NANO LETTERS
Plasmon Polaritons in Finite-Length Metal−Nanoparticle Chains: The Role of Chain Length Unravelled
2005 Vol. 5, No. 5 985-989
D. S. Citrin* School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0250, and Georgia Tech Lorraine, Metz Technopoˆ le, 2-3 rue Marconi, 57070 Metz, France Received March 17, 2005; Revised Manuscript Received April 14, 2005
ABSTRACT While the conceptual framework for nanoplasmonic waveguides composed of a chain of noncontacting metal nanoparticles usually neglects the effects of the ends, the long-range nature of the interparticle coupling underlying the electromagnetic transport means that finite chain length can play an important role. Here, the complex energies of the plasmon-polariton modes in finite-length nanoparticle chains are calculated to ascertain the effects of chain length on the mode dispersion and the radiative contribution to the attenuation. The results indicate that, for typical parameters, the infinite-chain limit is reached with ∼10 nanoparticles. Thus, even for chain lengths well exceeding the attenuation length, long-range coupling of distant nanoparticles is shown to impact the dispersion and radiative loss.
Electromagnetic energy transport along chains of noncontacting metal nanoparticles (NP) has attracted interest for possible applications in the guiding of light by structures with subwavelength dimensions.1 The transport mechanism is the coherent transfer of plasmonic excitation from NP to NP by means of the retarded multipole-multipole interactions. One way to understand the transport is that it is mediated by photons; a surface plasmon (SP) in one NP2 decays into a photon that is reabsorbed in another NP. The net coherent sum of all such processes leads to the formation of a hybrid SP-photon mode known as a plasmon polariton.3-9 A recent experimental study has explored the attenuation of the propagation of plasmon polaritons in finite-length NP chains (NPC).5 A typical inferred value of the attenuation constant in NPCs composed of N ) 80, 50-nm-diameter spherical Au NPs is RL ≈ 2.5 × 105 cm-1, giving an 5 attenuation length of R-1 L ≈ 40 nm, which is on the order of the interparticle spacing. This is in a series of NPCs with center-to-center interparticle spacings of d ) 75-125 nm. Thus, plasmon-polariton attenuation in NPCs can be extremely severe. To what extent that attenuation depends on the NPC length itself, and the role played by attenuation, is in part the aim of this letter. Attenuation of electromagnetic propagation in such NPCs has been simulated by means of the finite-difference timedomain (FDTD) method;5 however, this approach is necessarily numerically intensive, and thus may in practice be limited in the number of spatial dimensions included. * E-mail:
[email protected]. 10.1021/nl050513+ CCC: $30.25 Published on Web 04/26/2005
© 2005 American Chemical Society
Moreover, we should not underestimate the need for convenient models that are not computationally intensive in order to carry out rapid exploration of parameter space. A widely applied approach toward these ends is based on tight-binding models (TBM) with a limited number of neighbors included; most typically only the nearest neighbors (NN) are retained, and radiative decay is accounted for after the fact through a phenomenological parameter.8 We have recently shown10 that, though in some regimes the NN TBM may provide a reasonable basis for a description of the resonant frequencies, a correct understanding of the spontaneous emission by plasmon polaritons in NPCs, and thus ultimately of attenuation, requires a unified treatment of the mode frequencies and radiative decay as they both arise from the same physical mechanism.10,11 Moreover, within an NN TBM, the main effect of finite NPC length is to impose boundary conditions on the problem and thus to select discrete infinite-NPC states. The effect of long-range coupling, however, will have a nontrivial influence on the modes. These effects are dealt with by the dynamic coupled-dipole model. For example, in refs 12-14, the retarded interparticle dipole-dipole coupling was used to study the plasmonpolariton modes of NPCs. This model, the validity of which is discussed below, provides in its proper domain an excellent description of the plasmon-polariton modes of NPCs with a very small number of parameters and in a manner that can be applied to broad classes of materials. In this letter, we compute the resonant frequencies and radiative widths of plasmon-polariton modes in finite-length NPCs using the
