Hydrodynamic Drude Model Research Articles

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.

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.

Chern invariants of topological continua: A self-consistent nonlocal hydrodynamic model

Topological systems are characterized by integer Chern invariants. In a continuous photonic system characterized by a local Drude model, the material response is ill-behaved at large wavenumbers, leading to noninteger Chern invariants and ambiguity in the existence of topological edge modes. This problem has been solved previously by introducing an ad hoc material model including a spatial cutoff material response, which leads to a finite Brillouin zone and integer invariants. In this work, we calculate Chern numbers in magnetized continuous plasma systems by considering the effect of nonlocality using a hydrodynamic (HD) Drude model. Then we argue that this model presents several advantages compared with the previous models, e.g., introducing physical response at large wavenumbers and integer Chern invariants with sum to zero without the need for an interpolated material response. Therefore the HD model forms a complete and self-consistent model, which resolves the Chern number issues in topological photonic continua.

A posteriori error estimates for finite element discretizations of time-harmonic Maxwell’s equations coupled with a non-local hydrodynamic Drude model

We consider finite element discretizations of Maxwell’s equations coupled with a non-local hydrodynamic Drude model that accurately accounts for electron motions in metallic nanostructures. Specifically, we focus on a posteriori error estimation and mesh adaptivity, which is of particular interest since the electromagnetic field usually exhibits strongly localized features near the interface between metals and their surrounding media. We propose a novel residual-based error estimator that is shown to be reliable and efficient. We also present a set of numerical examples where the estimator drives a mesh adaptive process. These examples highlight the quality of the proposed estimator, and the potential computational savings offered by mesh adaptivity.

Optical second harmonic generation from plasmonic nanoshells using the nonlocal hydrodynamic model

In this paper, we investigate the linear and nonlinear optical response of spherical plasmonic nanoshells within the nonlocal hydrodynamical model for conduction electrons. The optical absorption and surface second-harmonic generation (SHG) associated with metallic core-shell nanoparticles is studied on the basis of the hydrodynamic Drude model. The influence of nonlocality on the optical SHG is investigated. We show that nonlocal effects can be applied to boost SHG from plasmonic nanoshells. It is found that the surface SHG radiation from the surface of nanoshells can be achieved by symmetry-breaking and is sensitive to nonlocality of the electron response. It is also shown that nonlocal effects lead to enhancement and shift of the resonant wavelengths of the optical absorption and SHG in metal nanoshells. Keywords: Optical absorption, Second harmonic generation, plasmonic nanoshell, nonlocal effects.

Second harmonic generation in metasurfaces with multipole resonant coupling

Abstract We report on the numerical demonstration of enhanced second harmonic generation (SHG) originating from collective resonances in plasmonic nanoparticle arrays. The nonlinear optical response of the metal nanoparticles is modeled by employing a hydrodynamic nonlinear Drude model implemented into Finite-Difference Time-Domain (FDTD) simulations, and effective polarizabilities of nanoparticle multipoles in the lattice are analytically calculated at the fundamental wavelength by using a coupled dipole–quadrupole approximation. Excitation of narrow collective resonances in nanoparticle arrays with electric quadrupole (EQ) and magnetic dipole (MD) resonant coupling leads to strong linear resonance enhancement. In this work, we analyze SHG in the vicinity of the lattice resonance corresponding to different nanoparticle multipoles and explore SHG efficiency by varying the lattice periods. Coupling of electric quadrupole and magnetic dipole in the nanoparticle lattice indicates symmetry breaking and the possibility of enhanced SHG under these conditions. By varying the structure parameters, we can change the strength of electric dipole (ED), EQ, and MD polarizabilities, which can be used to control the linewidth and magnitude of SHG emission in plasmonic lattices. Engineering of lattice resonances and associated magnetic dipole resonant excitations can be used for spectrally narrow nonlinear response as the SHG can be enhanced and controlled by higher multipole excitations and their lattice resonances. We show that both ED and EQ–MD lattice coupling contribute to SHG, but the presence of strong EQ–MD coupling is important for spectrally narrow SHG and, in our structure, excitation of narrow higher-order multipole lattice resonances results in five times enhancement.

