Geometric Dependence of the Line Width of Localized Surface

Letter pubs.acs.org/JPCL

Geometric Dependence of the Line Width of Localized Surface Plasmon Resonances Yang Li,†,‡,∥ Ke Zhao,†,§,∥ Heidar Sobhani,†,‡ Kui Bao,†,‡ and Peter Nordlander*,†,‡,§ †

Department of Physics and Astronomy, ‡Laboratory for Nanophotonics, §Department of Electrical and Computer Engineering, Rice University, Houston, Texas, United States ABSTRACT: For the same number of electrons and plasmon frequencies, longitudinal plasmon resonances in metallic nanorods exhibit narrower line widths than plasmon modes in spherical particles. We show that this property is a general feature of high aspect ratio nanostructures and can be explained very simply by incorporating retardation effects into a harmonic oscillator model. The origin of the effect is dynamic depolarization, which renormalizes the mass of the electrons and the oscillating electron liquid. The scattering spectrum derived from our model agrees very well with FDTD simulations. Because plasmon damping determines many important features and applications of LSPR, such as the Q factor of plasmonics devices and the magnitude of the induced field enhancements, our study will play an important role for the design of nanostructures with narrow plasmon resonances. SECTION: Plasmonics, Optical Materials, and Hard Matter

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influence of geometry on the plasmon line width, it is important to keep these variables fixed because both the radiative and nonradiative plasmon damping increase with Ne and ωLSPR.51 Here, we exploit the unique geometric tunability of gold nanoshells and nanorods to design nanoparticles possessing the same Ne and ωLSPR but distinctly different geometries. Using finite-difference time-domain (FDTD) simulations we show quite generally that spherical particles exhibit broader line width than high aspect ratio nanoparticles. We show that this difference originates from the geometry dependence of the radiative damping and that it can be explained intuitively by including retardation effect into a simple harmonic oscillator model for the LSPR. The geometries of the nanoparticles are shown to strongly influence the plasmon line width through the structural dependence of the dynamic depolarization, which renormalizes the electron mass and subsequently the total energy of the oscillating electron liquid. The scattering spectra derived from our model agree very well with FDTD simulations and clearly show that elongated nanoparticles possess narrower line widths than spherical particles. Our findings and insights will be important for the design of plasmonic nanostructures supporting narrow plasmon resonances. In Figure 1, we compare different pairs of gold ellipsoidal nanorod and spherical nanoshell, each pair possessing the same ωLSPR, and Ne is kept identical for all structures. This is achieved by adjusting the aspect ratio for the nanorods (ratio of long and short axis, a/b) and for the nanoshells (ratio of the inner and

etal nanoparticles with their unique light-capturing properties mediated through their localized surface plasmon resonances (LSPRs) are highly appealing candidates for nanoscale optical devices such as nanoantennas and substrates for surface-enhanced spectroscopies.1−6 The developments in nanofabrication techniques during the past decade have enabled accurate fabrication of nanoparticles of a variety of shapes exhibiting LSPRs that can be tuned across the visible and near-IR part of the spectrum.7−15 Much work has recently focused on the line width of plasmonic resonances, which determines the magnitude of the plasmon-induced electromagnetic field enhancements that are relevant for surfaceenhanced spectroscopies such as surface-enhanced Raman scattering.16−26 In this context, the development of structures exhibiting plasmonic Fano resonances as a result of radiative coherence effects has been particularly important.27−47 While a simple understanding for how the energies of LSPRs depend on the geometry and dielectric environment of the nanostructure has already been developed, for instance, using the plasmon hybridization concept,48 a simple understanding of how the geometrical shape of a nanoparticle influences the width of the plasmon resonance is still lacking. An LSPR excited by an external field quickly dephases, through both radiative and nonradiative channels, such as Landau damping and impurity scattering. The dephasing time is manifested as a line width broadening in the LSPR spectrum and plays a critical role in determining its Q factor and the induced electromagnetic field enhancements. While previous work has investigated the influence of geometry on the line width, such studies have typically involved nanoparticles with a different total number of electrons Ne and LSPR frequencies ωLSPR.49,50 However, for a direct and unambiguous assessment of the © XXXX American Chemical Society

Received: February 24, 2013 Accepted: April 2, 2013

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dx.doi.org/10.1021/jz4004137 | J. Phys. Chem. Lett. 2013, 4, 1352−1357

