Generalized circuit model for coupled plasmonic systems
Felix Benz, Bart de Nijs, Christos Tserkezis, Rohit Chikkaraddy, Daniel O. Sigle, Laurynas Pukenas, Stephen D. Evans, Javier Aizpurua, Jeremy J. Baumberg
GGeneralized circuit model for coupledplasmonic systems
Felix Benz, Bart de Nijs, Christos Tserkezis, Rohit Chikkaraddy, Daniel O. Sigle, Laurynas Pukenas, Stephen D. Evans, JavierAizpurua, and Jeremy J. Baumberg , ∗ NanoPhotonics Centre, Cavendish Laboratory, JJ Thompson Ave, University of Cambridge,Cambridge, CB3 0HE, UK Donostia International Physics Center (DIPC) and Centro de F´ısica de Materiales, CentroMixto CSIC-UPV/EHU, Paseo Manuel Lardizabal 4, 20018 Donostia/San Sebasti´an, Spain Molecular and Nanoscale Physics, School of Physics and Astronomy, University of Leeds,Leeds, LS2 9JT, UK ∗ [email protected] Abstract:
We develop an analytic circuit model for coupled plasmonicdimers separated by small gaps that provides a complete account ofthe optical resonance wavelength. Using a suitable equivalent circuit,it shows how partially conducting links can be treated and providesquantitative agreement with both experiment and full electromagneticsimulations. The model highlights how in the conducting regime, the kineticinductance of the linkers set the spectral blue-shifts of the coupled plasmon. © 2018 Optical Society of America. One print or electronic copy may be madefor personal use only. Systematic reproduction and distribution, duplication of anymaterial in this paper for a fee or for commercial purposes, or modifications of thecontent of this paper are prohibited.
OCIS codes: (250.5403) Plasmonics; (240.6680) Surface plasmons; (160.4236) Nanomateri-als; (160.3918) Metamaterials.
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1. Introduction
Plasmonic nanostructures are at the heart of many research fields ranging from biological appli-cations [1, 2] to quantum information processing [3, 4]. In recent years significant progress hasbeen made in both the chemical preparation of noble-metal nanostructures [5] and in simulationools to model them [6–10]. However, few attempts have been made to develop a comprehen-sive analytical model [11, 12] that could facilitate direct validation of experimental results andyield parameters that can guide the implementation of full electromagnetic simulations. Mostanalytical approaches approximate Maxwells equations and are limited to specific geometries,like large gap separations [13]. A promising alternative is to treat any nanoplasmonic systemas a high frequency circuit composed of capacitors, inductors, and resistors [14–17]. Such acircuit model is extremely versatile as generalized equations for the resonance condition canbe obtained, and their dependence on the geometry of the nanostructure understood explicitly.Here we use this circuit approach to develop a simple analytical expression for two coupledplasmonic systems in close proximity. Surprisingly this has been absent until now, despite itsextreme utility, because a number of subtle effects need to be considered. Our model is capableof describing both insulating and conductive gaps and applicable to arbitrary coupled plasmonicsystems. We stress that such models are extremely useful to guide experiments, as electromag-netic simulation tools do not give direct insights (like those shown here) or scaling rules on thedetails of the nano-geometry. In the ultrasmall gap regime used here (and of strong interest forexperiments), we show these models are accurate enough to be highly effective.In this work we first review the circuit model for single nanoparticles, and show how to ef-fectively account for nanoparticle size. We then show the full model for coupled plasmonicscomponents when the gap between them is insulating, and when the gap has a finite conduc-tance. We then show the origin of the scaling in the resonance coupled wavelength arising fromthe inductance of the gap region, producing a simple estimate for the blueshifts observed.
