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APS/123-QED Dynamics Analysis Of Plasmon Resonance Modes In Nanoparticles I. D. Mayergoyz, Z. Zhang Department of Electrical and Computer Engineering, Institute for Advanced Computer Studies, University of Maryland, College Park, Maryland, 20742, USA 7 G. Miano 0 Univ Naples Federico II, Dept Elect Engn, Via Claudio 21, Naples, I-80125 Italy 0 (Dated: February 6, 2008) 2 Anoveltheoreticalapproachtothedynamicsanalysisofexcitationofplasmonmodesinnanopar- n ticlesispresented. Thisapproachisbasedonthebiorthogonalplasmonmodeexpansionanditleads a to the predictions of time-dynamics of excitation of specific plasmon modes as well as their steady J state amplitude and their decay. Temporal characteristics of plasmon modes in nanoparticles are 9 expressedintermsoftheirshapes,permittivitydispersionrelationsandexcitationconditions. Inthe 2 case of theDrudemodel, analytical expressions for time-dynamics of plasmon modes are obtained. ] PACSnumbers: 41.20.Cv,42.25.Fx,42.68.Mj,78.67.Bf i c s - l Plasmonresonancesinnanoparticleshavebeenthe fo- InthecaseoftheDrudemodel,analyticalexpressionsfor r t cusofconsiderableexperimentalandtheoreticalresearch plasmon dynamics are obtained which suggest that the m lately. This researchis motivated by numerous scientific reciprocalof the Drude damping factor can be identified . andtechnologicalapplicationsoftheseresonancesinsuch with the decay (dephasing) time of plasmon modes. t a areas as near-field microscopy, nano-lithography, surface To start the discussion, consider a nanoparticle with m enhanced Raman scattering, nanophotonics, biosensors, boundaryS anddielectricpermittivityǫ(ω)infree-space - opticaldatastorage,etc. Steadystatepropertiesofplas- with dielectric constant ǫ . It has been demonstrated d 0 mon resonances in nanoparticles have been mostly stud- [8, 9, 10, 11] that the resonance values ǫ of dielectric n k o ied, while the dynamics of specific plasmon modes is the permittivity andcorrespondingresonantplasmonmodes c leastunderstoodareaofplasmonics. Thisstate ofaffairs can be found by solving the eigenvalue problems for the [ has prompted the current burst of activity in the exper- boundary integral equation 1 imental research of plasmon dynamics. This research is λ r n v exemplified by publications [1, 2, 3, 4, 5, 6, 7], where σ (Q)= k σ (M) MQ· QdS , (1) 1 the plasmon dynamics and plasmon decay (dephasing) k 2π IS k rM3 Q M 3 rates have been extensively studied by using advanced 7 or its adjoint equation femtosecond techniques. The theoretical temporal anal- 1 0 ysis of plasmon modes is also very important to fully λk rQM nM τ (Q)= τ (M) · dS , (2) 7 comprehend the time-dynamics of mode excitation and k 2π k r3 M 0 decayaswellastoestimatetheirsteadystateamplitude. IS QM / where σ (M) has the physical meaning of surface elec- t The temporal analysis of plasmon resonance modes can k a tric charges on S that produce electric field E of k-th beveryinstrumentalintheareaoflightcontrollabilityof k m plasmonmode,τ (M)hasthephysicalmeaningofdipole plasmonresonancesinsemiconductornanoparticles[8,9], k - densitiesonS thatproducedisplacementfieldD ofk-th d wherepropertimesynchronizationsofexcitationsofspe- k n cific plasmon modes may be needed. plasmonmode, whileallothernotationshavetheir usual o meaning. The resonance values of ǫk and λk are related c In this Letter, we present a self-consistent theoretical by v: technique for the temporal analysis of specific plasmon ǫ ǫ i modes in nanoparticles and compare (where it is possi- λ = k− 