Modified field enhancement and extinction by plasmonic nanowire dimers due to nonlocal response

Quantum mechanical limit to plasmonic enhancement as observed by surface-enhanced Raman scattering

Plasmonic nanostructures enable light to be concentrated into nanoscale 'hotspots', wherein the intensity of light can be enhanced by orders of magnitude. This plasmonic enhancement significantly boosts the efficiency of nanoscale light-matter interactions, enabling unique linear and nonlinear optical applications. Large enhancements are often observed within narrow gaps or at sharp tips, as predicted by the classical electromagnetic theory. Only recently has it become appreciated that quantum mechanical effects could emerge as the feature size approaches atomic length-scale. Here we experimentally demonstrate, through observations of surface-enhanced Raman scattering, that the emergence of electron tunnelling at optical frequencies limits the maximum achievable plasmonic enhancement. Such quantum mechanical effects are revealed for metallic nanostructures with gap-widths in the single-digit angstrom range by correlating each structure with its optical properties. This work furthers our understanding of quantum mechanical effects in plasmonic systems and could enable future applications of quantum plasmonics.

Surface plasmon dependence on the electron density profile at metal surfaces

We use an extension of the hydrodynamic model to study nonlocal effects in the collective plasmon excitations at metal surfaces and narrow gaps between metals, including the surface spill-out of conduction band electrons. In particular, we simulate metal surfaces consisting of a smooth conduction-electron density profile and an abrupt jellium edge. We focus on aluminum and gold as prototypical examples of simple and noble metals, respectively. Our calculations agree with the dispersion relations measured from planar surfaces for these materials. Systems involving small gaps display a regime of tunnelling electrons, which is partially captured by the overlap of electron densities. This extension of the hydrodynamic model to cope with inhomogeneous density profiles provides a relatively fast and accurate way of describing the optical response of metal surfaces at subnanometer distances.

Effect of surface roughness on self-assembled monolayer plasmonic ruler in nonlocal regime

Recently, self-assembled monolayers (SAMs) have been used for plasmonic rulers to measure the nonlocal influence on the Au nanoparticle - metal film resonance wavelength shift and probe the ultimate field enhancement. Here we examine the influence of surface roughness on this plasmonic ruler in the nonlocal regime by comparing plasmonic resonance shifts for as-deposited and for ultra-flat Au films. It is shown that the resonance shift is larger for ultra-flat films, suggesting that there is not the saturation from nonlocal effects previously reported for the spacer range from 0.7 nm to 1.6 nm. We attribute the previously reported saturation to the planarization of the as-deposited films by thinner SAMs, as measured here by atomic-force microscopy. This work is of interest both in probing the ultimate limits of plasmonic enhancement with SAMs for applications in Raman and nonlinear optics, but also in the study of SAMs planarization as a function surface roughness.

Refractive-index sensitivities of hybrid surface-plasmon resonances for a core-shell circular silver nanotube sensor

We study the scattering and absorption of an H-polarized plane electromagnetic wave by a circular silver nanotube in the visible range of wavelengths using the separation of variables. The computed spectra of the extinction cross section display several hybrid localized surface-plasmon resonances of the dipole and multipole type. Analytical equations are derived for their resonance wavelengths. Bulk refractive-index sensitivities of nanotube-based sensors are determined, showing higher values for multipole resonances.

Quantum plasmonics: optical properties of a nanomatryushka

Quantum mechanical effects can significantly reduce the plasmon-induced field enhancements around nanoparticles. Here we present a quantum mechanical investigation of the plasmon resonances in a nanomatryushka, which is a concentric core-shell nanoparticle consisting of a solid metallic core encapsulated in a thin metallic shell. We compute the optical response using the time-dependent density functional theory and compare the results with predictions based on the classical electromagnetic theory. We find strong quantum mechanical effects for core-shell spacings below 5 Å, a regime where both the absorption cross section and the local field enhancements differ significantly from the classical predictions. We also show that the workfunction of the metal is a crucial parameter determining the onset and magnitude of quantum effects. For metals with lower workfunctions such as aluminum, the quantum effects are found to be significantly more pronounced than for a noble metal such as gold.

Blueshift of the surface plasmon resonance in silver nanoparticles: substrate effects

We study the blueshift of the surface plasmon (SP) resonance energy of isolated Ag nanoparticles with decreasing particle diameter, which we recently measured using electron energy loss spectroscopy (EELS) [1]. As the particle diameter decreases from 26 down to 3.5 nm, a large blueshift of 0.5 eV of the SP resonance energy is observed. In this paper, we base our theoretical interpretation of our experimental findings on the nonlocal hydrodynamic model, and compare the effect of the substrate on the SP resonance energy to the approach of an effective homogeneous background permittivity. We derive the nonlocal polarizability of a small metal sphere embedded in a homogeneous dielectric environment, leading to the nonlocal generalization of the classical Clausius-Mossotti factor. We also present an exact formalism based on multipole expansions and scattering matrices to determine the optical response of a metal sphere on a dielectric substrate of finite thickness, taking into account retardation and nonlocal effects. We find that the substrate-based calculations show a similar-sized blueshift as calculations based on a sphere in a homogeneous environment, and that they both agree qualitatively with the EELS measurements.

