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Surface Plasmon Resonance Damping in Spheroidal Metal Particles: Quantum Confinement, Shape and Polarization Dependences Jean Lermé, Christophe Bonnet, Marie-Ange Lebeault, Michel Pellarin, and Emmanuel Cottancin J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b12298 • Publication Date (Web): 14 Feb 2017 Downloaded from http://pubs.acs.org on February 16, 2017

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The Journal of Physical Chemistry

Surface Plasmon Resonance Damping in Spheroidal Metal Particles: Quantum Confinement, Shape and Polarization Dependences

Jean Lermé*, Christophe Bonnet, Marie-Ange Lebeault, Michel Pellarin, Emmanuel Cottancin Univ Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière F-69622, VILLEURBANNE, France

* To whom correspondence should be addressed. E-mail: [email protected]

1 Environment ACS Paragon Plus


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Abstract A key parameter for optimizing nanosized optical devices involving small metal particles is the spectral width of their Localized Surface Plasmon Resonances (LSPR). In the small size range the homogeneous LSPR linewidth is to a large extent ruled by the spatial confinementinduced broadening contribution which, within a classical description, underlies the popular phenomenological limited-mean-free-path model. This unavoidable contribution to the LSPR linewidth is basically a quantum finite-size effect rooted in the finite extent of the electronic wavefunctions. This broadening reflects the surface-induced decay of the coherent collective plasmon excitations into particle-hole (p-h) excitations (Landau damping), the signature of which is a size-dependent fragmented LSPR band pattern which is clearly evidenced in absorption spectra computed within the Time-Dependent Local-Density-Approximation (TDLDA). In this work we analyze the spatial confinement-induced LSPR damping contribution in the framework on an exact Hamiltonian formalism, assuming for convenience a jellium-type ionic density. In resorting to the Harmonic Potential Theorem (HPT), theorem stating that in the case of a harmonic external interaction the electronic center-of-mass coordinates separates strictly from the intrinsic motions of the individual electrons, we derive a simple approximate formula allowing to (i) quantify the size dependence of the LSPR damping in spherical nanoparticles (1/R law, where R is the sphere radius), and, (ii) bring to the fore the main factors ruling the confinement-induced LSPR broadening. Then the modeling is straightforwardly generalized to the case of spheroidal (prolate or oblate) metal particles. Our investigations show that the LSPR damping is expected to depend strongly on both the aspect ratio of the spheroidal particles and the polarization of the irradiating electric field, that is, on the nature -transverse or longitudinal- of the collective excitation. It is found that the magnitude of the damping is tightly related to that of the LSPR frequency which rules the number of p-h excitations degenerate with the plasmon energy. Qualitative analysis


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suggests that the results are quite general and probably hold for other non-spherical particle shapes. In particular, in the case of elongated particles, as rods, the enlargement of the longitudinal LSPR band by the confinement effects is predicted to be much smaller than that of the transverse LSPR band.



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1. INTRODUCTION

Developing photonics devices involving plasmonic metal nanostructures and nanoparticles (NPs) is a major area of current research in plasmonics.1-7 In this general context, many experimental and theoretical works have been carried out in order to gain a detailed and thorough understanding of the factors governing the collective electronic excitations in small particles made of simple metals, in particular the alkali and noble metals (Ag, Au), which exhibit the most conspicuous plasmonic properties.8-10 These unceasing efforts are justified insofar as most technological, sensing, biological or medical applications require tailoring specific optical properties. Nowadays the effort is mainly focused on silver NPs, and more especially gold NPs which are biocompatible, and thus well suited for biological and medical applications,4-7 and much less subject to oxidization compared to silver.11 The plasmonic excitations, referred to as Localized Surface Plasmon Resonances (LSPR) in the literature, give rise to large and spectrally narrow absorption/scattering bands, and correspond to collective electronic excitations which can be resonantly driven by light over the near UV-visible-near IR spectral range by playing on the size, shape and dielectric environment of the particles.8,12,13 The quality factor (QF) of the LSPR bands is a key parameter in many labeling, imaging and sensing applications at the nanoscale, as well as in the context of surfaceenhanced fluorescence/Raman spectroscopies since the local field enhancement around the particle is tightly related to the QF-value. Studying the various dissipative mechanisms responsible for the plasmon dephasing time and their relative contributions to the overall LSPR bandwidth is also of tremendous interest from a fundamental point of view. This last decade great effort has been made in order to reduce the inhomogeneous broadening effects


