Finite-size and quantum effects in plasmonics: manifestations and theoretical modelling [Invited]

P. Elli Stamatopoulou; Christos Tserkezis

doi:10.1364/OME.456407

1. Introduction

Plasmonics is currently one of the most popular research themes in a combination of areas such as optics, solid-state physics and nanotechnology. The collective response of free electrons, with which plasmonics deals, was first predicted theoretically by Pines and Bohm [1], and manifested at the surface of a thin metallic film by Ritchie [2]. At first, surface plasmons were studied theoretically [3] and probed experimentally (with electron beams [4]) as an interesting solid-state problem, but their relevance to optics had not become clear, since they cannot be directly excited by light due to momentum mismatch (their dispersion relation always lies below the light line) [5]. This situation changed in the late 1960s, when two possibilities for evanescent coupling through prisms were presented, the Otto [6] and Kretschmann [7] configurations. Nevertheless, for the next couple of decades, plasmons —manifesting as surface plasmon polaritons (SPPs) in flat films, or as localised surface plasmons (LSPs) in metallic nanoparticles (NPs)— were still scarcely studied, either from a theoretical material-response point of view [8,9], or in relation to their absorptive properties [10] and their ability to modify light emission by molecules [11].

A major boost in interest in plasmonics —which established the connection to nano-optics— came from the observation of extraordinary optical transmission through subwavelength metallic hole arrays [12]. Investigations of the role of plasmons in surface-enhanced Raman scattering (SERS) were also starting to intensify [13,14], and the two activities converged to the realisation that plasmonic NPs can tightly confine and strongly enhance light at the nanoscale, and focus it in volumes considerably below the diffraction limit [5]. This extreme confinement had just become experimentally accessible to near-field microscopy, paving the way for a large diversity of schemes to exploit it, from sensing [15] and catalysis [16,17] to optical communications [18], lasing [19,20] and photovoltaics [21,22]. Another fundamental strength of plasmonic nanostructures is the way they influence the properties of their environment, drastically increasing the local density of optical states and thus affecting the coupling with nearby emitters in the weak-[23–25] and strong-coupling regimes [26,27]. The appeal of all these prospects is in fact so strong, that the nano-optics community usually finds a way to surpass the practical limitations imposed by losses [28–30], or seeks ways to work around them [31].

Since plasmonic NPs are known for their extreme concentration and manipulation of light [32], the obvious way to further increase these capabilities is by introducing interactions, in NP dimers and aggregates, where plasmon hybridisation [33] leads to new optical modes. The prototypical set-up for such interactions is the NP dimer, typically consisting of two identical, strongly interacting spheres [34], but also appearing in realisations of heterodimers of two spheres with different dimensions or of different materials [35], or even as NPs interacting with their mirror image in an underlying metallic film [36,37]. Shape is, of course, not restricted to spheres; wires [38], nanocubes [39] or nanorods [40] have proven equally attractive coupled systems. More NPs can be added to form long NP chains of either non-touching [41,42] or connected NPs [43]. The situation can keep increasing in complexity, eventually leading to NP clusters [44,45] or periodic arrays [46,47].

Whether it concerns applications relying on field enhancement, like SERS, or extreme light–matter interactions, a clear trend, right from the beginning, has been to minimise both the size of the particles involved and their separations. NPs with sizes of just a few nm have only very recently started being considered as candidates for novel applications [48,49], but have long been triggering our curiosity regarding the microscopic mechanisms involved in their optical response [50,51]. On the other hand, the cavities formed between NPs in dimers and aggregates can be of great importance, and a series of different methods to produce such narrow separations has been proposed, ranging from lithography [52] and binding with molecular linkers [53] to arranging NPs on a thin film with which they interact [54] or pushing them with electric forces [55]. But even before such nanogaps were actually produced, it had become clear that the classically predicted divergence of the field enhancement inside them could not be feasible, leading to the quest for understanding the physical mechanisms that regularise it, initiating thus the field of quantum plasmonics [56–58].

In solid-state physics, it has been known for decades that local scalar dielectric functions prove inadequate for the full description of any material, once the length scales involved become comparable to a characteristic intrinsic length of the system. In metals this length is directly related to the Fermi wavelength or, in a more classical description, the electron mean free path [59]. A wavevector-dependent —and thus nonlocal in space— permittivity is thus required [60], which, already in its most simple form, accounts for the effect of screening: the induced charges have some spatial extent, and inner electrons screen the interaction with the outer ones, leading to the NP behaving as effectively smaller. For some time, any disagreement between experimental spectra and calculations based on classical electrodynamics (CED) was quickly and intuitively attributed to nonlocality. The development of numerical methods that account for it with affordable computational cost, mostly within the framework of hydrodynamics [61–63], has shown that this is not always the case, and in many occasions CED provides an accurate description of the optical response down to the last couple of nm [64].

Once the size of the particles has become really small, not exceeding a couple of nm, computational methods developed for many-body problems in condensed-matter physics, like time-dependent density functional theory (TDDFT) [65,66] can shed more light on the underlying physics; but for larger structures with millions of electrons the computational cost remains prohibitive. It soon became clear that, in addition to the electron screening mechanism that is described by standard nonlocal models, electron spill-out might also become relevant when the work function of the metal is low [67–69] and electrons have non-negligible probability of surmounting the assumed NP boundaries. In ultranarrow plasmonic gaps (e.g. between two nearly touching NPs), this spill-out can even lead to direct electron tunnelling from one particle to the other, effectively creating a new, longer resonator [70,71]. These predictions would finally offer one first explanation of why the classically predicted divergence of the near-field in the gap is not achievable.

Electron spill-out and tunnelling prevail in good Drude metals characterised by free-electron bands and low work functions, and the fully quantum mechanical calculations of the absorption spectra of ultrasmall jellium nanoparticles like sodium has clarified many of the underlying mechanisms [72]. In the case of noble metals, however, d-electron screening is important, implying that a jellium treatment within TDDFT is inadequate, while more precise atomistic descriptions [73,74] still contain a certain degree of ambiguity (see, e.g. the review by Varas et al. [75] or the textbook by Marques et al. [66] for a detailed overview of the different levels of approximations for the choice of the exchange-correlation functional). In any case, experimental measurements have shown that screening must be the dominant mechanism all the way down to the quantum confinement regime [76,77]; but screening does not justify the increased damping and resonance broadening that is also observed. For that, the mechanism most likely responsible is that of surface-enhanced (or, more accurately, surface-enabled) Landau damping: conduction-band electrons acquire, from collisions with the NP boundaries, the additional momentum required to move to a higher energy within the same band, creating in the process electron-hole pairs and thus removing carriers from the plasmon [78–81].

In what follows, we first discuss the experimental manifestations of inadequacies of a strictly CED-based description due to the effects mentioned above, and then analyse the various phenomenological, analytic, or numerical models developed to advance our understanding and provide accurate yet computationally effective pictures of small plasmonic nanostructures. While the review discusses several quantum and finite-size effects —the latter term accounting for phenomena that become prominent as the structure’s surface to volume ratio increases— there is no obvious way to distinguish between the two terms; all effects are, after all, of quantum origin. We reserve, thus, the word “quantum” for those effects that cannot be treated with any other theory but quantum mechanics. To better serve intuition, we focus our analysis on how each of the models of interest can be analytically implemented in the description of spherical NPs; numerical solutions are readily available for arbitrary geometries.

