Plasmonic modes in cylindrical nanoparticles and dimers: Plasmons in cylindrical nanoparticles

royalsocietypublishing.org/journal/rspa Research Cite this article: Downing CA, Weick G. 2020 Plasmonic modes in cylindrical nanoparticles and dimers. Proc.R.Soc.A476: 20200530. https://doi.org/10.1098/rspa.2020.0530 Received: 7 July 2020 Accepted: 16 November 2020 Subject Areas: solid state physics, nanotechnology, optics Keywords: nanoplasmonics, nanoparticles, dimers, quantum-size effects Author for correspondence: Guillaume Weick e-mail: guillaume.w[email protected] Plasmonic modes in cylindrical nanoparticles and dimers Charles A. Downing1,2 and Guillaume Weick3 1Departamento de Física de la Materia Condensada, CSIC-Universidad de Zaragoza, 50009 Zaragoza, Spain 2Department of Physics and Astronomy, University of Exeter, Exeter EX4 4QL, UK 3Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, 67000 Strasbourg, France CAD, 0000-0002-0058-9746;GW,0000-0002-0617-5835 We present analytical expressions for the resonance frequencies of the plasmonic modes hosted in a cylindrical nanoparticle within the quasi-static approximation. 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. 1. Introduction The optical properties of small metal clusters have been studied throughout the twentieth 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 2020 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/ by/4.0/, which permits unrestricted use, provided the original author and source are credited. 2 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200530 ........................................................... light generation and biophotonics [4–6]. 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 quasi-particle in plasmonics is the localized surface plasmon (LSP), a collective oscillation of conduction band electrons, which arises when a metallic nanoparticle (NP) 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 NP is a fundamental task of the field [9–14]. Recently, a number of groundbreaking experiments [15–21] 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 within the quasi-static limit (valid when the dimensions of the NP is smaller than the wavelength associated with the LSP resonance frequency) 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 NP 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 [22]. 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 [23–25], or have required numerics [26–31]. In our model, the inevitable quantum corrections which arise at the nanoscale are addressed by accounting for the so-called spill-out effect [32]. In this quantum size effect, the resonance frequency is modified due to a proportion of electrons spilling outside of the small metallic NP, thus lowering the average electronic density inside the NP. This effect arises due to the ground-state many-body wave function, which determines the electronic density, having tails which leak outside of the sharp boundary of the NP 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 NPs [22], and more recently has been investigated for plasmons in ultra-sharp groove arrays [33]. Coulomb interactions between LSPs housed in different NPs can give rise to collective plasmons spread out over the combined nanostructure [34,35]. The study of collective plasmons in NP arrays, including architectures built from cylindrical NPs [36–40], has led to a wealth of diverse physics, from plasmonic waveguides [41,42] to light harvesters [43,44] to analogues of a topological insulator [45–48]. In this work, we are concerned with the simplest example of a coupled system, the NP dimer [49–52], 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 [53–55]. In order to account analytically for such collective plasmonic effects, we adapt our aforementioned theory to the case of a dimer of cylindrical metallic NPs. We derive simple expressions for the bright and dark mode resonance frequencies of the system as a function of the interparticle 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 require time-consuming numerical computations [56–62]. 