Direct Observation of Plasmon Band Formation and Delocalization in Quasi-Infinite Nanoparticle Chains
Martin Mayer, Pavel L. Potapov, Darius Pohl, Anja Maria Steiner, Johannes Schultz, Bernd Rellinghaus, Axel Lubk, Tobias A. F. König, Andreas Fery
DDirect Observation of Plasmon Band Formation and Delocalizationin Quasi-In fi nite Nanoparticle Chains Martin Mayer, † , § Pavel L. Potapov, ⊥ Darius Pohl, § , , ∥ Anja Maria Steiner, † , § Johannes Schultz, ⊥ Bernd Rellinghaus, § , , ∥ Axel Lubk, * , ⊥ Tobias A. F. Ko ̈ nig, * , † , § and Andreas Fery * , † , § , ‡ † Institute of Physical Chemistry and Polymer Physics and ‡ Department of Physical Chemistry of Polymeric Materials,Leibniz-Institut fu ̈ r Polymerforschung Dresden e.V., Hohe Strasse 6, 01069 Dresden, Germany § Cluster of Excellence Center for Advancing Electronics Dresden (cfaed) and ∥ Dresden Center for Nanoanalysis, TechnischeUniversita ̈ t Dresden, D-01062 Dresden, Germany ⊥ Institute for Solid State Research and
Institute for Metallic Materials, Leibniz-Institut fu ̈ r Festko ̈ rper und Werksto ff forschung,Helmholtzstrasse 20, 01069 Dresden, Germany * S Supporting Information
ABSTRACT:
Chains of metallic nanoparticles sustainstrongly con fi ned surface plasmons with relatively lowdielectric losses. To exploit these properties in applications,such as waveguides, the fabrication of long chains of lowdisorder and a thorough understanding of the plasmon-modeproperties, such as dispersion relations, are indispensable.Here, we use a wrinkled template for directed self-assembly toassemble chains of gold nanoparticles. With this up-scalablemethod, chain lengths from two particles (140 nm) to 20 particles (1500 nm) and beyond can be fabricated. Electron energy-loss spectroscopy supported by boundary element simulations, fi nite-di ff erence time-domain, and a simpli fi ed dipole couplingmodel reveal the evolution of a band of plasmonic waveguide modes from degenerated single-particle modes in detail. Instriking di ff erence from plasmonic rod-like structures, the plasmon band is con fi ned in excitation energy, which allows lightmanipulations below the di ff raction limit. The non-degenerated surface plasmon modes show suppressed radiative losses fore ffi cient energy propagation over a distance of 1500 nm. KEYWORDS:
Surface plasmons, nanoparticle, plasmonic polymer, template-assisted self-assembly, electron-energy loss spectroscopy L ocalized surface plasmon resonances (LSPR) are self-sustaining resonances appearing when delocalized con-duction-band electrons of a metal are con fi ned within ananoparticle. Plasmonic resonances are characterized bystrong and localized electromagnetic fi eld enhancement, whichis strongly sensitive to the geometry and composition of thenanoparticle and the environment. This makes them attractivefor a wide range of applications, in which sub-wavelength controlof electromagnetic fi elds from the infrared to ultraviolet range iscrucial. In particular, long metallic nanoparticle chains havebeen proposed for plasmonic waveguiding, i.e., photonictransport con fi ned to the submicron and, hence, subwavelengthlength scale, which is di ffi cult to realize with optical devices. Inthis length scale, plasmonic waveguides open up new strategiesfor signal transport due to their strong con fi nement and the highsignal speed. It has been predicted through analytical andnumerical studies that regular nanoparticle chains supportplasmon modes with distinct dispersion relations and, hence,signal transmission velocities depending on the geometricparameters of the chain. More recently, geometric modi fi ca-tions of the monopartite chain prototype, such as bipartitechains or zig-zag chains, are predicted to feature more complicated band structures, including plasmonic band gapsas well as topological edge states. For the realization of such waveguides and the experimentalproof of the predicted e ff ects, however, energy dissipation in themetal and precise positional control of the metallic nanoparticlesare still bottlenecks. In particular, the intrinsic losses of theemployed metals limit the overall performance of plasmonicwaveguides. However, smart assembly, resulting in fi nelytuned particle coupling, can further lower the dissipation. Forexample, Liedl et al. recently showed low-dissipative andultrafast energy propagation at a bimetallic chain with three 40nm particles. Various colloidal techniques, such as DNAorigami or self-assembly by chemical linkers, are available toassemble noble metal nanoparticles into chains. In principle,in fi nitely long particle chains can be fabricated with thesetechniques; however, trade-o ff s in particle spacing, particle size,or linear geometry must be accepted. In recent years, masks havebeen used for colloidal self-assembly to overcome the sizelimitation, which results into long-range energy transfer, even Received:
