Lorentz Friction for Surface Plasmons in Metallic Nanospheres - The

Article pubs.acs.org/JPCC

Lorentz Friction for Surface Plasmons in Metallic Nanospheres Witold A. Jacak* Institute of Physics, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland ABSTRACT: The Lorentz friction for surface plasmons in metallic nanospheres has been analyzed versus nanoparticle size upon random-phase approximation quasiclassical method. An accurate inclusion of the Lorentz friction to the plasmon damping has been performed. Total attenuation of surface plasmons in a metallic nanosphere has been calculated and minimized with respect to the radius of the nanosphere. Experimentally measured anomalous red-shift of plasmon resonance frequency, increasing with the metallic particle radius growth, especially pronounced for a large size limit, was explained in agreement with the experimental data for Au nanospheres of radii 10−75 nm (the measured quantity being the extinction of the light trans-passing through the water colloidal solution of metallic nanospheres with various radii). A very good agreement of the measured data and the theory with properly included Lorentz friction has been achieved.

P

nanospheres but within an approximation via reduction of the third-order time derivative (in perturbation approach) the firstorder one as in the ordinary damping term in the scheme of damped harmonic oscillator.11,18 Such an approximation resulted in ∼a3 increase of the irradiation damping rate for plasmons with growth of the metallic nanosphere radius a. Such fast growth of irradiation losses with nanoparticle radius suggests that the Lorentz friction is dominating over all other attenuation channels in the case of large nanospheres but also causes violation of the perturbation calculus necessary limits (smallness condition), which precludes usage of this approximation for nanosheres with larger radii. In order to describe the effect of the Lorentz friction beyond the scope of the previously used perturbation approach regime, linearization of the third-order derivative must be avoided, and the whole contribution has to be accounted for accurately. As shown further, adequate corrections are most important for nanospheres of radii above ca. 30 nm (for Au in vacuum). Accurate accounting for the Lorentz friction enables an explanation of size-dependent anomalous light extinction observed in large metallic nanospheres (as measured in water colloidal solutions of metallic nanoparticles with varying size11). Exact inclusion of the Lorentz friction is critical for improvement of the experimental data fitting, which was previously not highly accurate upon an approximate-perturbative approach with a growing discrepancy occurring especially for the larger nanospheres (a > 30 nm, for Au) when the perturbation approach fails. As mentioned above, small metallic nanoparticles in the form of clusters of size 1−5 nm do not exhibit irradiation efficiency as high as nanospheres with radii a > 10 nm. The small metallic clusters with radii on the order of a single nanometer were

lasmonic oscillations in metallic nanoparticles are recently in the center of attention due to prospective applications in photovoltaics and in photonics. Solar cells modified by deposition on surface metallic nanoparticles are regarded as prospective for a new generation of solar cells with enhanced efficiencies due to plasmonic mediation in solar energy transfer.1−5 The crucial role in this phenomenon is played by surface plasmons in metallic nanoparticles which can be easily excited by incident photons just to quickly transmit energy to the substrate semiconductor. Irradiation hampers plasmon oscillations, but the higher irradiation losses the better the energy transmittance. Radiation properties of plasmons are also determinant for the subdiffraction plasmon−polariton weakly damped propagation in plasmonic wave guides made of metallic nanochains6−10 with prospective applications in nanophotonics. As was observed experimentally and predicted theoretically, irradiation losses in plasmons oscillations are strongly dependent on nanoparticle dimensions.11 One should note that the ultrasmall metal clusters of radius a ∈ (1, 5) nm show very weak irradiation,12−15 and the plasmon damping in this scale is primarily due to the scattering of electrons as for Ohmic losses and electron collisions with cluster boundary. With increasing nanoparticle dimensions, plasmon irradiation intensity quickly grows due to strengthening of the plasmon oscillation amplitude, proportionally dependent on the number of electrons in the nanoparticle. Electron plasma oscillation radiative losses can be accounted for by the Lorentz friction.16,17 The Lorentz friction acts on accelerating charges which irradiate the electromagnetic wave. The irradiated energy can be treated as effective kinetic energy losses caused by the Lorentz friction hampering electron motion. In the case of oscillating dipole, as is the situation of a dipole-type surface plasmon in metallic nanosphere, the Lorentz friction is proportional to the third-order time derivative of this dipole.16,17 The Lorentz friction has been already included to describe surface plasmons in metallic © XXXX American Chemical Society

Received: November 18, 2014 Revised: February 27, 2015

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DOI: 10.1021/jp511560g J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

Damped Plasmonic Oscillations in Metallic Nanospheres Induced by Electric Field. For the model we will consider a metallic (Au) nanosphere of radius a immersed in a dielectric environment with the relative dielectric permittivity ε. We make an assumption of the presence of varying external electric field which can excite plasmon collective modes of oscillations of the electron liquid in the nanosphere. Such a situation can correspond to the dipole limit for interaction with light (i.e., in case when the wavelength of incident light well exceeds the nanosphere dimension). The surface plasmon resonance for metallic nanospheres (in noble metal case, Au, Ag) with the radius a ∈ (5, 100) nm falls for approximately 500 nm wavelength light; thus, conditions for the dipole approximation are fulfilled and the resonant incident electromagnetic signal can be treated as homogeneous along the entire nanoparticle. We will analyze a collective reaction of the electron liquid to this space-homogeneous but time-dependent field in the spherical metallic nanoparticle assuming the socalled jellium model, i.e., neglecting the dynamics of heavier ions. Their dynamics are inertial due to higher mass of ions in comparison to the electron mass. Within the jellium model, the charge of ion background is static and uniformly smeared over the whole nanosphere, with the density ne(r) = neΘ(a − r), ne = Ne/V, where Ne is the number of free electrons in the nanosphere, V is the volume of the nanosphere, Θ is the Heaviside step function, and |e|ne is the the averaged density of the positive charge compensating for the charge of free electrons. The plasmonic oscillations in metallic nanosphere can be described by the local fluctuation of electron density defined similarly as in the RPA approach in bulk metal developed by Pines and Bohm20

