Influence of Morphology on the Optical Properties of Metal

Delivered by Ingenta toAmanda Barnard

IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE

Copyright copy 2007 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofComputational and Theoretical Nanoscience

Vol 4 231ndash238 2007

Influence of Morphology on theOptical Properties of Metal Nanoparticles

A L Gonzaacutelez and Cecilia Noguezlowast

Instituto de Fiacutesica Universidad Nacional Autoacutenoma de Meacutexico Apartado Postal 20-364 Meacutexico DF 01000 Meacutexico

The influence of morphology on the optical properties of silver nanoparticles is studied A generalrelationship between the surface plasmon resonances and the morphology of each nanoparticle isestablished The optical response is investigated for cubes and decahedrons with different trunca-tions We found that polyhedral nanoparticles composed with less faces show more surface plas-mon resonances than spherical-like ones It is also observed that the vertices of the nanoparticlesplay an important role in the optical response because the sharpener they become the greaterthe number of resonances For all the nanoparticles a main resonance with a dipolar characterwas identified as well as other secondary resonances of less intensity It is also found that as thenanoparticle becomes more symmetric the main resonance is always blue shifted

Keywords Metal Nanoparticles Morphology Optical Properties Surface Plasmon Resonances

1 INTRODUCTION

New synthesis methods have been developed to fabricatenanoparticles (NPs) with specific size and shape that inturn have enabled us to control optical properties to revealnew aspects of their underlying science and to tailor themfor clearly defined applications For instance the opticalresponse of nanoparticles is now being investigated fortheir potential in optics magneto-optic photonics as ananoengineered substrate on which the Surface EnhancedRaman Scattering (SERS) response can be precisely con-trolled and optimized and for chemical and biosens-ing applications such as optical addressable diagnosticmethods devices and therapies based on the plasmonicresponse of metallic NPs1ndash3 These advances that allowmetals to be structured and characterized on the nanome-ter scale have renewed the interest in optical proper-ties from physicists chemists and materials scientists tobiologists

The optical properties of metal nanoparticles (NPs) canbe tuned by controlling their size and shape Indeedmetallic NPs support surface plasmon resonances whichare highly dependent on geometry and environment pro-viding a starting point for emerging research fields likeplasmonics1 Surface plasmons are collective excitations ofthe electrons at the interface between a conductor and aninsulator and are described by evanescent electromagneticwaves which are not necessarily located at the interface

lowastAuthor to whom correspondence should be addressed

Surface plasmon resonances (SPR) appear in a numberof different phenomena including the optical response ofmaterials at different scales and the Casimir and van derWaals forces between macroscopic bodies

Besides to the technological implications the opticalsignature of NPs can be also used as a tool of character-ization Optical techniques are non-destructive and witha proper implementation they can be used to performin situ and in real time measurements providing statisti-cal properties of the whole sample These attributes areimportant because the properties of nanoparticles dependon the environment45 When growth and characteriza-tion are made in different ambient conditions this can bean additional uncontrollable variable for their interpreta-tion Thus optical spectroscopies can be used as comple-mentary tools of structural characterization techniques likeAtomic Force Microscopy (AFM) Scanning TunnelingMicroscopy (STM) Transmission Electron Microscopy(TEM) etc which provide the image of a small piece ofthe sample giving information about local properties andcharacterizing a few NPs at a time

The interesting observation that NPs support SPR thatcan be tuned by controlling their size and shape offersa starting point for nanotechnology4ndash7 In this article wegive some insights of the SPR as a function of the mor-phology for silver NPs of different polyhedral shapes Inthe case of silver many results indicate the presence oficosahedra and decahedral shapes as well as other relatedmorphologies like cubes and truncated cubes7 A very sim-ilar pattern is found in gold and copper89

J Comput Theor Nanosci 2007 Vol 4 No 2 1546-198X20074231008 doi101166jctn2007005 231


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RESEARCHARTICLE

Influence of Morphology on the Optical Properties of Metal Nanoparticles Gonzaacutelez and Noguez

