Plasmonics (2019) 14:465–483https://doi.org/10.1007/s11468-018-0825-0

Nonradiative Energy Losses of Plasmon-Polariton in a MetallicNano-chain Deposited on a Semiconductor Substrate

Witold A. Jacak1

Received: 22 June 2018 / Accepted: 7 August 2018 / Published online: 3 September 2018© The Author(s) 2018

AbstractPropagation of plasmon-polariton in a metallic nano-chain in a dielectric surroundings is almost undamped and irradiation-less and such nano-chains are considered as quasi-perfect wave-guides for plasmon-polaritons. If, however, a chain isdeposited on a semiconductor substrate or embedded in a semiconductor surroundings, plasmon-polaritons are stronglyquenched due to the energy transfer from plasmon oscillations to band electrons in the semiconductor medium via the near-field coupling channel. For the reverse direction of this energy transfer the balance of the thermal losses of plasmon-polaritoncan be achieved. By a control of a dielectric spacing between the metallic nano-chain and the semiconductor substrate (anda type of the deposition), various regimes for coupling of two systems can be organized. We present the exact solution forpropagation of strongly damped plasmon-polaritons in metallic nano-chain due to coupling of plasmons with band electronsin the semiconductor substrate.

Keywords Plasmon-polariton · Metallic nano-chain · Semiconductor substrate

Introduction

Experimental and theoretical investigation of plasmon oscil-lations in metallic nanoparticles and metallic nano-arrayshas a growing significance for applications in nano-photonics and plasmonics. Metallic surface modificationsin nano-scale of solar cells exhibit growth of photo-voltaicefficiency due to plasmon mediation of sun light energy har-vesting [1–6]. Periodic structures of metallic nanoparticlesare considered as plasmon wave-guides with low damp-ing [7–9]. Propagation of plasmon-polaritons in metallicnano-chains [10–12] is prospective for sub-diffraction lightcircuits for new generation of opto-electronics via lightsignal transformation into plasmon-polariton with muchshorter wavelength [13–15]. Plasmon-polaritons do notinteract with photons due to the large incommensurability inmomentum for the same energy and this prevents radiationlosses of plasmon-polaritons. Nevertheless, when metallicnano-chains are placed on a semiconductor substrate or

� Witold A. Jacakwitold.aleksander.jacak@pwr.edu.pl

1 Department of Quantum Technologies, Wrocław Universityof Science and Technology, Wyb. Wyspianskiego 27,50-370 Wrocław, Poland

embedded in a semiconductor surroundings, then the strongcoupling of plasmon oscillations in each metallic nanocom-ponent causes the plasmon-polariton energy outflow to thesemiconductor electron system resulting in strong dampingof the collective plasmon modes. On the other hand, thesame near-field coupling may be used in opposite directionto supply energy to plasmon-polariton from the semicon-ductor substrate where the inverse electron occupation ofsemiconductor band system may be attained, e.g., by appli-cation of a lateral current. This might be convenient tobalance the dissipation of plasmon-polariton energy causedby electron scattering in metallic components and pro-vide a perfectly undamped plasmon-polariton propagationin metallic chains. For description of near-field coupling ofplasmon-polariton in a metallic nano-chain with the semi-conductor substrate, one can utilize the Fermi golden rule toderive the related damping rate.

Let us recall that plasmon properties in metallicnanoparticles depend on a nanoparticle size. For metallicultrasmall clusters of size 1–3 nm for radius the dominatingare quantum effects as, e.g., the spill-out of electronsbeyond the jellium rim [16–19] lowering electron densityin such small clusters. Plasmon oscillation energy isthus red-shifted as proportional to the square-root ofthe electron density. Damping of plasmons in this size-scale is caused by their decay into electron-hole pairs

466 Plasmonics (2019) 14:465–483

(Landau damping) [18]. Nevertheless, for larger metallicnanoparticles, both the spill-out and the landau damping areof lowering significance as proportional to the inverse ofthe nanoparticle radius. Damping of plasmons for particlesof size of several tens nm is linked to their pronouncedradiative properties [16, 17, 20]. For nanoparticles of noblemetals (gold, silver, and copper), their surface plasmonresonances overlap with the visible light spectrum.

Irradiation of surface plasmons in nanoparticles mani-fests itself in their collective specific propagation in metallicnano-chains [21–23]. A long range propagation of plasmon-polariton signal along these structures has been demon-strated; e.g., in [24], it has been reported observation of5 μm range propagation in gold nanoparticles with averagedradius a = 50 nm aligned in the equidistant chain with sep-aration between neighboring spheres d = 200 nm. In seriesof papers [10, 25–28], practically not damped propagationof plasmon-polaritons modes in gold and silver nano-chainshas been demonstrated over distances ∼ 0.5 μm. Plasmon-polaritons are referred to hybridization of plasmons withphotons resulting in lowering of their group velocity below0.1×c [10, 26, 28]. The related momentum incommensura-bility at the same energy of photons and plasmon-polaritonscauses that the latter do not irradiate energy and propagatealong metallic nano-chains over large distances [11, 24].

In the present paper, we describe the propagation ofplasmon-polaritons in a metallic nano-chain deposited ona semiconducor substrate or embedded in a semiconductormedium including a coupling of plasmons with semicon-ductor band system in near-field-zone and related energytransfer between plasmon-polaritons and semiconductorelectrons. Energy outflow from plasmon-polaritons to thesemiconductor band system enhances damping of plasmon-polaritons, whereas an opposite direction of the energy flowmay be utilized to cover plasmon-polariton Joule lossescaused by electron scattering in metallic components. First,we will analyze the surface dipole-type plasmons in a sin-gle metallic nanosphere (within the formerly formulatedRPA model [29, 30]) including damping effects, in partic-ular of the Lorentz friction of plasmons resulting in thefar-field-zone radiation and the additional energy outflowvia channel of near-field-zone coupling of dipole plasmonsto electrons in a substrate semiconductor. Next, we willdescribe collective surface plasmons excitations in a nano-chain of metallic particles, i.e., plasmon-polaritons, takinginto account all channels for energy losses. The propagationof damped plasmon-polaritons will be analyzed for longi-tudinal and transverse polarizations of plasmon oscillationswith respect to the propagation direction and for variety ofgeometry and size and material details of the whole systemconfiguration.

Plasmon Oscillations in a Single MetallicNanosphere Including Their Damping

A metallic nanosphere is conventionally modeled asthe spherical static positive jellium with electricallybalanced electron liquid which can locally fluctuate indensity resulting in regular charge oscillations: the surfaceplasmons—when oscillations have a translational characterwith the electrical local imbalance occurring on the spheresurface only, and the volume plasmons—of compressionalcharacter along the radius of a nanosphere. It is interestingthat the energy of nanosphere volume plasmons forparticular modes is always greater than the bulk plasmon

energy �ωp = �

√e2ne

mε0(m is the mass of electron, e is

the electron charge, ne is the density of free elrectronsin a metal, ε0 is the dielectric constant), whereas theenergy of surface plasmon modes is lower than �ωp

[29]. The latter property makes noble metal (Au, Ag, Cu)nanoparticles attractive for applications as their surfaceplasmon resonances overlap with the visible light spectrum.

Fluctuations of electron local density in the nanosphereof surface and volume parts [29]

δρ(r, t) ={

δρ1(r, t), f or r < a,

δρ2(r, t), f or r ≥ a, (r → a+),(1)

satisfy the equations derived in the framework of randomphase approximation (RPA) [29]:

∂2δρ1(r, t)∂t2

= 2

3

εF

m∇2δρ1(r, t) − ω2

pδρ1(r, t), (2)

and

∂2δρ2(r, t)∂t2

= − 2

3m∇

{[3

5εF ne + εF δρ2(r, t)

]rrδ(r− a)

}

−[

2

3

εF

m

rr∇δρ2(r, t) + ω2

p

4π

rr∇

∫d3r1

1

|r − r1|× (δρ1(r1, t)Θ(a − r1) + δρ2(r1, t)Θ(r1 − a))] δ(r − a),

(3)

where Θ is the Heaviside step function defining the staticjellium ne(r) = neΘ(a − r) and a is the nanosphere radius.The analysis and solution of the above equations have beenperformed in details in [29], resulting in determination ofplasmon self-mode spectra, both for the volume and surfacemodes.

This RPA treatment did not account, however, forplasmon damping. The damping of plasmons can beincluded in a phenomenological manner, by adding theterm,− 2

τ0

∂δρ(r,t)∂t

, to the r.h.s. of both Eqs. 2 and 3, takingadvantage of their oscillatory form [29]. The damping ratio

Plasmonics (2019) 14:465–483 467

1τ0

accounts for electron scattering losses (eventually Jouleheat losses in metal material of a nanosphere) [25],

1

τ0� vF

2λb

+ CvF

2a, (4)

where C is the constant of unity order, vF is the Fermivelocity in the metal, λb is the electron mean free path, thesame as in the bulk metal (including scattering of electronson other electrons, on impurities and on phonons in themetallic nanoparticle [25]); e.g., for Au, vF = 1.396 × 106

m/s and λb � 53 nm (at room temperature); the latter termin the formula (4) accounts for scattering of electrons onthe boundary of the nanoparticle, whereas the former onecorresponds to scattering processes similar as in bulk. Theother effects, as the so-called Landau damping (especiallyimportant in small clusters [19, 31]), corresponding todecay of plasmon for high energy particle-hole pair, are oflowering significance for nano-sphere radii larger than 2-3 nm [31] and are completely negligible for radii larger than5 nm (we will consider here the large nanospheres with radii≥ 5 nm). Note that the similarly lowering role with theradius growth plays also electron liquid spill-out effect [16,18], though it was of primary importance for small clusters[16, 20].

