Abstract
Metallic nanostructures exhibit a multitude of optical resonances associated with localized surface plasmon excitations. Recent observations of plasmonic phenomena at the subnanometre to atomic scale have stimulated the development of various sophisticated theoretical approaches for their description. Here instead we present a comparatively simple semiclassical generalized nonlocal optical response theory that unifies quantum pressure convection effects and induced charge diffusion kinetics, with a concomitant complexvalued generalized nonlocal optical response parameter. Our theory explains surprisingly well both the frequency shifts and sizedependent damping in individual metallic nanoparticles as well as the observed broadening of the crossover regime from bondingdipole plasmons to chargetransfer plasmons in metal nanoparticle dimers, thus unravelling a classical broadening mechanism that even dominates the widely anticipated short circuiting by quantum tunnelling. We anticipate that our theory can be successfully applied in plasmonics to a wide class of conducting media, including doped semiconductors and lowdimensional materials such as graphene.
Introduction
Studies of transport and wave dynamics in complex and confined geometries^{1} are now bridging several fields ranging from nanoplasmonics^{2,3,4} and metamaterials^{5} to molecular electronics^{6} and mesocopic quantum transport^{7,8}, with, for example, charge carriers responding to externally perturbing fields as well as exhibiting stochastic kinetics and entropic effects such as diffusion^{9}. When considering ultrafast responses of optically driven collective plasma oscillations in nanoscale geometries, it is expected that the optical response should exhibit both quantum properties of the electron gas as well as classical diffusion dynamics of the optically induced charge. The coexistence and interplay of quantum and classical effects have profound implications for our understanding of light–matter interactions at the nanoscale, with direct relevance to the emerging field of quantum plasmonics^{10}.
The behaviour of plasmon resonances of individual silver metallic nanoparticles (MNPs)^{11,12} and gold MNP dimers^{13} seems to be possible to understand only by invoking quantummechanical effects, that is, quantum electron transitions and quantum tunnelling, respectively. At the same time, one might question the necessity of considering numerous quantumlevel transitions in nmsized NPs (that is, consisting of thousands of atoms and with a size much exceeding the Fermi wavelength) in the case of monomers and the very possible existence of ultrafast tunnelling phenomena (that is, tunnelling currents oscillating at optical frequencies) in the case of dimers. While classical electrodynamics in a Drude local response approximation (LRA) unambiguously fails to explain the observed phenomena, we show that the possibility for semiclassical accounts has not been exhausted.
Linear response theory is inherent to our understanding of situations where matter is subject to externally perturbing fields. Common strategies assume a temporally instantaneous and spatially local response, while nature is rich in examples where underlying degrees of freedom are responsible for a much more complex response. Materials exhibiting frequency dispersion are well known for having complexvalued response functions due to Kramers–Kronig relations that originate from the ubiquitous principle of causality. By contrast, spatial dispersion can usually be neglected and most materials are well treated within LRA. Insulators represent a prime example since the polarization of one particular atom in the crystal is only weakly affected by coupling to neighbouring atoms. Conducting media constitute a clear exception to this picture^{14,15} and despite the widespread use of LRA approaches, the free carriers may mediate a response over finite distances that cannot necessarily be neglected in a nanoplasmonic context.
In terms of the Maxwell equations, the electrical field in a medium with nonlocal response is formally governed by
where ε(r, r′) is the nonlocal response function. This general concept of nonlocal response of conducting media originates from the competing mechanisms of pressuredriven convective flow of charge as well as disorder or entropydriven diffusion of charge^{14}. Quite surprisingly, while the literature is rich on discussions of the former effect within hydrodynamic models, the importance of the latter in nanoplasmonic systems remains unexplored, and, according to our knowledge, there is no unifying realspace description applicable to realistic plasmonic nanostructures. Pioneering works focused on pressuredriven convective flow of charge in ideal geometries^{16,17,18,19}, while the exploration of nonlocal response in arbitrarily shaped metal nanostructures has only recently been initiated^{20}, emphasizing realspace rigorous formulations of semiclassical hydrodynamic equations^{21} and different solution strategies^{22,23,24,25,26}. Thus, large blueshifts in nanoscale noble metal plasmonic structures^{11,27,28} have been interpreted in the context of the quantum pressurerelated nonlocal response^{27,28}, while quantum confinement^{11} and surfacescreening^{29} explanations have also been proposed.
