Resonant Field Enhancements from Metal Nanoparticle Arrays

Abstract

Theoretical and semiempirical studies of two-dimensional (2D) metal nanoparticle arrays under periodic boundary conditions yield quantitative estimates of their electromagnetic (EM) field factors, revealing a critical relationship between particle size and interparticle spacing. A new theory based on the RLC circuit analogy has been developed to produce analytical values for EM field enhancements within the arrays. Numerical and analytical calculations suggest that the average EM enhancements for Raman scattering can approach 2x10^11 for Ag nanodisks (5x10^10 for Au) and 2x10^9 for Ag nanosphere arrays (5x10^8 for Au). Radiative losses related to retardation or damping effects are less critical to the EM field enhancements from periodic arrays compared to that from other nanostructured metal substrates. These findings suggest a straightforward approach for engineering nanostructured arrays with direct application toward surface-enhanced Raman scattering (SERS).

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Purdue University
Purdue e-Pubs
Birck and NCN PublicationsBirck Nanotechnology Center
1-15-2004
Resonant Field Enhancements from Metal
Nanoparticle Arrays
Dentcho A. Genov
Purdue University
Andrey K. Sarychev
Purdue University
Vladimir M. Shalaev
Purdue University
Alexander Wei
Purdue University, alexwei@purdue.edu
This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for
additional information.
Genov, Dentcho A.; Sarychev, Andrey K.; Shalaev, Vladimir M.; and Wei, Alexander, "Resonant Field Enhancements from Metal
Nanoparticle Arrays" (2004).Birck and NCN Publications.Paper 75.
http://docs.lib.purdue.edu/nanopub/75

Page 2

Resonant Field Enhancements from
Metal Nanoparticle Arrays
Dentcho A. Genov,†Andrey K. Sarychev,†Vladimir M. Shalaev,†and
Alexander Wei*,‡
Departments of Chemistry and Electrical and Computer Engineering,
Purdue UniVersity, West Lafayette, Indiana 47907
Received June 5, 2003; Revised Manuscript Received Novem ber 17, 2003
ABSTRACT
Theoretical andsemiempirical studies oftwo-dimensional (2D) metal nanoparticlearrays underperiodic boundary conditions yieldquantitative
estimates of their electromagnetic (EM) field factors, revealing a critical relationship between particle size and interparticle spacing. A new
theory based on the RLC circuit analogy has been developed to produce analytical values for EM field enhancements within the arrays.
Numerical and analytical calculations suggest that the average EMenhancements for Raman scattering (G h) can approach 2 × 1011for Ag
nanodisks (5 × 1010for Au) and 2 × 109for Ag nanosphere arrays (5 × 108for Au). Radiative losses related to retardation or damping effects
are less critical to the EM field enhancements from periodic arrays compared to that from other nanostructured metal substrates. These
findings suggest a straightforward approach for engineering nanostructured arrays with direct application toward surface-enhanced Raman
scattering (SERS).