theory as mentioned above that accounts self-consistently for the transport and radiative decay of plasmon polaritons. A theoretical treatment of optical spectra of finite NPCs accounting for retarded dipole-dipole coupling was recently given in ref 12. There, a narrowing of the optical extinction spectrum in R ) 50 nm (R is the NP radius) Ag NPCs was seen with increasing particle number (and thus NPC chain length) for interparticle distance d ) 470 nm. Our focus here is on elucidating the origins of such chain-length dependences. Clearly, finite NPCs are of considerable potential practical importance for applications, and it is thus of interest to ascertain to what extent we can make inferences based on the properties of infinite NPCs. We will see that the key point is to include the retarded dipole-dipole coupling between all pairs of NPs in the NPC, i.e., we do not truncate the couplings between particles separated by distances greater than a given length. (Other multipoles are neglected in this letter; see below for the justification of this assumption.) We find the following: (i) The modes of a finite NPC approach those of an infinite NPC more quickly as NPC length is increased for polarization along the NPC axis (longitudinal) than for polarization perpendicular to the NPC (transverse), due to the shorter-range nature of the retarded dipole-dipole coupling in the former case than the latter. (ii) The range of the interparticle coupling is not limited by attenuation of polariton propagation in a simple fashion. (iii) NPCs for which RL has been measured (N ) 80)5,7 are well within the infinite-NPC limit. The results therefore show why the resonant frequencies (and also radiative widths) may depend on NPC length even when that length is considerably in excess of the attenuation length RL. Our treatment follows that of refs 10, 11, and 15. This technique has been used with great success to treat a number of polariton problems in low-dimensional systems.16 Though the salient results are given here, more details may be found in the references cited. We begin by considering a NPC of N equally spaced NPs arranged along a line. The NPs are assumed to be spherical and isotropic. The substrate is taken into account as follows. Consider the case when the NPs are in a dielectric medium (e.g., air), with background dielectric constant b, directly on top of the surface of the substrate with dielectric constant sub. From the method of images,17 the field emitted by the SP on a NP will be scaled by a factor 2sub/(b + sub) compared to the case when the NP is embedded in a uniform dielectric with dielectric constant b. For more complicated substrate geometry (e.g., if the NPCs are on a mesa or in a trench, etc.), the method of images can be suitably modified. We treat the NPs as point dipoles,11,18-20 an approximation that has proven useful in describing to some extent the plasmon-polariton modes in NPCs.8 Clearly, other multipole moments may play a role;21,22 however, it is assumed that the dominant modes that contribute to the optical spectra are dipolar, which is likely if d/(2R) well exceeds unity - a limit admittedly not well achieved in refs 5 and 7. Due to the short-range nature of these higher multipole couplings, however, a real NN coupling term (radiative decay in most 986