widely investigated in the 1980s, theoretically within the socalled local density approximation (LDA) or time-dependent local density approximation (TDLDA) methods12−14,19 and experimentally for ultrasmall particles of alkali metals like Na or K, which confirmed their poor irradiation efficiency. The electromagnetic irradiation of plasmons in these small systems turned out to be weak due to the relatively small amount of electrons in such clusters, but other unique properties were important on this scale. Special attention was paid to the socalled spill-out of electron liquid beyond the rim of ion lattice and the influence of this effect on plasmon oscillations in small clusters. This spill-out results in the dilution of electron density inside the system, and this is a reason for the red-shift of resonance frequency of surface plasmons experimentally observed for ultrasmall nanoparticles of K and Na. The spillout effect diminishes, however, with the increase of the cluster size because of lowering role of surface related phenomena when increasing the radius of the system. The spill-out takes place on the scale of the Thomas−Fermi length20 which is of subnanometer value and is lower by at least 1 order of magnitude in comparison to the nanosphere radius when a > 5 nm. In metallic nanoparticles with the size of several tens of nanometers the radiative plasmon effects are of primary significance and many quantum effects like spill-out or the so-called Landau damping (corresponding to the decay of plasmon into the high energy electron−hole pair with respect to the Fermi surface)21 are negligible in this size scale. To describe plasmon behavior in relatively large electron systems with 105−8 electrons, i.e., with the nanosphere radius of several tens of nanometers, the numerical accurate methods based on the Kohn−Sham approach (LDA and TDLDA) are not useful as limited to much smaller number of electrons, i.e., 200−300, as was the case for 2−3 nm scale clusters. With an increase of electron number above 300, these numerical methods are less effective because numerical-type constraints rapidly slow calculations even at high performance facilities. For the large nanoparticles various versions of random phase approximation (RPA) turned out to be useful.12,13,18,22 Of particular interest are large nanoparticles of noble metals (gold, silver, and also copper) because of location of surface plasmon resonances in particles of these metals within the visible light spectrum, which is important for photovoltaic applications. In the present paper, we focus on large metallic nanospheres analyzed in the framework of the RPA semiclassical method including damping effects. The following paragraph contains analysis of the surface dipole-type plasmons in a single nanosphere within our previously formulated model11,18 but including the damping effects of scattering and of irradiation type, the latter expressed by the Lorentz friction force for plasmons. The subsequent paragraph contains discussion of the relevant dynamical equation for plasmonic oscillations with damping and compares the new accurate formulation of accounting for the Lorentz friction with the previously developed linearized approximation. The next subsequent paragraph is addressed to comparing theoretical radiative effects of Lorentz friction dominated plasmon damping with the experimental results. Finally, the last paragraph is focused on the discussion of some conclusions drawn from the exact analysis of the Lorentz friction to the widely developed Mie theory of light properties in metallic nanoparticles highly useful and efficient for interpretation of many experiments in the field of nanophotonics and plasmonics.

ρ(r, t ) = (1)

j

where rj is the quasiclassically positions of electrons and |Ψe(t) > is the multiparticle wave function of the electron system (for Hamiltonian 4). The Fourier picture of 1 has the simple form ρ ̃ (k , t ) =

∫ ρ(r, t )e−ik·rd3r =

(2)

where the operator ρ̂ (k) = ∑j e−ik·rj. The dynamic equation for this local electron density operator can be found by the twicerepeating Heisenberg equation d 2ρ (̂ k) 1 [[ρ (̂ k), Ĥe], Ĥe] = (iℏ)2 dt 2

(3)

with the Hamiltonian for electrons in the metallic nanosphere upon the jellium model and with electron interaction expressed by local electron density operators including also interaction of electrons with jellium18 Ne

Ĥe = −∑ j=1

+

ℏ2∇2j 2m

e2 (2π )3

−

e2 (2π )3

∫ d3knẽ (k) 2kπ2 (ρ+̂ (k) + ρ (̂ k))

∫ d3k 2kπ2 (ρ+̂ (k)ρ (̂ k) − Ne)

(4)

−ik·r

where ñe(k) = ∫ d rne(r)e . Equation 3, after averaging over the multiparticle wave function and according to the scheme of RPA, i.e., neglecting sums of exponents in random phases, attains finally the form18 3

B


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The Journal of Physical Chemistry C σ (Ω , t ) =

⎡ ϵ ωp2 r 2 r − ⎢ F ∇δρ ̃(r, t ) + ∇ ⎢⎣ 3 m r 4π r

∫

+

⎤ 1 δρ ̃(r1, t )⎥δ(a − r ) d3r1 ⎥⎦ |r − r1|

∫0

In the above formula, the Thomas−Fermi approximation for the kinetic energy has been used;18,20 ωp is the bulk plasmon frequency, ωp2 = (4πne e2)/m (in SI ωp2 = (e2ne)/(ε0m)). It was taken into account that ∇Θ(a − r) =−(r/r)δ(a − r), which gives rise to the Dirac delta distinguishing surface plasmon fluctuations in eq 5. Due to the structure of eq 5, a general solution is thus decomposed in two parts related to the two distinct domains

∂t

2

2 ϵF 2 = ∇ δρ1̃ (r, t ) − ωp2δρ1̃ (r, t ) 3m

(6)

ω0l = ωp

∂t

2

⎤r 2 ⎧⎡ 3 ∇⎨⎢ ϵFne + ϵF δρ2̃ (r, t )⎥ δ ⎦r 3m ⎩⎣ 5 ⎫ ⎡2 ϵ r (a + ϵ − r )⎬ − ⎢ F ∇δρ2̃ (r, t ) ⎭ ⎢⎣ 3 m r ωp2 r ∇ 4π r

∫ d3r1 |r −1 r | (δρ1̃ (r1, t )Θ(a − r1)

δρ1̃ (r, t ) = neF(r, t ), for r < a ,

(9)

supplemented with the initial conditions F(r, t)|t=0 = 0, σ(Ω,t)|t=0 = 0 (Ω is the spherical angle) and with the neutrality condition, F(r, t)|r→a = 0, ∫ ρ(r,t)d3r = Ne. After some algebra (as presented in details in ref 18), one can find for the spherical symmetry of the nanoparticle (assumed as the nanosphere) ∞

F (r , t ) =

l

∞

∑ ∑ ∑ Almnjl (knlr)Ylm(Ω)sin(ωnlt ) l = 1 m =−l n = 1

dr1

r1l + 2 al + 2

(l + 1)ωp2 lωp2 − (2l + 1)ωnl2

jl (knlr1)sin(ωnlt )

Ylm(Ω)ne

(11)

l 2l + 1

(12)

