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

Curvature dependent onset of quantum tunneling in subnanometer gaps

The quantum tunneling in subnanometer gap sizes in gold dimers is studied in order to account for the dependency of the onset of quantum tunneling on the dimer's radius and accordingly the gap wall's curvature, realized in experiments. Several nanodimers both nanowires and nanospheres with various radii and gap sizes are modelled and simulated based on the quantum corrected model, determining the onset of the quantum tunneling. Results show that the onset of quantum tunneling is both dependent on the gap size as well as on the dimer's radius. As larger dimers result in larger effective conductivity volumes, the influence of the quantum tunneling begins in larger gap sizes in larger dimers.

An Optimized Schwarz Method for the Optical Response Model Discretized by HDG Method

An optimized Schwarz domain decomposition method (DDM) for solving the local optical response model (LORM) is proposed in this paper. We introduce a hybridizable discontinuous Galerkin (HDG) scheme for the discretization of such a model problem based on a triangular mesh of the computational domain. The discretized linear system of the HDG method on each subdomain is solved by a sparse direct solver. The solution of the interface linear system in the domain decomposition framework is accelerated by a Krylov subspace method. We study the spectral radius of the iteration matrix of the Schwarz method for the LORM problems, and thus propose an optimized parameter for the transmission condition, which is different from that for the classical electromagnetic problems. The numerical results show that the proposed method is effective.

Spatial nonlocality effect on the surface plasmon propagation in plasmonic nanospheres waveguide

Spatial nonlocality affects the plasmonic characteristics of nanostructures. We used the quasi-static hydrodynamic Drude model to obtain the surface plasmon excitation energies in various metallic nanosphere structures. The surface scattering and radiation damping rates were phenomenologically incorporated into this model. We demonstrate that spatial nonlocality increases the surface plasmon frequencies and total plasmon damping rates in a single nanosphere. This effect was amplified for small nanospheres and higher multipole excitation. In addition, we find that spatial nonlocality reduces the interaction energy between two nanospheres. We extended this model to a linear periodic chain of nanospheres. Then we obtain the dispersion relation of surface plasmon excitation energies using Bloch's theorem. We also show that spatial nonlocality decreases the group velocities and energy decay lengths of the propagating surface plasmon excitations. Finally, we demonstrated that the effect of spatial nonlocality is significant for very small nanospheres separated by short distances.

Particle Size-Dependent Onset of the Tunneling Regime in Ideal Dimers of Gold Nanospheres

We report on the nanoparticle-size-dependent onset of quantum tunneling of electrons across the subnanometer gaps in three different sizes (30, 50, and 80 nm) of highly uniform gold nanosphere (AuNS) dimers. For precision plasmonics, the gap distance is systematically controlled at the level of single C-C bonds via a series of alkanedithiol linkers (C₂-C₁₆). Parallax-corrected high-resolution transmission electron microscope (HRTEM) imaging and subsequent tomographic reconstruction are employed to resolve the nm to subnm interparticle gap distances in AuNS dimers. Single-particle scattering experiments on three different sizes of AuNS dimers reveal that for the larger dimers the onset of quantum tunneling regime occurs at larger gap distances: 0.96 ± 0.04 nm (C₆) for 80 nm, 0.83 ± 0.03 nm (C₅) for 50 nm, and 0.72 ± 0.02 nm (C₄) for 30 nm dimers. 2D nonlocal and quantum-corrected model (QCM) calculations qualitatively explain the physical origin for this experimental observation: the lower curvature of the larger particles leads to a higher tunneling current due to a larger effective conductivity volume in the gap. Our results have possible implications in scenarios where precise geometrical control over plasmonic properties is crucial such as in hybrid (molecule-metal) and/or quantum plasmonic devices. More importantly, this study constitutes the closest experimental results to the theory for a 3D sphere dimer system and offers a reference data set for comparison with theory/simulations.

Do We Really Need Quantum Mechanics to Describe Plasmonic Properties of Metal Nanostructures?

Optical properties of metal nanostructures are the basis of several scientific and technological applications. When the nanostructure characteristic size is of the order of few nm or less, it is generally accepted that only a description that explicitly describes electrons by quantum mechanics can reproduce faithfully its optical response. For example, the plasmon resonance shift upon shrinking the nanostructure size (red-shift for simple metals, blue-shift for d-metals such as gold and silver) is universally accepted to originate from the quantum nature of the system. Here we show instead that an atomistic approach based on classical physics, ωFQFμ (frequency dependent fluctuating charges and fluctuating dipoles), is able to reproduce all the typical "quantum" size effects, such as the sign and the magnitude of the plasmon shift, the progressive loss of the plasmon resonance for gold, the atomistically detailed features in the induced electron density, and the non local effects in the nanoparticle response. To support our findings, we compare the ωFQFμ results for Ag and Au with literature time-dependent DFT simulations, showing the capability of fully classical physics to reproduce these TDDFT results. Only electron tunneling between nanostructures emerges as a genuine quantum mechanical effect, that we had to include in the model by an ad hoc term.

