Structural Eﬀects in the Electromagnetic Enhancement ...€¦ · understanding of the electromagnetic interactions between nano-particles. It is widely accepted that a major component

Structural Effects in the Electromagnetic Enhancement Mechanismof Surface-Enhanced Raman Scattering: Dipole Reradiation andRectangular Symmetry Effects for Nanoparticle ArraysLogan K. Ausman, Shuzhou Li, and George C. Schatz*

Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, United States

*S Supporting Information

ABSTRACT: Surface-enhanced Raman scattering (SERS) enhancement factors for Ag andAu sphere array structures are determined by rigorously including dipole reradiation in aT-matrix formalism. Comparisons are made with the more commonly used local fieldenhancement due to plane-wave excitation, |E(r0;ω)|

2|E(r0;ωs)|2 which for zero Stokes shift is

|E(r0;ω)|4 to determine the errors associated with this approximation. Substantial errors

(factors of 10−100) are found for the peak enhancements at a scattering angle well awayfrom the incident direction, but for backscattering, the errors are negligible. We also present|E|4 enhancement factors using a periodic boundary discrete dipole approximation methodfor several metal strip array structures, and we show that a certain combination of rectangulararray structure and strip properties leads to electromagnetic enhancement factors for mixedphotonic-plasmonic resonances that are considerably higher than can be produced witheither square arrays or 1-D arrays based on the same particles and spacings.

I. INTRODUCTION

Since the discovery of surface-enhanced Raman scattering (SERS)in the 1970s1−3 there has been much interest in the design andfabrication of SERS substrates that optimize the enhancementfactor (EF). This has led to applications in biomolecule4−8 andchemical sensing9−11 with specific examples in the analysis ofanthrax,5,7 half-mustard agent,9 pigments in art restoration,10,11

and glucose.4,6,7 Further improvements in substrate fabricationhave been stimulated by the discovery of single-moleculeSERS.12−14

The prospect of employing SERS in sensing applications hasresulted in interest in nanoparticle array structures that supportSERS, as such structures are readily fabricated by a variety oftechniques, including soft-lithography methods,15 that lendthemselves to mass production. However the design of viablearray structures to be used as SERS sensors requires a thoroughunderstanding of the electromagnetic interactions between nano-particles.It is widely accepted that a major component of the SERS

enhancement is the result of locally enhanced electric fieldsproduced by the substrate at the location of the molecule. Themost common substrates are metal nanoparticles, usually Au orAg, and the local electric field enhancements arise because oflocalized surface plasmon resonance (LSPR) excitation6,16−19 inthese particles when they interact with light. Because the LSPR isa collective oscillation of conduction electrons, it behaves largelyaccording to the laws of classical electrodynamics. Therefore, itmakes sense that most models of SERS are purely classical,although there are examples in the literature that incorporatequantum mechanical methods.20−24

Past interest in nanoparticle array structures has includedstudies of 1-D linear chains and 2-D arrays that are both irregularand regular.25−44 In the papers by Zou and Schatz, linear chainsof spheres and sphere dimers were studied using both T-matrixtheory and the coupled dipole approximation. They found thatlong-range photonic resonances in the 1d array led to narrowfeatures in the extinction spectrum that are associated with a largeenhancement in the electric field near the particles. In onepaper,45 they observed that these photonic resonances (orphotonic lattice modes) added an additional factor of 17 to theenhancement over the isolated particle case. Zhao et al.conducted a similar study of EFs for a 1d array of silver nanoshelldimers.31 They observed an additional order of magnitudeenhancement of the optimal dimer array with respect to anisolated dimer. This work was later extended to 2d arrays ofnanoshell dimers by Song et al.33 where the results were similar tothe 1d array findings. Optimization of the EFs for mixedphotonic-plasmonic resonances through the use of rectangulararrays and anistropic nanoparticles has not been considered.Because the majority of classical electrodynamic methods

involves the scattering of an incident plane polarized wave it isnot surprising that the existing models of SERS due to substratesmade of arrays of nanoparticles have been based on the plane-wave (PW) approximation.27,28,45,46 This approximation as-sumes that the SERS EF is the product of the electric fieldintensity at the incident frequency, |Eloc(r0;ω)|

2, and the electricfield intensity at the Stokes shifted frequency, |Eloc(r0;ωs)|

2, both

Received: December 20, 2011Revised: July 12, 2012Published: July 12, 2012

Article

pubs.acs.org/JPCC

© 2012 American Chemical Society 17318 dx.doi.org/10.1021/jp2122938 | J. Phys. Chem. C 2012, 116, 17318−17327

pubs.acs.org/JPCC

evaluated at the molecular location. This means that |Eloc(r0;ω)|2

and |Eloc(r0;ωs)|2 are obtained from calculations that involve PW

scattering from the substrate. Often an additional assumption ismade that the plasmon resonance occurs over a broad range offrequencies and that for small Stokes shift the enhancement canbe taken to be |Eloc(r0;ω)|

