Abstract

We develop an analytic circuit model for coupled plasmonic dimers separated by small gaps that provides a complete account of the optical resonance wavelength. Using a suitable equivalent circuit, it shows how partially conducting links can be treated and provides quantitative agreement with both experiment and full electromagnetic simulations. The model highlights how in the conducting regime, the kinetic inductance of the linkers set the spectral blue-shifts of the coupled plasmon.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  28. P. Nordlander and E. Prodan, “Plasmon hybridization in nanoparticles near metallic surfaces,” Nano Lett. 4, 2209–2213 (2004).
    [Crossref]
  29. S. Hudlet, M. SaintJean, C. Guthmann, and J. Berger, “Evaluation of the capacitive force between an atomic force microscopy tip and a metallic surface,” Eur. Phys. J. B 2, 5–10 (1998).
    [Crossref]
  30. F.J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65, 115418 (2002).
    [Crossref]
  31. M. Hegner, P. Wagner, and G. Semenza, “Ultralarge atomically flat template-stripped Au surfaces for scanning probe microscopy,” Surf. Sci. 291, 39–46 (1993).
    [Crossref]
  32. B. de Nijs, R.W. Bowman, L.O. Herrmann, F. Benz, S.J. Barrow, J. Mertens, D.O. Sigle, R. Chikkaraddy, A. Eiden, A. Ferrari, O.A. Scherman, and J.J. Baumberg, “Unfolding the contents of sub-nm plasmonic gaps using normalising plasmon resonance spectroscopy,” Faraday Discuss. 178, 185–193 (2015).
    [Crossref] [PubMed]
  33. M. Staffaroni, J. Conway, S. Vedantam, J. Tang, and E. Yablonovitch, “Circuit analysis in metal-optics,” Photonics Nanostructures - Fundam. Appl. 10, 166–176 (2012).
    [Crossref]
  34. J.P. Folkers, P.E. Laibinis, and G.M. Whitesides, “Self-assembled monolayers of alkanethiols on gold: comparisons of monolayers containing mixtures of short- and long-chain constituents with methyl and hydroxymethyl terminal groups,” Langmuir 8, 1330–1341 (1992).
    [Crossref]
  35. C.D. Bain, E.B. Troughton, Y.T. Tao, J. Evall, G.M. Whitesides, and R.G. Nuzzo, “Formation of monolayer films by the spontaneous assembly of organic thiols from solution onto gold,” J. Am. Chem. Soc. 111, 321–335 (1989).
    [Crossref]

2015 (2)

F. Benz, C. Tserkezis, L.O. Herrmann, B. de Nijs, A. Sanders, D.O. Sigle, L. Pukenas, S.D. Evans, J. Aizpurua, and J.J. Baumberg, “Nanooptics of Molecular-Shunted Plasmonic Nanojunctions,” Nano Lett. 15, 669–674 (2015).
[Crossref]

B. de Nijs, R.W. Bowman, L.O. Herrmann, F. Benz, S.J. Barrow, J. Mertens, D.O. Sigle, R. Chikkaraddy, A. Eiden, A. Ferrari, O.A. Scherman, and J.J. Baumberg, “Unfolding the contents of sub-nm plasmonic gaps using normalising plasmon resonance spectroscopy,” Faraday Discuss. 178, 185–193 (2015).
[Crossref] [PubMed]

2014 (4)

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108, 462 (2014).
[Crossref]

R.W. Taylor, F. Benz, D.O. Sigle, R.W. Bowman, P. Bao, J.S. Roth, G.R. Heath, S.D. Evans, and J.J. Baumberg, “Watching individual molecules flex within lipid membranes using SERS,” Sci. Rep. 4, 5940 (2014).
[Crossref] [PubMed]

G. Di Martino, Y. Sonnefraud, M.S. Tame, S. Kéna-Cohen, F. Dieleman, Ş.K. Özdemir, M.S. Kim, and S.A. Maier, “Observation of quantum interference in the plasmonic Hong-Ou-Mandel effect,” Phys. Rev. Appl. 1, 034004 (2014).
[Crossref]

R. Trivedi, A. Thomas, and A. Dhawan, “Full-wave electromagentic analysis of a plasmonic nanoparticle separated from a plasmonic film by a thin spacer layer,” Opt. Express 22, 19970 (2014).
[Crossref] [PubMed]

2013 (2)

N. Liu, F. Wen, Y. Zhao, Y. Wang, P. Nordlander, N.J. Halas, and A. Alú, “Individual nanoantennas loaded with three-dimensional optical nanocircuits,” Nano Lett. 13, 142–147 (2013).
[Crossref]

J.A. Scholl, A. García-Etxarri, A.L. Koh, and J.A. Dionne, “Observation of quantum tunneling between two plasmonic nanoparticles,” Nano Lett. 13, 564–569 (2013).
[Crossref]

2012 (5)

K.J. Savage, M.M. Hawkeye, R. Esteban, A.G. Borisov, J. Aizpurua, and J.J. Baumberg, “Revealing the quantum regime in tunnelling plasmonics,” Nature 491, 574–577 (2012).
[Crossref] [PubMed]

M. Staffaroni, J. Conway, S. Vedantam, J. Tang, and E. Yablonovitch, “Circuit analysis in metal-optics,” Photonics Nanostructures - Fundam. Appl. 10, 166–176 (2012).
[Crossref]

C. Cirací, R.T. Hill, J.J. Mock, Y. Urzhumov, A.I. Fernández-Domínguez, S.A. Maier, J.B. Pendry, A. Chilkoti, and D.R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

G. Di Martino, Y. Sonnefraud, S. Kéna-Cohen, M. Tame, Ş.K. Özdemir, M.S. Kim, and S.A. Maier, “Quantum statistics of surface plasmon polaritons in metallic stripe waveguides,” Nano Lett. 12, 2504–2508 (2012).
[Crossref] [PubMed]

R. Esteban, A.G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun. 3, 825 (2012).
[Crossref] [PubMed]

2010 (1)