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which prevent experimental data from being reliably compared with theory. Thanks to various single-particle far-field experiments, combining both optical and imaging measurements on the same nano-object, as well as powerful lithographic techniques, homogeneous LSPR bandwidths of almost entirely characterized NPs can be measured nowadays.14-32 These impressive experimental developments provide stimulating challenges to theorists in placing severe modeling constraints. In the large-size domain the optical properties of NPs are well understood and can be suitably predicted by classical electrodynamics calculations, namely in solving the Maxwell equations with appropriate boundary conditions, assuming bulk dielectric functions.33-39 In this size range the evolution of the LSPR bandwidths can be analyzed in terms of size- and shape-dependent contributions stemming from nonradiative (intra- and inter-band components) and radiation damping mechanisms.30 In contrast, in the very small (roughly for radii less than 10-15 nm)-, and medium-size ranges, where the radiation damping is either negligible or not too dominating, noticeable discrepancies between experiment and theory can be observed. As a matter of fact different experimental results on -at first sight- similar size- and shape-selected NPs, are reported on in the literature.40-46 Indeed, when the particle size is decreased, the role played by the particle surface is of crucial importance and has a determining impact on the physical properties of the particle. Its involvement in the classical modeling, reduced to that of a perfect mathematical interface separating two homogeneous dielectric media, is clearly an oversimplified description. Probably all the morphological “details” over the surface (roughness, defects, local lattice shrinkage, residual surfactants, local porosity in the case of matrix-embedded particles), which have in most cases a negligible quantitative impact on the optical properties in the large-size domain, are expected to contribute to the net damping rate. These surface details probably explain the scatter of the data reported on by different research groups and



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observed in the statistics of the measurements performed on -apparently- quite identical NPs of a given sample.43,44 Nevertheless, in the case of small particles, the increase of the LSPR spectral width with decreasing size is a quasi-systematic observed trend. This feature, usually referred to as the spatial confinement -or finite size- effect, was empirically included in model calculations for a long time in a simple way, in noting that the conduction electron mean-free-path in small particles is reduced because of surface scattering events. Since all the finite size- and surfaceinduced effects are expected to lead to corrections scaling as S/V (S and V are the surface area and volume of the particle, respectively) the surface-induced increase of the scattering rate can be phenomenologically taken into account by introducing an additional contribution to the damping parameter Γ entering the Drude/Sommerfeld-type parametrization of the dielectric function corresponding to the free electrons, according to the scaling law8,47

h∆ω = hΓ( Leff ) = hΓ0 + gh

vF Leff

(1)

In the above equation hΓ0 the intrinsic bulk contribution, vF the Fermi velocity, Leff an appropriate shape-dependent length scale characterizing the particle, and g a dimensionless parameter of the order of unity, the value of which is thought to depend on the nature of the surface scattering process. The above parametrization, where the Fermi velocity enters explicitly, comes directly from the simple classical “billiard” picture which is invoked by many authors for introducing this phenomenological 1 / Leff -correction (Leff being defined in the classical ballistic picture as the average effective electron mean-free-path).47-50 It should be emphasized that -in general- hΓ ( Leff ) is not the full width at half-maximum (fwhm) of the LSPR band that is observed experimentally. In a classical modeling of the optical response of