4; in this article we refer to this case asthe PW approximation.An infinite PW is a reasonable model for most sources of

illumination in real experiments; however, the reradiation field bythe dipole induced in the molecule at the Stokes shiftedfrequency can be considerably different from that of a PW.Because the molecule is much smaller than the metal particles itinteracts with, it is appropriate to model this field as that arisingfrom a point dipole. There have been a few studies that incor-porate dipole reradiation (DR)47−51 rigorously into the evalua-tion of electromagnetic EFs, but the great majority including allstudies of nanoparticle arrays have used the PW approximation.However the comparisons between DR and PW results showsignificant errors in the PW approximation for certain scatteringgeometries, providing motivation for doing further studies of DReffects. For example, for a spherical dimer system,52 it wasobserved that the agreement between PW and DR results ishighly dependent on both detector and molecule location. Thesedifferences were found to depend on interferences between thedirectly emitted field and the field that scatters from thenanoparticles and on effects that arise from excitation of higherorder angular momentum components (multipole resonances)in the plasmon excitation of the nanoparticles. It can be expectedthat similar but more complicated processes will play a role inSERS on array structures, but this has not been considered so far.Here we model SERS for a molecule in the presence of a

substrate comprised of an array of Ag or Au nanospheres byemploying a T-matrix method that rigorously incorporates DR,and we compare this with the corresponding results of PWcalculations. From these results, we are able to establish when thePW approximation is adequate and when not. On the basis ofthese results, we then use the PW approximation within thecontext of the discrete dipole approximation (DDA) approachwith periodic boundary conditions, and we show how theparameters of the DDA calculation may be chosen such thatDDA and T-matrix results are in good agreement. Then, with theperiodic DDA method, we study mixed photonic/plasmonicresonances for 1-D and 2-D metal strip array structures, andwe find that for rectangular arrays with a 2:1 ratio of arraydimensions the SERS EFs are larger than those for square arraysand considerably larger (factor of >10) than can be obtainedfrom 1-D arrays or from isolated strips. Chemical contributionsto the SERS EF have been ignored in this work, and the arrays areassumed to be in a homogeneous dielectric, such as can beachieved by index matching if the array is on a surface.The T-matrix and DDA theories are outlined in Section II,

whereas Section III presents the results for arrays of spheres andfor rectangular metal strip arrays. In Section IV we present someconcluding remarks.

II. THEORY

A. Including Dipole-Reradiation in the T-MatrixMethod. The model that we employ here is based on theT-matrix method for PW scattering from an array of sphericalparticles but modified to include reradiation from a dipolesource. A thorough treatment of the T-matrix method is givenelsewhere.53−55 The basis for the T-matrix method is the Mie

theory framework, which employs a basis of vector sphericalharmonic functions to represent an arbitrary field. That is

∑ ∑= +σ σ

=

∞

=−

p k q kE N r M r[ ( ) ( )]l m l

l

lm lm lm lm1

( ) ( )

(1)

with coefficients plm and qlm, and vector spherical harmonic basisfunctions N and M that are solutions of the vector Helmholtzequation and are identically 0 for l = 0. In spherical polarcoordinates these basis functions are

= ∇ ×σ σkk

kN r M r( )1

( )lm lm( ) ( )

(2)

= ∇ ×σ σk u kM r r r( ) { ( )}lm lm( ) ( )

(3)

and the scalar function ulm(kr) satisfies the scalar Helmholtzequation. In spherical polar coordinates this is

θ

σ

σ

=

==

=

σ σ ϕ

σ⎧⎨⎪⎩⎪

u k z kr P

z krj kr

h kr

r( ) ( ) (cos )e

( )( ), for 1

( ), for 3

lm l lm im

ll

l

( ) ( )

( )(1)

(4)

The function zl is a spherical Bessel function that is chosen tosatisfy the appropriate boundary conditions of the scatteringproblem; that is, jl is a spherical Bessel function and hl

(1) is aspherical Hankel function. The Pl

m are associated Legendrefunctions and together with the exponential these form the scalarspherical harmonics.To incorporate a dipole electric field source in a vector

spherical harmonic basis, we begin by noting that the electric fieldof an oscillating induced dipole, p, can be obtained by using therelation

∫

∫

ω π ω ω ω

πεμ

ω δ

πεμ

ω

= ′ · ′ ′

= ′ · ′ − ′

= ·

ic

r

kr

k

E r G r r J r

G r r p r r

G r r p

( ; )4

( , ; ) ( ; ) d

4( , ; ) ( ) d

4( , ; )

2 03

2

0 03

2

0 0(5)

where in the above G0 is the free-space tensor Green’s functionthat can be obtained in a vector spherical harmonic basis via theOhm-Rayleigh method56 and is

∑ ∑ωπ

= − ++

+

> <=

∞

=−∓ > ± <

∓ > ± <

ik ll l

k k

k k

G r r N r N r

M r M r

( , ; )4

( 1)2 1( 1)

[ ( ) ( )

( ) ( )]

l m l

lm

l m l m

l m l m

01

(3) (1)

(3) (1)(6)

where |r>| > |r<|. When the observation point is at r<, then the topsign in front of them is taken. The bottom sign in front of them istaken when the observation point is r>. The first choice results inthe following expressions for the field coefficients for a dipolesource field incident on a single sphere

εμ= − +

+·−p

ik ll l

kN r p( 1)2 1( 1)

( )lmm

l m

3(3)

0(7)

εμ= − +

+·−q

ik ll l

kM r p( 1)2 1( 1)

( )lmm

l m

3(3)

0(8)

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp2122938 | J. Phys. Chem. C 2012, 116, 17318−1732717319

This single-sphere result can then be added into the T-matrixformalism, as has been previously done to incorporate theelectromagnetic coupling interactions between multiplespheres.47 The resulting electric field can then be used tocompute the SERS EF, which is

θ ϕθ ϕθ ϕ

=G rF r

F r( ; , )

( ; , )

( ; , )s ss s

s sDR 0

0

0 0

2

(9)

where F is the far-field amplitude of the SERS electric field,expressed as

θ ϕ =→∞

rkF r E r( ; , )

e( )s s

sikr s

kr0 SERS

ss (10)

and F0 is the far-field amplitude of the Raman scattered light froman isolated molecule.Eq 10 can also be used to develop a theory that describes

interference effects that arise in DR but not in the PWapproximation. Equation 10 can be rewritten in the simplifiedform