O. Pérez-González, N. Zabala, A.G. Borisov, N.J. Halas, P. Nordlander, and J. Aizpurua, “Optical spectroscopy of conductive junctions in plasmonic cavities,” Nano Lett. 10, 3090–3095 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (2)

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photonics Rev. 2, 136–159 (2008).
[Crossref]

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A.M. Funston, C. Novo, P. Mulvaney, L.M. Liz-Marzán, and F. Javier García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

2007 (2)

P.K. Jain, W. Huang, and M.A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: a plasmon euler equation,” Nano Lett. 7, 2080–2088 (2007).
[Crossref]

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).
[Crossref] [PubMed]

2006 (1)

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74, 075103 (2006).
[Crossref]

2005 (2)

N. Engheta, A. Salandrino, and A. Alú, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[Crossref] [PubMed]

C. Sönnichsen, B.M. Reinhard, J. Liphardt, and A.P. Alivisatos, “A molecular ruler based on plasmon coupling of single gold and silver nanoparticles,” Nat. Biotechnol. 23, 741–745 (2005).
[Crossref] [PubMed]

2004 (2)

H. Itoh, K. Naka, and Y. Chujo, “Synthesis of dold nanoparticles modified with ionic liquid based on the imidazolium cation,” J. Am. Chem. Soc. 126, 3026–3027 (2004).
[Crossref] [PubMed]

P. Nordlander and E. Prodan, “Plasmon hybridization in nanoparticles near metallic surfaces,” Nano Lett. 4, 2209–2213 (2004).
[Crossref]

2002 (1)

F.J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65, 115418 (2002).
[Crossref]

2000 (2)

A.K. Sarychev and V.M. Shalaev, “Electromagnetic field fluctuations and optical nonlinearities in metal-dielectric composites,” Phys. Rep. 335, 275–371 (2000).
[Crossref]

I.E. Mazets, “Polarization of two close metal spheres in an external homogeneous electric field,” Tech. Phys. 45, 1238–1240 (2000).
[Crossref]

1998 (1)

S. Hudlet, M. SaintJean, C. Guthmann, and J. Berger, “Evaluation of the capacitive force between an atomic force microscopy tip and a metallic surface,” Eur. Phys. J. B 2, 5–10 (1998).
[Crossref]

1997 (1)

F. J. García de Abajo and J. Aizpurua, “Numerical simulation of electron energy loss near inhomogeneous dielectrics,” Phys. Rev. B 56, 15873–15884 (1997).
[Crossref]

1993 (1)

M. Hegner, P. Wagner, and G. Semenza, “Ultralarge atomically flat template-stripped Au surfaces for scanning probe microscopy,” Surf. Sci. 291, 39–46 (1993).
[Crossref]

1992 (1)

J.P. Folkers, P.E. Laibinis, and G.M. Whitesides, “Self-assembled monolayers of alkanethiols on gold: comparisons of monolayers containing mixtures of short- and long-chain constituents with methyl and hydroxymethyl terminal groups,” Langmuir 8, 1330–1341 (1992).
[Crossref]

1989 (1)

C.D. Bain, E.B. Troughton, Y.T. Tao, J. Evall, G.M. Whitesides, and R.G. Nuzzo, “Formation of monolayer films by the spontaneous assembly of organic thiols from solution onto gold,” J. Am. Chem. Soc. 111, 321–335 (1989).
[Crossref]

Aizpurua, J.

F. Benz, C. Tserkezis, L.O. Herrmann, B. de Nijs, A. Sanders, D.O. Sigle, L. Pukenas, S.D. Evans, J. Aizpurua, and J.J. Baumberg, “Nanooptics of Molecular-Shunted Plasmonic Nanojunctions,” Nano Lett. 15, 669–674 (2015).
[Crossref]

K.J. Savage, M.M. Hawkeye, R. Esteban, A.G. Borisov, J. Aizpurua, and J.J. Baumberg, “Revealing the quantum regime in tunnelling plasmonics,” Nature 491, 574–577 (2012).
[Crossref] [PubMed]

R. Esteban, A.G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun. 3, 825 (2012).
[Crossref] [PubMed]

O. Pérez-González, N. Zabala, A.G. Borisov, N.J. Halas, P. Nordlander, and J. Aizpurua, “Optical spectroscopy of conductive junctions in plasmonic cavities,” Nano Lett. 10, 3090–3095 (2010).
[Crossref] [PubMed]

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photonics Rev. 2, 136–159 (2008).
[Crossref]

F. J. García de Abajo and J. Aizpurua, “Numerical simulation of electron energy loss near inhomogeneous dielectrics,” Phys. Rev. B 56, 15873–15884 (1997).
[Crossref]

Alivisatos, A.P.

C. Sönnichsen, B.M. Reinhard, J. Liphardt, and A.P. Alivisatos, “A molecular ruler based on plasmon coupling of single gold and silver nanoparticles,” Nat. Biotechnol. 23, 741–745 (2005).
[Crossref] [PubMed]

Alú, A.

N. Liu, F. Wen, Y. Zhao, Y. Wang, P. Nordlander, N.J. Halas, and A. Alú, “Individual nanoantennas loaded with three-dimensional optical nanocircuits,” Nano Lett. 13, 142–147 (2013).
[Crossref]

N. Engheta, A. Salandrino, and A. Alú, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[Crossref] [PubMed]

J. Shi, F. Monticone, S. Elias, Y. Wu, D. Ratchford, X. Li, and A. Alú, “Modular assembly of optical nanocircuits,” Nat. Commun.5, (2014).
[Crossref]

Bain, C.D.

C.D. Bain, E.B. Troughton, Y.T. Tao, J. Evall, G.M. Whitesides, and R.G. Nuzzo, “Formation of monolayer films by the spontaneous assembly of organic thiols from solution onto gold,” J. Am. Chem. Soc. 111, 321–335 (1989).
[Crossref]

Bao, P.

R.W. Taylor, F. Benz, D.O. Sigle, R.W. Bowman, P. Bao, J.S. Roth, G.R. Heath, S.D. Evans, and J.J. Baumberg, “Watching individual molecules flex within lipid membranes using SERS,” Sci. Rep. 4, 5940 (2014).
[Crossref] [PubMed]

Barrow, S.J.