2. Single particles
We start from the description of a single spherical nanoparticle, which is well character-ized by an LC circuit that has been shown to be the exact solution to Maxwell’s equa-tions. Engetha et al. [14] have determined the impedance contributions of the fringing field Z − = − i ω C s = − i2 πω R ε and the sphere Z − = − i πω R εε with nanoparticle radius R ,frequency ω , gold permittivity ε , vacuum permittivity ε , and the capacitance due to the fring-ing field C s . The resonance wavelength can be determined by ℑ ( / Z tot ) =
0, which yields thewell-known resonance condition in the electrostatic case for a plasmonic nanoparticle ε = − ε = ε ∞ − (cid:16) ω p ω (cid:17) = ε ∞ − (cid:18) λλ p (cid:19) . (1)This yields the single particle resonance frequency, λ s = λ p ( ε ∞ + ) . The plasma frequency andwavelength ( ω p and λ p respectively), and the background high frequency permittivity ( ε ∞ ) arematerial parameters which should not depend on nanoparticle size. However, electrodynamicaleffects, beyond the electrostatic approximation, lead to a weak particle size dependence of theresonance wavelength. There are several ways of correcting the resonance condition due toretardation [18, 19], however, a relatively simple phenomenological way to account for thisis to assume that ε ∞ depends slightly on the nanoparticle size. The resonance wavelength ofdifferent gold nanoparticle sizes in water (permittivity ε m = .
33) from extinction spectroscopy(Fig. 1) provides estimated values for ε ∞ ( R ) using ε = − ε m together with Eq. (1), as we showin Dataset 1 (Ref. [20]).
3. Coupled plasmonic systems
To describe purely capacitive-coupled resonances a capacitor (with capacitance C g ) can beplaced between two equivalent single particle circuits mimicking the capacitive coupling, s phe r e Z f r i nge R e s onan c e w a v e l eng t h ( n m ) H i gh f r equen cy pe r m i tt i v i t y ε ∞ Fig. 1. Resonance wavelength for different nanoparticle sizes and resulting values for ε ∞ ,determined from extinction spectroscopy in water. Inset: Single nanoparticle equivalentLC-circuit. Source data available in Dataset 1 (Ref. [20]). which reproduces the strongly red-shifted resonance [21–23]. However, for either a conductivespacer [24] or for very small gaps [25,26] effects due to the increasing conductivity between thetwo systems have to be considered, with two results: First, the coupled bonding dimer plasmonmode (BDP) blue-shifts due to screening of the local field in the gap, forming the screened cou-pled mode (SBDP). In this conductivity region the screening field reduces the voltage betweenthe two elements. Second, for higher conductivities a charge transfer mode, characterized by anet current between the two nanoparticles is formed [27]. Since both modes have different phys-ical origins they have to be described by different equivalent circuits. Here we restrict ourselvesto the screening of the coupled mode as this is directly accessible in our optical experiments.Previously we considered the ratio of the screened charge to the charge stored on a capaci-tor without any conductive link [24]. This simplified picture can be described by a resistor inseries with the coupling capacitor effectively reducing the voltage at the capacitor. While thisapproach works well for describing relative changes of the charge stored on the capacitor peroptical cycle it is not capable of predicting the absolute value of the resonance wavelength.Therefore we use here an equivalent parallel circuit, which is composed of a capacitor shuntedwith a resistor and an inductor, which reproduces the behaviour of conductive paths through thegap, in series (see Fig. 2(a)). For a sufficiently low resistance the inductor produces a screeningvoltage, while for high resistance the inductor plays no role and the coupling is purely domi-nated by the capacitor. If the gap inductance is large, then little screening is obtained becausecharge cannot short across the gap within an optical cycle. Hence the fully screened mode atlarge conductance is blue-shifted furthest if the gap inductance approaches the inductance givenby the individual sphere. The resonance frequency of this circuit found when ℑ ( / Z tot ) = Z tot = − πω R ε m ε − i πω R εε + − i ω C g + [ R g − i ω L g ] − . (2)Simplifying this equation (see appendix A) yields the resonance condition: (cid:101) ω + (cid:101) ω (cid:20) ω d (cid:18) ε m Γ − (cid:19) + δ (cid:21) − ω d δ = (cid:101) ω = ω / ω p is the normalised resonance frequency, the dimer mode ω d = / √ ε m + ε ∞ + ε m η , normalised gap capacitance η = C g / C s , normalised inductive coupling s C s C g L g R g L s C s (a) (b) gold nanoparticle Rd spacer layernanoparticle mirrorimage go l d f il m 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 couplingand conductive coupling. (b) Illustration of the nanoparticle on mirror (NPoM) geometry.Source data available in Dataset 1 (Ref. [20]). rate Γ = − L g C s ω p , and normalised loss rate δ = (cid:16) R g L g ω p (cid:17) . For the case of very low conductiv-ity ( R g →