Here we develop a semiclassical generalized nonlocal optical response (GNOR) theory that incorporates both quantum pressure effects and induced charge diffusion kinetics. We show that the GNOR approach can account for the main features observed in recent optical experiments with plasmonic nanostructures^{11,13,28,30} without accounts for quantizedenergy transitions and without invoking quantum tunnelling that should not, as we argue later on, be important at optical frequencies.
Results
Isotropic and shortrange nonlocal response
We take equation (1) as our starting point, while assuming a generic shortrange isotropic response. Irrespective of the detailed microscopic mechanism behind the nonlocal response, the wave equation in the metal can then be reframed as^{31}
where ε_{D} is the Drude dielectric function usually associated with Ohmic local response of the electron gas (possibly generalized to also include interband effects), while the GNOR parameter ξ represents a phenomenological length scale associated with the shortrange nonlocal correction to the local response Drude part. Importantly, the GNOR parameter accounts for mechanisms of very different origins that may compete or play in concert, while causing the same Laplaciantype correction to the LRA. For example, both convection and diffusion can lead to spatial dispersion in conducting media^{14}. Within LRA, the induced charge density Δn is a delta function at the surface of the metal and diffusion will naturally smear this charge density with the shorttime dynamics characteristic for pure diffusion (Fig. 1c). Also, convection tends to spread the charge density and within the common hydrodynamic Drude model, convection maps directly to the form of equation (2), that is, with a Laplacian correction to the local response Drude part^{24}. In the following, we treat both dynamical effects on an equal footing by considering both propagating longitudinal pressure waves (in a hydrodynamic model) and diffusion (in convectiondiffusion model). The main result of our analysis is the following expression for the GNOR parameter:
where β∝υ_{F} is a characteristic velocity associated with pressure waves in the electron gas (υ_{F} being the Fermi velocity) while D is the diffusion constant for the chargecarrier diffusion. The former is already known to cause frequency shifts (blueshifts)^{17,21,22}, while the latter turns out to cause line broadening, that is the GNOR parameter ξ is in general a complexvalued quantity. As also anticipated from more general discussions^{14}, our rigorous semiclassical treatment shows that nonlocal effects may manifest themselves over distances greatly exceeding atomic dimensions and become comparable to characteristic structure dimensions, such as the radius R of a nanoparticle (Fig. 1a) or the gap distance g in a dimer (Fig. 1b). As a main result, we show that the GNOR even dominates more pure quantummechanical effects in the electromagnetic response at optical frequencies, such as the anticipated effect of quantummechanical tunnelling currents in dimers with subnanometre gaps (Fig. 1b).
Semiclassical response models
We consider the standard equationofmotion for an electron in an external electrical field subject to the continuity equation. The common LRA simply neglects effects of quantum pressure as well as the diffusion contributions to the induced currents. We address these two aspects in turn and finally discuss how they play in concert to result in a complexvalued GNOR parameter.
Within the hydrodynamic model (including quantum pressure, but neglecting diffusion), the response is governed by a generalized constitutive equation^{21,22}
where σ_{D} is the usual Drude conductivity. Within Thomas–Fermi theory, and γ=1/τ is the damping rate also present in the Drude theory.
Taking the opposite standpoint (including diffusion, while neglecting quantum pressure), linearization of the problem gives (Supplementary Note 1)
as also derived recently in the context of metamaterial wire media^{32}. Our key observation is that this result is mathematically similar to equation (4), while the different physical origins cause different prefactors for the nonlocal correction to Ohm’s law.