March 12, 2019
Revised:
May 10, 2019
Published:
May 22, 2019
Letterpubs.acs.org/NanoLett
Cite This:
Nano Lett. − © 2019 American Chemical Society DOI: 10.1021/acs.nanolett.9b01031
Nano Lett. − This is an open access article published under a Creative Commons Non-Commercial NoDerivative Works (CC-BY-NC-ND) Attribution License, which permits copying andredistribution of the article, and creation of adaptations, all for non-commercial purposes. D o w n l o a d e d v i a LE I B N I Z I F W D R E S D E N on O c t ob e r , a t : : ( U T C ) . S ee h tt p s :// pub s . ac s . o r g / s h a r i nggu i d e li n e s f o r op ti on s on ho w t o l e g iti m a t e l y s h a r e pub li s h e d a r ti c l e s . round a micrometer-sized corner. It has been shown thatabove a rather unde fi ned chain length, the so-called “ in fi nitechain limit ” , the longitudinal plasmonic modes converge to anonzero asymptotic energy. Consequently, the plasmonicresponse above this lengths di ff ers from shorter chains becauseof the discrete nature of the chain and the fi nite couplingstrength. To make it easier to distinguish between them,chains above the in fi nite chain limit are referred to plasmonicpolymers, whereas shorter chains are called plasmonicoligomers, in close analogy to organic polymer synthesis. Although the energy transport is principally improved by darkmodes, which su ff er signi fi cantly less from radiation losses thanthe bright ones, it is still a matter of debate whether those darkmodes are responsible for the transport. Recently, methods forvisualizing localized plasmons on the nanometer scale haveemerged. These methods involve transmission electronmicroscopy (TEM) combined with electron energy-loss spec-troscopy (EELS) with high-energy resolution, which is nowreadily available in dedicated monochromated TEM instru-ments. Examples of successful application of TEM and EELSmethods includes mapping LSPR modes in metallic nano-cubes, nanorods, − and nanospheres. In this Letter, we fabricated long regular nanoparticle chains,which allow for individual probing of local fi elds. Bycomprehensive spatially resolved electron energy loss studies,the plasmonic response is characterized. Here, we will particularly address the transition from individual single particlemodes to plasmonic bands in quasi-in fi nite long chains, whichhas not been directly observed previously. Robust excitation ofthe plasmonic waveguide modes relies on recent developmentsin our groups: single crystalline wet-chemical synthesis andtemplate-assisted colloidal self-assembly as well as improvedEELS characterization in the TEM and electromagneticmodeling. Template-assisted colloidal self-assembly (Figure 1) was usedto fabricate colloidal nanoparticle chains (particle diameter of 70nm) on micrometer-scale carbon-coated TEM grids.
Because high optical quality, reproducibility, and narrow sizedistributions are crucial parameters, single-crystalline sphericalgold nanoparticles (AuNSp) were synthesized by seed-mediatedgrowth, which comply with these requirements.
The highlylinear assembly of these gold spheres into wrinkled elastomerictemplates results in closely packed chains with a homogeneousspacing of <2 nm.
The obtained interparticle distance relies,in this case, only on the well-de fi ned thickness of the employeddielectric spacer, i.e., in this case, on the protein shell. Theresulting spacing can also be seen in the TEM images in Figure1b. The directed self-assembly process followed by wet-contactprinting on the target structure (TEM grid) is outlined in Figure1a. Transfer to a TEM grid allows the spectroscopic study of thecoherent plasmonic coupling in chains of di ff erent lengths(Figure 1b) while maintaining the good fi lling rate (de fi ned by Figure 1.
Large-area template-assisted self-assembly of various chain lengths and wet transfer for spectroscopic (EELS) studies. (a) Schematictemplate-assisted colloidal self-assembly via spin coating followed by wet-transfer printing on a TEM grid. (b) A microscope image and (S)TEMimages and selective details (3, 6, and 10 particle chains) of the transferred gold particle chains.
Figure 2.
Theoretical and experimental plasmonic modes of a (a) dimer, (b) trimer, and (c) pentamer. Schematic descriptions, integrated surfacecharge images, and plots (black line) as well as corresponding dipole moments for simulated surface charge plots for the most-dominant longitudinalmodes. Experimental and simulated EELS maps of longitudinal and transversal modes.
