Radiative decay of plasmons in a metallic nanoshell

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PHYSICAL REVIEW B 69, 155402 ~2004!

Radiative decay of plasmons in a metallic nanoshell

T. V. Teperik* and V. V. PopovInstitute of Radio Engineering and Electronics (Saratov Division), Russian Academy of Sciences, Saratov 410019, Russia

F. J. Garcıá de AbajoCentro Mixto CSIC-UPV/EHU and Donostia International Physics Center, Apartado Postal 1072, 20080 San Sebastian, Spa

~Received 10 August 2003; revised manuscript received 17 November 2003; published 2 April 2004!

Retarded plasmon eigenmodes in metallic nanoshells are theoretically analyzed, and both plasmon eigen-frequencies and plasmon decay rates are calculated. Spherelike and voidlike plasmon modes are considered andtheir behavior with geometrical parameters is analyzed. Special attention is given to the problem of radiativedecay of different plasmon modes supported by such systems. It is concluded that by varying the shell-layerthickness, the voidlike plasmon decay time can be varied over more than two orders of magnitude throughoutthe femtosecond range. For shell layers thinner than the characteristic skin depth, the voidlike plasmon modesexhibit subfemtosecond radiative lifetimes and hence they become more radiative than spherelike ones. Forshell-layer thickness exceeding the characteristic skin depth, the decay time of the voidlike plasmons becomesof the order of tens of femtoseconds, yielding ultrahigh local-field enhancements. We predict local-field en-hancement factors that exceed 60 and 150 in gold and silver nanoshells, respectively. These results are sup-ported by calculations of absorption cross sections of these shells to external light. The results are applied toqualitatively explain strong coupling of plasmons with light in nanocellular metallic films, recently observed inlight reflection experiments.

DOI: 10.1103/PhysRevB.69.155402 PACS number~s!: 73.20.Mf, 61.46.1w, 78.47.1p

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I. INTRODUCTION

Electromagnetic eigenmodes of small metallic partichave attracted much attention in the past and are of gimportance in current technology since they can be eciently excited by light and their resonant frequencies cantuned by varying the geometrical structure of metal naclasters and the dielectric properties of the host mediumby choosing different metallic materials.1,2

Plasma oscillations excited in metallic particles causlarge field enhancement of the local field inside and nearparticle. This filed enhancement is currently discussed fogreat variety of potential applications.1,3,4 The local-field en-hancement factor is given in the harmonic oscillator mod2

by u f u5v res/2g, wherev res is the resonance frequency an2g is the homogeneous linewidth of resonance—thewidth at half maximum~FWHM! of the resonance~see alsoRefs. 5 and 6!. The linewidth of the resonance is relatedthe plasmon-energy decay timetpl51/2g, where g is theplasmon-field decay rate. The value oftpl can be measuredusing different nonlinear optical techniques6–11 or the spec-tral hole burning technique,12 or it can be extracted from thnear-field spectra of the individual particles based onnear-field optical antenna effect.5 Ultrafast decay times onthe sub-10-fs scale and local-field enhancement factors oto 15 have been reported for circularly shaped metallic naparticles.

Recently,13 plasmon excitations in gold nanoporous filmhave been experimentally observed and their remarkphotonic properties established. It was presumed in Refthat plasmon modes excited in spherical nanocavities comuch more effectively to light than those in metallic spherwhich results in strong reflectivity resonances observedexperiment.

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As an intuitive explanation of their observations, the athors of Ref. 13 developed a simple model of plasmon mosupported by a spherical void in an infinite metallic mediuAlthough that model gives the eigenfrequency values, whsomehow can be fitted to the frequencies of resonances inmeasured reflectivity spectra, it cannot describe the coupbetween plasmon modes in the nanocavities and the radiafield. The reason is that the plasmon modes in a voidnonradiative because their electromagnetic field cannot rate into the infinite metal having a negative dielectric funtion. At the same time, the enormous resonance peaks inreflectivity spectra observed in Ref. 13 suggest strong cpling of nanocavity plasmons to the incident light. Therefoan understanding of the effect of coupling between plasmin metallic nanocavities and the radiation field becomesgreat importance.