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 NP and discuss their respective decay rates. 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 NPs in §4. Finally, we draw some conclusions in §5. 2. Plasmonic modes in a single cylindrical nanoparticle We consider a cylindrical NP of radius aand length L, containing Nevalence electrons with charge −e<0andmassme(see the inset in figure 1). We start by neglecting the electronic spill-out 3 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200530 ........................................................... z a L 0 y x − L/ 2 + L/ 2 0.2 0.4 0.6 0.8 1.0 resonance frequency/ w p 0 2 4 6 8 10 L/a w 0, w 0, disk w 0, wire 0 0.2 0.4 0.6 0.8 2 4 6 8 10 L/a w 0, ^ w 0, ^ disk w 0, ^ wire 1/ 2 (a)(b) Figure 1. Resonance frequencies ω0,and ω0,⊥(solid red lines), in units of the plasma frequency ωp,asafunctionofthe aspect ratio L/afor both the (a) longitudinal (cf. (2.14)) and (b) transverse (cf. (2.20)) modes. Dashed green lines: the disk limit approximations,from(2.15)and (2.21)for panels(a)and(b),respectively. Dottedblue lines:the wirelimitapproximations,from (2.16) and (2.22) for (a)and(b), respectively. Horizontal dash-dotted line in (b): the asymptotic result ωp/√2, for L/a→∞. Inset: sketch of a cylindrical metallic nanoparticle of radius aand length L. (Online version in colour.) effect, and assume that the density n(r) of valence electrons is uniform (with density n0) inside the cylinder, and vanishing outside, i.e. n(r)=n0Θ(a−r)ΘL 2+zΘL 2−z, (2.1) 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 (ˆ z,α=) and transverse (ˆ r,α=⊥) directions1closely follows the one presented, for example, in [22] for a spherical NP, which yields for the LSP resonance frequency the well-known Mie result ωp/√3, with ωpthe plasma frequency.2We first impose a rigid shift uαof the electron distribution, which gives rise to the displaced density n(r−uα). Assuming that uα=|uα|is small with respect to the dimensions of the cylinder, we have n(r−uα)≃n(r)+δnα(r), with δnα(r)=−uα·∇n(r). (2.2) We then consider the resulting change in the Hartree energy (in cgs units) δEα=e2 2d3rd3rδnα(r)δnα(r) |r−r|, (2.3) with respect to the equilibrium situation. This quantity gives access to the restoring force Fα=− ∂ ∂uα (δEα)=−kαuα(2.4) and to the resulting spring constant kα. The latter quantity then provides an expression for the normal mode frequency ω0,α=kα Me , (2.5) where Me=Nemecorresponds 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 uα. 1Here and in what follows, hats designate unit vectors. 2Note that a similar phenomenological approach has been successfully applied by the authors of [63] to spin-dependent dipole excitations, and excellent agreement was obtained against time-dependent density functional theory numerical calculations. 4 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200530 ........................................................... (a) Longitudinal mode We assume the longitudinal displacement u=uˆ z, such that the change in density (2.2) is δn(r)=un0Θ(a−r)δz−L 2−δz+L 2, (2.6) 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 aclosing the cylinder at z=±L/2 (cf. the inset in figure 1). In order to evaluate the modification of the Hartree energy (2.3) due to the above density change, we shall exploit the Laplace expansion of the Newtonian kernel [64] 1 |r−r|=2 π +∞  m=−∞∞ 0 dkeim(θ−θ)cos k[z−z]Im(kr<)Km(kr>). (2.7) Here, Im(x)andKm(x) are modified Bessel functions of the first and second kinds, respectively, while r<=min(r,r)andr>=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: a 0 drra 0 drrI0(kr<)K0(kr>)=a2 2k2[1−2I1(ka)K1(ka)], (2.8) we find δE=8π(en0u)2a3∞ 0 dx x2sin2L 2ax[1−2I1(x)K1(x)], (2.9) which is harmonic in the displacement u. Evaluating the first term in the above integral using ∞ 0dtsin2(t)/t2=π/2, and integrating the second term