2 FORMALISM

When a particle is under the action of an electromagnetic(EM) field its electrons start to oscillate transformingenergy from the incident EM wave into for example ther-mal energy in a so-called absorption process The elec-trons can also be accelerated and then they can radiateenergy in a so-called scattering process The sum of botheffects absorption and scattering is known as the extinc-tion of light see Figure 1 In this work we consider NPswhich are large enough to employ the classical EM the-ory However they are small enough to observe the depen-dence of the optical properties with its size and shapeThis means that the inhomogeneities of the particle aremuch smaller as compared to the wavelength of the inci-dent EM field such that each point of the nanoparticlecan be described in terms of its macroscopic dielectricfunction which depends on the frequency only Here werestrict ourselves to the elastic or coherent case where thefrequency of the absorbed and scattered light is the sameas the frequency of the incident light

When the size of a homogeneous particle is muchsmaller than the wavelength of the incident light the NPsfeel a field spatially constant but with a time dependentphase This is known as the quasi-static approximationwhich is characterized by keeping the time but not the spa-tial dependence of the EM field The electric field causesthe conduction electrons to oscillate coherently displacingthe electron cloud from the nuclei The positive chargesare assumed to be immobile while the negative chargesare allowed to move under the influence of the incidentfield The long range correlation of electrons caused byCoulomb forces and the attraction between positive andnegative charges results in a restoring force that changesthe oscillation frequency of the electron cloud with respectto the positive background In metallic particles the col-lective motion of electrons produces surface modes whosenumber position and width are determined by the parti-cle shape and variations of the dielectric function Sincethe main effect producing the restoring force is the sur-face polarization the proper resonances depend on theNP shape For example ellipsoids with three differentaxes have three different oscillation frequencies depend-ing on the relative length of the axes such that the optical

Fig 1 Schematic model of light extinction due to scattering andabsorption effects

response of the NP is sensitive to the choice of the lightpolarization

For nanoparticles of less than 10 nm radiation pro-cesses are negligible then the particle only absorbs energythrough the excitation of surface plasmon resonances4

Altering the surface by modifying the size shape andorenvironment of the conductor the properties of SPR canbe tailored SPR of small particles can be studied in termsof the strength of the coupling to the applied field of theoptically active electromagnetic surface modes of the sys-tem These modes only depend on the morphology of theparticle and not on its material properties10

The optical response of a NP characterized by a dielec-tric function can be obtained by finding the solutionof the Maxwellrsquos equations In 1908 Gustav Mie foundthe exact answer for a homogeneous spherical particle ofarbitrary size11 However exact solutions of the Maxwellrsquosequations for particles with other arbitrary shapes are notstraight forward and only approximations are possibleBecause of the complexity of the systems being studiedhere efficient computational methods capable of treatinglarge size systems are essential In the last few years sev-eral numerical methods have been developed to determinethe optical properties of non-spherical particles such as theDiscrete Dipole Approximation T-matrix Spectral Repre-sentation Finite Difference methods etc12 In this workwe employ the Discrete Dipole Approximation

21 Discrete Dipole Approximation

The Discrete Dipole Approximation (DDA) is a well-suited technique for studying scattering and absorption ofelectromagnetic radiation by particles with sizes of theorder or less of the wavelength of the incident light DDAhas been applied to a broad range of problems includ-ing interstellar dust grains ice crystals in the atmosphereinterplanetary dust human blood cells surface featuresof semiconductors metal nanoparticles and their aggre-gates and more DDA was first introduced by Purcelland Pennypacker13 and has been subjected to severalimprovements in particular those made by Draine andcollaborators14ndash16 Below we briefly describe the maincharacteristics of DDA and its numerical implementationthe DDSCAT code17 For a more complete description ofDDA and DDSCAT the reader can consult Refs [12ndash16]

DDSCAT builds up a solid object using an array ofN polarizable entities located in a periodic lattice whichresembles the shape and size of the particle under studysee Figure 2 These polarizable entities are located at thepositions ri with i= 12 N DDSCAT assigns to eachentity a dipole moment given as

pi = harri middot Ei loc (1)

where harri is the dipolar polarizability of the entity at ri and

Ei loc is the total electric field acting on the i-th dipole also

232 J Comput Theor Nanosci 4 231ndash238 2007


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RESEARCHARTICLE

Gonzaacutelez and Noguez Influence of Morphology on the Optical Properties of Metal Nanoparticles

Fig 2 DDA approximates a solid scatterer by an array of polarizablepoint dipoles that mimics the morphology

called the local field The discretization of the particle is agood approximation when the distance d between adjacentpolarizable entities is much smaller than the wavelength of the incident electromagnetic field Once ri and harr

i aregiven and the condition d 1 is fulfilled it is possi-ble to predict the light absorption and scattering by freeand embedded particles In our case the silver nanoparti-cles of interest are suspended in a solvent in such a waythat both solvent and particles constitute a dilute colloidalsuspension

