Nonlocal Plasmons Research Articles

Classical vs. quantum plasmon-induced molecular transformations at metallic nanojunctions

Chemical transformations near plasmonic metals have attracted increasing attention in the past few years. Specifically, reactions occurring within plasmonic nanojunctions that can be detected via surface and tip-enhanced Raman (SER and TER) scattering were the focus of numerous reports. In this context, even though the transition between localized and nonlocal (quantum) plasmons at nanojunctions is documented, its implications on plasmonic chemistry remain poorly understood. We explore the latter through AFM-TER-current measurements. We use two molecules: i) 4-mercaptobenzonitrile (MBN) that reports on the (non)local fields and ii) 4-nitrothiophenol (NTP) that features defined signatures of its neutral/anionic forms and dimer product, 4,4'-dimercaptoazobenzene (DMAB). The transition from classical to quantum plasmons is established through our optical measurements: It is marked by molecular charging and optical rectification. Simultaneously recorded force and current measurements support our assignments. In the case of NTP, we observe the parent and DMAB product beneath the probe in the classical regime. Further reducing the gap leads to the collapse of DMAB to form NTP anions. The process is reversible: Anions subsequently recombine into DMAB. Our results have significant implications for AFM-based TER measurements and their analysis, beyond the scope of this work. In effect, when precise control over the junction is not possible (e.g., in SER and ambient TER), both classical and quantum plasmons need to be considered in the analysis of plasmonic reactions.

Nonlocal Soft Plasmonics in Planar Homogeneous Multilayers

Plasmonics is the study of resonant oscillations of free electrons in metals caused by incident electromagnetic radiation. Surface plasmons can focus and steer light on the subwavelength scale. Apart from metals, plasmonic phenomena can be observed in soft matter systems such as electrolytes which we study here. Resonant charge oscillations can be induced for ions in solution, however, due to their larger mass, they are plasmon-active in a lower frequency regime and on a larger wavelength scale. Our investigation focuses on spatial confinement which allows increasingly strong charge interactions and gives rise to nonlocality or spatial dispersion effects. We derive and discuss the nonlocal optical response of ionic plasmons using a hydrodynamic two-fluid model in a planar homogeneous three-layer system with electrolyte-dielectric interfaces. As in metals, we observe the emergence of additional longitudinal propagation modes in electrolytes which causes plasmonic broadening. Studying such systems enables us to identify and understand plasmonic phenomena in biological and chemical systems.

Nonlocal Optical Response of Particle Plasmons in Single Gold Nanorods.

The investigation of particle plasmons in metal nanoparticles has predominantly relied on local optical response approximations. However, as the nanoparticle size approaches the average distance of electrons to the metal surface, mesoscopic effects such as size-dependent plasmon line width broadening and resonance energy blue shifts are expected to become observable. In this work, we compared the experimental spectral characteristics with simulated values obtained by using a generalized nonlocal optical response theory-based local analogue model. Our results show that the nonlocal plasmon damping effects in single nanoparticles are less pronounced than those observed in plasmon-coupled systems. Furthermore, our research demonstrates that single-particle dark-field spectroscopy is an effective tool for investigating the nonlocal optical response of particle plasmons in single nanoparticles. These results have important implications for the rational design of novel nanophotonic devices.

OpenSANS: A Semi-Analytical solver for Nonlocal plasmonicS

Prompted by the renewed interest the past decade in the hydrodynamic model for the description of nanoplasmonic systems, we present OpenSANS, a MATLAB toolbox for the simulation of canonical (that is, standalone spherical and cylindrical) nanoparticles and planar stratified structures whose critical dimensions are of mere nanometres. The method comprises of a generalisation of Mie theory to properly address the hydrodynamic motion of free electrons in metals and a compact S matrix formulation for the efficient treatment of multiple layers. The toolbox is built in a modular manner; each part (routine) has its own important functionality and can be used independently from the others. Our primary goal is the presentation of a highly efficient and accessible solver for the nanoplasmonics community. This is achieved by the very semi-analyticity of the chosen method and by powerful vectorisation techniques optimised in a very popular environment, MATLAB. Program summaryProgram Title: OpenSANSCPC Library link to program files:https://doi.org/10.17632/hmk224xngb.1Code Ocean capsule:https://codeocean.com/capsule/2246190Licensing provisions: GPLv3Programming language: MATLAB R2018b and newer.Supplementary material: See Appendix A.Nature of problem: Solve the scattering problem from multilayered, dielectric, local or nonlocal metallic nanoparticles, excited by a plane wave and placed in a local (dielectric or metallic) background.Solution method: Mie theory, appropriately extended to incorporate nonlocal effects (longitudinal waves) as predicted by the Hard Wall-Hydrodynamic Model, combined with S matrix theory for efficient handling of the multilayered structure.