Nanostructured metal-dielectric interfaces often exhibit
enhanced optical phenomena at visible and near-infrared
(NIR) frequencies via excitation of surface plasmon modes.1,2
The enticing possibilities of engineering such properties for
applications in photonics and chemical sensing have led to
a resurgence of activity in the design of plasmonic materials
with subwavelength dimensions.3Enhanced electromagnetic
(EM) field effects can be generated either in a broad spectral
range, as is the case for disordered metal-dielectric com-
posites,2,4or at select frequencies from periodically ordered
metal nanostructures. Periodicity plays a key role in tuning
the optical response of the latter, and has been documented
in experimental and theoretical investigations of plasmon-
enhanced effects such as surface-enhanced Raman scattering
(SERS),5-7extraordinary optical transmission,8-10and robust
photonic band gaps at visible and NIR wavelengths.11-13
SERS has attracted widespread attention because of its
demonstrated potential for single-molecule spectroscopy and
chemical sensing with high information content.14-16The
rational design of optimized SERS substrates remains a
challenging goal, despite extensive efforts to elucidate the
physical basis of signal enhancement. Several theoretical
studies have described highly localized EM fields at the
junction of metal nanostructures,7,17-19with local EM
enhancement factors Gloc) |Eloc(λ)/E0(λ)|4as high as 1011-
1012for a two-particle system.20However, less attention has
been paid to the average EM enhancement factors (G h )
〈Gloc〉), which has greater relevance for the design and
optimization of SERS-based chemical sensors. In this regard,
Garcı ´a-Vidal and Pendry have provided electrodynamics
calculations on periodic nanostructured metal films with G h
values on the order of 106, a level of activity commonly
observed in many experimental systems.7
Here we provide numerical calculations and a simple
analytical theory for calculating EM field enhancements in
two-dimensional (2D) arrays of metal nanoparticles embed-
ded in a dielectric medium. The numerical simulations and
analytical values are in good agreement and yield G h values
as high as 2 × 1011for arrays of cylindrical Ag nanodisks
and 5 × 1010for arrays of Au nanodisks. Analytical values
can also be obtained for 2D arrays of metal nanospheres,
yielding G h values on the order of 2 × 109and 5 × 108for
Ag and Au nanoparticles, respectively. These activities,
which are up to several orders of magnitude greater than
those of aperiodic metal-dielectric composites or roughened
metal surfaces, are independent of morphology-dependent
resonances such as those responsible for microcavity-
enhanced SERS.21The enhancements of the nanostructured
arrays are strongly dependent on the ratio of particle diameter
to interparticle spacing, which determines both the intensity
of local field factors and the available cross-sectional area
for sampling chemical and biomolecular analytes. Our
models illustrate how the interplay between field enhance-
ment and interparticle spacing can impact the design of array-
based SERS sensors for trace chemical analysis.
Numerical calculations were first performed on 2D arrays
of cylindrical oblate disks with high aspect ratio and diameter
* Corresponding author. E-mail: alexwei@purdue.edu
†Department of Electrical and Computer Engineering.
‡Department of Chemistry.
NANO
LETTERS
2004
Vol. 4, No. 1
153-158
10.1021/nl0343710 CCC: $27.50
Published on Web 12/09/2003
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d , λ, arranged in square or hexagonal lattices in a dielectric
medium ?dusing periodic boundary conditions (see Figure
1).22Such arrays can be approximated as planar systems for
the purpose of estimating local field factors.2Here we apply
the current conservation law, which can be expressed in terms
of a local potential ?(r)
where E0is the incident electric field, σ(r) ) -iω?m/4π is
the local conductivity, and ?m) ?′m+ i?′′mis the complex
dielectric function of the metal. Describing the continuity
equation in these terms allows the collective plasmon
response to be determined under quasistatic conditions in a
scale-invariant manner.2In addition, Eloc(r)/E0 can be
calculated as a continuous function of packing density,
described by a single geometric parameter γ ) d/δ, where
d is the distance between the particles at the point of closest
approach. The parameter γ is fundamentally important,
because the resonance condition can be derived directly from
Poisson’s equation to yield the simple relationship ?′m≈
-γ?d.
It is important to note that the quasistatic approximation
is valid for nanoparticle arrays with periodicities below the
diffraction limit (λ/2). Radiative loss from elastic (Rayleigh)
scattering is negligible, and losses due to retardation effects
(a function of the finite skin depth of nanoparticles larger
than 30 nm) can be accounted for by a first-order correction
(see below).2Furthermore, for a 2D nanoparticle array where
γ . 1, the spatial localizations of the EM resonances
between particles are well within the quasistatic limit.