To examine the essential physics of coupling betwelight and plasmons in metallic nanocavities we analyze ha simple model of single metallic shell with a vacuum cosuspended in vacuum. The model of multilayered sphercluster has been developed earlier14 on the basis of classicaMie theory15 and it is widely used in nano-optics1,16–22

mostly for calculation of stationary extinction spectrananocomposite metallic materials. Since metallic nanoshdisplay unique structurally tunable optical properties,17,22 theinterest in the optical features of such complex particgrows steadily. However, the nonstationary process of rative decay of plasmons in metallic nanoshell has not baddressed earlier neither theoretically nor experimentally

In this paper we use the electromagnetic Mie theoryorder to calculate the eigenfrequencies and decay rateretarded plasmon modes in a metalic nanoshell. We specfocus on the problem of tunable radiative decay of plasmsupported by such a system, and show that the decay tim

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T. V. TEPERIK, V. V. POPOV, AND F. J. GARCIÁ DE ABAJO PHYSICAL REVIEW B 69, 155402 ~2004!

plasmons in metallic nanoshell can be effectively tuned otwo orders of magnitude in the femtosecond domain.estimate the local-field enhancement factor at plasmon rnances of a metallic nanoshell and show theoretically thatsome nanoshell parameters the local-field enhancementtor in voidlike plasmon mode can reach ultrahigh values. Ishown by calculation of the absorption cross section ofshell to external light that such ultrahigh resonant enhanment of the local field produces sharply enhanced lightsorption. We conclude that unique optical properties of mtallic nanoshells pave the way towards various useapplication of such particles in plasmon-resonance naoptics.

II. MODEL AND BASIC EQUATIONS

Let us consider a metallic shell having external and intnal radii a andb, respectively~see the inset of Fig. 1!. Sup-pose that inside the shell core and outside the shelvacuum. We describe the dielectric response of the metaan electric fieldE exp(2ivt) in the local Drude model as

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wherevp is the bulk plasmon frequency andge is the phe-nomenological bulk electron relaxation rate.

Being interested in exploring retarded plasmon eigmodes supported by such a metallic shell we seek a soluof the Maxwell equations in each medium. We decompthe electromagnetic field in each medium into spherical h

FIG. 1. ~a!, ~b! Normalized eigenfrequenciesv/vp and radiativedecay ratesg r /v of the fundamental (l 51) voidlike ~solid curves!and spherelike~dashed curves! modes vs the normalized internaradius of the shellR5bvp /cl for different values of normalizedshell-layer thicknessH5hvp /cl( l 51).

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monics, which, due to the spherical symmetry of the prolem, become uncoupled and, therefore, can be considindependently. Since the field has to be finite in all poininside the shell core, we use the spherical Bessel functiothe first kind of thel th order, j l(k0r ), wherel is the orbitalmomentum quantum number, for describing the radialpendence of field of thel th spherical harmonic inside thshell core.23 Herek05v/c andc is the speed of light. Insidethe shell layer we use a combination of the spherical Hanfunctions of the first and second kind,hl

(1)(kr) andhl(2)(kr),

respectively, wherek5vA«(v)/c. With allowance for thescattering condition, which requires that only outgoispherical waves exist at infinityr→`,23 we seek a solutionin the form of spherical Hankel functionshl

(1)(k0r ) in thehost medium outside the shell.

Satisfying the conditions of continuity of tangential components of the electric and magnetic fields at internal aexternal surfaces of the shell, we obtain the eigenfrequerelation for electrical modes~those with zero radial component of the magnetic field23! supported by the metallicshell:24

hl(1)~ra!a l

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ra5ak0 , rb5bk0 , ha5ak, hb5bk, and the prime denotedifferentiation with respect to the argument. The square rof «(v) is chosen here to have a non-negative imaginpart. Equation~2! essentially coincides with the corresponing relations for a coated sphere25 having a hollow core.

We treat the frequencyv in Eqs. ~1! and ~2! as acomplex-valued quantityv5v2 ig. The real part of thecomplex-valued frequency gives the plasmon eigenfrequewhile the imaginary partg is the total decay rate. It is obvious thatg5g r , the radiative decay rate, if we neglect thelectron relaxation processes@ge50 in Eq.~1!#. Eachqth ~inascending order in the eigenfequency value! solution of Eq.~2! for a given orbital momentum quantum numberl yieldsthe frequencies of electrical Mie modesElmq ,23 whereq isthe radial quantum number. It must be noticed that Eq.~2! isindependent of the azimuthal quantum numberm due to thespherical symmetry of the problem and, therefore, the eigmodes are degenerate with respect tom.

