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Surface-response functions obtained from equilibrium electron-density profiles

  • N. Asger Mortensen ORCID logo EMAIL logo , P. A. D. Gonçalves ORCID logo , Fedor A. Shuklin , Joel D. Cox , Christos Tserkezis , Masakazu Ichikawa and Christian Wolff
Published/Copyright: May 7, 2021
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Nanophotonics
From the journal Nanophotonics Volume 10 Issue 14

Abstract

Surface-response functions are one of the most promising routes for bridging the gap between fully quantum-mechanical calculations and phenomenological models in quantum nanoplasmonics. Among all currently available recipes for obtaining such response functions, the use of ab initio methods remains one of the most conspicuous trends, wherein the surface-response functions are retrieved via the metal’s non-equilibrium response to an external time-dependent perturbation. Here, we present a complementary approach to approximate one of the most appealing surface-response functions, namely the Feibelman d-parameters, yield a finite contribution even when they are calculated solely with the equilibrium properties of the metal, described under the local-response approximation (LRA) but with a spatially varying equilibrium electron density, as input. Using model calculations that mimic both spill-in and spill-out of the equilibrium electron density, we show that the obtained d-parameters are in qualitative agreement with more elaborate, but also more computationally demanding, ab initio methods. The analytical work presented here illustrates how microscopic surface-response functions can emerge out of entirely local electrodynamic considerations.

Keywords: electrodynamics; Landau damping; nonlocal response; quantum plasmonics; surface-response formalism

1 Introduction

The plasmonic response of metallic nanostructures is commonly explored within the framework of classical electrodynamics [1], typically describing the free electrons of metals classically within the Drude-like local-response approximation (LRA) [2]. The classical LRA prescription thus treats a metal as a homogeneous gas of noninteracting electrons confined by a hard wall at the metal’s surface. In this fashion, any aspects of nonlocal (i.e., q-dependent) response [3], [4], [5] are commonly neglected both in the bulk of the metal (e.g., finite compressibility of the Fermi gas) and at its surface (e.g., Friedel oscillations and electronic spill-out associated with a finite work function).

Despite neglecting quantum-mechanical effects, the LRA has constituted a critical theoretical framework in the overall development of plasmonics [2], [6], [7]. More recently, the importance of quantum phenomena has been pursued via classical accounts, including smooth equilibrium electron-density profiles [8], [9], [10], semiclassical hydrodynamic models [11], [12], [13], and ab initio studies [14], [15]. The former approaches can be criticized for only dealing with some quantum aspects semiclassically, while the latter are typically restricted by their complexity and by their practical applicability to small plasmonic systems [16], [17], [18], [19], [20]. In this context, surface-response functions aim to capture the dominant quantum phenomena and microscopic aspects of the surface, while still allowing for a (semi)classical treatment of the light–matter interactions in the bulk of the metal. As such, there has recently been a renewed interest in electrodynamic surface-response functions [21], [22], [23] in the context of plasmon-enhanced light–matter interactions [24], [25], [26], [27] and quantum plasmonics [28], [29], [30], emphasizing their importance in plasmon–emitter interactions in nanoscale environments [26], [27], plasmon-enhanced interactions with two-dimensional (2D) materials [27], [31], and in revealing the detailed spectral properties of plasmon resonances themselves [18], [26], [32], [33], [34], [35], [36].

Traditionally, surface-response functions have been obtained through first-principle calculations of the electrodynamics of metal surfaces subjected to time-varying electric fields [37], e.g., by employing time-dependent density-functional theory (TDDFT) [14], while they can, in some cases, also be analytically evaluated, e.g., from semiclassical hydrodynamic models [34], [38], [39], [40]. In all cases, the common strategy has been to first evaluate the non-equilibrium response to obtain the induced charge density, ρ ind ω ; r n ̂ , and extract from it the surface-response function(s), e.g., the Feibelman d ⊥-parameter (corresponding to the centroid of induced charge density [21]). Here, we explicitly show that even when using a local-response approach along with the equilibrium electron density-profile alone as input, there is a finite contribution to the metallic surface-response functions provided that the (equilibrium) electron density varies smoothly from its bulk value deep inside the metal to zero near the metal’s surface [41], [42], [43] (as opposed to terminating abruptly at it). Such an approach, despite its simplicity and inherent limitations, could nevertheless facilitate new physical insights into the electrodynamic fingerprints associated with quantum spill-out/spill-in, without resorting to computationally demanding ab initio methods. We should, however, emphasize that the approach presented below ignores the finite compressibility of the electron gas, already captured by semiclassical hydrodynamic models [5], as well as more complicated many-body effects and correlations that time-dependent ab initio methods seek to capture, while naturally still resorting to approximations [14].