Madelung Formalism for Electron Spill-Out in Nonlocal Nanoplasmonics

Current multiscale plasmonic systems pose a modeling challenge. Classical macroscopic theories fail to capture quantum effects in such systems, whereas quantum electrodynamics is impractical given the total size of the experimentally relevant systems, as the number of interactions is too large to be addressed one by one. To tackle the challenge, in this paper we propose to use the Madelung form of the hydrodynamic Drude model, in which the quantum effect electron spill-out is incorporated by describing the metal-dielectric interface using a super-Gaussian function. The results for a two-dimensional nanoplasmonic wedge are correlated to those from nonlocal full-wave numerical calculations based on a linearized hydrodynamic Drude model commonly employed in the literature, showing good qualitative agreement. Additionally, a conformal transformation perspective is provided to explain qualitatively the findings. The methodology described here may be applied to understand, both numerically and theoretically, the modular inclusions of additional quantum effects, such as electron spill-out and nonlocality, that cannot be incorporated seamlessly by using other approaches.

Orbital-Free Methods for Plasmonics: Linear Response

Plasmonic systems, such as metal nanoparticles, are widely used in different application areas, going from biology to photovoltaics.The modeling of the optical response of such systems is of fundamental importance to analyze their behavior and to design new systems with required properties.When the characteristic sizes/distances reach a few nanometers, non-local and spill-out effects become relevant and conventional classical electrodynamics models are no more appropriate. Methods based on the Time-Dependent Density-Functional Theory (TD-DFT) represent the current reference for the description of quantum effects. However, TD-DFT is based on knowledge of all occupied orbitals whose calculation is computationally prohibitive to model large plasmonic systems of interest for applications.On the other hand, methods based on the Orbital-Free (OF) formulation of TD-DFT, can scale linearly with the system size.In this Review, OF methods ranging from semiclassical models to the quantum hydrodynamic theory, will be derived from the linear response TD-DFT, so that the key approximations and properties of each method can be clearly highlighted. The accuracy of the various approximations will be then validated for the linear optical properties of jellium nanoparticles, the most relevant model system in plasmonics. OF methods can describe the collective excitations in plasmonic systems with great accuracy andwithout system-tuned parameters. The accuracy on these methods depends only on the accuracy on the (universal) kinetic energy functional of the ground-state electronic density. Current approximations and future development directions will be indicated.

Quantum surface effects in the electromagnetic coupling between a quantum emitter and a plasmonic nanoantenna: time-dependent density functional theory vs. semiclassical Feibelman approach

We use time-dependent density functional theory (TDDFT) within the jellium model to study the impact of quantum-mechanical effects on the self-interaction Green's function that governs the electromagnetic interaction between quantum emitters and plasmonic metallic nanoantennas. A semiclassical model based on the Feibelman parameters, which incorporates quantum surface-response corrections into an otherwise classical description, confirms surface-enabled Landau damping and the spill out of the induced charges as the dominant quantum mechanisms strongly affecting the nanoantenna-emitter interaction. These quantum effects produce a redshift and broadening of plasmonic resonances not present in classical theories that consider a local dielectric response of the metals. We show that the Feibelman approach correctly reproduces the nonlocal surface response obtained by full quantum TDDFT calculations for most nanoantenna-emitter configurations. However, when the emitter is located in very close proximity to the nanoantenna surface, we show that the standard Feibelman approach fails, requiring an implementation that explicitly accounts for the nonlocality of the surface response in the direction parallel to the surface. Our study thus provides a fundamental description of the electromagnetic coupling between plasmonic nanoantennas and quantum emitters at the nanoscale.

Tunable SERS Enhancement via Sub-nanometer Gap Metasurfaces

Raman sensing is a powerful technique for detecting chemical signatures, especially when combined with optical enhancement techniques such as using substrates containing plasmonic nanostructures. In this work, we successfully demonstrated surface-enhanced Raman spectroscopy (SERS) of two analytes adsorbed onto gold nanosphere metasurfaces with tunable subnanometer gap widths. These metasurfaces, which push the bounds of previously studied SERS nanostructure feature sizes, were fabricated with precise control of the intersphere gap width to within 1 nm for gaps close to and below 1 nm. Analyte Raman spectra were measured for samples for a range of gap widths, and the surface-affected signal enhancement was found to increase with decreasing gap width, as expected and corroborated via electromagnetic field modeling. Interestingly, an enhancement quenching effect was observed below gaps of around 1 nm. We believe this to be one of the few studies of gap-width-dependent SERS for the subnanometer range, and the results suggest the potential of such methods as a probe of subnanometer scale effects at the interface between plasmonic nanostructures. With further study, we believe that tunable sub-nanometer gap metasurfaces could be a useful tool for the study of nonlocal and quantum enhancement-quenching effects. This could aid the development of optimized Raman-based sensors for a variety of applications.

Efficient modal analysis of plasmonic nanoparticles: from retardation to nonclassical regimes

With recent developments in nanotechnologies, metal nanoparticles permeate a wide range of dimension scales, from light wavelength-scale domains down to a few nanometers approaching electronic scales. The electrodynamics at metal surfaces hosts a rich interplay between plasmon oscillations, retardation effects of light, and nonclassical (quantum) effects of electrons. Incorporating all these effects and modeling optical responses of nanoparticles generally rely on pure numerical methods, which are, however, disadvantageous in physical interpretations and computational speed. Herein, we establish a modal method that accurately predicts plasmon responses of metal nanoparticles, including both retardation and nonclassical corrections on an equal footing. The proposed method, based on electrostatic plasmon modes, is parameterized by a set of geometrically dependent factors, which, once computed, can be repeatedly used for same-shaped nanoparticles independent of size and material composition. The predictive accuracy of the method is examined for single nanoparticles, multi-scale plasmonic architectures-such as dimer structures with deep-nanometer gap-and geometrically deformed structures, with feature dimensions ranging from a few nanometers to hundreds of nanometers.