Discretization of eq 1 on a square-mesh lattice under
periodic boundary conditions yielded L2equations (L ) 120),
which were solved by the exact block elimination approach.23
This method provides solutions for site potentials corre-
sponding to Eloc(r)/E0in L4operations, an enormous savings
in computing time compared with the L6operations required
by Gaussian elimination methods.24G h is obtained simply as
the mean value of Gloc) |Eloc(r)/E0|4within a unit cell of
the periodic lattice. We note that these calculations are
equally valid for periodic arrays of nanowires at constant
depth as for oblate metal nanodisks with high aspect ratio,
whose electric fields and currents are confined to the plane
of the system.2
The intensities of the local and average EM field enhance-
ments depend greatly on both incident wavelength and
diameter-spacing ratio (see Figures 1 and 2 for Glocand G h
of Au nanodisk arrays at different values of γ). Au nanodisk
arrays with large diameter-spacing ratios (γ g 30) can
produce Glocvalues as high as 1010, whereas Ag nanodisk
arrays can produce Glocvalues as high as 1012. These optical
gains are the product of wavelength-selective resonant modes
within the periodically ordered arrays (see below).22With
respect to the average EM enhancements, G h can be described
as a resonance band whose width increases with γ. Arrays
with large γ can produce high G h over a greater range of
excitation wavelengths, which has important practical rami-
fications for SERS.
The highest G h values (G hmax) can be several orders of
magnitude greater than that generated by a random metal-
dielectric film at the percolation threshold (p ) pc), as well
as those produced by periodic gratings whose plasmonic
responses are at the saturation limit for continuous metal
films (see Figure 2).7In contrast, average field enhancements
outside of the resonance band are much lower than those
generated by broadband amplifiers such as disordered metal-
dielectric films. Introducing disorder into the 2D arrays also
substantially decreases G h, indicating that long-range order
is an important factor for generating high local EM resonant
enhancements, which in turn provide the greatest contribution
to G h.
The intensities of the local field enhancements that
comprise G h are also dependent on their spatial relationship
within the nanoparticle array, and also in their relationship
with the polarization of the incident wavevector. In our
calculations, values for Elocwere obtained at fixed diameter-
spacing ratios with the incident light polarized along the
x-axis, such that the local EM enhancements are greatest at
sites between particles along the direction of polarization
(see Figure 1 for γ ) 5 and 10). However, a change in the
polarization angle produces large shifts in the positions and
intensities of the maximum local field factors, with a small
reduction in the average EM enhancements (see Figure 3).
For example, G h values produced by square-lattice 2D arrays
decrease by a factor of 2 when θ changes from 0° to 45°.
The polarization angle is likely to be an important parameter
for single-molecule SERS and related spectroscopies, in
which the analytes of interest are thought to be localized
between particles.14,15,25In this scenario, the Raman signals
can be maximized by adjusting θ until the highest local fields
are coincident with the exact position of the molecule.
Figure 1. Cross-sectional view of local EM field factors pro-
duced by p-polarized light (l ) 647 nm, E0) Ex) within periodic
2D arrays of Au nanodisks embedded in a low-dielectric medium
(?d) 1.5). Two different lattice geometries are shown: (a, b) 2D
square lattice; (c, d) 2D hexagonal lattice. Two different diameter-
spacing ratios are shown: (a, c) γ ) 5; (b, d) γ ) 10. Note the
change in scale for |Eloc(r)/E0| for arrays with different values of
γ. Discretization was performed using a 120 × 120 square lattice.
∇‚[σ(r)(- ∇?(r) + E0)] ) 0(1)
154Nano Lett., Vol. 4, No. 1, 2004

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To confirm the validity of the numerical calculations under
quasistatic conditions, we have developed an analytical
approach for estimating G h. The EM coupling between metal
nanoparticles can be conceptualized as arrays of RLC circuits
across the interparticle gap, with each element i representing
a resonance defined by local spacing δi(see Figure 4).2The
negative permittivity of the metal (?′m< 0) and the dielec-
tric ?dare represented respectively by inductance R-L and
capacitance C. The RLC model suggests that the collective
plasmon resonances of the 2D nanoparticle arrays should
shift strongly toward lower frequencies (high LC) with
increasing γ, in accord with our numerical simulations and
recent calculations involving metal nanoparticle dimers (see
Figure 2).20In addition, the model is consonant with the
broadening of the plasmon bandwidths that occurs with red-
shifting, which reflects a greater degeneracy of EM states at
high γ due to the effectively greater radius of curvature at
the interface between particles.