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.

Bridging the hydrodynamic Drude model and local transformation optics theory

The recent ability of plasmonic nanostructures to probe subnanometer and even atomic scales demands theories that can account for the nonlocal dynamics of the electron gas. The hydrodynamic Drude model (HDM) captures much of the microscopic dynamics of the quantum mechanical effects when additional boundary conditions are considered. Here, we revisit the HDM under the Madelung formalism to reexpress its coupled system of equations as a single nonlinear Schr\"odinger equation in order to have a natural quantum mechanical description of plasmonics. Specifically, we study the response of two overlapping nanowires with this formalism. We ensure that an proposed frame concurs with classical electrodynamics when the local response approximation holds in the plasmonic system by finding the correction needed.

Plasmon enhanced second harmonic generation by periodic arrays of triangular nanoholes coupled to quantum emitters

Optical properties of periodic arrays of nanoholes of a triangular shape with experimentally realizable parameters are examined in both linear and nonlinear regimes. By utilizing a fully vectorial three-dimensional approach based on the nonlinear hydrodynamic Drude model describing metal coupled to Maxwell's equations and Bloch equations for molecular emitters, we analyze linear transmission, reflection, and nonlinear power spectra. Rigorous numerical calculations demonstrating second and third harmonic generation by the triangular hole arrays are performed. It is shown that both the Coulomb interaction of conduction electrons and the convective term contribute on equal footing to the nonlinear response of metal. It is demonstrated that the energy conversion efficiency in the second harmonic process is the highest when the system is pumped at the localized surface plasmon resonance. When molecular emitters are placed on a surface of the hole array line shapes, the second harmonic signal exhibits three peaks corresponding to second harmonics of the localized surface plasmon mode and upper and lower polaritonic states.

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.

Absence of unidirectionally propagating surface plasmon-polaritons at nonreciprocal metal-dielectric interfaces

In the presence of an external magnetic field, the surface plasmon polariton that exists at the metal-dielectric interface is believed to support a unidirectional frequency range near the surface plasmon frequency, where the surface plasmon polariton propagates along one but not the opposite direction. Recent works have pointed to some of the paradoxical consequences of such a unidirectional range, including in particular the violation of the time-bandwidth product constraint that should otherwise apply in general in static systems. Here we show that such a unidirectional frequency range is nonphysical using both a general thermodynamic argument and a detailed calculation based on a nonlocal hydrodynamic Drude model for the metal permittivity. Our calculation reveals that the surface plasmon-polariton at metal-dielectric interfaces remains bidirectional for all frequencies.

Influence of the Nonlocal Effect on the Optical Properties of Nonspherical Plasmonic Semiconductor Nanoparticles

Noble metals are commonly used as plasmon materials because of their high density of free electrons, but semiconductor materials are also becoming of interesting in this field because its electron density can be varied by doping. Metal nitrides can be an alternative to noble metals because of their low absorption loss and high electron density. Among others, TiN and ZrN seem to be most suitable as alternative plasmonic materials because their optical properties are dominated by conduction electrons near the plasmon frequency. There is the flame spray pyrolysis process, which is currently developed to produce such kind of nanoparticles. In this paper, based on an extension of the discrete sources method, the effect of the hydrodynamic Drude model of the quantum nonlocal effect on the optical characteristics of semiconductor nanoparticles is analyzed. The influence of accounting for the nonlocal effect (NLE) on the optical properties under spherical particles deformation has been investigated. It has been shown that accounting for the NLE leads to a plasmon resonance blue shift and a damping similar to noble metals. It was found that smaller particles demonstrate larger NLE influence than larger ones. Besides, the influence of polarization on the local and nonlocal responses of 3D nonspherical semiconductor particles has been investigated as well. Using simulation accounting for the nonlocal effect, it is shown that the extinction of a nonspherical ZrN particles exceeds that of a gold particle.