2. Local response approximation

The optical response of small metallic nanospheres can be calculated and understood within the framework of Mie theory [82], which was originally developed in the framework of the local response approximation (LRA). In practice, LRA, which is based on the assumption that the material responds only locally to the driving electromagnetic (EM) fields, describes efficiently the vast majority of EM phenomena in plasmonics. Essentially, all one has to do is solve the wave equation (expressed here in terms of the electric field $\mathbf {E}$)

(1)$$\nabla^{2} \mathbf{E} (\mathbf{r}) + \frac{\omega^{2}}{c^{2}} \varepsilon (\omega) \mu (\omega) \mathbf{E} (\mathbf{r}) = 0~$$

(where $\omega$ is the angular frequency, $c = 1/\sqrt {\varepsilon _{0} \mu _{0}}$ the speed of light in vacuum, $\varepsilon _{0}$ the vacuum permittivity, $\mu _{0}$ the vacuum permeability, and $\varepsilon$ and $\mu$ the corresponding response functions in a homogeneous medium) in each sub-space (inside and outside a given NP), subject to the appropriate boundary conditions; in the absence of surface charges and currents these collapse simply to the continuity of the tangential components of the electric and magnetic fields. In the case of spherical symmetry, Eq. (1) can be solved analytically by expanding the fields into spherical waves characterised by the usual angular momentum indices $\ell$ and $m$. Practically any observable —such as the extinction cross section or the electron energy-loss (EEL) probability— can then be expressed through a sum of sufficiently many multipoles, weighted by the corresponding Mie coefficients [82,83].

For metals at optical frequencies, $\mu$ is typically taken equal to $1$ [84], while for the permittivity it is often sufficient to assume a Drude form [59]

(2)$$\varepsilon_{\mathrm{m}} (\omega) = \varepsilon_{\infty} (\omega) - \frac{\omega_{\mathrm{p}}^{2}} {\omega \left(\omega + \mathrm{i} \gamma\right)}~,$$

where $\gamma$ is the Drude damping rate (due to collisions with the fixed ions), and $\omega _{\mathrm {p}} = \sqrt {n e^{2}/( m_{\mathrm {e}} \varepsilon _{0})}$ is the plasma frequency (with $e$ being the elementary charge, $m_{\mathrm {e}}$ the electron mass, and $n$ the electron density in the metal); taking $\varepsilon _{\infty } = 1$ describes a pure free-electron gas like that in alkali metals, while $\varepsilon _{\infty } \neq 1$ accounts for the contribution of bound electrons and interband transitions (e.g. in noble metals), and is usually determined experimentally.

Applying these material-response assumptions to the exact analytic solution, the spectra of small metallic NPs are shown to be dominated by resonances of electric dipolar, quadrupolar and so on (in general, $2^{\ell }$) character. The origin of these resonances can be straightforwardly understood by referring to the quasistatic approximation for the scattering Mie coefficients,

(3)$$a_{1} = \mathrm{i} \frac{2 \left(k R\right)^{3}}{3} \frac{\varepsilon_{\mathrm{m}} - \varepsilon} {\varepsilon_{\mathrm{m}} + 2\varepsilon}~, \quad \quad b_{1} = \mathrm{i} \frac{2 \left(k R\right)^{3}}{3} \frac{\mu_{\mathrm{m}} - \mu} {\mu_{\mathrm{m}} + 2\mu}~,$$

where $\varepsilon _{\mathrm {m}}$ and $\varepsilon$ are the relative permittivities of the NP (of radius $R$) and its environment respectively, and $\mu _{\mathrm {m}}$, $\mu$ the corresponding permeabilities, while $k$ is the wavenumber in the environment. A dipolar resonance of electric or magnetic type will be observed when $a_{1}$ or $b_{1}$, respectively, has a pole. Since the permeability differs only slightly from unity, small nanospheres do not, typically, support magnetic resonances. But electric resonances can easily be obtained, as all that is required is a negative permittivity (in general, an interface between two media with permittivities of different signs). Metals in the intraband regime (dominated by free electrons) are characteristic examples exhibiting such optical properties, and they are readily available in nature. Using the simple Drude model of Eq. (2) to describe the metal, one immediately obtains the frequency of the dipolar LSP at $\omega _{\mathrm {p}} /\sqrt {3}$, assuming no background contribution ($\varepsilon _{\infty } =1$) and air ($\varepsilon = 1$) as the surrounding medium. For higher-order multipoles (still in small NPs) this condition is generalised to

(4)$$\omega_{\ell} = \sqrt{ \frac{\ell} {\ell \varepsilon_{\infty} + \left(\ell +1\right) \varepsilon}} \; \omega_{\mathrm{p}} ~.$$

The above analysis, based on Eq. (3), holds for very small sizes, where the NP essentially behaves like a point dipole. In this quasistatic approximation, the frequency of the dipolar LSP does not depend on NP size. This result is exactly the electrostatic polarisability obtained easily with the method of images [5]. This crude approximation can be extended to any NP shape, and reflects the fact that the NP, due to its small size, experiences an external field that is practically constant. The situation will naturally change as the size increases, and experiments have shown that already for radii of the order of $5$ nm one observes deviations from the quasistatic prediction, with the resonances gradually redshifting [77]. Two mechanisms are responsible for this frequency change: retardation and radiation damping. Retardation means that as the NP becomes bigger, its free electrons do not respond to the exciting light simultaneously, but with a small phase difference in the propagation inside the NP. On the other hand, while the spectrum of a very small NP is dominated by absorption, as the size increases the ratio between the magnetic and kinetic inductance of the NP increases proportionally to it [85]; the NP becomes a better antenna, re-radiating part of the incident energy. The impact of these two mechanisms on the frequency of the dipolar LSP is shown in Fig. 1(a), for normalised (to the corresponding plasma frequency $\omega _{\mathrm {p}}$ and wavelength $\lambda _{\mathrm {p}} = 2 \pi c/\omega _{\mathrm {p}}$ respectively) frequencies and radii, for an NP described by Eq. (2) (with $\varepsilon _{\infty } = 1$) in air. The red axes show what these normalisations mean in practice, for a plasma frequency corresponding to energy $\hbar \omega _{\mathrm {p}} = 9$ eV (typical for noble metals).

Fig. 1. LRA response of plasmonic nanospheres. (a) Normalised frequency ($\omega / \omega _{\mathrm {p}}$) of the dipolar LSP of a spherical NP of radius $R$ (as shown schematically in the inset) described by the Drude model of Eq. (2) with $\varepsilon _{\infty } = 1$, in air ($\varepsilon =1$), as a function of its normalised radius $R/\lambda _{\mathrm {p}}$, obtained within LRA. The blue dashed line marks the resonance in the quasistatic approximation, $\omega _{\mathrm {p}}/ \sqrt {3}$. The corresponding energy in eV, and radius in nm, are given at the top and the right axis respectively, assuming a plasmon energy $\hbar \omega _{\mathrm {p}} = 9$ eV and a damping parameter $\hbar \gamma = 0.09$ eV. (b) Extinction cross section ($\sigma _{\mathrm {ext}}$) spectra (normalised to the geometric cross section $\pi R^{2}$) for the NP of (a), for three radii, $R/\lambda _{\mathrm {p}} = 0.145$ (red solid line), $R/\lambda _{\mathrm {p}} = 0.051$ (green dashed line), and $R/\lambda _{\mathrm {p}} = 0.007$ (blue dotted line); for $\hbar \omega _{\mathrm {p}} = 9$ eV these radii correspond to $20$, $7$, and $1$ nm, respectively.