employing special functions, we find δE=4π(eun0)2a3πL 2a+4 3−gL a. (2.10) In the expression above, the function g(x)isdefinedas g(x)=x 6x2+4K−4 x2−x2−4E−4 x2, (2.11) where K(x)=1 0 dt (1 −t2)(1 −xt2)and E(x)=1 0 dt1−xt2 1−t2(2.12) 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: g(x)≃4 3+(6ln2−1−2lnx)x2 4+O(x4), x1 (2.13a) and g(x)≃π 2x+1 2x−1 4x3+O(x−5), x1. (2.13b) 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: ω0,=ωp1+2a πL4 3−gL a. (2.14) Here, the plasma frequency of the considered metal is ωp=(4πn0e2/me)1/2, with the electron density n0=Ne/πa2Lfor the examined cylinder. We plot in figure 1athe longitudinal resonance frequency (2.14) as a function of the aspect ratio L/aof the cylinder as the solid red line. As one can see from the figure, ω0,is a monotonically decreasing function of the parameter L/a, with the limiting values limL/a→0{ω0,}=ωpand 5 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200530 ........................................................... limL/a→∞{ω0,}=0, which coincide with the well-known asymptotic results for a spheroidal NP [11]. 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 note that this trend has been confirmed experimentally [16]. 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 ωdisk 0,≃ωp1−L 4πa6ln2−1−2lnL a,L a1. (2.15) Clearly, this expression tends linearly towards the plasma frequency ωpin the extreme pancake limit (L→0); see the dashed green line in figure 1a. In the opposite limit of a wire, we obtain with (2.13b) ωwire 0,≃ωp8a 3πL,L a1, (2.16) which is plotted as a blue dotted line in figure 1a, showcasing the inverse square root decay to zero frequency. (b) Transverse mode In order to have access to the eigenfrequency of the transverse dipolar plasmonic mode, here we assume the arbitrary small displacement u⊥=uˆ x(see the inset in figure 1), such that the change in electron density (2.2) is δn⊥(r)=un0cos θδ(r−a)ΘL 2+zΘL 2−z. (2.17) Completing an analogous calculation as to that for the preceding case of the longitudinally polarized mode (cf. §2a) leads to the following equation for the change in the Hartree energy (2.3): δE⊥=8π(en0u)2a3∞ 0 dx x2sin2L 2axI1(x)K1(x). (2.18) Evaluating the above integral then yields δE⊥=2π(eun0)2a3gL a−4 3, (2.19) 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 (2.19), as ω0,⊥=ωpa πLgL a−4 3. (2.20) We plot the transverse resonance frequency (2.20) in figure 1bas the solid red line, as a function of the aspect ratio L/a. As is evident from the figure, ω0,⊥is a monotonically increasing function of the parameter L/a, bounded by the two limits limL/a→0{ω0,⊥}=0 and limL/a→∞{ω0,⊥}=ωp/√2 (the latter limit is denoted by the horizontal dash-dotted line in the figure). As is the case for the longitudinal plasmonic mode, such asymptotic limits are the same for a spheroidal NP [11]. Contrary to the longitudinal mode shown in figure 1a, 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 1b, 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 (2.20) 6 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200530 ........................................................... with the leading order expansions (2.13). Explicitly, one finds ωdisk 0,⊥≃ωpL 4πa6ln2−1−2lnL a,L a1 (2.21) and ωwire 0,⊥≃ωp √21−4a 3πL,L a1. (2.22) (c) Discussion: comparison to spheroids, screening effects, and damping rates of the plasmonic resonances In appendix A, we compare our analytical results (2.14) and (2.20) for the LSP resonance frequencies of a cylindrical NP to the closed-form expressions for a spheroidal particle with the same aspect ratio (e.g. [11]) and find an excellent agreement. Such a correspondence between both geometries as been previously pointed out by Venermo & Sihvola [65], who compared the polarizability of a cylinder calculated by means of numerical simulations to that of a spheroid, which is known analytically [11]. The comparison presented in appendix A thus confirms the relevance as well as the adequacy of our approach, which