Generalized electromagnetic theorems for nonlocal plasmonics

The ultraconfined light of plasmonic modes put their effective wavelength close to the mean free path of electrons inside the metal electron gas. The Drude model, which can not take the repulsive interactions of electrons into account, then clearly begins to show its limits. In an intermediate length scale where a full quantum treatment is computationally prohibitive, the semiclassical hydrodynamic model, instrinsically non-local, has proven successful. Here we generalize the expression for the absorption volume density and the reciprocity theorem in the framework of this hydrodynamic model. We validate numerically these generalized theorems and show that using classical expressions instead leads to large discrepancies.

Nonlocality Induced Cherenkov Threshold

AbstractCherenkov radiation is generally believed to be threshold‐free in hyperbolic metamaterials owing to the extremely large photonic density of states in classical local framework. Although recent advances in nonlocal and quantum plasmonics extend our understanding of light‐matter interactions in metamaterials, how such effects influence Cherenkov radiation in hyperbolic metamaterials still remains unknown. Here, it is demonstrated that effects of nonlocality add a new degree of freedom to engineer Cherenkov thresholds in hyperbolic metamaterials. The interplay between finite structural dimensions and nonlocal nature of metallic electrons results in a nonzero Cherenkov threshold. Counterintuitively, such nonlocality‐induced Cherenkov threshold can be significantly smaller than the classically predicted one if the metamaterial is designed to work around the epsilon‐near‐zero frequency. This phenomenon is attributed to the excitation of longitudinal plasmon modes which are absent in classical electromagnetic framework. These findings apply to a general class of hyperbolic materials, including metallodielectric layered structures, nanorod/nanoribbon arrays, hyperbolic van der Waals crystals, etc.

Role of diffusive surface scattering in nonlocal plasmonics

The recent generalised nonlocal optical response (GNOR) theory for plasmonics is analysed, and its main input parameter, namely the complex hydrodynamic convection–diffusion constant, is quantified in terms of enhanced Landau damping due to diffusive surface scattering of electrons at the surface of the metal. GNOR has been successful in describing plasmon damping effects, in addition to the frequency shifts originating from induced-charge screening, through a phenomenological electron diffusion term implemented into the traditional hydrodynamic Drude model of nonlocal plasmonics. Nevertheless, its microscopic derivation and justification is still missing. Here we discuss how the inclusion of a diffusion-like term in standard hydrodynamics can serve as an efficient vehicle to describe Landau damping without resorting to computationally demanding quantum-mechanical calculations, and establish a direct link between this term and the Feibelman d parameter for the centroid of charge. Our approach provides a recipe to connect the phenomenological fundamental GNOR parameter to a frequency-dependent microscopic surface-response function. We therefore tackle one of the principal limitations of the model, and further elucidate its range of validity and limitations, thus facilitating its proper application in the framework of nonclassical plasmonics.

Monitoring strong coupling in nonlocal plasmonics with electron spectroscopies

Plasmon-exciton polaritons provide exciting possibilities to control light-matter interactions at the nanoscale by enabling closer investigation of quantum optical effects and facilitating novel technologies based, for instance, on Bose-Einstein condensation and polaritonic lasing. Nevertheless, observing and visualising polaritons is challenging, and traditional optical microscopy techniques often lead to ambiguities regarding the emergence and strength of the plasmon-exciton coupling. Electron microscopy offers a more robust means to study and verify the nature of plexcitons, but is still hindered by instrument limitations and resolution. A simple theoretical description of electron beam-excited plexcitons is therefore vital to complement ongoing experimental efforts. Here we apply analytic solutions for the electron-loss and photon-emission probabilities to evaluate plasmon-exciton coupling studied either with the recently adopted technique of electron energy-loss spectroscopy, or with the so-far unexplored in this context cathodoluminescence spectroscopy. Foreseeing the necessity to account for quantum corrections in the plasmonic response, we extend these solutions within the framework of general nonlocal hydrodynamic descriptions. As a specific example we study core-shell spherical emitter-molecule hybrids, going beyond the standard local-response approximation through the hydrodynamic Drude model for screening and the generalised nonlocal optical response theory for nonlocal damping. We show that electron microscopies are extremely powerful in describing the interaction of emitters with the otherwise weakly excited by optical means higher-order plasmonic multipoles, a response that survives when quantum-informed models are considered.