The homogeneous (2) and (3) determine self-frequenciesof plasmon modes. One can write out also the dualinhomogeneous equations with the explicit driving factor.This factor would be the time-dependent electric field,e.g., the electric component of the incident e-m wave(laser beam or sun-light in photovoltaic applications [32]).The resonant light wavelength with surface plasmon inthe metallic nanosphere (Au, Ag, Cu) is of order of500 nm and highly exceeds the nanosphere size (withradius 5–50 nm); thus, the dipole-approximation regimeconditions are satisfied. Hence, the driving field E(t) of e-m wave almost homogeneous over the nanosphere (whichcorresponds to the dipole approximation) can only excitethe dipole surface mode in the electron liquid of the metallicnanosphere. The dipole type mode may be described by thefunction Q1m(t) with l = 1 and m = −1, 0, 1 being theangular momentum numbers related to spherical symmetryof the metallic nanoparticle. The function Q1m(t) satisfiesthe equation,

∂2Q1m(t)

∂t2+ 2

τ0

∂Q1m(t)

∂t+ ω2

1Q1m(t)

=√

4π

3

ene

m

[Ez(t)δm,0 + √

2(Ex(t)δm,1 + Ey(t)δm,−1

)],

(5)

where ω1 = ωp√3ε

(it is a dipole surface plasmon Mie-

type frequency [33], ε is the dielectric susceptibility of the

nanosphere surroundings). The electron density fluctuationscan be written as follows [29],

δρ(r, t) =⎧⎨⎩

0, r < a,1∑

m=−1Q1m(t)Y1m(Ω), r ≥ a, r → a+,

(6)

where Ylm(Ω) is the spherical function with l = 1. Theplasmon oscillations given by Eq. 6 define the dipole D(t),

⎧⎪⎪⎨⎪⎪⎩

Dx(t) = e∫

d3rxδρ(r, t) =√

2π√3

eQ1,1(t)a3,

Dy(t) = e∫

d3ryδρ(r, t) =√

2π√3

eQ1,−1(t)a3,

Dz(t) = e∫

d3rzδρ(r, t) =√

4π√3

eQ1,0(t)a3.

(7)

The dipole D(t) satisfies thus the equation (by virtue ofEq. 5),

[∂2

∂t2+ 2

τ0

∂

∂t+ ω2

1

]D(t) = a34πe2ne

3mE(t) = εa3ω2

1E(t).

(8)

The damping term in the above equation includes energydissipation due to electron scattering in metallic nanosphere,i.e., electron-electron, electron-phonon, electron-admixturescattering, as well contribution of the boundary scatteringeffect [25]. There is, however, also an important channel ofplasmon damping caused by irradiation losses, not includedinto formula for τ0. The radiative losses of plasmon energyin the dielectric surroundings can be expressed by theLorentz friction [34], i.e., by the fictitious electric fieldslowing down the motion of charges,

EL = 2√

ε

3c3

∂3D(t)

∂t3. (9)

Thus, one can rewrite (8) including the Lorentz frictionterm,

[∂2

∂t2+ 2

τ0

∂

∂t+ ω2

1

]D(t) = εa3ω2

1E(t) + εa3ω21EL, (10)

or more explicitly, for the case when E = 0,

[∂2

∂t2+ ω2

1

]D(t)= ∂

∂t

[2

τ0D(t)+ 2

3ω1

(ωpa

c√

3

)3∂2

∂t2D(t)

].

(11)

Applying now the perturbation procedure for solving ofEq. 11 and treating the r.h.s of this equation as theperturbation, one obtains in zeroth step of perturbation

468 Plasmonics (2019) 14:465–483

[∂2

∂t2 + ω21

]D(t) = 0, from which ∂2

∂t2 D(t) = −ω21D(t).

Therefore, within the first step of the perturbation, one cansubstitute the latter formula to the r.h.s. of Eq. 11, i.e.,[

∂2

∂t2+ 2

τ

∂

∂t+ ω2

1

]D(t) = 0, (12)

where

1

τ= 1

τ0+ ω1

3

(ωpa

c√

3

)3

. (13)

In this way, we have included the Lorentz friction intothe total attenuation rate 1

τ. This is justified for not

extreme large nanospheres, i.e., when the second term inEq. 13, proportional to a3, is sufficiently small to fulfill theperturbation procedure constraints. For nanospheres withradius 10–30 nm, this approximation is fulfilled and wasverified experimentally for Au and Ag nanospheres [30, 35].Inclusion of the Lorentz friction explains an experimentallyobserved red-shift of resonance surface plasmon frequencywith growing a. Indeed, the solution of Eq. 12 is of the form

D(t) = Ae−t/τ cos(ω′1t + φ), where ω′

1 = ω′1

√1 − 1

(ω1τ)2 ,

which gives the experimentally observed red-shift of theplasmon resonance due to ∼ a3 growth of plasmon dampingcaused by irradiation. The Lorentz friction term in Eq. 13dominates plasmon damping in a dielectric surroundings fora ≥ 12 nm (for Au and Ag) due to this a3 dependence.The plasmon damping grows rapidly with a and resultsin pronounced red-shift of resonance frequency in goodcoincidence with the experimental data for 10 < a < 30nm (Au and Ag) [30]. Taking into account that 1

τ0scales as

1a

, while the Lorentz friction contribution as a3 for 10 <

a < 30 nm, in this region for size of metallic particlesone encounters the cross-over of the attenuation rate withrespect to a. The minimum of damping is achieved at

a∗ =(

9

2

√εvF c3

ω4p

)1/4

, (14)

and for a < a∗, the damping ratio grows with lowering a

approximately as ∼ 1a

, while for a > a∗, this ratio growswith rising a as proportional to a3. The value of a∗ can beestimated for Au, Ag an Cu—cf. Table 1.

Table 1 Radius of the nanosphere corresponding to the minimal valueof surface plasmon damping in dielectric surroundings

n = √ε a∗

Au Ag Cu

n = 1 vacuum 8.4 8.44 6.46

n = 1.4 water 9.137 9.18 9.202

n = 2 9,989 10.037 10.04

Radiative Properties of Plasmon-Polaritonsin theMetallic Nano-chain Embeddedin the Dielectric Medium

When a metallic nanosphere is an element of the chain ofsimilar nanospheres equidistantly distributed along a line,one has to take into account that except of the irradiationlosses in a particular nanosphere, the simultaneous incomeof energy takes place from radiation of other nanosphereswhen in the total chain the collective plasmon excitation(plasmon-polariton) propagates.

The interaction between nanospheres in the chain canbe considered as of dipole-type coupling. The minimalseparation of nanospheres in the chan is d = 2a (d is themeasure of distance between neighboring sphere centers)and the dipole approximation of plasmon interaction inthe nanosphere-chain is sufficiently accurate for d > 3a,when multipole interaction contribution can be neglected.Various numerical large-scale calculations of e-m fielddistribution in such systems were done including dipole andalso multipole interactions between plasmonic oscillationsin metallic components [21–23]. The model of interactingdipoles [36, 37] was developed earlier for investigationof stellar matter [38, 39] and next, it has been adoptedto metal particle systems [40, 41]. The numerical studiesbeyond the dipole model [21, 23] indicated that the dipolemodel is sufficiently accurate when the particle separationis not lower than particle dimensions, when the multipolecontribution to interaction are small [42].

The oscillating dipole D(t) located in the point r causesin the other place r0 (assuming the vector r0 is fixed to theend of r) the electric field in the form as follows (includingrelativistic retardation of electromagnetic signals) [34, 43]:

E(r, r0, t) = 1

ε

(− ∂2

v2∂t2

1

r0− ∂

v∂t

1

r20

− 1

r30

)D(r, t − r0/v)

+1

ε

(∂2

v2∂t2

1

r0+ ∂

v∂t

3

r20

+ 3

r30

)n0(n0 ·D(r, t−r0/v)),

(15)

where n0 = r0r0

and v = c√ε. The above formula includes

the terms corresponding to the near-field-zone (denominatorwith r3

0 ), medium-field-zone (denominator with r20 ) and far-

field-zone (denominator with r0) contributions to dipolefield. Equation 15 allows for writing out the dynamicalequation for plasmon oscillations at each nano-sphere ofthe chain, which can be numbered by integer l (d denotesthe separation between nano-spheres in the chain, a is thenanosphere radius, the vectors r and r0 are collinear, if theorigin of coordinate system is associated with one of thenano-spheres in the chain).