Quantum effects and nonlocality in strongly coupled plasmonic nanowire dimers

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.

Surface plasmons and nonlocality: a simple model

Surface plasmons on metals can concentrate light into subnanometric volumes and on these near atomic length scales the electronic response at the metal interface is smeared out over a Thomas-Fermi screening length. This nonlocality is a barrier to a good understanding of atomic scale response to light and complicates the practical matter of computing the fields. In this Letter, we present a local analogue model and show that spatial nonlocality can be represented by replacing the nonlocal metal with a composite material, comprising a thin dielectric layer on top of a local metal. This method not only makes possible the quantitative analysis of nonlocal effects in complex plasmonic phenomena with unprecedented simplicity and physical insight, but also offers great practical advantages in their numerical treatment.

Electron-energy loss study of nonlocal effects in connected plasmonic nanoprisms

We investigate the emergence of nonlocal effects in plasmonic nanostructures through electron-energy loss spectroscopy. To theoretically describe the spatial dispersion in the metal permittivity, we develop a full three-dimensional nonlocal hydrodynamic solution of Maxwell's equations in frequency domain that implements the electron beam as a line current source. We use our numerical approach to perform an exhaustive analysis of the impact of nonlocality in the plasmonic response of single triangular prisms and connected bowtie dimers. Our results demonstrate the complexity of the interplay between nonlocal and geometric effects taking place in these structures. We show the different sensitivities to both effects of the various plasmonic modes supported by these systems. Finally, we present an experimental electron-energy loss study on gold nanoprisms connected by bridges as narrow as 1.6 nm. The comparison with our theoretical predictions enables us to reveal in a phenomenological fashion the enhancement of absorption damping that occurs in these atomistic junctions due to quantum confinement and grain boundary electron scattering.

Robust subnanometric plasmon ruler by rescaling of the nonlocal optical response

We present the optical response of two interacting metallic nanowires calculated for separation distances down to angstrom range. State-of-the-art local and nonlocal approaches are compared with full quantum time-dependent density functional theory calculations that give an exact account of nonlocal and tunneling effects. We find that the quantum results are equivalent to those from classical approaches when the nanoparticle separation is defined as the separation between centroids of the screening charges. This establishes a universal plasmon ruler for subnanometric distances. Such a ruler not only impacts the basis of many applications of plasmonics, but also provides a robust rule for subnanometric metrology.

Hyperbolic metamaterial lens with hydrodynamic nonlocal response

We investigate the effects of hydrodynamic nonlocal response in hyperbolic metamaterials (HMMs), focusing on the experimentally realizable parameter regime where unit cells are much smaller than an optical wavelength but much larger than the wavelengths of the longitudinal pressure waves of the free-electron plasma in the metal constituents. We derive the nonlocal corrections to the effective material parameters analytically, and illustrate the noticeable nonlocal effects on the dispersion curves numerically. As an application, we find that the focusing characteristics of a HMM lens in the local-response approximation and in the hydrodynamic Drude model can differ considerably. In particular, the optimal frequency for imaging in the nonlocal theory is blueshifted with respect to that in the local theory. Thus, to detect whether nonlocal response is at work in a hyperbolic metamaterial, we propose to measure the near-field distribution of a hyperbolic metamaterial lens.

Localized surface plasmon resonances in spatially dispersive nano-objects: phenomenological treatise

Nonlocal optical response of materials, important at the nanometric scale, influences numerous optical phenomena, such as electromagnetic field confinement and spectral characteristics of plasmonic resonances. Here, we present a general phenomenological approach to account for nonlocal material polarizabilities in nanoscale metal particles. The problem of nonlocal plasmonic resonances is formulated by an integro-differential equation in a space domain and solved by adopting its weak form, implemented in the finite element method, thus, dispensing with the requirements on additional boundary conditions. As an example, nonlocal smearing effects in plasmonic nanorods of various cross sections and nanotubes have been considered. Clear signature of nonlocality manifests itself in the interference fringes in the potential profile and a significant frequency shift of the localized surface plasmon resonances. These effects are especially important for nanoparticles with geometrical features comparable to the de Broglie wavelengths of electrons participating in the light-matter interactions. The proposed method provides a universal tool for phenomenological account of nonlocalities of any kind with the only requirement of linearity in system's response.

