Letter pubs.acs.org/NanoLett
The Dark Side of Plasmonics D. E. Gómez,*,†,‡,§ Z. Q. Teo,† M. Altissimo,§ T. J. Davis,‡,§ S. Earl,† and A. Roberts† †
School of Physics, The University of Melbourne, Parkville, Victoria, 3010, Australia Materials Science and Engineering, CSIRO, Private Bag 33, Clayton, Victoria, 3168, Australia § Melbourne Centre for Nanofabrication, Clayton, Victoria 3168, Australia ‡
S Supporting Information *
ABSTRACT: Plasmonic dark modes are pure near-field modes that can arise from the plasmon hybridization in a set of interacting nanoparticles. When compared to bright modes, dark modes have longer lifetimes due to their lack of a net dipole moment, making them attractive for a number of applications. We demonstrate the excitation and optical detection of a collective dark plasmonic mode from individual plasmonic trimers. The trimers consist of triangular arrangements of gold nanorods, and due to this symmetry, the lowest−energy dark plasmonic mode can interact with radially polarized light. The experimental data presented confirm the excitation of this mode, and its assignment is supported with an electrostatic approximation wherein these dark modes are described in terms of plasmon hybridization. The strong confinement of energy in these modes and their associated near fields hold great promise for achieving strong coupling to single photon emitters. KEYWORDS: Surface plasmons, plasmon hybridization, dark modes, plasmonic trimer, nanorods, radial polarization
C
bright modes of coupled nanoparticles. For the simplest case of a coupled dimer of nanoparticles, a dark mode can originate from an out-of-phase oscillation of the bright (dipolar) modes of the particles; for in this situation, the resulting collective plasmon mode has a net zero dipole moment. Dark plasmon modes cannot be directly excited with linearly polarized light, and in turn, one needs other illumination schemes to excite of these modes. In the past, excitation of dark plasmon modes has been achieved with focused electron beams,23,27 by using tailored far-field illumination techniques such as spatially inhomogeneous fields,28 evanescent excitation,29 non-normal incidence,30 spatial phase shaping,31 and more subtle retardation effects.32 Here we demonstrate the far-field excitation of a collective dark plasmon mode using radially polarized light. The dark mode is supported by a structure consisting of three interacting nanorods configured in a triangular geometry and is characterized by spectral peaks blue-shifted from the plasmon resonance of the noninteracting nanorods. We show the dependence of the spectral position of this dark mode on parameters such the nanorod separation distance and the symmetry-breaking of the coupled nanostructure. The experimental results are compared to a simple theoretical model that confirms the excitation of radially symmetric collective plasmon modes.
ollecting and storing energy at the nanoscale has tremendous implications in many areas including the creation of novel light sources and optoelectronic devices and the development of nonconventional heterogeneous catalysts for economically important chemical reactions. Metal nanoparticles are very efficient at extracting energy from light, a process that results in the excitation of surface plasmon polaritons: collective electron oscillations that are confined to the interface between the nanoparticle and its surrounding environment. This strong spatial localization leads to enormous energy densities which, to date, have been been employed in the creation of sub-wavelength optical photodiodes,1 as photocatalysts in artificial photosynthesis2−10 and hydrogen dissociation,11 as well as optically controllable dopants of graphene sheets12 in addition to sensitizers in photovoltaic applications.13 Of particular interest are metallic nanostructures that exhibit dark plasmon modes.14−25 These modes by virtue of having vanishing dipole moments can store electromagnetic energy more efficiently than modes with dipolar character (bright modes) due to an inhibition of radiative losses. This results in narrower spectral line widths making them ideal candidates for developing enhanced biological and chemical sensors, subwavelength high-Q optical cavities, and lossless nanoscale waveguides.19,26 For individual metal nanoparticles, these dark modes (in a quasi-static approximation) are those that arise from high-order multipole resonances supported by the nanoparticle such as the quadrupolar modes that exist in, for instance, metal nanoparticles with spherical symmetry. However, dark modes also arise from the interaction of the © XXXX American Chemical Society
Received: May 7, 2013 Revised: June 10, 2013
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around Z axis located at the center of the trimer, leaves it unchanged). Equation 1 predicts that the three lowest-energy plasmon modes that arise from LSP interactions in the symmetric trimer can be characterized by the irreducible representations of the point group D3h.36,43 These modes are the eigenvectors of the coupling matrix of eq 1 and are given by linear superpositions of the eigenmodes of the uncoupled nanoparticles as indicated in Figure 1, akin to a hybridization of plasmon modes.