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RADIATIVE DECAY OF PLASMONS IN A METALLIC . . . PHYSICAL REVIEW B 69, 155402 ~2004!

At large distance from the shellr @a the Hankel functionhl

(1)(k0r ) can be approximated as exp$i@k0r2p(l11)/2#%/k0r and the electromagnetic field away from thshell becomes proportional to exp@ik0(r2ct)#/r. Note that, forcomplex-valued frequencyv, the wave vectork0 has a com-plex value as well,k05k082 ik09 . The scattering conditions ar→` are satisfied by choosing a solution withk08.0 corre-sponding to outgoing spherical waves atr @a. Such a solu-tion with k08.0 also has the propertyk09.0, meaning thatthe electromagnetic field grows in amplitude away frommetallic shell as}exp@k09(r2ct)#/r, which describes thepropagating amplitude front of the decaying mode. Tamplitude-phase pattern of the electromagnetic field is tycal of radiative eigenoscillations in open electrodynamsystems.23 Higher amplitudes of an electromagnetic fieldpoints located farther away from the shell arise from radtion arriving at these points at a given instant of time dueradiative decay of plasmon oscillations in the shell at earinstants of time. Similar time-space dependence of the fiof radiatively decaying modes was established in varisystems: e.g., radiative cyclotron-polaritons in a twdimensional electron plasma,26 leaky modes in photoniccrystal-slab waveguide,27 and radiative exciton-polaritons ia quantum-well.28 Divergence of the solution for outgoinwaves atr→`, which is commonly referred to asexponen-tial catastrophe,29 is discarded on the physical groundconsidering the initial conditions. If we conceive the decaing process that begins at timet50 then at a later timet.0, the exponentially growing solution~with r→`) hasphysical meaning only in a finite space domainr ,ct occu-pied by the decaying mode~see also relevant argumentsRefs. 27 and 28!.

III. RESULTS AND DISCUSSION

To obtain the radiative decay rate of eigenmodes inmetallic shell, which is basically the parameter controllithe coupling between eigenoscillations and radiation fiel30

we assume in our calculationsge50 for a while. In thispaper, only the electrical modes withq51 will be consid-ered. Only such plasmon modes can be excited in a swith radius that is smaller than the light wavelength.

Figure 1~a! shows the calculated eigenfrequencies for fudamental plasmon Mie modesE1m1 ( l 51, q51) in metallicshell versus the normalized internal radius of the shelR5bvp /cl for different values of the normalized shell-laythicknessH5hvp /cl, whereh5a2b. It is seen in Fig. 1~a!that, for a metallic shell, each modeE1m1 splits into twomodes, voidlikeE1m1

(v) and spherelikeE1m1(s) ones. The split-

ting originates from the coupling of plasmons bound to dferent~internal and external! surfaces of the shell. This doublet structure of optical spectra of metallic shells has bdemonstrated theoretically1,17,18 and observedexperimentally.1,17 Similar effects occur in a planar metafilm as well, where the resonance splitting is induced bycoupling of surface plasmon modes propagating at differboundaries of the film.31

As one can see in Fig. 1, the eigenfrequencies of void

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and spherelike plasmon modes do not change essenwith shell-layer thickness in the range ofH under consider-ation. However, the radiative decay rates of these mochange drastically withH @cf. Figs. 1~a! and 1~b! and payattention to the logarithmic scale over the ordinate axis, F1~b!#.

In Fig. 2, the eigenfrequencies and radiative decay raare represented as a function of the normalized thicknesthe shell layer for different plasmon modesElm1

(v,s) ( l51,2,3) and forR56. For a gold [email protected] eV ~Ref.32!# this value ofR corresponds to an actual internal radib.150 nm. It is seen in Fig. 2~b! that the radiative decayrate of thel th mode decreases with increasingl. Althoughvoidlike plasmons are bound to the internal surface ofshell, these modes can be more radiative than sphereones if the shell-layer thicknessh is small enough as compared with the characteristic screening length,d l5 ld, whered5c/vp is the characteristic skin depth (d.25 nm forgold!. At the normalized shell-layer thicknessH,0.4, thefundamental voidlike mode exhibits the highest radiativecay rate. It is worth mentioning that one hasv l /c,1 forspherelike modes andv l /c.1 for voidlike modes, wherev l5bv/ l is the orbital phase velocity. Such a fast orbitphase velocity peculiar to voidlike modes causes theirhanced radiative decay~see background arguments, e.g.,Ref. 23!.

