Abstract

Using a fully quantum mechanical approach we study the optical response of a strongly coupled metallic nanowire dimer for variable separation widths of the junction between the nanowires. The translational invariance of the system allows to apply the time–dependent density functional theory (TDDFT) for nanowires of diameters up to 10 nm which is the largest size considered so far in quantum modeling of plasmonic dimers. By performing a detailed analysis of the optical extinction, induced charge densities, and near fields, we reveal the major nonlocal quantum effects determining the plasmonic modes and field enhancement in the system. These effects consist mainly of electron tunneling between the nanowires at small junction widths and dynamical screening. The TDDFT results are compared with results from classical electromagnetic calculations based on the local Drude and non-local hydrodynamic descriptions of the nanowire permittivity, as well as with results from a recently developed quantum corrected model. The latter provides a way to include quantum mechanical effects such as electron tunneling in standard classical electromagnetic simulations. We show that the TDDFT results can be thus retrieved semi-quantitatively within a classical framework. We also discuss the shortcomings of classical non-local hydrodynamic approaches. Finally, the implications of the actual position of the screening charge density at the gap interfaces are discussed in connection with plasmon ruler applications at subnanometric distances.

© 2013 OSA

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2013 (5)

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]

L. Stella, P. Zhang, F. J. García-Vidal, A. Rubio, and P. García-González, “Performance of nonlocal optics when applied to plasmonic nanostructures,” J. Phys. Chem. C117, 8941–8949 (2013).
[CrossRef]

T. V. Teperik, P. Nordlander, J. Aizpurua, and A.G. Borisov, “Robust subnanometric plasmon ruler by rescaling of the nonlocal optical response,” Phys. Rev. Lett.110, 263901 (2013).
[CrossRef] [PubMed]

K. Andersen, K. L. Jensen, N. A. Mortensen, and K. S. Thygesen, “Visualizing hybridized quantum plasmons in coupled nanowires: From classical to tunneling regime,” Phys. Rev. B87, 235433 (2013).
[CrossRef]

R. C. Monreal, T. J. Antosiewicz, and P. Apell, “Competition between surface screening and size quantization for surface plasmons in nanoparticles,” New J. Phys.15, 083044 (2013).
[CrossRef]

2012 (13)

R. T. Hill, J. J. Mock, A. Hucknall, S. D. Wolter, N. M. Jokerst, D. R. Smith, and A. Chilkoti, “Plasmon ruler with angstrom length resolution,” ACS Nano6, 9237–9246 (2012).
[CrossRef] [PubMed]

X. Ben and H. S. Park, “Size-dependent validity bounds on the universal plasmon ruler for metal nanostructure dimers,” J. Phys. Chem. C116, 18944–18951 (2012).
[CrossRef]

J. Kern, S. Großmann, N. V. Tarakina, T. Häckel, M. Emmerling, M. Kamp, J.-S. Huang, P. Biagioni, J. C. Prangsma, and B. Hecht, “Atomic-scale confinement of resonant optical fields,” Nano Lett.12, 5504–5509 (2012).
[CrossRef] [PubMed]

H. Duan, A. I. Fernández-Domínguez, M. Bosman, S. A. Maier, and J. K. W. Yang, “Nanoplasmonics: classical down to the nanometer scale,” Nano Lett.12, 1683–1689 (2012).
[CrossRef] [PubMed]

D. C. Marinica, A. K. Kazansky, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum plasmonics: nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer,” Nano Lett.12, 1333–1339 (2012).
[CrossRef] [PubMed]

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

C. Ciracì, 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,” Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett.108, 106802 (2012).
[CrossRef] [PubMed]

A. I. Fernández-Domínguez, P. Zhang, Y. Luo, S. A. Maier, F. J. García-Vidal, and J. B. Pendry, “Transformation-optics insight into nonlocal effects in separated nanowires,” Phys. Rev. B86,241110(R) (2012).
[CrossRef]

G. Toscano, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction by plasmonic nanowire dimers due to nonlocal response,” Optics Express20, 4176–4188 (2012).
[CrossRef] [PubMed]

G. Toscano, S. Raza, S. Xiao, M. Wubs, A.-P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Surface-enhanced Raman spectroscopy (SERS): nonlocal limitations,” Opt. Lett.37, 2538–2540 (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,” Nature491, 574–577 (2012).
[CrossRef] [PubMed]

M. Banik, P. Z. El-Khoury, A. Nag, A. Rodriguez-Perez, N. Guarrottxena, G. C. Bazan, and V. A. Apkarian, “Surface-enhanced Raman trajectories on a nano-dumbbell: transition from field to charge transfer plasmons as the spheres fuse,” ACS Nano6, 10343–10354 (2012).
[CrossRef] [PubMed]