The local field due to an array of point dipoles under anapplied electromagnetic field is given as

Ei loc = Ei app + Ei ind (2)

where Ei app is the applied field and Ei ind is the inducedfield acting on the i-th entity due to the radiation of all theothers N minus1 dipoles that compose the nanoparticle

Let us consider the applied field as a monochromaticplane wave

Ei app = E0 expik middot rminustwhere E0 denotes the magnitude of the incident electricfield k the wave vector the frequency and t meanstime On the other hand the induced field is given by

Ei ind =Nsumj=1

primeharrAij middot pj (3)

where the symbol prime means i = j andharrAij is the matrix that

couples the electromagnetic interaction among dipolesThis interaction matrix is expressed as

harrAij middot pj =

eikrij

r3ij

k2 rij times rij times pj

+ 1minus ikrij r2ij

[r2ij pj minus3rij rij middot pj

](4)

where rij is a vector from the position of dipole i-th todipole j-th rij denotes its magnitude and k = k

Substituting the local field from Eqs (2) and (3) into theexpression for the dipolar moment in Eq (1) we obtain asystem of 3N complex coupled equations

pi = harri middot

(Eiinc +

Nsumj=1

primeharrAij middot pj)

(5)

From the above expression we can calculate the set ofdipole moments that mimic the optical response of theparticle and once we know each pi it is possible to obtain

the light extinction and absorption cross sections using thefollowing expressions13

Cext =4k

E02Nsumj=1

Im Ej inc middot plowastj (6)

Cabs =4k

E02Nsumj=1

Im pj middot minus1j

lowast plowastj minus23k3 pj 2 (7)

where (lowast) means complex conjugate The scattering crosssection Csca is defined as the difference between extinctionand absorption cross sections Csca = Cext minusCabs

DDSCAT17 creates a cubic lattice array of dipoles andassigns to each one a polarizability given by the LatticeDispersion Relation (LDR)15

LDR = CM

1+CMb1 +b2+b3Sk2d

(8)

where is the macroscopic dielectric function of the par-ticle S b1 b2 and b3 are the coefficients of the expansionto third order in k to incorporate radiation effects and CM

is the polarizability given by the well known Clausius-Mossotti relation18

minus1 = 4nCM

1minus4nCM3(9)

Here we have assumed that the polarizability is isotropicand is the same for all the entities harr

i = LDR A keyfactor in determining the level of accuracy that can bereached for a given number of dipoles is the prescriptionfor assigning dipole polarizabilities15 Besides the finitewavelength it is also important to include the influence ofthe geometry of the particle has to be considered in thedipole polarizabilities19

It is also important to select a ldquorealisticrdquo dielectric func-tion which better resembles the material properties of theparticle at the nanometer scale As starting point we canemploy dielectric functions measured experimentally forbulk metals bulk and then incorporate the main effectsat the appropriate scale The experimental dielectric func-tion has contributions due to interband (inter) and intra-band (intra) electron transitions and assuming that bothcontributions are additive we have

bulk= inter+ intra (10)

Interband contributions are due to electron transitions fromoccupied to empty bulk bands separated by an energy gapThe electrons are bound by a restoring force given bythe energy difference between ground and excited electronstates usually at the ultraviolet (UV) region for metals11

Intraband contributions come from electron transitions atthe Fermi level in incompletely filled bands or an other-wise when a filled band overlaps in energy with an emptyband These transitions also provide an absorption mecha-nism but at lower energies in metals from infrared (IR) tovisible light Electrons at the Fermi level in metals can

J Comput Theor Nanosci 4 231ndash238 2007 233


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RESEARCHARTICLE


be excited by photons with very small energies such thatwe say that electrons are ldquofreerdquo and their contribution tobulk can be approximated by the Drude model of freeelectrons11

intra= 1minus 2p

+ i (11)

Here p = 4e2m is the plasma frequency with thenumber of electrons per volume unit e the electron chargem the electron mass and 1 a damping constant due tothe dispersion of the electrons At low temperatures col-lision times are determined by impurities and imperfec-tions in the lattice while at room temperatures they aredispersed by the ions of the system