Plasmonics (2019) 14:465–483 469

Therefore, the equation for the collective surfaceplasmon oscillation in the lth sphere of the chain is asfollows,

[∂2

∂t2+ 2

τ0

∂

∂t+ ω2

1

]Dα(ld, t)

= εω21a

3m=∞∑

m=−∞, m �=l

Eα

(md, t − |l − m|d

c

)

+εω21a

3ELα(ld, t) + εω21a

3Eα(ld, t). (16)

The first term of the r.h.s. in Eq. 16 describes the dipolecoupling between nano-spheres in the chain and the nexttwo terms correspond to contribution to plasmon attenuationdue to the Lorentz friction (as described in the previousparagraph) and the driving field of an external electric field.The index α enumerates polarization, the longitudinal, α =z, and the transverse, α = x(y), with respect to the chainorientation (assumed in z direction). According to Eq. 15we have:

Ez(md, t) = 2

εd3

(1

|m − l|3 + d

v|m − l|2∂

∂t

)Dz(md, t−|m− l|d/v),

Ex(y)(md, t) = − 1

εd3

(1

|m − l|3 + d

v|m − l|2∂

∂t+ d2

v2|l − d|∂2

∂t2

)Dx(y)(md, t − |m − l|d/v). (17)

Taking advantage of the chain periodicity, one canassume in analogy to Bloch states in 1D crystals with thereciprocal lattice of quasi-momentum,

Dα (ld, t) = Dα (k, t) e−ikld ,

0 ≤ k ≤ 2πd

.(18)

The above can be asserted in a more formal manner takingthe Fourier picture of Eq. 16. As dipoles are localized onnanospheres in their centers, the system is discrete similarlyas in the case of phonons in 1D crystal. One can thusapply the discrete Fourier transform (DFT) with respectto the positions, whereas the continuous Fourier transform(CFT) with respect to time. DFT is defined for a finite setof numbers, so one can consider the chain with 2N + 1nanospheres, i.e., the chain of length L = 2Nd . Thus forany discrete characteristics f (l), l = −N, ..., 0, ..., N ofthe chain, like a dipole distribution, one deals with DFT

picture f (k) =N∑

l=−N

f (l)eikld , where k = 2π2Nd

n, n =0, ..., 2N . Hence, kd ∈ [0, 2π ] due to periodicity of theequidistant chain. On the whole system, the Born-Karmanboundary condition is imposed resulting in the above formof k = 2π

2Ndn,. In order to account for the infinite length

of the chain, one can finally take the limit N → ∞ whichcauses that the variable k is quasi-continuous, but still kd ∈[0, 2π ].

The Fourier picture of Eq. 16, DFT for positions and CFTfor time, is derived in the Appendix 1 and gives,

(−ω2 − i

2

τ0ω + ω2

1

)Dα(k, ω)

= ω21a3

d3Fα(k, ω)Dα(k, ω) + εa3ω2

1E0α(k, ω), (19)

with

Fz(k, ω) = 4∞∑

m=1

(cos(mkd)

m3cos(mωd/v) + ωd/v

cos(mkd)

m2sin(mωd/v)

)

+2i

[1

3(ωd/v)3 + 2

∞∑m=1

(cos(mkd)

m3sin(mωd/v) −ωd/v

cos(mkd)

m2cos(mωd/v)

)],

Fx(y)(k, ω) = −2∞∑

m=1

(cos(mkd)

m3cos(mωd/v) + ωd/v

cos(mkd)

m2sin(mωd/v) −(ωd/v)2 cos(mkd)

mcos(mωd/v)

)

−i

[−2

3(ωd/v)3 + 2

∞∑m=1

(cos(mkd)

m3sin(mωd/v) + ωd/v

cos(mkd)

m2cos(mωd/v)

−(ωd/v)2 cos(mkd)

msin(mωd/v)

)]. (20)

470 Plasmonics (2019) 14:465–483

Fig. 1 The damping rateImωx(y)(k), the self-frequencyReωx(y)(k) and the groupvelocity vx(y)(k) of plasmonpolariton for transversepolarization in infinite chains ofAu nanospheres with radiusa = 10 nm and chain-separationd = 3a for a vacuumsurroundings (upper) and asemiconductor Si surrounding(lower)

transverse plasmon-polariton mode

0 200 400 600 800 10000.030.040.050.060.070.08

kd

xy

1/

0 200 400 600 800 10000.9850.9900.9951.0001.0051.0101.0151.020

kd

xy

0 200 400 600 800 1000

5 10

0

5 10

kd

x

0 200 400 600 800 1000

0.01

0.02

0.03

0.04

0.05

kd

xy

1/ 0

0 200 400 600 800 10000.98

0.99

1.00

1.01

1.02

kd

0 200 400 600 800 1000

5 10

0

5 10

kd

x

The direct calculation of functions ImFz(k, ω) andImFx(y)(k, ω) corresponding to the radiative damping forlongitudinal and transverse plasmon-polariton polariza-tions, respectively, is explicitly done in the Appendix 1—(50) and (52). We have shown there that both these functionsperfectly vanish when 0 < kd ±ωd/v < 0 (the correspond-ing region—the light cone—is indicated in Fig. 5). Outsidethis region, radiative damping expressed by ImFα(k, ω)

functions is nonzero, which for longitudinal and transversemodes is illustrated in Figs. 6 and 7, correspondingly.

Plasmon-Polariton Self-modes in the Chainin the Dielectric Surroundings

The real part of the functions Fα renormalizes the self-frequency of plasmon-polaritons in the chain, whereas

the imaginary part renormalizes damping of these modes.ReFα(k, ω) and ImFα(k, ω) are functions of k and ω.Applying the perturbation method of solution, within thefirst order approximation one can put ω = ω1 in ReFα

and also in the residual nonzero ImFα outside the region0 < kd ± ω1d/v < 2π . Let us emphasize, however,that vanishing of ImFα(k, ω) inside the region 0 < kd ±ωd/v < 2π holds for any value of ω [44], thus also for alsofor exact (nonperturbative) solution as shown numericallyin Figs. 1, 2, 3, and 4.

Upon the perturbation scheme, one can rewrite thedynamic (19) for plasmon-polariton modes in the chain inthe following form:

(−ω2− i

2

τα(k)ω+ ωα(k)2

)Dα(k, ω) = εa3ω2

1E0α(k, ω),

(21)

Fig. 2 The damping rateImωx(y)(k), the self-frequencyReωx(y)(k) and the groupvelocity vx(y)(k) of plasmonpolariton for transversepolarization in infinite chains ofAu nanospheres with radiusa = 5 nm and chain-separationd = 5a for a vacuumsurroundings (upper) and asemiconductor (Si) surrounding(lower)

0 200 400 600 800 1000

0.0220.0240.0260.0280.030

kd

xy

0 200 400 600 800 1000

0.995

1.000

1.005

1.010

kd

xy

0 200 400 600 800 10004 102 10

02 104 10

kd

x

0 200 400 600 800 1000

0.1380.1400.1420.1440.1460.148

kd

xy

0 200 400 600 800 1000

0.985

0.990

0.995

kd

xy

0 200 400 600 800 10004 102 10

02 104 10

kd

x

transverse plasmon-polariton mode

1/ 0

1/

Plasmonics (2019) 14:465–483 471

Fig. 3 The damping rateImωx(y)(k), the self-frequencyReωx(y)(k) and the groupvelocity vx(y)(k) ofplasmon-polariton forlongitudinal polarization ininfinite chains of Aunanospheres with radius a = 10nm and chain-separation d = 4a

for a vacuum surroundings(upper) and a semiconductor(Si) surrounding (lower)

0 200 400 600 800 1000

0.0150.0200.0250.0300.035

kd0 200 400 600 800 1000

0.96

0.98

1.00

1.02

1.04

kd

0 200 400 600 800 10004 10

2 10

0

2 10

4 10

kd

z

1/ 0

longitudinal plasmon-polariton mode

-Im

0 200 400 600 800 10000.0400.0450.0500.0550.0600.065

kd

1/

0 200 400 600 800 1000

0.96

0.98

1.00

1.02

1.04

kd

0 200 400 600 800 10004 10

2 10

0

2 10

4 10

kd

z

where the renormalized attenuation rate,

0 200 400 600 800 10000.02300.0235

0.02400.02450.0250

kd0 200 400 600 800 1000

0.98

0.99

1.00

1.01

kd

0 200 400 600 800 1000

1 105 10

05 101 10

kd

z

longitudinal plasmon-polariton mode

1/ 0

0 200 400 600 800 1000

0.0305

0.03100.0315

0.0320

0.0325

kd

1/

0 200 400 600 800 1000

0.98

0.99

1.00

1.01

kd

0 200 400 600 800 1000

1 105 10

05 101 10

kd

z

Fig. 4 The damping rate Imωx(y)(k), the self-frequency Reωx(y)(k)

and the group velocity vx(y)(k) of plasmon-polariton for longitudinalpolarization in infinite chains of Au nanospheres with radius a = 5 nm

and chain-separation d = 5a for a vacuum surroundings (upper) and asemiconductor (Si) surrounding (lower)

1

τα(k)=

{1τ0

, f or 0 < kd ± ω1d/v < 2π,

1τ0

+ a3ω12d3 ImFα(k, ω1), f or kd − ω1d/v < 0 or kd + ω1d/a > 2π,

(22)

and the renormalized bare self-frequency (due to dampingthe true resonance is, however, red-shifted as for dampedoscillator—not written here),

ω2α(k) = ω2

1

(1 − a3

d3ReFα(k, ω1)

). (23)

Equation 21 can be easily solved both for inhomogeneousand homogeneous (when E0α = 0) case. The generalsolution of Eq. 21 has a form of sum of the generalsolution of homogeneous equation and of single particularsolution of inhomogeneous equation. The first one includes

472 Plasmonics (2019) 14:465–483

initial conditions and describes damped self-oscillationswith frequency,

ω′α =

√ω2

α(k) − 1

τ 2α(k)

, (24)

i.e., for each k and α,

D′α(k, t) = Aα,ke

i(ω′αt+φα,k)e−t/τα(k), (25)

with constants Aα,k and φα,k adjusted to the initialconditions.