Towards realistic modeling of plasmonic nanostructures: a comparative study to determine the impact of optical effects on solar cell improvement

Plasmonic structures may improve cell performance in a variety of ways. More accurate determining of the optical influence, unlike ideal simulations, requires modeling closer to experimental cases. In this modeling and simulation, irregular nanostructures were chosen and divided into three groups and some modes. For each mode, different sizes of nanoparticles were randomly selected, which could result in pre-determined average particle size and standard deviation. By 3D finite-difference time-domain (3D-FDTD), the optical plasmonic properties of that mode in a solar cell structure were investigated when the nanostructure was added to the buffer/active layer of the organic solar cell. The far- and near-field results were used to compare the plasmonic behavior, relying on the material and geometry. By detailed simulations, Al and Ag nanostructure at the interface of the ZnO/active layer can improve organic solar cell performance optically, especially by the near-field effect. Unlike Au and relative Ag, the Al nanostructured sample showed less parasitic absorption loss.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s10825-021-01829-x.

Impact of Nonlocality on Group Delay and Reflective Behavior Near Surface Plasmon Resonances in Otto Structure

In this work, we study the effects of nonlocality on the optical response near surface plasmon resonance of the Otto structure, and such nonlocality is considered in the hydrodynamic model. Through analyzing the dispersion relations and optical response predicted by the Drude’s and hydrodynamic model in the system, we find that the nonlocal effect is sensitive to the large propagation wavevector, and there exists a critical incident angle and thickness. The critical point moves to the smaller value when the nonlocal effect is taken into account. Before this point, the absorption of the reflected light pulse enhances; however, the situation reverses after this point. In the region between the two different critical points in the frequency scan calculated from local and nonlocal theories, the group delay of the reflected light pulse shows opposite behaviors. These results are explained in terms of the pole and zero phenomenological model in complex frequency plane. Our work may contribute to the fundamental understanding of light–matter interactions at the nanoscale and in the design of optical devices.

Optomagnets in nonmagnetic plasmonic nanostructures

Using a simplified hydrodynamic model of the free electron gas of a metal, we theoretically investigate optically induced DC current loops in a plasmonic nanostructure. Such current loops originate from an optical rectification process relying on three electromotive forces, one of which arises from an optical spin-orbit interaction. The resulting static magnetic field is found to be maximum and dramatically confined at the corners of the plasmonic nanostructure, which reveals the ability of metallic discontinuities to concentrate and tailor static magnetic fields on the nanoscale. Plasmonics can thus generate and tune static magnetic fields and strong magnetic forces on the nanoscale, potentially impacting small scale magnetic tweezing and sensing as well as the generation of magneto-optical effects and spin waves.

Plasmonic modes in cylindrical nanoparticles and dimers

We present analytical expressions for the resonance frequencies of the plasmonic modes hosted in a cylindrical nanoparticle within the quasi-static approximation. Our theoretical model gives us access to both the longitudinally and transversally polarized dipolar modes for a metallic cylinder with an arbitrary aspect ratio, which allows us to capture the physics of both plasmonic nanodisks and nanowires. We also calculate quantum mechanical corrections to these resonance frequencies due to the spill-out effect, which is of relevance for cylinders with nanometric dimensions. We go on to consider the coupling of localized surface plasmons in a dimer of cylindrical nanoparticles, which leads to collective plasmonic excitations. We extend our theoretical formalism to construct an analytical model of the dimer, describing the evolution with the inter-nanoparticle separation of the resultant bright and dark collective modes. We comment on the renormalization of the coupled mode frequencies due to the spill-out effect, and discuss some methods of experimental detection.

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 Feibelmandparameter 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.

Terahertz and infrared nonlocality and field saturation in extreme-scale nanoslits

With advances in nanofabrication techniques, extreme-scale nanophotonic devices with critical gap dimensions of just 1-2 nm have been realized. The plasmonic response in these extreme-scale gaps is significantly affected by nonlocal electrodynamics, quenching field enhancement and blue-shifting the resonance with respect to a purely local behavior. The extreme mismatch in lengthscales, ranging from millimeter-long wavelengths to atomic-scale charge distributions, poses a daunting computational challenge. In this paper, we perform computations of a single nanoslit using the hybridizable discontinuous Galerkin method to solve Maxwell's equations augmented with the hydrodynamic model for the conduction-band electrons in noble metals. This method enables the efficient simulation of the slit while accounting for the nonlocal interactions between electrons and the incident light. We study the impact of gap width, film thickness and electron motion model on the plasmon resonances of the slit for two different frequency regimes: (1) terahertz frequencies, which lead to 1000-fold field amplitude enhancements that saturate as the gap shrinks; and (2) the near- and mid-infrared regime, where we show that narrow gaps and thick films cluster Fabry-Pérot (FP) resonances towards lower frequencies, derive a dispersion relation for the first FP resonance, in addition to observing that nonlocality boosts transmittance and reduces enhancement.