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Selected extinction spectra are shown in Fig. 1(b), for three values of the normalised radius, corresponding to $1$, $7$ and $20$ nm. The metal is again described by a simple Drude model, with relatively low damping rate ($\gamma = 0.01 \omega _{\mathrm {p}}$). The smallest particle lies indeed in the quasistatic regime, and its resonance is exactly at $\omega _{\mathrm {p}}/\sqrt {3}$. As the size increases, the extinction cross section becomes larger in magnitude and starts redshifting. This range of sizes, close to the quasistatic regime, is the main target of this review. For even larger sizes (which are in fact the ones most frequently encountered in experiments), the dipolar LSP has redshifted and broadened quite significantly, and a higher-order, quadrupolar LSP is also efficiently excited.

3. Evidence of the inadequacy of LRA

The first experimental proof that LRA, like any approximation, would inevitably fail at some limit —the limit here emerging from decreasing the NP size— came already in the 1960s, in the works of Doremus [86] and Kreibig [87], who observed considerable broadening of the optical spectra of small gold and silver NPs, strongly dependent on NP size [Fig. 2(b)]. Within a classical description, this size-dependent broadening (SDB) was attributed to the reduction of the electron mean free path: with the NP size becoming comparable to the mean free path of conduction electrons in the bulk of the metal, collisions with the metal surfaces cannot be neglected, essentially affecting the relaxation rate that enters Eq. (2). This response has been verified repeatedly, see e.g. Refs. [51,88–90].

Fig. 2. Experimental measurements showing non-classical effects in the response of plasmonic systems. (a) TEM images of small silver nanoparticles, scale bars: $2$ (upper) and $5$ nm (lower) in diameter [Reprinted (adapted) by permission from [76], Copyright 2012, Springer Nature]. (b) SDB measurements in silver NPs (Reprinted by permission from [87], Copyright 1969, Springer Nature). (c) LSP shifts of small Ag NPs in silica measured with EELS and optical spectroscopy (open symbols), and theoretical simulations in different embedding environments (solid symbols) (Reprinted by permission from [91], Copyright 2019, Springer Nature). (d) LSP blueshifts in Ag NPs including higher-order multipoles (reproduced from [77]). (e) LSP redshifts (with respect to the Mie prediction) for small sodium clusters (Reprinted by permission from [92], Copyright 1995 by the American Physical Society). (f) EELS spectra of NP dimers with separations ranging from a few Angstrom to physical contact (Reprinted with permission from [55], Copyright 2013, American Chemical Society).

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In addition to the resonance broadening, significant deviations from the predictions of LRA have been measured for the LSP energies as well. For spherical NPs made of the most common metals in plasmonics, i.e., gold and silver [see Fig. 2(a) for a typical transmission electron microscope (TEM) image], an anomalous size dependence of the optical absorption was measured already in the 1970s [93,94]. Thereafter, several experiments have shown either blueshift or redshift of the plasmon peak in small metallic clusters [95,96] [see Figs. 2(c–e)]. The blueshift has been observed mostly with electron energy-loss spectroscopy (EELS) which, unlike optical microscopy, can efficiently capture nonradiative modes, also including higher-order multipoles [50,76,77,91,97]. It has been observed optically as well, mostly for film-coupled NPs studied with dark-field microscopy [98,99], in the reflectivity of NP arrays [100], and with photodepletion spectroscopy for small clusters [101]. On the other hand, in the case of small sodium and lithium clusters, experiments have shown a redshift [92,102,103] [see Fig. 2(e)], in agreement with TDDFT calculations for jellium metals [68,104,105]. Different theoretical approaches predict different directions of this size-dependent shift (a comprehensive discussion on the matter is presented by Kreibig et al. [89]). Admittedly, such systems are governed by a plethora of mechanisms that may shift the LSP energy either to higher (e.g. screening) or lower (e.g. spill-out) energies, and identifying the underlying size effect is not always straightforward [106]. An important source of spectral shifts is related to the embedding environment of the NP under study, as illustrated in Fig. 2(c); while many theoretical studies focus on NPs in vacuum, experimentally they are usually either supported by a substrate or embedded in a matrix [91,107,108]. At this point, it is useful to keep in mind that the tendency of the photonics community to assign specific quantum and finite-size effects on certain spectral features, while overlooking other external factors, together with the overall ease in using the “quantum” label, has raised considerable criticism from the side of cluster and surface physics, making clear the need for a complementary bottom-up perspective on the matter [91,109,110]. Deviations from the response expected within LRA can also emerge due to surface chemical reactions [111,112] or doping [113], while higher-order multipoles shift only little before eventually being completely damped [77].

While size-dependent damping and small resonance shifts hint that LRA can be inadequate already for individual NPs, the situation becomes even more dramatic in the case of interacting systems, most typically in NP dimers and film-coupled particles. In such situations, LRA predicts that, as the NPs approach each other, their optical modes —originating from hybridisation among the LSPs of the individual NPs [33]— will keep redshifting [34]. Yet what carefully conducted experiments with either NP-covered atomic-force microscopy tips [70] or individual NPs manipulated with electron beams [55] have shown is that for separations less than $\sim 1$ nm this modal redshift ceases and gives place to a blueshift of the corresponding resonance [114]. Even more striking is the fact that a new mode emerges at lower energies; this seems to correspond to the first LSP mode of the elongated NP resulting from the merging of the initial individual NPs, yet the resonance appears even before physical contact, at the last few Angstrom, as shown in Fig. 2(f). This response is in fact reminiscent of what is observed in NP dimers connected with conductive molecules [115], strongly suggesting that a charge-transfer path might be established in the former case as well. For a more comprehensive overview of similar experimental works, we refer the reader to the review by Zhu et al. [57].

On top of the far-field optical response, the LRA finding that most strongly suggest that something goes wrong is the divergence of the near-field enhancement as the gap between the NPs closes [34]. While experimental evidence for the contrary cannot be quantified, since the field enhancement can only be measured indirectly, this prediction goes against our well-established belief that nature dislikes divergences; it is much more likely that the mechanisms responsible for preventing this are simply not included in LRA, and both theoretical and experimental efforts are needed to provide descriptions that remedy all these inconsistencies. In practice, given the experimental challenges and the ambiguities that always emerge —even when interpreting measured far-field spectra— the number of experimental works focusing on such effects is relatively limited (for a detailed account, see the recent review by Shi et al. [116]); nevertheless, they have inspired a large volume of theoretical efforts to understand those observations and model them in the most efficient way, as we describe in the next sections.

