Plasmonic modes in cylindrical nanoparticles and dimers

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

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

INTRODUCTION
The optical properties of small metal clusters have been studied throughout the 20th century [1], in a field which is now referred to as plasmonics [2]. Modern nanoplasmonics aims to confine and control light at the nanoscale, in an amalgamation of photonics and electronics [3]. It is envisaged that applications will arise in areas from data storage and microscopy to light generation and biophotonics [4][5][6]. In the last few years, the subfield of quantum plasmonics has branched away, whereby quantum mechanical phenomena play a crucial role [7].
An intensively studied quasiparticle in plasmonics is the localized surface plasmon (LSP), a collective oscillation of conduction band electrons, which arises when a metallic nanoparticle is irradiated by light [2] or hot electrons [8]. Exploring how the resonance frequency of the plasmon changes depending on the geometry of its hosting nanoparticle is a fundamental task of the field [9][10][11][12][13][14]. Recently, a number of groundbreaking experiments [15][16][17][18][19][20] have probed the plasmonic response of metallic cylinders, and in particular the limiting cases of nanodisks and nanowires. Inspired by these experiments, in this work we derive simple, analytical expressions for the dipolar plasmon resonances in both the longitudinal and transverse polarizations (that is, along the cylindrical axis and perpendicular to it). Our model is based upon a calculation of the change in Hartree energy of the nanoparticle due to the collective displacement of the valence electrons. We assume that the electrons in the nanostructure form a body of approximately uniform density, which allows us to employ continuum mechanics and set up a simple equation of motion [21]. Importantly, our analytic theory is valid for any aspect ratio of the cylinder, and as such is of relevance for a wide range of experiments. Our work therefore complements previous theoretical studies of plasmonic cylinders, which have either employed the nanowire approximation [22], or have required numerics [23][24][25][26][27].
In our model, the inevitable quantum corrections which arise at the nanoscale are addressed by accounting for the so-called spill-out effect [28]. In this quantum size effect, the resonance frequency is modified due to a proportion of electrons spilling outside of the small metallic nanoparticle, thus lowering the average electronic density inside the nanoparticle. This effect arises due to the ground-state many-body wavefunction, which determines the electronic density, having tails which leak outside of the sharp boundary of the nanoparticle surface, so that a non-negligible number of electrons reside outside of the cluster. The spill-out effect has been studied historically in relation to spherical nanoparticles [21], and more recently has been investigated for plasmons in ultra-sharp groove arrays [29].
Coulomb interactions between LSPs housed in different nanoparticles can give rise to collective plasmons spread out over the combined nanostructure [30,31]. The study of collective plasmons in nanoparticle arrays, including architectures built from cylindrical nanoparticles [32][33][34][35][36], has led to a wealth of diverse physics, from plasmonic waveguides [37,38] to light harvesters [39,40] to analogues of a topological insulator [41][42][43][44]. In this work, we are concerned with the simplest example of a coupled system, the nanoparticle dimer [45][46][47][48], which constitutes the building block of more complex metastructures, and where insight into the nature of coupled plasmons can be achieved.
A series of experiments on nanoplasmonic dimers in the near-field coupling regime have revealed both bright and dark plasmonic modes, where the dipole moments are oriented in-phase or out-of-phase, respectively [49][50][51]. In order to account analytically for such collective plasmonic effects, we adapt our aforementioned theory to the case of a dimer of cylindrical metallic nanoparticles. We derive simple expressions for the bright and dark mode resonance frequencies of the system as a function of the inter-nanoparticle separation, which allows for a clear description of how the plasmonic coupling scales with distance. Our results supplement theories of cylindrical dimers in the literature, which predominately involve assumptions about the aspect ratio of the cylinder, or requires time-consuming numerical computations [52][53][54][55][56][57][58]. We also comment on the spill-out effect in the dimer, and suggest some methods for the experimental detection of our predicted effects. This paper is organized as follows: in §2, we calculate the dipolar resonances of a single cylindrical nanoparticle. We find the modifications to the resonance frequencies due to the spill-out effect in §3. The theory is extended to describe collective effects in a dimer of cylindrical nanoparticles in §4. Finally, we draw some conclusions in §5.