Unveiling the effect of electron tunneling on the plasmonic resonance of closely spaced gold particles

Recent experimental techniques enable the nanoparticles' (NPs) ensembles to be made with an angstrom-level of interparticle gap widths. The theoretical description of their optical properties becomes more challenging because of the nonnegligible quantum mechanism (QM) effects such as electron tunneling and nonlocal screening. To demonstrate the microscopic mechanism of QM effects and quantitatively elucidate their impact on a variety of surface plasmon resonance (SPR) models of gold particle oligomers, in this work we performed a theoretical study on closely spaced Au cluster dimers and NP oligomers by the first-principles approach, and the classical or quantum-corrected electromagnetic models (CEM or QCM), respectively. Through the first-principles calculation on a series of Au_N dimers with different interparticle distances d and N, we depicted the variation of the possibility of direct electron tunneling (DET) across the junction constructed by two nearest NPs with d and N, and found that it exactly follows an exponential decay and the decay rate linearly varies with 1/N. The impact of the QM effect on the SPR models excited along the dimer axis is much more profound than those perpendicular to the dimer axis. CEM/QCM calculations on strongly coupled NP dimers and symmetric and asymmetric trimers demonstrated the evolution of their optical properties with variable NP sizes, gap separations and light polarization, as well as the QM effect on the major SPR modes. The side-by-side comparison between the results from time-dependent density functional theory and CEM/QCM models sheds light on understanding the origin of a variety of SPR models of gold NP oligomers and the QM effect on those modes, and makes a connection between the calculations of small cluster and large NP oligomers.

Towards rational design and optimization of near-field enhancement and spectral tunability of hybrid core-shell plasmonic nanoprobes

In biology, sensing is a major driver of discovery. A principal challenge is to create a palette of probes that offer near single-molecule sensitivity and simultaneously enable multiplexed sensing and imaging in the “tissue-transparent” near-infrared region. Surface-enhanced Raman scattering and metal-enhanced fluorescence have shown substantial promise in addressing this need. Here, we theorize a rational design and optimization strategy to generate nanostructured probes that combine distinct plasmonic materials sandwiching a dielectric layer in a multilayer core shell configuration. The lower energy resonance peak in this multi-resonant construct is found to be highly tunable from visible to the near-IR region. Such a configuration also allows substantially higher near-field enhancement, compared to a classical core-shell nanoparticle that possesses a single metallic shell, by exploiting the differential coupling between the two core-shell interfaces. Combining such structures in a dimer configuration, which remains largely unexplored at this time, offers significant opportunities not only for near-field enhancement but also for multiplexed sensing via the (otherwise unavailable) higher order resonance modes. Together, these theoretical calculations open the door for employing such hybrid multi-layered structures, which combine facile spectral tunability with ultrahigh sensitivity, for biomolecular sensing.

Numerical Analysis of Nonlocal Optical Response of Metallic Nanoshells

Nonlocal and quantum effects play an important role in accurately modeling the optical response of nanometer-sized metallic nanoparticles. These effects cannot be described by conventional classical theories, as they neglect essential microscopic details. Quantum hydrodynamic theory (QHT) has emerged as an excellent tool to correctly predict the nonlocal and quantum effects by taking into account the spatial dependence of the charge density. In this article, we used a QHT to investigate the impact of nonlocality and electron spill-out on the plasmonic behavior of spherical Na and Au nanoshells. We adopted a self-consistent way to compute the equilibrium charge density. The results predicted by QHT were compared with those obtained with the local response approximation (LRA) and the Thomas–Fermi hydrodynamic theory (TFHT). We found that nonlocal effects have a strong impact on both the near- and far-field optical properties of nanoshells, in particular, for the antibonding resonant mode. We also investigated the optical response of these systems for different thicknesses of the shell, both for Na and Au metals.

Electromagnetic analysis of the lasing thresholds of hybrid plasmon modes of a silver tube nanolaser with active core and active shell

Results from the electromagnetic modeling of the threshold conditions of hybrid plasmon modes of a laser based on a silver nanotube with an active core and covered with an active shell are presented. We study the modes of such a nanolaser that have their emission wavelengths in the visible-light range. Our analysis uses the mathematically grounded approach called the lasing eigenvalue problem (LEP) for the set of the Maxwell equations and the boundary and radiation conditions. As we study the modes exactly at the threshold, there is no need to invoke nonlinear and quantum models of lasing. Instead, we consider a laser as an open plasmonic resonator equipped with an active region. This allows us to assume that at threshold the natural-mode frequency is real-valued, according to the situation where the losses, in the metal and for the radiation, are exactly balanced with the gain in the active region. Then the emission wavelength and the associated threshold gain can be viewed as parts of two-component eigenvalues, each corresponding to a certain mode. In the configuration considered, potentially there are three types of modes that can lase: the hybrid localized surface plasmon (HLSP) modes of the metal tube, the core modes, and the shell modes. The latter two types can be kept off the visible range in thin enough configurations. Keeping this in mind, we focus on the HLSP modes and study how their threshold gain values change with variations in the geometrical parameters of the nanotube, the core, and the shell. It is found that essentially a single-mode laser can be designed on the difference-type HLSP mode of the azimuth order m = 1, shining in the orange or red spectral region. Furthermore, the threshold values of gain for similar HLSP modes of order m = 2 and 3 can be several times lower, with emission in the violet or blue parts of the spectrum.