The Journal of Physical Chemistry Letters

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diminish when the aspect ratio of the nanorod becomes close to one because then the two structures are becoming similar. Because of the generally broad resonances, the peaks do not have perfect Lorentzian line shapes, and we extract the line width (to be discussed below) as the difference between the two half-maximum points on the spectra. We now proceed to provide a physical explanation for the difference in the line widths. For simplicity, we limit the discussion to dipolar LSPR, where all electrons undergo harmonic oscillations when driven by an external electric field

⎯⇀ ⎯

⎯⇀ ⎯

E ext(t) = E 0e−iωt. For simplicity but without losing generality we only consider axially symmetric nanoparticles with the external electric field polarized along its long axis, which is ⎯⇀ ⎯

⎯⇀ ⎯

defined to be in the z direction, E 0 = E 0êz, where êz is the unit vector along the z direction. The equation of motion for each electron can be written: ...

mz(̈ t )eẑ = −Kz(t )eẑ − mγz(̇ t )eẑ + mγradz (t )eẑ ⎯⇀ ⎯

⎯⇀ ⎯

− e[Eext(t ) + Ep(t )]

(1)

where m and e are the electron mass and charge respectively, z(t) is the deviation of electrons from their equilibrium position, K is the geometry-dependent restoring force for electrons due to positive ionic background and the induced surface charges, γ and γrad are the intrinsic and radiative ⎯⇀ ⎯

⎯⇀ ⎯

damping, and E p (t) = E pe−iωt is the retarded electric depolarization field induced by the local dynamic dipole moments of the nanoparticle. As a result, the time dependence ⎯⇀ ⎯⇀ of z is also harmonic: z (t) = z e−iωt. For the dipolar LSPR, we take the depolarization field to be uniform and equal to the field induced at the center of the nanoparticle from all local dipoles across the nanoparticle.54 This turns out to be a reasonable approximation as verified below by our FDTD simulation. The consequence of this approximation is that the motions of all electrons and their associated dynamic dipole moments in the nanoparticle are identical, which fits into the physical picture of a dipolar plasmon mode as a rigidly oscillating electron liquid. The intrinsic damping γ can be either measured experimentally or derived from quantum mechanical calculations accounting for different intrinsic damping mechanisms. The radiation damping coefficient is:

Figure 1. Scattering spectra for matched pairs of nanorods (solid lines) and nanoshells (dashed lines). For the nanorods, the polarization is longitudinal. (a) Illustration of the line width extraction for a matched nanorod (a = 330, b = 55) nm and nanoshell (r1,r2) = (245,250) nm pair. The dotted line is calculated by Mie theory including only the dipolar mode. (b) Normalized scattering spectrum calculated using full-wave simulations. The geometries of the nanorods are (120,90), (131,87), (160,80), (183,73), (205,68), (256,64), (330,55), (400,50), and (450,45) nm from the bottom to the top spectrum. The corresponding inner and outer radii of nanoshells are (80,115), (97,124), (120,140), (145,159), (165,176), (192,201), (245,250), (286,290), and (318,321) nm.

outer radii, r1/r2). For simplicity, we use a Drude dielectric model ε(ω) = ε∞ − ω2B/ω(ω + iγ) with ωB = 8.94 eV, γ = 0.069 eV, and a background permittivity ε∞ =9.5 for all calculations presented in this study. None of the conclusions presented in the study would be changed if a more realistic permittivity had been used. Our FDTD simulations are carried out by the commercial software FDTD Solutions,52 and full Mie theory is used for the spherical particles.53 Because the dipolar and higher order multipolar LSPRs lie very close to each other for the nanoshells, the line width extraction for the nanoshells (to be discussed below) has been performed for only the dipolar components, as illustrated with the dotted line in Figure 1a. The incident light is assumed to be polarized along the long axis of the nanorods. A systematic comparison of different pairs of nanorods and nanoshells is shown in Figure 1b. For easy visualization, each spectrum is normalized so that the dipolar resonance peak is unity, and the spectra of different pairs are shifted from each other. The full-wave simulation in Figure 1 shows that for each pair the line width of the nanoshell is broader than that of the nanorod, and this difference tends to

