CONTRIBUTED TALKS REVISTA MEXICANA DE FISICA S56 (2) 56–58 JUNIO 2010

Out-of-plane plasmon-polariton modes in two-dimensional nanoparticle arrays

T.D. BackesSchool of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta,

Georgia 30332-0250,e-mail: backes@gatech.edu

D.S. CitrinUnite Mixte Internationale 2958 Georgia Tech-CNRS, Georgia Tech Lorraine,

Metz Technopole, 2 rue Marconi, 57070 Metz, France,e-mail: david.citrin@ece.gatech.edu

Recibido el 14 de julio de 2008; aceptado el 23 de octubre de 2008

Plasmon-polaritons in a square, two-dimensional lattice of noncontacting nanoparticles are studied using the coupled dipole model. Theapproach considers diffractive losses out of the plane and also for the vector nature of the electromagnetic field, and is thus a three-dimensionaltheory of the two-dimensional nanoparticle array. We consider the out-of-plane polarization, where long range coupling is present, andpresent the case where the plasmon energy crosses the light cone near the M-point in the Brillouin Zone. In this case, only a restricted set ofnonradiative modes are present.

Keywords:Surface plasmon; nanofabrication; ultrasmall.

Se estudian los plasmares-polaritones en un arreglo cuadrado de partıculas, sin tocarse unas con otras, usando el modelo del dipolo acoplado.La aproximacion considera perdidas difractivas fuera del plano y por la naturaleza del vector del campo electromagnetico, siendo un modelotridimensional de un arreglo tridimensional de nanopartıculas. Consideramos la polarizacion fuera del plano, donde se presenta acoplamientode largo alcance y presentamos el caso donde la energıa del plasmon cruza el cono de luz cerca del punto-M de la zona de Brillouin. En estecaso soloesta presente un conjunto restringido de modos no radiactivos.

Descriptores:Plasmon superficial; nanofabricacion; ultrasmall.

PACS: 73.20.Mf; 42.25.Fx; 87.80.Cc

1. Introduction

Arrays on noncontacting noble metal nanoparticles (NP) havegarnered a great amount of recent interest. The Plasmon po-lariton (PP), a hybrid plasmonic optical mode, has the po-tential to allow for a number of applications, such as inter-connects, detectors, and lenses. In this letter, we examinethe square 2D nanoparticle array (NPA) in particular. Thesetypes of arrays are of interest due to the defined surface plas-mon (SP) resonances that occur in nanoparticles, in compar-ison to the broad resonances of metal sheets.

Some studies of 2D arrays have appeared in the litera-ture. Sakoda [3] simulated using the finite-difference time-domain method the dispersion curves of a square arrays ofmetal cylinders. The finite-element method was used in [4]to explore negative refraction in arrays. In Ref. 5, the multi-ple multipole method was used to analyze triangular latticesin addition to the square lattice. In all of these studies, thearrays were of composed infinite cylinders, and out-of-planelosses were not accounted for.

This paper examines the out-of plane-Plasmon-polaritonmodes in square-lattice NPA’s using a coupled-dipole model.This model follows the work done with respect to 1D NPchains [13, 14]. We treat a NPA, with the NP diameterδ andthe center-to-center inter-NP spacingd. The dipole approx-imation has been shown to be accurate except in the case ofNP’s that are nearly in contact [17]. The NP’s are treated

as oscillating point dipole and coupling within the array isaccounted for by the the retarded dipole field. More specifi-cally, we apply this technique to show that square arrays canbe suitably designed such that all SPPs are radiative with theexception of a small region in the Brillouin Zone near theM-point.

2. Calculations

The dipolar Green’s function (GF)Gα(q, ε) for excitationwavevectorq and modeα, can be used to describe the PP’s.This treatment captures both radiative PP’s [12], the fields

FIGURE 1. Diagram of the nanoparticle array.

OUT-OF-PLANE PLASMON-POLARITON MODES IN TWO-DIMENSIONAL NANOPARTICLE ARRAYS 57

scattered by the NPA, and nonradiative surface polaritons, thenearfield standing waves. The PP’s are given by the singular-ities ofGα(q, ε),

G−1α (q, ε)δqq′

=

(ε2 − ε2

p − iγp

2εp− Σ0 − Σα(q, ε)

)δqq′ , (1)

with εp and γp the the single-NP surface-plasmon reso-nance energy and nonradiative damping, respectively.Σ0

is the radiative decay of a single NP SP, and the re-tarded dipole-dipole coupling is contained within the radia-tive self-energy (SE)

Σα(q, ε) =∞ ′∑

h,j=−∞Σh,j;α(ε)ei(qxhd+qyjd),

which is simply the retarded dipole-dipole energy per NP (theprime indicating that the(h, j)=(0, 0) term is not included).We divide the solution toG−1

α (q, ε) = 0 into real and imag-inary componentsEα(q) and iΓα(q) and use the pole ap-proximations

Eα(q) ≈ εp + ReΣα(q, εp)

and

Γα(q) ≈ −ImΣα(q, εp)/~.