Optical properties of plasmonic core-shell nanomatryoshkas: a quantum hydrodynamic analysis

Plasmonic response of the metallic structure characterized by sub-nanometer dielectric gaps can be strongly affected by nonlocal or quantum effects. In this paper, we investigate these effects in spherical Na and Au nanomatryoshka structures with sub-nanometer core-shell separation. We use the state-of-the-art quantum hydrodynamic theory (QHT) to study both near-field and far-field optical properties of these systems: results are compared with the classical local response approximation (LRA), Thomas-Fermi hydrodynamic theory (TF-HT), and the reference time-dependent density functional theory (TD-DFT). We find that the results obtained using the QHT method are in a very good agreement with TD-DFT calculations, whereas other LRA and TF-HT significantly overestimate the field-enhancements. Thus, the QHT approach efficiently and accurately describes microscopic details of multiscale plasmonic systems whose sizes are computationally out-of-reach for a TD-DFT approach; here, we report results for Na and Au nanomatryoshka with a diameter of 60 nm.

The effect of nonlocal dielectric response on the surface-enhanced Raman and fluorescence spectra of molecular systems

We present a theoretical study on the influence of the nonlocal dielectric response on surface-enhanced resonant Raman scattering (SERRS) and fluorescence (SEF) spectra of a model molecule confined in the center of a Ag nanoparticle (NP) dimer. In the simulations, the nonlocal dielectric response caused by the electron-hole pair generation in Ag NPs was computed with the d-parameter theory, and the scattering spectra of a model molecule representing the commonly used fluorescent dye rhodamine 6G (R6G) were obtained by density-matrix calculations. The influence of the separation between Ag NP dimers on the damping rate and scattering spectra with and without the nonlocal response were systematically analyzed. The results show that the nonlocal dielectric response is very sensitive to the gap distance of the NP dimers, and it undergoes much faster decay with the increase of the separation than the radiative and energy transfer rates. The Raman and fluorescence peaks as simulated with the nonlocal dielectric response are relative weaker than that without the nonlocal effect for smaller NP separations because the extra decay rates of the nonlocal effect could reduce both the population of the excited state and the interband coherence between the ground and excited states. Our result also indicates that the nonlocal effect is more prominent on the SEF process than the SERRS process.

An eigenvalue approach to quantum plasmonics based on a self-consistent hydrodynamics method

Plasmonics has attracted much attention not only because it has useful properties such as strong field enhancement, but also because it reveals the quantum nature of matter. To handle quantum plasmonics effects, ab initio packages or empirical Feibelman d-parameters have been used to explore the quantum correction of plasmonic resonances. However, most of these methods are formulated within the quasi-static framework. The self-consistent hydrodynamics model offers a reliable approach to study quantum plasmonics because it can incorporate the quantum effect of the electron gas into classical electrodynamics in a consistent manner. Instead of the standard scattering method, we formulate the self-consistent hydrodynamics method as an eigenvalue problem to study quantum plasmonics with electrons and photons treated on the same footing. We find that the eigenvalue approach must involve a global operator, which originates from the energy functional of the electron gas. This manifests the intrinsic nonlocality of the response of quantum plasmonic resonances. Our model gives the analytical forms of quantum corrections to plasmonic modes, incorporating quantum electron spill-out effects and electrodynamical retardation. We apply our method to study the quantum surface plasmon polariton for a single flat interface.

Perfectly matched layers for nonlocal media with hydrodynamic-Drude description: a transformation optics approach

We develop a transformation optics theory for the nonlocal media in the hydrodynamic Drude model by generalizing the free-electron current density equation to a transformation invariant form. Applying the transformation optics theory, perfectly matched layers (PMLs) for the nonlocal media are theoretically formulated and implemented in frequency domain with finite element method. The nonlocal PMLs are shown to absorb outgoing surface and volume plasmons without inducing unphysical reflections. The effectiveness of the nonlocal PMLs is quantitatively demonstrated by the behaviors that the numerical errors continuously approach zero with increasing linear mesh density.

Optical Observation of Plasmonic Nonlocal Effects in a 2D Superlattice of Ultrasmall Gold Nanoparticles

The advances in recent nanofabrication techniques have facilitated explorations of metal structures into nanometer scales, where the traditional local-response Drude model with hard-wall boundary conditions fails to accurately describe their optical responses. The emerging nonlocal effects in single ultrasmall silver nanoparticles have been experimentally observed in single-particle spectroscopy enabled by the unprecedented high spatial resolution of electron energy loss spectroscopy (EELS). However, the unambiguous optical observation of such new effects in gold nanoparticles has yet not been reported, due to the extremely weak scattering and the obscuring fingerprint of strong interband transitions. Here we present a nanosystem, a superlattice monolayer formed by sub-10 nm gold nanoparticles. Plasmon resonances are spectrally well-separated from interband transitions, while exhibiting clearly distinguishable blueshifts compared to predictions by the classical local-response model. Our far-field spectroscopy was performed by a standard optical transmission and reflection setup, and the results agreed excellently with the hydrodynamic nonlocal model, opening a simple and widely accessible way for addressing quantum effects in nanoplasmonic systems.