Here we are interested in nanoparticles whose sizes aresmaller than 10 nm such that physical phenomena asso-ciate with radiation effects like scattering and radiationdamping are negligible ie Csca asymp 0 such that Cext =Cabs

4 However we have to consider that the conductionelectrons suffer an additional damping effect due to surfacedispersion or finite size In such case we have to make anadditional adaptation to bulk including an extra damp-ing term a which depends on the size a of the parti-cle This surface dispersion is present when the mean freepath of the ldquofreerdquo electrons is comparable or larger thanthe dimension of the particle such that the electrons arescattered by the surface rather than the ions The smallerthe particle the more important are the surface dispersioneffects The surface dispersion not only depend on the par-ticlersquos size but also on its shape20

To include surface dispersion we need modify the freeelectron or intraband contributions by changing the damp-ing term Assuming that the susceptibilities are additivewe can obtain the effect of the bound charges by sub-tracting the free electron contribution intra from thebulk dielectric function The free electron or intrabandcontributions are obtained using the Drude model andthe theoretical values of p Then we include the sur-face dispersion by adding the extra damping term ato the Drude model Finally we obtain a dielectric func-tion which also depends on the NPrsquos size and includesthe contributions of(i) the free electrons(ii) surface damping and(iii) interband or bound electrons effects and is given by

a=bulkminusintra+

1minus 2p

+i+ia

(12)

In this work we will consider for all the cases the surfacedispersion of a sphere of radius a is given by 1a =vfa21 where vf is the Fermi velocity of the electron cloudWe will show that surface dispersion effects do not changethe location of the surface modes but they affect the cou-pling of the proper modes with the applied field makingwider and less intense

3 RESULTS AND DISCUSSION

To understand the influence of morphology on the SPR westudy the extinction efficiency Qext for different polyhedralNPs defined as the extinction cross section per unit areaA as follows

Qext =Cext

A(13)

We consider silver particles with an effective volume4a3

eff3 with aeff = 10 nm In all cases the spectra werecalculated within DDA using more than 105 dipoles toassure convergence We employ the dielectric function asmeasured on bulk silver by Johnson and Christy22 andmodified according to Eq (12) We assume that the NPsare immersed in a media with a refraction index n = 1and are well dispersed at a low concentration In this diluteregime the interactions between particles are negligible23

such that the absorbance can be modeled as the opticalresponse of one immersed NP times the concentration ofparticles7

31 Cubic Morphology

We study the extinction efficiency of cubic particles andcompare it with those obtained for different truncatedcubes and the sphere The truncated cubes are obtainedby symmetrically edging the eight vertices of the cube byltimes r where l is the length of the cubersquos side and 0lt r 12 We label the different truncations with the number r When r = 12 a cuboctahedron is obtained Six octagonsand eight triangles compose all the truncated cubes whilethe cuboctahedron is composed by six planar squares andeight triangles All the truncated cubes have fourteen facesFinally if we performed a symmetric truncation of thecube with an infinite number of planes one could arriveto the sphere as shown in Figure 3

In Figure 4 we show the extinction efficiency of a sil-ver nanocube As we will show later the structure below335 nm is independent of the morphology of the particlebecause the main absorption mechanism at those wave-lengths is the interband component At larger wavelengthsthe spectrum is very sensitive to the morphology of eachNP For the cubic NP the spectrum shows a rich struc-ture of peaks contrary to the case of the sphere that has

Fig 3 Cube nanoparticle and two different truncated cubes and thesphere



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Fig 4 Extinction efficiency of a silver cube nanoparticle as a functionof the wavelength of the incident light The main six surface plasmonresonances are indicated by arrows

a single resonance These peaks are associated to the SPRinherent to the cubic geometry Fuchs24 found nine SPRfor the cubic geometry where only six of them account formore than the 95 of the spectrum as seen in Figure 4The two most intense SPRs correspond to the dipolarand quadrupolar charge distributions and are located at406 nm and 383 nm respectively The other SPRs are atsmaller wavelengths (lt370 nm) making wider the spec-trum Let us remind you that the small nanosphere showsa single peak because only a homogeneous arrangementof the charges is possible giving rise to a dipolar chargedistribution On the other hand small cubes have more res-onances because the charges are not longer able to arrangein a homogeneous distribution resulting in many differentways beside the dipolar charge distribution even in thelong wavelength approximation24