B. de Nijs, R.W. Bowman, L.O. Herrmann, F. Benz, S.J. Barrow, J. Mertens, D.O. Sigle, R. Chikkaraddy, A. Eiden, A. Ferrari, O.A. Scherman, and J.J. Baumberg, “Unfolding the contents of sub-nm plasmonic gaps using normalising plasmon resonance spectroscopy,” Faraday Discuss. 178, 185–193 (2015).
[Crossref] [PubMed]

Baumberg, J.J.

B. de Nijs, R.W. Bowman, L.O. Herrmann, F. Benz, S.J. Barrow, J. Mertens, D.O. Sigle, R. Chikkaraddy, A. Eiden, A. Ferrari, O.A. Scherman, and J.J. Baumberg, “Unfolding the contents of sub-nm plasmonic gaps using normalising plasmon resonance spectroscopy,” Faraday Discuss. 178, 185–193 (2015).
[Crossref] [PubMed]

F. Benz, C. Tserkezis, L.O. Herrmann, B. de Nijs, A. Sanders, D.O. Sigle, L. Pukenas, S.D. Evans, J. Aizpurua, and J.J. Baumberg, “Nanooptics of Molecular-Shunted Plasmonic Nanojunctions,” Nano Lett. 15, 669–674 (2015).
[Crossref]

R.W. Taylor, F. Benz, D.O. Sigle, R.W. Bowman, P. Bao, J.S. Roth, G.R. Heath, S.D. Evans, and J.J. Baumberg, “Watching individual molecules flex within lipid membranes using SERS,” Sci. Rep. 4, 5940 (2014).
[Crossref] [PubMed]

K.J. Savage, M.M. Hawkeye, R. Esteban, A.G. Borisov, J. Aizpurua, and J.J. Baumberg, “Revealing the quantum regime in tunnelling plasmonics,” Nature 491, 574–577 (2012).
[Crossref] [PubMed]

Benz, F.

F. Benz, C. Tserkezis, L.O. Herrmann, B. de Nijs, A. Sanders, D.O. Sigle, L. Pukenas, S.D. Evans, J. Aizpurua, and J.J. Baumberg, “Nanooptics of Molecular-Shunted Plasmonic Nanojunctions,” Nano Lett. 15, 669–674 (2015).
[Crossref]

B. de Nijs, R.W. Bowman, L.O. Herrmann, F. Benz, S.J. Barrow, J. Mertens, D.O. Sigle, R. Chikkaraddy, A. Eiden, A. Ferrari, O.A. Scherman, and J.J. Baumberg, “Unfolding the contents of sub-nm plasmonic gaps using normalising plasmon resonance spectroscopy,” Faraday Discuss. 178, 185–193 (2015).
[Crossref] [PubMed]

R.W. Taylor, F. Benz, D.O. Sigle, R.W. Bowman, P. Bao, J.S. Roth, G.R. Heath, S.D. Evans, and J.J. Baumberg, “Watching individual molecules flex within lipid membranes using SERS,” Sci. Rep. 4, 5940 (2014).
[Crossref] [PubMed]

Berger, J.

S. Hudlet, M. SaintJean, C. Guthmann, and J. Berger, “Evaluation of the capacitive force between an atomic force microscopy tip and a metallic surface,” Eur. Phys. J. B 2, 5–10 (1998).
[Crossref]

Borisov, A.G.

R. Esteban, A.G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun. 3, 825 (2012).
[Crossref] [PubMed]

K.J. Savage, M.M. Hawkeye, R. Esteban, A.G. Borisov, J. Aizpurua, and J.J. Baumberg, “Revealing the quantum regime in tunnelling plasmonics,” Nature 491, 574–577 (2012).
[Crossref] [PubMed]

O. Pérez-González, N. Zabala, A.G. Borisov, N.J. Halas, P. Nordlander, and J. Aizpurua, “Optical spectroscopy of conductive junctions in plasmonic cavities,” Nano Lett. 10, 3090–3095 (2010).
[Crossref] [PubMed]

Bowman, R.W.

B. de Nijs, R.W. Bowman, L.O. Herrmann, F. Benz, S.J. Barrow, J. Mertens, D.O. Sigle, R. Chikkaraddy, A. Eiden, A. Ferrari, O.A. Scherman, and J.J. Baumberg, “Unfolding the contents of sub-nm plasmonic gaps using normalising plasmon resonance spectroscopy,” Faraday Discuss. 178, 185–193 (2015).
[Crossref] [PubMed]

R.W. Taylor, F. Benz, D.O. Sigle, R.W. Bowman, P. Bao, J.S. Roth, G.R. Heath, S.D. Evans, and J.J. Baumberg, “Watching individual molecules flex within lipid membranes using SERS,” Sci. Rep. 4, 5940 (2014).
[Crossref] [PubMed]

Bryant, G.

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photonics Rev. 2, 136–159 (2008).
[Crossref]

Chikkaraddy, R.

B. de Nijs, R.W. Bowman, L.O. Herrmann, F. Benz, S.J. Barrow, J. Mertens, D.O. Sigle, R. Chikkaraddy, A. Eiden, A. Ferrari, O.A. Scherman, and J.J. Baumberg, “Unfolding the contents of sub-nm plasmonic gaps using normalising plasmon resonance spectroscopy,” Faraday Discuss. 178, 185–193 (2015).
[Crossref] [PubMed]

Chilkoti, A.

C. Cirací, R.T. Hill, J.J. Mock, Y. Urzhumov, A.I. Fernández-Domínguez, S.A. Maier, J.B. Pendry, A. Chilkoti, and D.R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

Chujo, Y.

H. Itoh, K. Naka, and Y. Chujo, “Synthesis of dold nanoparticles modified with ionic liquid based on the imidazolium cation,” J. Am. Chem. Soc. 126, 3026–3027 (2004).
[Crossref] [PubMed]

Cirací, C.

