced substrate-induced coupling in two-dimensional gold nanoparticle arrays,” Phys. Rev. B 66(24), 245407 (2002).
[CrossRef]

Schultz, D. A.

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116(15), 6755–6759 (2002).
[CrossRef]

Schultz, S.

J. J. Mock, D. R. Smith, and S. Schultz, “Local refractive index dependence of plasmon resonance spectra from individual nanoparticles,” Nano Lett. 3(4), 485–491 (2003).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

(Color online) (a) Measured reflectance of the nanoparticle network with Au filling factor of 21% on the PAM substrate. (b) Resistance of nanoparticle networks versus the Au filling factor using four-point probe method. Inset: the SEM micrograph of the sample in (a).

Fig. 2
Fig. 2

(Color online) Optical transmission spectra of nanoparticle film with Au filling factor of 21% calculated for non-interacting Au spheres (a), Au clusters of particles arranged in a 12-membered ring geometry (b), and our cluster model (c). The calculation is based on the dielectric function of bulk Au from [26]. The experimental data measured with a Tungsten lamp are also plotted in (c) (open circles).

Fig. 3
Fig. 3

(Color online) Optical transmission spectra of a series of Au nanoparticle networks. The open blue circles are the experimental data (associated with the left Y axis) measured with a Tungsten lamp and the solid red curves are the calculated data using our effective cluster model. The theoretical curves are normalized with respect to the experimental values by multiplying a factor of 0.7 to 0.85. With the effective cluster model, the region fraction of Phase-I ( V s i n g l e ) in the above samples are calculated to be: (from top to bottom) 66%, 48%, 37%, 34%, and 32%, respectively.

Fig. 4
Fig. 4

(Color online) Normalized conductivity σ / σ 0 as a function of the Au filling factor. The green solid line is calculated from the Bruggeman’s equation in the general effective medium theory. The red dashed line is calculated from our effective cluster model (Eq. (10-11). The blue scatters are the experimental data measured by the four-point probe method.

Equations (13)

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ε M G = ε h + 3 ε h f Λ 1 f Λ , with Λ = ε i ε h ε i + 2 ε h = α 3 ε h V
V 1 ε 1 ε B R ε 1 + 2 ε B R + V 2 ε 2 ε B R ε 2 + 2 ε B R = 0 ,
Γ = Γ b u l k + A ν F / R ,
ε ( ω ) = ε b u l k ( ω ) + ω p 2 ω 2 + i Γ b u l k ω ω p 2 ω 2 + i Γ ω ,
P = α ( E + E r a d ) ,
E r a d = 2 3 i k 3 P + k 2 R P ,
α i = ε 0 ε i ε h ε h + L i ( ε i ε h ) V , i = x , y , z
V s i n g l e ε s i n g l e ε B R ε s i n g l e + 2 ε B R + V c l u s t e r ε c l u s t e r ε B R ε c l u s t e r + 2 ε B R = 0 ,
ε s i n g l e = ε a i r ( 1 + 2 p s Λ s 1 p s Λ s ) , Λ s = ε m e t a l ε a i r ε m e t a l + 2 ε a i r
ε c l u s t e r = ε a i r ( 1 + 2 3 p c Λ c 1 2 3 p c Λ c ) , Λ c = 1 3 i = 1 2 A i ε m e t a l ε a i r ε a i r + L i ( ε m e t a l ε a i r )
V s i n g l e = ( p c f ) / ( p c p s ) , V c l u s t e r = 1 V s i n g l e
V s i n g l e σ s i n g l e σ e σ s i n g l e + 2 σ e + V c l u s t e r σ c l u s t e r σ e σ c l u s t e r + 2 σ e = 0.
σ e = 0 V c l u s t e r < V c σ e = 2 3 σ c l u s t e r ( V c l u s t e r V c ) V c l u s t e r > V c

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