cases will be dominated by dipole-dipole coupling) may be added to the present treatment and will not substantially modify our conclusions since the effects of such a term will have saturated already at very small values of N. In fact, recent theoretical work directly addresses the accuracy of retaining only retarded dipole-dipole couplings by comparison with calculations carried out involving higher multipoles (T-matrix) as well.13 It was found for R ) 5 and 30 nm Ag NPCs that the results assuming only dipolar terms were quite close to those obtained from the T-matrix approach for d/(2R) g 1.25, but even for d/(2R) g 1.01 (i.e., NPs almost in contact) the discrepancies in the extinction spectra were only on the order of or less than 10%. Recent work within a quasistatic multipolar approach shows that (in the static case) the retention of only dipole-dipole coupling provides an excellent description of the plasmonpolariton modes for d/(3R) J 1.23 To proceed, the plasmon polaritons may be identified with the poles of the retarded dipole Green function (GF) D() (ref 16), which we can write in the site representation as Dn,m(), with the energy and m, n ∈{1, 2, ..., N} denoting two NP sites. The components of the inverse of D() are Dnη,mβ-1() )
[
]
2 - p2 - iγnr() δηβδnm - Σnη,mβ() 2p
(1)
where η and β refer to the directions of the polarization in NPs n and m, respectively, (with η,β ) L, T1, T2 indicating the polarization in the longitudinal or the two orthogonal transverse directions with respect to the chain axis), p is the single-NP SP energy (which may be polarization dependent due to image-charge effects associated with the substrate), the superscript -1 denotes the matrix inverse, and γnr() ) sign()γnr is the nonradiative part of the single-NP homogeneous width of the SP.24 Here, Σnη,mβ() is the radiative self-energy (SE). An explicit expression for the SE, accounting only for dipole-dipole coupling, is25,26 Σnη,mβ() ) 2sub κ3 b + sub b
[
∫ d3r1 ∫ d3r2 pnη(r1) - (16 - rˆ rˆ )κr1 eiκr +
]
1 4π (1 - iκr)(1 6 - 3rˆ rˆ ) 3 3eiκr + 3δ3(r) pmβ(r2) (2) κr 3κ where pnη(r1) is the polarization density of the plasmonic excitation on NP n in direction η, r ) |r1 - r2|, κ ) xb/pc, and c is the speed of light in vacuo.11 Due to symmetry, the L-, T1-, and T2-modes are decoupled, and the two transverse modes T1 and T2 may have their degeneracy lifted by the presence of the substrate (henceforth denoted T without a subscript). That the SE is a complex quantity is due to the fact that it is computed using freespace electromagnetic modes that propagate away from the NPC (evanescent modes are also correctly included by analytic continuation), as is dictated by causality.27 The instantaneous (nonretarded or long-wavelength) limit is obtained when κr f 0. Note that the retarded dipole-dipole coupling is of Nano Lett., Vol. 5, No. 5, 2005
qualitatively different nature for L- and T-modes; for the former, the long-range term (that proportional to r-1 is absent p‚rˆ ) 0 for n * m, η ) L), though for the later case it is present. The results below will show that long-range coupling is an important feature of the T-modes. There are four important parameters in the problem, viz. the squared dipole moment (proportional to parameter γ defined shortly), the resonant freespace optical wavevector κ, the interparticle spacing d, and the number N of particles in the NPC. Because the dispersion and radiative widths are proportional to the squared dipole moments, we can conveniently scale the dispersion and radiative width by this quantity. In addition, because κ and d only enter into the theory in an essential way through their product χ, the relevant results depend only on χ and N. One can trivially rescale these results in a suitable fashion to fit the problem at hand. We thus scale the SEs by a parameter associated with the single-NP radiative width 2γ/3 (ref 28) in the limit that the NP radius goes to zero, γ)
2sub κ3 | b + sub b
∫ d3r p(r)|2
(3)
Figure 1. (a) Re σλ,T(χ) and (b) -Im σλ,T(χ) for the T-modes plotted as functions of dimensionless frequency χ for N ) 7. The curves of the same line type (e.g., solid, dashed, etc.) in (a) and (b) correspond to the same mode of the NPC.