2 Results

We consider a metallic nanostructure where n 0(r) is the equilibrium electron density (see Figure 1a), which is spatially inhomogeneous in the vicinity of the metal’s surface, possibly including, e.g., quantum spill-out and/or Friedel oscillations [44] due to a finite work function [45]. In the presence of time-harmonic electromagnetic fields, the electrodynamics of the system is governed by the integro-differential wave equation

(1) ∇ × ∇ × E ( r ) = ω 2 c 2 ∫ d r ′ ε ( r , r ′ ) E ( r ′ ) ,

where ω is the angular frequency, c is the speed of light in vacuum, and ɛ(r, r′) is the nonlocal linear-response function, i.e., the (nonlocal) dielectric function of the quantum electron gas (here assumed to be isotropic, for the sake of simplicity). The microscopic and analytical understanding of ɛ(r, r′) is in general limited to bulk considerations within the random-phase approximation (RPA) or the hydrodynamic model (HDM) [4], [5], [27], [41], [46], [47].

Figure 1: Schematic representation of the microscopic features of a metal–vacuum interface. (a) Metal–vacuum interface, indicating the surface region where the electron density varies from its asymptotic, bulk values ɛ m ≡ ɛ LRA(z < z 1) and ɛ d ≡ ɛ LRA(z > z 2) = 1 (where |z 1,2| ≫ 0). (b) Top: schematic of the (normalized) equilibrium electron-density profile n ̄ 0 ( z ) ${\bar{n}}_{0}\left(z\right)$ characterized by a smearing length a in the vicinity of the surface (here defined by the z = 0 plane). Bottom: Real part of the system’s dielectric function Re ɛ LRA(z) [Eq. (8)] associated with n ̄ 0 ( z ) = [ 1 − tanh ( z / a ) ] / 2 ${\bar{n}}_{0}\left(z\right)=\left[1-\mathrm{tanh}\left(z/a\right)\right]/2$ , along with the ensuing Re E z (z) and Re ρ ind(z) [note that ρ ind ∝ ∂ z ε LRA − 1 ${\rho }_{\text{ind}}\propto {\partial }_{z}{\varepsilon }_{\text{LRA}}^{-1}$ in the long-wavelength regime]. All quantities are in arbitrary units. Parameters: ω = ω p / 3 $\omega ={\omega }_{\text{p}}/\sqrt{3}$ , and for visualization purposes a Drude-type bulk damping of γ/ω p = 0.3.
Figure 1:

Schematic representation of the microscopic features of a metal–vacuum interface. (a) Metal–vacuum interface, indicating the surface region where the electron density varies from its asymptotic, bulk values ɛ m ≡ ɛ LRA(z < z 1) and ɛ d ≡ ɛ LRA(z > z 2) = 1 (where |z 1,2| ≫ 0). (b) Top: schematic of the (normalized) equilibrium electron-density profile n ̄ 0 ( z ) characterized by a smearing length a in the vicinity of the surface (here defined by the z = 0 plane). Bottom: Real part of the system’s dielectric function Re ɛ LRA(z) [Eq. (8)] associated with n ̄ 0 ( z ) = [ 1 − tanh ( z / a ) ] / 2 , along with the ensuing Re E z (z) and Re ρ ind(z) [note that ρ ind ∝ ∂ z ε LRA − 1 in the long-wavelength regime]. All quantities are in arbitrary units. Parameters: ω = ω p / 3 , and for visualization purposes a Drude-type bulk damping of γ/ω p = 0.3.

2.1 Local-response approximation

In order to proceed with the nonlocal, integro-differential wave equation (1), it is common to invoke further approximations — in the context of plasmonics, the prevailing one being the LRA, epitomized by

(2a) ε ( r , r ′ ) ≈ ε LRA ( r ) δ ( r − r ′ ) .

Here, the inherently finite-range associated with the nonlocal response of the electron gas is neglected in favor of a zero-range, local response (mathematically represented by the Dirac delta function in the previous expression). Physically, this is equivalent to neglecting spatial dispersion represented by a finite wave vector dependence of the dielectric function [4], [5], [27], and thus ignoring, for instance, the finite dynamic compressibility of the electron gas [4], [5]. In spite of this — and as we show in what follows — some quantum aspects associated with an inhomogeneous electron gas (Figure 1a), like electronic spill-out, can still be incorporated to some extent in the LRA. In particular, the LRA reduces the nonlocal wave equation (1) to the familiar local-response one:

(2b) ∇ × ∇ × E ( r ) = ω 2 c 2 ε LRA ( r ) E ( r ) ,

which is conceptually simpler and computationally more tractable [48].