On the Non-Local Surface Plasmons’ Contribution to the Casimir Force between Graphene Sheets

Herein we demonstrate the dramatic effect of non-locality on the plasmons which contribute to the Casimir forces, with a graphene sandwich as a case study. The simplicity of this system allowed us to trace each contribution independently, as we observed that interband processes, although dominating the forces at short separations, are poorly accounted for in the framework of the Dirac cone approximation alone, and should be supplemented with other descriptions for energies higher than 2.5 eV. Finally, we proved that distances smaller than 200 nm, despite being extremely relevant to state-of-the-art measurements and nanotechnology applications, are inaccessible with closed-form response function calculations at present.

Edge-dependent reflection and inherited fine structure of higher-order plasmons in graphene nanoribbons

We investigate higher-order plasmons in graphene nanoribbons, and we present how electronic edge states and wave-function fine structure influence the graphene plasmons. Based on nearest-neighbor tight-binding calculations, we find that a standing-wave model based on nonlocal bulk plasmon dispersion is surprisingly accurate for armchair ribbons of widths even down to a few nanometers, and we determine the corresponding phase shift upon edge reflection and an effective ribbon width. Wider zigzag ribbons exhibit a similar phase shift, whereas the standing-wave model describes few-nanometer zigzag ribbons less satisfactorily, to a large extent because of their edge states. We directly confirm that also the larger broadening of plasmons for zigzag ribbons is due to their edge states. Furthermore, we report a prominent fine structure in the induced charges of the ribbon plasmons, which for armchair ribbons follows the electronic wave-function oscillations induced by intervalley coupling. Interestingly, the wave-function fine structure is also found in our analogous density-functional theory calculations, and both these and tight-binding numerical calculations are explained quite well with analytical Dirac theory for graphene ribbons.

Nonlocal Plasmonic Modes and Plasmonic Band Structures in Cylindrically Curved Graphene

In this paper, hydrodynamic model has been analytically solved to investigate the nonlocal plasmons in cylindrically curved graphene layers. Within the quasi-static approximation, the dispersion relations for both local and nonlocal cases have been derived; the nonlocal effect is found to shift the dispersion relations upwards. High-order azimuthal modes possess different cutoff frequencies due to such nonlocality, which may occur even in large-scale highly doped structures. In periodically doped cases, the nonlocal effect can modify the corresponding plasmonic band structures, i.e., moving the locations of the bandgaps. The periodicity has made the material more sensitive to the plasmon nonlocality. Our investigations may lead to more attentions to the nonlocal plasmonic responses in graphene which are important for graphene-plasmon-based photonic devices.

Nonlocal and Nonlinear Surface Plasmon Polaritons and Optical Spatial Solitons Induced by the Thermocapillary Effect.

We study the propagation of surface plasmon polaritons (SPPs) on a metal surface which hosts a thin film of a liquid dielectric. The Ohmic losses that are inherently present due to the coupling of SPPs to conductors' electron plasma, induce temperature gradients and fluid deformation driven by the thermocapillary effect, which lead to a nonlinear and nonlocal change of the effective dielectric constant. The latter extends beyond the regions of highest optical intensity and constitutes a novel thermally self-induced mechanism that affects the propagation of the SPPs. We derive the nonlinear and nonlocal Schrödinger equation that describes propagation of low intensity SPP beams, and show analytically and numerically that it supports a novel optical spatial soliton excitation.

Quantum plasmonics: longitudinal quantum plasmons in copper, gold, and silver

The propagation of plasmonic waves in various metallic quantum nanostructures has received considerable attention for its applications in technology. The quantum plasmonic properties of metallic nanostructures in the quantum size regime have been difficult to describe using an appropriate model. Here nonlocal quantum plasmons are investigated in the most important metals of copper, gold, and silver. The dispersion properties of these metals and the propagation of longitudinal quantum plasmons in the high photon energy regime are studied using a new model of nonlocal quantum dielectric permittivity. The epsilon-near-zero properties are investigated and the spectrum and the damping rate of the longitudinal quantum plasmons are obtained in these metals. The quantum plasmons’ wave function is shown for both the classical and quantum limits. It is shown that silver is the most appropriate for quantum metallic structures in the development of next-generation quantum optical and sensing technologies, due to its low intrinsic loss.

Effect of Temperature and Doping on Plasmon Excitations for an Encapsulated Double‐Layer Graphene Heterostructure

We perform a comprehensive analysis of the spectrum of graphene plasmons which arise when a pair of sheets are confined between conducting materials. The associated enhanced local fields may be employed in the manipulation of light on the nanoscale by adjusting the separation between the graphene layers, the energy band gap as well as the concentration of charge carriers in the conducting media surrounding the two‐dimensional (2D) layers. We present a theoretical formalism, based on the calculation of the surface response function, for determining the plasmon spectrum of an encapsulated pair of 2D layers and apply it to graphene. We solve the coupled equations involving the continuity of the electric potential and discontinuity of the electric field at the interfaces separating the constituents of the hybrid structure. We have compared the plasmon modes for encapsulated gapped and gapless graphene. The associated nonlocal graphene plasmon spectrum coupled to the “sandwich” system show a linear acoustic plasmon mode as well as a low‐frequency mode corresponding to in‐phase oscillations of the adjacent 2D charge densities. These calculations are relevant to the study of energy transfer via plasmon excitations when graphene is confined by a pair of thick conducting materials.