provides an analytical understanding of plasmonic modes for the cylinder geometry. Thus far, our approach has neglected the possible dielectric screening of the valence electrons by the delectrons (characterized by a dielectric constant d), which is of relevance for noble metal NPs, as well as the presence of a dielectric embedding medium (with constant m). For the sphere geometry, the presence of screening and the resulting dielectric mismatch notoriously renormalizes [1] the Mie frequency from ωp/√3toωp/(d+2m)1/2. Within our theoretical approach, it is straightforward to realize that when d≈m=, since the Hartree energy (2.3) is renormalized by a factor −1, the resonance frequencies in (2.14) and (2.20) take on the same expressions, up to a replacement of ωpby ωp/√, leading to a redshift of the resonances. The case d=mis much more involved due to the complicated form of the Coulomb interaction in cylindrical coordinates, even within the wire limit [66], and is out of the scope of the present work. A final comment is here in order about the damping of the plasmonic excitations which we have elucidated thus far. Metallic nano-objects are subject to radiative and non-radiative damping mechanisms which broaden the resonance of the collective excitation, such that the total decay rate of the LSP modes are given by γα=γr α+γnr (α=,⊥). Within our dipolar approximation, the radiative decay rates γr αcan be readily estimated from the electromagnetic field generated in the far-field by a point dipole [64] carrying a charge −eNeand oscillating at the LSP resonance frequency ω0,α. Evaluating the total power radiated by the dipole and the energy initially stored in it, we find γr α=ω2 pω2 0,α 6c3a2L, (2.23) with cthe speed of light in vacuum. The radiative damping rates γr αthus depend on the cylinder dimensions through the explicit dependence a2Ldisplayed by the equation above, but also through the aspect-ratio dependence of ω0,α(figure 1), and increases with the dimensions of the cylinder. Using the expansions (2.15), (2.16), (2.21) and (2.22), we find for the longitudinal mode γr,disk ≃ω4 pa2L/6c3in the disk limit and γr,wire ≃4ω4 pa3/9πc3in the wire limit, which, interestingly, does not depend on L,sinceωwire 0,goes to zero for L/a1 (see (2.16)). For the transverse mode, we find γr,disk ⊥≃(6 ln 2 −1)ω4 paL2/24πc3and γr,wire ⊥≃ω4 pa2L/12c3. The non-radiative contribution γnr =γO+γLto the total LSP linewidth, which is modeindependent in a first approximation, can be divided into two parts. The first part corresponds to the Ohmic, bulk-like contribution γOwhich essentially arises from electron–phonon and electron–electron scattering. The experiments on single gold nanorods protected by a silica shell of [16] report a value γO≈65 meV/¯ h. The second part is the Landau damping decay rate γL, a purely quantum-mechanical effect [22,32] which comes from the confinement of the 7 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200530 ........................................................... electronic eigenstates within the NP, and which reads γL=AvF/eff, with Aa (material and dielectric environment-dependent) constant of order 1, vFthe Fermi velocity, and eff an effective confinement length. The experiments of [16] have shown that eff =(aL)1/2provides a good fit to the measured data. In these experiments on individual gold nanorods having lengths Lin between 32 nm and 70 nm and radii ain the range 4.3 nm to 11 nm, Landau damping was shown to largely dominate the size-dependent part of the total linewidth (which is in the 80–140 meV/¯ h range), while the maximal value of the radiative damping decay rate reported is only 15 meV/¯ h. 3. 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 NP. This approximation follows from our assumed hard wall meanfield 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 NPs of only a few nanometres in size [32]. 