2.2 Piecewise-constant approximation (PCA)

Inspired by the long-established traditions in the electrodynamics of composite dielectric problems [49], it is common in plasmonics [2] to invoke yet another approximation: the step-like, abrupt surface termination of the metal, thereby neglecting any microscopic inhomogeneities in the vicinity of the surface [herein defined by z = 0, without loss of generality, with the metal and the dielectric each occupying the z < 0 and z > 0 half-spaces, respectively (Figure 1a)]. Under this approximation, ɛ LRA(z) → ɛ PCA(z), with

(3a) ε PCA ( z ) ≡ ε LRA ( − ∞ ) Θ ( − z ) + ε LRA ( ∞ ) Θ ( z )

(3b) ≡ ε m Θ ( − z ) + ε d Θ ( z ) ,

where Θ is the Heaviside step function, and the system’s dielectric function is constructed from two interfacing piecewise-constant (bulk) local-response functions, ɛ m ≡ ɛ m(ω) and ɛ d ≡ ɛ d(ω) (and Eq. (2b) is then solved by invoking the classical pillbox arguments at the interface [1]). Here, ɛ m is the Drude-like dielectric function of the free-electron gas [2], [11]

(4) ε m = ε + − ω p 2 ω 2 + i ω γ ,

with ɛ + ≡ ɛ +(ω) allowing for the incorporation of the polarization due to the positive ionic background or for a heuristic account of interband transitions. It should be emphasized that the PCA has been tremendously successful in advancing the field of plasmonics, being sufficient to interpret the majority of experimentally observed phenomena [2]. What makes the PCA legitimate in most cases is the fact that the electron density is only non-uniform across an extremely small region in the vicinity of the metal surface, typically spanning only a few ångströms (i.e., on the order to the metal’s bulk Fermi wavelength, λ F). In spite of this, such a “classical” PCA is currently being challenged by the recent developments in nanoscale plasmonics and plasmon-empowered light–matter interactions at nanometric scales [15], [25], [26], [27], [31], [35], [50].

2.3 Surface-response formalism

In the PCA, the induced charge is strictly a (singular) surface charge, i.e., ρ ind(z) ∝ δ(z) [1], [21], [27], while in reality, the induced charge actually assumes a nonsingular density of a finite, surface-peaked nature (Figure 1b). In this context, the Feibelman d-parameters, d ⊥ ≡ d ⊥(ω) and d ∥ ≡ d ∥(ω), are dynamical surface-response functions that correspond to the first moment (i.e., the centroid) of the induced charge density and of the normal derivative of the tangential current density, given, respectively, by (ω-dependence implicit) [21]

(5) d ⊥ = ∫ − ∞ ∞ d z z ρ ind ( z ) ∫ − ∞ ∞ d z ρ ind ( z ) , d ∥ = ∫ − ∞ ∞ d z z ∂ ∂ z J x ind ( z ) ∫ − ∞ ∞ d z ∂ ∂ z J x ind ( z ) ,

which are complex-valued surface-response functions, i.e., d α (ω) = d α ′(ω) + id″ α (ω) with α ∈ {⊥, ∥}. The general appeal of the d-parameters is that, once they are obtained, the system’s optical response can be calculated by solving a d-parameter-modified electrodynamic problem, namely, the LRA wave Eq. (2b) together with the “classical” PCA [recall Eq. (3a)] but now subjected to the d-parameter-corrected, mesoscopic boundary conditions [26], [27], [33], [34], [35]. Computationally, this is clearly more attractive than having to solve the more complex integro-differential problem typified by Eq. (1), while at the same time such reformulation into a quantum-informed “classical-equivalent” electrodynamic problem also paves the way for further analytical work [26], [27], [34]. Naturally, different mechanisms can be incorporated (together or separately) via the d-parameters, e.g., nonlocality, quantum spill-out/spill-in, Landau damping, etc. [21], [51]. In the following, we limit our consideration to the LRA contribution to the d-parameters emerging solely from a spatially varying dielectric function, i.e., ɛ LRA(z).

Alternatively to Eq. (5), the d-parameters can also be written in terms of surface integrals associated with the difference between the actual, microscopic fields and the classical, “Fresnel” fields stemming from the PCA [21], [22], [52], [53], [54], [55], [56], [57], [58], specifically (see Supplementary Material):

(6a) d ⊥ = − ε d ε m − ε d ∫ − ∞ ∞ d z E z ( z ) − E z PCA ( z ) E z PCA ( 0 − ) ,

(6b) d ∥ = 1 ε m − ε d ∫ − ∞ ∞ d z D x ( z ) − D x PCA ( z ) ε 0 E x PCA ( 0 − ) ,

where E x , z PCA , D x PCA are fields obtained within the classical, piecewise-constant approach. In the long-wavelength regime and to leading-order in q|z 2 − z 1|, the Feibelman d-parameters (6) associated with a local, but smoothly varying dielectric function ɛ LRA(z) can be written as [55], [59], [60], [61], [62] (see Supplementary Material)

(7a) d ⊥ = 1 ϵ m − 1 − ϵ d − 1 ∫ − ∞ ∞ d z ε LRA − 1 ( z ) − ε PCA − 1 ( z ) ,

(7b) d ∥ = 1 ϵ m − ϵ d ∫ − ∞ ∞ d z ε LRA ( z ) − ε PCA ( z ) .