=r

E Feikr

(11)

and for a collection of spheres, one can decompose the field intooutgoing spherical wave contributions from each sphere andfrom the molecular dipole emitter. Therefore, the overall far-fieldamplitude behaves as

∑= −rr

ik r rF F exp( [ ])i

s

ii i stot

(12)

where ri is the distance from the center of the ith sphere, or themolecule, to the observation point, and rs is the distance from theuniversal origin to the point of observation. Eq 12 and eq 9indicate that for an array of spheres there can be interferencepatterns in the far-field amplitudes of the form

*· − − *· −k r r k r rF F F FRe{ } cos( [ ]) Im{ } sin( [ ])i j j i i j j i

(13)

where i ≠ j. These phase interference effects contribute todifferences between the PW and DR results because PW onlyconsiders near-field enhancements where phase interferences donot occur. As we will see below, these interference effects averageout when the nanospheres are arranged symmetrically withrespect to the detector location such as a backscatteringgeometry, where emissions from the different particles, and themolecular dipole, are all in-phase.B. Discrete Dipole Approximation with Periodic

Boundary Conditions. In the DDA method, the particles arereplaced by cubic dipole arrays with polarizabilities that aredetermined by the dielectric constant of the particles. Althoughthis is an approximate method that does not precisely satisfyMaxwell’s equations, calculations of the optical properties ofcollections of finite metallic nanoparticles using DDA lead toresults that are in good agreement with experimental measure-ments.57−59 Recently, the DDA method has been extended totreat periodic structures,60 which we outline in the following. Westart by noting that a monochromatic incident PW electric fieldcan be described as

ω= · −t i tE r E k r( , ) exp[ ( )]inc 0 0 (14)

The extension to periodic structures60 is made by considering acollection of N polarizable points that define a target unit cell(TUC). We note that the index j = 1, ..., N runs over all of thedipoles in a single TUC. Now let the indices m, n run overreplicas of the TUC in a 2-D periodic array of identical TUCs, sothat the (m,n) replica of dipole j is located at

= + +m nr r L Ljmn j u v00 (15)

with Lu and Lv being the lattice vectors for the array. In thepresence of a PW field, the polarizations of the dipoles in thetarget will oscillate coherently. Each dipole will be affected byinteractions with the incident electric field and also interactionswith the electric fields generated by all of the other point dipoles.The replica dipole polarization Pjmn(t) is phase-shifted relativeto Pj00(t)

= · +t t i m nP P k L L( ) ( ) exp[ ( )]jmn j u v00 0 (16)

We define a matrix A such that−Aj,kmnPkmn gives the electric fieldE at rj00 produced by an oscillating dipole Pkmn located at rkmn.The 3 × 3 tensor A depends on the target geometry andwavelength of the incident radiation but not on the targetcomposition nor on the direction or polarization state of theincident wave.Using the relationship between rjmn and rj00, we may construct

a matrix A such that for j ≠ k −Aj,kPk00 gives the electric field atrj00 produced by a dipole Pk00 and all of its replica dipoles Pkmn.For j = k it gives the electric field at rj00 produced only by thereplica dipoles

∑ ∑ δ δ δ = − · +=−∞

∞

=−∞

∞

i m nA A k L L(1 ) exp[ ( )]j km n

jk m n j kmn u v, 0 0 , 0

(17)

where δij is the Kronecker delta. As |m| and |n| approach ∞, thelocation j00 is in the radiation zone of dipole kmn, and theelectric-field magnitude dependence is 1/r. The sums in eq 17would be divergent were it not for the oscillating phases of theterms, which ensure convergence.Eq 17 is evaluated numerically by introducing a factor

exp[−(γk0r)4] to suppress smoothly the contributions fromlarge r and truncate the sums

∑ γ ≈ ′ · + −i m n k rA A k L Lexp[ ( ) ( ) ]j km n

j kmn u v j kmn,,

, 0 0 ,4

(18)

where rj,kmn = |rkmn−rj00| and the summation is over (m,n) withrj,kmn ≤ 2/γk0, that is, out to distances where the suppressionfactor exp[−(γk0r)4] ≈ e−16. More terms are included in thesummation when the interaction cutoff parameter γ is small.The polarizations Pj00 of the dipoles in the TUC must satisfy

the system of equations

∑α= − ≠

P E r A P[ ( ) ]j j jk j

j k k00 inc , 00(19)

If there are N dipoles in one TUC, then eq 19 is a system of 3Nlinear equations where the polarizability tensors, αj, are obtainedfrom lattice-dispersion theory.61 After A has been calculated, eq19 can be solved for Pj00 using iterative techniques when N≫ 1.The electric fields near the target can be calculated by the exact

expression for E from a point dipole, modified by a function ϕ



∑ ∑ω

ϕ

= · − + ′

× × ×

+−

· −

ω−

⎪

⎪

⎪

⎪

⎧⎨⎩

⎫⎬⎭

t i t

ik R

RR k

ik R

RR

E r E k r

R P R

R R P P

( , ) exp[ ( )] e

exp( )( ) ( )

(1 )[3 ( ) ]

i t

j m n

jmn

jmnjmn jmn jmn jmn

jmn

jmnjmn jmn jmn jmn jmn

0 0,

002

02

2

≡ −R r rjmn jmn

ϕ γ≡ − ·≥

<⎪

⎪

⎧⎨⎩R k R

R d

R d R d( ) exp[ ( ) ]

1 for

( / ) for0

44

The function ϕ(R) smoothly suppresses the oscillatingcontribution from distant dipoles to allow the summations tobe truncated. When r is near the target surface, the summationsover (m,n) are limited to |Rjmn| ≤ 2/γk0. The (R/d)4 factorsuppresses the R−3 divergence of E as r approaches the locationsof individual dipoles and at the dipole locations results in E that isexactly equal to the field that is polarizing the dipoles in the DDAformulation. The value of γ needs to be chosen with some care, assmaller values lead to large computational effort, whereas largervalues lead to less accurate results. Later we show that bycomparing DDA and T-matrix theory results it is possible todetermine the largest value of γ such that the two sets of resultsare in good agreement.For the metal strip array studies, DDA calculations were

performed using DDSCAT7.0.62 The grid spacing is 2 nm in allcalculations. The dielectric constants for both gold and silver arefrom Johnson and Christy.63 The DDA electromagnetic EF inSERS is taken to be g4 (with g = |E|/|E0|), where E and E0 are thelocal and incident fields, respectively, at the particle surfaces. Theprecise location of the surface is described later.