In Figure 5 the extinction efficiencies of truncatednanocubes with r from 18 to 12 (cuboctahedron) areshown in solid lines The spectra for spherical (dotted line)and cubic (dashed line) NPs are also included for compar-ison It is observed that even for the smallest truncation ofr = 18 the spectrum is very sensitive to the morphologyThe dipolar resonance is blue shifted by about 20 nm and

300 350 400 4500

2

4

6

Wavelength (nm)

Ext

inct

ion

effic

ienc

y

cubetruncated 18truncated 16truncated 14truncated 13truncated 12sphere

Fig 5 Extinction efficiencies as a function of the wavelength of theincident light of a silver cube different truncated cubes and a sphericalnanoparticle

now becomes the most intense resonance The location ofthe dipolar and quadrupolar resonances are now very closesuch that only one wide peak is observed around 386 nmwhile the structure below 370 nm remains almost identicalto the spectrum of the cube The same trend is observedfor larger truncations and from Figure 5 we find that asthe length-size of the truncation increases(i) the main resonance is always blue shifted(ii) the peaks at smaller wavelength are closer to the dom-inant resonance such that they are hidden and(iii) the width of the main resonance increases

For instance the full width at the half maximum (FWHM)of the 18 truncated cube is about 20 nm while the one forthe cuboctahedron is 40 nm This means that the secondaryresonances do not disappear but they are hidden by thedominant resonance For comparison we have includedthe spectrum of a silver nanosphere of 10 nm of diame-ter In this case the sphere shows a single SPR locatedat 356 nm and shows a FWHM of 15 nm For icosahedraNPs (not shown here) we have found the main SPR at363 nm with a FWHM of 25 nm We can conclude that asthe number of faces of the NP increases the energy rangeof the spectrum becomes smaller the main resonanceis blue shifted the FWHM decreases and fewer reso-nances are observed Therefore with small modificationsof the morphology it is possible tune SPRs at differentwavelengths

32 Decahedral Morphology

Another important morphology present in metal NPs is thedecahedron or pentagonal bipyramid which is obtainedby different synthesis methods25ndash29 It is known that metalnanoparticles of a few nanometers in size show differentstructural motifs depending on their size composition andenergetic conditions89 The regular decahedron is com-posed with ten planar triangular faces which resemble twopentagons as seen in Figure 6 where three different ori-entations are shown The decahedron is an asymmetricparticle such that the optical response depends of the ori-entation of the incident electromagnetic field In Figure 6we show the three different orientations of the regulardecahedron with respect to the incident electromagneticfield where in (a) the electromagnetic field E is paral-lel to the pentagonal motif and E is along the verticesin (b) E is also parallel but is along the edges while in(c) E is perpendicular to the pentagonal motif

In Figure 7(a) the extinction efficiency of the decahe-dral particle for the three different polarizations as wellas their average (inset) are shown The dashed and solidlines correspond to the case of parallel polarization (a) and(b) respectively while the dotted line corresponds to theperpendicular polarization (c) We find a large anisotropyof the extinction when the light incidence is such that theelectric field is parallel and perpendicular to the pentagonalmotif When the electric field is parallel to the pentagon



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(a) (b)

K

E

(c)

Fig 6 Regular decahedral or pentagonal bipyramid nanoparticle andits three different orientations to the incident electromagnetic field(a) Parallel to the pentagon along the vertices (b) parallel to the pentagonalong the edges and (c) perpendicular to the pentagon

the corresponding spectra are very wide with a FWHM of90 nm and a maximum at about 403 nm On the otherhand when the electric field is perpendicular to the pen-tagon the spectrum shows a maximum at about 343 nm isabout three times less intense and has a FWHM of 45 nmThe spectra for both parallel polarizations are almost iden-tical except near the maxima where small differences areobserved These differences would be discussed below Onthe other hand the maxima of the average spectrum is at410 nm and the FWHM is about 90 nm In conclusion

300 400 500 600

1

3

5

Wavelength (nm)

Ext

inct

ion

effic

ienc

y

parallel

parallel

perpendicular

non-dissipationlimit

300 400 500 600

1

2

3

Ext

inct

ion

effic

ienc

y

parallel

parallel

perpendicular

350 450 5500

1

2 average

(a)

(b)