The SE is the dynamic dipole-dipole energy associatedwith NP’s (h, j) and(h′, j′),

Σ(h,j)−(h′,j′)(ε) = E(h,j),(h′,j′) ·p(h′,j′)

with E(h,j),(h′,j′) the oscillating [asexp(−iωt)] electric fieldassociated with the SP of NP(h, j) at the position of NP(h′, j′) andp(h′,j′) is the dipole moment of the SP of NP(h′, j′). A schematic diagram of the NPA is shown in Fig.1. Based on the approach of Refs. [13,14,18,19], the nonva-nishing SE for the mode transverse to the plane of the arrayis

Σ(q, ε) = nγ0

∞∑

h,j=−∞

′{1

k3(h2 + j2)3/2

− i

k2(h2 + j2)− 1

k(h2 + j2)1/2

}

× eik(h2+j2)1/2ei(qxh+qyj), (2)

whereγ0 = ω3p2/(4πε0c3), ~ω = ε, ε0 is the electric con-

stant,k = dωε1/2/c is the dimensionless optical wavevec-tor in the background medium,q = qd is the dimensionlessexcitation wavevector,ε is the dielectric constant, which in-cludes the gain of the medium, andn =

√ε is the complex

index of refraction. It should be noted that the out-of-planemode SE is completely decoupled from modes that exist inthe plane of the NPA, and the PP’s are purely transverse. Inorder to compute the SE in dimensionless form, we divide theSE byγ0 such thatσ(q, k) = γ−1

0 Σ(q, ε).

FIGURE 2. Z-polarized Plasmon-polariton real dispersionReσ(q, k) as functions ofq for a Au δ=50-nm square-latticenanoparticle array with interparticle center-to-center spacingd = 75 nm embedded in a semiconductor with complex refractiveindexn = 3.5− 0.05i.

FIGURE 3. Z-polarized Plasmon-polariton radiative loss−Imσ(q, k) as functions of q for a Au δ=50-nm square-lattice nanoparticle array with interparticle center-to-center spacingd = 75 nm embedded in a semiconductor with complex refractiveindexn = 3.5− 0.05i.

3. Results

We turn our focus on the complex SE. We assume reason-able values for interparticle spacingd=75 nm and nanopar-ticle sizeδ = 50-nm. We consider Au NP’s since the radia-tive decay time is on the order of the nonradiative dephas-ing time [22]. The embedding medium is chosen to haveRen = 3.5, with complex dielectric constantε = n2, and asmall amount of loss is added for numerical stability. Weapproximate the shift ofεp due to the embedding medium:

Rev. Mex. Fıs. S56 (2) (2010) 56–58

58 T.D. BACKES AND D.S. CITRIN

εp =(εp)air

( 23 + 1

3Ren), [23]

and definekp as the optical wavevector evaluated atε = εp.The SP energy in air is taken to be(εp)air =2 eV [20–22,24].

Figure 2 shows the dispersion Reσ(q, k), and Fig. 3shows the radiative loss−Imσ(q, k) as functions ofq for theZ-mode, which is purely transverse. In this case, the photondispersion has undergone zone folding almost everywhere inthe BZ once it crosses the SP dispersion, except in a smallregion close to the M-point. We see that Reσ(q, k) exhibitsoverall negative phase velocity with an additional feature atthe crossing with the light coneq= k. This appears as a sharpdecline beyond which the dispersion is mostly constant. Theradiative width in a small region close to the M-point is suchthat the SP-like SPPs are nonradiative; elsewhere in the BZ,

radiative decay is prevalent. At this particular interparticlespacing, beam forming can occur.

4. Conclusion

We have presented a theory for the out-of-plane polarizedPP’s in NPA’s with square symmetry. We have examined aregion where the light cone has undergone zone folding nearthe X-point, but not in a small region near the M-point. Thisq dependence allows for the restriction of nonradiative darkmodes that do not have radiative losses to a small region inthe Brillouin Zone.

Acknowledgments

This work was supported in part by the National ScienceFoundation grant NSF DMR-0305524, and by NASA grantNNX07-AR54H.

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