C. Cirací, R.T. Hill, J.J. Mock, Y. Urzhumov, A.I. Fernández-Domínguez, S.A. Maier, J.B. Pendry, A. Chilkoti, and D.R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

Conway, J.

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G. Di Martino, Y. Sonnefraud, M.S. Tame, S. Kéna-Cohen, F. Dieleman, Ş.K. Özdemir, M.S. Kim, and S.A. Maier, “Observation of quantum interference in the plasmonic Hong-Ou-Mandel effect,” Phys. Rev. Appl. 1, 034004 (2014).
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C. Cirací, R.T. Hill, J.J. Mock, Y. Urzhumov, A.I. Fernández-Domínguez, S.A. Maier, J.B. Pendry, A. Chilkoti, and D.R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337, 1072–1074 (2012).
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N. Liu, F. Wen, Y. Zhao, Y. Wang, P. Nordlander, N.J. Halas, and A. Alú, “Individual nanoantennas loaded with three-dimensional optical nanocircuits,” Nano Lett. 13, 142–147 (2013).
[Crossref]

R. Esteban, A.G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun. 3, 825 (2012).
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O. Pérez-González, N. Zabala, A.G. Borisov, N.J. Halas, P. Nordlander, and J. Aizpurua, “Optical spectroscopy of conductive junctions in plasmonic cavities,” Nano Lett. 10, 3090–3095 (2010).
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Novo, C.

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C.D. Bain, E.B. Troughton, Y.T. Tao, J. Evall, G.M. Whitesides, and R.G. Nuzzo, “Formation of monolayer films by the spontaneous assembly of organic thiols from solution onto gold,” J. Am. Chem. Soc. 111, 321–335 (1989).
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G. Di Martino, Y. Sonnefraud, M.S. Tame, S. Kéna-Cohen, F. Dieleman, Ş.K. Özdemir, M.S. Kim, and S.A. Maier, “Observation of quantum interference in the plasmonic Hong-Ou-Mandel effect,” Phys. Rev. Appl. 1, 034004 (2014).
[Crossref]

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[Crossref] [PubMed]

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V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A.M. Funston, C. Novo, P. Mulvaney, L.M. Liz-Marzán, and F. Javier García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
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O. Pérez-González, N. Zabala, A.G. Borisov, N.J. Halas, P. Nordlander, and J. Aizpurua, “Optical spectroscopy of conductive junctions in plasmonic cavities,” Nano Lett. 10, 3090–3095 (2010).
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P. Nordlander and E. Prodan, “Plasmon hybridization in nanoparticles near metallic surfaces,” Nano Lett. 4, 2209–2213 (2004).
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F. Benz, C. Tserkezis, L.O. Herrmann, B. de Nijs, A. Sanders, D.O. Sigle, L. Pukenas, S.D. Evans, J. Aizpurua, and J.J. Baumberg, “Nanooptics of Molecular-Shunted Plasmonic Nanojunctions,” Nano Lett. 15, 669–674 (2015).
[Crossref]

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J. Shi, F. Monticone, S. Elias, Y. Wu, D. Ratchford, X. Li, and A. Alú, “Modular assembly of optical nanocircuits,” Nat. Commun.5, (2014).
[Crossref]

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C. Sönnichsen, B.M. Reinhard, J. Liphardt, and A.P. Alivisatos, “A molecular ruler based on plasmon coupling of single gold and silver nanoparticles,” Nat. Biotechnol. 23, 741–745 (2005).
[Crossref] [PubMed]

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V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A.M. Funston, C. Novo, P. Mulvaney, L.M. Liz-Marzán, and F. Javier García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
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R.W. Taylor, F. Benz, D.O. Sigle, R.W. Bowman, P. Bao, J.S. Roth, G.R. Heath, S.D. Evans, and J.J. Baumberg, “Watching individual molecules flex within lipid membranes using SERS,” Sci. Rep. 4, 5940 (2014).
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S. Hudlet, M. SaintJean, C. Guthmann, and J. Berger, “Evaluation of the capacitive force between an atomic force microscopy tip and a metallic surface,” Eur. Phys. J. B 2, 5–10 (1998).
[Crossref]

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A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74, 075103 (2006).
[Crossref]

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F. Benz, C. Tserkezis, L.O. Herrmann, B. de Nijs, A. Sanders, D.O. Sigle, L. Pukenas, S.D. Evans, J. Aizpurua, and J.J. Baumberg, “Nanooptics of Molecular-Shunted Plasmonic Nanojunctions,” Nano Lett. 15, 669–674 (2015).
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A.K. Sarychev and V.M. Shalaev, “Electromagnetic field fluctuations and optical nonlinearities in metal-dielectric composites,” Phys. Rep. 335, 275–371 (2000).
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K.J. Savage, M.M. Hawkeye, R. Esteban, A.G. Borisov, J. Aizpurua, and J.J. Baumberg, “Revealing the quantum regime in tunnelling plasmonics,” Nature 491, 574–577 (2012).
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B. de Nijs, R.W. Bowman, L.O. Herrmann, F. Benz, S.J. Barrow, J. Mertens, D.O. Sigle, R. Chikkaraddy, A. Eiden, A. Ferrari, O.A. Scherman, and J.J. Baumberg, “Unfolding the contents of sub-nm plasmonic gaps using normalising plasmon resonance spectroscopy,” Faraday Discuss. 178, 185–193 (2015).
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J.A. Scholl, A. García-Etxarri, A.L. Koh, and J.A. Dionne, “Observation of quantum tunneling between two plasmonic nanoparticles,” Nano Lett. 13, 564–569 (2013).
[Crossref]

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M. Hegner, P. Wagner, and G. Semenza, “Ultralarge atomically flat template-stripped Au surfaces for scanning probe microscopy,” Surf. Sci. 291, 39–46 (1993).
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A.K. Sarychev and V.M. Shalaev, “Electromagnetic field fluctuations and optical nonlinearities in metal-dielectric composites,” Phys. Rep. 335, 275–371 (2000).
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J. Shi, F. Monticone, S. Elias, Y. Wu, D. Ratchford, X. Li, and A. Alú, “Modular assembly of optical nanocircuits,” Nat. Commun.5, (2014).
[Crossref]

Sigle, D.O.