Equations (7) unambiguously illustrate how ɛ LRA(x) ≠ ɛ PCA(x) contributes to a finite d ⊥ and d ∥. Naturally, in general, there will be further contributions to the d-parameters stemming from the nonlocal response of the electron gas (e.g., treated within the nonlocal RPA or the HDM); nevertheless, it is important to emphasize that there is a nonzero contribution to the surface-response already within the LRA once the PCA is relaxed. In the following, we shall illustrate this in more detail with an elementary model that elucidates the physics — within the constraints associated with the LRA — of both spill-out and spill-in of the metal’s electron density. Despite its inherent simplicity, the strength of the simple model adopted below lies in its ability to render analytical results in closed form.

2.4 Metal surface with a smoothly varying electron density

As mentioned previously, a more realistic representation of a metal surface is to abandon the assumption of an infinitely sharp dielectric–metal interface and instead allow the metal’s electron density to vary smoothly from its value deep inside the metal, n 0 bulk ≡ n 0 ( z → − ∞ ) , to zero well inside the vacuum (Figure 1). This can be modeled through a simple generalization [21], [38], [41], [42], [43], [63], [64] of Eq. (4), that is

(8) ε LRA ( z ) = ε ∞ ( z ) − ω p 2 ω 2 + i ω γ n ̄ 0 ( z ) , n ̄ 0 ( z ) = n 0 ( z ) n 0 bulk .

where n 0(r) ≡ n 0(z) is the spatial profile of the equilibrium electron density. Here, ɛ ∞(z) takes into account the variation from the background polarization, subjected to the requirement that deep inside the metal (dielectric) it converges to the polarization due to the jellium background of positive ions, ɛ ∞(z → −∞) = ɛ + (to the dielectric’s permittivity ɛ ∞(z → +∞) = ɛ d). As a complementary perspective, this can also be interpreted as the common local response of the Drude kind, but with a spatially varying plasma frequency, ω p ( z ) ≡ ω p n ̄ 0 ( z ) . In passing, we note that Eq. (8) has been used widely over the years, including Refs. [8], [9], [65], [66], [67], [68], [69], [70], while smoothly varying profiles have also been considered in model discussions of local-field corrections [71]. Finally, we note how the PCA mathematically emerges upon replacing n ̄ 0 ( z ) by a Heaviside function in Eq. (8), i.e., n ̄ 0 ( z ) → Θ ( − z ) , corresponding to the classical, step-like termination of the equilibrium electron density.

2.5 Transition from spill-in to spill-out

To illustrate the transition from spill-in to spill-out, we consider a model electron-density profile of the form [62]

(9) n ̄ 0 ( z ) = tanh 2 z − z 0 a Θ ( z 0 − z ) ,

which is smooth and has the desired properties lim z → − ∞ n ̄ 0 ( z ) = 1 and lim z → + ∞ n ̄ 0 ( z ) = 0 [in fact, the latter can be made more stringent, e.g., lim z → z 0 n ̄ 0 ( z ) = 0 ]. The value z 0 indicates the position where the metal’s electron density vanishes, whereas the quantity a charactokerizes the steepness of the spatial profile of the (normalized) equilibrium electron density [with lim a → 0 n ̄ 0 ( z ) = Θ ( z 0 − z ) ]. The quantity z 0, in particular, governs whether the induced electron density spills inwards or outwards. For bulk electron densities of typical plasmonic metals, both a and z 0 amount to a few ångströms, and the model qualitatively captures the main results of self-consistent jellium considerations [45], while more refined models are needed to also represent finer details, e.g., Friedel oscillations [44], [72].

Further, we assume that the transition from the jellium background (i.e., the metal’s positively charged ions) to the dielectric remains infinitely sharp because these only contain tightly bound electrons which are thus essentially immobile[1] when compared with the conductive free-electrons; hence, in the following we take

(10) ε ∞ ( z ) = ε + Θ ( − z ) + ε d Θ ( z ) ,

where we have assumed, without loss of generality, that the edge of jellium background is located at z b = 0. In passing, we note that if we enforce charge neutrality, then a and z 0 are not independent, and d ∥ = 0 (for the electron-density profile considered here (9), this would set z 0 = a). In spite of this, in what follows we assume that a and z 0 can be varied independently, as this may facilitate the treatment of charged metal surfaces (i.e., arising from either surface roughness, molecular adsorption, or the presence of Shockley surface states).