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 NP is given by V(r)=V0Θ(r−a)Θ|z|−L 2, (3.1) where V0=F+Wis the height of the potential, with Fand Wthe Fermi energy and the work function of the NP, respectively. Such a hypothesis has been tested using density functional ab initio calculations using the local density approximation in [67], and is a fairly good approximation to the realistic mean-field potential. Due to the finite height V0of the mean-field potential (3.1), some part of the valence electrons can spill out of the cylindrical NP, effectively increasing its length and radius according to the replacements L→˜ L=L+2,a→˜ a=a+⊥. (3.2) Here, the small spill-out lengths Land ⊥ain the longitudinal (ˆ z) and transverse (ˆ r) directions, respectively, can be estimated from the average number of spill-out electrons Nand N⊥in both of these directions according to =1 2 N Ne Land ⊥=1 2 N⊥ Ne a. (3.3) 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 (2.20), which will then provide us with an estimate of the renormalized resonance frequencies. (a) Average number of spill-out electrons and spill-out lengths At zero temperature, the average numbers of spill-out electrons in the longitudinal and transverse directions are given by N= occ  λr<a |z|>L/2 d3r|ψλ(r)|2and N⊥= occ  λr>a |z|<L/2 d3r|ψλ(r)|2, (3.4) respectively. Here, λlabels 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 wave function ψλ(r) obeys the 8 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200530 ........................................................... time-independent Schrödinger equation −¯ h2 2me∇2+V(r)ψλ(r)=λψλ(r), (3.5) with λthe corresponding eigenenergies. Note that in (3.4), we disregard the negligible number of spill-out electrons arising at the corners of the cylindrical NP. The choice of mean-field potential (3.1) leads to a non-separable Schrödinger equation (3.5). However, the replacement V(r)≃V0Θ(r−a)+Θ|z|−L 2 (3.6) is both an excellent approximation for the original V(r), with only the corners of the cylinder deviating from the non-separable potential (3.1), and leads to an exactly solvable problem. Decomposing the separable potential (3.6) into V(r)=Vr(r)+Vz(z) with Vr(r)=V0Θ(r−a)and Vz(z)=V0Θ(|z|−L/2), the stationary Schrödinger equation (3.5) then reads ∂2 ∂r2+1 r ∂ ∂r+1 r2 ∂2 ∂θ2+∂2 ∂z2+k2−2me ¯ h2Vr(r)+Vz(z)ψnm˜ n(r)=0, (3.7) where k=2me/¯ h2, and where mis the magnetic quantum number and n(˜ n) is the principal quantum number due to the transverse (longitudinal) motion. We separate the variables in (3.7) using ψnm˜ n(r)=Fnm(r,θ)Z˜ n(z), (3.8) and are thus led to two Schrödinger equations in the reduced eigenvalues krand kz, respectively, where k2=k2 r+k2 z, whose solutions are given explicitly in appendix B. Using the results presented there (see in particular (B10)), we are then able to evaluate the integrals entering (3.4), which are approximately given in the high-energy, semiclassical limit of k0a1andk0L1 (with k0= (2meV0/¯ h2)1/2)by r<a |z|>L/2 d3rψnm˜ n(r) 2≃2 κzL k2 z k2 0 and r>a |z|<L/2 d3rψnm˜ n(r) 2≃1 2κra, (3.9) where κr=(k2 0−kr)1/2and κz=(k2 0−kz)1/2. Upon substituting the expressions (3.9) into (3.4), we then replace the summation over the set of quantum numbers n,mand ˜ nby an integral over wave vector k. We take for the density of states the leading-in-¯ hWeyl term [68], which is appropriate in the semiclassical limit kFa1and kFL1 (with kFthe Fermi wave vector). For typical noble metals, such as Ag or Au, one has kFa≃10 a[nm], so that the semiclassical approximation is suitable even for nanometre-sized NPs [69]. The aforementioned prescription leads to N≃2V (2π)3k<kF d3k2 κzL k2 z k2 0 and N⊥≃2V (2π)3k<kF d3k1 2κra, (3.10) where the prefactor of 2 accounts for the spin degeneracy and V=πa2Lis the volume of the cylinder. Performing the above integrals in spherical coordinates, we arrive at N=k2 0a2 πkF/k0 0 dxx 4+1 −1 dtt2 1−x2t2and N⊥=k2 0aL 4πkF/k0 0 dxx 2+1 −1 dt1 1−x2(1 −t2), (3.11) where k0>kF, 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 N=(kFa)2 4πhF V0and N⊥=k2 FaL 4πh⊥F V0. (3.12) 9 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200530 ........................................................... –1.0 −0.5 0 0.5 1.0 0 2 4 6 8 10 L/a j j ^ 0.94 0.96 0.98 1.00 ˜ w 0, a / w 0, a 0 0.025 0.050 0.075 0.100 1/k F a 0 0.1 0.2 0.3 0.4 0.25 0.50 0.75 1.00 k F ^ k F  F /V 0  (a)(b)(c) Figure 2. (a) Spill-out lengths (3.15), scaled with the Fermi wave vector kF, as a function of the ratio of the Fermi energy to mean-field potential strength, F/V0.