4. Nonlocal effects

To identify the possible origins of discrepancies and deviations from LRA, one should revisit the entire modelling approach, starting with Maxwell’s equations and the derivation of the wave equation Eq. (1). Maxwell’s equations in matter, and the constitutive relations connecting the response fields inside a material with the externally applied ones, form the backbone of any optical response study; all the effects we discuss here are related to the displacement field $\mathbf {D}$ and its origin from the electric field $\mathbf {E}$ of an incident EM wave. Within linear response theory, and in the frequency domain, the relevant constitutive relation reads

(5)$$\mathbf{D} (\mathbf{r}, \omega) = \varepsilon_{0} \int \mathrm{d} \mathbf{r}' \; \varepsilon(\mathbf{r}, \mathbf{r}', \omega) \mathbf{E} (\mathbf{r}'; \omega)~,$$

where $\varepsilon (\mathbf {r}, \mathbf {r}'; \omega )$ is the nonlocal relative permittivity. Here we have assumed an isotropic medium for simplicity, meaning that $\varepsilon$ is scalar (zero-order tensor). Equation (5) implies that the displacement field at position $\mathbf {r}$ depends on the electric field at every other point $\mathbf {r}'$ in space. An interesting way to experimentally probe this nonlocal response through the changes it causes to plasmons in graphene monolayers attached to a metallic film was proposed in Ref. [117] and materialised in Ref. [118].

Within LRA, the nonlocal dependence is relaxed through the introduction of a local bulk permittivity $\varepsilon (\mathbf {r}, \mathbf {r}'; \omega ) = \varepsilon (\omega ) \delta (\mathbf {r} - \mathbf {r}' )$, which largely simplifies Eq. (5) to

(6)$$\mathbf{D} (\mathbf{r}, \omega) = \varepsilon_{0} \varepsilon (\omega ) \mathbf{E} (\mathbf{r}, \omega).$$

As discussed in the previous section, as plasmonic systems (or their separations) are gradually reduced to sizes comparable to intrinsic length scales associated with the electron dynamics, such as the Fermi wavelength, LRA can no longer capture the underlying electron motion and interactions. It is important to note here that this failure does not reflect any limitations of Maxwell’s equations, from which Eq. (1) emerges, but rather our poor modelling of the medium they are asked to describe, as a local, homogeneous and isotropic medium obeying Eq. (2). Neither does resorting to experimentally measured permittivities solve the problem, since these are also local quantities describing the bulk of the metal.

To remedy this situation, we highlight in what follows the basic steps in the derivation of one of the most widespread models for a nonlocal permittivity, namely the hydrodynamic Drude model (HDM), and refer the interested reader to a more detailed discussion in Refs. [62,119–121]. In order to amend some of the inadequacies of LRA, HDM —proposed already in the 1930s [122] and being essentially a limiting case of the Lindhard permittivity [60,123]— extends the local Drude model by treating the dynamics of the material as that of a compressible electron gas, and encompasses nonlocal corrections while maintaining a CED core. Herein, the electron dynamics is expressed via the electron density $n (\mathbf {r}; t)$ and velocity $\mathbf {v} (\mathbf {r}; t)$ as entailed by the continuity equation, together with a generalised equation of motion

(7a)$$\frac{\partial n}{\partial t} + \nabla \cdot (n \mathbf{v}) = 0$$

(7b)$$\frac{\partial \mathbf{v}}{\partial t} + \left( \mathbf{v} \cdot \nabla \right) \mathbf{v} ={-} \frac{e}{m_{\mathrm{e}}} \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right) - \gamma \mathbf{v} - \frac{1}{m_{\mathrm{e}}} \nabla \frac{\delta G[n]}{\delta n},$$

where $G[n]$ is a total energy functional of the density that contains all interactions and correlations between electrons; as previously, $\gamma$ is the simple Drude damping rate due to (instantaneous) collisions with the ions (electron–electron collisions are disregarded) [59]. Taking only the internal kinetic energy of the electrons into account, $G$ simplifies to the Thomas-Fermi functional $T_{\mathrm {TF}}[n]$,

(8)$$G[ n(\mathbf{r}; t) ] \approx T_{\mathrm{TF}} [ n(\mathbf{r}; t)] = \frac{3 \hbar^{2}}{10 m_{\mathrm{e}}} \left(3 \pi^{2} \right)^{2/3} \int \mathrm{d} \mathbf{r} \; n(\mathbf{r}; t)^{5/3}~.$$

In this case the term $\nabla (\delta G[n]/\delta n)$ in Eq. (7b) corresponds to a hydrodynamic pressure ($p$) gradient describing the convective flow of the electron gas. In the absence of an external field, the electron gas is in equilibrium, with its density $n = n_{0}$ being constant. For small deviations from equilibrium the density comprises the homogeneous equilibrium part $n_{0}$ and an out-of-equilibrium part $n_{1}$, containing dynamic and inhomogeneous quantities, which translates to $n = n_{0} + n_{1}$ and $p = p_{0} + p_{1}$ accordingly. The linearised equations of motion become

(9a)$$\frac{\partial n_{1}}{\partial t} + n_{0} \nabla \cdot \mathbf{v} = 0$$

(9b)$$\frac{\partial \mathbf{v}}{\partial t} + \frac{e}{m_\mathrm{e}} \mathbf{E} + \frac{\beta^{2}}{n_{0}} \nabla n_{1} ={-} \gamma \mathbf{v}~,$$

where we have set the hydrodynamic parameter $\beta ^{2} = 1/m_{\mathrm {e}} \cdot \partial p_{1}/\partial n_{1}$. Using $\mathbf {J}= e n_{0} \mathbf {v}$ to define the current density, together with the linearised Eqs. (9), we arrive at the generalised Ohm’s law for HDM,

(10)$$\frac{\beta^{2}}{\omega \left(\omega +\mathrm{i} \gamma \right)} \nabla \left[\nabla \cdot \mathbf{J}(\mathbf{r}; \omega) \right] + \mathbf{J}(\mathbf{r}; \omega) = \sigma (\omega) \mathbf{E} (\mathbf{r}; \omega)~,$$

where $\sigma$ is the Drude conductivity, related to Eq. (2) through $\varepsilon _{\mathrm {m}} = \varepsilon _{\infty } + \mathrm {i} \sigma /(\varepsilon _{0} \omega )$. The first term in Eq. (10) embodies the nonlocal correction to Ohm’s law, that naturally vanishes as $\beta \to 0$. For a three-dimensional electron gas, the hydrodynamic parameter takes the value $\beta ^{2} = v_{\mathrm {F}}^{2}/3$ in the low-frequency limit ($\omega \ll \gamma$) (with $v_{\mathrm {F}}$ being the Fermi velocity), while $\beta ^{2} = 3 v_{\mathrm {F}}^{2}/5$ in the high-frequency limit ($\omega \gg \gamma$) that is relevant to plasmonics [124], as one can easily derive through the total energy of the free-electron gas [125].