III. RESULTSA. Dipole Reradiation Effects for 2d Arrays of Spheres.

Just as with single-sphere chains,26−28,30,31,45 2d arrays of nano-particles can also support photonic resonances. The differencebetween the 1d chains and 2d arrays is that there can now beinteractions between particles that are no longer normal to theincident field position, and as a result, the intensities are modifiedfrom the 1d case. Here we present arrays that are made up ofspheres placed on a rectangular grid. We present the arrange-ment in Figures 1 and 2 and define the distance dx to be thecenter-to-center distance between neighboring spheres along thex direction (axis labeled E) and dy to be similarly defined forspheres along the y direction (axis labeledH).We consider arraysmade up of 112 nm diameter spheres on a 40 × 10 grid. Thischoice of sphere diameter was dictated by a result from Zouet al.,27 who found that large spheres (∼100 nm) are optimal forstrong plasmon/photonic resonance coupling. The maximumnumber of particles we can consider is limited by computationalconsiderations to 400, so our goal in choosing a 40 × 10 grid wasto include 40 particles in the y direction so that photonicresonances would be well-defined (as was clear from our previousstudies of 1-D arrays27) and 10 particles in the x direction so thatwe could see what effect that the x dimension has on the results.We also consider two different materials for the spheres, whereone set of arrays contains all Ag spheres and the other contains allAu spheres.63 To make the photonic resonances be to the redof the plasmonic resonances, we will fix dy at 450 nm for Ag and

Figure 1. Scheme of the relevant molecule and detector coordinates andthe arrangement of the spheres for 2-D arrays. The actual arraysemployed contain 40 spheres along the y(H)-direction and 10 spheresalong the x(E)-direction.

Figure 2. Depiction of a 2-D array showing the location (in red) of thesphere at the origin of the overall coordinate system. The molecule islocated on this sphere along the axis parallel with E. We note that thedimensions are not to scale but have been condensed.



540 nm for Au while varying dx. Also note that in this section ofthe paper the medium surrounding the spheres is assumed to bevacuum.To calculate SERS enhancements, we examine a molecule

located along the E-field direction 0.25 nm from the surface ofthe sphere at the origin, as previously described. This gives us EFsthat are appropriate for a single-molecule SERS experiment.Later in this paper we also consider what happens when weaverage over all locations on the surface of the particles.The extinction spectra are presented in Figures 3 and 4 for Ag

and Au, respectively. Figure 3 shows broad maxima at 360 and

385 nm that correspond to the single-particle plasmon reso-nances (quadrupole and dipole, respectively) for the Ag particleswe are considering. In addition, there is a sharp maximum at465 nm (slightly shifted from the 450 nm spacing) that corre-sponds to the photonic resonance that is coupled to the tail of theplasmon resonance. In this case, the extinction is maximized for adx roughly equal to the value of dy, but there is little variation ofthe extinction with dx. For Au, the results in Figure 4 show a peaknear 560 nm (also shifted from the 540 nm spacing), corre-sponding to a case where the single-particle plasmon resonanceand the photonic resonance are nearly coincident. In this case, wesee that the magnitude of the peak extinction efficiency variessomewhat with dx, with a maximum when dx is roughly 1/2 dy.Note also that the extinction peak for the Ag spheres is alsonarrower than that for the Au spheres, and as a result the maxi-mum extinction is larger for Ag. This is due to the presence of

interband transitions in Au that are more important (for thewavelengths considered) than for Ag. Also, the resonancefrequencies of the Ag arrays do not shift with dx, whereas there isa slight red shift in Figure 4 as the Au spheres are brought closertogether. To explain this, we note that the photonic resonance isat 465 nm (2.67 eV) for the Ag sphere in Figure 3, whereas thedipole plasmon resonances are at 385 (3.22 eV) and 405 nm(3.06 eV) for single sphere and arrays, respectively. This meansthat the energy gaps between photonic resonance and dipoleplasmon resonances are 0.55 or 0.39 eV for single sphere andarrays, respectively. The counterparts for the gold sphere andarrays are 0.20 or 0.17 eV in Figure 4, so the energy gaps aresmaller and therefore the coupling of resonances will be stronger.A consequence of this is that for Ag the photonic resonance ismore perfectly geometrical in nature, meaning that interparticlespacings in both directions should be the same, whereas for Au,the mixing of plasmonic and photonic resonances changes withdx, leading to the red shift noted above.When examining the SERS enhancement, we consider two

detector locations. The first is for a backscattering arrangement(θs = π) where we have typically seen good agreement betweenPW and DR results.64 The other location is for a detector at θs =7π/18 and ϕs = π/2, which corresponds to a location that is 70°from the surface normal of the array with components along kand H. Whereas this second detector location is somewhatarbitrary, it is not untypical of directions used for widefieldmeasurements, and based on a previous work,64 we expect to seedifferences between PW and DR for this location.Figure 5 presents the SERS excitation profiles for the Ag array

with dy and dx both 450 nm (close to values that produce the peak

enhancement). The molecule is located 0.25 nm away from thesurface of the sphere depicted in Figure 2. The Figures showmany similar features to the extinction spectra, with maximumEFs on the order of 106. We note that this is about one order ofmagnitude larger than the largest EF for a 1d chain of 40 spheresof the same diameter along H and is 103 times larger than theisolated single sphere EF (see results presented in SupportingInformation). We also see that for both detector locationsconsidered, the DR result is different from PW in the regionbetween 325 and 375 nm. This is at least partially due todifferences in the multipoles that are excited by plane-waves withrespect to multipoles that are accessed by a dipole emitter.Between 360 and 370 nm, the backscattering configuration

Figure 3. Extinction spectra of 40 × 10 arrays containing 112 nmdiameter Ag spheres. The value of dy is fixed at 450 nm.