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Nano Lett. −3862 hains per area) and close packing of the nanoparticles withinchains. To obtain close-packed chains with varying lengths and aslightly reduced density of chains on the sample, the pH of thecolloidal nanoparticle solution was slightly reduced, as describedin the experimental section. Thus, such assemblies mark theperfect test system to study the e ff ects of various particle lengthswith comparable properties in a combinatorial approach.The following investigates the transition from plasmonicmonomer to polymer (i.e., beyond the in fi nite chain limit) indetail. Figure 2 shows plasmonic oligomers consisting of anumber of nanoparticles well-below the in fi nite chain limit. Theplasmonic properties for such short particle chains with up to fi ve particles have been studied extensively by Mulvaney et al. EELS in the TEM is nowadays a common approach to spatiallyimage plasmons. In this method, the energy loss of a focusedelectron beam upon crossing the electric fi eld of a plasmon(excited by the evanescent fi eld of the very same electron) isspectroscopically mapped.For short particle chains, several plasmon modes can beobserved in the EELS maps and spectra (see also Figure S4). Bydirectly comparing experimentally observed maps of energy losswith the electromagnetic simulations, the nature of the inducedcoupling interactions can be elucidated. This comparison allowsfor the identi fi cation of the plasmonic modes by theircorresponding surface charges. Starting from a single particle(Figure S2), the addition of a second particle (forming a dimer,Figure 2a) induces hybridization. Hence, the longitudinalmodes split into a symmetric (L2) and antisymmetric (L1) onewith higher and lower energy, respectively, compared to thefundamental (transversal) mode.
The maps of energy-lossand the derived surface charges reveal the lower energyantisymmetric mode. The higher-energy L2 mode is notunambiguously detectable because of its lower interactionwith the electron beam (see below for details) and its largerdamping by interband transitions.By forming a trimer (Figure 2b), the L1 mode shifts further tolower energies. Additionally, the next order longitudinal L2mode, with a node in the central particle, can be discerned fromthe transversal and L1 mode. The mutual cancellation ofinduced dipoles generates a net dipole moment of zero (evennumbers of surface charge waves) rendering this L2 mode dark(i.e., non-radiatively interacting with photons). The higher-energy L3 mode above the transversal mode is again damped byinterband transitions. As the number of particles increases to fi veparticles, the energetically lowest mode (L1) approaches thein fi nite chain limit already (see Figure 2c). However, theinduced longitudinal modes L1 and L2 can still be discerned,and the surface charge waves cover the complete length of thechain.By exciting such particle chains with an electromagnetic fi eld,e.g., a light wave or the evanescent fi eld of a focused electronbeam, collective localized surface plasmons are induced in theparticle chain. The induced electron density oscillation results ina localized dipole moment of the single particles (de fi ned bysurface charges). At large interparticle distances, theseoscillations do not couple and are energetically degenerate.When the interparticle distance is decreased far below theexcitation wavelength, the localized dipole moments of thesingle particles couple, lifting the degeneracy. By extension,higher-order multipole moments also couple, which becomesincreasingly important at small particle distances. The followingshows that the principal plasmonic behavior of nanoparticlechains can be well-described on the dipolar coupling level. In particular the dipoles oriented along the chain axis stronglycouple coherently leading to a set of distinguishable longitudinalmodes (Figure 3).Therefore, similar to the electron-wave function in a diatomicmolecule, the energetically lowest coherent plasmonic mode in aplasmonic dimer (L1 at 1.4 eV/885 nm) is an antisymmetricbonding mode (Figure 3b). For longer chains, several harmonicsof the longitudinal mode can be excited (see also Figure 3a),which are de fi ned by the number of nodes ( n ). In this context, anode is de fi ned by a zero dipole moment at a speci fi c chainposition and the longitudinal modes (L m ) are indexed by m = n+ 1 . Using this de fi nition, bright modes (nonzero overalldipole moment) occur at odd m and dark modes (zero overalldipole moment) at even m , respectively. Figure 3b exemplarilyshows two selected EEL spectra for chains with 2 and 12particles. In contrast to rod like structures, the excitationenergy of the L1 mode in long chains converges at low energies,described as the in fi nite chain limit. In literature, this limit istypically de fi ned somewhere between 8 and 12 particles (seealso a spectral visualization in Figure S1). In contrast,nanoparticle dipole moments, perpendicular to the chain axis,couple only weakly, resulting in (almost) degenerate transversalmodes (marked as T at 2.2 eV/560 nm).Approaching the in fi nite chain limit, the intensity of thefundamental longitudinal mode L1 shifts further to smallerenergies and gradually vanishes (Figure 4). Electromagneticsimulations reveal that this decrease can be ascribed to anincreased damping due to intraband transitions (see Figure S7for respective spectra). Above L2, the L3 mode becomesobservable as identi fi ed by the number of nodes ( n = 4).When increasing the number of particles in a chain beyond thein fi nite chain limit, it becomes increasingly di ffi cult or evenimpossible to discern pure L m modes (Figures 5 and S5 for the Figure 3. De fi nitions of the longitudinal plasmonic modes and theirspectroscopic response. (a) Schematic descriptions of the plasmonicmodes along the geometric axis by surface charge ( ± ), dipole moment(black line), net-dipole moment, node, and surface charge wave(schematically representing the polarization fi eld of the respectiveplasmonic modes). (b) Selected electron energy loss spectra (EELS)for 2 and 12 particles. The blue and red lines show the EELS responseaveraged over 10 −
40 pixels, and the black line represents the valuesaveraged over all spectra of the scanned map.