Surface Plasmon Coupling of Compositionally Heterogeneous Core-Satellite Nanoassemblies

Understanding plasmon coupling between compositionally heterogeneous nanoparticles in close proximity is intriguing and fundamentally important because of the energy mismatch between the localized surface plasmons of the associated nanoparticles and interactions beyond classical electrodynamics. In this Letter, we explore surface plasmon coupling between silver nanoparticles (AgNPs) and gold nanoparticles (AuNPs), assembled in the form of core-satellite structures. A recently developed assembly method allows us to prepare ultrapure core-satellite nanoassemblies in solution, where 50 nm AgNPs are surrounded by 13 nm AuNPs via alkanedithiol linkers. We observe changes in the plasmon coupling between the AgNP core and AuNP satellites as the core-to-satellite gap distance varies from 2.3 to 0.7 nm. Comparison with theoretical studies reveals that the traditional hybridized plasmon modes are abruptly replaced by charge-transfer plasmons at a ∼1 nm gap. Changes with the number of satellites are also discussed.

Far-field analysis of axially symmetric three-dimensional directional cloaks

Axisymmetric radiating and scattering structures whose rotational invariance is broken by non-axisymmetric excitations present an important class of problems in electromagnetics. For such problems, a cylindrical wave decomposition formalism can be used to efficiently obtain numerical solutions to the full-wave frequency-domain problem. Often, the far-field, or Fraunhofer region is of particular interest in scattering cross-section and radiation pattern calculations; yet, it is usually impractical to compute full-wave solutions for this region. Here, we propose a generalization of the Stratton-Chu far-field integral adapted for 2.5D formalism. The integration over a closed, axially symmetric surface is analytically reduced to a line integral on a meridional plane. We benchmark this computational technique by comparing it with analytical Mie solutions for a plasmonic nanoparticle, and apply it to the design of a three-dimensional polarization-insensitive cloak.

Nonlocal effects in a hybrid plasmonic waveguide for nanoscale confinement

The effect of nonlocal optical response is studied for a novel silicon hybrid plasmonic waveguide (HPW). Finite element method is used to implement the hydrodynamic model and the propagation mode is analyzed for a hybrid plasmonic waveguide of arbitrary cross section. The waveguide has an inverted metal nano-rib over a silicon-on-insulator (SOI) structure. An extremely small mode area of~10⁻⁶λ² is achieved together with several microns long propagation distance at the telecom wavelength of 1.55 μm. The figure of merit (FoM) is also improved in the same time, compared to the pervious hybrid plasmonic waveguide. We demonstrate the validity of our method by comparing our simulating results with some analytical results for a metal cylindrical waveguide and a metal slab waveguide in a wide wavelength range. For the HPW, we find that the nonlocal effects can give less loss and better confinement. In particular, we explore the influence of the radius of the rib's tip on the loss and the confinement. We show that the nonlocal effects give some new fundamental limitation on the confinement, leaving the mode area finite even for geometries with infinitely sharp tips.

Perfect imaging, epsilon-near zero phenomena and waveguiding in the scope of nonlocal effects

Plasmons in metals can oscillate on a sub-wavelength length scale and this large-k response constitutes an inherent prerequisite for fascinating effects such as perfect imaging and intriguing wave phenomena associated with the epsilon-near-zero (ENZ) regime. While there is no upper cut-off within the local-response approximation (LRA) of the plasma polarization, nonlocal dynamics suppress response beyond ω/v(F), where v(F) is the Fermi velocity of the electron gas. Nonlocal response has previously been found to pose limitations to field-enhancement phenomena. Accounting for nonlocal hydrodynamic response, we show that perfect imaging is surprisingly only marginally affected by nonlocal properties of a metal slab, even for a deep subwavelength case and an extremely thin film. Similarly, for the ENZ response we find no indications of nonlocal response jeopardizing the basic behaviors anticipated from the LRA. Finally, our study of waveguiding of gap plasmons even shows a positive nonlocal influence on the propagation length.

Atomic-scale confinement of resonant optical fields

In the presence of matter, there is no fundamental limit preventing confinement of visible light even down to atomic scales. Achieving such confinement and the corresponding resonant intensity enhancement inevitably requires simultaneous control over atomic-scale details of material structures and over the optical modes that such structures support. By means of self-assembly we have obtained side-by-side aligned gold nanorod dimers with robust atomically defined gaps reaching below 0.5 nm. The existence of atomically confined light fields in these gaps is demonstrated by observing extreme Coulomb splitting of corresponding symmetric and antisymmetric dimer eigenmodes of more than 800 meV in white-light scattering experiments. Our results open new perspectives for atomically resolved spectroscopic imaging, deeply nonlinear optics, ultrasensing, cavity optomechanics, as well as for the realization of novel quantum-optical devices.