Turning to the Maxwell wave equation, the above nonlocal ∇(∇·J) corrections to Ohm’s law can be rewritten as a Laplacian correction to the Drude dielectric function^{24,31}, as anticipated in equation (2). The convection and diffusion components of the current are of the same mathematical form and subject to the same boundary conditions. Thus, in the linear response the considered nonlocal contributions add up (as in semiconductor driftdiffusion theory), as confirmed by linearizing the full hydrodynamic diffusionconvection problem (Supplementary Note 1), revealing how both quantum and classical kinetic effects can play in concert contributing to the nonlocal response. Therefore, we arrive at equation (2) with
which becomes equation (3) when neglecting damping. Note that for higher ω, diffusion becomes relatively more important compared with convection. Another important practical observation is that diffusion effectively causes the following modification of the nonlocal βparameter appearing in prior hydrodynamic work: β^{2}→β^{2}+D(γ−i ω). We explicitly neglect electron spillout and the possible existence of associated quantumtunnelling phenomena (see also Supplementary Note 2). Consequently, boundary conditions remain unchanged in the presence of diffusion (n·J=0 on the metal surfaces so that no electrons escape the metal volumes). Thus, existing numerical schemes and methods^{22,23,24,25,26} can readily be exploited to implement the GNOR approach for various plasmonic configurations.
The diffusion constant is generally interlinked with other transport parameters such as the scattering time, that is, D∝τ=γ^{−1}. For ω≫γ, we thus recover equation (3) for the GNOR parameter that now explicitly exposes the two competing length scales that were previously discussed only qualitatively^{14}, that is, the convection length β/ω on the one hand and the diffusion length on the other hand. The damping associated with the latter is an important new finding that turns out to be crucial when approaching the nanoscale. Diffusion degrades plasmonic excitations, providing an additional broadening mechanism that, mathematically, is enacted by an imaginary contribution to ξ^{2}.
Validity domain
For classical gases (such as dilute plasmas, electrolytes and weakly doped semiconductors), the velocity distribution is governed by Maxwell–Boltzmann statistics and D is proportional to the temperature, as given by the Einstein relation. For metals on the other hand, Fermi–Dirac statistics implies a narrow transport velocity distribution^{14} with a characteristic velocity υ_{F}. As a result, the diffusion constant is simply , corresponding to a meanfree path of ℓ=υ_{F}τ. We point out that our diffusive model is valid for structural dimensions exceeding the meanfree path that in pure single crystals^{33} can be of the order of 100 nm for Ag and Au, down to ~3 nm for Na (see Supplementary Table 1). Moreover, in realistic plasmonic nanostructures, ℓ depends on actual materialprocessing conditions, becoming shorter than in singlecrystalline bulk metals. This enlarges the validity domain of a diffusion description to include structures with dimensions of only a few nanometres. For even smaller dimensions, electrons will move ballistically between the surfaces of the structure, and surface scattering might become important. For metals, equation (6) simplifies to . This result leads to an important insight into the interplay of different broadening mechanisms: the lower the Ohmic loss and absorption, the more important is the nonlocal response due to longrange diffusion of the induced charge. Reintroducing the diffusion constant and the βparameter we arrive at equation (3) that holds if the meanfree path significantly exceeds the convective length, that is, ℓ>>υ_{F}/ω. The existence of spatial dispersion in homogenous media, as for example, appearing within the common hydrodynamic Drude model, can be derived from higherlevel descriptions such as the Boltzmann equation or the randomphase approximation (RPA). Beyond hydrodynamics, confined structures with broken translational invariance constitute a largely unexplored territory, with few attempts dealing with RPA aspects of localized plasmon resonances^{34}. In support of our prediction of diffusive broadening, recent RPA studies reveal an increased plasmon linewidth associated with Landau damping^{35}, that is, electronhole pair excitation near the surface of nanostructures as also observed in studies based on timedependent density functional theory (TDDFT)^{36}.
We provide in the following two key examples of the GNOR approach, demonstrating that the interplay of quantum pressure and diffusion has a remarkable impact on the optical response of plasmonics nanostructures and solving longstanding open problems.