Modal analysis for nanoplasmonics with nonlocal material properties

Plasmonic devices with feature sizes of a few nanometers exhibit effects which can be described by the nonlocal hydrodynamic Drude model. We demonstrate how to exploit contour integral methods for computing eigenfrequencies and resonant states of such systems. We propose an approach for deriving the modal expansion of relevant physical observables. We use the methods to perform a modal analysis for a metal nanowire. All complex eigenfrequencies in a large frequency range and the corresponding resonant states are computed. We identify those resonant states which are relevant for the extinction cross section of the nanowire.

Hydrodynamic acoustic plasmon resonances in semiconductor nanowires and their dimers

The hydrodynamic Drude model known from metal plasmonics also applies to semiconductor structures of sizes in between single-particle quantum confinement and bulk. But contrary to metals, for semiconductors two or more types of plasma may have to be taken into account in order to properly describe their plasmonic properties. In this combined analytical and computational study, we explore predictions of the recently proposed two-fluid hydrodynamic Drude model for the optical properties of plasmonic semiconductor nanowires, in particular for thermally excited InSb nanowires. We focus on the low-frequency acoustic surface and bulk plasmon resonances that are unique fingerprints for this model and are yet to be observed. We identify these resonances in spectra for single nanowires based on analytical calculations, and they are in complete agreement with our numerical implementation of the model. For dimers of nanowires we predict substantial increase of the extinction cross section and field enhancement of the acoustic localized surface plasmon resonance, which makes its observation in dimers more likely.

The numerical scheme of the discrete sources method to analyze 3D plasmonic nanostructures accounting for the non-local effect

We present an extension of the discrete sources method for solving the time-harmonic Maxwell equations accounting for the non-local effect. The developed numerical scheme coupling the non-local hydrodynamic Drude model and the generalized non-local optical response model is described in detail. Numerical results computed for a single non-spherical particle and a linear dimer are presented. Our results demonstrate that the discrete sources method can be efficiently used to model the non-local effect in a wide range of 3D scattering problems.

Circular Dichroism in Nanoparticle Helices as a Template for Assessing Quantum-Informed Models in Plasmonics

As characteristic lengths in plasmonics rapidly approach the sub-nm regime, quantum-informed models that can capture those aspects of the quantum nature of the electron gas that are not accessible by the standard approximations of classical electrodynamics, or even go beyond the free-electron description, become increasingly more important. Here we propose a template for comparing and validating the predictions of such models, through the circular dichroism signal of a metallic nanoparticle helix. For illustration purposes, we compare three widely used models, each dominant at different nanoparticle separations and governed by its own physical mechanism, namely the hydrodynamic Drude model, the generalised nonlocal optical response theory, and the quantum-corrected model for tunnelling. Our calculations show that indeed, each case is characterised by a fundamentally distinctive response, always dissimilar to the predictions of the local optical response approximation of classical electrodynamics, dominated by a model-sensitive absorptive double-peak feature. In circular dichroism spectra, the striking differences between models manifest themselves as easily traceable sign changes rather than neighbouring absorption peaks, thus overcoming experimental resolution limitations and enabling efficient evaluation of the relevance, validity, and range of applicability of quantum-informed theories for extreme-nanoscale plasmonics.

Exciton behavior under the influence of metal nanoparticle near fields: Significance of nonlocal effects