The RLC model provides a theoretical framework for
describing changes in EM Raman enhancement as a function
of γ and ?m. Under low-loss conditions the dielectric function
can be expressed as ?m) ?′m(1 - iκ), where ?′m< 0 and loss
factor κ , 1. If γ . 1 and the surrounding dielectric ?dis
taken into account, one can derive an expression for an
effective dielectric function ?eff
where parameters W ) |?′m|/?dand ∆ ) (W/γ - 1)/κ (see
Supporting Information for more details). This composite
dielectric function reaches a maximum when ∆ f 0, so that
?eff
real and larger than ?d, such that the array response is
dominated by interparticle capacitance (C). If ∆ < 0, the
imaginary part of ?effprevails, resulting in large losses and
subsequently large fluctuations in local field. It is worth
noting that the effective absorption of the metal-dielectric
composite, ?′′eff, is proportional to?|?m|/κ near resonance,
which increases as κ goes to zero. A high effective absorption
is necessary to produce giant fluctuations in the local fields
between particles and determines the capacity of the 2D
arrays to accumulate and release EM energy.
The EM enhancement factor G h (averaged over the unit
cell, minus metal particles) can now be expressed analytically
as
max= |?′m|(1 + i)π/?4κ(W + 1). If ∆ > 1, ?eff is mostly
(see Supporting Information for more details). The maximum
EM enhancement G hmaxis also achieved when ∆ f 0; if a
Drude free-electron response is assumed in which ?′m(ω) )
1 - (ωp/ω)2and κ ) ωτ/ω, with plasma frequency ωp. ω
and relaxation frequency ωτ , ω, parameter W can be
approximated as (ωp/ω)2?d-1and G hmaxcan then be estimated
as
Figure 2. Average EM enhancements (G h) from 2D arrays of Au nanodisks as a function of incident wavelength (λ) at fixed particle
diameter-spacing ratios (γ ) 5, 10, and 30): (a) 2D square lattice; (b) 2D hexagonal lattice. A plot of G h from random metal-dielectric
films at the percolation threshold (p ) pc) is included for comparison.
Figure 3. Local EM field distributions and intensities at different
polarization angles. Left, θ ) 0° (E0) Ex); Right, θ ) 45°.
Figure 4. Frequency-dependent plasmon response depicted as an
array of RLC circuits.
?eff= |?′m|(
π
?2κ(∆ - i)(W + 1)
-(1 + π/2)
(W + 1))
(2)
G h =
π(W + 1)7/2
2((4 - π)W + 4)κ7/2×
?
4∆2+ 9
(∆2+ 1)3/2-∆(4∆4+ 15∆2+ 15)
(∆2+ 1)3
(3)
G hmax= π(ωp/??d)5ωτ
-7/2ωres
-3/2
(4)
Nano Lett., Vol. 4, No. 1, 2004 155

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in which the resonance frequency is defined as ωres ≈
ωp(γ?d)-1/2. For the case of Au nanodisk arrays, we use ωp
) 9.3 eV and ωτ) 0.03 eV26to obtain an approximate G hmax
value of 5 × 1010?d-5/2λres3/2, where λres ) 2πc/ωres is the
resonance wavelength expressed in µm. For the case of Ag
(ωp) 9.1 eV, ωτ) 0.021 eV26) we obtain a G hmaxvalue of
2 × 1011?d-5/2λres3/2, which is four times larger than that of
gold.
The analytical expressions derived from the RLC model
are in excellent agreement with the numerical calculations
of G h for the 2D nanowire arrays at different values of γ
(see Figure 5). It is particularly noteworthy that the analytical
G hmaxvalues are essentially identical with those obtained by
numerical calculations under quasistatic conditions. It should
be noted, however, that the numerical calculations exhibit
additional maxima at shorter wavelengths that are not present
in the analytical solutions. These peaks are likely related to
artifacts resulting from the discretization procedure.