2011 (11)

O. Pérez-González, N. Zabala, and J. Aizpurua, “Optical characterization of charge transfer and bonding dimer plasmons in linked interparticle gaps,” New J. Phys.13, 083013 (2011).
[CrossRef]

M. Hentschel, D. Dregely, R. Vogelgesang, H. Giessen, and N. Liu, “Plasmonic oligomers: the role of individual particles in collective behavior,” ACS Nano5, 2042–2050 (2011).
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2010 (8)

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2009 (8)

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2008 (3)

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2007 (3)

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2006 (2)

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2005 (5)

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1983 (1)

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1970 (1)

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S. S. Aćimović, M. P. Kreuzer, M. U. González, and R. Quidant, “Plasmon near-field coupling in metal dimers as a step toward single-molecule sensing,” ACS Nano3, 1231–1237 (2009).
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D. C. Marinica, A. K. Kazansky, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum plasmonics: nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer,” Nano Lett.12, 1333–1339 (2012).
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M. Schnell, A. Garcia-Etxarri, A. J. Huber, K. Crozier, J. Aizpurua, and R. Hillenbrand, “Controlling the near-field oscillations of loaded plasmonic nanoantennas,” Nature Photonics3, 287–291 (2009).
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N. Liu, M. Hentschel, T. Weiss, A. P. Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science332, 1407–1410 (2011).
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R. Alvarez-Puebla, L. M. Liz-Marzán, and F. J. García de Abajo, “Light concentration at the nanometer scale,” J. Phys. Chem. Lett.1, 2428–2434 (2010).
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P. Apell, Å. Ljungbert, and S. Lundqvist, “Non-local effects at metal surfaces,” Physica Scripta30, 367–383 (1984).
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M. Banik, P. Z. El-Khoury, A. Nag, A. Rodriguez-Perez, N. Guarrottxena, G. C. Bazan, and V. A. Apkarian, “Surface-enhanced Raman trajectories on a nano-dumbbell: transition from field to charge transfer plasmons as the spheres fuse,” ACS Nano6, 10343–10354 (2012).
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T. Atay, J.-H. Song, and A. V. Nurmikko, “Strongly interacting plasmon nanoparticle pairs: from dipole-dipole interaction to conductively coupled regime,” Nano Lett.4, 1627–1631 (2004).
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S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nature Materials2, 229–232 (2003).
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D. Y. Lei, A. Aubry, Y. Luo, S. A. Maier, and J. B. Pendry, “Plasmon interaction between overlapping nanowires,” ACS Nano5, 597–607 (2011).
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Aussenegg, F. R.

Bachelier, G.

S. Marhaba, G. Bachelier, Ch. Bonnet, M. Broyer, E. Cottancin, N. Grillet, J. Lerme, J.-L. Vialle, and M. Pellarin, “Surface plasmon resonance of single gold nanodimers near the conductive contact limit,” J. Phys. Chem. C113, 4349–4356 (2009).
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K.-D. Tsuei, E. W. Plummer, A. Liebsch, K. Kempa, and P. Bakshi, “Multipole plasmon modes at a metal surface,” Phys. Rev. Lett.64, 44–47 (1990).
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M. Banik, P. Z. El-Khoury, A. Nag, A. Rodriguez-Perez, N. Guarrottxena, G. C. Bazan, and V. A. Apkarian, “Surface-enhanced Raman trajectories on a nano-dumbbell: transition from field to charge transfer plasmons as the spheres fuse,” ACS Nano6, 10343–10354 (2012).
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J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nature Materials9, 193–204 (2010).
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Baumberg, J. J.

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the quantum regime in tunnelling plasmonics,” Nature491, 574–577 (2012).
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R. W. Taylor, T.-Ch. Lee, O. A. Scherman, R. Esteban, J. Aizpurua, F. M. Huang, J. J. Baumberg, and S. Mahajan, “Precise subnanometer plasmonic junctions for SERS within gold nanoparticle assemblies using cucurbit[n]uril ”glue”,” ACS Nano5, 3878–3887 (2011).
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M. Banik, P. Z. El-Khoury, A. Nag, A. Rodriguez-Perez, N. Guarrottxena, G. C. Bazan, and V. A. Apkarian, “Surface-enhanced Raman trajectories on a nano-dumbbell: transition from field to charge transfer plasmons as the spheres fuse,” ACS Nano6, 10343–10354 (2012).
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S. Berciaud, L. Cognet, P. Tamarat, and B. Lounis, “Observation of intrinsic size effects in the optical response of individual gold nanoparticles,” Nano Lett.5, 515–518 (2005).
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Biagioni, P.