To investigate dark plasmonic modes, we have chosen a triangular configuration of metal nanorods. Individual rods exhibit a strong (nondegenerate) longitudinal dipolar plasmon mode, making them ideal for studying the effects of dipole− dipole plasmon interactions. This is contrasted with the case of spherical nanoparticles,20,33 where the dipolar modes of the noninteracting particles have three-fold degeneracy (when excluding the substrate effect), complicating a theoretical description, for in this case one needs to invoke symmetryadapted combinations of these degenerate modes when considering the interparticle interactions.20 The interaction between localized surface plasmons (LSPs) arises from their electric near-fields and can, under certain circumstances, be interpreted as a Coulomb-type interaction between dipoles. To model these interactions, we resort to an electrostatic approximation33−41 wherein we implicitly neglect retardation effects on the basis of the sub-wavelength dimensions of the nanoparticles (that is, we assume ka ≪ 1, where a is the size of the nanoparticle and k the wavevector in free space of the incident illumination). According to this model, the coupling between two particles a and b is described by a coupling coefficient Cab = −Gab/(ω − ωa + iΓ/2)35,42 where ω is the frequency of the applied field, ωa − iΓ/2 is the complex resonance frequency of the LSP (here Γ is a Drude damping term), and Gab is a factor describing the geometrical coupling between the particles. To the lowest order, this geometrical factor can be expressed as a dipole−dipole coupling term: Gab ∝ d−3[3(p⃗a·d̂)(p⃗b·d̂) − (p⃗a·p⃗b)],35 where d is the interparticle distance and p⃗a,b are the dipole moments (d̂ is a unit vector along the line separating the particles). For smaller separations, the expansion of the geometric factor can include higher-order multipoles such that the coupling coefficient is represented by a multipolar series Cab = Σl,jCl,jab that includes all multipole terms l and j of the isolated nanoparticles. For an isolated nanoparticle a, an incident light field will excite a LSP creating a surface charge distribution σ(r)⃗ . This charge distribution can be written in terms of the eigenmodes σma (r)⃗ (or self-sustained oscillations) of the particle according to σ(r)⃗ = Σmaam(ω)σam(r)⃗ , where aam(ω) are the excitation amplitudes which describe the coupling of the incident light field with a particular eigenmode (here m is an index that indicates the m-th mode sustained by particle a). Modes with strong dipolar character couple strongly with uniform electric fields and have thus large values of ama (ω). Interparticle interactions modify these excitation amplitudes, which are mathematically described by a matrix multiplication involving a coupling matrix C that contains in its nondiagonal elements all of the details about the interparticle interactions. For a triangular structure where all three rods are identical and placed at equidistant positions, the excitation amplitudes that result from interparticle interactions are given by the following equation:36 ⎛ a1̃ ⎞ ⎛ 1 −C −C ⎞−1 ⎛ a1 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ a 2̃ ⎟ = ⎜−C 1 −C ⎟ ·⎜ a 2 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎝ − C − C 1 ⎠ ⎝ a3 ⎠ ⎝ a3̃ ⎠
Figure 1. Three lowest-order modes of an interacting triangle of particles according to the EEM.35,36 Mode a is the highest in energy of the three and is a mode with radial symmetry (A′ representation of the point group D3h) which has no net dipole moment and is thus the lowest-energy dark mode of the structure. Modes b and c are degenerate modes (lower energy than the A′ mode) which are characterized by having a net dipole moment and perpendicular polarizations.