0. (3) X k ǫ +ǫ ble) the theoreticalresults with experimental data. This k 0 r a techniqueisbasedonthebiorthogonalplasmonmodeex- Afterλk arecomputedandresonancevaluesofpermit- pansion and it leads to the predictions of time-dynamics tivityǫ arefoundbyusing(3),resonancefrequenciesare k ofexcitationofspecificplasmonmodesandtheirdecayas determined from the dispersion relation: wellastheirsteadystateamplitude intermsofnanopar- ′ ticleshapes,permittivitydispersionrelationsandexcita- ǫk =ǫ(ωk)=Re[ǫ(ωk)]. (4) tion conditions. For instance, an explicit formula is de- It turns out that eigenfunctions σ (M) and τ (M) are rivedforthesteadystateamplitudeofplasmonmodesin k i biorthogonal: termsofrealandimaginarypartsofdielectricpermittiv- ity, amplitude ofincidentfieldandits spatialorientation σ (M)τ (M)dS =δ . (5) with respect to dipole moments of the plasmon modes. k i M ki IS 2 Thus, the set of eigenfunctions σ (M) can be used for The last integral can be simplified, because the incident k the biorthogonal expansion of actual boundary charges field E(0)(Q,t) is practically uniform within nanoparti- σ(M,t) induced on particle boundary during the excita- cles. Thus, this field can be represented as E(0)(Q,t) = tion process: E f(t). By using this formof E(0)(Q,t) in (13) and tak- 0 ∞ inginto accountformula(3)andthe factthatdue to the σ(M,t)= a (t)σ (M), (6) “scaling” freedom provided by normalization condition k k (5)thefollowingexpression n τ (Q)dS =(ǫ ǫ )p kX=1 S Q k Q k− 0 k forthe dipole momentp ofk-thplasmonmode (see[9]) where,accordingto(5),theexpansioncoefficientak(t)is can be used, the last forkmuHla can be represented in the given by the formula: form: ak(t)= σ(M,t)τk(M)dSM. (7) a˜ (ω)=(E p )g˜ (ω)f˜(ω), (14) k 0 k k IS · It is clear that the time evolution of the expansion coef- where ficient a (t) reveals the time-dynamics of k-th plasmon k ǫ(ω) ǫ 0 mode corresponding to the eigenfunction σk(M). Since g˜k(ω)= ǫ −ǫ(ω), (15) k themediumofresonantnanoparticleisdispersiveandex- − hibitsnonlocalintimeconstitutiverelationD(t)vsE(t), and f˜(ω) is the Fourier transform of f(t). the frequencydomaintechnique willbe employedforthe Formulas (14)-(15) are remarkably simple and they calculations of ak(t). This means that the equation (6)- clearlysuggestthatg˜k(ω)canbeconstruedasnormalized (7) will be Fourier transformed (byE p )transferfunctionofk-thplasmonmode. The 0 k · ∞ formula(15)revealstheresonancenatureofexcitationof σ˜(M,ω)= a˜k(ω)σk(M), (8) k-thplasmonmode atthe frequencyωk. Indeed, accord- k=1 ingto(4),atthisfrequencyǫk ǫ(ωk)= iIm[ǫ(ωk)]and X − − ′′ the magnitude of g˜ (ω) will be narrow peaked if ǫ (ω ) k k is sufficiently small, that is when the specific plasmon a˜ (ω)= σ˜(M,ω)τ (M)dS , (9) k k M resonance is strongly pronounced. IS By using formula (14), the following expression is de- and the equation for a˜ (ω) will be first derived and k rived for a (t): solved. Subsequently,a (t)willbefoundthroughinverse k k Fourier transform. t To derive the equation for a˜k(ω), the Fourier trans- ak(t)=(E0·pk) gk(t−t′)f(t′)dt′, (16) formed boundary condition for the normal components Z0 of electric field on S is invoked: with gk(t) being the inverse Fourier transform of g˜k(ω). Thus, the algorithm of analysis of time-dynamics of k- ǫ(ω)E˜+(Q,ω) ǫ E˜−(Q,ω)=[ǫ ǫ(ω)]E˜(0)(Q,ω). (10) n − 0 n 0− n th plasmon mode can be stated as follows. First, the Here: E˜+(Q,ω) and E˜−(Q,ω) are the limiting values of eigenvalue problem (1) and (3) is solved and resonance n n the normal components of Fourier transformed electric values ǫk of dielectric