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a small metal sphere (assuming the quasistatic approximation) the analytical expression of the LSPR-fwhm depends on both the real and imaginary components of the complex dielectric function, ε (ω ) = ε 1 (ω ) + iε 2 (ω ) , of the metal (see ref 8 page 32). In many cases of interest, in particular for alkali and for noble metals when the interband absorption does not overlap the LSPR band (LSPR frequency ω 0 ), a simplified expression can be obtained, namely h∆ω = 2hε 2 (ω 0 ) / ∂ε 1 / ∂ω ω ∝ hΓ . In the specific case of a simple metal (free electrons 0

only; the polarization arising from the core electrons can be neglected) the full dielectric function can be suitably modeled through the Drude-Sommerfeld formula, namely

ε (ω ) = 1 − ω P2 /[ω (ω + iΓ)] , where ω P is the plasma frequency and Γ the conduction (free) electron scattering rate. Since Γ 0, H(x) = 0 for x < 0] and the jellium sphere radius, respectively (the positive charge density is en+ (r ) , where e is the elementary charge). The number of conduction electrons, N, is related to the jellium radius R through the equation R = rs N 1 / 3 where rs is the Wigner-Seitz radius characterizing the bulk conduction electron density [ ( 4π / 3) rs3 = 1 / n+ ]. This simple model was for many decades successfully applied to compute the electronic structure in small simple metal particles.9-10 It involves the simplest ionic distribution allowing to generate, in the frame of a fully self-consistent quantum modeling, a flat-bottom electron confining potential well with steep walls at the particle surface.



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Figure 1 : Illustration of the homogeneous jellium model for the parameters N = 440 (number of conduction electrons) and rs = 3.02 bohr [Wigner-Seitz radius corresponding to the bulk silver density67 ; n+ = 3 /( 4π rs ) ]. 3

n+(r) is the ionic density [ n+ (r ) = n+ for r < R, and zero for r > R]. Upper figure: self-consistent ground-state electron density computed within the Kohn-Sham (KS) Density Functional Theory (DFT) formalism. Lower figure: self-consistent KS-DFT confining potential (Veff). The electron-jellium (Vjel ; eq 3 ) and harmonic (Vhar ; eq 5) interactions, divided by 50, are also shown. The dashed vertical line indicates the location of the jellium sphere radius R = rs N

1/ 3

.

The Hamiltonian representing N interacting electrons of effective mass me in the presence of the jellium sphere is

N

H = ∑[ i =1

N p i2 + V jel (ri )] + ∑Vc (ri − r j ) 2me i< j


(2)

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Vc (ri − r j ) = e 2 / 4πε 0 ri − r j

is the bare electron-electron Coulomb interaction and

V jel (r ) ≡ V jel ( r ) is the electron-jellium interaction, that will be referred to as the “external

interaction”. V jel (r ) can be expressed in the following convenient form

V jel (r ) =

1 R3 meω02 (r 2 − 3R 2 ) H ( R − r ) − meω02 H (r − R) 2 r

(3)

where ω0 = ω p / 3 . ω p = (e2 n+ / meε 0 )1 / 2 is the bulk metal plasma frequency.67 Note that the above external interaction is harmonic inside the jellium sphere and Coulombic outside. Selfconsistent calculations show that the occupied electronic states in the ground-state lie essentially in the quadratic region, the “amount of electrons” experiencing the Coulombic region being very small (see Figure 1). Therefore an excellent zero-order approximation ( H 0 ) and a tiny effective correction ( ∆V ) can be defined in splitting the Hamiltonian according to

N

H = H 0 + ∆V = ∑ [ i =1

N p i2 + Vhar (ri )] + ∑Vc (ri − r j ) + ∆V 2me i< j

(4)

with

Vhar (r ) =

1 meω02 (r 2 − 3R 2 ) 2

(5)

N

∆V = ∑ g (ri ) H ( ri − R)