Optical signatures of nonlocal plasmons in graphene

We theoretically investigate under which conditions nonlocal plasmon response in monolayer graphene can be detected. To this purpose, we study optical scattering off graphene plasmon resonances coupled using a subwavelength dielectric grating. We compute the graphene conductivity using the Random Phase Approximation (RPA) obtaining a nonlocal conductivity and we calculate the optical scattering of the graphene-grating structure. We then compare this with the scattering amplitudes obtained if graphene is modeled by the local RPA conductivity commonly used in the literature. We find that the graphene plasmon wavelength calculated from the local model may deviate up to $20\%$ from the more accurate nonlocal model in the small-wavelength (large-$q$) regime. We also find substantial differences in the scattering amplitudes obtained from the two models. However, these differences in response are pronounced only for small grating periods and low temperatures compared to the Fermi temperature.

Nonlocal nonlinear plasmonics

Nonlocal nonlinear plasmonics

Plasmon excitations for encapsulated graphene

We have developed an analytical formulation to calculate the plasmon dispersion relation for a two-dimensional layer which is encapsulated within a narrow spatial gap between two bulk half-space plasmas. This is based on a solution of the inverse dielectric function integral equation within the random-phase approximation (RPA). We take into account the nonlocality of the plasmon dispersion relation for both gapped and gapless graphene as the sandwiched two-dimensional (2D) semiconductor plasma. The associated nonlocal graphene plasmon spectrum coupled to the ‘sandwich’ system is exhibited in density plots, which show a linear mode and a pair of depolarization modes shifted from the bulk plasma frequency.

Plasmon Excitations of Multi-layer Graphene on a Conducting Substrate.

We predict the existence of low-frequency nonlocal plasmons at the vacuum-surface interface of a superlattice of N graphene layers interacting with conducting substrate. We derive a dispersion function that incorporates the polarization function of both the graphene monolayers and the semi-infinite electron liquid at whose surface the electrons scatter specularly. We find a surface plasmon-polariton that is not damped by particle-hole excitations or the bulk modes and which separates below the continuum mini-band of bulk plasmon modes. The surface plasmon frequency of the hybrid structure always lies below , the surface plasmon frequency of the conducting substrate. The intensity of this mode depends on the distance of the graphene layers from the conductor’s surface, the energy band gap between valence and conduction bands of graphene monolayer and, most importantly, on the number of two-dimensional layers. For a sufficiently large number of layers the hybrid structure has no surface plasmon. The existence of plasmons with different dispersion relations indicates that quasiparticles with different group velocity may coexist for various ranges of wavelengths determined by the number of layers in the superlattice.

Nonlocal plasma spectrum of graphene interacting with a thick conductor

A mean-field theory formalism is employed to analyze the nonlocal plasmon dispersion relation of monolayer graphene which is Coulomb coupled to a thick conductor. We calculate numerically the undamped plasmon excitation spectrum for arbitrary wave number. For gapped graphene, both the low-frequency (acoustic) and high-frequency (surface) plasmons may lie within an undamped opening in the particle-hole region. Furthermore, we find undamped plasmon excitations in a region of frequency-wave vector space which has no counterpart for free-standing gapped graphene.

Quantum Effects of Nonlocal Plasmons in Epsilon-Near-Zero Properties of a Thin Gold Film Slab

Dispersion properties of metals and propagation of quantum plasmons in the high photon energy range are studied. The nonlocal dielectric permittivity of a metal is determined by the quantum plasma effects and is calculated by taking into account collisions between free charge carriers and the lattice. The properties of epsilon-near-zero material are investigated in a thin gold film slab. The spectrum and the damping rate of the quantum plasmons are obtained for a wide range of energies, and the electron’s wave function is calculated in both classical and quantum limits. It is shown that the quantum plasmons exist with a propagation length of 1–10 nm, which strongly depends on the electron energy. The propagation length is found to be much larger than the propagation length in the classical regime, where the former is comparable to the atomic radius and the average inter-particle distance. It is found that the spatial localization of the electron wave function is extended due to the quantum effects. It is also shown that the damping of electromagnetic waves in the high photon energy range decreases when the photon energy decreases which is opposite to the conclusions obtained from the classical Drude model in this range.