permittivity and surface electric field at Q S from inside and outside S, respectively, charges σk(M) of the corresponding plasmon mode are while E˜(0)(∈Q,ω)is the Fouriertransformednormalcom- found. Next, by using σk(M), the dipole moment pk is n computed. Then,g (t)isdeterminedthroughtheinverse ponent of the incident field on S. k By using formula σ˜(Q,ω)=ǫ [E˜−(Q,ω) E˜+(Q,ω)], Fouriertransformofg˜k(ω)givenby(14). Finally,formula 0 n − n (16) is employed to evaluate the time evolution of a (t) theboundarycondition(10)canberearrangedasfollows: k which reveals the time-dynamics of k-th plasmon mode. σ˜(Q,ω) E˜+(Q,ω)=E˜(0)(ω). (11) It is worthwhile to mention that the outlined computa- ǫ(ω) ǫ0 − n n tions can be performed by using actual, experimentally − measured ǫ(ω). Now it can be recalled [12, 13] that Expressions (14)-(16) can be useful for analytical cal- σ˜(Q,ω) 1 r n E˜+(Q,ω)= + σ˜(Q,ω) MQ· QdS . culations as well. To demonstrate this, consider the n − 2ǫ0 4πǫ0 IS rM3 Q M steady state case when f(t) = sinωkt, where ωk is the (12) resonance frequency defined by the formula (4). Then, By substituting (12) into (11) and then by using ex- f˜(ω)=i π [δ(ω ω ) δ(ω+ω )] and, by using (14)- 2 − k − k pansion (8) and formulas (1) and (9), the following ex- (15)aswellastheinverseFouriertransformofa˜ (ω),we k p pression for a˜k(ω) is derived: arrive at: ′ 2ǫ λ [ǫ(ω) ǫ ] ǫ(ω ) ǫ a˜k(ω)= 2ǫ λ +0[ǫk(ω) ǫ−](λ0 1) Eñ(0)(Q,ω)τk(Q)dSQ. ak(ss)(t)=−(E0·pk) ǫ′′k(ω−) 0cosωkt+sinωkt , 0 k − 0 k− IS (cid:20) k (cid:21) (13) (17) 3 wheresuperscript(ss)indicatesthesteadystateofa (t). 8 k (a) average plasmon Inthecaseofstrongplasmonresonanceswhen ǫ′(ω ) field k 6 laser pulse ǫ >> ǫ′′(ω ), the last formula is simplified as|follows−: field 0 k | | | 4 As typaik(csas)l(tfo)r=m−o(sEt0r·epsokn)aǫ′n(ǫcω′e′k(s)ω,−kt)hǫe0csotseωadkty. stat(e18is) Electric field [arb.units]−022 shifted by almost ninety degree in time with respect to −4 theincidentfieldE (t). Itisalsonaturalthatthemagni- 0 −6 tude of the steady state is controlled by the ratio of real and imaginaryparts of dielectric permittivity at the res- −80 50 100 150 200 250 Time [fs] onance frequency. It is also revealing that the resonance 8 magnitude depends on the spatial orientation of the in- (b) afieveldr a g e p lasmon 6 laser pulse cident field E0(t) with respect to the dipole moment pk field of plasmon resonance mode. 4 1146 (1) Electric field [arb. units]−022 (2) −4 12 nt C10 −6 e effici 8 −80 50 100 150 200 250 Co Time [fs] 6 FIG.2: DynamicsofplasmonresonancesforAunanoringson 4 aglasssubstratecomputedfortheresonancewavelength1102 nm(see[14])byusing(a)Durdemodel(γ =1.0753×1014 s−1 2 [15]) and (b) Au dispersion relation from [15]. 400 600 800 1000 1200 1400 Wavelength [nm] Indeed, by using (21) in (15) we obtain: FIG. 1: C(ω0) computed for: (1) gold nanocylinders placed on aglass substratethat resonate at wavelength 774nm (see ω2+γ2 [2]) and (2) gold nanorings placed on a glass substrate that g˜k(ω)= k , (22) −(iω α )(iω α ) resonateatwavelength1102nm(see[14]). TheAudispersion − 1 − 2 relation from [15] havebeen used in computations. where: α = γ+iβandα = γ iβ,β = √4ωk2+3γ2. Now 1 2 2 2− 2 the inverse Fourier transform of (22) yields for t>0: Foroff-resonanceexcitationf(t)=sinω t, similarcal- 0 culations lead to the expression: gk(t)= 2(ωk2+γ2) e−γ2tsinβt. (23) ak(ss)(t)=(E0·pk)C(ω0)cos(ω0t+ϕ), (19) − 4ωk2+3γ2 Formulas (16) and (2p3) can be used for analytical cal- where culationsofa (t) for“rectangular”laserpulses E(0)f(t). k Indeed, in the case when: [ǫ′(ω ) ǫ ]2+[ǫ′′(ω )]2 