F. Benz, C. Tserkezis, L.O. Herrmann, B. de Nijs, A. Sanders, D.O. Sigle, L. Pukenas, S.D. Evans, J. Aizpurua, and J.J. Baumberg, “Nanooptics of Molecular-Shunted Plasmonic Nanojunctions,” Nano Lett. 15, 669–674 (2015).
[Crossref]

B. de Nijs, R.W. Bowman, L.O. Herrmann, F. Benz, S.J. Barrow, J. Mertens, D.O. Sigle, R. Chikkaraddy, A. Eiden, A. Ferrari, O.A. Scherman, and J.J. Baumberg, “Unfolding the contents of sub-nm plasmonic gaps using normalising plasmon resonance spectroscopy,” Faraday Discuss. 178, 185–193 (2015).
[Crossref] [PubMed]

R.W. Taylor, F. Benz, D.O. Sigle, R.W. Bowman, P. Bao, J.S. Roth, G.R. Heath, S.D. Evans, and J.J. Baumberg, “Watching individual molecules flex within lipid membranes using SERS,” Sci. Rep. 4, 5940 (2014).
[Crossref] [PubMed]

Smith, D.R.

C. Cirací, R.T. Hill, J.J. Mock, Y. Urzhumov, A.I. Fernández-Domínguez, S.A. Maier, J.B. Pendry, A. Chilkoti, and D.R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

Sonnefraud, Y.

G. Di Martino, Y. Sonnefraud, M.S. Tame, S. Kéna-Cohen, F. Dieleman, Ş.K. Özdemir, M.S. Kim, and S.A. Maier, “Observation of quantum interference in the plasmonic Hong-Ou-Mandel effect,” Phys. Rev. Appl. 1, 034004 (2014).
[Crossref]

G. Di Martino, Y. Sonnefraud, S. Kéna-Cohen, M. Tame, Ş.K. Özdemir, M.S. Kim, and S.A. Maier, “Quantum statistics of surface plasmon polaritons in metallic stripe waveguides,” Nano Lett. 12, 2504–2508 (2012).
[Crossref] [PubMed]

Sönnichsen, C.

C. Sönnichsen, B.M. Reinhard, J. Liphardt, and A.P. Alivisatos, “A molecular ruler based on plasmon coupling of single gold and silver nanoparticles,” Nat. Biotechnol. 23, 741–745 (2005).
[Crossref] [PubMed]

Staffaroni, M.

M. Staffaroni, J. Conway, S. Vedantam, J. Tang, and E. Yablonovitch, “Circuit analysis in metal-optics,” Photonics Nanostructures - Fundam. Appl. 10, 166–176 (2012).
[Crossref]

Taflove, A.

A. Taflove and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Tame, M.

G. Di Martino, Y. Sonnefraud, S. Kéna-Cohen, M. Tame, Ş.K. Özdemir, M.S. Kim, and S.A. Maier, “Quantum statistics of surface plasmon polaritons in metallic stripe waveguides,” Nano Lett. 12, 2504–2508 (2012).
[Crossref] [PubMed]

Tame, M.S.

G. Di Martino, Y. Sonnefraud, M.S. Tame, S. Kéna-Cohen, F. Dieleman, Ş.K. Özdemir, M.S. Kim, and S.A. Maier, “Observation of quantum interference in the plasmonic Hong-Ou-Mandel effect,” Phys. Rev. Appl. 1, 034004 (2014).
[Crossref]

Tang, J.

M. Staffaroni, J. Conway, S. Vedantam, J. Tang, and E. Yablonovitch, “Circuit analysis in metal-optics,” Photonics Nanostructures - Fundam. Appl. 10, 166–176 (2012).
[Crossref]

Tao, Y.T.

C.D. Bain, E.B. Troughton, Y.T. Tao, J. Evall, G.M. Whitesides, and R.G. Nuzzo, “Formation of monolayer films by the spontaneous assembly of organic thiols from solution onto gold,” J. Am. Chem. Soc. 111, 321–335 (1989).
[Crossref]

Taylor, R.W.

R.W. Taylor, F. Benz, D.O. Sigle, R.W. Bowman, P. Bao, J.S. Roth, G.R. Heath, S.D. Evans, and J.J. Baumberg, “Watching individual molecules flex within lipid membranes using SERS,” Sci. Rep. 4, 5940 (2014).
[Crossref] [PubMed]

Thomas, A.

Trivedi, R.

Troughton, E.B.

C.D. Bain, E.B. Troughton, Y.T. Tao, J. Evall, G.M. Whitesides, and R.G. Nuzzo, “Formation of monolayer films by the spontaneous assembly of organic thiols from solution onto gold,” J. Am. Chem. Soc. 111, 321–335 (1989).
[Crossref]

Tserkezis, C.

F. Benz, C. Tserkezis, L.O. Herrmann, B. de Nijs, A. Sanders, D.O. Sigle, L. Pukenas, S.D. Evans, J. Aizpurua, and J.J. Baumberg, “Nanooptics of Molecular-Shunted Plasmonic Nanojunctions,” Nano Lett. 15, 669–674 (2015).
[Crossref]

Urzhumov, Y.

C. Cirací, R.T. Hill, J.J. Mock, Y. Urzhumov, A.I. Fernández-Domínguez, S.A. Maier, J.B. Pendry, A. Chilkoti, and D.R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

Vedantam, S.

M. Staffaroni, J. Conway, S. Vedantam, J. Tang, and E. Yablonovitch, “Circuit analysis in metal-optics,” Photonics Nanostructures - Fundam. Appl. 10, 166–176 (2012).
[Crossref]

Vo-Dinh, T.

Wagner, P.

M. Hegner, P. Wagner, and G. Semenza, “Ultralarge atomically flat template-stripped Au surfaces for scanning probe microscopy,” Surf. Sci. 291, 39–46 (1993).
[Crossref]

Wang, Y.