Figure 4. Extinction spectra of 40 × 10 arrays containing 112 nmdiameter Au spheres. The value of dy is fixed at 540 nm. Figure 5. SERS excitation profile for 112 nm Ag sphere 40 × 10 array

when dx = 450 nm and dy = 450 nm. The PW result is in black, thebackscattering result is red dashed, and the 70° from normal result is theblue dashed-dotted line.



results are in close agreement between DR and PW. This changeswhen the detector is located at θs = 7π/18 and ϕs = π/2. At thislocation, the differences between DR and PW are significant dueto phase interference effects that were previously described,52 asshown in Figure 6. In this case, the phase interference effects

diminish the magnitude of the photonic effect, and the result is amaximum EF that is lower than is obtained from a backscatteringconfiguration.The error associated with the PW result at the maximum

enhancement in Figure 5 for dx = 450 nm is 1.5% in thebackscattering case, but this increases to 6400% when thedetector orientation is θs = 7π/18 and ϕs = π/2. Although thereason for the large errors observed for the second detectorlocation is due to the same effects described elsewhere,52 the 2dnature of the array means that there are more of theseinteractions and the errors are larger.Now we investigate the SERS excitation profiles for an array

made from 112 nmAu spheres with dx = 280 nm and dy = 540 nm(parameters that lead to the largest enhancements). The resultsare shown in Figure 7. The maximum EF for Au spheres is ∼105,

which is an order of magnitude smaller than was seen for Agspheres. This enhancement is still one order of magnitude largerthan a 40 sphere chain of 112 nm Au spheres and is 102 times

larger than the isolated single sphere result. The Figure shows asimilar trend to the results for the Ag particles near the photonicresonance, but these results are in better agreement with eachother in the blue region of the spectrum. This is likely due to theinterband transitions in Au that mask out any photonic effects atshorter wavelengths. For the wavelength that gives the maximumEF, the PW result closely matches the backscatteringconfiguration DR result, with the error for dx = 280 nm being2.9% in Figure 7. This error increases to 1400% when thedetector is located at θs = 7π/18 and ϕs = π/2. The errorsassociated with the Au sphere arrays are similar to those seenfromAg sphere arrays, which confirms that the source of error is afar-field effect and not due to modifications from the dipoleemission or a property of the material composition of thespheres.As was previously mentioned, one difference between the Ag

and Au sphere arrays is that for Ag the enhancement ismaximized when dx is close to dy, whereas for Au dx is about 1/2dy. PW results that show this effect for Ag and Au are presented inthe Supporting Information, and it is found that for Ag the dx = dypeak enhancement is nearly a factor of 10 higher than dx = 1/2 dy,whereas for Au the dx = 1/2 dy result is three times higher. Thisarises because the resonance frequency of Ag spheres is relativelyinsensitive to dx, whereas the resonance frequency of Au spheresshows a significant red shift as the spheres are pushed together asa result of stronger mixing of plasmonic and photonic resonancesfor Au. This red shifting leads to larger enhancements due tostrong variation of the imaginary dielectric constant withwavelength, whereas silver does not show this behavior.

B. Electric-Field Enhancements Due to Au Strip Arrays.In this section, we first validate the DDA method for arrays ofparticles by comparing DDA and T-matrix results for the 2-Dsilver sphere array considered in Section IIIA. Here we comparethe average PW EFs instead of maximum EFs to reduce thenumerical uncertainty in the results. The average valuedetermined from T-matrix theory was evaluated by calculating|Eloc|

4 at discrete θ and ϕ points 0.25 nm from the marked spherein Figure 2. The total number of points included in the averagewas 3171. To model Raman measurements for the nanostruc-tures, the EF is calculated using EF = ∫ g4dS/∫ dS, which averagesthe electromagnetic enhancement over the available surface area.The diameter of the spheres is 112 nm, and both dx and dy are450 nm in this array. Because of computational constraints, theT-matrix calculations refer to a 40 × 10 array, whereas the DDAcalculation is for an infinite array, so perfect agreement is notexpected. The surrounding medium for this comparison isassumed to be vacuum.In Figure 8, the blue line represents the average EFs from the

T-matrix method, which defines a benchmark for this system. Allother lines in this Figure are from DDA calculations usingdifferent values of the interaction cutoff parameter γ. For thisarray system, the EF curves for both the DDA and T-matrixmethods show two sharp peaks, one near 360 nm correspondingto the single particle plasmon resonance and one near 460 nmcorresponding to the photonic resonance. The peak positionsfrom two methods are almost the same when γ in the DDAcalculations is taken to be 0.01. At this γ value, the cutoff radius ofthe summation in eq 17 is about 32 times larger than the incidentwavelength. The excellent agreement between these twomethods for the 460 nm resonance gives us confidence that wecan extend the DDAmethod to the gold strip arrays that we nowconsider. We also note that Au has broader resonances that are atlonger wavelengths thanAg, so the differences in results in Figure 8

Figure 6. Schematic showing the induced dipole in the molecule and thedipole induced in the particle by the molecule. The interference ofemission between these dipoles determines the dependence of the far-field emission on detection angles.