PLASMONIC MODES IN A SINGLE CYLINDRICAL NANOPARTICLE
We consider a cylindrical nanoparticle of radius a and length L, containing N e valence electrons with charge −e < 0 and mass m e (see figure 1). We start by neglecting the electronic spill-out effect, and assume that the density n(r) of valence electrons is uniform (with density n 0 ) inside the cylinder, and vanishing outside, i.e., where (r, θ, z) are the usual cylindrical coordinates, and where Θ(x) is the Heaviside step function. Our strategy to obtain the frequencies of the plasmonic normal modes along the longitudinal (ẑ, α = ) and transverse (r, α =⊥) directions 1 closely follows the one presented, e.g., in reference [21] for a spherical nanoparticle. We first impose a rigid shift u α of the electron distribution, which gives rise to the displaced density n(r − u α ). Assuming that u α = |u α | is small with respect to the dimensions of the cylinder, we have n(r − u α ) n(r) + δn α (r), with δn α (r) = −u α · ∇n(r). (2.2) We then consider the resulting change in the Hartree energy (in cgs units) with respect to the equilibrium situation. This quantity gives access to the restoring force and to the resulting spring constant k α . The latter quantity then provides an expression for the normal mode frequency where M e = N e m e corresponds to the total electronic mass. Let us now consider the longitudinal (α = ) and transverse (α =⊥) polarizations each in turn, which arise from different electronic distribution displacements u α .

Longitudinal mode
We assume the longitudinal displacement u = uẑ, such that the change in electron density (2.2) is where δ(x) is the Dirac delta function. Equation (2.6) corresponds to a charge imbalance that is located at the two disks of radius a closing the cylinder at z = ±L/2 (cf. figure 1). In order to evaluate the change in Hartree energy (2.3) due to the above density change, we shall exploit the Laplace expansion of the Newtonian kernel [59] 1 Here, I m (x) and K m (x) are modified Bessel functions of the first and second kinds, respectively, while r < = min(r, r ) and r > = max(r, r ). Upon inserting (2.6) and (2.7) into (2.3), we arrive at a seven-dimensional integral. After carrying out the straightforward angular and Cartesian integrals, and using the following result for the double radial integral, which is harmonic in the displacement u. Evaluating the first term in the above integral with the result ∞ 0 dt sin 2 (t)/t 2 = π/2, and integrating the second term with the aid of special functions, we find (2.10) In the expression above, the function g(x) is defined as and are the complete elliptic integrals of the first and second kinds, respectively. The monotonically increasing function (2.11) has the following asymptotic expansions for small and large arguments: The result (2.10), together with (2.4) and (2.5), then yields the following analytic expression for the resonance frequency of the dipolar longitudinal mode of the cylinder: (2.14) Here the plasma frequency of the considered metal is with the electron density n 0 = N e /πa 2 L for the examined cylinder. We plot in figure 2(a) the longitudinal resonance frequency (2.14) as a function of the aspect ratio L/a of the cylinder as the solid red line. As one can see from the figure, ω 0, is a monotonically decreasing function of the parameter L/a, with the limiting values lim L/a→0 {ω 0, } = ω p and lim L/a→∞ {ω 0, } = 0. Physically, the longitudinal mode softens when the aspect ratio of the cylinder increases, since the ratio of uncompensated charges to the compensated ones (by the ionic background) decreases with increasing L/a.
We now consider the two limiting cases of (2.14), namely when the cylinder can be treated as a nanodisk (L/a 1) or a nanowire (L/a 1), and where insightful expressions can be obtained. Let us first examine the disk limit. Using the expansion (2.13a), (2.14) becomes Clearly, this expression tends linearly towards the plasma frequency ω p in the extreme pancake limit (L → 0), see the dashed green line in figure 2(a). In the opposite limit of a wire, we obtain with (2.13b) which is plotted as a blue dotted line in figure 2(a), showcasing the inverse square root decay to zero frequency.

Transverse mode
In order to have access to the eigenfrequency of the transverse dipolar plasmonic mode, here we assume the arbitrary small displacement u ⊥ = ux (see figure 1), such that the change in electron density (2.2) is Completing an analogous calculation as to that for the preceding case of the longitudinally polarized mode [cf. § §22 2.1] leads to the following equation for the change in the Hartree energy (2.3), Evaluating the above integral then yields where g(x) is defined in (2.11). We thus obtain an analytic expression for the resonance frequency of the transverse dipolar plasmonic mode, using (2.4) and (2.5) with (2.20), as in units of the plasma frequency (2.15). We plot the transverse resonance frequency (2.21) in figure 2(b) as the solid red line, as a function of the aspect ratio L/a. As is evident from the figure, ω 0,⊥ is a monotonically increasing function of the parameter L/a, bounded by the two limits lim L/a→0 {ω 0,⊥ } = 0 and lim L/a→∞ {ω 0,⊥ } = ω p / √ 2 (the latter limit is denoted by the horizontal dash-dotted line in the figure). Contrary to the longitudinal mode shown in figure 2(a), the transverse mode gets harder when the aspect ratio of the cylinder increases, since the ratio of uncompensated charges that sit on the longitudinal surface of the cylinder to the compensated ones increases with increasing L/a. In figure 2(b), the limiting cases of a nanowire (L/a 1, dashed green line) and nanodisk (L/a 1, dotted blue line) are also displayed, and have functional forms which arise directly from (2.21) with the leading order expansions (2.13). Explicitly, one finds and (2.23)