This enables us to write results in dimensionless form. The SE normalized by γ is σn,m,η(χ) ) Σn,m,η()/γ; normalized quantities are plotted throughout. One has 2 σn,m,T(χ) ) - iδn,m + 3 1 i 1 - 2 eiχ|n-m| (4) (1 - δn,m) 3 3 2 χ|n m| χ |n - m| χ (n - m)
[
2 σn,m,L(χ) ) - iδn,m 3 2(1 - δn,m)
]
[
]
1 i - 2 eiχ|n-m| (5) 3 2 χ |n - m| χ (n - m) 3
Note in eq 5 the absence of the long-range |n - m|-1 term that is present in eq 4. In eqs 4 and 5, any on-site real renormalization of the single-NP SP energy is assumed to have been folded into p. The dipolar modes labeled λ ∈{1, ..., N} of the NPC are found by diagonalizing the matrix Dnm,η() in the site indices n and m. Inasmuch as |Σnm,η(p)| , p, we can make the pole approximation Σnm,η() ≈ Σnm,η(p), and hence diagonalize σnm,η(χ) to yield the normalized frequency shift as Re σλ,η(χ) and the radiative width of mode λ as -Im σλ,η(χ). Thus, as discussed above, eqs 4 and 5 apply to any set of NPC parameters, so long as the assumptions of the model are satisfied. As an example, in Figure 1 are plotted (a) Re σλ,T(χ) and (b) -Im σλ,T(χ) for the T-modes of a N ) 7 NPC. In Figure 1a, we see that for χ < 1, the bandwidth of the set of modes (the dispersion) falls off rapidly with increasing χ as successive particles are in the near-field of their neighbors, and the r-3 term in the SE plays a major role; as χ increases beyond unity, the dispersion is dominated by the r-1 term. Now refer to Figure 1b, which shows the radiative width. Nano Lett., Vol. 5, No. 5, 2005
Figure 2. (a) Re σλ,T(χ) and (b) -Im σλ,T(χ) T-modes as functions of the number N of nanoparticles in the chain for χ ) 1.
For χ f 0 (i.e., the resonant optical wavelength is much longer than any other spatial dimension in the problem, namely Nd), one mode dominates the radiative properties. This mode has the dipoles in phase along the NPC. As χ increases to unity and beyond, the wavelength (in the background medium) decreases to d. As this happens, different modes can more easily access a density of final photon modes into which to decay while conserving both energy and momentum. Note that for the NPCs of refs 5 and 7, χ ) 0.8-1.3. Figure 2 shows the behavior of the dispersion and radiative width as a function of N [(a) Re σλ,T(χ), (b) -Im σλ,T(χ)] for the T-modes for χ ) 1. In Figure 2a, the dispersion increases with N and reaches the bandwidth of the infinite NPC at 987
Figure 3. Bandwidth B(N) (difference between maximum and minimum normalized mode frequencies) as a function of χ for N ) 7 plotted as lnB(N). Solid (dashed) curve is for T- (L-)modes.
Figure 4. Dependence of bandwidth B(N) on N for χ ) 1 plotted as χ-1ln[B(N) - B(N - 1)]. Solid (dashed) curve is for T- (L-) modes.
N∼10. Figure 2b shows that the modes can be classified as dark and light, with one intermediate-strength mode. The way in which the dispersion approaches the infinite-NPC limit is governed by the range of the interparticle coupling, while that in which the radiative loss approaches this limit is governed by interference between the emitted electromagnetic fields of the various dipoles, and thus by the resonant optical wavelength. Computations for the L-mode exhibit similar gross behavior. Figure 3 shows ln[B(7)] as a function of χ (i.e., as for fixed N, but varying d) with B(N) the bandwidth [difference between the maximum and the minimum values over λ of Re σλ,L(χ) or Re σλ,T(χ)]. The oscillations appear to be due to interference effects associated with the radiative widths and the mode frequencies on B as χ is varied. Note that for χ > 1, the T-modes show less effective sensitivity (smoothing out the oscillations) to d, since the long-range coupling tends to average the polarization over several NPs, while the L-modes show more sensitivity to d, since the long-range coupling is absent. In Figure 4 is shown the scaling of B(N) as a function of N with χ ) 1. The results are plotted as χ-1ln[B(N + 1) - B(N)]. Note that the infinite-NPC limit is achieved more rapidly for the L-modes than for the T-modes. At this point, we can make a semiquantitative comparison with the experimental work of ref 5 in which χ ∼ 1. First, based on Figure 4, we can conclude that structures with N ) 80 NPs, as in refs 5 and 7, are within the infinite-NPC limit. The fact that the computed bandwidths of the L- and T-modes above are constant for N J 10 is in close agreement with the experimentally found result of ref 5 where it was found that the LT splitting saturated at N ) 7. We note, however, that the optical extinction of Ag NPCs has been predicted to show continued changes for N . 10 (refs 12 and 29). These further changes in the optical properties with 988