Figure 7. SERS excitation profile for 112 nm Au sphere 40 × 10 arraywhen dx = 280 nm and dy = 540 nm. The PW result is in black, thebackscattering result is red dashed, and the 70° from normal result is theblue dashed-dotted line.



for Ag are likely to be less important for Au. The value of γ wasfixed at 0.01 in the DDA results that we present.The comparison of the intensities for the 360 nm resonance

shows less good agreement for the localized plasmon resonance.Here we note that the T-matrix value is calculated 0.25 nm awayfrom the surface, whereas the DDA value is half the grid spacingfrom the surface. In Figure 8, the DDA grid spacing is 2 nm, so

the field is calculated 1 nm away from the surface. The DDA fieldincreases with decreasing grid spacing, so one would expect thetwo methods to be different given the difference in grid spacings(and this would be less important at 460 nm, where theresonance is more delocalized). We did some test calculationswith different grids that suggest that the agreement would beimproved if the electric fields were calculated at the sameposition. However, it is not feasible to choose a grid spacing of0.5 nm for the systems we are considering using the DDA sothere is not a way of providing a quantitative result.To examine the effect of nanoparticle anisotropy on light

scattering from arrays, we examine electromagnetic scatteringdue to a single Au strip and to the Au strip arrays that are picturedin Figure 9. This provides a convenient platform for studyingarray effects because such strips can easily be fabricatedexperimentally.41 Also, the plasmon wavelength for strips canbe tuned to sufficiently long wavelengths by varying structure sothat Au has SERS EFs that are comparable to those for Ag. In thepresent study, we chose the strips to be 200 nm long, 50 nmwide,and 20 nm thick and with ends that are rounded to result inparticles that are shaped like a hockey rink. For such particles, ifthe incident electric field is polarized along the length of the stripsand the gold strip is assumed to be in a homogeneous envi-ronment with refractive index of 1.33, then its dipole resonance isat ∼1232 nm, which is a wavelength relevant to NIR SERSmeasurements.For a single gold strip, we find that the average EF is 5 × 105 at

the incident wavelength of 1232 nm. For a 1-D array of thesestrips, with array axis perpendicular to the long side of strips(denoted 1-D array (I) in Figure 9), another peak shows up thatis due to the photonic resonance of the array structure. The newresonance is at 1272 nm when the center-to-center distance is940 nm, which is roughly the incident wavelength divided by therefractive index of 1.33. This observation is consistent withprevious studies.27 The average EF for this kind of 1-D arrayvaries with the center-to-center distances, as shown in Figure 10a.The average EF reaches a maximum of 5 × 106 for a center-to-center distance of 940 nm. This value is 10 times larger than thatof a single gold strip. For a 1-D array whose axis is parallel to thelong side of strips (denoted 1-D array (II)), no new peaks appearin its extinction spectra. The average EF for this kind of 1-Darray also varies with the center-to-center distance, as shown inFigure 10b. However, the EF value is only slightly larger than thatof a single gold strip even at the optimized center-to-centerdistance of 930 nm.

Figure 10. (a) Dependence of average EF on interparticle distance A in 1-D array (I). (b) Dependence of average EF on interparticle distance B in 1-Darray (II). The polarization of incident light is along the axis of gold strips.

Figure 9. Scheme of 1-D gold strip arrays and 2-D arrays. Gold strips are200 nm long, 50 nm wide, and 20 nm thick.

Figure 8. Average enhancement factors of a square array of silverspheres with different cut off value in DDA calculations. The diameter ofspheres is 112 nm. The center-to-center distances in both directions are450 nm.



Now consider 2-D arrays of strips, with center-to-centerdistances denoted as A and B corresponding to directionsperpendicular to and parallel to the long side of strips,respectively, as noted in Figure 9. In Figure 11a, the maximumextinction efficiency is plotted as contours versus the distances Aand B. Figure 11b shows the corresponding results for the EF. Ingeneral, the extinction efficiency is larger when either distance Aor distance B approaches 900 nm. However, the maximumextinction, and the maximum enhancement, occurs for an arraywithA of 1000 nm and B of 500 nm rather thanA = B= 900 nm asmight be expected. In addition, the average EF of this array has amaximum value of 5× 107, which is 100 times larger than that of asingle gold strip and 10 times larger than the optimum 1-D array.The extinction spectra of the single strip, the optimized two

sets of 1-D strip arrays, and the optimized 2-D strip array areshown in Figure 12a. Contours of the electric fields around theparticles are presented for the single strip, optimized 1-D array,and optimized 2-D array in Figure 12b−d. These Figures showthat the spatial profile is the same for the different structures butwith much greater intensity as one goes from isolated particle to1-D to 2-D structures.

These results are very much analogous to what we found forthe Au sphere structures, where the optimized rectangular 2-Darray structures had higher EFs than those for the 1-D arrays,with particle spacings that correspond to approximately theexpected photonic resonance condition in one direction but toabout half that distance for the other direction.

IV. CONCLUSIONS

This article has presented a study of SERS electromagnetic EFsfor 2-D arrays of nanoparticles in which we have probed crucialaspects of the meaning of the EF and how it can be optimized byvarying array structure. One aspect was the importance ofDR effects for these structures, where we found that phaseinterference effects and multipole resonances can in some caseslead to important differences (factors of 10−100) betweenresults calculated using the rigorous DR expression and the moreapproximate PW expression. Fortunately, for the special case ofback-scattered measurements, where the incident wavevectorand scattered wavevector are parallel, the local PW approx-imation is adequate.

Figure 11. (a) Extinction efficiency and (b) average EF for localized surface plasmon resonances of 2-D gold strip arrays.