Sizedependent damping
In general, the hydrodynamic corrections give a blueshift of resonances as the characteristic dimensions are reduced^{17,22,27,28}, which is in qualitative accordance with recent optical spectroscopy on goldparticle gap structures^{27} and electronenergy loss spectroscopy (EELS) studies of silver nanoparticles^{11,28}. With the complexvalued GNOR parameter ξ at hand, we now anticipate the blueshift to occur along with broadening of the resonant response when decreasing characteristic structure dimensions. In the case of a spherical particle, the blueshift has a β/R dependence^{28}, leading us to foresee that the linebroadening scales as 1/R as well. In the quasistatic limit (λ>>R), one can straightforwardly work out the complexvalued resonance frequency ω=ω′+iω′′ by considering the polarizability pole (Supplementary Note 3). As usual, the real part ω′ gives the surface plasmon resonance frequency, while the imaginary part ω′′ is related to the resonance linewidth. For simplicity, we consider the case of a particle in vacuum with no interband effects and find (to second order in 1/R)
It transpires clearly that the 1/R sizedependent nonlocal effects are present in both the resonance frequency and linewidth. It should be emphasized that, until now, line shifts have been explained by nonlocal response (and competing theories), whereas the line broadening was ‘put in by hand’. Here by using the GNOR theory we have arrived at a unified explanation of both experimentally observed phenomena by nonlocal effects. Line broadening has been seen experimentally in the extinction of small particles^{37,38,39,40,41} and EELS measurements on plasmons in thin nanowires and bowtie antennas have also revealed plasmon losses exceeding the expectations based on bulkdamping parameters^{42,43}. In the literature such line broadening has often been phenomenologically accounted for by a sizedependent damping rate^{37,38,39,40,41}, but without placing it in the context of nonlocal semiclassical equations of motion.
The phenomenology introduced by Kreibig^{37,38} describes the linewidth broadening by introducing a sizedependent correction to the damping rate: γ→γ+Aυ_{F}/R. Equipped with the GNOR theory, one does not need to assume the 1/R dependence: it comes immediately out as a consequence of the GNOR correction to the dipolar sphere polarizability. By comparison with the Kreibig model, we formally find that . For metals this implies that A~ω_{p}τ/4. Use of bulk values for noble metals (see Supplementary Table 1) would estimate a too high A parameter compared with experiments where for spheres A is found to be of the order unity^{37,38}.
So far, we have assumed that the transport time is given by the bulk relaxation time τ_{0}. However, in the experiments dealing with plasmonic nanostructures, it has been found that one has to increase the collision frequency^{44,45} or the imaginary part of permittivity^{46} by several times as compared with the bulk metal (gold) parameters for the simulations to better correspond to the experimental observations. This sizeindependent correction factor was ascribed to the influence of the surface scattering and grain boundary effects in nanostructures^{44,45,46}. In our case, we can simply introduce the characteristic relaxation time τ_{s} associated with these effects, which can be estimated from the condition that A~1 as τ_{s}~4/ω_{p}.
With the above refinement at hand we can demonstrate using numerical simulations (Fig. 2) that, in agreement with equations (7) and (8), the hydrodynamic response causes a blueshift (Fig. 2a), whereas the diffusion causes an additional broadening (Fig. 2b). In the latter figure, it is also shown that a similar resonance broadening (but not the blueshift) is predicted by the LRA with additional Kreibig damping.
Our realspace nonlocal wave equation (equation (2)) along with the GNOR parameter (equation (3)), which unravels the fundamental link between diffusive broadening and the Kreibiglike surface scattering thereby enables one to solve a longstanding open problem: surfacerelated scattering can now be computationally accounted for also in complexshaped geometries beyond that of spherical symmetry and low radius of curvature.
Nanowire dimers
Plasmonic dimers (Fig. 1b) are rich on hybridization phenomena as the gap distance g is reduced^{47} and nonlocal hydrodynamic effects on both hybridization and field enhancement have been anticipated^{22}. To elucidate the diffusion contribution to nonlocal effects, we may consider dimers of nanowires where the nanowire radius R itself is too large to cause either nonlocal hydrodynamic effects or increased damping of the Kreibig kind. Nevertheless, we expect that the dimer would exhibit nonlocal effects once the dimer gap distance g turns comparable in magnitude to ξ, resulting in additional broadening in the vicinity of the gap as υ_{F}/g increases. With our present formalism, this can now be quantified without any need to invoke ad hoc assumptions specifically for dimers.