increasing N may occur because the optical properties depend how the oscillator strength is distributed within the plasmonpolariton band, which may be particularly sensitive to N when χ ∼ 1. This is the topic of further study. An important aspect of plasmon-polariton propagation in NPCs is attenuation management. Recent work30 shows that Au single-NP dephasing is dominated by radiative or nonradiative processes, depending on particle diameter; nonradiative for 2R ∼ 10 nm, radiative for J 200 nm. While nonradiative dissipation is not likely to be strongly affected by the geometrical arrangement of the NPs, as we have seen the radiative contribution is. This observation was made in ref 14 in which it is noted that while the dispersion is governed by short-range coupling, the radiative properties are determined by all terms in the interparticle coupling. Moreover, even given the observed large homogeneous widths of smaller NPs, our pole approximation nonetheless remains valid. In the following, we make several remarks related to attenuation, its management, the role of long-range coupling, and the validity of TBMs. The relationship between the total homogeneous width γtot ) γnr -ImΣλ,η(p) and the attenuation Rη of plasmon-polariton propagation for polarization η was discussed in ref 5. In general, one has Rη ) 2γtot/ Vg where Vg is the group velocity for polarization η. (The factor of 2 is due to our definition of the γ’s.) In ref 5, the relation at the wavevector at half the Brillouin-zone boundary (the point of highest group velocity in a NN TBM) was found to satisfy Rηd ) 2γtot/Bη, where here Bη is the bandwidth available for transport for modes of polarization η. A typical value of RL ≈ 2.5 × 105 cm-1 was found. One might also expect γtot to play an important role in determining the range of the effective retarded dipole-dipole matrix elements. Invoking again a pole approximation, but now making an effort to include γnr self-consistently, we have Σn,m,η() f Σn,m,η(λ) ≈ Σn,m,η(p - iγnr). We thus have for the exponential factors within the SE in the site representation in eq 2 exp(iχ|n - m|) f exp(iχ|n - m|) exp[-γnrxbd|n - m|/(pc)], where we have assumed that the nonradiative contribution to the homogeneous width dominates over the radiative contribution. Substituting parameters typical of Au NPs,5 we obtain γnrxb/(pc) ∼ 102 cm-1. This implies an effective exponential cutoff of the interparticle coupling of ∼102 µm. In other words, the exponential range of the coupling is not Rη, but rather RηVgxb/c ) 2γtotxb/c. The range of the coupling is much longer than the typical NP spacing d ∼ 100 nm (i.e., a range of ∼103 particles) and also much longer than R-1 η . Thus, it is important to include the long-range coupling even in the presence of attenuation. In conclusion, we presented numerical computations of the plasmon-polariton modes in finite-length NPCs of noncontacting metal NPs assuming point dipoles coupled via the retarded dipole-dipole interaction. The important role of long-range coupling for the T-modes in the approach to the infinite-NPC limit is elucidated. We point out that the computed results are universal in the sense that they can be scaled for any values of γ, κ, and d. The method accounts for radiative decay on the same basis as the transport. We Nano Lett., Vol. 5, No. 5, 2005
also show that despite the fact that the attenuation is shorter than the range of the retarded dipole-dipole coupling, interactions between distance NPs are important in determining the plasmon-polariton modes and their associated radiative widths. Acknowledgment. The author would like to thank S. E. Ralph for helpful discussions. This work was supported in part by the National Science Foundation through grant NSFDMR-0303969. References (1) Quinten, M.; Leitner, A.; Kren, J. R.; Aussenegg, F. R. Opt. Lett. 1998, 23, 1331. (2) Kreibig, U.; Vollmer, M. Optical Properties of Metal Clusters; Springer: Berlin, 1995. (3) Maier, S. A.; Brongersma, M. L.; Kik, P. G.; Meltzer, S.; Requicha, A. A. G.; Atwater, H. A. AdV. Mater. 2001, 13, 1501. (4) Maier, S. A.; Brongersma, M. L.; Atwater, H. A. Mater. Res. Soc. Symp. Proc. 2001, 637, E2.9.1. (5) Maier, S. A.; Kik, P. G.; Atwater, H. A. Appl. Phys. Lett. 2002, 81, 1714. (6) Maier, S. A.; Brongersma, M. L.; Atwater, H. A. Appl. Phys. Lett. 2001, 78, 16. (7) Maier, S. A.; Brongersma, M. L.; Kik, P. G.; Atwater, H. A. Phys. ReV. B 2002, 65, 193408. (8) Brongersma, M. L.; Hartman, J. W.; Atwater, H. A. Phys. ReV. B 2002, 62, R16356. (9) Maier, S. A.; Barclay, P. E.; Johnson, T. J.; Friedman, M. D.; Painter, O. Appl. Phys. Lett. 2004, 84, 3990. (10) Citrin, D. S. Nano Lett. 2004, 4, 1561. (11) Citrin, D. S. Opt. Lett. 1995, 20, 901. (12) Zou, S.; Janel, N.; Schatz, G. C. J. Chem. Phys. 2004, 120, 10871.