N. Liu, F. Wen, Y. Zhao, Y. Wang, P. Nordlander, N.J. Halas, and A. Alú, “Individual nanoantennas loaded with three-dimensional optical nanocircuits,” Nano Lett. 13, 142–147 (2013).
[Crossref]

Wen, F.

N. Liu, F. Wen, Y. Zhao, Y. Wang, P. Nordlander, N.J. Halas, and A. Alú, “Individual nanoantennas loaded with three-dimensional optical nanocircuits,” Nano Lett. 13, 142–147 (2013).
[Crossref]

Whitesides, G.M.

J.P. Folkers, P.E. Laibinis, and G.M. Whitesides, “Self-assembled monolayers of alkanethiols on gold: comparisons of monolayers containing mixtures of short- and long-chain constituents with methyl and hydroxymethyl terminal groups,” Langmuir 8, 1330–1341 (1992).
[Crossref]

C.D. Bain, E.B. Troughton, Y.T. Tao, J. Evall, G.M. Whitesides, and R.G. Nuzzo, “Formation of monolayer films by the spontaneous assembly of organic thiols from solution onto gold,” J. Am. Chem. Soc. 111, 321–335 (1989).
[Crossref]

Wu, Y.

J. Shi, F. Monticone, S. Elias, Y. Wu, D. Ratchford, X. Li, and A. Alú, “Modular assembly of optical nanocircuits,” Nat. Commun.5, (2014).
[Crossref]

Yablonovitch, E.

M. Staffaroni, J. Conway, S. Vedantam, J. Tang, and E. Yablonovitch, “Circuit analysis in metal-optics,” Photonics Nanostructures - Fundam. Appl. 10, 166–176 (2012).
[Crossref]

Zabala, N.

O. Pérez-González, N. Zabala, A.G. Borisov, N.J. Halas, P. Nordlander, and J. Aizpurua, “Optical spectroscopy of conductive junctions in plasmonic cavities,” Nano Lett. 10, 3090–3095 (2010).
[Crossref] [PubMed]

Zhao, Y.

N. Liu, F. Wen, Y. Zhao, Y. Wang, P. Nordlander, N.J. Halas, and A. Alú, “Individual nanoantennas loaded with three-dimensional optical nanocircuits,” Nano Lett. 13, 142–147 (2013).
[Crossref]

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V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A.M. Funston, C. Novo, P. Mulvaney, L.M. Liz-Marzán, and F. Javier García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
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Faraday Discuss. (1)

B. de Nijs, R.W. Bowman, L.O. Herrmann, F. Benz, S.J. Barrow, J. Mertens, D.O. Sigle, R. Chikkaraddy, A. Eiden, A. Ferrari, O.A. Scherman, and J.J. Baumberg, “Unfolding the contents of sub-nm plasmonic gaps using normalising plasmon resonance spectroscopy,” Faraday Discuss. 178, 185–193 (2015).
[Crossref] [PubMed]

J. Am. Chem. Soc. (2)

C.D. Bain, E.B. Troughton, Y.T. Tao, J. Evall, G.M. Whitesides, and R.G. Nuzzo, “Formation of monolayer films by the spontaneous assembly of organic thiols from solution onto gold,” J. Am. Chem. Soc. 111, 321–335 (1989).
[Crossref]

H. Itoh, K. Naka, and Y. Chujo, “Synthesis of dold nanoparticles modified with ionic liquid based on the imidazolium cation,” J. Am. Chem. Soc. 126, 3026–3027 (2004).
[Crossref] [PubMed]

Langmuir (1)

J.P. Folkers, P.E. Laibinis, and G.M. Whitesides, “Self-assembled monolayers of alkanethiols on gold: comparisons of monolayers containing mixtures of short- and long-chain constituents with methyl and hydroxymethyl terminal groups,” Langmuir 8, 1330–1341 (1992).
[Crossref]

Laser Photonics Rev. (1)

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photonics Rev. 2, 136–159 (2008).
[Crossref]

Nano Lett. (7)

N. Liu, F. Wen, Y. Zhao, Y. Wang, P. Nordlander, N.J. Halas, and A. Alú, “Individual nanoantennas loaded with three-dimensional optical nanocircuits,” Nano Lett. 13, 142–147 (2013).
[Crossref]

G. Di Martino, Y. Sonnefraud, S. Kéna-Cohen, M. Tame, Ş.K. Özdemir, M.S. Kim, and S.A. Maier, “Quantum statistics of surface plasmon polaritons in metallic stripe waveguides,” Nano Lett. 12, 2504–2508 (2012).
[Crossref] [PubMed]

P.K. Jain, W. Huang, and M.A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: a plasmon euler equation,” Nano Lett. 7, 2080–2088 (2007).
[Crossref]

F. Benz, C. Tserkezis, L.O. Herrmann, B. de Nijs, A. Sanders, D.O. Sigle, L. Pukenas, S.D. Evans, J. Aizpurua, and J.J. Baumberg, “Nanooptics of Molecular-Shunted Plasmonic Nanojunctions,” Nano Lett. 15, 669–674 (2015).
[Crossref]

J.A. Scholl, A. García-Etxarri, A.L. Koh, and J.A. Dionne, “Observation of quantum tunneling between two plasmonic nanoparticles,” Nano Lett. 13, 564–569 (2013).
[Crossref]

O. Pérez-González, N. Zabala, A.G. Borisov, N.J. Halas, P. Nordlander, and J. Aizpurua, “Optical spectroscopy of conductive junctions in plasmonic cavities,” Nano Lett. 10, 3090–3095 (2010).
[Crossref] [PubMed]

P. Nordlander and E. Prodan, “Plasmon hybridization in nanoparticles near metallic surfaces,” Nano Lett. 4, 2209–2213 (2004).
[Crossref]

Nat. Biotechnol. (1)

C. Sönnichsen, B.M. Reinhard, J. Liphardt, and A.P. Alivisatos, “A molecular ruler based on plasmon coupling of single gold and silver nanoparticles,” Nat. Biotechnol. 23, 741–745 (2005).
[Crossref] [PubMed]

Nat. Commun. (1)

R. Esteban, A.G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun. 3, 825 (2012).
[Crossref] [PubMed]

Nature (1)