It can easily be shown that the nonlocal corrections in HDM emerge only via the presence of longitudinal waves in the electric field, which calls for an additional boundary condition. Assuming a step profile for the equilibrium electron density, i.e. $n_{0}$ constant inside the metal and abruptly dropping to zero at the metal boundary $\partial \Omega$, leads to the condition [126]

(11)$$\mathbf{J} \cdot \hat{\mathbf{n}} \big|_{\mathbf{r}\, \in\, \partial \Omega} = 0~,$$

with $\hat {\mathbf {n}}$ denoting the unit vector normal to the metal surface. This essentially “hard wall” boundary assumption is by construction responsible for the inability of HDM to describe electron spill-out beyond the assumed surface of the metal (as shown schematically in Fig. 3), while it is also not always clear if and how it can be employed to describe an interface with another material with some collective electronic response, e.g. metallic or excitonic. In the case of spherical particles in a simple dielectric environment, one can solve Eq. (10), subject to the boundary condition of Eq. (11), to obtain the HDM-modified Mie coefficients. In the limit of small radii (which is now the main focus anyway), one can then extract an expression equivalent to that of Eq. (4) for the frequencies of all multipoles [62],

(12)$$\omega_{\ell} = \sqrt{ \frac{\ell}{\ell \varepsilon_{\infty} + \left(\ell +1\right) \varepsilon + \ell \left(\ell + 1\right) \varepsilon \delta_{\mathrm{NL}\ell} }} \; \omega_{\mathrm{p}}~,$$

where

(13)$$\delta_{\mathrm{NL} \ell} = \left( \frac{\varepsilon_{m}} {\varepsilon_{\infty}} -1\right) \frac{j_{\ell} (k_{\mathrm{mL}} R)}{k_{\mathrm{mL}} R j_{\ell}^{'} (k_{\mathrm{mL}} R)}~.$$

Here, $j_{\ell }$ is the spherical Bessel function of order $\ell$, the prime denotes the derivative with respect to its argument, and $k_{\mathrm {mL}}$ is the longitudinal wavenumber in the metal, obtained from the longitudinal dispersion relation [127]

(14)$$\varepsilon_{m\mathrm{L}} \equiv \varepsilon_{\infty} - \frac{\omega_{\mathrm{p}}^{2}} {\omega \left(\omega + \mathrm{i} \gamma\right) - \beta^{2} k_{\mathrm{mL}}^{2} } = 0~.$$

Fig. 3. The path from a purely classical to a quantum description. From left to right, the local-response, hydrodynamic, and surface-response approaches, and, on the right end, level quantisation in ultrasmall systems. The bottom row also contains effects that can be captured with extensions of these models, with GNOR being an extension of HDM, and the Feibelman formalism and QCM being DFT-based.

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HDM has been used in analytic calculations of small nanospheres already since the 1970s [124,128–133], but it became extremely popular around the end of the 2000s, when it was successfully implemented in various numerical methods, from the finite-difference time-domain [134] and the finite-element method [135,136] to the boundary-element method [137] and hybrid schemes [138–140], proving itself a useful tool in analytical and numerical studies of small NPs and aggregates thereof [141–148]. The model has also been implemented for the description of plasmonic nanostructures excited by electron beams, in EELS or cathodoluminescence studies [149–151]. Nonlocal screening, as captured —or perhaps one should say mimicked— by HDM, has also been explored with regards to its role on the coupling of plasmonic NPs with quantum emitters [152–158], the performance of plasmonic lasers [159], optical nonlinearities [160], or Förster energy transfer [161] and the Casimir force [162]. In most cases, HDM has been so successful that it inspired efforts to effectively implement it into CED calculations without in fact solving the corresponding wave equation, just by appropriately rescaling the system [163,164]. Recently, the model was also extended for semiconductors [165] and polar dielectrics [166], characterised by dispersion relations similar to that of Eq. (14).

Despite its merits, HDM still fails to describe crucial nonlocal phenomena, Landau damping being a characteristic example. To account for the latter, the generalised nonlocal optical response model (GNOR) developed by Mortensen et al. [167] adds to the convective contribution of HDM a diffusion term, described by the diffusion constant $\mathcal {D}$, that acts as a relaxation mechanism, emulating surface-scattering phenomena. This term modifies the continuity equation in the frequency domain to

(15)$$-\mathrm{i} \omega e n (\mathbf{r}; \omega) = \mathcal{D} \nabla^{2} \left[en (\mathbf{r}; \omega) \right] + \nabla \cdot \left[{-}e n_{0} \mathbf{v} (\mathbf{r}; \omega) \right] = \nabla \cdot \mathbf{J} (\mathbf{r}; \omega)~.$$

Fick’s law acquires, therefore, an additional diffusion term

(16)$$\mathbf{J} (\mathbf{r}; \omega) ={-}e n_{0} \mathbf{v} (\mathbf{r}; \omega) + e \mathcal{D} \nabla n(\mathbf{r}; \omega)~,$$

which leads to the generalised Ohm’s law within GNOR,

(17)$$\Bigg[ \frac{\beta^{2}}{\omega \left(\omega + \mathrm{i} \gamma\right)} + \frac{\mathcal{D}}{\mathrm{i} \omega} \Bigg] \nabla \left[ \nabla \cdot \mathbf{J} (\mathbf{r}; \omega) \right] + \mathbf{J} (\mathbf{r}; \omega) = \sigma (\omega) \mathbf{E }(\mathbf{r}; \omega)~.$$

This looks remarkably similar to Eq. (10), except for the $\mathcal {D}/\mathrm {i}\omega$ term in the square brackets on the left-hand side. Any computational tool developed for HDM can thus be extended to GNOR, simply by introducing the complex hydrodynamic convective-diffusive parameter $\eta ^{2} = \beta ^{2} +\mathcal {D} \left (\gamma - \mathrm {i} \omega \right )$ in place of $\beta ^{2}$. In Ref. [167] it was shown that the mere substitution of $\eta ^{2}$ yields the observed spectral shifts and size-dependent resonance broadening, predicting thus both induced-charge screening and surface-enabled Landau damping [168–170]. The model has been used to describe how weak [156,157] and strong coupling [171,172] between plasmonic nanostructures and quantum emitters are affected by this damping, and was successfully implemented in one of the quasinormal mode schemes [173]. Nonetheless, GNOR still relies on the hard-wall boundary conditions; therefore, electron spill-out effects are overlooked, while the appropriate choice for $\mathcal {D}$ is not always intuitively obvious [174].

One way to circumvent the rigid hard-wall assumption, as proposed by Zaremba and Tso [175], is to treat the equilibrium density of the electrons in the metal self-consistently within the hydrodynamic theory itself. This observation inspired the development of the self-consistent HDM (SC-HDM), where an exchange-correlation functional term that depends on the gradient of the electron density is added to the internal energy functional of Eq. (8). Thus, corrections in regions of intense variation of the equilibrium density, such as close to the structure boundaries, are taken into account and electron spill-out can be predicted successfully [176–178], albeit at an increased computational cost that starts resembling that of TDDFT calculations.

A more efficient treatment of the hard-wall constraint, that intrinsically contains most relevant size effects, is to invoke surface-response functions at the metal boundary. The step profile of the electron density, prior consisting of the metal and the host environment parts, now breaks down into an additional domain between the two: a transition region characterised by an evanescent electron-density tail and Friedel oscillations. There are certainly more routes than one to treat the transition domain, for instance as an infinitely thin layer of dipoles oriented normally to the NP surface [179] or a nonlocal–local metal composite [164]. In the framework of Feibelman’s work in surface science [180–182], nonlocality is imposed via the surface-response parameters $d_{\perp }$ and $d_{\parallel }$, also called Feibelman $d$ parameters, entering the ordinary boundary conditions, while the bulk material is still described by local response functions [183–185]. In what follows, we only provide a brief overview of the Feibelman surface-response formalism (SRF), as detailed derivations can be found elsewhere [186,187]. As illustrated in Fig. 3, $d_{\perp }$ and $d_{\parallel }$ represent the frequency-dependent centroids of the induced charge $\rho$ and the normal derivative of the tangential current $J_{y}$, respectively, evaluated at a planar metal–dielectric interface (assumed to lie in the $yz$ plane below). In mathematical notation this translates to

(18)$$d_{{\perp}} (\omega) = \frac{\int \mathrm{d} x\; x \rho (x, \omega)} {\int \mathrm{d} x\; \rho (x, \omega)}~, \quad \quad d_{{\parallel}} (\omega) = \frac{\int \mathrm{d} x\; x \partial_{x} J_{y} (x, \omega)} {\int \mathrm{d} x\; \partial_{x} J_{y} (x, \omega)}~,$$

with $d_{i} (\omega ) = d_{i}^{'} (\omega ) + \mathrm {i} d_{i}^{''} (\omega )$ ($i = \perp, \parallel$) being in general complex-valued (one prime denotes the real and two the imaginary part, as usually). For $d_{\perp }$, the real part describes the position of the centroid of induced charge with respect to the assumed interface, while the imaginary part is a measure of surface-enabled damping [188]. Usually, positive $d_{\perp }^{'}$ means spill-out while negative values correspond to spill-in, although this can depend on the initial assumption of where the interface is located [189].