Third-harmonic generation from gold nanowires of rough surface considering classical nonlocal effect

We demonstrate that the nonlocal dielectric response of metal, comparing with the traditional local model, will significantly boost the third-order harmonic generation (THG) from gold nanowires of rough surface by a factor of several orders of magnitude. The enhancement is probably due to the penetrated field into the fine nanostructures on the metal surface in nonlocal model. The anisotropy THG efficiency versus the angle of incidence is also demonstrated due to the inhomogeneous surface morphology. The possible ways to verify the nonlocal effect to the THG are demonstrated. The results have a general significance in explaining the experimental observations.

Enhancement of two-photon photoluminescence and SERS for low-coverage gold films

Electromagnetic field enhancement (FE) effects occurring in thin gold films 3-12-nm are investigated with two-photon photoluminescence (TPL) and Raman scanning optical microscopies. The samples are characterized using scanning electron microscopy images and linear optical spectroscopy. TPL images exhibit a strong increase in the level of TPL signals for films thicknesses 3-8-nm, near the percolation threshold. For some thicknesses, TPL measurements reveal super-cubic dependences on the incident power. We ascribe this feature to the occurrence of very strongly localized and enhanced electromagnetic fields due to multiple light scattering in random nanostructures that might eventually lead to white-light generation. Raman images exhibit increasing Raman signals when decreasing the film thickness from 12 to 6-nm and decreasing signal for the 3-nm-film. This feature correlates with the TPL observations indicating that highest FE is to be expected near the percolation threshold.

Robustness of the far-field response of nonlocal plasmonic ensembles

Contrary to classical predictions, the optical response of few-nm plasmonic particles depends on particle size due to effects such as nonlocality and electron spill-out. Ensembles of such nanoparticles are therefore expected to exhibit a nonclassical inhomogeneous spectral broadening due to size distribution. For a normal distribution of free-electron nanoparticles, and within the simple nonlocal hydrodynamic Drude model, both the nonlocal blueshift and the plasmon linewidth are shown to be considerably affected by ensemble averaging. Size-variance effects tend however to conceal nonlocality to a lesser extent when the homogeneous size-dependent broadening of individual nanoparticles is taken into account, either through a local size-dependent damping model or through the Generalized Nonlocal Optical Response theory. The role of ensemble averaging is further explored in realistic distributions of isolated or weakly-interacting noble-metal nanoparticles, as encountered in experiments, while an analytical expression to evaluate the importance of inhomogeneous broadening through measurable quantities is developed. Our findings are independent of the specific nonclassical theory used, thus providing important insight into a large range of experiments on nanoscale and quantum plasmonics.

Quantum mechanical effects in plasmonic structures with subnanometre gaps

Metallic structures with nanogap features have proven highly effective as building blocks for plasmonic systems, as they can provide a wide tuning range of operating frequencies and large near-field enhancements. Recent work has shown that quantum mechanical effects such as electron tunnelling and nonlocal screening become important as the gap distances approach the subnanometre length-scale. Such quantum effects challenge the classical picture of nanogap plasmons and have stimulated a number of theoretical and experimental studies. This review outlines the findings of many groups into quantum mechanical effects in nanogap plasmons, and discusses outstanding challenges and future directions.

Classification of scalar and dyadic nonlocal optical response models

Nonlocal optical response is one of the emerging effects on the nanoscale for particles made of metals or doped semiconductors. Here we classify and compare both scalar and tensorial nonlocal response models. In the latter case the nonlocality can stem from either the longitudinal response, the transverse response, or both. In phenomenological scalar models the nonlocal response is described as a smearing out of the commonly assumed infinitely localized response, as characterized by a distribution with a finite width. Here we calculate explicitly whether and how tensorial models, such as the hydrodynamic Drude model and generalized nonlocal optical response theory, follow this phenomenological description. We find considerable differences, for example that nonlocal response functions, in contrast to simple distributions, assume negative and complex values. Moreover, nonlocal response regularizes some but not all diverging optical near fields. We identify the scalar model that comes closest to the hydrodynamic model. Interestingly, for the hydrodynamic Drude model we find that actually only one third (1/3) of the free-electron response is smeared out nonlocally. In that sense, nonlocal response is stronger for transverse and scalar nonlocal response models, where the smeared-out fractions are 2/3 and 3/3, respectively. The latter two models seem to predict novel plasmonic resonances also below the plasma frequency, in contrast to the hydrodynamic model that predicts standing pressure waves only above the plasma frequency.

Electronic spill-out induced spectral broadening in quantum hydrodynamic nanoplasmonics

The hydrodynamic theory is a powerful tool to study the nonlocal effects in metallic nanostructures that are too small to obey classical electrodynamics while still too large to be handled with a full quantum-mechanical theory. The existing hydrodynamic model can give accurate quantitative predictions for the plasmonic resonance shifts in metallic nanoplasmonics, yet is not able to predict the spectral width which is usually taken as a pre-set value instead. By taking account the fact that due to electron density spill-out from a surface, the Coulomb interaction screening is less efficient close the surface thus leads to a higher electron-electron scattering rate in this paper, we study how the electron-density-related damping rate induced by such Coulomb interaction will affect the plasmonic spectral broadening. We perform the simulation on a Na nanowire, which shows that the absorption spectra width is wider when the size of the nanowire becomes smaller. This result is consistent well with the reported experiment. Therefore, our theoretical model extends the existing hydrodynamic model and can provide much more quantum insight about nonlocal effects in metallic nanostructures.