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Nano Lett. −3862 aps of 10 and 15 particles) because more modes occupy thesame spectral region. Indeed, instead of spectrally well-separatedlongitudinal modes, distinguished by their node number, abroad band of surface plasmon modes emerges (denoted by L * ).This L * mode is characterized by two broad nodes close to theedge and a large maximum in between. The impact of the bandformation on the surface charges is summarized in Figure S6.The longest studied chain covers a distance of almost 1.5 μ m,although there are much longer chains on the TEM grid (>30particles) available. However, those particles tend to contain acertain degree of disorder, such as non-colinear alignment,distance variations, etc. These irregularities increasinglyin fl uence the mode formation by giving rise to localizatione ff ects, as was already visible in Figure 5a (see below for a more-detailed discussion of this e ff ect).Figure 6a summarizes the evolution of the excitation energyupon the transition from plasmonic oligomers to plasmonicpolymers by plotting their peak position as a function of thechain length j . Signi fi cant scattering of the mode energies isobservable, which is predominantly due to slight variations(disorder) of the observed chains. For instance, examining theL1 mode of the dimer, the latter consists of “ kissing ” spheres(Figure 2a), thus leading to an enhanced interaction and, hence,energy shift. The transversal mode is independent from thechain length within the error of the measurement. This indicates small coupling interactions between transversal dipoles and,hence, the degeneracy of the corresponding single particleoscillations. In contrast, more longitudinal modes, separated inenergy, appear with increasing chain length. Their energydecreases as the number of particles increases, approaching alower boundary in the in fi nite chain limit. Eventually, thelongitudinal modes energetically approach each other forplasmonic polymers and superimpose (i.e., degenerate) due totheir fi nite lifetimes and, hence, energy widths (Figure 6b).Consequently, they cannot be discerned anymore (Figure 5),forming an e ff ective L * mode. This degenerated L * mode isde fi ned by its characteristic in-phase excitation at the chain endsand its characteristic superposition of multiple harmoniclongitudinal modes in the central region. In the simpli fi edsketch in Figure 6b, the transition from isolated modes to thesuperimposed L * mode is visualized by an oscillating wavemodel.The behavior of long particle chains di ff ers from the reportedplasmonic properties of long metallic rods, which support high-order harmonics with well-distinguishable energies. Tounderstand the plasmonics of long nanoparticle chains, simplecoupled dipole models have been used in the past (seeDowning et al. for a more elaborate quantum description).Accordingly, nanoparticle chains may be approximated by anassembly of discrete individual dipoles P i for each particle (of Figure 4.
Particle chain length approaching the in fi nite chain limit. Schematic description, simulated surface charge plots of the lowest-order L1 mode,and maps of energy loss for all resolvable plasmonic modes (panels a and b, 6 and 8 particles, respectively; experimental results are shown on the left andsimulated results on the right; the intensity of L1 was experientially below the detection limit). Figure 5.