FREQUENCY RENORMALIZATION DUE TO THE SPILL-OUT EFFECT
So far, our approach has been purely classical, and has neglected the spill out of the electronic wave functions outside of the nanoparticle. This approximation follows from our assumed hard wall mean-field potential, resulting in the approximate density of valence electrons given by (2.1). However, the quantum-mechanical spill-out effect is known to renormalize the LSP resonance frequencies, and is particularly prominent for nanoparticles of only a few nanometers in size [28]. We thus relax the above hard-wall approximation, and assume that the mean-field potential (including both the ionic positive background and the electron-electron interactions) seen by the valence electrons of the nanoparticle is given by where V 0 = F + W is the height of the potential, with F and W the Fermi energy and the work function of the nanoparticle, respectively. Such a hypothesis has been tested using density functional ab initio calculations using the local density approximation in [60], and is a fairly good approximation to the realistic mean-field potential. Due to the finite height V 0 of the mean-field potential (3.1), some part of the valence electrons can spill out of the cylindrical nanoparticle, effectively increasing its length and radius according to the replacements Here, the small spill-out lengths L and ⊥ a in the longitudinal (ẑ) and transverse (r) directions, respectively, can be estimated from the average number of spill-out electrons N and N ⊥ in both of these directions according to In the following, we will estimate N and N ⊥ using semiclassical expansions, which will give us access to the spill-out lengths and ⊥ . We will then incorporate the prescription (3.2) into the mode frequencies (2.14) and (2.21), which will then provide us with an estimate of the renormalized resonance frequencies.

Average number of spill-out electrons and spill-out lengths
At zero temperature, the average number of spill-out electrons in the longitudinal and transverse directions are given by respectively. Here, λ labels the bound states in the mean-field potential (3.1) and the summations run over occupied states up to the Fermi level. The single-particle wavefunction ψ λ (r) obeys the time-independent Schrödinger equation with the eigenenergies λ . Note that in (3.4), we disregard the negligible number of spill-out electrons arising at the corners of the cylindrical nanoparticle. The choice of mean-field potential (3.1) leads to a nonseparable Schrödinger equation (3.5). However, the replacement is both an excellent approximation for the original V (r), with only the corners of the cylinder deviating from the nonseparable potential (3.1), and leads to an exactly solvable problem. Decomposing the separable potential (3.6) into V (r) = V r (r) + V z (z) with V r (r) = V 0 Θ(r − a) and V z (z) = V 0 Θ(|z| − L/2), the stationary Schrödinger equation (3.5) then reads where k = 2m e / 2 , and where m is the magnetic quantum number and n (ñ) is the principal quantum number due to the transverse (longitudinal) motion. We separate the variables in (3.7) using and are thus led to two Schrödinger equations in the reduced eigenvalues k r and k z , respectively, where k 2 = k 2 r + k 2 z , whose solutions are given explicitly in the appendix. Using the results presented there [see in particular (A.12)], we are then able to evaluate the integrals entering (3.4), which are approximately given in the high-energy, semiclassical limit of k 0 a 1 and where κ r = (k 2 0 − k r ) 1/2 and κ z = (k 2 0 − k z ) 1/2 . Upon substituting the expressions (3.9) into (3.4), we then replace the summation over the set of quantum numbers n, m, andñ by an integral over wavevector k. We take for the density of states the leading-in-Weyl term [61], which is appropriate in the semiclassical limit k F a 1 and k F L 1 (with k F the Fermi wavevector). For typical noble metals, such as, e.g., Ag or Au, one has k F a 10 a[nm], so that the semiclassical approximation is suitable even for nanometer-sized nanoparticles [62]. The aforementioned prescription leads to where the prefactor of 2 accounts for the spin degeneracy, and V = πa 2 L is the volume of the cylinder. Performing the above integrals in spherical coordinates, we arrive at where k 0 > k F , and where the integrals with respect to the dimensionless radial (x) and polar (t) coordinates are yet to be performed. Evaluating the above integrals (3.11), we find the expressions Here we have introduced the auxiliary functions Scaling the results (3.12) with the total number of electrons in the nanoparticle N e = La 2 k 3 F /3π, we obtain Thus, the fraction of spill-out electrons in both the longitudinal and transverse directions scale with the inverse of the spatial extent of the cylinder (∝ 1/a, 1/L), and so becomes increasingly important for particles with nanometric dimensions.
With the above results (3.14), we can now evaluate the spill-out lengths (3.3), which read Importantly, these two quantities do not depend on the nanoparticle dimensions L and a, and only on the Fermi energy F (or the Fermi wavevector k F ) and the depth V 0 of the mean-field potential (3.1). The spill-out lengths (3.15) are plotted in figure 3 as a function of F /V 0 . As one can see from the figure, both of these quantities smoothly increase with the above-mentionned ratio. Since k F is typically of the order of 10 8 cm −1 for alkaline or noble metals, and since F /V 0 is usually of the order of 0.5 [28], the spill-out lengths (3.15) are only of a few tenths of an angstrom. However, as we will see in the next section, such a tiny spread of the electronic wave functions outside of the nanoparticle may have a non-negligible effect on the LSP resonance frequency.