Fig 7 (a) Extinction efficiency as a function of the wavelength of theincident light of regular decahedral nanoparticles for different light polar-izations The inset shows the orientational average (b) The same but inthe non-dissipation limit

we find that the parallel polarization dominates the averagespectrum The morphology of the decahedral NP showsseveral SPRs in a wide range of wavelengths Howeverthey are not observed because (a) they are close of eachother such that the most intense hides the others andor(b) dissipation effects make wider the resonances and thedetailed structure is ldquowashed outrdquo4

We have already mentioned that the location of the res-onant frequencies of the proper modes of the system is notimmediate because it requires taking the non-dissipationlimit Here to find the location of each one of the SPRs foreach morphology we have considered that the constants and a in Eq (12) tend to infinity to achieve this limit InFigure 7(b) we show the extinction efficiency correspond-ing to the three different polarizations described above buttaking the non-dissipation limit Here we clearly observethat the peaks become more pronounced as well as newpeaks appear such that we can distinguish the differ-ent resonances that compose the spectra For instance theoptical response for the perpendicular polarization is com-posed of at least two different SPRs at 343 and 357 nm ofabout the same magnitude For the spectra correspondingto parallel polarized light we find several SPRs In thecase when the electric field is along the vertices of the pen-tagon as shown in Figure 6(a) we identify at least eightdifferent resonances The first one is at 353 nm that givesrise to a small shoulder as well as other resonances at 372437 447 464 478 and 492 nm The SPRs at 437 and447 are also present when the electric field is parallel andalong the edges of the pentagons For the case along thevertices (dashed line) the main resonances are at 405 nmand 430 nm and their intensity are very similar While forthe polarization along the edges the main SPRs are redshifted to 409 nm and 447 nm respectively The first one isalmost twice as intense as the other These dissimilaritiesin location and intensity of the resonances are responsiblefor the small difference observed in the spectra Howeverthe main difference is between light polarized parallel andperpendicular to the pentagon when SPRs of very differ-ent energy or wavelength and intensity can be tuned justby changing the light polarization

When the size of the NP is in the range of 1 nm to5 nm the regular decahedron is never observed The mostcommon shapes are the truncated ones the Marks dec-ahedron and the round decahedron The first structurewas introduced by Marks30 and is remarkably stable andcontains extra 111 facets In very clean growth condi-tions or with weak interactions with substrates this is oneof the predominant shapes for the discussed size inter-val An alternative way to describe the Marks decahe-dron is as a regular decahedron which has truncations onits facets as shown in Figure 8(b) When the truncationreaches a maximum value a morphology with the shapeof a star decahedron is formed see Figure 8(c) Anothertype of decahedral particle which is often observed cor-responds to the round pentagonal particle An example of



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RESEARCHARTICLE


Fig 8 Regular decahedron and its truncated morphologies and thesphere

these particles is shown in Figure 8(d) This kind of par-ticle can be described as a truncated decahedron in whichthe truncation has a minimum possible value producinga contrast reduction in the borders This type of particleis frequently formed when colloidal growth methods areused26 Here we discuss the optical response of the Marksdecahedra with a truncation of r = 16 and the maximumtruncation of r = 12 which corresponds to the star deca-hedron and we also discuss the rounded decahedron witha truncation of r = 18

In Figure 9 the extinction efficiencies of the regular(dotted line) Marks (solid line) star (dashed line) androunded (dashed-dotted line) decahedral nanoparticles forthe three different polarizations are shown Figure 9(a) cor-responds to the case of parallel polarization along the ver-tices and (b) to parallel polarization along the edges while(c) corresponds to the perpendicular one We observe forthe perpendicular polarization Figure 9(c) that the opticalresponse of the regular decahedron does not change forsmall truncations in both cases the Marks and roundeddecahedra On the other hand the response of the stardecahedron is totally different since it shows a sharp res-onance at 380 nm with a FWHM of 20 nm while for theother morphologies the maxima is at 343 nm about seventimes less intense and has a FWHM of 45 nm For bothparallel polarization cases all the spectra of the truncateddecahedra show differences to respect to the regular oneFor the rounded and Marks decahedra the same effect isobserved as in the case of truncated cubes The main reso-nance is blue shifted and becomes the most intense reso-nance after truncation and also its FWHM decreases as aresult of the increment of the faces On the other hand thestar decahedral shows the opposite behavior In this casethe main resonance is red shifted to around 550 nm andthe spectra becomes very wide since a lot of resonancesare present Comparing the star decahedron with the cubewe find some similarities such as (i) a large numberof resonances located in a wide range of wavelengths(ii) the main resonance is located at the right of the spec-tra We also observe that these two morphologies presentthe sharpest vertices such that the charge distribution atthe tips becomes very inhomogeneous leading to extremefield localization24