These modes consist of doubly degenerate bright modes spanning the irreducible representation E′, which are redshifted in frequency from the plasmon mode of the uncoupled rods and are characterized by the surface charge distributions of Figure 1b and c which have a collective dipole moment p⃗ given by: p⃗ ∝
cos(ψ )x ̂ + sin(ψ )y ̂ G + (ω − ωa + i Γ/2)
(2)
where ψ is the angle of the incident linearly polarized light measured with respect to one of the nanorods as indicated in Figure 1. G is a factor describing the interparticle coupling (full details in the Supporting Information). The other eigenmode of the symmetric trimer is one which has zero net dipole moment (a dark mode) corresponding to the A′ representation of the D3h point group and is blue-shifted relative to the resonance frequency of the dipole mode of the non−interacting nanorods (for a detailed derivation, see Supporting Information). The surface charge distributions associated with these modes are shown in Figure 1, where it is clear that the dark mode A′ in addition to lacking a net dipole moment, has radial symmetry and consequently, can be excited by a radially polarized light beam. These theoretical results predict that the scattering spectrum of a D3h symmetric trimer, a quantity proportional to |p⃗|2, consists of a single peak located at the resonance wavelength of the bright degenerate modes, which is given by the root of the denominator of eq 2 (real part).
(1)
where a1,2,3 are the excitation amplitudes of particles 1, 2, and 3, whereas ã1,2,3 are the excitation amplitudes modified by the interparticle interactions. C is the single coupling constant required to describe the D3h symmetric trimer, given the C3 rotation axis of this spatial configuration (a rotation of π/3 B
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corresponding to the resonance frequency of the bright modes ωb = ωa − G(d), which depends on the magnitude of the interparticle coupling G(d). This coupling decreases with increasing separation distance between the particles, and this translates to blue shifts in the measured spectra, in accordance with the experimental evidence. Figure 3a shows that, under radially polarized illumination conditions, the scattering spectrum measured for one of the Au trimers of Figure 2 consists of two bands: one located at the position corresponding to the doubly degenerate bright mode and another located to the blue side of the spectrum. According to the model described in relation to eq 1, the nanoparticle trimer should exhibit a resonance with a radially symmetric surface charge distribution that has a resonance located at a higher energy than those corresponding to either the bright modes or the uncoupled nanoparticles (dipole mode). Bearing these considerations in mind, we therefore assign the observed spectral band at ∼550 nm to the excitation of the dark plasmonic mode A′ of the trimer. In Figure 2b we show how the scattering spectrum of the same trimer changes with the angle ψ of incident linearly polarized light (arbitrarily measured with respect to the position of rod 1 of Figure 1 or the x-axis). In addition to the obvious changes in intensity, there are also spectral changes that occur when rotating the angle ψ of polarization, indicating that the trimer is not perfectly D3h symmetric, for as predicted by eq 2, changes in ψ should not lead to changes in the wavelength of the observed resonance. Careful inspection of the SEMs shown in Figure 2 reveals systematic asymmetries in the nanorod trimers that arise from artifacts during the electron beam lithography. To semiquantitatively account for these imperfections, we consider only two types of asymmetries which transform the D3h symmetric trimer into a C2v one (that is, we only consider asymmetries where the trimer no longer possesses a C3 rotation axis but a C2 one). A C2v symmetric trimer can be described within the electrostatic approximation, by modifying the coupling matrix of eq 1 (for a detailed derivation, see Supporting Information) as follows:
Nanorod structures with triangular symmetry were fabricated by electron beam lithography (Vistec EBPG 5000 plus ES) using a double-layer positive resist consisting of 50 nm of poly(methyl methacrylate) (Micro-Chem, 950k A2) on top of 110 nm of methyl methacrylate (Micro-Chem, MMA(8.5) MAA EL6) on n-type ⟨100⟩ silicon substrates with a 100 nm SiO2 layer. After EBL exposure, the patterns were developed with a solution consisting of 1:3 methyl isobutyl ketone/ isopropanol for 90 s, rinsed with isopropanol, and dried with a nitrogen gun. Some 40 nm of Au were deposited by electron beam evaporation, using 1 nm of chromium as the adhesion layer. A subsequent lift off step with acetone produced the nanostructures. Normal incidence images of the resulting structures were obtained by scanning electron microscopy (FEI, NovaNanoSEM 430). The scattering spectra from the nanostructures were measured in a near-normal incidence dark-field microscope44 (more details in Supporting Information), equipped with a supercontinuum white light source. We measured the scattering spectra of individual trimers using unpolarized, linearly polarized, and radially polarized light which was obtained by illuminating a commercially available radial polarization converter (Arcoptix S.A.) with linearly polarized light. In Figure 2a, we show the scattering spectrum measured with unpolarized light for a series of Au trimers that differ in the
⎛ a1̃ ⎞ ⎛ 1 −C1 −C1 ⎞−1 ⎛ a1 ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ a 2̃ ⎟ = ⎜−C1 1 −C2 ⎟ ·⎜ a 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ a3̃ ⎠ ⎝−C1 −C2 1 ⎠ ⎝ a3 ⎠
(3)
where now instead of using a single coupling coefficient to describe the interparticle interactions, as was done in relation to eq 1, two types of coupling constants are introduced to account for the C2v type of asymmetry: one nanorod, by virtue of being either of a different size or by being slightly displaced from an ideal isosceles position, is coupled to the remaining two rods, which are assumed to remain in symmetric positions with respect to a reflection through the x axis of Figure 1, with a coupling strength described by C1, while the remaining pair has a mutual interaction accounted for by the constant C2. Likewise, this equation can also be used to account for the case where small angular deviations in the positions of the nanorods leads to asymmetric couplings. In either case, we assume the deviations to be small and write C2 = αC1, where α is a parameter that accounts for the asymmetry in the trimers (α ∼ 1, but it exactly equals 1 for a D3h−symmetric trimer).