0) which gives the typical coupled mode of the bonding dimer plasmon (BDP) thiscan be simplified to (cid:101) ω = ω d giving resonant wavelength λ BDP = λ d = λ p (cid:112) ε m + ε ∞ + ε m η (4)for coupled plasmonic particles. In the extreme case of high conductivity ( R g → ∞ so δ − → λ SBDP = λ d (cid:113) − ε m Γ = λ p (cid:115) ε m + ε ∞ + ε m η − ε m Γ (5)which is a central result of our paper.We now use the nanoparticle on mirror geometry to test this model. In this geometry a goldnanoparticle is placed on a thin spacer layer above a gold film (Fig. 2(b)). Dipoles induced inthe nanoparticle can couple to their image charges forming a tightly confined coupled plasmonmode [28]. The gap capacitance for this geometry can be obtained following Hudlet et al. [29](see appendix B): C g = πε ( n g ) χ R × ln (cid:20) + R d θ max (cid:21) (6)with refractive index of the material in the gap n g , separation between nanoparticle and gold film d , and parameters θ max (which arises from the laterally localised electric field) and χ = . − ( CH ) x − SH with x from 3 to 17 on atomicallyflat template stripped gold [31]. The thickness of each SAM is measured using spectroscopicellipsometry. Subsequently, 80nm gold nanoparticles are assembled on top of these SAMs. Theresonance wavelengths in air ( ε m =
1) of more than 1,000 particles per sample were recordedusing a fully automated darkfield microscope and cooled spectrometer. Further experimentaldetails can be found in the appendix C and elsewhere [32]. The average resonance wavelengthsand standard errors were determined by fitting Gaussian distributions to the histograms (seeappendix C). (a) (b)(d)(c)
GRR dimer R NPoM G c R e s onan c e w a v e l eng t h ( n m ) ) SAM exp. BEM sim. circuit model900850800750700650 R e s onan c e w a v e l eng t h ( n m ) d (nm) SAM exp. BEM sim. n g = 1.8 2.0 1.8 1.6 1.4 c ir cu it m ode l: n g d d = 1 nm d = 2 nm d /nm .5 1 2 3circuit modelBEM simulations n g R e s onan c e w a v e l eng t h ( n m ) n g F i e l d l o c a li s a t i on , θ ( º ) R e f r a c t i v e e x ponen t, χ Fig. 3. Comparison of the presented circuit model to experimental and simulated data. (a,b)Coupled plasmon mode with R = θ max , see appendix B) and refractive indexexponent, χ (empty squares). (d) Blue-shift of the coupled plasmon mode as gap conduc-tance increases, for R = G (see main text). Source data available in Dataset 1 (Ref. [20]). Comparing our analytical circuit model, BEM simulations, and experimental data shows ex-cellent agreement. The characteristic continuous blueshift of the coupled plasmon resonancefor increasing gap spacing and different refractive indices is clearly shown in Fig. 3(a). Sim-ilarly the expected linear redshift with increasing refractive index is completely reproducedFig. 3(b)).The gap capacitance parameters are found to be weakly dependent on the nanoparticle sizeand geometry (Fig. 3(c)). This is because the field localisation (described here by θ max ) isslightly different between NPoM and two nanoparticles in a dimer, in particular in the effectsof increasing the spacer refractive index (see appendix B). Most crucially, the model predictsfor the first time analytically the blue shift of the coupled plasmon when the gap starts conduct-ing (Fig. 3(d)). While such effects from mixed conducting SAM layers have only previouslybeen computed in full simulations [27], our analytic solution indeed captures three importantproperties of such a conductive system: (i) the need to exceed a critical conductance G c , (ii)continuous blue shifts with increasing conductance, and (iii) the saturation of the screening forlarge conductances. The exact critical conductance is overestimated in our model (Fig. 3(d)solid line), and fits much better when shifted by − G (dashed Fig. 3(d)), however the orderof magnitude is acceptable. Using Eq. (3), we obtain an analytical estimate for the critical con-ductance, G c G (cid:39) π G Z R λ p λ d Γ where Z = Ω is the impedance of free space (see appendix A).