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Opt. Express (2)

Photonics Nanostructures - Fundam. Appl. (1)

M. Staffaroni, J. Conway, S. Vedantam, J. Tang, and E. Yablonovitch, “Circuit analysis in metal-optics,” Photonics Nanostructures - Fundam. Appl. 10, 166–176 (2012).
[Crossref]

Phys. Rep. (1)

A.K. Sarychev and V.M. Shalaev, “Electromagnetic field fluctuations and optical nonlinearities in metal-dielectric composites,” Phys. Rep. 335, 275–371 (2000).
[Crossref]

Phys. Rev. Appl. (1)

G. Di Martino, Y. Sonnefraud, M.S. Tame, S. Kéna-Cohen, F. Dieleman, Ş.K. Özdemir, M.S. Kim, and S.A. Maier, “Observation of quantum interference in the plasmonic Hong-Ou-Mandel effect,” Phys. Rev. Appl. 1, 034004 (2014).
[Crossref]

Phys. Rev. B (3)

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74, 075103 (2006).
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F. J. García de Abajo and J. Aizpurua, “Numerical simulation of electron energy loss near inhomogeneous dielectrics,” Phys. Rev. B 56, 15873–15884 (1997).
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F.J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65, 115418 (2002).
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Phys. Rev. Lett. (1)

N. Engheta, A. Salandrino, and A. Alú, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
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Sci. Rep. (1)

R.W. Taylor, F. Benz, D.O. Sigle, R.W. Bowman, P. Bao, J.S. Roth, G.R. Heath, S.D. Evans, and J.J. Baumberg, “Watching individual molecules flex within lipid membranes using SERS,” Sci. Rep. 4, 5940 (2014).
[Crossref] [PubMed]

Science (2)

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).
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C. Cirací, R.T. Hill, J.J. Mock, Y. Urzhumov, A.I. Fernández-Domínguez, S.A. Maier, J.B. Pendry, A. Chilkoti, and D.R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337, 1072–1074 (2012).
[Crossref] [PubMed]

Surf. Sci. (1)

M. Hegner, P. Wagner, and G. Semenza, “Ultralarge atomically flat template-stripped Au surfaces for scanning probe microscopy,” Surf. Sci. 291, 39–46 (1993).
[Crossref]

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A. Taflove and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

J. Shi, F. Monticone, S. Elias, Y. Wu, D. Ratchford, X. Li, and A. Alú, “Modular assembly of optical nanocircuits,” Nat. Commun.5, (2014).
[Crossref]

F. Benz, B. de Nijs, C. Tserkezis, R. Chikkaraddy, D.O. Sigle, L. Pukenas, S.D. Evans, J. Aizpurua, and J.J. Baumberg, Data for Generalized circuit model for coupled plasmonic systems, DSpace@Cambridge (2015), https://www.repository.cam.ac.uk/handle/1810/252428 .

Supplementary Material (1)

NameDescription
» Dataset 1       Data for generalized circuit model for coupled plasmonic systems

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Figures (9)

Fig. 1
Fig. 1

Resonance wavelength for different nanoparticle sizes and resulting values for ε, determined from extinction spectroscopy in water. Inset: Single nanoparticle equivalent LC-circuit. Source data available in Dataset 1 (Ref. [20]).

Fig. 2
Fig. 2

Illustration of the used circuit model and the nanoparticle on mirror geometry. (a) Equivalent circuit for a coupled plasmon mode with both purely capacitive coupling and conductive coupling. (b) Illustration of the nanoparticle on mirror (NPoM) geometry. Source data available in Dataset 1 (Ref. [20]).

Fig. 3
Fig. 3

Comparison of the presented circuit model to experimental and simulated data. (a,b) Coupled plasmon mode with R = 40nm NPoM for different gap separations and refractive indices. (c) Nanoparticle size and geometry dependence of the lateral field localisation (solid circles, expressed in terms of the angle θmax, see appendix B) and refractive index exponent, χ (empty squares). (d) Blue-shift of the coupled plasmon mode as gap conductance increases, for R = 30nm, simulated and experimental data from [24]. The dashed line is shifted by 170G0 (see main text). Source data available in Dataset 1 (Ref. [20]).

Fig. 4
Fig. 4

Interpretation of the introduced gap inductance. (a) Gap inductance vs neck radius, extracted by using Eq. (5) on simulation results, and compared to kinetic inductance model (solid line). Filaments are either gold, or near-perfect conductor. (b) Schematic of sheath currents confined to penetration depth Λ down conducting filament, which gives kinetic inductance. Source data available in Dataset 1 (Ref. [20]).

Fig. 5
Fig. 5

Resonance wavelength as a function of the gap conductivity for several gap inductances (80 nm NPoM, d = 1.1 nm, n = 1.45). Source data available in Dataset 1 (Ref. [20]).

Fig. 6
Fig. 6

Field penetration depth in NPoM. Exponential decay length of optical field calculated from simple model above, and from full BEM simulations. Source data available in Dataset 1 (Ref. [20]).

Fig. 7
Fig. 7

Illustration of the used capacitance model. (a) By using a θmax much smaller than π, only a part of the sphere is taken into account for the capacitance. Throughout the paper θmaxπ/5 is used for the nanoparticle on mirror geometry. (b) Refractive index dependency for both nanoparticle on mirror and nanoparticle dimers, illustrating the different dependencies on the refractive index. Source data available in Dataset 1 (Ref. [20]).

Fig. 8
Fig. 8

Refractive index exponent χ for different geometries. (a,b) Nanoparticle on mirror (NPoM) and dimer geometry. The grey shaded area shows the approximate lateral dimension of the plasmon. The dashed lines indicate the exponent value for our two reference geometries: a film covering the whole gold substrate for the nanoparticle on mirror geometry and a nanoparticle dimer submerged in a medium with the respective refractive index for the dimers. Source data available in Dataset 1 (Ref. [20]).