Although diffusion is of a classical origin, the discussion of its effect in dimers ties up with very recent experiments on dimers in the quantumtunnelling regime^{13}. Ab initio approaches show a crossing from the classical hybridization of localized surface plasmon resonances to tunnellingmediated chargetransfer plasmons (CTPs)^{36,48,49,50}. Being able to push experiments into this intriguing regime^{13,30,51}, commonly associated with expectations of quantum physics, leaves an open question: Can this regime be adequately described with semiclassical models? While nonlocal response within the hydrodynamic semiclassical model has been found unsuccessful in explaining features from TDDFT^{48,49,50}, there have been phenomenological attempts of classically modelling the crossover regime. The ‘quantumcorrected model’ (QCM)^{52} adds an artificial conducting and lossy material in the gap to mimic shortcircuiting currents associated with quantum tunnelling. While apparently successful in qualitatively fitting results of ab initio simulations^{49,50,52}, the model raises concerns regarding its physical foundation. The wellestablished understanding of mesoscopic quantum electron transport^{7} is that the tunnelling through the classically forbidden gap region is elastic (ballistic transport) while energy relaxation takes place inside the metallic contact regions. Opposite to that, the artificial gap material introduced in the QCM causes dissipation within the gap, while there is no associated relaxation occurring on the metal sides of the junction.
While ab initio works emphasize tunnelling^{52,49,50}, recent experiments on dimers^{13,30} do not offer explicit evidence that the broadening is associated with quantum tunnelling. The formation of a subnanometre gap is evident from the observed DC voltagedriven tunnelling current^{13}, while there is no explicit confirmation of AC tunneling currents caused by the optical driving. Optical rectification phenomena are possible by photonassisted inelastic tunnelling currents^{53}, while earlier timeresolved elastictunnelling experiments have reported tunnelling RC times (i.e. the characteristic time scale of the equivalent resistorcapacitor circuit) in the picosecond range^{54}, thus suggesting a suppression of optical frequency tunneling currents that would take place in femtoseconds. This apparently makes quantumtunnelling dynamics too slow and a less likely mechanism to explain the broadening of dimer modes at optical frequencies. Applying the GNOR framework, we demonstrate in the following that the diffusion offers a strong competing damping mechanism. In fact, for fast driving of the junction^{55}, diffusion may completely dominate the dissipation of the dimer junction as we illustrate by a circuit analysis (see Supplementary Note 2).
The diffusiondriven damping occurs right inside the surface of the metals (not in the gap), becoming progressively more pronounced for smaller gaps and vanishing for large gaps. To exemplify this in detail, we revisit recent nonlocal (hydrodynamic) simulations^{49}, while making sure to account for the diffusive broadening as well by use of the complexvalued GNOR parameter (Fig. 3). For relatively large gaps, one observes the bondingdipole plasmon (BDP) along with higherorder modes known to appear below the plasma frequency within the LRA^{56} (for a larger radius this becomes particularly clear, see Supplementary Fig. 1). As the gap is reduced to the nonlocal regime g≲υ_{F}/ω, as considered in Fig. 3 for R=4.9 nm, resonances are slightly blueshifted with respect to the LRA result^{22}. When the gap shrinks further, progressively stronger hybridization^{47} and accordingly larger BDP redshifts are clearly seen. At the same time, the BDP is gradually suppressed owing to the increasing role of diffusion as the contact point, g=0, is approached. This is in strong contrast to predictions from both the LRA^{56} (with even diverging field enhancement) and from previous nonlocal theories that treated the βparameter realvalued^{22,49,50,57} (see Supplementary Fig. 1). As we enter the contact regime, the BDP fades away, vanishing completely for g<0. For touching wires, the CTP appears, whose resonance blueshifts and grows in strength for larger wire overlaps. We note that, for g~0, the diffusive broadening is so strong that only higherorder modes persist (as the induced surface charge is located away from the contact point), while both BDP and the CTP are strongly suppressed. This makes a discussion on their possible coexistence problematic^{51}. Finally, we note that in the anticipated tunnelling regime the extinction spectra are strongly broadened by the complex nonlocal response. In fact, our semiclassical approach is in remarkable agreement with the TDDFT results^{48,49,50}, with the diffusion contribution being responsible for ‘repairing’ the apparent incompatibility of TDDFT calculations and earlier hydrodynamic predictions^{49,50}. Our semiclassical GNOR theory thereby pinpoints induced charge diffusion as the dominant broadening mechanism in recent EELS and optical experiments on plasmonic dimers^{13,30}, thus challenging tunnellingcurrent interpretations for which the phenomenological QCM was constructed.