4. The role of the gap inductance G ap i ndu c t an c e L g ( f H ) Neck radius (nm)1.2 1.8 n g Au no loss d =1 nm L k (a) sc r een i ng c u rr en t s Λ a (b) 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 sheathcurrents confined to penetration depth Λ down conducting filament, which gives kineticinductance. Source data available in Dataset 1 (Ref. [20]). The maximum blue shift when the gap becomes conducting (saturating the screening) is setby the inductance of this filament, which screens charges in the gap expelling the electric field.To understand what sets the value of this inductance we perform BEM simulations for bothinsulating and completely conducting gaps. The conductive region was modelled as a cylinderconnecting the gold spheres. Varying the neck radius a reveals that L g ∝ a − for gold while foran ideal conductor (with extremely low damping) we find L g ∝ a − (Fig. 4(a)). This dependencearises because L g is dominated by the kinetic inductance L k produced by the magnetic field inthe gap encircling the current acting to drag back the electrons. To obtain L k we equate thekinetic energy of N moving electrons with the energy of an inductor N mv = L k I withvelocity v , current I = nevA , area A , and N = nAd , giving L k = dA mne = dA ε ω p with density ofmobile carriers in the conducting link n and electron mass m [33]. The electric field howevercan only penetrate a certain distance Λ into the connecting filament so that current flows onlyn an outer sheath (Fig. 4(b)) giving L k = ε ω p d + Λ π Λ a = ε ω p fa (7)with the geometry factor f = π (cid:2) + d Λ (cid:3) . We find screening depths of Λ ∼ f → π which then accounts extremely well for the a − dependencein the full simulations of L g and even produces quantitative agreement with the magnitude ofthe gap inductance (Fig. 4(a)). For an imperfect metal the screening depth cannot be ignored,giving a modified form which depends also on n g (appendix A). The blue-shift of the coupledplasmon seen when conducting links are inserted between plasmonic elements can thus beunderstood as arising from their kinetic inductance. Inserting Eq. (7) into Eq. (5), we find ananalytic form of the blue-shift ∆ λλ = ε m aR (cid:18) + d Λ (cid:19) − (8)which for small gaps depends linearly on the ratio of neck to nanoparticle radius. These shiftscan indeed be large (we obtain 70nm spectral shifts experimentally, see Fig. 3(d)). We also notethe interesting ramification that ∆ λ will depend on the morphology of links across a plasmonicgap. This opens great opportunities in understanding nanoscale conducting elements purelythrough optical spectroscopy, which can then follow dynamic processes.
5. Conclusion
In conclusion, we present a circuit model which can be used to describe coupled plasmonresonances in both capacitive and conductive coupling regimes. It provides an exceptionallystraightforward way to calculate analytically the resonance wavelength for different gap sizes,nanoparticle sizes, refractive indices, and linker conductivities. We showed how a test geometrybased on a nanoparticle on gold mirror gives excellent agreement of the model with both ex-periment and simulation. This model allows us to identify the crucial properties of the system,and resolves the key role of kinetic inductance. Further geometries can easily be incorporatedby adjusting the capacitances of both the gap region and the individual plasmonic systems.
AppendixA. Derivation of the model