(13)

where C is the constant of unity order, a is the nanosphere radius, vF is the Fermi velocity in the metal, and λb is the electron mean free path in bulk metal (including scattering of electrons on other electrons, on impurities and on phonons24); for example, for Au vF = 1.396 × 106 m/s and λb ≃ 53 nm (at room temperature); the latter term in formula 13 accounts for scattering of electrons on the boundary of the nanoparticle, whereas the former one corresponds to scattering processes similar to bulk (Ohmic irreversible energy losses). The other quantum effects, as, for example, the so-called Landau damping (especially important in small clusters19,21), corresponding to the decay of plasmon to a high energy particle−hole pair, are of lower significance for nanosphere radii larger than 2−3 nm21 and are completely negligible for radii larger than 5 nm. Electron response to the driving field E(t) (which is homogeneous over the nanosphere, corresponding to the dipole approximation) resolves to single dipole type mode, effectively described by the three-component function denoted here as Q1m(t) (l = 1 and m = −1, 0, 1). The dynamical eq 8 reduces in this case to

(8)

For solution of eqs 7 and 8 let us represent both parts of the electron fluctuation in the following manner

δρ2̃ (r, t ) = σ(Ω, t )δ(r + ϵ − a), ϵ = 0+, for r ≥ a , (→ a + )

a

v Cv 1 ≃ F + F τ0 2λb 2a

1

⎤ + δρ2̃ (r1 , t )Θ(r1 − a))⎥δ(a + ϵ − r ) ⎥⎦

∞

∑ ∑ ∑ Almn

which, for l = 1, gives the dipole type surface oscillation frequency originally found by Mie,23 ω01 = ωp/31/2 (for simplicity, hereafter we will denote it as ω1 = ω01). The advantage of this RPA approach consists of the oscillatory form of both eqs 7 and 8. The damping of plasmons can be thus included in a phenomenological manner by addition of an attenuation term to plasmon dynamic equations, i.e., of the term, −(2/τ0)(∂δρ1(2)(r,t))/(∂t), added to the rhs of eqs 7 and 8. The damping ratio 1/τ0 accounts for electron scattering losses and can be approximated by the formula24

=−

+

Ylm(Ω)sin(ω0lt )

(7)

and (here ϵ = 0+ is introduced formally to ensure the correct definition of the Dirac delta) ∂ 2δρ2̃ (r, t )

a2

where jl(ξ) = (π/(2ξ))1/2Il+1/2(ξ) is the spherical Bessel function, Ylm(Ω) is the spherical function, ωnl = ωp(1 + x2nl/ (kT2 a2))1/2 are the frequencies of volume plasmon selfoscillations, kT = ((6nee2)/(ϵF))1/2 is the Thomas−Fermi length (ϵF is Fermi energy), xnl are the nodes of the Bessel function jl(ξ), and knl = xnl/a, ω0l = ωp(l/(2l + 1))1/2 are the frequencies of electron surface self-oscillations (surface plasmon frequencies). The function F(r,t) describes the volume plasmon oscillations, while σ(Ω,t) describes the surface plasmon oscillations. In the formula for σ(Ω,t), eq 11, the first term corresponds to surface self-oscillations, while the second term describes the surface oscillations induced by the volume plasmons. The frequencies of the surface selfoscillations corresponding to multipole modes enumerated by l have the form

corresponding accordingly to the volume and surface charge fluctuations. These two parts of local electron density fluctuation satisfy the equations (for particularities of the relevant derivation cf. ref 18) ∂ 2δρ1̃ (r, t )

Blm

l = 1 m =−l n = 1

(5)

⎧ ⎪ δρ1̃ (r , t ), for r < a , δρ ̃(r, t ) = ⎨ ⎪ ⎩ δρ2̃ (r, t ), for r ≥ a , (→ a + )

∑∑ l = 1 m =−l ∞ l

⎫ ⎤r 2 ⎧⎡ 3 ∇⎨⎢ ϵFne + ϵFδρ ̃(r, t )⎥ δ(a − r )⎬ ⎦r ⎭ 3m ⎩⎣ 5

−

l

∞

⎤ ∂ 2δρ ̃(r, t ) ⎡ 2 ϵF 2 =⎢ ∇ δρ ̃(r, t ) − ωp2δρ ̃(r, t )⎥Θ(a − r ) ⎣3 m ⎦ ∂t 2

(10)

and C


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The Journal of Physical Chemistry C ∂ 2Q 1m(t ) ∂t

2

+

4π ene [Ez(t )δm ,0 + 3 m

=

while an unbalanced charge density fluctuation occurs only on the surface. One can note that in the case of the volume plasmons the noncompensated charge density fluctuations are present inside the sphere because the volume plasmon modes are related to compressional modes resulting in volume charge fluctuations not balanced by the jellium along the radius inside the nanosphere. The damping term 1/τ0 accounts for dissipation energy of plasmons due to scattering of electrons similar as for Ohmic losses. This channel of energy dissipation includes all types of scattering phenomena: electron−electron, electron−phonon, electron−admixture collisions as for Ohmic losses, and additionally the contribution related to scattering of electrons on the nanoparticle boundary.24 All of these scattering processes cause attenuation of plasmons, and their energy is dissipated irreversibly and finally converted into heat. By reducing structural imperfections in the metal one can diminish these losses to some limiting value. The scattering channel of plasmons attenuation turns out to be especially important in small metal clusters, for which it is a predominant mechanism of plasmon damping together with the Landau damping.19,21 In small metallic clusters the irradiation effects play a negligible role because of the low total number of electrons (∼100−300 for radius a ∼ 2−3 nm). For ultrasmall clusters the nanosphereedge scattering also plays a significant role as its contribution to damping time ratio is proportional to 1/a and thus is large for small a but quickly diminishes when the radius increase. The scattering of electrons by the boundary of the nanosphere may undergo along different regimes depending of microscopic details of the surface. One can refer this regime to two limiting model scattering types: reflection and diffusive types which differ in the resulting attenuation rate. In order to model various types of boundary scattering of electrons one can introduce in eq 13 the effective constant C of order of unity which can be helpful in accommodation of experimental data utilizing the 1/a size-scaling of this attenuation channel. Below we will show that radiation losses which also contribute to the overall attenuation of plasmons start to be important in the case of larger metallic nanoparticles with sufficiently large number of electrons. Radiation losses scale as a3 (for not too large nanospheres, e.g., for Au up to ca. 30 nm) and, therefore, the radiative losses quickly dominate plasmon damping in the range of 10 nm (for Au, Ag, and Cu). Therefore, at the immediate size of nanospheres, we encounter crossover in the size dependence of the plasmon damping: for lower radii sizedependence scales as 1/a whereas for larger radii it scales as a3.11 This pronounced crossover in size dependence of surface plasmon damping is illustrated in Figure 2. Irradiation energy losses of oscillating electrons related to surface plasmon time-dependent dipole can be expressed by the so-called Lorentz friction,16 i.e., by the effective electric field hampering movement of charges and thus lowering the amplitude of dipole oscillations:

2 ∂Q 1m(t ) + ω12Q 1m(t ) τ0 ∂t 2 (Ex(t )δm ,1 + Ey(t )δm , −1)] (14)

where ω1 = ωp/(3ε) is the dipole surface plasmon frequency found within the semiclassical RPA approach18 (ωp = (4πne2)/ m is the plasmon resonance frequency in bulk metal and ε is the relative dielectric permittivity of the nanosphere surrounding material). For ε = 1, this frequency coincides with the classical Mie frequency ωMie = ωp/(2ε + 1)1/2.23 The semiclassical RPA dipole frequency ω1 is a bit smaller than ωMie for ε > 1, as illustrated in Figure 1, which is supported, on the other hand, by a more exact numerical TDLDA calculus indicating some overestimation of the plasmon resonance frequency by the Mie formula.25 1/2

Figure 1. Comparison of the RPA semiclassical formula for the dipole resonance frequency of surface plasmons in the metallic nanosphere, ω1 = ωp/(3ε)1/2, with the classical formula by Mie for this frequency, ωMie = ωp/(2ε + 1)1/2.

The electron density fluctuations described by eq 14 follow ⎧ 0, r < a , ⎪ δρ(r, t ) = ⎨ 1 ⎪ ∑ Q 1m(t )Y1m(Ω) r ≥ a , r → a + ⎩ m =−1

(15)

For dipole-type plasmonic oscillations as presented by eq 15, the dipole of electron system d(t) can be written as follows d(t ) = e =

∫ d3r rδρ(r, t ) 2π 3 ea [Q 1,1(t ), Q 1, −1(t ), 3

2 Q 1,0(t )]

(16)

and the dipole d(t) satisfies following equation (obtained via rewriting eq 14): ⎡ ∂2 ⎤ a34πe 2ne 2 ∂ + ω12 ⎥d(t ) = E(t ) = εa3ω12 E(t ) ⎢ 2 + τ0 ∂t 3m ⎣ ∂t ⎦ (17)

EL =

According to eq 16, we notice that the dipole of plasmon oscillations scales with the nanosphere radius as ∼a3, which indicates that all electrons in the sphere contribute to the surface plasmon excitations. This can be referred to the fact that the surface plasmon modes correspond to uniform translationtype oscillations of the whole electron liquid. Inside the sphere the uniformly shifted charge of electrons is still exactly compensated by static and also uniform positive jellium,

2 ∂ 3d(t ) 3c 3 ∂t 3

(18)

Hence, one can rewrite eq 17 including the Lorentz friction term as ⎡ ∂2 ⎤ 2 ∂ + ω12 ⎥d(t ) = εa3ω12 E(t ) + εa3ω12 E L ⎢ 2 + τ0 ∂t ⎣ ∂t ⎦ (19) D


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measurement of light extinction spectra11 in agreement with the perturbation approach in this size window. In this case, solution of eq 21 has the following form d(t) = d0e−t/τcos(ω1′t + ϕ), where ω1′ = ω1(1 − 1/(ω1τ)2)1/2 gives the experimentally observed red-shift of the plasmon resonance owing to ∼a3 increase of damping caused by radiation losses. The Lorentz friction term in eq 22 starts to dominate plasmon damping for approximately a ≥ 12 nm (for Au and Ag, almost independently of the relative dielectric permittivity of the surrounding medium ε as illustrated in the insets to Figure 2) due to this strong a3 dependence; cf. Figure 2. The plasmonic oscillation damping grows sharply with a, thus causing red-shift of the surface dipole plasmon resonance in very good coincidence with experimental data for 10 < a < 30 nm (Au and Ag).11 Attenuation of Dipole Surface Plasmons with Exact Inclusion of the Lorentz Friction. Let us now consider the dynamic equation for surface plasmons in the metallic nanosphere 20 with the Lorentz friction but without the substitution of the Lorentz friction term (2/(3ω1ε1/2))((ωpa)/ (c31/2))3((∂3d(t))/(∂t3)) by the approximate-perturbative formula, −(2ω1)/(3ε1/2((ωpa)/(c31/2))3∂d(t)/(∂t)). Such an approximate linearization of the third-order derivative of d(t) is justified only when the total attenuation term ∂/(∂t)[−(2/τ0) + (2/3ω1ε1/2((ωpa)/(c31/2))3(∂2/(∂t2))]))(dt) is sufficiently small as needed by the perturbation calculus conditions. To compare different contributions to eq 20 one must change to dimensionless time variable t→ t′ = ω1t. Then eq 20 attains the form

Figure 2. Comparison of different contributions to plasmonic oscillations damping (Au nanoparticles in vacuum and in water) in the size regions of a ∼ 10−20 nm (in insets) where the crossover in size-dependence for damping rate takes place; in upper right inset the comparison of an impact of C constant in eq 13 is illustrated modeling various regimes of scattering of electrons on nanosphere boundary.

or in a more explicit form, limiting to the case when E = 0, as ⎡ ∂2 ⎤ ⎢ 2 + ω12 ⎥d(t ) ⎣ ∂t ⎦ 3 ⎡ ⎤ 2 2 ⎛ ωpa ⎞ ∂ 2 ∂ = ⎢ − d(t ) + ⎟ 2 d(t )⎥ ⎜ 3ω1 ε ⎝ c 3 ⎠ ∂t ∂t ⎢⎣ τ0 ⎦

3 ∂ 2d(t ′) 2 ∂d(t ′) 2 ⎛ ωpa ⎞ ∂ 3d(t ′) + d(t ′) = + ⎜ ⎟ τ0ω1 ∂t ′ 3 ε ⎝ c 3 ⎠ ∂t ′3 ∂t ′2

(20)

(23)

One can solve eq 20 by application of the perturbation calculus; i.e., one can treat the rhs of this equation as a small perturbation. Therefore, in the zeroth step of the perturbation calculus one gets [∂2/(∂t2) + ω12]d(t) = 0 and, thus, ∂2/(∂t2) d(t) = −ω12d(t). Next, upon the first step of the perturbation method, one can put the latter formula to the rhs of eq 20, i.e.