We analytically characterize the influence of a neighboring metal nanoparticle (MNP) on the behavioral trends of a quantum dot (QD) using a generalized nonlocal optical response (GNOR) method based approach, taking the MNP distance dependent modifications to the QD population relaxation and dephasing rates into account. The GNOR model is a recent generalization and an extension of the hydrodynamic Drude model (HDM), which goes beyond HDM by taking into account both the convection current and electron diffusion in the MNPs. It allows unified theoretical explanation of some experimentally observed plasmonic phenomena which otherwise would require ab initio analysis as the conventional local response approximation (LRA) fails to account for them. For example, it has been demonstrated in literature that the GNOR model captures size dependent resonance shifts of small MNPs which are unrevealed by the conventional LRA based methods, and it has proven to yield results displaying better agreement with the experimental observations for plasmonic experiments. Attempts to incorporate MNP nonlocal effects in the analytical characterization of vicinal excitons found in literature utilize the phenomenological hydrodynamic model and assume the absence of MNP interband effects. Moreover, they are only applicable to narrow parameter regions. In this paper we present a complete analytical characterization which overcomes these drawbacks and lends to the perusal of the system over wide continua of various parameters, enabling us to get an elevated view at a much lesser level of complexity compared to the conventional LRA based numerical methods or the conventional ab initio methods of accounting for the nonlocal effects. Our proposed GNOR based model predicts strong modifications to various QD properties such as population difference, absorption, MNP induced shifts to excitonic energy and F\orster enhanced broadening, coherent plasmonic field enhancement, and quantum state purity, compared to the conventional LRA based predictions. Such modifications are prominent with small MNP radii, high QD dipole moments, small detunings (of the coherent external illumination from the bare excitonic resonance), and near parameter regions exhibiting plasmonic meta resonance (PMR)-like behavior. Moreover, our complete analytical characterization enables optimization of the large system parameter space for different applications, a luxury not fully offered by the methods currently available in literature.

Analysis of the scattering properties of 3D non-spherical plasmonic nanoparticles accounting for non-local effects

Based on an extension of the discrete sources method (DSM) our newly developed computer model enables us to analyze polarised light scattering by 3D non-spherical particles accounting for non-local effects (NLE) for plasmonic nanoparticles. Using this theory, computer simulations for spheroid particles of different shape have been conducted. The influence of the NLE both considering the Hydrodynamical Drude model and the generalised non-local optical response on the spectral scattering properties has been analyzed. It has been found that there is a strong influence of particle shape and polarisation of the incident radiation on the scattering properties of a plasmonic particle. Additionally, it has been demonstrated that NLE for non-spherical 3D particles leads to a decrease in intensity of the corresponding localised surface plasmon resonance and provides a blue shift of the plasmon resonance for the non-spherical particles.

Two-fluid hydrodynamic model for semiconductors

The hydrodynamic Drude model (HDM) has been successful in describing the optical properties of metallic nanostructures, but for semiconductors where several different kinds of charge carriers are present, an extended theory is required. We present a two-fluid hydrodynamic model for semiconductors containing electrons and holes (from thermal or external excitation) or light and heavy holes (in $p$-doped materials). The two-fluid model predicts the existence of two longitudinal modes, an acoustic and an optical, whereas only an optical mode is present in the HDM. By extending nonlocal Mie theory to two plasmas, we are able to simulate the optical properties of two-fluid nanospheres and predict that the acoustic mode gives rise to peaks in the extinction spectra that are absent in the HDM.

Plasmonic modes in nanowire dimers: A study based on the hydrodynamic Drude model including nonlocal and nonlinear effects

A combined analytical and numerical study of the modes in two distinct plasmonic nanowire systems is presented. The computations are based on a Discontinuous Galerkin Time-Domain approach and a fully nonlinear and nonlocal hydrodynamic Drude model for the metal is utilized. In the linear regime, these computations demonstrate the strong influence of nonlocality on the field distributions as well as on the scattering and absorption spectra. Based on these results, second-harmonic generation efficiencies are computed over a frequency range that covers all relevant modes of the linear spectra. In order to interprete the physical mechanisms that lead to corresponding field distributions, the associated linear quasi-electrostatic problem is solved analytically via conformal transformation techniques. This provides an intuitive classification of the linear excitations of the systems that is then applied to the full Maxwell case. Based on this classification, group theory facilitates the determination of the selection rules for the efficient excitation of modes, both in the linear and the nonlinear regime. This leads to significantly enhanced second-harmonic generation via judiciously exploiting the system symmetries. These results regarding the mode structure and second-harmonic generation are of direct relevance to other nano-antenna systems.