The resonance frequency of the circuit shown in Fig. 2(a) (main text) is found when ℑ ( / Z tot ) = Z tot = − πω R ε m ε − i πω R εε + − i ω C g + [ R g − i ω L g ] − (9) Z tot = π R ε × ωω ( ε m + ε ) + i C s × ω C g C s + − ω C s L g − i R g C s (10)Using the Drude model Eq. (1) for ε and replacing all frequencies ω by (cid:101) ω = ω / ω p yields: Z tot = i2 π R ω p ε (cid:32) ω ( ε m + ε ∞ ) (cid:101) ω − + ε m (cid:101) ωη + (cid:101) ω Γ − i γ (cid:33) (11)with η = C g C s , Γ = − L g C s ω p , and γ = R g C s ω p . Rearranging Eq. (11) yields: tot = − i2 π R ω p ε (cid:101) ωε m (cid:16) (cid:101) ωη + (cid:101) ω Γ − i γ (cid:17) + ( ε m + ε ∞ ) (cid:101) ω − ε m (cid:16) (cid:101) ωη + (cid:101) ω Γ − i γ (cid:17) [( ε m + ε ∞ ) (cid:101) ω − ] (12) ℑ (cid:18) Z tot (cid:19) = ℑ − i2 π R ω p ε ε m (cid:16) (cid:101) ωη + (cid:101) ω Γ − i γ (cid:17) (cid:2) ( ε m + ε ∞ ) (cid:101) ω − (cid:3) (cid:101) ωε m (cid:16) (cid:101) ωη + (cid:101) ω Γ − i γ (cid:17) + ( ε m + ε ∞ ) (cid:101) ω − (13) ℑ (cid:18) Z tot (cid:19) = ℜ − π R ω p ε ε m (cid:16) (cid:101) ωη + (cid:101) ω Γ − i γ (cid:17) (cid:2) ( ε m + ε ∞ ) (cid:101) ω − (cid:3) (cid:101) ωε m (cid:16) (cid:101) ωη + (cid:101) ω Γ − i γ (cid:17) + ( ε m + ε ∞ ) (cid:101) ω − ! = ℜ (cid:20) ε m (cid:18) (cid:101) ωη + (cid:101) ω Γ + i γ (cid:101) ω Γ + γ (cid:19) (cid:2) ( ε m + ε ∞ ) (cid:101) ω − (cid:3)(cid:20) (cid:101) ωε m (cid:18) (cid:101) ωη + (cid:101) ω Γ − i γ (cid:101) ω Γ + γ (cid:19) + ( ε m + ε ∞ ) (cid:101) ω − (cid:21)(cid:21) ! = ℜ (cid:20) ε m (cid:18) (cid:101) ωη + (cid:101) ω Γ + i γ (cid:101) ω Γ + γ (cid:19) (cid:20) (cid:101) ωε m (cid:18) (cid:101) ωη + (cid:101) ω Γ − i γ (cid:101) ω Γ + γ (cid:19) + ( ε m + ε ∞ ) (cid:101) ω − (cid:21)(cid:21) ! = ε m (cid:18) (cid:101) ωη + (cid:101) ω Γ (cid:101) ω Γ + γ (cid:19) (cid:20) ( ε m + ε ∞ ) (cid:101) ω − + (cid:101) ωε m (cid:18) (cid:101) ωη + (cid:101) ω Γ (cid:101) ω Γ + γ (cid:19)(cid:21) + (cid:18) γε m (cid:101) ω Γ + γ (cid:19) (cid:18) (cid:101) ωγε m (cid:101) ω Γ + γ (cid:19) ! = ε m (cid:101) ω (cid:18) η (cid:0) (cid:101) ω Γ + γ (cid:1) + Γ (cid:19) (cid:20) ( ε m + ε ∞ ) (cid:101) ω − + (cid:101) ωε m (cid:18) (cid:101) ωη + (cid:101) ω Γ (cid:101) ω Γ + γ (cid:19)(cid:21) + γε m (cid:18) (cid:101) ωγε m (cid:101) ω Γ + γ (cid:19) ! = (cid:101) ω ε m (cid:2) η (cid:0) (cid:101) ω Γ + γ (cid:1) + Γ (cid:3) + γ ε m +( ε m + ε ∞ ) (cid:101) ω − (cid:2) η (cid:0) (cid:101) ω Γ + γ (cid:1) + Γ (cid:3) (cid:0) (cid:101) ω Γ + γ (cid:1) ! = γ << (cid:101) ω Γ , simplifying Eq. (19) further. Using ω d = / √ ε m + ε ∞ + ε m η = λ − d and δ = (cid:0) γ Γ (cid:1) yields: (cid:101) ω + (cid:101) ω (cid:20) ω d (cid:18) ε m Γ − (cid:19) + δ (cid:21) − ω d δ = .1. Conductive gaps In the case of conductive gaps, the resistance R g controls the crossover between the coupledresonance and the screened coupled resonance. The inductance L g determines the spectral po-sition of the fully screened mode ( i.e. highly conducting gap where δ becomes small), so that λ SBDP λ p = λ d (cid:16) − ε m Γ (cid:17) − / . We write this in terms of the ratio between gap and sphere induc-tance, Γ = − L g L s ( + ε ∞ ) , with L s taken at the single particle resonance. If the gap inductanceis large, then little screening is obtained because charge cannot short across the gap withinan optical cycle. Hence the fully screened mode at large conductance is blue-shifted furthestif the gap inductance approaches the inductance given by the individual sphere ( L g L s = γ = − Γ ω d (cid:113) − Γ which impliesa critical conductance G c = ω p C s λ d √ Γ + Γ = R λ p λ d √ Γ + Γ G . This predicts indeed the rightmagnitude of the critical conductance compared to the full BEM simulations. Changing thegap inductance thus modifies both the wavelength of the screened mode, as well as the criticalconductance (Fig. 5). r e s onan c e w a v e l eng t h ( n m ) conductance ( G ) L g / L s0 Fig. 5. Resonance wavelength as a function of the gap conductivity for several gap induc-tances (80 nm NPoM, d = 1.1 nm, n = 1.45). Source data available in Dataset 1 (Ref. [20]). For the kinetic inductance as described in the main text, we can substitute into λ SBDP = λ d (cid:113) − ε m Γ = λ d (cid:113) − ε m C s L g ω p = λ d (cid:114) − ε m ε ω p a πε R f ω p = λ d (cid:113) − ε m a π R f (21)using Γ = L g C s ω p with C s = πε R , L g = ε ω p fa , and then using a binomial expansion we find: λ SBDP = λ d (cid:18) + ε m a π R f (cid:19) (22)which gives the result for the blue shift in the main text. .2. Screening depths