(6)

i =1



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where

g (r ) ≡ g ( r ) = − me ω 02

R3 1 − meω 02 (r 2 − 3R 2 ) r 2

(7)

2.2 Corrections in energy and red-shift of the LSPR. Eq 4 is the basic starting point for quantifying, in the framework of this simple model, the finite-size corrections affecting the optical response of small metal particles (spectral red-shift and bandwidth of the LSPR) and deriving quasi-analytical formulas thanks to standard first-order perturbation theory. If the correction ∆V is disregarded (H = H0), the external interaction acting on the electrons is strictly parabolic over the whole space. In such a case it can be shown that: (i) the motion of the electronic CM separates perfectly from the intrinsic internal motions (electronic motions in the CM reference frame), and, in addition, (ii) the dipole excitation in the nonretarted limit (uniform exciting field) occurs at the frequency ω0 entering the quadratic external potential (see Section 4 in ref 61, where these fundamental properties have been illustrated through self-consistent TDLDA calculations). Moreover it can be proved that (iii) the entire electron distribution moves as a whole (rigid displacement), as in a simple dynamical classical picture. These features (i-iii) characterize what is referred to as “the Generalized Kohn Theorem (GKT)” or the “Harmonic Potential Theorem” (HPT) in the literature.68,70 Actually, the second property (ii) was experimentally confirmed for a long time in the case of parabolic quantum wells involving finite electron systems confined in semiconductor nanostructures.6870

Thanks to a suitable change of variables H 0 can be expressed as the sum of two terms. The

first one depends on the electronic CM coordinates and expresses as the Hamiltonian of a three-dimensional isotropic harmonic oscillator. The second one describes the intrinsic -or


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relative- internal electron motions in the CM frame (see the Supporting Information, sections A1 and A2, for details)

H 0 = H CM + H int

H CM

(8)

2 PCM 1 2 = + ( Nme )ω02 R CM 2( Nme ) 2

N

H int ≈ ∑ [ i =1

N p i '2 + Vhar (ri ' )] + ∑ Vc (ri '−r j ' ) 2me i< j

(9)

(10)

N

with R CM = (∑ ri ) / N and ri ' = ri − R CM . i =1

The energy spectra of the Hamiltonians H CM and H int will underlie the collective and single particle-hole excitations, respectively. The eigen-wavefunctions and -energies of H0 are of the form (obvious notations are used)

ψ i = nx , n y , nz

CM

ϕk

(11)

Ei = (n x + n y + n z + 3 / 2)hω 0 + Ekint = EiCM + E kint

(12)

The integer number set ( n x , n y , n z ) characterizes the degree of excitation of the degenerate plasmon modes (collective electron oscillations) along the three Cartesian axes. In the presence of a monochromatic electromagnetic field (frequency ω, amplitude E0), linearly



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polarized along the z-axis for instance, such as λ >> R (quasistatic limit), the particle-light interaction writes as

N

W (t ) = eE 0 [∑ ri ] . e z e −iωt = eNE 0 R CM . e z e −iωt

(13)

i =1

showing therefore that the light couples directly and solely to the CM z-coordinate. If the particle is initially in the ground state the absorption spectrum will consist of a single narrow peak at the energy hω0 = hω p / 3 , corresponding to the transition nz = 0 → nz = 1 (absorption of one photon of energy hω0 ; see eqs A27-A28 in the Supporting Information). On the other hand the intrinsic motion state ϕk

is left unchanged. Note that the value

ω0 = ω p / 3 is the LSPR frequency obtained from classical electrodynamics, in the quasistatic limit, in the case of a simple metal sphere in vacuum when a Drude-Sommerfeld parametrization is assumed for modeling the dielectric function of the conduction electron gas ( ε (ω ) = 1 − ω p2 /[ω (ω + iΓ)] ).8 This value is also the asymptotic (i.e. large radii) LSPR frequency computed in the frame of self-consistent quantum formalisms, such as the Random Phase Approximation (RPA) or the TDLDA. It is worthwhile noting also that the collective excitations, that is the CM energy spectrum, is governed by the external forces only (parameter ω0), as expected from classical mechanics, since the electron-electron interaction Vc contributes only to the intrinsic Hamiltonian Hint. These preliminaries, which have been very shortly sketched out in ref 61, point out that the perturbation operator ∆V is responsible for all the finite-size corrections. Selfconsistent numerical RPA/TDLDA computations can quantify accurately these corrections, either within the simple jellium model10 or in more realistic models involving additional