The surface response functions of Eqs. (18) correspond to surface polarisation and current terms, $\mathbf {P} (\mathbf {r}) = \boldsymbol {\pi } (\mathbf {r}) \delta (\mathbf {r} - \mathbf {r}_S)$ and $\mathbf {J} (\mathbf {r}) = \mathbf {K} (\mathbf {r}) \delta (\mathbf {r} - \mathbf {r}_S)$, respectively, where $S$ denotes the metal–dielectric interface. The surface polarisation and current densities are each proportional to a Feibelman $d$ parameter and are driven by the discontinuities of the field components [186],

(19a)$$\boldsymbol{\pi} = \varepsilon_{0} d_{{\perp}} \big[ \hat{\mathbf{n}} \cdot \left( \mathbf{E}_{\mathrm{e}} - \mathbf{E}_{\mathrm{m}} \right) \hat{\mathbf{n}} \big]$$

(19b)$$\mathbf{K} = \mathrm{i} \omega d_{{\parallel}} \big[ \hat{\mathbf{n}} \times \left( \mathbf{D}_{\mathrm{e}} - \mathbf{D}_{\mathrm{m}} \right) \times \hat{\mathbf{n}} \big]~,$$

where the subscript $\mathrm {e}$ refers to the corresponding field in the environment, and $\mathrm {m}$ to that in the metal. The densities can be absorbed into the traditional boundary conditions, leading to [187]

(20a)$$\mathbf{E}_{\mathrm{e} \parallel} - \mathbf{E}_{\mathrm{m} \parallel} ={-} d_{{\perp}} \nabla_{{\parallel}} \left( E_{\mathrm{e}\perp} - E_{\mathrm{m}\perp} \right)$$

(20b)$$\mathbf{H}_{\mathrm{e} \parallel} - \mathbf{H}_{\mathrm{m}\parallel} = \mathrm{i} \omega d_{{\parallel}} \left( \mathbf{D}_{\mathrm{e}\parallel} - \mathbf{D}_{\mathrm{m} \parallel} \right) \times \hat{\mathbf{n}}~,$$

where $\nabla _{\parallel }$ is the surface nabla operator. For the evaluation of the $d$ parameters several methods have been proposed, employing, for instance, jellium TTDFT calculations [179,187,190], a smoothly varying electron density function within LRA [189], semiclassical hydrodynamic-based models [63], or with an atomic-layer potential as the starting point [191]. While TDDFT —most commonly in the time-dependent local density approximation (TDLDA)— can, in principle, offer a mesoscopic passage from classical local to quantum plasmonics [75,192–195], it inevitably comes with high computational cost and low applicability. Feibelman parameter-based descriptions offer thus a more intuitive and tractable route for capturing multiple nonlocal phenomena, while still operating on the safe ground of CED. To compare with the previous models, SRF predicts that the frequencies of the LSPs of a small sphere are given, in analogy with Eq. (4), by [187]

(21)$$\varepsilon_{\mathrm{m}} + \frac{\ell + 1}{\ell} \varepsilon - \left(\varepsilon_{\mathrm{m}} - \varepsilon \right) \frac{\ell + 1}{R} \left(d_{{\perp}} - d_{{\parallel}} \right) = 0~.$$

Unlike Eq. (4), this expression cannot be straightforwardly solved for the LSP frequencies $\omega _{\ell }$, since the $d$ parameters are also frequency-dependent. One can solve it numerically or graphically for given surface-response functions, or try to obtain a perturbative solution for small shifts from the LRA result which, in the case of a sphere (and assuming $\varepsilon = \varepsilon _\infty = 1$) leads to [187]

(22)$$\omega_{\ell} \simeq \omega_{\ell,\mathrm{LRA}} \left[ 1 - \frac{1}{2} \left( \frac{\ell + 1}{R} d_{{\perp}}^{(0)} - \frac{\ell + 1}{R} d_{{\parallel}}^{(0)} \right) \right]~,$$

where $\omega _{\ell,\mathrm {LRA}}$ are the frequencies given by Eq. (4), and $d_{\perp,\parallel }^{(0)}$ corresponds to the real part of the corresponding function calculated at $\omega _{\ell,\mathrm {LRA}}$.

As a final comment on SRF, it should be noted that the appropriate $d$ parameters to describe any specific problem must be obtained for the corresponding metal–dielectric interface, so as to take environment effects into account. But once this has been done, the obtained parameters can subsequently be used as tabulated data [191], in the same way measured permittivities are used in LRA calculations. In order to be practically useful, such calculations should be performed for flat interfaces, rather than depending on the shape and curvature of the specific NPs. Assuming a “local flatness” for NPs is a good approximation as long as the curvature does not exceed $\sim 1$ nm [179]; but further work is still needed to establish the exact capabilities and limitations of the method, including very recent endeavours in modal analyses [196,197].

5. Landau damping

Although Landau damping stems from nonlocality, the richness of the physical implications that are associated with it requires an individual discussion of the matter. Figure 3 provides a simplistic picture of the band structure in a metal and the origin of this damping mechanism; let us assume two states with energies $E_{1}$ below the Fermi energy, and $E_{2} = \hbar \omega + E_{1}$ above it, characterised by wavevectors $\mathbf {k}_{1}$ and $\mathbf {k}_{2} = \mathbf {k}_{1} + \Delta \mathbf {k}$, respectively. Typically, an incident photon carrying a wavenumber $k \approx 0$, or an SPP, would not suffice for the direct transition from state 1 to state 2 [198]. But collisions with the boundaries of the particle provide the momentum $\Delta k \geq \omega / v_{\mathrm {F}}$ required for the photon absorption and the subsequent decay of the plasmon to electron-hole pairs. As might be expected, the additional energy dissipation channel manifests itself as a broadening of the plasmon resonance in the spectra.

Since Landau damping is associated with electron-boundary collisions, it goes without saying that it most likely occurs near the surface of the metal, rather than in bulk, justifying, thus, the full term surface-enhanced/enabled Landau damping. Naturally, it becomes increasingly more apparent as the NP size reduces, coexisting with —and maybe dominating over— other relaxation mechanisms. References [80,198] provide an interesting discussion regarding the role of Landau damping in small metallic structures as the physical limit of plasmonic field confinement and enhancement. In plasmonic dimers, however, this feature has also been attributed to electron tunnelling, as we will briefly discuss in Sec. 7.