For the inhomogeneous case, the particular solution is asfollows:

D′′α(k, t)=εa3ω2

1E0α(k)ei(γ t+ηα,k)1√

(ω2α(k)−γ 2)2+ 4γ 2

τ 2α (k)

,

(26)

suitably to assumed single Fourier time-component ofE0α(k, t) = E0α(k)eiγ t , and tg(ηα,k) = 2γ /τ(α,k)

ω(α,k)2−γ 2 asfor a driven oscillator. Let us emphasize that E0α(k) isthe real function for E0α(ld)∗ = E0α(ld) = E0α(−ld).An appropriate choice of the latter function, in practicea choice of the number of externally excited nanospheresin the chain, e.g., by suitably focused laser beam, allowsfor modeling of its Fourier picture E0α(k). This givesthe envelope of the wave packet if one inverts theFourier transform in solution given by Eq. 26 back tothe position variable. For the case of external excitationof only single nnanosphere, the wave packet envelopeincludes homogeneously all wave vectors k ∈ [0, 2π ]. Thelarger number of nanospheres is simultaneously excited thenarrower in k wave packet envelope can be selected. ForE0α(ld)∗ = E0α(ld) = E0α(−ld) the Fourier transformhas the same properties, i.e., E0α(k)∗ = E0α(k) = E0α(−k)

and the latter equality can be rewritten, due to the period 2πd

for k, as, E0α(−k) = E0α( 2πd

− k) = E0α(k) (to shift k tothe equivalent positively-valued domain k ∈ [0, 2π ]). Theinverse Fourier picture of Eq. 26 (its real part) is,

D′′α(ld, t) =

2π/d∫

0

dkcos(kld − γ t − ηα,k)εa3ω2

1E0α(k)

× 1√(ω2

α(k) − γ 2)2 + 4γ 2

τ 2α (k)

. (27)

This integral can rewritten by virtue of mean value theoremin the form,

D′′α(ld, t) = 2π

dcos(k∗ld − γ t − ηα,k∗)εa3ω2

1E0α(k∗)

× 1√(ωα(k∗)2 − γ 2)2 + 4γ 2

τ 2α (k∗)

. (28)

The above expression describes the undamped wave motionwith frequency γ and the velocity, amplitude, and phaseshift determined by k∗. The energy losses are supplementedcontinuously by the driving force as for any steady state ofthe damped and driven oscillator. The amplitude attains itsmaximal value at the resonance, when

γ = ωα(k∗)

√1 − 2

(τα(k∗)ωα(k∗))2. (29)

In the chain being the subject of a persistent drivingforce in the form of the timedependent external electricfield applied to some number (even small number) ofnanospheres, one deals with undamped wave packed prop-agation along the whole chain—the energy supply by adriving factor covers damping loses. These modes depend-ing of particular shaping of the wave packet by specificchoice of the chain excitation, may be responsible forexperimentally observed long range practically undampedplasmon-polariton propagation [11, 24, 27, 28].

The self-modes of Eq. 25 are damped and theirpropagation depends on appropriately prepared initialconditions admitting nonzero values of Aα,k . The resultingwave packet may embrace the wave-numbers k from someregion of [0, 2π ]. If only wave-numbers k, for which 0 <

kd±ω1d/v < 2π contribute to the wave packet, its dampingis only of Ohmic-type (as is shown in the Appendix 1).The value of 1

τ0lowers with growing a (cf. Eq. 4)—thus

for longer range of these damped excitations in the chain,the larger spheres are more suitable. The limiting value of1τ0

= vf

2λB∼ 1013 1/s, which gives the maximal range

of propagation for these modes of plasmon-polariton ∼0.1cτ0 ∼ 10−6 m; for the group velocity of the wavepacked, we assumed ∼ 0.1c as its maximum value (thoughdepending on radius and separation of nanospheres in thechain). The group velocity calculated for both polarizationsare presented in right panels of Figs. 1, 2, 3, and 4 (foraccurate solution).

Though the presented above analysis is addressed tochains consisting of ideal nanospheres, the conclusionshold also for other shape particle chains and meet withexperimental observations at least qualitatively. In [28],propagation of plasmon-polariton in nano-chain of silverrod-shaped particles (90:30:30 nm oriented with longeraxis perpendicularly to the chain in order to enhance near-field coupling [27], with separation face to face, 70 nm)is evidenced by observing of luminescence of dye particlelocated in proximity to transmitting e-m signal but distantlyfrom the point-like excitation source over the range of 0.5μm. The observed behavior has been supported by FDTDnumerical simulations. Several samples of the chain were

Plasmonics (2019) 14:465–483 473

fabricated by the electron beam lithography in the formof 2D matrix with sufficiently well-separated individualchains. The energy blue-shift of plasmon resonance forthe nano-rods in the chain in comparison to the singleparticle is observed by ca. 0.1 eV (cf. Fig. 4 in [28]). Thisagrees with our estimation of reducing the radiation lossesin the chain in comparison to strong Lorentz friction forsingle metallic nano-particle and related smaller red-shift ofdamped oscillations. The position of resonance maximumin the chain is located at higher energy for transversemode than for the longitudinal one [26], which also agreeswith theoretical predictions. In ref. [27], it is indicatedthat FDTD simulations give lower values of the groupvelocities for both polarizations and higher attenuation ratesin comparison to these quantities previously estimated [25,26] upon simplified point-dipole-model with near field-fieldinteraction only and neglecting retardation effects. Let usnote, however, that the simplified approach including onlynear-field contribution to electric field of interacting dipolesleads to an artifact, i.e., for some values of d and a chainparameters the instability of collective dynamics occurs[45]. This instability is completely removed by inclusion ofmedium- and far-field contributions to electric field of thedipole including relativistic retardation [44]. Nevertheless,the dipole interaction model even if including besides thenear-field contribution also medium- and far-field ones andall retardation effects, still suffers from the absence ofmagnetic fields component needed for complete descriptionof far-field wave propagation. As it is demonstrated inrefs. [46, 47], for large separations in the chain, thescattering of e-m radiation dominates the signal behavior inmetallic nano-chains which then acts as the Bragg gratingfor plasmon-polaritons. For ellipsoidal gold nanoparticles(210:80 nm) deposited on the top of silicon wave-guide, thechange of regime from collective plasmon-polariton guidingto the Bragg scattering scenario takes place at distancesbetween nanoparticles exceeding ca 1 μm [46]. This provesthat the model of dipole coupling in the nano-chain worksquite well in a wide region of chain parameters, in practiceup to micron order for distances between metallic elementsin the chain, which supports the qualitative argument thatthe Bragg grating regime is not efficient for sub-wave-length distances and justifies applicability of the modelconsidered in the present paper. The SNOM measurementsof near-field coupled plasmon modes in metallic nano-chain [10] interpreted within classical field-susceptibilitiesformalism in ref. [48] also supports the sufficient levelof accuracy of dipole approximation for interaction in thechain for the considered here scale of nanosphere radii a oforder of 10–30 nm and the chain separation d not exceeding∼ 10 × a.

Damping of Surface Plasmons in a SingleMetallic Nanoparticle Depositedon a Semiconductor Substrate

The interaction of band electrons with surface plasmon inthe metallic nanoparticle deposited on the semiconductortop surface (or embedded in a semiconductor medium)causes the energy transfer resulting in an additionaldamping of plasmons. The perturbation of electron bandsystem in the substrate semiconductor due to the presence ofdipole surface plasmon oscillations in metallic nanosphere(with a radius a) deposited on the semiconductor surface(or emedded in) has the form of the potential of the e-mfield of an oscillating dipole. The Fourier components of theelectric Eω (the Fourier component of Eq. 15) and magneticBω fields produced in the distance R from the center ofconsidered nanosphere with the dipole of surface plasmonwith the frequency ω, have the form [34],

Eω = 1

ε

{D0

(k2

R+ ik

R2− 1

R3

)

+n(n · D0)

(−k2

R− 3ik

R2+ 3

R3

)}eikR (30)

and

Bω = ik√ε[D0 × n]

(ik

R− 1

R2

)eikR, (31)

(ε is the dielectric permittivity). In the case of the sphericalsymmetry, the dipole of plasmon is considered as pinnedto the center of the nanosphere, D = D0e

−iωt . In Eqs. 30and 31, we used the notation for the retarded argument,iω

(t − R

c

) = iωt − ikR, n = RR

, ω = ck, momentump = �k. Because we consider the interaction with a closelyadjacent layer of the substrate semiconductor, the terms withdenominators R2 and R we neglect as small in comparisonto the term with R3 denominator—this is the near-field-zone approximation (the magnetic field disappears and theelectric field is of the form of a static dipole field [34]).Therefore, the related perturbation potential added to thesystem Hamiltonian attains the form,

w = eψ(R, t) = e

εR2 n · D0sin(ωt + α) = w+eiωt + w−e−iωt .