Going beyond jellium models, we note that care should be taken when turning to atomistic representations of the surface, where the choice of origin is reflected in the corresponding surface-response functions [36] (although the overall quantum surface-response is unchanged provided that both d ⊥ and d ∥ are considered).

2.6 Simple jellium next to vacuum

For clarity purposes, we first ignore background polarization effects or interband transitions and consider a simple jellium–vacuum interface, so that ɛ + = ɛ d = 1. In this case, the integrals in Eqs. (7) yield

(11a) d ⊥ ( Ω ) = z 0 − a Ω ̃ a r c t a n h Ω ̃ − 1 ,

(11b) d ∥ ( Ω ) = z 0 − a .

where Ω = ω/ω p and Ω ̃ = Ω ( Ω + i Γ ) , with Γ = γ/ω p. As we shall see, the frequency-independent result for d ∥ is a particular consequence of having assumed ɛ + = ɛ d. In the absence of bulk damping (Γ → 0+), Eq. (11a) can be written as [62]

(12) d ⊥ ( Ω ) = z 0 + a Ω 2 ln Ω − 1 Ω + 1 + i π Θ ( 1 − Ω ) ,

with the low-frequency behavior of d ⊥ given by

(13a) R e d ⊥ ( Ω ≪ 1 ) ≃ z 0 ,

(13b) I m d ⊥ ( Ω ≪ 1 ) ≃ a π 2 Ω .

Notice that, even in the absence of bulk damping, there is a nonzero contribution of surface-assisted damping embodied through Im d ⊥ ≠ 0 [see Eq. (13b)]. More fundamentally, this is a consequence of Kramers–Kronig relations (wherein a dispersive Re d ⊥ renders Im d ⊥ ≠ 0) [73]. Moreover, we emphasize that the asymptotic limits (13) are in agreement with results emerging from sum-rule considerations [74], [75]. Interestingly, in the above result, z 0 resembles the so-called static image-plane position that emerges from a self-consistent solution of the jellium perturbed by a static field [74], [75], [76], being a quantity of interest in surface science at large (a particular example being that of the van der Waals interaction of an atom near a metallic surface [52], [74]). Recently, acoustic graphene plasmons have been proposed as a mean to probe the quantum surface-response of metals [31] by placing a graphene sheet only a few nanometers away from a metal surface [77], [78]. In particular, the static surface-response, d ⊥(0) [which, within our simple treatment, amounts to z 0; see Eq. (13a)], dependence could be experimentally probed in this way [31].

The results [Eqs. (11)–(12)] for a simple jellium surface next to vacuum are presented in Figure 2, showing how Re d ⊥ is always negative for z 0 = 0 (Figure 2a; black curve). Increasing z 0/a brings the low-frequency part of Re d ⊥ to positive values (Figure 2a; light-red and red curves), potentially extending into the frequency regime ω p / 3 ≤ ω < ω p supporting semiclassical (specifically, within the HDM) localized surface plasmon (LSP) resonances in metal nanoparticles [79]. Consistent with causality and Kramers–Kronig relations, the dispersiveness of Re d ⊥ is accompanied by a finite Im d ⊥ (green, Figure 2a; orange, Figure 2b) [73], [74], [75].

Figure 2: Feibelman d-parameters in the LRA for a jellium–vacuum interface (ɛ + = ɛ d = 1) characterized by a smooth electron-density profile. (a) Real, Re d ⊥ (black, light-red, red), and imaginary part, Im d ⊥ (green) [Eq. (11a)] of the d-parameters for the electron-density profile described in Eq. (9) with varying z 0/a (whose effect is a simple vertical shift of the Re d ⊥ curve); we assume a Drude bulk damping of Γ = γ/ω p = 0.1. (b) Effective surface-response function d eff ≡ d ⊥ − d ∥ [from Eq. (11)]. The dashed curves depict the result in the lossless case [62] [using Eqs. (11b) and (12)]. The grey-shaded region indicates the frequency window supporting semiclassical localized plasmon resonances in metallic nanoparticles.
Figure 2:

Feibelman d-parameters in the LRA for a jellium–vacuum interface (ɛ + = ɛ d = 1) characterized by a smooth electron-density profile. (a) Real, Re d ⊥ (black, light-red, red), and imaginary part, Im d ⊥ (green) [Eq. (11a)] of the d-parameters for the electron-density profile described in Eq. (9) with varying z 0/a (whose effect is a simple vertical shift of the Re d ⊥ curve); we assume a Drude bulk damping of Γ = γ/ω p = 0.1. (b) Effective surface-response function d eff ≡ d ⊥ − d ∥ [from Eq. (11)]. The dashed curves depict the result in the lossless case [62] [using Eqs. (11b) and (12)]. The grey-shaded region indicates the frequency window supporting semiclassical localized plasmon resonances in metallic nanoparticles.