In eq 23 the dimensionless damping ratio caused by scattering, 2/(τ0ω1) = (2/ω1)[(vF/2λb) + (vF/2a)] ≃ 0.027 for a = 10 nm (other parameters are taken for Au and Ag as collected in Table 1) and diminishes with increasing radius a; cf. Figure 2 insets.

⎡∂ ⎤ 2 ∂ + ω12 ⎥d(t ) = 0 ⎢ 2 + τ ∂t ⎣ ∂t ⎦

Table 1. Nanosphere Parameters Assumed for Calculation of Damping Rates for Surface Plasmons

2

(21)

where the effective attenuation rate is 3 ω1 ⎛ ωpa ⎞ 1 1 = + ⎜ ⎟ τ τ0 3 ε ⎝c 3 ⎠

(22)

Upon these steps the Lorentz friction is accounted for in total attenuation rate 1/τ utilizing the approximate-perturbative linearization of the third order derivative in eq 18. Such an approximation leads to the effective model of radiation losses within the scheme of harmonic damped oscillator. Nevertheless, as mentioned, this approximation is justified only for small perturbations, i.e., when the radiation term, the second one in eq 22 proportional to a3, is sufficiently small, as required by the perturbation calculus limits. For the nanospheres of radii a < 30 nm (Au, vacuum) the perturbation constraints are not violated, which was also independently confirmed experimentally for Au and Ag nanospheres.11 For nanosphere radii 5−30 nm the perturbation approximation allows for explanation of an experimentally observed red-shift of resonance surface plasmon frequency with the increase of the radius a, recordable from the

material

Au

Ag

bulk plasmon energy, ℏωp (eV) bulk plasmon frequency, ωp (1/s) Mie dipole plasmon energy, ℏω1 (eV) Mie frequency, ω1 = ωp/(3)1/21/2 (1/s) constant in eq 13, C Fermi velocity, vF (m/s) bulk mean free path (rt), λb (nm)

8.57 1.302 × 1016 4.94 0.752 × 1016 1, 4 1.396 × 106 53

8.56 1.3 × 1016 4.93 0.75 × 1016 1, 4 1.4 × 106 57

One can also estimate the dimensionless coefficient of the Lorentz friction term in eq 23. It equals 2/(3ε1/2)((ωpa)/ (c31/2))3 and rapidly increases with the radius a growth because it is proportional to a3 (for a = 10 nm is rather small, ≃0.0104, but for a = 25 nm increases more than ten times to ≃0.1638; the example is taken for Au nanospheres in vacuum). When looking for a solution to the perturbation method eq 23 one gets the renormalized attenuation rate for the effective damping term in the form, 1/(ω1τ0) + (1/3ε1/2)((ωpa)/ (c31/2))3. Because of the rapid increase of the factor a3 with the radius a growth, the effective damping term obtained by the perturbation approach quickly reaches the value 1, for which E


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Figure 3. Real and imaginary parts of the exact oscillating solution of eq 23, i.e., the resonance frequency and the damping rate given by eqs 24, are plotted as the functions of the nanosphere radius a and for two distinct dielectric surrounding media (vacuum and water); the lines corresponding to the perturbative solutions are presented for comparison. In the right panel, the resonance wavelength is depicted versus the radius a for the vacuum (ε = 1) and water surroundings (ε = 2). In the inset in the central panel an impact of C constant in eq 13 is illustrated.

Figure 4. Damping rate and the resonance frequency comparison of the oscillating solutions of eq 23. The exact and the approximate perturbative approach dependences are presented, both with respect to the nanosphere radius a and for selected dielectric surroundings; the termination of perturbative solutions displays an artifact of the perturbation method.

the oscillator enters an overdamped regime. The value 1 of this renormalized attenuation rate is attained at a ≃ 57 nm (for Au in vacuum). Nevertheless, this is an apparent artifact of the perturbation calculus, and one can check that the exact solution of eq 23 sufficiently well agrees with the solution obtained in the perturbation approach only up to a ≈ 20−30 nm. For the higher values of a the rapid increase of the effective attenuation rate causes unacceptable discrepancy between the perturbation approximation and the accurate solution, as illustrated in Figures 3 and 4. In Figure 4 there is plotted a comparison of the selffrequency and the damping rate for dipole-type surface plasmon as functions of the nanosphere radius a for the perturbation approximation (in this approximate case ω = ω1[1 − [(1/(ω1τ0)) + (1/3ε1/2)((ωpa)/(c31/2))3]2]1/2) and for the exact solution of eq 23. The lines corresponding to the approximate perturbation solution end at a ≃ 57, 64, 69, 72, 75 nm (Au, for relative dielectric permittivity of surroundings ε = 1, 2, 3, 4, 5, respectively). At these values of a the effective attenuation rate found upon the perturbation calculus reaches a limiting value of 1 (then the corresponding resonance frequency vanishes and resonance wavelength λ = c2π/ω → ∞). This limiting behavior corresponds to the termination of the existence of oscillating solutions of the perturbative oscillatory equation which in this case falls into an overdamped regime with an aperiodic solution. This is, however, an artificial property caused by the perturbation solution method which cannot be applied for too high of a perturbation term value. For an accurate solution of eq 23 this singular behavior disappears and the realistic oscillating solution exists for larger a as well. Thus, one can notice that the plasmon resonance red-shift is

strongly overestimated within the framework of the perturbative calculus based approach to the Lorentz friction unless the nanosphere radius is lower than ∼20 nm (Au, vacuum). The dynamical equation for dipole surface plasmon excitations including the exact form for the Lorentz friction, i.e., eq 23, is a third-order linear differential equation. This equation can be solved exactly and the exponents of its solutions, ∼eiΩt′, can be listed in the following analytical form: Ω1= −

i 21/3(1 + 6lq) i i( − − 1/3 = iα ∈ Im 3l 3l ( 2 3l

i(1 + i 3 )(1 + 6lq) i(1 − i 3 )( i + + 3l 22/33l ( 21/36l 1 =ω+i τ 1 Ω3= − ω + i = −Ω*2 τ Ω 2= −