(32)

The term w+ = (w−)∗ = e

εR2eiα

2in · D0 describes emission,

i.e., the case of our interest.According to the Fermi golden rule (FGR) scheme, the

inter-band transition probability is proportional to,

w(k1, k2)= 2π

�

∣∣<k1|w+|k2 >∣∣2

δ(Ep(k1)−En(k2)+�ω),

(33)

474 Plasmonics (2019) 14:465–483

where the Bloch states in the conduction and valencebands are assumed as planar waves (for simplicity), Ψk =

1(2π)3/2 eik·R−iEn(p)(k)t/�, Ep(k) = −�

2k2

2m∗p

− Eg, En(k) =�

2k2

2m∗n

(indices n, p refer to electrons from the conduction andvalence bands, respectively, Eg is the forbidden gap).

The matrix element,

< k1|w+|k2 >= 1

(2π)3

∫d3R

e

ε2ieiαn·D0

1

R2e−i(k1−k2)·R.

(34)

can be found analytically by a direct integration, whichgives the formula (q = k1 − k2),

< k1|w+|k2 >= −1

(2π)3

eeiα

εD0cosΘ(2π)

∫ ∞

a

dR1

q

d

dR

sinqR

qR

= 1

(2π)2

eeiα

ε

D0 · qq2

sinqa

qa. (35)

Next, we must sum up over all initial and final states in bothbands. Thus, for the total interband transition probability,we have,

δw =∫

d3k1

∫d3k2 [f1(1 − f2)w(k1, k2)

−f2(1 − f1)w(k2, k1)] , (36)

where f1, f2 assign the temperature dependent distributionfunctions (Fermi-Dirac distribution functions) for initial andfinal states, respectively. For room temperatures f2 � 0 andf1 � 1, which leads to,

δw =∫

d3k1

∫d3k2 · w(k1, k2). (37)

After some also analytical integration in the aboveformula, we arrive at the expression,

δw = 4

3

μ2(m∗n + m∗

p)2(�ω−Eg)e2D20√

m∗nm

∗p2π�5ε2

∫ 1

0dx

sin2(xaξ)

(xaξ)2

√1−x2

= 4

3

μ2

√m∗

nm∗p

e2D20

2π�3ε2ξ2

∫ 1

0dx

sin2(xaξ)

(xaξ)2

√1 − x2, (38)

according to assumed band dispersions, m∗n and m∗

p denote

the effective masses of electrons and holes, μ = m∗nm∗

p

m∗n+m∗

pis

the reduced mass, the parameter ξ =√

2(�ω−Eg)(m∗n+m∗

p)

�. In

limiting cases for a nanoparticle radius a, we finally obtain,

δw =⎧⎨⎩

43

μ√

m∗nm∗

p(�ω−Eg)e2D20

�5ε2 , f or aξ � 1,

43

μ3/2√

2√

�ω−Ege2D20

a�4ε2 , f or aξ � 1.(39)

In the latter case in Eq. 39, the following approximation wasapplied,∫ 1

0dx

sin2(xaξ)

(xaξ)2

√1 − x2 ≈ (f or aξ � 1)

1

aξ

∫ ∞

0

× d(xaξ)sin2(xaξ)

(xaξ)2= π

2aξ,

whereas in the former one,∫ 1

0 dx√

1 − x2 = π/4.With regard to two limiting cases, aξ � 1 or

aξ � 1, ξ =√

2(�ω−Eg)(m∗n+m∗

p)

�, we see that a �

1/ξ �⎧⎨⎩

> 2 × 10−9[m] f or�ω−Eg

Eg< 0.02

< 2 × 10−9[m] f or�ω−Eg

Eg> 0.02

, and this

range weakly depends on effective masses and Eg . Thusfor nanoparticles with radii a > 2 nm, the first regimeholds only close to Eg (less than the 2% distance tolimiting Eg), whereas the second regime holds in therest of the ω domain. For comparison, a � 1/ξ �⎧⎨⎩

> 0.5 × 10−9[m] f or�ω−Eg

Eg< 0.5

< 0.5 × 10−9[m] f or�ω−Eg

Eg> 0.5

, the first region

widens considerably (to ca. 50% relative distance to Eg),but holds only for ultra-small size of nanoparticles (a < 0.5nm). For larger nanospheres, e.g., with a > 10 nm, thesecond regime is thus dominating.

Assuming that the energy acquired by the semiconductorband system, A, is equal to the output of plasmon oscillationenergy (resulting in plasmon damping), one can estimatethe corresponding damping rate of plasmon oscillations.Namely, at the lowering in time plasmon amplitude D0(t) =D0e

−t/τ ′, one finds for a total transmitted energy,

A = β

∞∫

0

δw�ωdt = β�ωδwτ ′/2 =⎧⎨⎩

23

βωτ ′μ√

m∗nm∗

p(�ω−Eg)e2D20

�4ε2 , f or aξ � 1,

23

βωτ ′μ3/2√

2√

�ω−Ege2D20

a�3ε2 , f or aξ � 1,

(40)

where τ ′ is the damping time-rate, β accounts for lossesnot included in the model, especially to reduce the energytransfer for a realistic deposition type on the top of

semiconductor layer instead of the fully embedded case.Comparing the value of A given by the formula (40) withthe energy loss of damping plasmon estimated in [29] (the

Plasmonics (2019) 14:465–483 475

initial energy of the plasmon oscillations which has been

transfered step-by-step to the semiconductor, A = D20

2εa3 ),one can find,

1

τ ′ =⎧⎨⎩

4βωμ√

m∗nm∗

p(�ω−Eg)e2a3

3�4ε, f or aξ � 1,

4βωμ3/2√

2√

�ω−Ege2a2

3�3ε, f or aξ � 1.

(41)

By τ ′, we denote here a large damping of plasmons due toenergy transfer to the semiconductor highly exceeding theinternal damping, characterized by τ0, due to scattering of

electrons inside the metallic nanoparticle [29](

1τ0

� 1τ ′

).

For example, for nanospheres of Au deposited on the Silayer, we obtain for Mie self-frequency ω = ω1,

1

τ ′ω1=

⎧⎪⎨⎪⎩

44.092β(

a[nm]1[nm]

)3μm

√m∗

nm∗p

m, f or aξ � 1,

13.648β(

a[nm]1[nm]

)2 (μm

)3/2, f or aξ � 1,

(42)

for light(heavy) carriers in Si, m∗n = 0.19(0.98) m, m∗

p =0.16(0.52) m, m is the bare electron mass, μ = m∗

nm∗p

m∗n+m∗

pand

Eg = 1.14 eV, �ω1 = 2.72 eV. For these parameters andnanospheres with the radius a in the range of 5–50 nm, thelower case of Eq. 42 applies (at ω = ω1). The parameter β

fitted from the experimental data [3, 29] equals to ca 0.001.

Plasmon-Polariton Dynamics in theMetallicChain Deposited on a SemiconductorMedium

Assuming that the metallic nano chain is deposited on(or embedded in), the semiconductor medium, plasmonoscillations at each nanoparticle of the chain are additionallydamped due to energy transfer to the semiconductorband system as described above. Thus, Eq. 19 should begeneralized to the form:

(−ω2 − i

(2

τ0+ 2

τ ′

)ω + ω2

1

)Dα(k, ω)

= ω21a3

d3Fα(k, ω)Dα(k, ω) + εa3ω2

1E0α(k, ω), (43)

where τ ′ is given by Eq. 41. Equation 20 still holds.The damping rate in the case of the complete embedded

chain in the semiconductor medium does not depend on thepolarization of plasmon oscillations and let us consider firsthis simplest case. Upon the perturbation scheme of solutionof Eq. 43, we can consider the real part of the r.h.s. of thisequation as the correction to the frequency of the plasmon-polariton whereas the imaginary part as the correction to thedamping rate. In the explicite form, one can thus deal witha perturbative solution,

ω(k) =√

ω21(1 − a3

d3ReFα(k, ω1)) −

(1

τ0+ 1

τ ′(ω1)− a3

2d3ImFα(k, ω1)

)2

,

1

τ= 1

τ0+ 1

τ ′(ω1)− a3

2d3ImFα(k, ω1). (44)

where τ ′(ω1) is given by Eq. 42. The exact solution ofEq. 43 is presented in Figs. 1, 2, 3, and 4 and exhibit minorcorrections of perturbative solution, though the singularitypoints (indicated with circles) need to thorough exactsolution to avoid the singular misbehavior of perturbativesolution (cf. Appendix 2).