Particle chains beyond the in fi nite chain limit. Schemes, simulated surface charge plots, and maps of energy loss for all selected plasmonicmodes (panels a and b show 12 and 20 particles, respectively; experimental results are shown on the left, and simulated results are shown on the right).Note the localized increase of fi eld strength around the small kink of the chain in panel a at the ninth particle of the chain. Nano Letters
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Nano Lett. −3862 ndex i ). These dipoles couple with their neighbors by dipole − dipole interaction facilitated by the electric fi eld propagator G ij (see section S1 of the Supporting Information for an explicitexpression of the propagator), as described by the following self-consistent polarization model: ikjjjjjjj y{zzzzzzz ∑ ω α ω ω ω ω = + ≠ P E G P ( ) ( ) ( ) ( ) ( ) i j i ij j ext (1)
Here, α denotes the (isotropic) single sphere polarizabilityand E ext the external electric fi eld associated with, e.g., theelectron beam. A plasmonic mode occurs at the poles of ( α − − G ) − (with G ≡ { G ij }), which can be found by searching thezeros of det( − α ( ω ) G ( ω )). However, α ( ω ) G ( ω ) is generallynon-Hermitian due to retardation and loss (i.e., the complexnature of α ), and α is a nonlinear function in the complex plane.Therefore, exact zeros (i.e., exact resonances) generally do notexist and maximal responses to external fi elds occur at theminima of the determinant in the complex plane ω , i.e., atfrequencies with imaginary part accounting for the fi nite lifetimeof the plasmon mode. The above model is a simpli fi ed version ofthe more general multiple elastic scattering of multipoleexpansions model (MESME), which has been previouslyused to simulate SPR in nanoparticle chains. Indeed, the smallinterparticle distance also leads to signi fi cant quadrupole andeven higher-order multipole interactions. Especially, the shortdistance interaction between nearest neighbors is increased incomparison to the simple dipole interaction model.Here, the simple structure of the dipole model is exploited toanalytically discuss several aspects of the long chain limit, notaccessible by a full-scale numerical solution of Maxwell equation.First of all, by evaluating the eigenvalue problem GP ( ω ) = a ω P ( ω ) (see section S1 in the Supporting Information), eq 1 admits analytical solutions for the mode structure in arbitrary chainlengths, if the interaction to nearest neighbors ( i.e . j = i ±
1) isrestricted, the inverse polarizability in the considered energy lossregime ( i.e . α − ≈ a ω ) is linearized, and a ω -averaged propagator(i.e., G ( ω ) ≈ G ) is used. In particular, the longitudinal modeenergies are given by ω ω = − ϵ{ } π + ( ) G l n l xx ln ,with n denoting the number of particles. The correspondingpolarizations read = ϵ{ } π + ( ) P l n sin , 1, ..., xj l ljn , 1 , with j beingthe particle index. Inserting the dipole fi eld propagator andemploying a polarizability derived from the dielectric function ofgold, the transverse mode is, however, too high in energy at 2.5eV, and the energy band is too narrow (0.5 eV) compared to theexperiment. To obtain the band widths in reasonable agreementwith the experimental results (i.e., approximately that of takinghigher-order multipole interactions into account), the couplingparameters were adapted to G xx ≈
800 THz (3.3 eV/375 nm) forthe dipole aligned along the chain (see Figure 6a). An indirectproof for the existence of higher-order multipole couplings isprovided by the full-scale numerical solution of Maxwell ’ sequation, which correctly reproduced the bandwidth. Theenergy shift can be explained by the interaction with the carbonsubstrate. The latter acts as a re fl ecting half-plane for the electric fi eld of the plasmon mode, introducing a second propagator (seesection S1 in the Supporting Information for an explicitexpression): ikjjjjjjj y{zzzzzzz ∑ ∑ ω α ω ω ω ω ω ω = + + ≠ P E G P G P ( ) ( ) ( ) ( ) ( ) ( ) ( ) i j i ij j j ij j ext ref (2)
Primarily, the re fl ected partial wave directly acts back on theemitting particle, giving rise to an additional self-interactionterm α ( ω ) G ii ref ( ω ) P i ( ω ). This term leads to a renormalization ofthe polarization and, hence, to plasmon energy according to: ikjjjjjjj y{zzzzzzz ∑ ω α ωα ω ω ω ω ω = − + ≠ P G E G P ( ) ( )1 ( ) ( ) ( ) ( ) ( ) i ii j i ij j ref ext (3)
Numerical calculations using the dielectric function of carbonand the geometry of the gold particles show that this e ff ect issu ffi cient to explain the red shift of approximately 0.2 eV (seealso ref 38). In the full-scale solution of the Maxwell equations,an e ff ective medium approach was employed to reproduce thered shift. Considering the various involved approximations, theagreement among full-scale simulations, the dipole model, andthe experimental results is rather good (see Figure 6a). Theremaining di ff erences to the experimental observed energies areascribed to long-range coupling of the re fl ected wave (notincluded in our models) and deviations from the idealizedgeometry in the experiment, such as fl uctuations in distance,shape, alignment, surface, etc. The latter can be subsumed underdisorder e ff ects. While the accurate description of disordere ff ects requires elaborate perturbation and renormalizationschemes (for example, see ref 39 for a treatment within the self-consistent theory of Anderson localization), it is has beenestablished that disorder in 1D systems always tends to localizethe wave fi eld with a characteristic exponential damping | E | ≈ exp( | x − x | / ξ ( η )) depending on the disorder strength η . Upon close inspection of the investigated long chains, indeed,the localized excitations are close to geometric perturbations of
Figure 6.