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(13) Zhao, L.; Kelly, K. L.; Schatz, G. C. J. Phys. Chem. B 2003, 107, 7343. (14) Markel, V. A. J. Mod. Opt. 1993, 40, 2281. (15) Citrin, D. S. J. Opt. Soc. Am. B, in press. (16) Citrin, D. S. Phys. ReV. B 1993, 47, 3832. (17) Jackson, J. D. Classical Electrodynamics; Wiley: New York, 1999. (18) Yaghjian, A. D. IEEE Trans. Antennas Propagat. 2002, 50, 1050. (19) Shore, R. A.; Yaghjian, A. D. TraVeling Electromagnetic WaVes on Linear Periodic Arrays of Small Lossless Penetrable Spheres, Air Force Research Laboratory In-House Report AFRLSN-HS-TR2004-044, 2004. (20) Shore, R. A.; Yaghjian, A. D. Scattering-Matrix Analysis of Linear Periodic Arrays of Short Electric Dipoles, Air Force Research Laboratory In-House Report AFRL-SN-HS-TR-2004-045, 2004. (21) Shahbazyan, T. V.; Perakis, I. E.; Bigot, J.-Y. Phys. ReV. Lett. 1998, 81, 3120. (22) Shahbazyan, T. V.; Perakis, I. E. Phys. ReV. B 1999, 60, 9090. (23) Park, S. Y.; Stroud, D. Phys. ReV. B 2004, 69, 125418. (24) Which may be much larger than the radiative width, and may thus be the dominant cause of attenuation in the propagation of plasmonpolaritons in these structures. In particular, 2γnr ) 180 meV was measured for 50 nm Au spherical nanoparticles [Maier, S. A.; Kik, P. G.; Atwater, H. A.; Meltzer, S.; Requincha, A. A. G.; Koel, B. E. Proc. SPIE 2002, 4810, in press]; p/2γnr of 3.6 fs has been measured in similar nanoparticles. For larger nanoparticles (∼200 nm), radiative dissipation dominates [Klar, T.; Perner, M.; Grosse, S.; von Plessen, G.; Spirkl, W.; Feldmann, J. Phys. ReV. Lett. 1998, 80, 4249]. (25) Avery, J. S. Proc. Phys. Soc. 1966, 88, 1. (26) Craig, D. P.; Thirunamachandran, T. Molecular Quantum Electrodynamics; Academic: London, 1984. (27) Mahan, G. D. Many-Particle Physics; Plenum: New York, 1993. (28) This corrects a minor error in ref 11. (29) Zou, S.; Schatz, G. C. J. Chem. Phys. 2005, 122, 97102. (30) So¨nnichsen, C.; Franzl, T.; Wilk, T.; von Plessen, G.; Feldmann, J.; Wilson, O.; Mulvaney, P. Phys. ReV. Lett. 2002, 88, 77402.
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