In Figs. 1, 2, 3, and 4, we present the numerical solutionof the Eq. 43 for full domain of k for the real and imaginaryparts of ω(k) with corresponds to the perturbation solutiongiven by Eq. 44. The numerical solution of the nonlinear(43) is done by the Newton-type high accuracy methodapplied in 1000 points of the domain kd ∈ [0, 2π ] point bypoint (the derivative of Reω(k) with respect to k, in order tofind the group velocity, has been taken from the interpolatedReω(k) function).

Because ImFα(k, ω) = 0 in majority of the domainfor k as shown in Appendix 1, one can expect that

damping of plasmon-polariton k mode will go via thechannels τ0 and τ ′(ω1) only and not via the Lorentzfriction which is perfectly balanced by the radiation incomeof energy from other nanoparticles in the chain (cf.Appendix 1). The comparison of contributions of bothactive channels for plasmon-polariton damping is presentedin Figs. 1, 2, 3, and 4 for different geometry and sizeparameters of the chain and for the longitudinal andtransverse polarization of plasmon-polariton oscillationswith respect to its propagation direction. In the caseof a solution of nonhomogeneous (43), the stationarysolution for k-mode corresponds to the usual damped drivenoscillations, when the energy of the driving incident light istransferred to the semiconductor substrate in part whereasthe rest is dissipated into the Joule heat in metal material ofthe chain. The energy transferred to the semiconductor maybe transformed eventually into photo-current, reemitted as

476 Plasmonics (2019) 14:465–483

light via exciton recombination, and also partly convertedinto the Joule heat via scattering of band electrons withphonons, defects and admixtures in the semiconductormedium.

If we deal with two channels of plasmon damping, thusthe energy dissipation per time unit for these channels is

equal to∫ T

0dtT

2τ0

(∂Dα

∂t

)2and

∫ T

0dtT

2τ ′

(∂Dα

∂t

)2, respectively.

After averaging over the period of the stationary solutionof the driven and damped oscillator, we arrive withthe energy loss per time unit, 1

τ0ω2A2 and 1

τ ′ ω2A2,where ω is the frequency of the driving force and

A = εa3ω21E0√

(ω21−ω2)2+4ω2(1/τ0+1/τ ′)2

. This outflow of energy is

compensated during each period T of driving force by thepower of this force averaged over the period. Therefore,the ratio of energy losses via two channels equals to(1/τ0)/(1/τ ′). Because the part of energy transferred tothe semiconductor surroundings (substrate) may not beconverted into the Joule heat or irradiation, some deficit inenergy balance might be occurred not exceeding, howeverthe indicated above ratio. When this ratio, may of order oforder 1/3 (for realistic value of the parameter β [32], cf.Figures 1, 2, 3, and 4) one could observe, e.g., an ostensibleshortage in the energy balance on the level of ca. 75% ofincoming energy.

Conclusions

We have focused attention on energy losses of plasmon-polaritons caused by coupling of collective plasmons withanother charged system in near-field zone. Coupling ofdipole plasmon mode with closely located band electronsin a semiconductor opens a very quick and thus effectivechannel for energy transfer, which results in strong dampingof plasmon-polaritons in metallic nano-chains depositedon or embedded in a semiconductor surrounding/substrate.It should be emphasized that this channel of plasmon-polariton energy leakage is not of radiative type becauseit undergoes on the sub-diffraction scale with regardto wave-length of plasmon resonance. Depending onnanosphere radii in the chain different regimes for near-fieldcoupling with semiconductor substrate occur. For ultra-small nanoparticles with radii 2–3 nm the dipole couplingwith substrate electrons breaks translation invariance andlifts mementum conservation constraints imposed ontointerband electron transitions in semiconductor. Thishighly enhances the indirect (not conserving momentum)interband transition probability resulting in strong dampingof plasmon-polariton in a nano-chain. With increase of

nanoparticle size, this effect is gradually weakening and atca. 5 nm for nanoparticle radius (Au or Ag) the interbandtransition probability again grows up but due to increaseof dipole amplitude proportional to number of oscillatingelectrons. Again plasmon-polaritons are strongly dampedin this scale for nanoparticle size. Damping of plasmon-polaritons due to near-field coupling with semiconductorsurroundings typically three times exceeds Joule heat lossesdue to electron scattering in metallic nanoparticles, whichmeans that relatively easy the latter energy dissipationcan be balanced by inverted energy transfer from thesurrounding/substrate semiconductor to metallic nano-chain resulting in arbitrary long range plasmon-polaritonpropagation in such a system. This effect might be ofsome significance for prospective opto-electronic plasmon-polariton circuit applications.

Open Access This article is distributed under the terms of theCreative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricteduse, distribution, and reproduction in any medium, provided you giveappropriate credit to the original author(s) and the source, provide alink to the Creative Commons license, and indicate if changes weremade.

Appendix 1: Calculation of RadiativeDamping of Plasmon-Polariton in the ChainLocated in a Dielectric Surroundings

Both sides of Eq. 16 can be multiplied by ei(kld−ωt)

2π, and

next one can perform summation with respect to nanospherepositions and integration over t . Taking into account that,

1

2π

∞∫

∞

N∑l=−N

Dα

(±md + ld; t − md

v

)e−i(kld−ωt)

= ei(∓kmd+ω md

v

)Dα(k, ω), (45)

one obtains thus the following equation in Fourier rep-resentation (the discrete Fourier transform for nanospherepositions and the continuous Fourier transforms for time),

(−ω2 − i

2

τ0ω + ω2

1

)Dα(k, ω)

= ω21a3

d3Fα(k, ω)Dα(k, ω) + εa3ω2

1E0α(k, ω), (46)

where k = 2πn2Nd

, n = 0, 1, ..., 2N , i.e., kd ∈ [0, 2π ]due to periodicity of the chain with equidistant separation d

of nanospheres, and the form of k is due to Born-Karmanboundary condition with the period L = 2Nd . For N → ∞

Plasmonics (2019) 14:465–483 477

(infinite chain limit) k is a quasi-continuous variable. InEq. 46,

Fz(k, ω) = 4∞∑

m=1

(cos(mkd)

m3cos(mωd/v) + ωd/v

cos(mkd)

m2sin(mωd/v)

)

+2i

[1

3(ωd/v)3 + 2

∞∑m=1

(cos(mkd)

m3sin(mωd/v) −ωd/v

cos(mkd)

m2cos(mωd/v)

)],

Fx(y)(k, ω) = −2∞∑

m=1

(cos(mkd)

m3cos(mωd/v) + ωd/v

cos(mkd)

m2sin(mωd/v) −(ωd/v)2 cos(mkd)

mcos(mωd/v)

)

−i

[−2

3(ωd/v)3 + 2

∞∑m=1

(cos(mkd)

m3sin(mωd/v)ωd/v

cos(mkd)

m2cos(mωd/v)

−(ωd/v)2 cos(mkd)

msin(mωd/v)

)]. (47)

Some summations in the above equations can be doneanalytically [49]:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∞∑m=1

sin(mz)m

= π−z2 , f or 0 < z < 2π,

∞∑m=1

cos(mz)m

= 12 ln

(1

2−2cos(z)

),

∞∑m=1

cos(mz)

m2 = π2

6 − π2 z + 1

4z2, f or 0 < z < 2π,

∞∑m=1

sin(mz)

m3 = π2

6 z − π4 z2 + 1

12z3, f or 0 < z < 2π .

(48)

Using the above formulae, one can show that if 0 < kd ±ωd/v < 2π , then:

ImFz(k, ω) = 2∞∑

m=1

[sin(m(kd + ωd/v)) − sin(m(kd − ωd/v))

m3

−(ωd/v)cos(m(kd + ωd/v)) + cos(m(kd − ωd/v))

m2

]+ 2

3(ωd/v)3

= 2

[π2

6(kd + ωd/v) − π

4(kd + ωd/v)2 + 1

12(kd + ωd/v)3

]

−2

[π2

6(kd − ωd/v) − π

4(kd − ωd/v)2 + 1

12(kd − ωd/v)3

]

−2(ωd/v)

[π2

6− π

2(kd + ωd/v) + 1

4(kd + ωd/v)2

]

−2(ωd/v)

[π2

6− π

2(kd − ωd/v) + 1

4(kd − ωd/v)2

]+ 2

3(ωd/v)3 ≡ 0. (49)

However, if kd−ωd/v < 0 or kd+ωd/v > 2π for somevalues of wave vector k, then a more general formula mustbe used (by usage of Heaviside step function one can extend

the formulae (48) to the second period of their left sides;note that for d/a ∈ [2, 10], a < 25 nm the next periodsover the second one are not reached for kd ∈ [0, 2π)). This

478 Plasmonics (2019) 14:465–483

Fig. 5 The region (blue,0 < kd ± ω1d/v < 2π ) inwhere the radiation lossesvanish for infinite chains of Aunanospheres with radiusa = 10, 15, 20 nm andchain-separation d/a ∈ [3, 6](ω = ω1, v = c)

extended form for ImFz(k, ω) is as follows (here, we usedimensionless variables x = kd, y = d/a):