J. Kern, S. Großmann, N. V. Tarakina, T. Häckel, M. Emmerling, M. Kamp, J.-S. Huang, P. Biagioni, J. C. Prangsma, and B. Hecht, “Atomic-scale confinement of resonant optical fields,” Nano Lett.12, 5504–5509 (2012).
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H. Xu, E. Bjeneld, M. Käll, and L. Börjesson, “Spectroscopy of single hemoglobin molecules by surface enhanced Raman scattering,” Phys. Rev. Lett.83, 4357–4360 (1999).
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B. Fazio, C. D’Andrea, F. Bonaccorso, A. Irrera, G. Calogero, C. Vasi, P. G. Gucciardi, M. Allegrini, A. Toma, D. Chiappe, C. Martella, and F. B. de Mongeot, “Re-radiation enhancement in polarized surface-enhanced resonant Raman scattering of randomly oriented molecules on self-organized gold nanowires,” ACS Nano5, 5945–5956 (2011).
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S. Marhaba, G. Bachelier, Ch. Bonnet, M. Broyer, E. Cottancin, N. Grillet, J. Lerme, J.-L. Vialle, and M. Pellarin, “Surface plasmon resonance of single gold nanodimers near the conductive contact limit,” J. Phys. Chem. C113, 4349–4356 (2009).
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J. Borggreen, P. Chowdhury, N. Kebaïli, L. Lundsberg-Nielsen, K. Lützenkirchen, M. B. Nielsen, J. Pedersen, and H. D. Rasmussen, “Plasma excitations in charged sodium clusters,” Phys. Rev. B48, 17507–17516 (1993).
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Borisov, A. G.

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

D. C. Marinica, A. K. Kazansky, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum plasmonics: nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer,” Nano Lett.12, 1333–1339 (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,” Nature491, 574–577 (2012).
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Figures (7)

Fig. 1
Fig. 1

Sketch of the geometry of the nanowire dimer. Two identical cylindrical nanowires are infinite along the z-axis and have a diameter D of the circular cross-section in the (x, y)-plane. The nanowires are separated by a junction of separation width S. The incident radiation is linearly polarised with the electric field along the x-axis.

Fig. 2
Fig. 2

Extinction coefficient for the single jellium nanowire of diameter D = 6.2 nm (left) and D = 9.8 nm (right). Results are shown as function of the frequency ω of the incident radiation. The incoming field is the x-polarised plane wave. The TDDFT calculations are compared with results of the classical electromagnetic calculations using local (Drude) and NLHD. See the legend for definition of the different symbols used in the Fig.

Fig. 3
Fig. 3

Waterfall plot of the extinction cross section per length for a nanowire dimer in vacuum. The dimer consists of two Na nanowires of diameter D = 6.2 nm (left) and D = 9.8 nm (right) separated by a junction of variable width S. The incoming field is an x-polarised plane wave. The centers of the wires are at x = ±(D + S)/2), and S is negative for overlapping cylinders. S = −D would correspond to the limit of a single cylinder. TDDFT results are given as function of the frequency ω of the incident radiation for different separations S between the nanowires. For clarity a vertical shift proportional to the separation distance is introduced for each absorption spectrum. The red curves are used each 5 a0 ≈ 2.65 Å of S-change. These are labeled with corresponding S-values each 10 a0 ≈ 5.3 Å of S-change. The plasmonic modes responsible for the peaks in the absorption cross-section are labelled. These are: Bonding Dipole Plasmon (DP), Bonding Quadrupole Plasmon (QP), high order hybridised mode close to ωsp (HM), the lowest (dipole) Charge Transfer Plasmon (C1), and the higher energy Charge Transfer Plasmon (C2). On the right panel the blue dotted curve at S = 26.5 Å shows results of the classical Drude calculation with adjusted parameters and the green dotted curve represents results of the NLHD model. Further details are give in the main text.

Fig. 4
Fig. 4

Panels (a)–(k) Detailed analysis of the plasmon dynamics in the D = 9.8 nm nanowire dimer system. The incident x-polarised laser pulse is at resonance with the lowest (DP at S > 0 or C1 at S ≤ 0) plasmon mode. Panels (a)–(j) present snapshots of the induced charge density Δn, current density Jx, and electric field Ex for different junction widths S as indicated to the left of each row. The induced currents and fields are measured along the interparticle x-axis. Positive (negative) values correspond to the red (blue) color code. The induced densities are shown at the instant of time corresponding to the maximum dipole moment of the dimer. The induced currents and fields are shown at the instants of time when the induced fields in the junction reach the maximum. Panel (k): Conductivity analysis. The current Jx measured on the x-axis in the middle of the junction is plotted as a function of the normalized electric field at the same position. Different colors correspond to different junction widths S as labeled in the insert.