2.7 Dipolar resonance of a metallic nanosphere

To illustrate how the surface-response functions d ⊥ and d ∥ jointly influence the optical response of a metallic nanostructure (Figure 2b), we consider the prototypical case of a spherical nanoparticle of radius R; for simplicity, we take ɛ + = 1 and assume that the nanosphere is in vacuum (ɛ d = 1). Within the classical quasistatic LRA description the spectrum of LSP resonances is dominated by a size-independent dipole resonance at the frequency ω = ω p / 3 [2], [79], [80]. Accounting for nonclassical surface effects in a generalized Clausius–Mossotti relation, the pole associated with the dipolar LSP resonance is, to leading-order in d ⊥,∥/R, given by [26], [27], [34]

(14) 0 = ε m + 2 − ε m − 1 2 d ⊥ − d ∥ R ,

which illustrates how the smearing of the jellium near the surface of the particle causes nonclassical a/R size-dependent redshifts relative to the classical dipole resonance frequency (Figure 2b). Crucially, in this case, i.e., with ɛ + = ɛ d = 1, the “effective” surface-response function d eff ≡ d ⊥ − d ∥ [22], [27], [34] has a “universal” behavior, namely, it is (i) independent of z 0, and (ii) proportional to the smearing of the spatially varying electron-density profile, characterized by the length a. Thus, interestingly, these behaviors indicate that, independently of z 0, the smearing itself contributes to a net nonclassical redshift (Re d eff > 0; spill-out) of the dipolar LSP resonance position of a jellium nanosphere in vacuum [81].

In the following, we simultaneously relax the assumptions of ɛ d = 1 and of ɛ + = 1. Allowing the latter quantity to be larger than unity is commonly used to heuristically incorporate semiclassical accounts of background polarization effects or contributions arising from interband transitions in noble metals [2], [11].

2.8 Background and dielectric screening contributions

Turning to the general case of arbitrary ɛ + and ɛ d, the effort required to perform the integrals (7) becomes somewhat more elaborate, but nevertheless these integrals can still be evaluated analytically, reading (assuming z 0 ≥ 0)

(15a) d ⊥ a = C ⊥ 1 − Ω ̃ 2 ε + 1 − Ω ̃ 2 ε d z 0 a − ε d ε + Ω ̃ a r c t a n h Ω ̃ − 1 ε + − a r c t a n h Ω ̃ − 1 ε + tanh z 0 a   − ε m ε d ε m + ( ε d − ε + ) Ω ̃ a r c t a n h Ω ̃ − 1 ε d tanh z 0 a ,

(15b) d ∥ a = C ∥ z 0 − a a ,

where C ⊥ ≡ 1 + ( ε d − ε + ) Ω ̃ 2 − 1 and C ∥ ≡ (ɛ m − ɛ +)(ɛ m − ɛ d) are both being resonantly enhanced in the vicinity of ω = ω p / ε + − ε d ; this Bennett-type resonance [82], [83] should not be confused with the common surface plasmon resonance occurring at ω = ω p / ε + + ε d . Moreover, contrasting with the previous case (where ɛ + = ɛ d = 1), now both d ⊥ and d ∥ are dispersive (i.e., exhibit frequency dependence). Finally, we note that these factors reduce to C ⊥ = C ∥ = 1 in the ɛ + = ɛ d case. Additionally, in this particular case, d ⊥ and d ∥ are given by Eq. (11) upon replacing Ω ̃ − 1 → Ω ̃ − 1 / ε , where ɛ ≡ ɛ + = ɛ d.