Figure 12. (a) Extinction spectra of single gold strip, 1-D array (I), 1-D array (II), and 2-D array. The interparticle distances of the 1-D array (I) and 1-Darray (II) are 940 and 930 nm, respectively. The interparticle distances A and interparticle distance B of the 2-D array are 1000 and 500 nm, respectively.(b−d) EF contour plots of single strip, 1-D array (I), and 2-D array. The incident wavelengths are 1232, 1272, and 1360 nm, respectively.



We also demonstrated that photonic resonances in 2d arrays ofparticles can result in EFs that are an order of magnitude largerthan that of an optimized 1-D array and several orders ofmagnitude larger than for the corresponding isolated particle.Surprisingly, for some of the arrays, the interparticle spacings forthe optimized 2-D array correspond to rectangular rather thansquare array structures in which the spacing perpendicular to thefield polarization direction satisfies the photonic resonancecondition but that parallel is about half of the resonancecondition spacing. Whereas these additional enhancements forphotonic resonances can be quite large, we note that these effectswill be smaller when nonzero Stokes shifts are included in theRaman intensity evaluation.

■ ASSOCIATED CONTENT

*S Supporting InformationMaximum enhancement factor for isolated 112nm Ag and Auspheres with molecule 0.25 nm from surface and comparison of|Eloc|

4 when dx ∼ 1/2 λ (solid black) and dx ∼ λ (red dashed) in40 × 10 112 nm Ag and Au sphere arrays. This material isavailable free of charge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATION

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

This work was supported by AFOSR MURI grant FA9550-11-1-0275 and by the Northwestern University Materials ResearchCenter (NSF grant DMR-1121262).

■ REFERENCES(1) Albrecht, M. G.; Creighton, J. A. J. Am. Chem. Soc. 1977, 99, 5215.(2) Fleischmann, M.; Hendra, P. J.; McQuillian, A. J. Chem. Phys. Lett.1974, 26, 163.(3) Jeanmaire, D. L.; Van Duyne, R. P. Electroanal. Chem. 1977, 84, 1.(4) Shafer-Peltier, K. E.; Haynes, C. L.; Glucksberg, M. R.; Van Duyne,R. P. J. Am. Chem. Soc. 2003, 125, 588.(5) Zhang, X.; Young, M. A.; Lyandres, O.; Van Duyne, R. P. J. Am.Chem. Soc. 2005, 127, 4484.(6) Dieringer, J. A.; Mcfarland, A. D.; Shah, N. C.; Stuart, D. A.;Whitney, A. V.; Yonzon, C. R.; Young, M. A.; Zhang, X.; Van Duyne, R.P. Faraday Discuss. 2006, 132, 9.(7) Shah, N. C.; Lyandres, O.; Yonzon, C. R.; Zhang, X.; VanDuyne, R.P. ACS Symp. Ser. 2007, 963, 107.(8) Hudson, S. D.; Chumanov, G. Anal. Bioanal. Chem. 2009, 394, 679.(9) Stuart, D. A.; Biggs, K. B.; Van Duyne, R. P. Analyst 2006, 131, 568.(10) Whitney, A. V.; Casadio, F.; Van Duyne, R. P. Appl. Spectrosc.2007, 61, 994.(11) Brosseau, C. L.; Gambardella, A.; Casadio, F.; Grzywacz, C. M.;Wouters, J.; Van Duyne, R. P. Anal. Chem. 2009, 81, 3056.(12) Nie, S.; Emory, S. Science 1997, 275, 1102.(13) Kneipp, K.; Wang, W.; Kneipp, H.; Perelman, L. T.; Itzkan, I.;Dasari, R. R.; Feld, M. S. Phys. Rev. Lett. 1997, 78, 1667.(14) Camden, J. P.; Dieringer, J. A.; Wang, Y.; Masiello, D. J.; Marks, L.D.; Schatz, G. C.; Van Duyne, R. P. J. Am. Chem. Soc. 2008, 130, 12616.(15) Henzie, J.; Shuford, K. L.; Kwak, E.-S.; Schatz, G. C.; Odom, T.W.J. Phys. Chem. B. 2006, 110, 14028.(16) Campion, A.; Kambhampati, P. Chem. Soc. Rev. 1998, 27, 241.(17) Moskovits, M. J. Raman Spectrosc. 2005, 36, 485.(18) Haynes, C. L.; Mcfarland, A. D.; Van Duyne, R. P. Anal. Chem.2005, 77, 338A.(19) Schatz, G. C.; Young, M. A.; Van Duyne, R. P. ElectromagneticMechanism of SERS. In Surface Enhanced Raman Scattering: Physics and