Fig. 5
Fig. 5

Extinction cross section per length of a nanowire dimer as obtained with the full TDDFT calculations, with the quantum corrected model (QCM), with classical Drude electromagnetic calculations (Drude), and with calculations based on the nonlocal hydrodynamic model (NLHD). The dimer consists of two D = 9.8 nm Na nanowires in vacuum. The incoming plane wave is polarized along the dimer axis x. Upper panels: Waterfall plots of the dipole absorption cross-section as a function of the width of the junction S. Red curves correspond to S = −5.3 Å, −2.65 Å, 0 Å, 2.65 Å, 5.3 Å, 7.95 Å, and 10.6 Å. For further details see the caption of Fig. 3. Lower panels: Color plots of the local field enhancement at the center of the junction for positive separations. Results are shown as a function of the frequency ω of the incident radiation and junction width S. The color code is displayed at the bottom of the corresponding panels. In the Drude case, because of the divergence of the fields, the color scale has been saturated (enhancement > 200) for junction widths below 1.25 Å.

Fig. 6
Fig. 6

Detailed comparison between TDDFT and QCM calculations. The extinction cross-section per length of the D = 9.8 nm Na nanowire dimer is shown for small separations S. This S-range corresponds to the strong tunneling regime and the transition from separated to conductively coupled nanowires. The frequency range is zoomed at the transition from the bonding dipole (DP) to the lowest charge transfer (C1) plasmon. Waterfall plots of the optical absorption cross-section are shown for the junction widths changing from S = −2.65 Å (lowest blue line) to S = 4.77 Å (upper black line) in steps of 1 a0 (0.53 Å). For further details see the caption of Fig. 3.

Fig. 7
Fig. 7

Dynamic screening. (a) Time evolution of the density Δn induced by the ω = 3.16 eV laser pulse at the surface of the left cylinder facing the S = 13.25 Å wide junction. The data is shown as a function of the x-coordinate along the dimer axis for the D = 9.8 nm dimer. x = 0 corresponds to the center of the junction. Different curves correspond to instants of time spanning 1/2 optical period starting from t0. For further details see the text. (b) Schematic representation of the location of the plasmon induced screening charges in the junction. Within the local classical approach the screening charges are at the geometrical surfaces of the cylinders (here equivalent to the jellium edges) separated by the junction of width S. Within the TDDFT, the centroids of the screening charges (red areas) are located at Re[d(ω)] in front of the jellium edges so that the effective separation is S −2Re[d(ω)]. In the NLHD approach the centroids of the screening charges (blue areas) are located at a distance δ below the geometrical surface so that the effective separation is S + 2δ. (c) Energy of the dipole plasmon resonance as function of the junction width S. Dots: TDDFT results obtained for the D = 9.8 nm nanowire dimer. Dashed red (gray) lines show results of classical Drude (NLHD) calculations where the separation S is measured between the jellium edges. The solid red line shows the results of the classical Drude calculations performed for an effective separation S − 2Re[d(ω)]. The dotted gray line shows the results of the NLHD calculations performed for an effective separation S − 2Re[d(ω)] − 2δ. (d) Energy of the dipole plasmon resonance as function of the junction width S. Dots: TDDFT results obtained for nanowire dimers formed by D = 6.2 nm and D = 9.8 nm nanowires (see the legend). Solid and dashed lines show results of classical Drude calculations for D = 6.2 nm (blue) and D = 9.8 nm (red) dimers. Dashed lines: calculations performed for the junction width S measured between the jellium edges. Solid lines: calculations performed for a corrected effective separation S − 2Re[d(ω)].

Equations (7)

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i Ψ j ( x , y , t ) t = ( Δ 2 m + V eff ( x , y , t ; [ n ] ) ) Ψ j ( x , y , t ) ,
n ( x , y , t ) = 2 j occ . χ j | Ψ j ( x , y , t ) | 2 .
χ j = 1 π 2 ( E F E j ) ,
ε ( ω ) = 1 ω p 2 ω ( ω + i γ ) ,
ε L ( ω ) = 1 ω p 2 ω ( ω + i γ ) β 2 k 2 .
ε eff ( x , y , S , ω ) = 1 ω p 2 ω ( ω + i γ eff ( x , y , S ) ) .
ω res / ω sp = 1 Re [ d ( ω sp ) ] / R cl + O ( R cl 2 ) .

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