Discussion
In this article we have presented a semiclassical (GNOR) approach that is offering a longsought unification of nonlocal response mechanisms having both quantummechanical and classical origins. The GNOR theory places established observations of sizedependent damping into the context of nonlocal response and offers an accurate classical explanation of spectral broadening in MNP monomers and dimers without invoking quantummechanical tunnelling, whose efficiency at optical frequencies is questionable. We have so far considered degenerate electron systems such as metals, where screening is strong and nonlocal effects manifest themselves in the nanometre to subnanometre regime. In the search for a new mesoscopic regime, where plasmons potentially exhibit both semiclassical dynamics and quantum effects, lowdensitydoped semiconductors and tunable lowdimensional materials (including the graphene family of twodimensional materials) appear attractive^{58}. Quantum light–matter interactions^{59} and nonlocal response^{60} were already considered for graphene plasmonics, and, for such nondegenerate systems, our theory anticipates temperaturedependent nonlocal response that might lead to novel nonlocal effects accessible via experimental observations.
Methods
The optical response of dimers was obtained by solving the nonlocal wave equation with the aid of a commercially available finiteelement method. The numerical code is an extension to the COMSOL 4.2a RF Module, which incorporates nonlocal hydrodynamic effects in the optical response of arbitrarily shaped nanoplasmonic structures, possessing one dimension with translational symmetry. The code is freely available from http://www.nanopl.org and the implementation, testing and performance are reported elsewhere^{22,24}. Prior results on dimers^{49} were obtained with the same code, while only utilizing a realvalued βparameter. All numerical results can alternatively be calculated with the aid of a nonlocal boundaryelement method^{26}. Similarly, the complexvalued GNOR parameter can be substituted in nonlocal transformation optics approaches originally developed with a realvalued βparameter in mind^{57}.
Additional information
How to cite this article: Mortensen, N. A. et al. A generalized nonlocal optical response theory for plasmonic nanostructures. Nat. Commun. 5:3809 doi: 10.1038/ncomms4809 (2014).
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Acknowledgements
We acknowledge Giuseppe Toscano for use of his finiteelement code and we thank Jørn Hvam, AnttiPekka Jauho and Wei Yan for stimulating discussions. The Center for Nanostructured Graphene (CNG) is funded by the Danish National Research Foundation, Project DNRF58. The A P Møller and Chastine McKinney Møller Foundation is gratefully acknowledged for the contribution towards the establishment of the Center for Electron Nanoscopy. N.A.M. and M.W. acknowledge financial support by Danish Council for Independent Research—Natural Sciences, Project 132300087.
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N.A.M, S.I.B, T.S. and M.W. conceived the basic idea. N.A.M. derived the complex response model. S.R. performed all numerical simulations and the analytical analysis of spherical particles. Figures were prepared by N.A.M. and S.R. All authors interpreted and discussed the results and the writing of the manuscript was done in a joint effort.
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Supplementary Figures 12, Supplementary Table 1, Supplementary Notes 13 and Supplementary References (PDF 480 kb)
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Mortensen, N., Raza, S., Wubs, M. et al. A generalized nonlocal optical response theory for plasmonic nanostructures. Nat Commun 5, 3809 (2014). https://doi.org/10.1038/ncomms4809
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