Returning to the discussion surrounding Eq. (14), we note that, in addition to the nonclassical a/R-dependent redshift of the resonance frequency, the term ∝(d ⊥ − d ∥)/R emerging in the pole of the polarizability [26], [34] [the generalized version of Eq. (14) for arbitrary ɛ + and ɛ d] now acquires a finite contribution also from z 0, which may lead to a net blueshift of the dipolar LSP resonance. This behavior is also in line with recent experimental observations of the dependence of quantum size effects on the local dielectric environment of the interface [84] (notice that Eqs. (15) also enable further explorations of situations where ɛ d > 1). As illustrated in Figure 3, the combined effects of a non-unity interband permittivity, ɛ +, and of a finite z 0 may render the redshift of the classical dipolar LSP resonance frequency into a net blueshift, depending on both ɛ + and z 0/a (and also on the particular value of the bulk-damping parameter, γ, which “softens” the sharp feature at ω = ω p; see Figure 2b). In this way, the model conceptually explains how different metals may exhibit contrasting 1/R size-dependencies of their surface plasmon resonances [22], [51], [85], towards the blue for d eff < 0 (spill-in) and toward the red for d eff > 0 (spill-out). An example of the former is silver (characterized by significant interband and valence band screening contributions to the optical response) [22], [36], [86], while an example of the latter is sodium (whose optical response is well described by a simple jellium treatment) [22]. The imaginary part I m d ⊥ − d ∥ is a source of nonclassical 1/R size-dependent broadening [26], [34]. To experimentally resolve nonclassical size-dependent shifts, it is naturally preferable that | R e d ⊥ − d ∥ | ≫ I m d ⊥ − d ∥ , so that the spectral shift is not rendered unobservable due to damping.

Figure 3: Density plot of Re d ⊥ − d ∥ / a $\left({d}_{\perp }-{d}_{\parallel }\right)/a$ in the (ɛ +, z 0)-parameter space, computed at the classical quasistatic dipole LSP resonance frequency, ω = ω p / ε + + 2 $\omega ={\omega }_{\text{p}}/\sqrt{{\varepsilon }_{+}+2}$ , of a spherical particle of radius R. The black dashed line indicates Re d ⊥ − d ∥ = 0 $\left({d}_{\perp }-{d}_{\parallel }\right)=0$ , thus separating regimes with nonclassical 1/R size-dependent spectral redshifts [reddish regions; Re d ⊥ − d ∥ > 0 $\left({d}_{\perp }-{d}_{\parallel }\right){ >}0$ ] from blueshifts [bluish regions; Re d ⊥ − d ∥ < 0 $\left({d}_{\perp }-{d}_{\parallel }\right){< }0$ ]. We have assumed: ɛ d = 1 and γ/ω p = 0.1.
Figure 3:

Density plot of Re d ⊥ − d ∥ / a in the (ɛ +, z 0)-parameter space, computed at the classical quasistatic dipole LSP resonance frequency, ω = ω p / ε + + 2 , of a spherical particle of radius R. The black dashed line indicates Re d ⊥ − d ∥ = 0 , thus separating regimes with nonclassical 1/R size-dependent spectral redshifts [reddish regions; Re d ⊥ − d ∥ > 0 ] from blueshifts [bluish regions; Re d ⊥ − d ∥ < 0 ]. We have assumed: ɛ d = 1 and γ/ω p = 0.1.

3 Discussion and conclusions

In this article, we have revisited the concept of surface-response functions, highlighting that a finite contribution to the Feibelman d-parameters emerges even in an LRA-treatment with a spatially varying equilibrium electron-density profile. While this insight has appeared in some form within the early literature [59], [60], [61], [62], it has seemingly remained unnoticed in the more recent revival of surface-response functions and the widespread use of ab initio accounts for quantum plasmonics. In working out the equilibrium contribution to the dynamic surface-response functions, we have deliberately omitted nonlocal corrections. In this context, the bulk nonlocal hydrodynamic response associated with the quantum compressibility of the electron gas would contribute with a negative Re d ⊥ (well below the plasma frequency, and for a hard-wall jellium–vacuum interface), namely d ⊥ = − β / ω p 2 − ω 2 1 / 2 and d ∥ = 0 [21], [26], [34], [40], with β ∝ v F [5], [27], [87] being a characteristic velocity of longitudinal plasmons. Qualitatively, this could enhance regimes in Figure 3 with a net blueshift, while consequently also reducing the spectral shift in regimes with a net redshift. This possible interplay of quantum compressibility and quantum spill-out is manifested in self-consistent hydrodynamic treatments [13], [88], [89].

In conclusion, our analytical solution of the electrodynamics at metal surfaces transparently and unambiguously illustrates how the microscopic surface-response functions have a finite contribution originating entirely from equilibrium and local-response considerations as input. We believe that this finding offers important insights for the understanding and further advancement of first-principle methods for the computation of accurate surface-response functions, as well as for the experimental exploration of mesoscopic optical phenomena at metal surfaces [35], [84], [86], [90], [91]. The latter is now becoming even more tangible with the advent of ultraconfined acoustic graphene plasmons [27], [31], [77], [92], [93], [94]. Beyond the fundamental interest in surface-response functions, we note that the underlying quantum nonlocal response of metals should also pose fundamental limitations for many light–matter interaction phenomena [95], ranging from plasmon-emitter interaction dynamics [26], [96], [97], through surface-enhanced Raman spectroscopy [98] and plasmon-exciton strong-coupling dynamics [99], to hyperbolic metamaterials [100], non-reciprocal plasmon propagation [101], and the perfect-lens concept [102] — the final example also illustrating the many insightful contributions by Mark Stockman.