Figure 2. Scattering spectra of nanorod trimers measured with unpolarized light shown as a function of center-to-center interparticle separation d (w = 40 nm, L = 100 nm). ∞ corresponds to a single nanorod. All of the data have been normalized and vertically offset for clarity.
interparticle separation distance. In all cases the spectra consist of a single peak in accordance with the result presented in eq 2. This band blue-shifts as the interparticle separation distance is increased, indicating that it originates from the excitation of the doubly degenerate bright modes E′ discussed in relation to Figure 1a−b. According to the result of eq 2, the scattering spectrum should exhibit a maximum located at a wavelength C
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Figure 3. (a) Scattering spectra of a Au nanorod trimer (w = 40 nm, L = 100 nm, d = 110 nm) measured with unpolarized light (unp) and radially polarized light (rad). The latter was multiplied by 20 × for clarity. (b) Scattering spectra of the same trimer measured as a function of angle ψ of linearly polarized light.
Figure 4. Symmetry breaking: we show the spectra measured for trimers where the position of rod 1 is displaced in the horizontal direction away from the ideal isosceles triangle position by the amounts indicated in the figure. (a) Spectra measured with unpolarized light (b) with radially polarized light.
Instead of the simple expression for the total dipole moment given by eq 2, the trimer in this case has a dipole moment
which for linearly polarized light is given by (details in the Supporting Information): D
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Figure 5. Symmetry breaking: we vary the angle between rods 2 and 3 to the horizontal axis (for reference see Figure 1). (a) Spectra measured with unpolarized light (b) with radially polarized light.
pl⃗ ∝ −
where, for reference, we have also included the resonance frequencies for the case of D3h symmetry. These equations make a number of predictions in relation to the effect of symmetry on the optical properties of the trimer. First, according to eq 6, the bright modes are no longer degenerate, and according to eq 4, these two bright modes can be selectively excited with linearly polarized light at angles ψ = 0 or ψ = π/2. This will be manifest in the scattering spectra as spectral changes with the angle ψ, which accounts for the spectral shifts experimentally observed in Figure 3b. Second, these equations also predict the existence of a radially symmetric dark mode with a nonzero net dipole moment (thus technically not a dark mode). Furthermore, this dark mode can be excited with linearly polarized light incident at ψ = 0 (further proof given in Supporting Information). Lastly, eq 5 predicts that, under radially polarized illumination, the strength of the dipole moment (and thus of the measured scattered light intensity) scales linearly with both the coupling strength G1 and the asymmetry factor α. To test some of these predictions, we have also fabricated two sets of structures that are deliberately aimed to be C2v symmetric: one set consists of a trimer where the position of rod 1 (as referenced in Figure 1) is increased away from the ideal isosceles triangle position, leading to the results of Figure 4, and another set where the angle subtended by particles 2 and 3 was varied as indicated in Figure 5. In Figure 4a the spectra show a clear increase in the intensity of the band located at ∼560 nm with increasing the horizontal displacement Δd of rod 1 from the ideal isosceles position. This change is accompanied by small spectral shifts in this band. Figure 4b contains the spectra obtained with radially polarized light for this series of C2v symmetric rods, where it is clear that,
[δω(cos(2θ) + 2) − AG1(α + 4 cos(θ))cos(ψ )] x̂ AG1(αδω + 2AG1) − δω2 2 sin 2(θ) sin(ψ ) ŷ αAG1 + δω
(4)