Applications Topics App. Phys.; Kneipp, K., Moskovits, M., Kneipp, H.,Eds.; Springer-Verlag: Berlin, 2006; Vol. 103, p 19.(20) Jensen, L.; Aikens, C. M.; Schatz, G. C. Chem. Soc. Rev. 2008, 37,1061.(21) Masiello, D. J.; Schatz, G. C. Phys. Rev. A 2008, 78, 042505/1.(22) Chen, H. N.; McMahon, J. M.; Ratner, M. A.; Schatz, G. C. J. Phys.Chem. C 2010, 114, 14384.(23)Morton, S. M.; Silverstein, D.W.; Jensen, L.Chem. Rev. 2011, 111,3962.(24) Lombardi, J. R.; Birke, R. L. Acc. Chem. Res. 2009, 42, 734.(25) Malynych, S.; Chumanov, G. J. Am. Chem. Soc. 2003, 125, 2896.(26) Zou, S.; Schatz, G. C. J. Chem. Phys. 2004, 121, 12606.(27) Zou, S.; Janel, N.; Schatz, G. C. J. Chem. Phys. 2004, 120, 10871.(28) Zou, S.; Schatz, G. C. Chem. Phys. Lett. 2005, 403, 62.(29) Kumbhar, A. S.; Kinnan, M. K.; Chumanov, G. J. Am. Chem. Soc.2005, 127, 12444.(30) Zou, S.; Schatz, G. C. Nanotechnology 2006, 17, 2813.(31) Zhao, K.; Xu, H.; Gu, B.; Zhang, Z. J. Chem. Phys. 2006, 125,081102/1.(32) Kinnan, M. K.; Chumanov, G. J. Phys. Chem. C. 2007, 111, 18010.(33) Song, Y.; Zhao, K.; Jia, Y.; Hu, X.; Zhang, Z. J. Chem. Phys. 2008,129, 204506/1.(34) Yang, Z.; Li, Y.; Li, Z.; Wu, D.; Kang, J.; Xu, H.; Sun, M. J. Chem.Phys. 2009, 130, 234705/1.(35) Chu, Y.; Schonbrun, E.; Yang, T.; Crozier, K. B. Appl. Phys. Lett.2008, 93, 181108.(36) Khurgin, J. B.; Sun, G. Appl. Phys. Lett. 2009, 94, 221111.(37) Felidj, N.; Laurent, G.; Aubard, J.; Levi, G.; Hohenau, A.; Krenn,J. R.; Aussenegg, F. R. J. Chem. Phys. 2005, 123, 221103.(38) Yang, T.; Crozier, K. B. Opt. Express 2008, 16, 13070.(39) Vecchi, G.; Giannini, V.; Gomez Rivas, J. Phys. Rev. B 2009, 80,201401.(40) Auguie, B.; Barnes, W. L. Phys. Rev. Lett. 2008, 101, 143902.(41) Vecchi, G.; Giannini, V.; Gomez Rivas, J. Phys. Rev. Lett. 2009,102, 146807.(42) Liu, N.; Hentschel, M.; Weiss, T.; Alivisatos, A. P.; Giessen, H.Science 2011, 332, 1407.(43) García de Abajo, F. J. Rev. Mod. Phys. 2007, 79, 1267.(44) Halas, N. J.; Lal, S.; Chang, W. S.; Link, S.; Nordlander, P. Chem.Rev. 2011, 111, 3913.(45) Zou, S.; Schatz, G. C. Coupled Plasmonic Plasmon/PhotonicResonance Effects in SERS. In Surface-Enhanced Raman Scattering-Physics and Applications, Topics Appl. Phys.; Kneipp, K., Moskovits, M.,Kneipp, H., Eds.; Springer-Verlag: Berlin, 2006; Vol. 103, p 67.(46) Carron, K. T.; Fluhr, W.;Meier, M.; Wokaun, A.; Lehmann, H.W.J. Opt. Soc. Am. B 1986, 3, 430.(47) Kerker, M.; Wang, D.-S.; Chew, H. Appl. Opt. 1980, 19, 4159.(48) Kerker, M. J. Colloid Interface Sci. 1987, 118, 417.(49) Suh, Y. D.; Schenter, G. K.; Zhu, L.; Lu, H. P. Ultramicroscopy2003, 97, 89.(50) Zhu, L.; Schenter, G. K.; Micic, M.; Suh, Y. D.; Klymyshyn, N.; Lu,H. P. Appl. Spectrosc. 2003, 4962, 70.(51) Ausman, L. K.; Schatz, G. C. J. Chem. Phys. 2008, 129, 054704/1.(52) Ausman, L. K.; Schatz, G. C. J. Chem. Phys. 2009, 131, 084708.(53) Mackowski, D. W. Proc. R. Soc. London, Ser. A 1991, 433, 599.(54) Mackowski, D. W. J. Opt. Soc. Am. A 1994, 11, 2851.(55) Mackowski, D. W.; Mishchenko, M. I. J. Opt. Soc. Am. A 1996, 13,2266.(56) Tai, C.-T. Dyadic Green’s Functions in Electromagnetic Theory;Intext Educational Publishers: San Francisco, 1971.(57) Hicks, E. M.; Zou, S. L.; Schatz, G. C.; Spears, K. G.; Van Duyne,R. P.; Gunnarsson, L.; Rindzevicius, T.; Kasemo, B.; Kall, M. Nano Lett.2005, 5, 1065.(58) Zhang, J.; Li, S.; Wu, J.; Schatz, G. C.; Mirkin, C. A. Angew. Chem.,Int. Ed. 2009, 48, 7787.(59) Li, S.; Pedano, M. L.; Chang, S.-H.; Mirkin, C. A.; Schatz, G. C.Nano Lett. 2010, 10, 1722.(60) Draine, B. T.; Flatau, P. J. J. Opt. Soc. Am. A 2008, 25, 2693.(61) Draine, B. T.; Goodman, J. Astrophys. J. 1993, 405, 685.



http://pubs.acs.org

(62) Draine, B. T.; Flatau, P. J. User Guide for the Discrete DipoleApproximation Code DDSCAT 7.0, 2008. http://arxiv.org/abs/0809.0337v4.(63) Johnson, P. B.; Christy, R. W. Phys. Rev. B 1972, 6, 4370.(64) Ausman, L. K.; Schatz, G. C. In The Mie Theory; Springer Series inOptical Sciences 169, Hergert, W., Wriedt, T., Eds.; Springer, in 2012,Chapt. 5, 135−155.



http://arxiv.org/abs/0809.0337v4

http://arxiv.org/abs/0809.0337v4

Documents

Structural Eﬀects in the Electromagnetic Enhancement ...€¦ · understanding of the electromagnetic interactions between nano-particles. It is widely accepted that a major component