Fig. 9
Fig. 9

Darkfield spectroscopy of aliphatic SAM samples. (a) SAM thickness as a function of the number of carbon atoms, measured by ellipsometry. (b) Typical distribution of plasmon resonance wavelengths for over 1,000 nanoparticles on a mirror with butanethiol (C4) as a spacer. (c) Resonance wavelength for different aliphatic SAM spacer. Source data available in Dataset 1 (Ref. [20]).

Equations (29)

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ε = ε ( ω p ω ) 2 = ε ( λ λ p ) 2 .
Z t o t = 2 2 i π ω R ε m ε 0 i π ω R ε ε 0 + 1 i ω C g + [ R g i ω L g ] 1 .
ω ˜ 4 + ω ˜ 2 [ ω d 2 ( 4 ε m Γ 1 ) + δ ] ω d 2 δ = 0
λ B D P = λ d = λ p 2 ε m + ε + 4 ε m η
λ S B D P = λ d 1 4 ε m Γ = λ p 2 ε m + ε + 4 ε m η 1 4 ε m Γ
C g = 2 π ε 0 ( n g ) χ R × ln [ 1 + R 2 d θ max 2 ]
L k = 1 ε 0 ω p 2 d + 2 Λ 2 π Λ a = 1 ε 0 ω p 2 f a
Δ λ λ = ε m a R ( 1 + d 2 Λ ) 1
Z t o t = 2 2 i π ω R ε m ε 0 i π ω R ε ε 0 + 1 i ω C g + [ R g i ω L g ] 1
Z t o t = 4 i 2 π R ε 0 × ω ω 2 ( 2 ε m + ε ) + i C s × 1 ω C g C s + 1 ω C s L g i R g C s
Z t o t = i 2 π R ω p ε 0 ( 4 ω ( 2 ε m + ε ) ω ˜ 2 1 + 1 ε m 1 ω ˜ η + 1 ω ˜ Γ i γ )
Z t o t = 1 i 2 π R ω p ε 0 ( 4 ω ˜ ε m ( ω ˜ η + 1 ω ˜ Γ i γ ) + ( 2 ε m + ε ) ω ˜ 2 1 ε m ( ω ˜ η + 1 ω ˜ Γ i γ ) [ ( 2 ε m + ε ) ω ˜ 2 1 ] )
( 1 Z t o t ) = [ i 2 π R ω p ε 0 ( ε m ( ω ˜ η + 1 ω ˜ Γ i γ ) [ ( 2 ε m + ε ) ω ˜ 2 1 ] 4 ω ˜ ε m ( ω ˜ η + 1 ω ˜ Γ i γ ) + ( 2 ε m + ε ) ω ˜ 2 1 ) ]
( 1 Z t o t ) = [ 2 π R ω p ε 0 ( ε m ( ω ˜ η + 1 ω ˜ Γ i γ ) [ ( 2 ε m + ε ) ω ˜ 2 1 ] 4 ω ˜ ε m ( ω ˜ η + 1 ω ˜ Γ i γ ) + ( 2 ε m + ε ) ω ˜ 2 1 ) ] = ! 0
[ ε m ( ω ˜ η + ω ˜ Γ + i γ ω ˜ 2 Γ 2 + γ 2 ) [ ( 2 ε m + ε ) ω ˜ 2 1 ] [ 4 ω ˜ ε m ( ω ˜ η + ω ˜ Γ i γ ω ˜ 2 Γ 2 + γ 2 ) + ( 2 ε m + ε ) ω ˜ 2 1 ] ] = ! 0
[ ε m ( ω ˜ η + ω ˜ Γ + i γ ω ˜ 2 Γ 2 + γ 2 ) [ 4 ω ˜ ε m ( ω ˜ η + ω ˜ Γ i γ ω ˜ 2 Γ 2 + γ 2 ) + ( 2 ε m + ε ) ω ˜ 2 1 ] ] = ! 0
ε m ( ω ˜ η + ω ˜ Γ + i γ ω ˜ 2 Γ 2 + γ 2 ) [ ( 2 ε m + ε ) ω ˜ 2 1 + 4 ω ˜ 2 ε m ( ω ˜ η + ω ˜ Γ ω ˜ 2 Γ 2 + γ 2 ) ] + ( γ ε m ω ˜ 2 Γ 2 + γ 2 ) ( 4 ω ˜ γ ε m ω ˜ 2 Γ 2 + γ 2 ) = ! 0
ε m ω ˜ ( η ( ω ˜ 2 Γ 2 + γ 2 ) + Γ 1 ) [ ( 2 ε m + ε ) ω ˜ 2 1 + 4 ω ˜ 2 ε m ( ω ˜ η + ω ˜ Γ ω ˜ 2 Γ 2 + γ 2 ) ] + γ ε m ( 4 ω ˜ γ ε m ω ˜ 2 Γ 2 + γ 2 ) = ! 0
ω ˜ 2 ε m [ η ( ω ˜ 2 Γ 2 + γ 2 ) + Γ ] 2 + γ 2 ε m + ( 2 ε m + ε ) ω ˜ 2 1 4 [ η ( ω ˜ 2 Γ 2 + γ 2 ) + Γ ] ( ω ˜ 2 Γ 2 + γ 2 ) = ! 0
ω ˜ 4 + ω ˜ 2 [ ω d 2 ( 4 ε m Γ 1 ) + δ ] ω d 2 δ = 0
λ S B D P = λ d 1 4 ε m Γ = λ d 1 4 ε m C s L g ω p 2 = λ d 1 4 ε m ε 0 ω p 2 a 2 π ε 0 R f ω p 2 = λ d 1 2 ε m a π R f
λ S B D P = λ d ( 1 + ε m a π R f )
S m = 2 ε d d ε m
E z exp ( S m z )
Λ = 1 ( S m ) = d 2 ε d ( 1 / ε m )
Λ = d | ε λ 2 λ p 2 | 2 n g 2
η = C g C s = ε 0 n g 2 A / d 2 π R ε 0 = n g 2 A 2 π R d
η = 2 π ε 0 n g 2 R 2 π ε 0 R 0 π sin 2 θ θ [ d R + 1 cos θ ] d θ
η = n g 2 ln ( 1 + R 2 d θ max 2 )

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