Fig. 6 The functionImFz(k;ω = ω1) for infinitechains of Au nanospheres withradius a = 10, 15, 20 nm andchain-separationd = 3a, 4a, 5a

0 1 2 3 4 5 60

5

10

15

kd

z

d=3ad=4ad=5a

0 1 2 3 4 5 60

5

10

15

kd

z

0 1 2 3 4 5 60

5

10

15

kd

z

ImFz(k, ω) = Θ(2π − x − ωya/v)2

[π2

6(x + ωya/v) − π

4(x + ωya/v)2 + 1

12(x + ωya/v)3

]

Θ(−2π + x + ωya/v)2

[π2

6(x + ωya/v − 2π) − π

4(x + ωya/v − 2π)2 + 1

12(x + ωya/v − 2π)3

]

−Θ(x − ωya/v)2

[π2

6(x − ωya/v) − π

4(x − ωya/v)2 + 1

12(x − ωya/v)3

]

−Θ(−x + ωya/v)2

[π2

6(x − ωya/v + 2π) − π

4(x − ωya/v + 2π)2 + 1

12(x − ωya/v + 2π)3

]

−Θ(2π − x − ωya/v)2(ωay/v)

[π2

6− π

2(x + ωya/v) + 1

4(x + ωya/v)2

]

−Θ(−2π + x + ωay/v)2(ωay/v)

[π2

6− π

2(x + ωya/v − 2π) + 1

4(x + ωya/v − 2π)2

]

−Θ(−x + ωay/v)2(ωay/v)

[π2

6− π

2(x − ωya/v + 2π) + 1

4(x − ωya/v + 2π)2

]

−Θ(x − ωay/v)2(ωay/v)

[π2

6− π

2(x − ωya/v) + 1

4(x − ωya/v)2

]+ 2

3(ωay/v)3. (50)

The function given by Eq. 50 is depicted in Fig. 6. Theexpression (50) allows to account for the inconsistence ofperiodic functions given by the sums of sines and cosineswith the Lorentz friction term and not coincidence ofarguments kd ± ωd/v of trigonometric functions (out ofthe first period, which is sufficient for d/a ≤ 10, a < 25nm). In Fig. 5, we have plotted the solution of the equation

(kd − ωd/v)(kd + ωd/v − 2π) = 0, which determinesthe region for kd (denoted by x) versus d/a (denoted by y)inside which the exact cancellation of the Lorentz frictionby radiative energy income from other nanospheres takesplace. In Fig. 6, the comparison of this cancellation forvarious nanosphere diameters is presented, for longitudinalpolarization of plasmon collective excitations.

Plasmonics (2019) 14:465–483 479

The similar analysis can be done for transversalpolarization, i.e., for ImFx(y)(k, ω). This function is exactly

zero only for the region for arguments 0 < kd−ωd/v < 2π

and 0 < kd + ωd/v < 2π , where one can write:

ImFx(y)(k, ω) = −∞∑

m=1

[sin(m(kd + ωd/v)) − sin(m(kd − ωd/v))

m3

−(ωd/v)cos(m(kd + ωd/v)) + cos(m(kd − ωd/v))

m2

−(ωd/v)2 sin(m(kd + ωd/v)) − sin(m(kd − ωd/v))

m

]+ 2

3(ωd/v)3

= −[π2

6(kd + ωd/v) − π

4(kd + ωd/v)2 + 1

12(kd + ωd/v)3

]

+[π2

6(kd − ωd/v) − π

4(kd − ωd/v)2 + 1

12(kd − ωd/v)3

]

+(ωd/v)

[π2

6− π

2(kd + ωd/v) + 1

4(kd + ωd/v)2

]

+(ωd/v)

[π2

6− π

2(kd − ωd/v) + 1

4(kd − ωd/v)2

]

1

2(ωd/v)2[π − kd − ωd/v] − 1

2[π − kd + ωd/v] + 2

3(ωd/v)3 ≡ 0. (51)

Nevertheless, outside the region 0 < kd ± ωd/v < 2π ,the value of ImFx(y) is not zero, as it is demonstrated in

Fig. 7, and can be accounted for by the formula (x = kd,y = d/a):

ImFx(y)(k, ω) = −Θ(2π − x − ωya/v)

[π2

6(x + ωya/v) − π

4(x + ωya/v)2 + 1

12(x + ωya/v)3

]

−Θ(−2π + x + ωya/v)

[π2

6(x + ωya/v − 2π) − π

4(x + ωya/v − 2π)2 + 1

12(x + ωya/v − 2π)3

]

+Θ(x − ωya/v)

[π2

6(x − ωya/v) − π

4(x − ωya/v)2 + 1

12(x − ωya/v)3

]

+Θ(−x + ωya/v)

[π2

6(x − ωya/v + 2π) − π

4(x − ωya/v + 2π)2 + 1

12(x − ωya/v + 2π)3

]

+Θ(2π − x − ωya/v)(ωay/v)

[π2

6− π

2(x + ωya/v) + 1

4(x + ωya/v)2

]

+Θ(x − ωay/v)(ωay/v)

[π2

6− π

2(x − ωya/v) + 1

4(x − ωya/v)2

]

+Θ(−2π + x + ωay/v)(ωay/v)

[π2

6− π

2(x + ωya/v − 2π) + 1

4(x + ωya/v − 2π)2

]

+Θ(−x + ωay/v)(ωay/v)

[π2

6− π

2(x − ωya/v + 2π) + 1

4(x − ωya/v + 2π)2

]

+Θ(2π − x − ωya/v)1

2(ωay/v)2[π−x−ωya/v]+Θ(−2π+x+ ωya/v)

1

2(ωay/v)2[3π−x−ωya/v]

−Θ(x − ωya/v)1

2(ωay/v)2[π−x + ωya/v]−Θ(−x + ωay/v)2[−π−x + ωya/v]+ 2

3(ωay/v)3. (52)

480 Plasmonics (2019) 14:465–483

Fig. 7 The functionImFx(y)(k;ω = ω1) for infinitechains of Au nanospheres withradius a = 10, 15, 20 nm andchain-separationd = 3a, 4a, 5a

0 1 2 3 4 5 60

5

10

15

kd

xy

d=3ad=4ad=5a

0 1 2 3 4 5 60

5

10

15

kd

xy

0 1 2 3 4 5 60

5

10

15

kd

xy

Fig. 8 Sums∞∑

n=1

sin(nx)n

(left),

∞∑n=1

cos(nx)

n2 (center),∞∑

n=1

sin(nx)

n3

(right), for x ∈ (−1, 15)

5 10 15

1.51.00.5

0.51.01.5

---

5 100.5

0.51.01.5

-

5 10

1.0

0.5

0.5

-

-

This function is plotted in Fig. 7. The discontinuity jumpon the border between the regions with vanishing radiativedamping and with nonzero radiative attenuation is caused

by discontinuous function∞∑

n=1

sin(nz)n

(cf. Fig. 8) entering

ImFx(y) but not ImFz, cf. Equation 47.

Appendix 2: Exact Solution of Eq. 43—theResolution of the Problem of LogarithmicDivergence of Far-Field-Zone Contribution

The imaginary part of the solution ω(k) of Eq. 43 definesplasmon-polariton attenuation, while the real part of ω(k)

gives self-frequency of these oscillations (in the caseof homogeneous equation, i.e., when E0α(k, ω) = 0).The derivative of this self-frequency with respect to thewave vector k defines the group velocity of particularmodes. Because of logarithmic singular term in the far-fieldtransversal contribution to dipole interaction in the chain (cf.Equation 20),

∞∑m=1

cos(m(x + ωya/v)) + cos(m(x − ωya/v))

m

= −1

2ln[(2−2cos(x+ωya/v))(2−2cos(x − ωya/v))],

(53)

one cannot apply the perturbation method for solution ofdynamical equation, at least in the region close to thesingularity. Note that within the perturbation approach onecan substitute ω with ω1 (Mie frequency) in r.h.s. ofEq. 43. This produces, however, the hyperbolic singularityin transversal group velocity resulted by virtue of Eq. 53.Moreover, within the perturbation approach, the logarithmicsingularity occurs also for both polarizations, which is

noticeable if one takes the derivative with respect to k

from the perturbative expressions for Reωα(k). All thesesingularities occur in isolated points for which kd ±ω1d/v = lπ (l is an integer). Both hyperbolic andlogarithmic divergences in perturbation formula for groupvelocities in this points would result in artificial localexceeding c by corresponding group velocities. To resolvethe problem of this unphysical divergence, the exact solutionof Eq. 43 must be found, because of divergence of theexpression (53) the corresponding contribution cannot betreated still as perturbation. The exact solution of Eq. 43,found numerically, is plotted in Figs. 1, 2, 3, and 4, for bothpolarizations of plasmon-polaritons.