Study showing the transition from plasmonic oligomers toplasmonic polymers. (a) Experimental EELS ( fi lled circles) and BEMsimulated (hollow squares) energies of the various longitudinal modesas a function of particle length. Dashed lines show the results of thecoupled dipole model. (b) Schematic description for the formation ofthe merged longitudinal plasmonic modes L * beyond the in fi nite chainlimit. Nano Letters
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Nano Lett. −3862 he chain (e.g., close to the kink in Figure 5a); however, a largersample base would be required to further elaborate this e ff ect.Next, the in fi nite chain limit is discussed using the discretedipole model. The dipole coupling model describes analyticallythe formation of a continuous band of longitudinal modes with adispersion relation (wave vector q ) ω ( q ) = ω − G xx cos qd inthe in fi nite chain limit ( d denotes the interparticle distance).However, due to the fi nite energy resolution and the fi nitelifetime (energy width) of the plasmon modes, only asupposition of harmonics can be excited (L * mode). Thus,the highest group velocity v max = [ ∂ ω / ∂ q ] max ≈ c is reachedaround q = 2 π /4 d , which corresponds to longitudinal plasmonmodes close to the energy of the (nondispersive) transversemode. The group velocities decrease toward the longerwavelengths (and, therefore, lower excitation energies).Consequently, the group velocity and hence, the transport inthe transversal mode is almost zero as the coupling G yy,zz between the transverse dipoles is very small. Finally, theanalytical expressions for the net dipole moment are = − p q qL ( ) (1 cos ) x qd of the q -dependent modes (chainlength of L ), which is a measure for the optical coupling strengthof individual modes (e.g., optically dark modes have a netmoment zero). Accordingly, dark modes appear whenever q =2 π n / L , and the largest net dipole moment is realized in the longwavelength limit q →
0. Furthermore, from the analytical banddispersion, the plasmonic density of states can be computedaccording to ikjjjjjjjj ikjjjjj y{zzzzz y{zzzzzzzz ω ωω ω ω = = − − − dqd G d G DOS( ) 2 ( ) 1 2 xx xx which also grows toward smaller excitation energies (moregenerally toward the band edge, which coincides the bandminimum in this case). Consequently, we may conclude that theoptical coupling ( ∼ p × DOS) is maximal for lower excitationenergies, where it is eventually bounded by increasing losses.Thus, treating more complicated chains (e.g., with multi-partite basis) including beyond-nearest neighbor interactionsrequires numerical schemes. However, for periodic arrange-ments such as the long chain, the use of Bloch ’ s theorem greatlyfacilitates their implementation (see section S1 in theSupporting Information). In that case, eq 1 reads as; ∑ ω ω ω = μ ν μν ν c q D q c q ( , ) ( , ) ( , ) a b ab b (4) with c a μ (nanoparticle index in unit cell a and Cartesian index μ )denoting the excitation coe ffi cients of the Bloch waves P a μ = c a μ ( q, ω ) e iqxn and D a μ bv a dynamical coupling matrix. Thisformulation has been recently used to compute topologicale ff ects in Su − Schrie ff er − Heeger model chains. To highlight the substantial importance of the degenerated L * mode in the plasmonics of linear particle chains, its impact onenergy transport along such chains is brie fl y studied in thefollowing. These considerations will also shine light on animportant consequence of the non-Hermiticity (i.e., lossy)nature of the plasmonic system by providing a characteristiclength scale for the coherent coupling of the plasmonic modes.Plasmonic waveguiding is chosen because it is typicallydescribed as one of the most promising applications for linearparticle chains. Furthermore, the e ff ect is dominated bynear- fi eld e ff ects. As it has been suggested earlier, the previously observed degenerated modes have suppressed radiative losses,which may be further tuned by higher multipole order near fi eldcoupling terms (beyond dipole). − Finite-di ff erence time-domain (FDTD) electromagnetic modeling were employed toquantify the energy transport properties (Figure 7).As a simpli fi ed waveguiding experiment, a dipole source at oneend of the particle chain is used as source. Figure 7a visualizesthe energy transport e ffi ciency along the chain for all excitablemodes by the integrated electric fi eld (red indicating within andblue indicating out of the energy band).The energetically highest mode (transversal, 2.42 eV, darkblue) exhibits a double exponential decay resulting in atheoretical damping factor of − ffi cient waveguiding with damping factors as low as − and can beexplained by an optimal balance of group velocity (increasingtoward higher energies) and radiative losses (decreasing towardhigh energies) in the sub-radiant level above the super-radiantL1. In comparison to a comparable DNA-assembled waveguide,which is below the in fi nite chain limit, the herein observeddamping is lower by a factor of 3. Table 1 lists the properties of the supported waveguidingmodes as well as the integrated electric fi elds E at the fi rst andlast particle for each mode (necessary to calculate the damping Figure 7.