Frequency redshifts due to the spill-out effect
We are now in a position to calculate the renormalized resonance frequency in the longitudinal (transverse) polarizationω 0, (ω 0,⊥ ) due to the spill-out effect. We account for the spill-out of the electrons by treating the cylindrical nanoparticle with the effective dimensionsL andã as in (3.2). It follows from the direct substitution of these effective dimensions into the resonance frequencies (2.14) and (2.21), which assumed hard-wall confinement of the valence electrons, that the renormalized resonance frequencies are, to leading order in the scaled spill-out lengths /L and ⊥ /a [cf. (3.15)], given byω 0, Here, we defined the two functions is the derivative with respect to x of the function g(x) defined in (2.11). We plot in figure 4(a) the auxiliary functions (3.17), which are both monotonically increasing functions of the aspect ratio L/a of the cylinder sketched in figure 1, with 0 < j < 1 and −1 < j ⊥ < 0. Thus, the prefactors of the terms ∝ 1/L and ∝ 1/a in (3.16) are negative, such that the LSP resonance frequencies of the cylinder experience a redshift as the nanoparticle size decreases, as it is the case for the sphere geometry [28]. This is exemplified in figure 4(b), which displays the renormalized resonance frequencies (3.16) (in units of the bare ones) as a function of 1/k F a, for an aspect ratio L/a = 1, and for increasing values of F /V 0 . As one can see from the figure, the deviation due to the spill-out effect from the bare resonance frequencies can reach between ca. 1.5 % to 4.5 % (depending on the ratio F /V 0 , which is essentially material-dependent) for k F a = 10 (which typically corresponds to a 1 nm for normal metals). Progress in nanofabrication techniques should allow one to vary the cylindrical nanoparticle size and thus to measure the predicted size dependence of the resonance frequencies (3.16).

COUPLED MODES IN A CYLINDRICAL DIMER
We now consider a dimer of cylindrical nanoparticles composed of the same metal, which are aligned along the z axis, and are separated by a distance d (see figure 5). The two cylinders, denoted left (L) and right (R), are both of a radius a and length L, and contain N e valence electrons each. In the following, we study the resultant coupled plasmonic modes in the dimer, for both the longitudinal (α = ) and transverse (α =⊥) polarizations, in a similar fashion to the single cylinder calculation of §2.
Notably, we focus on the dipolar modes and do not take into account higher-order contributions to the collective plasmon-plasmon interactions, which are only relevant for small inter-nanoparticle separations. 2 We also focus on separation distances such that the near-field coupling between the LSPs on each nanoparticle dominates, and thus disregard retardation effects. The latter only lead to small frequency renormalization effects in nanoparticle dimers [64] (although retardation effects can be significant in long nanoparticle chains [43,65,66]).
We start by neglecting the spill-out of the electronic wave functions outside of each cylinder. As in §2, we assume that the density n(r) of valence electrons is uniform within each cylinder (with density n 0 ), and vanishing outside. We thus have n(r) = n L (r) + n R (r), (4.1) where, in cylindrical coordinates, the single nanoparticle contributions are In order to have access to the frequencies of the coupled dipolar modes, we shall impose two types of rigid displacements u τ α on the total density (4.1) for each polarization α, distinguished by the index τ = ±. To first order in the displacement field u τ α , the electronic density is perturbed like n(r − u τ α ) n(r) + δn τ α (r), where δn τ α (r) = −u τ α · ∇n(r). This density change gives rise to the two possible normal modes of the system (labelled by τ ) for each polarization α, which we now consider in turn.