The exact solutions presented in Figs. 1, 2, 3, and 2 do notdiffer significantly from those obtained in the perturbationmanner out of the singularity points, but the change sufficesto remove out the logarithmic divergence from derivativeof the self-frequencies. For the transverse polarization(Figs. 3 and 4), the difference is also not significantout of the singularity points. However, for the transversepolarization self-frequency in the case of exact solution, wedeal with quenching of the logarithmic divergence (53) incontrary to the approximated its version (obtained withinthe perturbation approach). Instead of infinite singularity,we observe in the exact plot for the transverse polarizationself-frequencies only relatively small minimum resultingthen conveniently in finite group velocity (not greater than5×107 m/s, for a = 10−20 nm). This quenched logarithmicsingularity marked in Fig. 9 into small local minimumis presented in Fig. 10—right (on the left it is presentedalso the correction to the discontinuity step in the dampingof transverse mode caused by logarithmic contribution toEq. 43).

The exact solution of the dynamic (19) (and of Eq. 43)resolves thus the problem of risky logarithmic divergentcontribution of transversal far-field part of dipole interaction

Plasmonics (2019) 14:465–483 481

0 200 400 600 800 1000

0.99

1.00

1.01

1.02

kd

xy

0 200 400 600 800 1000

0.01

0.02

0.03

0.04

0.05

kd

xy

a 10 nm Au d 4 a

0 200 400 600 800 1000

- 5 ´ 106

0

5 ´ 106

kd

xy

-

2 /1000π2 /1000π

2 /1000π

x

x

Fig. 9 The exact solution for self-frequency and damping rate oftransverse mode of plasmon-polariton in the nano-chain (ω in ω1units; solution of Eq. 19 (similar as of Eq. 43) in 1000 points onthe sector [0,2π); in plot of damping rate a local lowering belowthe Ohmic attenuation is marked; in the middle the plot for group

velocity of transverse mode is presented with hyperbolic-type singu-larities corresponding to logarithmic-type singularities odf selfenergy(left); singularities are truncated what is visualized in marked regionsin Fig. 10

of nano-spheres in the infinite chain and regularizes thefinal solution for corresponding characteristics of plasmon-polariton modes. In the vicinity of singularity points(in the domain for kd), the group velocity, though stillwell lower than light velocity, is, however, greater incomparison to group velocities in other regions of the k

wave vector domain. This elucidates the former numericalobservations [50] of long-range propagating mode fortransversal polarization of plasmon-polariton in the nano-chain.

Even though the real part of the function Fz andthus perturbative ωz(k) is given by a continuous function,the corresponding perturbative group velocity will havelogarithmic singularity as the derivative of ωz(k) withrespect to k will contain the divergent sum

∑m(cos(m(x +

ω1ya/v)) − cos(m(x − ω1ya/v)))/m. The similar term ispresent also in perturbative formula for ωx(y)(k). The origin

of these terms for both polarization is the medium-field-zone contribution to dipole interaction in the chain. In pointsx ± ω1ya/v = p2π , p integer, the logarithmic singularityproduces an artifact of the group velocity vz exceedingc. This precludes applicability of the perturbation solutionapproach, at least close to singularity points. Therefore,instead of putting ω = ω1 in function Fα in r.h.s. of Eq. 16(as was done for the perturbation method of solution ofthis equation), one must solve exactly the nonlinear (43).As was already mentioned above, this exact solution can befound numerically and both real and imaginary parts of ω

can be determined in the whole region kd ∈ [0, 2π)—inFigs. 1, 2, 3, and 4. For the longitudinal polarization of thegroup velocity, the exact solution is detailed on an exampleclose to the singular point in Fig. 11. The exact solutionsfor vz do not exhibit any singularities—the logarithmicsingularity in perturbation term is quenched to only small

157.556 157.558 157.560 157.562 157.5640.00

0.01

0.02

0.030.04

0.05

kd 2p 1000

xy

a 10 nm Au d 4 a

157.556 157.558 157.560 157.562 157.564

0.9820.9840.9860.9880.990

kd 2p 1000

xy

157.556157.558157.560157.562157.564- 4 ´ 108- 2 ´ 108

02 ´ 1084 ´ 1086 ´ 108

kd 2p 1000

xy

155 156 157 158 1590.00

0.01

0.02

0.03

0.04

0.05

kd 2p 1000

xy

155 156 157 158 159

0.9820.9840.9860.9880.9900.992

kd 2p 1000

xy

155 156 157 158 159

- 1 ´ 1080

1 ´ 1082 ´ 108

kd 2p 1000

xy

--

2 /1000π 2 /1000π2 /1000π

2 /1000π 2 /1000π

-

2 /1000π

x

x

x

xxx

xx

Fig. 10 Two level magnification of the scale of view on the truncatedsingularity region for transverse plasmon-polariton mode in the nano-chain for the exact solution of Eq. 19 (or similar of Eq. 43) ω in ω1units, v in m/s; vertical lines are artifact of numerical interpolation)(in

close vicinity of the singular point the solution has been found by theNewton-type method in 2000 points kd, point by point, for the sectorsindicated in the figure)

482 Plasmonics (2019) 14:465–483

0 200 400 600 800 1000

- 5 ´ 107

0

5 ´ 107

kd 2p 1000

z

a 10 nm Au d 3 a

0 200 400 600 800 10000.90

0.95

1.00

1.05

kd 2p 1000

z

0 200 400 600 800 1000- 6 ´ 107- 4 ´ 107- 2 ´ 107

02 ´ 1074 ´ 1076 ´ 107

kd 2p 1000

z

0 200 400 600 800 1000

0.96

0.98

1.00

1.02

1.04

kd 2p 1000

z

236.4 236.6 236.8 237.0 237.2 237.40.9910

0.9915

0.9920

0.9925

0.9930

kd 2p 1000

z

236.4 236.6 236.8 237.0 237.2 237.40

5.0 ´ 1071.0 ´ 1081.5 ´ 1082.0 ´ 1082.5 ´ 1083.0 ´ 108

kd 2p 1000

z

110.0110.1 110.2 110.3 110.4110.5 110.60.9240

0.9245

0.9250

0.9255

kd 2p 1000

z

110.0110.1110.2110.3110.4110.5110.60

5.0 ´ 1071.0 ´ 1081.5 ´ 1082.0 ´ 1082.5 ´ 1083.0 ´ 1083.5 ´ 108

kd 2p 1000

z

----

2 /1000 2 /1000 2 /1000 2 /1000

2 /1000 2 /1000 2 /10002 /1000

x

x

xxx

xx

xxxxxx

xxxxxxx

x

Fig. 11 The exact solution for the group velocity vz of longitudinalmode of plasmon-polariton in the nano-chain for a = 10, 15 nm andd = 3a, 4a, respectively, and the corresponding exact solutions forReωz (left); exact solution of dynamic (19) (or of Eq. 43) removes thelogarithmic singularity – the remaining local very narrow extremes are

truncated exactly at value of |c| (for magnification in close vicinityof the singular point the solution has been found by the Newton-typemethod in 2000 points kd, point by point, for the sectors indicatedin the bottom panels corresponding to marked fragments in the upperpanels)

local extrema similarly as it was demonstrated above for thetransversal polarization.

The logarithmic-type singularity in the self-energy fortransverse polarized plasmon-polaritons in the chain is thefeature which essentially differentiates these modes fromthe longitudinally polarized ones. This singularity is causedby the sum of far-field-zone part of the electric field of allnanoparticle dipoles which influence a charge oscillationsin each component of the chain and produces hyperbolic-type discontinuity in perturbative group velocity exclusivelyfor transversal polarization. Besides this discontinuity themedium-field-zone component of electric interaction ofdipoles additionally produces a logarithmic-type singularityin the perturbative group velocities for both polarizations(though any singularity in self-energies). We use herethe terms logarithmic-type or hyperbolic-type singularitiesto distinguish the exact behavior of the group velocitiesobtained by the exact solution for self-energies for bothpolarizations, which are sharpened and truncated at c dueto relativistic constraints imposed on the dynamic equationand manifesting themselves in the form of its solution.The retardation of electric signals prohibits the collectiveexcitation group velocity to exceed the light velocity. Thisquenching concerns infinite singularities which occur inthe perturbation expressions for self-energy and next inthe perturbation formulae for group velocities at singularpoints. The relativistic invariance of dynamic equation forcollective dipole plasmon oscillations in the chain prevents,however, exceeding the light velocity by the group velocityof particular plasmon-polariton modes. Thus, the exactsolution of this equation inherently posses also this property.Exact self-energies as solutions of nonlinear (43) have

suitably regularized their dependence with respect to k, thattheir derivatives do not exceed c.

It is worth emphasizing for the sake of completeness ofthe description, that inclusion of magnetic field of dipolesdoes not modify this scenario, because the magnetic fieldcontribution to self-energies is at least two order lower incomparison to electric field contribution due to the Fermivelocity of electrons being two order lower in comparison tolight velocity, which significantly reduces the Lorentz force.Therefore, the magnetic field of the dipoles [34, 43],

Bω = ik(Dω × n)

(ik

r0− 1

r20

)eikr0 , (54)

though contributing to the far-field and medium-fieldparts of plasmon-polariton self-energies, does not changesignificantly the similar terms caused by the electric fieldand causes only by two-orders lower corrections.

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