Figure 2~a! shows that the splitting between sphereliand voidlike modes increases with decreasing shell-lathickness, starting with a normalized thicknessH.0.2. Theeigenfrequencies of spherelike and voidlike modes exhredshifts and blue shifts, respectively. This fact agrees w

FIG. 2. ~a!, ~b! Normalized eigenfrequenciesv/vp and radiativedecay ratesg r /v of different voidlike~solid curves! and spherelike~dashed curves! modesElm1

(s,v) ( l 51,2,3) of a metallic shell vs thenormalized shell-layer thicknessH5hvp /cl for normalized inter-nal radiusR5bvp /cl56.

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the blue shift ~redshifts! of voidlike ~spherelike! plasmonresonances in optical spectra of a metallic shell with decreing shell-layer thickness as reported in Ref. 1.

To obtain the total decay rateg, which includes both ra-diative and electron relaxation~dissipative! contributions, wechose the parametersvp57.9 eV andge50.09 eV~Ref. 32!in Eq. ~1!, which provide an optimum fit to gold bulk opticaconstants as reported in Ref. 33 within the investigated sptral region. In Fig. 3 the total decay rates of voidlike aspherelike plasmon modes are presented. The plasmon drates of a metallic sphere of radiusa in vacuum and of a voidof radiusb in infinite metal are also shown by dash-dottlines for comparison. Note that the decay rate of plasmonthe void is entirely of nonradiative origin and caused excsively by electron relaxation processes. It is obvious thatg!v the difference between the total and radiative derates has to be attributed to the dissipative damping of pmon modes due to electron relaxation processesgd5g2g r . Note that plasmon frequencies shift only slightly~lessthan 1%!, when an actual nonradiative damping is taken inaccount.

It is seen in Fig. 3~a! that for a small shell-layer thicknesh,d ~i.e., H,1) the voidlike plasmons are overwhelmingradiative, while they are predominantly dissipative for larthicknessesh@d. On the contrary, the electron relaxatiocontribution to the total decay rate of spherelike plasmincreases with decreasing shell-layer thickness down th

FIG. 3. Normalized total~solid lines! and radiative~dashedlines! decay rates of different~a! voidlike modesElm1

(v) and ~b!spherelike modesElm1

(s) ( l 51,2,3) of a gold shell vs the normalizeshell-layer thicknessH5hvp /cl for normalized internal radiusR5bvp /cl514. The total decay rates of plasmons in a gold sphof normalized radiusR85R1H in vacuum and the decay rates,g5gd , of plasmons in a void of normalized radiusR in infinite goldmetal are shown by dash-dotted lines. Vertical arrows markpoints whereg r5gd .

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,d, while the spherelike plasmons are overwhelminglydiative for shell-layer thicknessh>d @see Fig. 3~b!#. There-fore, the total decay rateg is controlled almost completelyby ultrafast radiative processes with characteristic time inrange of a few femtoseconds for voidlike plasmons withh,d and for spherelike plasmons withh>d.

Figure 3 shows that the total decay rates of plasmonmetallic nanoshells vary dramatically over more than torders of magnitude with varying shell-layer thicknessH.Moreover, the decay rates of the voidlike plasmons are msensitive to the shell-layer thickness than those of sphereplasmons, and the range of variation of the total decay ralarger for voidlike modes by one order of magnitude@seeFigs. 3~a! and 3~b!#. Notice, however, that the total decarate of plasmons in metallic spheres varies very little wradiusR8 @dash-dotted lines in Fig. 3~b!#.