Longitudinal modes
Firstly, let us consider the symmetric mode in the longitudinal polarization (→→), which we denote with the index τ = − since it constitutes the low-energy mode. We enforce the displacement u − = uẑ in both cylinders, such that the shift in density (4.3) is δn − (r) = δn L, (r) + δn R, (r), with Secondly, let us consider the antisymmetric, high-energy mode (→←), which we mark with the index τ = +. Imposing the rigid displacements u + = ±uẑ, which act in different directions (±) in each cylinder (L relative to R), the change in density (4.3) reads δn + (r) = δn R, (r) − δn L, (r), in terms of the quantities (4.4). The change in electrostatic interactions between the electronic clouds is described by the change in the Hartree energy where we have used the decomposition where i, j ∈ {L, R}, α ∈ { , ⊥} and δn i, (r) are given in (4.4). The first two terms on the right-hand side of (4.5) describe the contributions from each cylinder in isolation, δE LL = δE RR = δE , where δE is given in (2.10). The effect of electrostatic coupling is contained in the final two terms of (4.5). An analogous calculation to that which led to (2.10) yields for the remaining two terms in (4.5) the expression where the function g(x) is defined in (2.11). The restoring force F τ α = −∂ u (δE τ α ) = −k τ α u then provides us with a relation to the effective spring constant k τ α of the problem, from which the resonance frequencies ω τ,α = k τ α /2N e m e are revealed. 3 Hence the longitudinal eigenfrequencies of a dimer of cylindrical nanoparticles are given by which accounts for both the high-energy (τ = +, →←) and the low-energy (τ = −, →→) modes. Here, ω 0, is the resonance frequency of an isolated cylinder (2.14), which depends on the aspect ratio L/a, while the effective coupling constant of the problem Ω has the functional form and depends on both L/a and d/a. Equation (4.8) analytically describes the longitudinal resonance frequencies of the pair of coupled dipolar modes of the cylindrical dimer, provided the separation d is not too small (i.e., where one can neglect multipolar effects and/or the quantum tunneling of electrons between the two nanoparticles comprising the dimer). To leading order in d/a 1 and d/L 1, using (2.13b) we have from which one immediately notices that in the extreme isolation limit (d → ∞), the coupling constant (4.9) vanishes and thus the single nanoparticle resonance frequency (2.14) is recovered. We plot the dimer resonance frequencies (4.8) as a function of the interparticle separation d in figure 6(a), for both the high-energy (black line) and the low-energy (red line) modes, for the aspect ratio L/a = 1. Evidently, for separations of just a few multiples of the cylinder size, the two resonance frequencies converge onto the one of a single cylinder ω 0, (see the gray dashed line). As one can see from the figure, for separations below the cylinder size, the pair of plasmonic levels become increasingly distinct and hence experimentally detectable. The bright mode (→→) may be accessed optically, whereas the dark mode (→←) should be probed via electron energy loss spectroscopy (EELS) [8].
For completeness, we now consider the two limiting cases of equation (4.8), i.e., the nanodisk (L/a 1) and nanowire (L/a 1) limits, and for large separation distances (d/a 1, d/L 1). Using (4.10), the full eigenspectrum (4.8) of the coupled nanodisks reads where ω disk 0, is given in (2.16), analytically demonstrating the expected inverse cubic decay with separation (∝ 1/d 3 ), which is characteristic of a quasistatic dipolar interaction [59]. In the opposite regime of a dimer of wire-like nanoparticles, and further assuming L 2 a/d 3 1, we have where an expression for ω wire 0, can be found in (2.17), again showcasing the characteristic scaling typical of the dipoledipole interaction regime (∝ 1/d 3 ).