In the small-gap metal-insulator-metal structures here, we find the propagation wavevector in agap of width d S m = ε d d ε m (23)which gives the field in the metal E z ∝ exp ( S m z ) (24)Solving for the penetration depth of optical field inside the metal Λ = ℜ ( S m ) = d ε d ℜ ( / ε m ) (25)We compare below this penetration depth (Fig. 6 blue squares) with full BEM simulationsof the exact NPoM structures (spherical 80nm Au NP) from which we also extract the fieldpenetration (Fig. 6 red circles). This gives a very good match for the various ( d [nm], n g ) =(1,1.3), (1,1.8), (0.5,1.8) which shift the resonant mode wavelength across the near infrared. E d r op o ff ( n m ) theory simulation Fig. 6. Field penetration depth in NPoM. Exponential decay length of optical field calcu-lated from simple model above, and from full BEM simulations. Source data available inDataset 1 (Ref. [20]).
We can also substitute in the Drude model, and providing the damping is not too large wefind Λ = d | ε ∞ − λ λ p | n g (26) B. Nanoparticle on mirror capacitance model
Modelling the gap capacitance is key to obtain reliable values for the resonance frequency andfor obtaining the correct dependencies on distance and refractive index. The simplest approxi-mation is a plate capacitor, with area A , filled with a material with refractive index n g : = C g C s = ε n g A / d π R ε = n g A π Rd (27)A more precise description of the capacitance of a spherical nanoparticle on a metallic mirroris [29]: η = πε n g R πε R (cid:90) π sin θθ [ dR + − cos θ ] d θ (28)By considering that the gap occupies only a small area of the full sphere ( i.e. restricting theupper integration boundary to a small angle θ max ) and using small angle approximations yields(see illustration in Fig. 7(a)): η = n g ln (cid:18) + R d θ max (cid:19) (29) θ max area contributingto the capacitance (a) (b) r e s onan c e w a v e l eng t h ( n m ) n g dimer (FDTD) d = 1nm NPoM (BEM) d = 1nm d = 2nm Fig. 7. Illustration of the used capacitance model. (a) By using a θ max much smaller than π , only a part of the sphere is taken into account for the capacitance. Throughout the paper θ max (cid:39) π / By comparing the refractive index dependency obtained from electromagnetic simulations(finite-difference time-domain - FDTD & boundary element method - BEM) we find that de-pending on the exact geometry the exponent of the refractive index deviates from two. Wehave performed simulations for different geometries and different refractive indices. Plottinglog ( η ) (calculated from the resonance wavelength using Eq. (4)) vs log ( n ) allows to extractthe refractive index exponent χ . Fig. 8 shows χ for both nanoparticle dimers and the nanopar-ticle on mirror geometry.Fig. 8 shows that using a cylindrical material in the gap can strongly influence the refractiveindex dependency; we observe two regions: For cylinder diameters smaller than the lateral fieldlocalization a steep increase is observed indicating that both the single particle mode and thecoupled mode shift their spectral position. A second region is observed for diameter larger thanthe plasmon dimension, here the dependence is shallow as mostly the single particle mode ischanged. For an extended film covering the whole gold substrate we find χ = .