0 0 0 C(ω )= − . (20) 0 s[ǫk ǫ′(ω0)]2+[ǫ′′(ω0)]2 0 for t<0, − f(t))= (24) (sinωkt for t 0, Figure 1 illustrates that C(ω ) is narrow peaked at the ≥ 0 resonancefrequencyωk for the twocases ofplasmonres- from (16), (23) and (24) we derive: onances experimentally studied in [2] and [14]. forNgex(tt,)wine(p1o6i)ntcaonutbethoabttasiimnepdlefoarntahlyetDicrauldeexpmreosdseiolnosf ak(t)=−(E0·pk)ωke−γ2t γ1cosβt−β2sinβt +ak(ss)(t), k (cid:18) (cid:19) ǫ(ω): (25) where ω2 ǫ(ω)=ǫ0"1− ω(ω+p iγ)#. (21) ak(ss)(t)=(E0·pk) ωγkcosωkt−sinωkt . (26) (cid:18) (cid:19) 4 In the case of finite rectangular laser pulse mon mode (thin line). The computed plasmon time- dynamics was used to compute the third order autocor- 0 for t<0, relation function (ACF) which was compared with the f(t)= sinω t for T t 0, (27) experimentally measured ACF from [2]. The results of  k ≤ ≤ 0 for t>T, comparison presented in Figure 4. Figure 5 presents the comparisonbetweenour calculationsof the second-order from (16), (23) and(27) we derive for t>T: ACFfornoncentrosymmetricL-shapegoldnanoparticles from [6] and the experimentally measured in [6] second- ak(t)=−(E0·pk)e−γ2(t−T) ωke−γ2T γ1cosβt− β2sinβt oexrdpeerriAmCenFt.aFl digautraesw4ithainndt5ensupgegrceestntt.he agreement with h (cid:18) (cid:19) ω k cosω t sinω t . (28) k k − γ − (cid:18) (cid:19)i It is instructive to compare the computational results obtained by using the Drude model with those obtained by using the experimentally measured dispersion relation ǫ(ω). This comparison is presented by Figures 2a and 2b for Au rings subject to finite rectangular laser pulses. ThesefiguressuggestthattheDrudemodelleads to quantitatively similar results as the use of actual dispersion relation. Formula (28) also implies that 1/γ can be identified as the decay (dephasing) time for the light intensityofplasmonmodes. Inaccordancewiththeavail- able data for γ (see [7, 15]), this suggests that formula (28) predicts the decay (dephasing) time for plasmon FIG.4: EnvelopofthecalculatedthirdorderACF(filledcir- modes in gold (and silver) nanoparticles in the range of cles) superimpose on the measured third order ACF (solid 5-12fs, which is consistent with the experimental results line)from [2] for Au cylinders on a glass substrate at wavelength 774 nm (maximum is normalized to 32). reported in [1, 2, 3, 4, 5, 6, 7]. average plasmon field 3 laser pulse field 2 Electric field [arb.units.] −011 −2 −3 20 40 60 80 100 120 140 Time [fs] FIG.3: Dynamicsofplasmon resonancesforAucylinderson aglass substratecomputedfor theresonance wavelength 774 nm (see [2]). The Au dispersion relation from [15] have been FIG. 5: The calculated second order ACF superimpose on used in computations. the envelop of measured second order ACF from [6] for non- centrosymmetric L-shape Au nanoparticles on a tantalum- dioxide substrate at wavelength 838 nm (maximum is nor- Finally, we present the comparison between computa- malized to 8). tional results based on the technique outlined above and experimental results presented in [2, 6]. By using equa- tions (15)-(16) we have computed the time-dynamics of the 774 nm resonance wavelength plasmon mode for Au cylinders on glass substrate [2]. Figure 3 presents the time variation of the incident electric field of the laser [1] B. Lamprecht, A. Leiner,and F. R Aussenegg, Appl. pulse (bold line) used in experiments reported in [2] and Phys. B 64, 269 (1997). the corresponding computed time-dynamics of the aver- [2] B. Lamprecht, J. R. Krenn, A. Leitner and F. 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