Corresponding author: N. Asger Mortensen, Center for Nano Optics, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark; Danish Institute for Advanced Study, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark; and Center for Nanostructured Graphene, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark, E-mail:

Acknowledgement

This paper is dedicated to Mark I. Stockman in appreciation of his pioneering contributions to the broad area of Nano Optics. We thank T. Christensen for valuable discussions and P. M. Frederiksen for facilitating the writing of the manuscript.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: N. A. M. is a VILLUM Investigator supported by VILLUM FONDEN (Grant No. 16498) and Independent Research Fund Denmark (Grant No. 7026-00117B). J. D. C. is a Sapere Aude research leader supported by Independent Research Fund Denmark (Grant No. 0165-00051B). C. W. acknowledges funding from a MULTIPLY fellowship under the Marie Skłodowska-Curie COFUND Action (grant agreement No. 713694). The Center for Nano Optics is financially supported by the University of Southern Denmark (SDU 2020 funding). The Center for Nanostructured Graphene is sponsored by the Danish National Research Foundation (Project No. DNRF103).

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2021-0084).

Received: 2021-03-01
Revised: 2021-04-19
Accepted: 2021-04-19
Published Online: 2021-05-07

© 2021 N. Asger Mortensen et al., published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

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  2. Editorial
  3. A tribute to Mark Stockman
  4. Perspectives
  5. Plasmons compressing the light – a jewel in the treasure chest of Mark Stockman’s legacy
  6. Novel non-plasmonic nanolasers empowered by topology and interference effects
  7. Nanofocusing: reaching out
  8. Spaser or plasmonic nanolaser? – Reminiscences of discussions and arguments with Mark Stockman
  9. Plasmonic nanolasers: fundamental properties and applications
  10. Research Articles
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  12. Surface-response functions obtained from equilibrium electron-density profiles
  13. Effect of nanoscale dielectric environments on concentration quenching
  14. Optical spin–orbit coupling in the presence of magnetization: photonic skyrmion interaction with magnetic domains
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  16. Exploiting space-time duality in the synthesis of impedance transformers via temporal metamaterials
  17. Optical current generation in graphene: CEP control vs. ω + 2ω control
  18. Edge detection with meta-lens: from one dimension to three dimensions
  19. Coherent control at gold needle tips approaching the strong-field regime
  20. Tailoring exceptional points in a hybrid PT-symmetric and anti-PT-symmetric scattering system
  21. Transition to strong coupling regime in hybrid plasmonic systems: exciton-induced transparency and Fano interference
  22. Single-nanoantenna driven nanoscale control of the VO2 insulator to metal transition
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https://doi.org/10.1515/nanoph-2021-0084
Keywords for this article
electrodynamics; Landau damping; nonlocal response; quantum plasmonics; surface-response formalism
Creative Commons
BY 4.0

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. A tribute to Mark Stockman
  4. Perspectives
  5. Plasmons compressing the light – a jewel in the treasure chest of Mark Stockman’s legacy
  6. Novel non-plasmonic nanolasers empowered by topology and interference effects
  7. Nanofocusing: reaching out
  8. Spaser or plasmonic nanolaser? – Reminiscences of discussions and arguments with Mark Stockman
  9. Plasmonic nanolasers: fundamental properties and applications
  10. Research Articles
  11. Space- and time-resolved second harmonic spectroscopy of coupled plasmonic nanocavities
  12. Surface-response functions obtained from equilibrium electron-density profiles
  13. Effect of nanoscale dielectric environments on concentration quenching
  14. Optical spin–orbit coupling in the presence of magnetization: photonic skyrmion interaction with magnetic domains
  15. Anomalous ultrafast all-optical Hall effect in gapped graphene
  16. Exploiting space-time duality in the synthesis of impedance transformers via temporal metamaterials
  17. Optical current generation in graphene: CEP control vs. ω + 2ω control
  18. Edge detection with meta-lens: from one dimension to three dimensions
  19. Coherent control at gold needle tips approaching the strong-field regime
  20. Tailoring exceptional points in a hybrid PT-symmetric and anti-PT-symmetric scattering system
  21. Transition to strong coupling regime in hybrid plasmonic systems: exciton-induced transparency and Fano interference
  22. Single-nanoantenna driven nanoscale control of the VO2 insulator to metal transition
  23. High laser induced damage threshold photoresists for nano-imprint and 3D multi-photon lithography
  24. Onset of charge interaction in strong-field photoemission from nanometric needle tips
  25. Massive surface-plasmon polaritons
  26. VCSEL with multi-transverse cavities with bandwidth beyond 100 GHz
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