Plane wave scattering from a plasmonic nanowire-film system with the inclusion of non-local effects

In this paper we present a theoretical analysis of the electromagnetic response of a plasmonic nanowire-film system. The analytical solution accounts for both the dispersive as well as non-local nature of the plasmonic media. The physical structure comprises of a plasmonic nanowire made of a plasmonic metal such as gold or silver placed over a plasmonic film of the same material. Such a nanostructure exhibits a spectrum that is extremely sensitive to various geometric parameters such as spacer thickness and nanowire radius, which makes it favorable for various sensing applications. The non-locality of the plasmonic medium, which can be captured using the hydrodynamic model, significantly affects the resonant wavelength of this system for structures of small dimensions (~ less than 5 nm gap between the nanowire and the film). We present an analytical method that can be used to predict the effect of non-locality on the resonances of the system. To validate the analytical method, we also report a comparison of our analytical solution with a numerical Finite Difference Time Domain analysis (FDTD) of the same structure with the plasmonic medium being treated as local in nature.

Numerical tool to take nonlocal effects into account in metallo-dielectric multilayers

We provide a numerical tool to quantitatively study the impact of nonlocality arising from free electrons in metals on the optical properties of metallo-dielectric multilayers. We found that scattering matrices are particularly well suited to take into account the electron response through the application of the hydrodynamic model. Though effects due to nonlocality are, in general, quite small, they, nevertheless, can be important for very thin (typically below 10 nm) metallic layers, as in those used in structures characterized by exotic dispersion curves. Such structures include those with a negative refractive index, hyperbolic metamaterials, and near-zero index materials. Higher wave vectors mean larger nonlocal effects, so that it is not surprising that subwavelength imaging capabilities of hyperbolic metamaterials are found to be sensitive to nonlocal effects. We find in all cases that the inclusion of nonlocal effects leads to at least a 5% higher transmission through the considered structure.

Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics

The standard hydrodynamic Drude model with hard-wall boundary conditions can give accurate quantitative predictions for the optical response of noble-metal nanoparticles. However, it is less accurate for other metallic nanosystems, where surface effects due to electron density spill-out in free space cannot be neglected. Here we address the fundamental question whether the description of surface effects in plasmonics necessarily requires a fully quantum-mechanical ab initio approach. We present a self-consistent hydrodynamic model (SC-HDM), where both the ground state and the excited state properties of an inhomogeneous electron gas can be determined. With this method we are able to explain the size-dependent surface resonance shifts of Na and Ag nanowires and nanospheres. The results we obtain are in good agreement with experiments and more advanced quantum methods. The SC-HDM gives accurate results with modest computational effort, and can be applied to arbitrary nanoplasmonic systems of much larger sizes than accessible with ab initio methods.

Anisotropy Effects on the Plasmonic Response of Nanoparticle Dimers

We present an ab initio study of the anisotropy and atomic relaxation effects on the optical properties of nanoparticle dimers. Special emphasis is placed on the hybridization process of localized surface plasmons, plasmon-mediated photoinduced currents, and electric-field enhancement in the dimer junction. We show that there is a critical range of separations between the clusters (0.1-0.5 nm) in which the detailed atomic structure in the junction and the relative orientation of the nanoparticles have to be considered to obtain quantitative predictions for realistic nanoplasmonic devices. It is worth noting that this regime is characterized by the emergence of electron tunneling as a response to the driven electromagnetic field. The orientation of the particles not only modifies the attainable electric field enhancement but can lead to qualitative changes in the optical absorption spectrum of the system.

Nonlocal optical response in metallic nanostructures

This review provides a broad overview of the studies and effects of nonlocal response in metallic nanostructures. In particular, we thoroughly present the nonlocal hydrodynamic model and the recently introduced generalized nonlocal optical response (GNOR) model. The influence of nonlocal response on plasmonic excitations is studied in key metallic geometries, such as spheres and dimers, and we derive new consequences due to the GNOR model. Finally, we propose several trajectories for future work on nonlocal response, including experimental setups that may unveil further effects of nonlocal response.

The Coupling between Gold or Silver Nanocubes in Their Homo-Dimers: A New Coupling Mechanism at Short Separation Distances

Using the DDA method, we investigated the near-field coupling between two excited Au or Ag 42 nm nanocubes in a face-to-face dimer configuration at small separation distances where the exponential coupling behavior distinctly changes. This could be due to the failure of the dipole approximation at short distances or a change in the electromagnetic field distribution between the adjacent monomers. A detailed calculation of the plasmonic field distribution strongly suggests that the latter mechanism is responsible for the failure of the expected exponential coupling behavior at small separation distances. The results suggest that the observed optical properties of the pair of Au or Ag nanocubes separated by distances larger than 6 nm, result from the electromagnetic coupling between the oscillating dipoles at the corners of the adjacent facets of the nanocubes. At separations smaller than 6 nm, the distribution of the plasmonic dipoles along both the facets and the corners of the adjacent monomers control the plasmonic spectra and the distance dependent optical properties of the dimer.