Energy-transport properties of plasmonic polymers(consisting of 20 nanoparticles). (a) Integrated electric fi eld alongthe particle chain in respect to various dipole energies. (b)Corresponding integrated surface charge images (red − blue colorscale) and surface charge waves (black line). Nano Letters
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Nano Lett. −3862 actor). The induced surface charge waves (Figure 7b)propagate with an e ff ective energy along the particle chain, asvisualized by the respective length of the wave package. Theenergy propagation for a selected mode (1.59 eV) is highlightedin Video S1. Finally, the mode size perpendicular to the particlechains (energy E at 1/ e of the cross-section) is critical forpractical applications to avoid cross-talk between neighboringwaveguides. The mode size for degenerated modes is below 300nm, while for the non-degenerated modes, it is 140 nm.However, in the real system, disorder e ff ects produce anadditional localization (see the discussion above), which mayfurther increase the damping factor.In summary, we directly observed the formation of hybridizedplasmonic modes on linear nanoparticle chains as a function ofthe chain length ranging from one nanoparticle to the in fi nitechain limit. Plasmonic oligomers and polymers have beenfabricated by template-assisted colloidal self-assembly yieldinghighly ordered chains with low disorder. Electron energy-lossspectroscopy and theoretical modeling allowed us to character-ize the plasmonic mode transition between short chains(plasmonic oligomers) and long chains (plasmonic polymers).Plasmonic oligomers show well-separated, non-degeneratedlongitudinal modes and degenerated transverse modes. Beyondroughly 10 particles, a band of longitudinal modes eventuallyemerges. This band exhibits a degenerated plasmonic waveguidemode signature with a large dispersion, which makes it suitablefor long distance energy transport in the optimal band region.Theoretical simulations of the waveguiding performance suggestthat the L * mode with a mode wavelength of 730 nm is able totransport energy with a damping of − fi ndings pave the way forfurther exploitation of plasmonic nanoparticle chains aswaveguides and photonic devices. For example, the waveguidemode can be intervened at each contact point by insertingmaterials such as fl uorescent emitters and organic or smartpolymers to further modulate and enhance the radiativeproperties of the plasmonic waveguide. Of particular interest isthe coupling at end points of the chain, which sustain particularedge states that may be tuned by complex chain geometries (e.g.,bipartite chains). Furthermore, the presented approach opensup new possibilities to study combinatorically the plasmonics ofmultimetallic chains or chains with multiply particle morphol-ogies within one chain.
Methods.
Synthesis, Template-Assisted Colloidal Self-Assembly, and Wet Transfer to Carbon-Coated TEM Grids.
The synthesis of single-crystalline spherical gold nanoparticleswith a diameter of 70.5 ± ]) aspublished elsewhere. The pH of the nanoparticle solution was adjusted to pH 10 to produce closely packed particle lines inincompletely fi lled templates.Transfer in the grooves of the template assembled particlechains was performed by wet-contact printing. The 3 nmcarbon- fi lm coated TEM grid (copper, 300 square meshes) wasincubated in PEI solution (1 mg/mL, 1800 g/mol, linear) for 1 hand subsequently washed with puri fi ed water (Milli-Q-grade,18.2 M Ω cm at 25 ° C). For the transfer, a 2 μ L water droplet(pH 10) was placed on the TEM grid, the particle- fi lled PDMSstamp was pressed onto the grid with a constant pressure of 100kPa, and the grid was left to dry under environmental conditions(23 ° C, 55% relative humidity). After drying (4 h), the stampwas carefully removed, leaving the nanoparticle chains on thecarbon fi lm of the grid. STEM EELS Characterization.