(24)

where ( = (2 + 27l2 + 18lq + (4(−1 − 6lq)3 + (2 + 27l2 + 18lq)2)1/2)1/3, q = 1/(τ0 ω1) and l = 2/(3ε1/2)(aωp/(c31/2))3. The functions ω and 1/τ (in dimensionless units, i.e., divided by ω1) are plotted in Figure 3 versus nanosphere radius a for illustration. One must note that eq 23 has two types of particular solutions, eiΩt′, with complex self-frequencies Ω as written out in eq 24. Solutions given by Ω2 and Ω3 describe the oscillating behavior with the real frequency and positive damping (note that iΩ2 and iΩ3 are mutually conjugated; thus, Ω2 and Ω3 have their real parts of opposite sign, while the same imaginary parts, these latter being positive correspond to the ordinary damping F


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The Journal of Physical Chemistry C rate for positive time). The self-frequency given by Ω1 turns out to be related with an unstable exponentially rising function (as Ω1 is purely negative imaginary solution). Such an unstable solution is a well-known artifact in Maxwell electrodynamics (cf., e.g., S75 in ref 16 for particulars) and corresponds to the infinite self-acceleration of the free charge exposed to the action of the Lorentz friction force. This solution corresponds to the singular solution of the equation mv̇ = constant × v̈, which is associated with a formal renormalization of the so-called fieldmass of the charge. The field-mass in Maxwell electrodynamics is infinite for a pointlike charge and is assumed to be canceled by arbitrarily assumed negative infinite nonfield mass, resulting in the ordinary mass, e.g., of the electron (this is, however, not defined mathematically in a proper way and links with some inconsistency of the classical electrodynamics of pointlike charges). Therefore, this unphysical singular particular solution should be discarded. The other of oscillatory type solutions describe dumped oscillations which resemble solution of the ordinary damped harmonic oscillator. It must be, however, emphasized that eq 23 does not describe a damped harmonic oscillator and therefore the attenuation rate and frequency (1/τ and ω as given by eq 24) do not agree with corresponding quantities for the ordinary damped harmonic oscillator. They are expressed by analytical formulas for Ω2 (or Ω3) by eq 24. They can be easy evaluated for various a and compared with corresponding quantities found within the perturbation calculus based approach. Such comparison is illustrated in Figures 3 and 4. From this comparison (presented in Figure 3)one can note that the application of the perturbation approach leads to a highly overestimated damping rate for a > 30 nm (Au, vacuum). For 30 < a < 50 nm, for gold nanoparticles, the small underestimation of the resonance wavelength occurs, whereas for a > 50 nm again a rapid and very strong overestimation of the resonance wavelength takes place, as illustrated in the right panel in Figure 3. Moreover, as has already been mentioned above, the approximate solution terminates at a = 57, 64, 69, 72, 75 nm (Au, ε = 1, 2, 3, 4, 5, correspondingly), above which the approximate oscillatory-type solution does not exist within the perturbative calculus approach. One can finally conclude that the utilization of the approximate formula to account for the Lorentz friction damping in a form as given by 22 is justified up to a ≃ 20 nm (Au, vacuum) and causes a small discrepancy (practically negligible error for damping rate and only a small underestimation of the resonance wavelength) at 20 < a < 30 nm, while for a > 30 nm, for gold, these approximate-perturbative parameters of the oscillatory plasmon excitations strongly deviate from the exact ones. In Figures 3 and 4 there is illustrated also the dependence of the listed above limiting values for radius ranging accuracy of the perturbation approach to radiative losses for plasmons with respect to the relative permittivity of the dielectric surroundings of the metallic nanosphere. The accurate value of the damping rate as plotted in Figure 3 reaches its minimum at a certain value of the nanosphere radius ⎛ 9 v c 3 ⎞1/4 a* = ⎜⎜ Cε F 4 ⎟⎟ ωp ⎠ ⎝2

are listed in Table 2 for ε = 1, 2, 4 (a** weakly depends on the material (Au or Ag), whereas strongly on the relative Table 2. Radius of the Nanosphere Corresponding to the Minimal (a*) and Maximal (a**) Values of Surface Plasmon Damping a*, a** (nm) n = (ε)

1/2

n = 1 vacuum n = 1.4 n=2

Au

Ag

8.77, 57.4 10.4, 64.3 12.4, 72

8.78, 57.42 10.4, 64.4 12.4, 72.1

permittivity of the dielectric surroundings and is located approximately at 57 nm (for Au, vacuum)). Comparison of Surface Plasmon Oscillation Features Including Lorentz Friction with the Experimental Data. For comparizon of the model results of surface plasmon damping in metallic nanosphere with the experimentally observed behavior, one can use former experimental data11 for extinction of light after trans-passing water colloidal solution of Au nanospheres with varying radii. Extinction peaks that correspond to plasmon excitations in metal nanospheres are plotted in Figure 7, left panel (in the right pannel of this figure analogous data for Ag nanoparticles are also presented for comparison, though the fitting is not so good as for Au, most probably due to surface imperfections of silver nanospheres and their possible reshaping). For Au nanoparticles coincidence of the peak positions, their widths and the evolution with increase of the nanosphere radius is visible by comparison of Figure 5 with Figure 7 (left).

Figure 5. Extinction spectra for nanoparticles of Au calculated as Lorentzian functions with center positions and widths given by the resonance frequencies and damping rates according to eq 24 for nanosphere radii as indicated in circles. Similarity to the attenuation curves obtained by measurement11 of extinction spectra for light passing the same small glass container filled with the water colloidal solution of metallic nanospheres with various radii for each filling and with same volume density of metallic components is noticeable (left panel in Figure 7).