Now we can estimate the local-field enhancement facat plasmon resonances asu f u5v/2g. The value ofu f u isabout 60 for the fundamental voidlike mode of gold shwith H51.5 ~the corresponding plasmon eigenfrequency1.39 eV!. This value ofH corresponds to the case wheg r5gd , i.e., g52g r . We predict even larger local-field enhancement factors of up to 150 atH51.95 ~the correspond-ing plasmon eigenfrequency is 1.52 eV! for a silver shell,which are characterized by a smaller electron relaxationge50.045 eV and higher bulk plasmon frequencyvp58.6 eV ~Ref. 34! ~see Fig. 4!. Note, for comparison, thathe local-field enhancement factor obtained at plasmon renance in a gold nanoball is below 15.5,6 The total decay rateof the fundamental voidlike mode of metallic shells of dferent radii is shown in Fig. 5. One can see that the derate of voidlike plasmons ath@d is smaller for metallicshells with larger core radius. Hence, the effect of local-fienhancement increases gradually with increasingR. It shouldbe noted that, in the harmonic oscillator model, the relatu f u5v/2g, whereg is the plasmon decay rate, is equivaleto the relationu f u5Q, whereQ is the quality factor of thestationary plasmon resonance excited by incoming radiatThe local-field enhancement factor defined in such a wrelates the amplitude of driven plasmon oscillations exciin the shell by incoming radiation at the resonance freque

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FIG. 4. Normalized total~solid lines! and radiative~dashed line!decay rates of the fundamental (l 51) voidlike mode of gold andsilver shells vs the normalized shell-layer thicknessH for R514.The decay rates of plasmons in a void of normalized radiusR ininfinite metal are shown by dash-dotted lines. Vertical arrows mthe points whereg r5gd .

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to that far away from resonance frequency. However, itdoesnot relate the amplitude of the local field inside or near tshell to the amplitude of incoming radiation~see also rel-evant arguments in Ref. 6!.

We also have supported our estimates of the local-fienhancement factors by calculations of the excitation oftionary plasmon resonances in the shell by incoming ligThe absorption cross section~see Fig. 6! has been obtainedas the difference between the extinction and scattering csections calculated using the optical theorem and Mie cficients corresponding to metallic shells.2,25 The parametersused in the calculation correspond to those marked by arrin Fig. 4. The local-field enhancement factors as extracfrom the absorption cross section spectra as the ratio betwthe resonance frequency and FWHM agree very well wthose estimated above from the eigenmode theory withinerror below 2%. The height of the resonance peaks in Fiis measured from the smooth nonresonant backgroundtribution. It should be noted that there is also an excellagreement between the frequencies of the resonances sin Fig. 6 and the corresponding frequencies of the plasmeigenmodes, which are specified above, because the ction g!v is well satisfied in this particular case.

FIG. 5. Normalized total~solid lines! and radiative~dashedlines! decay rates of the fundamental (l 51) voidlike mode of asilver shell vs the normalized shell-layer thicknessH at differentnormalized radii:R54, 6, 10, and 14, which correspond to actuinternal radiib.92.4, 138.6, 231, and 323.4 nm, respectively. Tdecay rates of plasmons in voids of normalized radiusR in infinitesilver metal~dash-dotted lines! are presented for comparison.

FIG. 6. Absorption cross sections for the voidlike plasmonresonance vs photon energy for gold and sliver shells withR514that are calculated for the shell parameters marked with arrowFig. 4. The cross section has been normalized to the projectedof the shell core,pb2.

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For a shell-layer thickness that is smaller than sevenanometers, surface scattering of electrons should play ain addition to bulk electron scattering. In this case, the ovall electron relaxation rate can be estimated in the framewof the limitation of electron mean-free-path model,1,2 accord-ing to which the bulk electron relaxation ratege should bereplaced byge1vF /L in Eq. ~1!. Here vF is the electronvelocity at the Fermi surface~which is of the order of 108

cm/s in metals! and L is the effective mean-free path foelectron collisions with the particle boundaries. In the aproximation of diffuse scattering of electrons from the shboundaries,18 one has

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not change appreciably the total decay rate of voidlike plmons, whereas it drastically enhances the total decay rathe spherelike plasmons at small values ofH. As a result, thedecrease in the radiative decay rate of spherelike plasmodes forH,1 ~i.e., h,d) is effectively canceled by theincrease in surface scattering, so that the total decay ratspherelike plasmons falls down only slightly and it exhibitsshallow dip atH.0.2, and then rises steeply with decreasiH. Notice the nontrivial behavior of the total decay ratethe voidlike plasmon mode at small values ofH. Althoughthe surface electron scattering opens an additional relaxachannel, the net total decay rate of the voidlike plasmodecreases slightly. To resolve this apparent paradox one mkeep in mind that at large values of the decay rateg.vradiative and dissipative decay processes are not indepenof each other anymore. Then Fig. 7 suggests that thehanced electron relaxation slows down the radiative decathe voidlike plasmons at small values ofH.