(b) Auxiliary functions jα(α=,⊥) from (3.17) as a function of the aspect ratio L/a. (c) Renormalized resonance frequencies ˜ω0,(violet lines) and ˜ω0,⊥(orange lines) from (3.16) in units of the bare frequencies ω0,αas a function of the inverse size of the nanoparticle (dotted lines: F/V0=0.25; dashed lines: F/V0=0.50; solid lines: F/V0=0.75), for L/a=1. (Online version in colour.) Here, we have introduced the auxiliary functions h(x)=3 2−x1 x−1+2−3 2xarcsin √xand h⊥(x)=1 √x+1−1 xarctanh √x. (3.13) Scaling the results (3.12) with the total number of electrons in the NP Ne=La2k3 F/3π, we obtain N Ne=3 4kFLhF V0and N⊥ Ne=3 4kFah⊥F V0. (3.14) Thus, the fraction of spill-out electrons in both the longitudinal and transverse directions scales 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 kF=3 8hF V0and kF⊥=3 8h⊥F V0. (3.15) Importantly, these two quantities do not depend on the NP dimensions Land a, and only on the Fermi energy F(or the Fermi wave vector kF) and the depth V0of the mean-field potential (3.1). The spill-out lengths (3.15) are plotted in figure 2aas a function of F/V0.Asonecanseefrom the figure, both of these quantities smoothly increase with the above-mentioned ratio. Since kFis typically of the order of 108cm−1for alkaline or noble metals, and since F/V0is roughly of the order of 0.5 [32], 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 NP may have a non-negligible effect on the LSP resonance frequency. (b) Frequency redshifts due to the spill-out effect We are now in a position to calculate the renormalized resonance frequency in the longitudinal (transverse) polarization ˜ω0,(˜ω0,⊥) due to the spill-out effect. We account for the spill-out of the electrons by treating the cylindrical NP with the effective dimensions ˜ Land ˜ aas in (3.2). It follows from the direct substitution of these effective dimensions into the resonance frequencies (2.14) and (2.20), which assumed hard-wall confinement of the valence electrons, that the renormalized resonance frequencies are, to leading order in the scaled spill-out lengths /Land ⊥/a(cf. (3.15)), given by ˜ω0,α≃ω0,α1−1+jαL a L−1−1 2jαL a⊥ a,α=,⊥. (3.16) 16 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200530 ........................................................... Appendix B. Schrödinger equation with a cylindrical step potential In this appendix, we provide details about the bound-state solutions to the Schrödinger equation (3.7), which enable us to evaluate semiclassically the average number of spill-out electrons (3.4), in both the longitudinal and transverse directions. Separating the variables as in (3.8), the transverse wave functions Fnm(r,θ) are subject to the following Schrödinger equation: ∂2 ∂r2+1 r ∂ ∂r+1 r2 ∂2 ∂θ2+k2 r−2me ¯ h2Vr(r)Fnm(r,θ)=0. (B 1) With the ansatz Fnm(r,θ)=Rnm(r)e imθ/(2π)1/2, where the quantum number m∈Z, and with the notation κr=(k2 0−k2 r)1/2,wherek0=(2meV0/¯ h2)1/2, one finds the following bound state solutions: Rnm(r)=Cnm ⎧ ⎨ ⎩ Jm(krr), r≤a, Jm(kra) Km(κra)Km(κrr), r>a,(B 2) where Jm(x)andKm(x) are the Bessel functions of the first and second kinds, respectively. The normalization constant in (B 2) is given by Cnm =√2 aJm(kra) Km(κra)2 Km+1(κra)Km−1(κra)−Jm+1(kra)Jm−1(kra)−1/2 ,(B3) while the transverse motion is subject to energy quantization via the transcendental equation krJm+1(kra)/Jm(kra)=κrKm+1(κra)/Km(κra), whose solutions are labelled with the quantum number n. The longitudinal wave functions Z˜ n(z) entering (3.8) obey d2 dz2Z˜ n(z)+k2 z−2me ¯ h2Vz(z)Z˜ n(z)=0, (B 4) which is equivalent to the textbook quantum mechanics exercise of a one-dimensional particle in a square box [74]. The solutions of (B 4) have either a symmetric (s) or an antisymmetric (a) parity, which we specify as Z˜ n(z)=Z˜ n,p(z), where the index p =(s, a). The even bound state solutions are given by Z˜ n,s(z)=κz 1+κzL/2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ cos kzL 2eκz(L/2+z),z≤−L 2, cos (kzz),|z|<L 2, cos kzL 2eκz(L/2−z),z≥L 2, (B 5a) where κz=(k2 0−k2 z)1/2>0. Similarly, the odd solutions read Z˜ n,a(z)=κz 1+κzL/2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −sin kzL 2eκz(L/2+z),z≤−L 2, sin (kzz),|z|<L 2, sin kzL 2eκz(L/2−z),z≥L 2. (B 5b) Both sets of eigenfunctions (B 5a)and(B5b) are associated with an individual transcendental equation describing the quantization of energy due to the longitudinal confinement, explicitly tan (kzL/2) =κz/kz(s modes) and tan (kzL/2) =−kz/κz(a modes). The solutions of these equations are labelled with the third quantum number of the problem, ˜ n. 17 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200530 ........................................................... Now that the full Schrödinger equation (3.7) is solved, we proceed with the evaluation of the integrals entering (3.4), namely Rin nm =a 0 drr|Rnm(r)|2,Rout nm =∞ a drr|Rnm(r)|2(B 6) and Zin ˜ n,p =+L/2 −L/2 dz|Z˜ n,p(z)|2,Zout ˜ n,p =−L/2 −∞ ++∞ +L/2dz|Z˜ n,p(z)|2,(B7) which describe the probability of finding the electrons inside or outside the cylindrical NP, in either the transverse or longitudinal directions. With (B 2), we obtain for the transverse integrals (B 6) the results Rin nm =Jm+1(kra)Jm−1(kra)−J2 m(kra) Jm+1(kra)Jm−1(kra)−J2 m(kra)Km+1(κra)Km−1(κra)/K2 m(κra)(B 8a) and Rout nm =Km+1(κra)Km−1(κra)−K2 m(κra) Km+1(κra)Km−1(κra)−K2 m(κra)Jm+1(kra)Jm−1(kra)/J2 m(kra).(B8b) Similarly, using (B 5), we find for the longitudinal integrals (B 7) Zin ˜ n,p =1 1+κzL/2κzL 2+κ2 z k2 0and Zout ˜ n,p =k2 z/k2 0 1+κzL/2.(B9) In the high-energy semiclassical limit (k0a1, k0L1), which is well suited for the problem at hand [69], we find that the expressions (B 8) and (B 9) are well approximated by Rin nm ≃1, Rout nm ≃1 2 1 κra,Zin ˜ n,p ≃1andZout ˜ n,p ≃2 κzL k2 z k2 0 , (B 10) which then lead to (3.9).5 Appendix C. Toy model: two coupled oscillating dipoles In this appendix, we demonstrate that the results (4.8) and (4.13) for the resonance frequencies of the coupled modes in a dimer of cylindrical NPs in the limit of large interparticle separation distance (i.e. d/a1andd/L1) can be recovered from a toy model of two coupled anisotropic oscillating dipolar moments. Let us consider two ideal electric dipoles pi=−Neeri(i=1, 2), with rithe associated displacement of the electronic cloud with charge −Neeand mass Neme. The dimer (with interparticle distance d) is aligned along the z-direction and each dipole oscillates at the frequency ωdip 0,in the longitudinal (z) direction and ωdip 0,⊥in the transverse (x,y) directions. The Lagrangian for the system described above reads L=Neme 2 2  i=1˙ r2 i−ωdip 0,⊥ 2x2 i+y2 i−ωdip 0, 2z2 i−N2 ee2 d3r1·r2−3r1·ˆ zr2·ˆ z.(C1) Using that Nee2/me=ω2 pV/4π, with Vthe volume of the electronic cloud, the Euler–Lagrange equations of motion for the toy model (C 1) lead to the coupled mode resonance frequencies ωdip τ,=ωdip 0, 2+2τω2 p V 4πd3,ωdip τ,⊥=ωdip 0,⊥ 2+τω2 p V 4πd3,τ=±.(C2) 5In this semiclassical limit, the leading order expressions (B 10) do not satisfy unitarity, which requires the inclusion of highorder terms. However, since the absent terms are of negligible importance for the range of parameters we consider in this work we may omit them. Notably, this semiclassical limit has been shown to be an excellent approximation for a spherical NP [69] and it has the significant advantage of providing additional physical insight. 18 royalsocietypublishing.org/journal/rspa Proc. R. Soc. 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