For periodic 2D arrays of metal nanospheres, the analytical
expression for the effective dielectric function is rather
different from eq 2:
As in the case of metal nanodisk arrays, the imaginary part
of ?effprevails when ∆ < 0, producing large fluctuations in
local field. The average EM enhancement between nano-
spheres is then expressed as
where G hsphis further averaged in the z-dimension between
+d/2 and -d/2. Assuming a Drude free-electron response,
the maximum EM enhancement G hsph
maxcan be estimated as
in which the resonance frequency is again defined as ωres≈
ωp(γ?d)-1/2. Substituting in the same optical constants as
before, we find that the G hsph
nanosphere arrays are approximately 2 × 109?d-2λres and
5 × 108?d-2λres, respectively. The nearly two order-of-
magnitude difference between eqs 4 and 7 can be ascribed
to the change in unit particle geometry. In the case of the
2D disk arrays, plasmonic contributions to the local EM field
strength are constant with respect to depth, whereas in the
case of the 2D nanosphere array the field strength rapidly
decreases with distance from the particle surfaces. Neverthe-
less, the resonance behaviors of the two systems are entirely
analogous with respect to γ and ?eff.
The analyses thus far have been performed using quasi-
static approximations in which retardation effects are as-
sumed to be negligible. However, such effects may become
significant when the particle size or lattice constant is no
longer small compared with the incident wavelength. Here
we estimate the extent to which retardation effects influence
G as a function of lattice periodicity D, for a given diameter-
spacing ratio (γ ) 30; see Figure 6). Local field factors were
calculated by solving the coupled-dipole equations both with
and without retarded potentials for very large square-lattice
arrays of Au nanodisks (N∼106; see Supporting Information
for more details).27A comparison of these factors reveals
no effects of retardation on Glocfor periodicities up to 40
nm, after which a gradual decrease ensues. Even so, the
retarded G values remain relatively high, indicating that
periodic order can significantly reduce radiation losses in
arrays of particles which otherwise experience significant
retardation effects.
Finally, we collate the theoretical relationship between EM
enhancement and periodic structure with some recent ex-
perimental SERS measurements, taken from hexagonally
close-packed 2D arrays of colloidal gold nanoparticles whose
diameter-spacing ratios range from approximately 15-100
(see Figure 7).5The empirical Raman signal enhance-
ments (GSERS) are amplified by local EM field factors and
are proportional to G h with the modification that GSERS )
|E(λ)/E0(λ)|2|E(λ′)/E0(λ′)|2, where λ and λ′ are the incident
and Stokes-shifted wavelengths, respectively.2,28In our
theoretical considerations we take into account the skin effect
by renormalizing the dielectric permittivity (see Supporting
Information for more details). We also take into account the
loss of incident field due to reflection and absorption at the
maxvalues for Ag and Au
Figure 5. Analytical solutions of average EM enhancements (G h)
as a function of λ for 2D square-lattice arrays of Au nanodisks and
nanospheres (solid and dashed curves, respectively), calculated for
γ ) 5 and 10. The corresponding numerical solution for the 2D
nanodisk array (dotted curve) is included for comparison.
?eff,sph= 2?d[(1 +
κ∆
W + 1)log(
W + 1
κ(∆ - i))- 1]
(5)
G hsph=3(W + 1)3
(W + 3)κ3[
π
2-
∆
1 + (∆ + κ)2- tan-1(∆ + κ)]
(6)
G hsph
max= (3π/2)ωp
4?d
-2ωτ
-3ωres
-1
(7)
Figure 6. Ratio of Gloc values (located midway between gold
nanodisks) calculated with and without retardation effects as a
function of lattice period D. Nanoparticles are arranged in a square-
lattice array with γ ) 30 and incident wavelength λ ) 1 µm.
156Nano Lett., Vol. 4, No. 1, 2004