One can see from Fig. 6 that ultrahigh resonant enhanment of the local field, ultimately caused by the resonexcitation of voidlike plasmon oscillations, produces sharenhanced light absorption. We suggest that enhanced abtion of similar origin results in sharp and deep extinctiresonances in the reflectivity spectra observed in the expments of Ref. 13 on nanoporous gold films. However,

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FIG. 7. Normalized total decay rates of the fundamentall51) voidlike and spherelike modes of a gold shell vs the normized shell-layer thicknessH for R56 with ~solid lines! and without~dashed lines! surface electron scattering.

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should be noted that it is difficult to compare quantitativethe results of our calculations with experimental data of R13 because we are considering a different kind of obje~metallic nanoshells!, which can only qualitatively model theactual experimental situation. More sophisticated theoretanalysis including multiple scattering of the light betwedifferent shells is underway, with a view to quantitativeexplain the results of Ref. 13.

IV. CONCLUSIONS

Radiatively decaying plasmon eigenmodes in a metananoshell has been theoretically studied. The eigenfreqcies and decay rates of voidlike and spherelike plasmhave been calculated. It has been shown that, although vlike plasmons are bound to the internal surface of the shthese modes can be more radiative than spherelike onesufficiently thin shell layers as compared with the characistic skin depth.

By varying the shell-layer thicknessh, the plasmon decaytime, tpl51/2g, can be varied over two orders of magnituin voidlike modes throughout the femtosecond range. Sdecay times can be measured using one of well-develotime-resolved measurement techniques.5–12 The voidlikemodes exhibit ultrashort radiative lifetimes of the order ofs whenh is much less than the characteristic skin depth.h of the order of a few characteristic skin depths, tplasmon-energy decay time in the voidlike mode becomethe order of tens of femtoseconds, which yields ultrahvalues of the local-field enhancement factor exceedingand 150 for gold and silver nanoshells, respectively. Thefect of local-field enhancement increases gradually withcreasing the shell-core radius.

It is worth noting in this connection that, although thmagnetic modes do not couple to plasma oscillationsspherical metallic particles, they should manifest themse

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within the investigated spectral region if the shell-core radgrows to be of the order of the light wavelength. Hence thmodes, as well as high-order (q.1) electrical modes, can, inprinciple, show up in optical spectra of nanocellular metamaterials with cavities of large enough radius. These effedeserve further investigations. The interaction between liand plasmons in periodic ensembles of metallic nanoshellwell as in periodic ensembles of voids in metallic substralso demands specific theoretical treatment.

We suggest that ultrahigh enhancement of the local-fiin voidlike plasmon modes is responsible for deep extinctresonances in reflectivity spectra observed recently in noporous gold films.13 We conclude that a metallic nanosherepresents a unique object whose radiative and local-fiproperties can be effectively tuned over a wide rangenanoengineering the shell parameters. This makes sunanoparticle very attractive for various applications in futusubmicron light technology.

ACKNOWLEDGMENTS

We thank S. V. Gaponenko, V. G. Golubev, and S.Tikhodeev for inspiring conversations. Helpful discussiowith A. N. Ponyavina and O. Stengel are gratefully appreated. This work was supported by the Russian FoundationBasic Research~Grant No. 02-02-81031! and the RussianAcademy of Science Program ‘‘Low-Dimensional QuantuNanostructures.’’ T.V.T. acknowledges the support fromPresident of Russia through the grant for young scientMK-2314.2003.02, from the National Foundation for Promtion of Science, and from INTAS~Grant No. YSF 2002-153!.F.J.G.A. acknowledges help and support from the Univerof the Basque Country UPV/EHU~Contract No. 00206.21513639/2001! and the Spanish Ministerio de Ciencia y Tecnlogıa ~Contract No. MAT2001-0946!.

J.

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Radiative decay of plasmons in a metallic nanoshell

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