From the experimentally collected extinction peaks presented in Figure 7 (left panel) one can extract the central their positions with respect to the nanosphere radius a. These points perfectly fit to the resonant self-frequencies defined by an exact solution of eq 23 exactly including the Lorentz friction. This coincidence of the exact solution (red line) with experimental data points is visualized in Figure 6 (left panel). In Figure 6 the

(25)

and the maximum at a** (one can note that the latter might be defined as the zero point of the appropriate derivative expressed analytically using eq 24, though the related formula for this derivative is cumbersome due to a complicated analytical expression for 1/τ function given by eq 24). Both a* and a** G


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Figure 6. (Left) Comparison of the resonance self-energies (expressed in the wavelength terms) for radiatively damped surface plasmons with the experimental data from extinction measurements (data after ref 11) for Au nanospheres in colloidal water solution with the radii 10, 15, 20, 25, 30, 40, 50, 75 nm. Blue points, experiment; green line, perturbative approximation; red line, exact inclusion of the Lorentz friction (ε = 2.6 and ωp renormalized by factor 0.78, the same as in Figure 5). (Right) Similar comparison with the surface plasmon resonances in colloidal gold particles of different radii (experimental data for radii 2.6, 10, 20, 25, 30, 40, 50, 60 nm after Figure 12.17 in ref 15).

radiative damping and in the extrinsic limit for the size effect in the Mie theory the too high red-shift is reduced by inclusion of the multipole components in Mie response, which lowers the red shift for larger nanosphere radii. This is due to the increasing contribution of multipole plasmon excitations to the overall Mie response when radius grows beyond the dipole approximation limit and the related increase of the resonance energy15 mitigating the red-shift. Similar growth of the multipole surface plasmon resonance frequency is observed also in the framework of the RPA approach,18 cf. Equations 11 and 12, where the multipole-mode frequencies ω0l=ωp(l/(2l +1))1/2 and they increase with l. To compare these explanations with the Lorentz friction induced corrections, let us first address the mutual links between the Mie theory and the microscopic RPA method. Note, after ref 27., that the major drawback of the Mie theory and its modifications is that the underlying electron relaxation mechanisms responsible for absorption of incident light are all included by the macroscopic material dielectric function, which does not distinguish between several possible energy dissipation processes. A microscopic picture for the plasmon absorption is therefore lacking within the Mie theory. It means that the Mie theory, which is a classical solution of the Maxwell equations for the incident planar e-m wave with imposed boundary spherical conditions, does not include the internal metal electron dynamics particularities except for the phenomenologically modeled dielectric function. The electron scattering processes, i.e., the dissipation due to electron−electron scattering, electron−phonon collisions, electron−admixtures or crystal imperfections including the nanoparticle boundary scattering, can be in principle calculated on the many body background typical for solids (like the collision term in Boltzmann equation or more generally by Green function approach) and they all lead to the scattering time rate modeled by eq 13. This dissipation channel time rate is used also in the Mie theory for small nanoparticle radii for which in experiment it is observed the red-shift proportional to 1/a (note that similarly scale also the Landau damping and the red-shift due to spill-out important in ultrasmall metallic clusters (1−3 nm)12 negligible, however, for radii of 10 nm order). For this range of radii (∼10 nm, Au) the dipole approximation holds which simplifies the Mie-type responsethus inclusion of the scattering as approximated by eq 13 gives reasonable agreement with experimental observations both in Mie theory and upon

approximate perturbative curve is also plotted (green line) for comparison. The increasing discrepancy between the approximated (perturbative) solution and the experimental data for larger radii of nanospheres is clearly visible, contrary to the satisfactory agreement with the exact solution. Better fitting is obtained after inclusion of the skin effect renormalization for the Lorentz friction term by slightly reducing number of electrons accessible to incident light wave to the activation layer of thickness h, i.e., by the factor (a3 − (a − h)3)/a3, when h < a, and by accompanying reducing of plasmon frequency due to lowering effective density. For fitting presented in Figure 6 we assumed the realistic skin-effect layer thickness and the corresponding reducing of the plasmon frequency. Comparison with the Size Effect in the Mie Theory. The size dependence of the red-shift of surface plasmon resonances observed experimentally11,26,27 was also interpreted with the Mie approach.15,28,29 To explain the crossover in the size dependence of the experimentally observed red-shift of plasmon resonance with growing nanosphere radius, within the Mie theory two regimes are considered:28 the intrinsic size effect (for a < 20 nm, for Au) and the extrinsic one (for a > 20 nm). The intrinsic size effect is associated with the dipole approximation of the Mie response which is justified for nanosphere radius well smaller than the surface plasmon resonance wavelength. To account for the absorption in the framework of the Mie approach, the dielectric function of the metallic nanosphere is phenomenologically assumed. For relatively small particles, as in the case of intrinsic size effect, this dielectric function includes electron scattering (the same as given by eq 13) which results in ∼1/a red-shift of the dipole Mie resonance. Nevertheless, for larger nanospheres (approximately at a > 20 nm, for Au) for which in experiment it is observed the resonance red-shift rising with a growth, the second mechanism is suggested upon the Mie theory referred as to the extrinsic size effect.27,28 The extrinsic regime resolves itself in Mie theory to the inclusion of the multipole mixing in e-m response which leads to the related shift of resonance energy with the radius growth. To obtain a coincidence with the experimentally observed size effect in the red-shift of plasmon resonance in larger nanospheres the irradiation correction to the dielectric function has been introduced in the form of a model damping term proportional to the number of electrons, thus ∼a3.29 This overestimates, however, the H


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Figure 7. Attenuation curves for nanoparticles of Au (left) and Ag (right) obtained by measurement of extinction spectra for light passing the water colloidal solution of metallic nanospheres with various radii (after our results11); the deformation of the regular Lorentzian extinction feature shape is noticeable at a = 75 nm (Au) and at a = 30, 40 nm (Ag).

losses dominate plasmon damping with a factor higher than 1 (ca. 5 at a = 20 nm). This was not included in the conventional Mie theory upon the intrinsic size effect. The radiative damping of plasmons (i.e., the Lorentz friction) must be thus included (for a > 12 nm, at least) to the modeled (even by the simplified Lorentzian) dielectric function of the metallic nanosphere; this will give the required red-shift in corrected intrinsic size effect regime even upon the dipole approximation only. The irregular size effect of plasmon damping caused by the Loretz friction, if accounted for accurately, well explains the experimentally observed also irregular (i.e., not proportional to a3) size effect for the red-shift. This Lorentz friction induced correction mixes, however, with the extrinsic size effect due to the multipole contributions but rather for radii a > 60 nm (Au) significantly exceeding the previously suggested limiting 20 nm.27−29 The quadrupole contribution (and the higher multipoles at larger radii) results in the deformation and larger broadening of the extinction features not allowing its Lorentzian form any longer (the higher energy quadrupole assistant broad peak occurs first at smaller wavelength in association to the dipole peak broadened and red-shifted by the Lorentz friction). In the experiment it is visible for Au (at 75 nm) and Ag (at 30−40 nm).11,26 Remarkably that for Ag this deformation of extinction features occurs at smaller radii than for Au,11,26 cf. Figure 7, probably due to lower resonance wavelength for Ag nanospheres (ca. 400 nm) and moreover by scattering due to cluster faceting, charges or molecular contaminations.26 The role of stabilizing molecules attached the surface of nanoparticles to prevent them from aggregation or precipitation in solution and other contaminations of the nanoparticle surface which affect the scattering and absorption of light in the extinction experiment with colloidal solutions of more chemically reactive metals (like Ag in comparison to Au) is also raised in ref 28. By the accurate treatment of the Lorentz friction term we have shown that the ∼a3 growth of the irradiation induced damping of plasmons, phenomenologically accounted for in the Mie theory, quickly saturates, which is quite reasonable and what is more important; it well coincides with the experimental observations (it is clear that the cubic growth cannot continue and must saturate and actually, it even falls down for larger a, as shown in Figures 3 and 4). Thus, the inclusion of the exact form of the Lorentz friction damping rate (as given by eq 24) into the modeled dielectric function for nanosystems (needed