The increased damping in small spheres was observed quite a few decades ago and studied more extensively by Kreibig and colleagues, who ascribed it to a decreased electron mean free path due to scattering off the NP surface [87,89,93], describing essentially the same process as Landau damping. Kreibig’s postulation boils down to the addition of a phenomenological correction to the damping parameter [199], that describes the rate at which electrons collide to the metal walls,

(23)$$\gamma \to \gamma_{\mathrm{bulk}} + A\,\frac{v_{\mathrm{F}}}{R}~,$$

where $A$ is a dimensionless constant (empirically $A \approx 1$) and $R$ the radius of the spherical NP, or any other characteristic dimension of a non-spherical configuration. In practice, the inverse proportionality relation $\gamma \propto 1/R$ reflects the surface-to-volume ratio $S/V$; in the same vein as Landau damping, in the bulk limit the surface collision damping rate is generally negligible, but it is enhanced near the surface. In sufficiently small particles, where the mean free path of the electrons is comparable to the dimension of the structure $R \sim v_F/\gamma _\mathrm {bulk}$, this term becomes comparable to the other “bulk” contributions. Alternatively, one can try to absorb the additional damping into an altogether modified permittivity for the bulk metal [93,200]. In nonlocal models, Landau damping can be described by the diffusion term in GNOR, or by the imaginary part of the Feibelman parameters in SRF, as discussed in the previous section.

6. Quantum confinement

The conduction-electron energy spectrum of a metallic particle is typically treated macroscopically, and thus considered to form a continuous band. For small metallic systems, however, that accommodate only a limited number of conduction electrons, the continuous band breaks into discrete energy levels, and the macroscopic approach is no longer valid. Quantum confinement manifests itself through the formation of a fine structure, owing to transitions between the energy levels, and a size-dependant resonance broadening [201,202]. Fröhlich [203] first discussed the consequences of the energy-state discreteness as manifested in the thermodynamic properties of a small metallic system, while Kubo and Kawata [204,205] were the ones to systematically address the problem.

As the system dimensions are reduced down to sizes comparable to the distance travelled by electron during one optical cycle ($R \sim fv_F$, where $f$ is the optical frequency [206]), the Drude permittivity gradually fails to describe the system, and a quantum approach for the dielectric function need be implemented. A simple model to treat the electrons is that of an electron gas constrained within an infinite cubic potential well of side $L$ [76,93,207,208]. To account for all allowed transitions from each initial state $i$ to each final state $f$ of $N$ conduction electrons, the dielectric function takes the form

(24)$$\varepsilon_{\mathrm{m}} (\omega) = \varepsilon_{\infty} + \frac{\omega_{\mathrm{p}}^{2}}{N} \sum_{i, f} \frac{S_{if} (F_{i} - F_{f})} {\omega_{if}^{2} -\omega^{2} - \mathrm{i} \omega \gamma_{if}}~,$$

where $S_{if}$, $\omega _{if}$, and $\gamma _{if}$ are the oscillator strength, eigenfrequency, and damping rate for each transition respectively, and $F_{i,f}$ denote the values of the Fermi-Dirac distribution function for the initial and final states.

The quantum box model effectively serves as an alternative and more refined approach for predicting SDB to the coarse approximation of the Kreibig term, while also describing quantum effects not taken into account by the latter [209,210] [see Fig. 4(a)]. In practice, the NP sizes for which fine structure due to quantum confinement prevails are of the order of $1$ nm [211], but examples of such quantisation have been reported experimentally in gold clusters [212,213] and theoretically not only for small metallic NPs [194], but even for larger graphene nanostructures [214]. For a description that goes beyond this quantum box model, one can always employ atomistic quantum models, typically in various incarnations and variations of TDDFT [215–218], which also allow the validation of the semiclassical models described above [68,219]. Recently, the generalised plasmonicity index was introduced as a means to distinguish between different excitations of quantum origin in small NPs, and clearly identify which resonances can be attributed to LSPs [220].

Fig. 4. TDDFT studies of quantum effects. (a) Optical response of a $100$ electron gold-jellium sphere, $0.74$ nm in radius. In addition to the peaks corresponding to the classical LSPs, the spectrum exhibits several others, revealing a second type of collective oscillations, the quantum core plasmons (Reprinted with permission from [194], Copyright 2012, American Chemical Society). (b) Maximum field enhancement in a plasmonic dimer for various interparticle separations (A to D), as calculated within LRA (red) and TDLDA (blue), and (c) field distributions calculated within LRA (top panels) and TDLDA (bottom panels) for the same interparticle separations [Reprinted (adapted) with permission from [193], Copyright 2009, American Chemical Society].

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7. Quantum tunnelling

Metallic dimers have proven an intriguing platform for quantum-plasmonic studies, exposing the limits of classical descriptions and challenging our understanding of the dominating underlying mechanisms [61,70,72,80,221]. Plasmon-induced surface charges in metallic NPs separated by a small enough nanogap can overcome the classically prohibited potential barrier and tunnel across the gap between the individual structures, a process known as quantum tunnelling. In practice, these screened dimer LSPs may manifest as a gap size-dependent resonance broadening in the spectra, owing to the tunnelling currents obstructing the capacitive mechanism governing the dimer by reducing the Coulomb coupling of charges of opposite signs between the two constituents [222,223], while charge-transfer plasmons (CTPs) emerge at longer wavelengths, as the result of the newly-formed conducting path and the corresponding effective elongation of the resonator.

Studies on plasmonic tunnelling have only recently commenced, with Zuloaga et al. [193] employing TDDFT to examine the optical properties of plasmonic dimers separated by small gaps, and showing the transition of the system from the classical to the quantum regime as the interparticle distance reduces from a few nm to touching NPs [see Figs. 4(b) and (c)]. A method for emulating the effect of the tunnelling mechanism to the optical properties of dimers of mesoscopic size is offered by the quantum corrected model (QCM) by means of a fictitious conductive material filling the interstitial space, incorporating thus quantum information in a CED framework [224,225]. In this context, the conductivity $\sigma _{0}$ of the junction is given by

(25)$$\sigma_{0} (l) = \frac{l}{2\pi^{2}} \int_{0}^{E_{\mathrm{F}}} \mathrm{d} E \, T(E, l),$$

where $E_{\mathrm {F}}$ is the Fermi energy of the metal and $T(E, l)$ the probability for the electrons to tunnel through the gap —treated as a flat-plane junction— as a function of energy $E$ and the separation distance $l$, as illustrated in Fig. 3. The tunneling damping rate $\gamma _{\mathrm {g}}$ characterising the junction is then obtained from

(26)$$\gamma_{\mathrm{g}} = \frac{\omega^2_{\mathrm{g}}} {4 \pi \sigma_{0} (l)}~.$$

Here, $\omega _{\mathrm {g}}$ is the plasma frequency of the junction; typically, it is assumed equal to that of the bulk metal. With these two parameters, $\omega _{\mathrm {g}}$ and $\gamma _{\mathrm {g}}$, one can readily describe the junction as a Drude metal in any CED calculation, and thus effectively treat tunnelling phenomena [226,227]. Either ab initio calculations or a model potential approach can be implemented to derive the tunnelling probability $T(E, l)$. Experimental observations have successfully captured the gap-dependent broadening, revealing at the same time the natural limit of plasmonic field enhancement; the maximum attainable near-field enhancement is suppressed significantly, as the electrons enter the tunnelling regime [55,70,228,229].