STEM scanning and EELSspectrum-imaging was performed in the probe-corrected Titan operating at 300 kV. The microscope was equipped with aTridiem energy fi lter and a Wien monochromator operating inthe accelerating mode, which ensured the energy resolution of100 −
120 meV. EELS was performed under the convergenceangle of 22 mrad and the collection angle of 8 mrad with theenergy dispersion of 0.01 eV per channel. The spectrum imageswere acquired with a beam current 200 pA and dwell time 25 ms.For each chain of gold particles, 4 − fi lecollected in a separate run without any sample object. Anexample of spectra treatment is shown in Figure S3. Finally, thedistinct peaks in the low-loss region of spectra were recognizedand their energy positions and magnitude were fi tted using thenonlinear least-squares procedure. The integrated area undereach fi tted peak was plotted as a function of the beam positiongiving rise to maps of the probability for plasmon excitation, i.e.,maps of energy-loss. Electromagnetic Simulations.
Simulations of electronenergy loss spectra and mappings were performed using theMatlab MNPBEM13 toolbox, which is based on the boundaryelement method (BEM) by Garci ́ a de Abajo. Each sphere ofthe chain was approximated by triangulation (400 vertices perparticle). The dimensions and positions of the gold nanosphereswere selected according to the experimental size (70.5 ± Ane ff ective medium of n = 1.2 is selected to emulate the air − substrate interface. The energy of the simulated electron beamwas set to the experimental accelerating voltage of the TEM(300 kV). For each particle chain several spectra were evaluatedin the energy range from 1 − ff erent (not overlapping)electron-beam positions to ensure excitation of all possibleplasmonic modes EELS mappings were performed at selected Table 1. Plasmonic Waveguide Properties of Di ff erent Plasmonic Modes (Transverse Mode: 2.42 eV and Longitudinal Modes:>2.42 eV; E ff ective Mode Size Describes the Radial Drop of the Integrated E perpendicular to the Particle Chains to 1/ e ) excitation energy, eV 2.42 2.07 1.89 1.79 1.70 1.59 1.47integrated E ( x = 0 nm) 1071 1513 1925 1758 1679 1519 1171integrated E N2 ( x N = 1380 nm) <0.01 <0.01 0.06 28 218 284 114damping in decibels per 1380 nm − − − − − − − − − − − − − − ff ective mode energy, eV 8.86 4.43 4.13 2.82 2.58 2.07 1.70e ff ective mode size, nm 137 141 145 173 198 229 261 Nano Letters
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DOI: 10.1021/acs.nanolett.9b01031
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Nano Lett. −3862 nergy levels and were simulated by 1.5 nm meshing of theelectron beam. The respective spectra are summarized in FigureS7.For energy-transfer and waveguiding simulations, a commer-cial-grade simulator based on the fi nite-di ff erence time-domain(FDTD) method was used to perform the calculations(Lumerical Inc., Canada, version 8.16). For the broadbandand single-mode excitation for energy transport, a dipole sourcefor the speci fi c wavelength range (short pulse length of 3 fs; E ≈ − λ = 300 − fi rst particle,respectively. To extract the exact peak positions from thespectra, the internal multipeak- fi tting function of IGOR Pro 7(WaveMetrics) was used. The mode size of the modes wasde fi ned by integrating of E along the particle line and radiallyde fi ning the 1/e decay of the E fi eld. For the FDTDsimulations, the same refractive index data, nanoparticledimension, inter particle distance, and materials constantswere used as in the BEM simulations. Perfectly matched layers inall principal directions as boundary conditions, zero-conformal-variant mesh re fi nement, and an isotropic mesh overwrite regionof 1 nm were used. All simulations reached the auto shut-o ff levelof 10 − before reaching 150 fs of simulation time. ■ ASSOCIATED CONTENT * S Supporting Information
The Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.nano-lett.9b01031.Additional simulations, additional EELS maps, exper-imental EELS spectra including data treatment, and adetailed description of the discrete dipole model (PDF)A movie showing the electric fi eld during energypropagation along a particle chain (MPG) ■ AUTHOR INFORMATION
Corresponding Authors * E-mail: [email protected]. * E-mail: [email protected]. * E-mail: [email protected].
ORCID
Martin Mayer:
Tobias A. F. Ko ̈ nig: Andreas Fery:
Notes
The authors declare no competing fi nancial interest. ■ ACKNOWLEDGMENTS
This project was fi nancially supported by the VolkswagenFoundation through a Freigeist Fellowship to T.A.F.K. A.L. hasreceived funding from the European Research Council (ERC)under the Horizon 2020 research and innovation program of theEuropean Union (grant agreement no. 715620). P.L.P.acknowledges funding from DFG “ Zukunftskonzept ” (F-003661-553-U ̈ ■ REFERENCES (1) Willets, K. A.; Van Duyne, R. P.
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