γrad =

1 e2 6πε0 mc 3

(2)

where ε0 and c are the dielectric constant and speed of light in vacuum.55 Note that the radiation reaction force is of Abraham−Lorentz type and proportional to the time derivative of the acceleration. The depolarization field in the center of the ⎯⇀ ⎯

nanoparticle can be written E p = Epêz, with the explicit expression for Ep derived from a Hertzian dipole field:56 ik·r

Ep =

⎡

2

⎛

∫ 4eπε ⎢⎣ kr sin2 θ + ⎝ r13 0

⎜

−i

⎤ k⎞ 2 ⎟(3 cos θ − 1)⎥P dV 2⎠ ⎦ r (3)

In this expression, k = ω/c, and r and θ are the coordinates of a local dipole of moment PdV in a spherical coordinate system with its origin at the center of the nanoparticle. P is the dipole moment per unit volume, and the integration is over the 1353



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volume V of the nanoparticle. Retaining contributions up to third order in k, Ep takes the form: ⎛ V ⎞P Ep = ⎜ −L + k 2D + ik3 ⎟ ⎝ 6π ⎠ ε0

−
nV = Rad LSPR |P|2 n2V 2 >Rad = nV = |P|2 nV 2 2 2ne

(11)

The time-averaged intrinsic and radiative dissipation powers are −
Intr = nV = |P|2 nV dt 2ne 2

(12)

and 1354



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(eq 16) for both nanoshells and nanorods. The effect of the shape of a nanoparticle on D and subsequently its effect on the line width eq 14 can be conceptually understood directly from the expression of the depolarization field in eq 5, where the contribution from each local dipole is weighted by a term cos2 θ, with θ being the azimuthal angle between the local dipole and the center of the nanoparticle. In a simple analogy, we can represent the nanoshell and nanorod by four local dipoles, as illustrated in Figure 4. For the nanoshells, the azimuthal angle θ

2 α(ω) = (Vε0ωB2 /(1 + ωLSPR D/c 2)) /((ω02 + ωB2L)

/(1 + ωB2D/c 2) − ω 2 − iω(γ + nVω 2γRad) /(1 + ωB2D/c 2))

(15)

This expression is equivalent to the modified long-wavelength approximation previously derived using quasi-electrostatic approaches for oblate spheroids.49,56,58,60,62 However, our approach based on the harmonic oscillators model is more intuitive and can be easily extended to the more general geometries, such as the nanoshell application pursued in this letter. The corresponding scattering cross section is:53 σscat(ω) =

ω4 |α(ω)|2 2 4 6πε0 c

(16)

Note that the line widths and resonance energies obtained from eq 15 in the limit of small damping agree exactly with eqs 10 and 14. Figure 3a,b compares the scattering spectra calculated using full-wave simulations and the result from our analytic approach Figure 4. Illustration of how nanorods and nanoshells can be represented using four local dipoles to show the effect of different geometry on the dynamic depolarization factor D.

is π/2 for two of the dipoles and 0 for the other two, while for the nanorod, θ = 0 for all dipoles, thus contributing more strongly to D. This simple picture illustrates that elongated nanoparticles always will have a narrower line width than more isotropic ones. In conclusion, using FDTD and Mie theory simulation, we find that for the given total number of electrons and resonant frequency, spherical particles possesses a broader line width than longitudinal resonances in ellipsoidal particles. We explain this phenomenon by incorporating the retardation effect into the harmonic oscillator model for LSPR. We show that the geometries of nanoparticles strongly affect their line width through the geometric dependence of the dynamic depolarization, which renormalizes the electron mass and energy of the oscillating electron gas. The scattering spectrum derived from our model agrees very well with the FDTD and Mie theory simulation, which supports our conclusion that elongated nanoparticles tend to have narrower line widths.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions ∥

Yang Li and Ke Zhao contributed equally

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the Robert A. Welch Foundation under grant C-1222 and by the U.S. Army Research Laboratory and Office under grant/contract number W911NF-12-1-0407. We also thank Dr. Nicolas Large for his help with the TOC graphic.

Figure 3. Scattering spectrum for nanoshells (a) and nanorods (b) of different geometric parameters (legends) calculated using full-wave simulations (solid lines) and eq 16 (dashed lines). 1355



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Geometric Dependence of the Line Width of Localized Surface

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