Despite the undeniable success of QCM in predicting tunnelling effects and the emergence of CTPs, it has received some criticism regarding its role on (and whether it is sufficient to explain) the broadening effect in dimers, triggering a fruitful discussion in the photonics community. An extensive discussion on the matter can be found in Ref. [63], where it is argued that relaxation associated with the tunnelling currents coexists with the relaxation due to Landau damping, with the latter mechanism being more likely to dominate the dimer dynamics. To that aim, Ref. [230] showed that, at optical frequencies, the gap-dependent broadening can indeed be solely attributed to Landau damping. At the same time, one should take into consideration whether the charge transfer tunnelling time is sufficiently small to keep ahead of the polarity reversal of the optical field. On the other hand, it is not unreasonable to assume that the damping rate of Eq. (26), calculated with TDDFT, already contains Landau damping. Existing theoretical models alone do not seem capable of resolving this debate, and additional efforts, combined with experimental testings, are required to shed more light on the dominant mechanism. In any case, both approaches have their own merits, and the true reason for damping will most likely contain a combination of these effects.

8. A simple example

Before ending this discussion, let us examine briefly how all the aforementioned models apply to the simplest possible example, the extinction spectra of a small metallic nanosphere. Our metal of choice will be sodium, because it is one of those metals with low work function where spill-out is significant and not obscured by screening, and qualitative differences are expected between the traditional hydrodynamic models (e.g. HDM, but not SC-HDM) and SRF. In what follows, sodium is described by the Drude dielectric function of Eq. (2), with $\hbar \omega _{\mathrm {p}} = 5.89$ eV, $\hbar \gamma = 0.1$ eV, $\varepsilon _{\infty } = 1$. In the HDM/GNOR descriptions, we take $v_{\mathrm {F}} = 1.06 \times 10^{6}$ m s$^{-1}$, $\beta = \sqrt {3/5} v_{\mathrm {F}}$, and $D = 2.67 \times 10^{-4}$ m$^{2}$ s$^{-1}$ [62], while the Feibelman $d_{\perp }$ parameter is taken from Ref. [190] and $d_{\parallel } = 0$.

In Fig. 5(a) we first compare the extinction spectra of this sodium sphere (sketched at the centre of the figure) as obtained within LRA (grey dashed line) and SDB, with $A$ in Eq. (23) taken equal to $1$. As expected for a modified damping rate, which has left the dielectric function and overall material response otherwise unaffected, the extinction spectrum is significantly broadened, but the frequency of the dipolar LSP remains unchanged. Figure 5(b) shows the corresponding spectrum obtained within HDM, in its most simple implementation, i.e. the hard-wall boundary condition. Here, the presence of longitudinal field components means that the induced charges have a spatial extent, rather than being accumulated exactly at the metal-dielectric interface; the NP responds as effectively smaller, leading to the blueshift of LSPs as compared to the LRA prediction. In terms of their magnitude, the plasmonic resonances remain nearly unaltered because, for shifts of the order of $0.1$ eV or less that are usually observed, the dielectric function of the metal has changed only slightly. HDM provides a quite successful description of the response of noble-metal nanostructures, made of silver or gold, because these metals have high work functions and significant screening due to d-band electrons [231] —which HDM mimics by smearing the induced charges. The screening effect is also reflected in the negative real part of the $d$ parameters of noble metals, where the frequency blueshifts predicted by hydrodynamic models, unlike in the case of sodium, are verified with SRF [191].

Fig. 5. Influence of non-classical models on the optical response of a sodium nanosphere (central schematic). The sphere has a radius $R = 5$ nm, and is described by a Drude model with $\hbar \omega _{\mathrm {p}} = 5.89$ eV, $\hbar \gamma = 0.1$ eV, $\varepsilon _{\infty } = 1$, while $v_{\mathrm {F}} = 1.06 \times 10^{6}$ m s$^{-1}$. In all panels, the grey dashed line corresponds to the LRA calculation. (a) Broadening of the spectrum according to SDB [black line, Eq. (23)]. (b) Nonlocal screening, as described by HDM (light blue line), with $\beta = \sqrt {3/5} v_{\mathrm {F}}$. (c) Convective–diffusive screening and broadening, as described by GNOR (purple line), with $D = 2.67 \times 10^{-4}$ m$^{2}$ s$^{-1}$. (d) Electron spill-out and surface-enabled Landau damping, as described by SRF, with $d_{\perp }$ obtained from Ref. [190].

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As it becomes immediately clear, the additional Landau damping is absent in HDM. To compensate for this, GNOR includes not only convection but also diffusion in the description of the electronic fluid; diffusion can indeed be considered as an absorption mechanism, since random collisions between electrons enhance losses. In Fig. 5(c) the GNOR spectrum for the sodium sphere of interest shows both the nonlocal blueshift of HDM, and the same resonance broadening as in the corresponding SDB calculation. But the biggest criticism to hydrodynamic models concerns the trends of the resonance shifts in alkali metals, which are in qualitative disagreement with both experimental measurements [92] and TDDFT calcuations [68]. SRF manages to address this issue efficiently, as shown in Fig. 5(d), where frequency redshifts, together with a significant degree of broadening, can be observed. Direct comparison of non-classical models with experiments can now clarify to which extent nonlocal damping should be attributed to surface effects, and how much of it originates from the bulk [232].

9. Conclusion

We have described finite-size and quantum effects that appear in plasmonics, becoming increasingly more relevant as nanofabrication makes metallic NPs of sizes of just a few nm, or NP aggregates with separations even below $1$ nm readily available. We discussed nonlocal screening, Landau damping, electron spill-out, tunnelling, and quantum confinement; what are their origins, how they manifest experimentally, and how one can implement them in hybrid, mesoscopic models that unify quantum and classical calculations. While exploration of these phenomena first emerged from the need to explain deviations between theoretical predictions within LRA and experimental measurements, and subsequently out of curiosity to establish what the dominant mechanisms governing the optical response of small plasmonic systems truly are, various applications are now being influenced by this progress. Tunnelling effects, for instance, are the basis for the design of efficient plasmonic tunnel junctions for controlling photocurrents [233], for biosensing [234], or for generating hot carriers [235]. On the latter topic, Landau damping probably constitutes the main driving force [236], with significant impact on applications for photocatalysis and photodetectors [237], as well as in the design of photovoltaics and sensors [238]. We hope that this review, though not comprehensive, and influenced by our own research interests, will provide a good introduction to newcomers to this area of plasmonics, prove useful as a tutorial, and inspire more people to get involved in this research, which combines elements from optics, solid-state physics, nanotechnology and computational physics.

Funding

Villum Fonden (16498).

Acknowledgments

We thank Asger Mortensen for discussions and feedback on the manuscript. P. E. S. is the recipient of the Zonta Denmark’s Scholarship for female PhD students in Science and Technology 2021.

Disclosures

The authors declare no conflicts of interest.

Data availability

The data that support all original results in this review are available from the corresponding author upon reasonable request.

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Finite-size and quantum effects in plasmonics: manifestations and theoretical modelling [Invited]

Abstract

1. Introduction

2. Local response approximation

3. Evidence of the inadequacy of LRA

4. Nonlocal effects

5. Landau damping

6. Quantum confinement

7. Quantum tunnelling

8. A simple example

9. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (30)

Optical Materials Express

P. Elli Stamatopoulou	https://orcid.org/0000-0001-9121-911X
Christos Tserkezis	https://orcid.org/0000-0002-2075-9036