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dielectric media (polarizable ionic core background and embedding matrix).60-61,71-72 The advantage of the present Hamiltonian approach is its ability to bring to the fore the physical ingredients in which are rooted the confinement-induced corrections. The first information already gained follows directly from the previous analysis: the spilling out of the electron cloud beyond the classical radius of the particle (R), where the Coulombic electron-jellium interaction is probed, is responsible for the quantum size corrections. This feature was supported by numerous theoretical works reported on in the literature. They indicate moreover that the magnitude of the finite-size corrections is intimately related to that of the extent of the electron spillout tail beyond the particle surface. This correlation is clearly evidenced in refs 60-61 through several illustrative examples, in the frame of various models. From a straightforward numerical estimation, it can be shown that the CM displacements are very tiny in the lower part of the CM spectrum, typically a small fraction of one Bohr radius a0 (see the note ref 7 in the Supporting Information). Taylor expansion of the perturbation ∆V up to the second order with respect to the CM coordinate RCM will allow to grasp the major corrections evidenced by numerical RPA/TDLDA calculations. Since g (r ) and dg (r ) / dr vanish for r = R one obtains (let us keep in mind that, in the following equation, the coordinates are actually quantum operators)

N

N

i =1

i =1

∆V = ∑ g (ri '+ R CM ) H ( ri '+ R CM − R ) ≈ ∑ g (ri ') H ( r 'i − R )

N

+ R CM . ∑ [∇g (ri )]0 H ( ri '− R ) + i =1

∂ 2 g (ri ) 1 U CM VCM ∑ [ ] 0 H (ri '− R ) ∑ 2 U ,V ∂u i ∂vi i


(14)


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where U and V stand for X, Y and Z, u and v stand for x, y or z , and the notation [..]0 means that the first and second derivatives of g(ri) are evaluated with RCM = 0 (these derivatives depend thus on the relative coordinates ri’). The first term in the right hand side of eq 14, which does not depend on the CM coordinates, can be added to the intrinsic Hamiltonian Hint, restoring thus the complete electron-jellium interaction in Hint. The second term, which involves both the CM and intrinsic motion coordinates (RCM and ri’), is the coupling operator responsible for the Landau damping mechanism (operator denoted Wc in the following). This term will be addressed specifically in the following subsection. The second-order terms can be handled in applying the standard first-order perturbation theory64 and replaced by an effective change of the CM Hamiltonian HCM, through a tiny correction of the LSPR frequency ω0. It can be shown that, for large N, only the square second-order terms (U = V) contribute to the first-order correction in energy due to the spherical symmetry of the electron density. One obtains (see the Supporting Information, section A.3 for details and relevant comments)

∆E i =

∂ 2 g (ri ) 1 1 ∆N CM 2 ψ i U CM [ ]0 H ( ri '− R ) ψ i = − Ei ∑ ∑ 2 2 U 2 N ∂u i i

(15)

where ∆N is the “amount” of electrons located beyond the jellium radius in the intrinsic many-electron state ϕk (electron density ρ e , k (r ) ≅ ρ e ,k ( r ) )

∞

∆N = 4π ∫ r 2 ρ e ,k ( r ) dr

(16)

R


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When N is very large (say N > 103), and as long as the low energy spectrum is concerned [intrinsic states involving a few particle-hole excitations (Nex

Surface Plasmon Resonance Damping in ... - ACS Publications

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