where θ is the angle subtended by particles 2 and 3 to the xaxis. G1,2 are the parts of the coupling constant that describe the geometrical aspects of the coupling, A is a constant35 and δω = ω − ωa + iΓ/2. Whereas for radial polarization, the dipole moment of the trimer is given by: pr ⃗ ∝
AG1(α + 2 cos(θ) − 2) + δω(2 cos(θ) − 1) AG1(αδω + 2AG1) − δω2
(5)
These equations predict that the resonance frequencies for the three lowest-energy hybrid modes of the trimer are given by: ⎧ ωa − AG1 D3h symmetry, 2× degenerate ⎪ ⎪ ⎪ ⎧ ωa − αAG1 ωb = ⎨ ⎪ ⎪ ⎡ ⎤ ⎪⎨ α α 2 + 8 ⎥ C2v symmetry ⎪ ⎪ ωa + AG1⎢ − 2 ⎢⎣ 2 ⎥⎦ ⎪⎪ ⎩⎩ (6)
⎧ ωa + 2AG1 ⎪ ⎪ ⎡ ωd = ⎨ α ⎪ ωa + AG1⎢ + ⎪ ⎣⎢ 2 ⎩
D3h symmetry ⎤ α2 + 8 ⎥ C2v symmetry 2 ⎥⎦
(7) E
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after a displacement of Δd = 20 nm, the intensity of the band located at ∼560 nm decreases. In these cases θ = π/3 and according to eq 5, the dipole moment p⃗r is proportional to both (α − 1) and the coupling constant G1, which is in itself a function of the separation distance between rod 1 and the remaining pair. As the distance Δd is increased, α is expected to increase due to the introduction of the asymmetry (α = 1 for a D3h-symmetric trimer), but this is also concomitant with a decrease in G1 (due to the increase in separation distance). These two effects combined lead to the observations made in reference to Figure 4b. Similar effects are obtained when changing the angle θ. In Figure 5a the spectra are dominated by a single spectral feature irrespective of the value of θ. This band red-shifts as θ increases from 40° to 70°, and it then blue-shifts at 80°. Radial illumination reveals again the band at ∼560 nm which is strongest in intensity for θ = 70° and decreases steadily as the angle is decreased. In this series of trimers, variations in the angle θ are expected to modify α and also G1, the geometrical coupling parameter between rod 1 and the remaining pair, accounting for the changes observed experimentally. We have demonstrated the excitation of a collective surface plasmon mode with radial symmetry. This mode is supported by three coupled Au nanorods arranged in either D3h or C2v symmetric configurations. Both symmetries can support radially symmetric hybrid plasmon modes which in the case of the D3h configuration lack a net dipole moment and are thus dark modes. For structures unintentionally less symmetric or structures with a C2v symmetry, theses radial modes have a small dipole moment (see eq 5) and resonance frequencies that are shifted to the blue of the noninteracting surface plasmon resonance. The strong suppression of radiative decay in these collective plasmon modes that originates from their low dipole moments results in narrower spectral bands. This characteristic makes these modes attractive for creating high−quality factor optical nanocavities, which have enormous near-fields. Theoretically, it has been predicted that these modes can also be excited by carefully positioned dipole emitters,24 opening thus the possibility of demonstrating strong coupling between electronic excitations in these emitters and localized plasmon resonances, a phenomenon that could be exploited for creating novel non-linear plasmonic devices.45
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We would also like to acknowledge M. MacDonald for her help with Labview.
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ASSOCIATED CONTENT
S Supporting Information *
Details on the electrostatic eigenmode description of the dark modes and the near-normal incidence dark-field illumination method. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
ACKNOWLEDGMENTS
D.E.G. would like to thank the ARC for support through a Discovery Project DP110101767 and the Melbourne Materials Institute for support. T.J.D and A.R. would like to thank the ARC for support through a Discovery Project DP110100221. F
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