Antiboding and bonding lasing modes with low gain threshold in nonlocal metallic nanoshell

Based on a full-wave nonlocal Mie theory, we establish the spaser generation condition for compact plasmonic nanolasers in the long-wavelength limit for dielectric-metal core-shell nanoparticles. We found that there exist two lasing states arising from the hybridized antibonding and bonding modes for this coated nanolaser. By varying the surrounding medium and the gain materials, we can achieve low gain threshold for each mode with flexible radii ratios on the purpose of realistic easy fabrication. Numerical results show that nonlocal effects have different influences on the required gain threshold and gain refractive index of these two lasing modes, which may be of great importance in the design of such kind of ultrasmall nanoparticle lasers.

Quantum effects in the optical response of extended plasmonic gaps: validation of the quantum corrected model in core-shell nanomatryushkas

Electron tunneling through narrow gaps between metal nanoparticles can strongly affect the plasmonic response of the hybrid nanostructure. Although quantum mechanical in nature, this effect can be properly taken into account within a classical framework of Maxwell equations using the so-called Quantum Corrected Model (QCM). We extend previous studies on spherical cluster and cylindrical nanowire dimers where the tunneling current occurs in the extremely localized gap regions, and perform quantum mechanical time dependent density functional theory (TDDFT) calculations of the plasmonic response of cylindrical core-shell nanoparticles (nanomatryushkas). In this axially symmetric situation, the tunneling region extends over the entire gap between the metal core and the metallic shell. For core-shell separations below 0.5 nm, the standard classical calculations fail to describe the plasmonic response of the cylindrical nanomatryushka, while the QCM can reproduce the quantum results. Using the QCM we also retrieve the quantum results for the absorption cross section of the spherical nanomatryushka calculated by V. Kulkarni et al. [Nano Lett. 13, 5873 (2013)]. The comparison between the model and the full quantum calculations establishes the applicability of the QCM for a wider range of geometries that hold tunneling gaps.

Nonlocal study of ultimate plasmon hybridization

Within our recently proposed generalized nonlocal optical response (GNOR) model, where nonlocal response is included by taking into account both convective and diffusive currents of the conduction electrons, we revisit the fundamental problem of an optically excited plasmonic dimer. We consider the transition from separated dimers via touching dimers to finally overlapping dimers. In particular, we focus on the touching case, showing a fundamental limit on the hybridization of the bonding plasmon modes due to nonlocality. Using transformation optics, we determine a simple analytical equation for the resonance energies.

A classical treatment of optical tunneling in plasmonic gaps: extending the quantum corrected model to practical situations

The optical response of plasmonic nanogaps is challenging to address when the separation between the two nanoparticles forming the gap is reduced to a few nanometers or even subnanometer distances. We have compared results of the plasmon response within different levels of approximation, and identified a classical local regime, a nonlocal regime and a quantum regime of interaction. For separations of a few Ångstroms, in the quantum regime, optical tunneling can occur, strongly modifying the optics of the nanogap. We have considered a classical effective model, so called Quantum Corrected Model (QCM), that has been introduced to correctly describe the main features of optical transport in plasmonic nanogaps. The basics of this model are explained in detail, and its implementation is extended to include nonlocal effects and address practical situations involving different materials and temperatures of operation.

van der Waals interactions at the nanoscale: the effects of nonlocality

Calculated using classical electromagnetism, the van der Waals force increases without limit as two surfaces approach. In reality, the force saturates because the electrons cannot respond to fields of very short wavelength: polarization charges are always smeared out to some degree and in consequence the response is nonlocal. Nonlocality also plays an important role in the optical spectrum and distribution of the modes but introduces complexity into calculations, hindering an analytical solution for interactions at the nanometer scale. Here, taking as an example the case of two touching nanospheres, we show for the first time, to our knowledge, that nonlocality in 3D plasmonic systems can be accurately analyzed using the transformation optics approach. The effects of nonlocality are found to dramatically weaken the field enhancement between the spheres and hence the van der Waals interaction and to modify the spectral shifts of plasmon modes.

Nonlocal effects: relevance for the spontaneous emission rates of quantum emitters coupled to plasmonic structures

The spontaneous emission rate of dipole emitters close to plasmonic dimers are theoretically studied within a nonlocal hydrodynamic model. A nonlocal model has to be used since quantum emitters in the immediate environment of a metallic nanoparticle probe its electronic structure. Compared to local calculations, the emission rate is significantly reduced. The influence is mostly pronounced if the emitter is located close to sharp edges. We suggest to use quantum emitters to test nonlocal effects in experimentally feasible configurations.

Quantum mechanical origin of the plasmon: from molecular systems to nanoparticles

The surface plasmon resonance (SPR) of noble metal nanoparticles is reviewed in terms of both classical and quantum mechanical approaches. The collective oscillation of the free electrons responsible for the plasmon is well described using classical electromagnetic theory for large systems (from about 10 to 100 nm). In cases where quantum effects are important, this theory fails and first principle approaches like time-dependent density functional theory (TDDFT) must be used. In this paper, we give an account of the current understanding of the quantum mechanical origin of plasmon resonances. We provide some insight into how the discrete absorption spectrum of small noble metal clusters evolves into a strong plasmon peak with increasing particle size. The collective character of the plasmon is described in terms of the constructive addition of single-particle excitations. As the system size increases, the number of single-particle excitations increases as well. A configuration interaction (CI) approach can be applied to describe the optical properties of particles of all shapes and sizes, providing a consistent definition of plasmon resonances. Finally, we expand our analysis to thiolate-protected nanoparticles and analyze the effects of ligands on the plasmon.