Theory of three-dimensional nanocrescent light harvesters

The optical properties of three-dimensional crescent-shaped gold nanoparticles are studied using a transformation optics methodology. The polarization insensitive, highly efficient, and tunable light harvesting ability of singular nanocrescents is demonstrated. We extend our approach to more realistic blunt nanostructures, showing the robustness of their plasmonic performance against geometric imperfections. Finally, we provide analytical and numerical insights into the sensitivity of the device to radiative losses and nonlocal effects. Our theoretical findings reveal an underlying relation between structural bluntness and spatial dispersion in this particular nanoparticle configuration.

Plasmon ruler with angstrom length resolution

We demonstrate a plasmon nanoruler using a coupled film nanoparticle (film-NP) format that is well-suited for investigating the sensitivity extremes of plasmonic coupling. Because it is relatively straightforward to functionalize bulk surface plasmon supporting films, such as gold, we are able to precisely control plasmonic gap dimensions by creating ultrathin molecular spacer layers on the gold films, on top of which we immobilize plasmon resonant nanoparticles (NPs). Each immobilized NP becomes coupled to the underlying film and functions as a plasmon nanoruler, exhibiting a distance-dependent resonance red shift in its peak plasmon wavelength as it approaches the film. Due to the uniformity of response from the film-NPs to separation distance, we are able to use extinction and scattering measurements from ensembles of film-NPs to characterize the coupling effect over a series of very short separation distances-ranging from 5 to 20 Å-and combine these measurements with similar data from larger separation distances extending out to 27 nm. We find that the film-NP plasmon nanoruler is extremely sensitive at very short film-NP separation distances, yielding spectral shifts as large as 5 nm for every 1 Å change in separation distance. The film-NP coupling at extremely small spacings is so uniform and reliable that we are able to usefully probe gap dimensions where the classical Drude model of the conducting electrons in the metals is no longer descriptive; for gap sizes smaller than a few nanometers, either quantum or semiclassical models of the carrier response must be employed to predict the observed wavelength shifts. We find that, despite the limitations, large field enhancements and extreme sensitivity persist down to even the smallest gap sizes.

Probing the ultimate limits of plasmonic enhancement

Metals support surface plasmons at optical wavelengths and have the ability to localize light to subwavelength regions. The field enhancements that occur in these regions set the ultimate limitations on a wide range of nonlinear and quantum optical phenomena. We found that the dominant limiting factor is not the resistive loss of the metal, but rather the intrinsic nonlocality of its dielectric response. A semiclassical model of the electronic response of a metal places strict bounds on the ultimate field enhancement. To demonstrate the accuracy of this model, we studied optical scattering from gold nanoparticles spaced a few angstroms from a gold film. The bounds derived from the models and experiments impose limitations on all nanophotonic systems.

Surface-enhanced Raman spectroscopy: nonlocal limitations

Giant field enhancement and field singularities are a natural consequence of the commonly employed local-response framework. We show that a more general nonlocal treatment of the plasmonic response leads to new and possibly fundamental limitations on field enhancement with important consequences for our understanding of surface-enhanced Raman spectroscopy (SERS). The intrinsic length scale of the electron gas serves to smear out assumed field singularities, leaving the SERS enhancement factor finite, even for geometries with infinitely sharp features. For silver nanogroove structures, mimicked by periodic arrays of half-cylinders (up to 120 nm in radius), we find no enhancement factors exceeding 10 orders of magnitude (10(10)).

Nonlocal effects in the nanofocusing performance of plasmonic tips

The nanofocusing performance of plasmonic tips is studied analytically and numerically. The effects of electron-electron interactions in the dielectric response of the metal are taken into account through the implementation of a nonlocal, spatially dispersive, hydrodynamic permittivity. We demonstrate that spatial dispersion only slightly modifies the device parameters which maximize its field enhancement capabilities. The interplay between nonlocality, tip bluntness, and surface roughness is explored. We show that, although spatial dispersion reduces the field enhancement taking place at the structure apex, it also diminishes the impact that geometric imperfections have on its performance.

Transformation-optics description of nonlocal effects in plasmonic nanostructures

We develop an insightful transformation-optics approach to investigate the impact that nonlocality has on the optical properties of plasmonic nanostructures. The light-harvesting performance of a dimer of touching nanowires is studied by using the hydrodynamical Drude model, which reveals nonlocal resonances not predicted by previous local calculations. Our method clarifies the interplay between radiative and nonlocal effects in this nanoparticle configuration, which enables us to elucidate the optimum size that maximizes its absorption and field enhancement capabilities.