14, for dimerssubmerged in a medium with a certain refractive χ = . .81.71.61.51.41.31.2 806040200 cylinder diameter (nm) l a t e r a l f i e l d l o c a li s a t i on (a) (b) r e f r a c t i v e i nde x e x ponen t, χ r e f r a c t i v e i nde x e x ponen t, χ 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 dimen-sion of the plasmon. The dashed lines indicate the exponent value for our two referencegeometries: a film covering the whole gold substrate for the nanoparticle on mirror geom-etry and a nanoparticle dimer submerged in a medium with the respective refractive indexfor the dimers. Source data available in Dataset 1 (Ref. [20]). C. Experimental details & results
As described in the main text self-assembled monolayers of different length aliphatic thiolswere used as experimental spacer. For each SAM the thickness of three samples (three spots persample) were measured by phase modulated ellipsometry. Fig. 9(a) shows the determined SAMthickness as a function of the number of carbon atoms. For each SAM a sample was coveredwith 80 nm gold nanoparticles. The resonance wavelength of over 1,000 nanoparticles wasmeasured per sample using an automated Olympus BX51 darkfield microscope. Further detailson the spectroscopy and evaluation can be found elsewhere [32]. Fig. 9(b) shows exemplarilyan obtained resonance wavelength distribution for butanethiol (C ) self-assembled monolayeras a spacer layer. The resonance wavelengths for all samples as a function of the SAM thicknesscan be found in Fig. 9(c). C.1. Sample Preparation
A 100 nm thick gold film was evaporated on a silicon (100) wafer (Kurt J. Lesker Company,PVD 200) using an evaporation rate of 1 ˚A/s with a base pressure of ≈ × − mbar. Toobtain atomically smooth gold surfaces a standard template stripping method was applied:small silicon (100) pieces were glued to the evaporated gold surface using Epo-Tek 377 epoxyglue. After curing the glue the sample can be pealed of the silicon wafer revealing a fresh,clean surface with excellent surface roughness [31]. 1-butanethiol (C ), 1-pentanethiol (C ), 1-nonanethiol (C ), 1-dodecanethiol (C ), and 1-octadecanethiol (C ) (Sigma-Aldrich, ≥ ≈
22 h. Subsequently, the samples were thoroughly rinsed with ethanol,briefly cleaned in an ultrasonic bath to remove excess unbound thiols, and blown dry. The sam-ples were stored under a constant nitrogen flow until they were used. 80 nm gold nanoparticleswere used as received from BBI Solutions. To deposit the nanoparticles on the SAM the previ-ously prepared samples were immersed in the nanoparticle solution. The time was adjusted in .00.50.0 f r equen cy ( ) c u m u l a t i v e f r eq . (-) C SA M t h i ck ne ss ( Å ) (a) (b) r e s onan c e w a v e l eng t h ( n m ) (c) Fig. 9. Darkfield spectroscopy of aliphatic SAM samples. (a) SAM thickness as a func-tion of the number of carbon atoms, measured by ellipsometry. (b) Typical distribution ofplasmon resonance wavelengths for over 1,000 nanoparticles on a mirror with butanethiol(C ) as a spacer. (c) Resonance wavelength for different aliphatic SAM spacer. Source dataavailable in Dataset 1 (Ref. [20]). order to reach a uniform but sparse coverage that allows spectroscopic investigations of indi-vidual nanoparticles. Excess nanoparticles were rinsed off with distilled water and the sampleswere dried with nitrogen. C.2. Experimental
Darkfield spectra of over 1,000 nanoparticles were recorded using a fully automated OlympusBX51 microscope in a reflective dark field geometry. The scattered light was collected with a ×
100 long working distance objective (NA 0.8) and analysed with a TEC-cooled Ocean OpticsQE65000 spectrometer. Further details can be found elsewhere [32].
C.3. Ellipsometry
The thickness of the self-assembled monolayers was measured using a Jobin-Yvon UVISELspectroscopic ellipsometer at angle of incidence 70 ◦ and wavelengths from 300-800 nm. Thebase layer permittivity is obtained from a clean gold substrate and the dielectric propertiesof the SAMs described using a Cauchy model with refractive index of 1.45 (as determinedby [34, 35]). cknowledgmentscknowledgments