Transverse modes
The calculation for the transverse polarized coupled modes (α =⊥) proceeds in a similar manner to that of the previous subsection. The main difference is that the symmetric mode (↑↑) is now the high-energy mode and is associated with τ = +, while the antisymmetric mode (↑↓) corresponds to the low-energy mode and has the label τ = −.
We again consider two separate displacements in order to characterize the two coupled dipolar modes. Firstly we assume u + ⊥ = ux, which gives rise to the symmetric mode, and secondly we let u − ⊥ = ±ux, which distinguishes the antisymmetric mode (in the right hand side of u − ⊥ , the ± refers to the displacements being in opposite directions for cylinder L as compared to R). The shift in electronic density across the dimer then reads δn τ ⊥ (r) = δn L,⊥ (r) + τ δn R,⊥ (r), where as follows from (4.2) and (4.3). The change in Hartree energy accounts for the transverse electrostatic interactions via δE τ where we utilized the decomposition of (4.6). The first two terms in δE τ ⊥ are single nanoparticle contributions, and so are exactly that of (2.20), explicitly δE LL ⊥ = δE RR ⊥ = δE ⊥ . The remaining coupling terms in δE τ ⊥ are also equal by symmetry, and are given by where g(x) was introduced in (2.11). As in the previous subsection, the analytic expression for the transverse resonance frequencies soon follows from the restoring force F τ ⊥ (see below (4.7) for details) as where the coupling constant Ω is defined in (4.9), and with the single cylinder resonance frequency ω 0,⊥ of (2.21).
Notably, compared to the longitudinal result ω τ, of (4.8), the second term under the square root in (4.15), the so-called coupling term, is half as large. This property is familiar from the canonical case of a dimer of spherical nanoparticles, where the longitudinal modes are also (approximately) twice as widely split as the transverse modes [48,64]. The transverse polarized dimer resonance frequencies (4.15) are plotted as a function of the interparticle separation d in figure 6(b), with the aspect ratio fixed at L/a = 1. The plot illustrates increasingly large inter-mode splittings for decreasing separations d, such that ω τ,⊥ become significantly detuned from the single cylinder result (dashed gray line). In stark contrast to the longitudinal coupled modes of figure 6(a), in panel (b) it is the high-energy mode which strongly couples to light (red line), and as such may be detected straightforwardly by optical means, while the low-energy mode is dark (black line), and as such requires EELS probing techniques.
A remark is now in order about the influence of the spill-out effect onto the coupled mode frequencies shown in figure 6. Since the effective coupling constant Ω of (4.9) already represents a rather small correction to the single particle results ω 0, and ω 0,⊥ in (4.8) and (4.15), respectively, we can safely assume that Ω is not renormalized by the spill-out lengths (3.15). Therefore, the renormalized coupled mode frequencies areω τ, = (ω 2 0, + 2τ Ω 2 ) 1/2 and ω τ,⊥ = (ω 2 0,⊥ +τ Ω 2 ) 1/2 respectively, whereω 0, andω 0,⊥ are given in (3.16). Henceforth, the only effect of the spill-out electrons is a global frequency shift towards the red end of the spectrum, which does not depend on the interparticle distance d.

CONCLUSION
We have considered the fundamental problem of characterizing analytically the longitudinal and transverse dipolar modes supported by a cylindrical nanoparticle with an arbitrary aspect ratio. Inspired by the trend of increasing miniaturization, we have derived semiclassical expressions for the spill-out lengths in metallic cylinders, and have shown that this quantum phenomenon can significantly renormalize the plasmonic frequencies. Recent experiments on plasmonic cylinders, including with nanodisks and nanowires, suggests that both the resonance frequency dependence on the cylinder aspect ratio and the quantum size effect we consider may be probed in cutting edge laboratories [15][16][17][18][19][20].
We have also developed an analytical theory of a dimer of cylindrical nanoparticles, and have derived a simple expression for the bright and dark mode eigenfrequencies of the collective plasmons, for both the longitudinal and transverse polarizations. These results provide insight into the evolution of the splitting of the coupled modes as a function of the inter-nanoparticle separation, and act as a benchmark for future experimental and numerical studies of collective plasmonic behavior [31,46,47].

ACKNOWLEDGMENTS
We are grateful to Rodolfo Jalabert and Dietmar Weinmann for stimulating discussions. We acknowledge support from Agence Nationale de la Recherche (Grant No. ANR-14-CE26-0005 Q-MetaMat). CAD acknowledges support from the Juan de la Cierva program of the Spanish MINECO. This research project has been supported by the University of Strasbourg IdEx program. In the high-energy semiclassical limit (k 0 a 1, k 0 L 1), which is well suited for the problem at hand [62], we find that the expressions (A.9) and (A.11) are well-approximated by