other approaches, including RPA. Within Mie approach this is conveyed to the intrinsic size effect. Nevertheless, for larger radii (appr. for radii a > 20 nm, for Au) the observed experimentally resonance red-shift is not proportional to 1/a, but rather is proportional to the radius a or to some power of a, e.g., a3 (for not too large nanosphere size, however). As the intrinsic mechanism including only electron scattering does not offer such a behavior, the radiation losses are also taken into account,29 and simultaneously the mixing of multipole components is proposed in the Mie response. One can thus explain the experimentally observed inverted (irregularly rising with a) red-shift for larger nanospheres due to the increase of the irradiation-induced damping reduced, however, by multipole resonances which are shifted with l-multipole mode number toward higher energies. Nevertheless, the frontier between the dipole and multipole limits is not sharply defined and is rather accommodated to the experiment explanation needs (because for the resonance surface plasmon wavelength being of ca. 500 nm (for Au) and even shifted with radii growth to ca. 650 nm, the radii e.g., of 20 or 30 nm do not differ significantly on the wavelength scale and the indeed important multipole contribution occurs rather for larger nanospheres, at approximately a > 60 nm, as (2a)/650 = 0.18 for a = 60 nm is well lower than 1/4 and the electric field of an incident e−m wave is still sufficiently homogeneous along the whole nanosphere which justifies the dipole approximation accuracy). The Lorentz friction induced irradiation losses which for radii a < 10 nm (for Au) were much lower than the scattering term given by eq 13, but they strongly contribute to plasmon energy dissipation for a > 12 nm (for Au in vacuum and for a little larger radius in the case of water surroundings) as visualized in Figure 1. At this radius, the inverse time for dissipation caused by the irradiation of energy (Lorentz friction) is similar to this one caused by scattering (indeed, if one estimates it according to eq 13 for a = 12 nm one obtains 7 × 1013 1/s, whereas the second term in eq 22 (the Lorentz friction term) is of the same magnitude at 12 nm). For larger a the Lorentz friction losses quickly dominate the plasmon damping as depicted in Figure 2. Thus, for nanospheres with radii of order of 5−10 nm (and even up to 15 nm, cf. Figure 2) the Mie approach gives reasonably intrinsic size effect neglecting the Lorentz friction losses. It is, however, not so, when we well pass the crossover point and arrive at a ∼ 15−25 nm or at larger scale for the radius. For such radii the irradiation I



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ACKNOWLEDGMENTS The work was supported by the NCN project no. 2011/03/D/ ST3/02643 and NCBiR PfPV project no. PT/1/2013/23.

in Mie theory and especially important in its regime for the intrinsic size effect) is an interesting correction to the Mie approach and contributes to the discussion of the interpretation of light extinction measurements in the region 10−80 nm for the radius of Au nanospheres. Thus, we have shown that the intrinsic regime with inclusion of the Lorentz friction can be extended to larger radii (up to ca. 60 nm, for Au) where the extrinsic multipole effect starts to contribute with gradually growing intensity for higher radii, but up to 60 nm the extrinsic effect rather weakly mixes with the saturation and lowering of the damping rate due to the Lorentz friction. The multipole corrections cannot be avoided for radii a > 60 nm, as is evident by deformation of the Lorentzian shape describing extinction features accompanied by their significant broadening, for Au observed at a = 75 nm.11,26 The accurate microscopic analysis of the Lorentz friction carried out within the RPA approach supports modeling of the dielectric function needed in the Mie theory and thus can well be utilized to prolong the intrinsic size-effect upon Mie approach beyond the 20 nm size limit to larger radii for which the dipole approximation still holds (up to approximately 60 nm for Au). One should note, however, that since dielectric function simultaneously represents all energy dissipation channels in a combined manner, details of the size effect caused by the exact form of the Lorentz friction (out of the scope of the Mie theory as of yet29) are required to be included to the modeled dielectric function. Thus, the specific size dependence of the damping time rate for plasmons due to the Lorentz friction derived upon exact solution of the relevant RPA equation for electron density dynamics in nanospheres (as depicted in Figures 3 and 4) may be employed to explain the observed irregular size-effect of plasmon resonance in the radius window 10−60 nm (for Au). In summary, we can conclude that the damping of surface plasmons in large metallic nanospheres is overwhelmed by radiation energy losses which strongly exceed electron scattering attenuation of plasmons for nanosphere radii larger than a critical one (apprimately 12 nm for Au in vacuum). The radiation losses can be expressed by the so-called Lorentz friction which hampers plasmon oscillations in metallic nanoparticle. The related damping rate is proportional initially to the volume of the naposphere, which results from the perturbative approach to the Lorentz friction contribution to plasmon damping. This rapid increase of the radiation losses saturates, however, for larger nanospheres and then slowly drops down. The deviation of the perturbative approximation in respect to the exact Lorentz friction term grows quickly for a > 30 nm (Au, vacuum) and dismiss the harmonic damped oscillator approximation as the model of plasmon damping. The solution of the dynamical equation for local electron density including the exact form of the Lorentz friction determines both the resonance frequency of plasmons and their damping beyond the damped harmonic oscillator model. The related irregular size-dependent red shift of plasmon dipole resonance very well coincides with the experimental data for metallic nanospheres with large radii (10−75 nm for Au).

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The authors declare no competing financial interest. J


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Lorentz Friction for Surface Plasmons in Metallic Nanospheres - The

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