0

2

4

Ext

inct

ion

effic

ienc

y

(a) parallel along vertices

regularMarks

starrounded

0

2

4

Ext

inct

ion

effic

ienc

y

(b) parallel along edges

regularMarks

starrounded

300 400 500 600 7000

2

4

6

Wavelength (nm)

Ext

inct

ion

effic

ienc

y

(c) perpendicular

regularMarks

starrounded

Fig 9 Extinction efficiency as a function of the wavelength of the inci-dent light of the regular decahedral nanoparticle and its truncated mor-phologies for different light polarizations

Recently single-crystal polyhedral NPs with uniformsizes have been synthesized31 From TEM images it wasfound that these polyhedral NPs exhibit defined facets withsharp edges and corners Tao and collaborators31 reportedthat small silver NPs develop into cubes of 80 nm and astheir size grows the NPs evolve from cubes to truncatedcubes cuboctahedra and finally to octahedra of 300 nmThe different stages were characterized by TEM and opti-cal spectroscopy where the extinction spectra were mea-sured The optical spectra for cubes cuboctahedra andoctahedra show highly complex plasmon signatures as aresult of their geometries For such large NPs the spectrashow a red shift with increasing size as a consequenceof the radiation effects Taking into account the later theoverall behavior of the optical spectra as a function of theNPs geometry agrees well with our theoretical results



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4 CONCLUSIONS

The influence of morphology on the optical properties ofmetal nanoparticles is studied theoretically using the dis-crete dipole approximation The location of the surfaceplasmon resonances of silver nanoparticles of differentpolyhedral shapes was obtained We found that as the sizetruncation of the cubic nanoparticle becomes larger themain resonance is blue shifted overlapping secondary res-onances and therefore increasing the full width at halfmaximum of the main resonance For decahedral particlesthe truncation to Marks and rounded decahedra shows thesame blue shift effect However the full width at half max-imum of the main resonance decreases maybe becausethe secondary resonances no longer exist It is also foundthat nanoparticles with fewer faces like the star decahe-dron show resonances in a wider range of wavelengthsperhaps because these geometrical shapes have sharpervertices as compared to the others It is expected thatthis information would be useful to motivate the develop-ment of more complex nanostructures with tunable surfaceplasmon resonances

Acknowledgments This work has been done with thepartial financial support from CONACyT grant No 44306-F and DGAPA-UNAM grant No IN101605

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14 B T Draine Astrophys J 333 848 (1988)15 B T Draine and J Goodman Astrophys J 405 685 (1993)16 B T Draine and P J Flatau J Opt Am A 11 1491 (1994)17 B T Draine and P J Flatau Source code DDSCAT 60

httpwwwastroprincetonedusimdraineDDSCAThtml18 E M Purcell Electricity and Magnetism Mc Graw-Hill (1963)19 A Rahmani P C Chaumet and G W Bryant Opt Lett 27 2118

(2002)20 E A Coronado and G C Schatz J Chem Phys 119 3926 (2003)21 U Kreibig J Phys F Met Phys 4 999 (1974)22 P B Johnson and R W Christy Phys Rev B 6 4370 (1972)23 R G Barrera C Noguez and E V Anda J Chem Phys 96 1574

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J Vac Sci Technol B 19 1091 (2001)27 C-H Kuo T-F Chiang L-J Chen and M H Huang Langmuir

20 7820 (2004)28 G Wei H Zhou Z Liu Y Song L Wang L Sun and Z Li

J Phys Chem B 109 8738 (2005)29 N Nilius N Ernst and H-J Freund Phys Rev Lett 84 3994

(2000)30 L D Marks Rep Prog Phys 57 603 (1994)31 A Tao P Sinsermsuksakul and P Yang Angew Chem Int Ed

45 4597 (2006)

Received 1 May 2006 Accepted 10 July 2006



IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE


2 FORMALISM

When a particle is under the action of an electromagnetic(EM) field its electrons start to oscillate transformingenergy from the incident EM wave into for example ther-mal energy in a so-called absorption process The elec-trons can also be accelerated and then they can radiateenergy in a so-called scattering process The sum of botheffects absorption and scattering is known as the extinc-tion of light see Figure 1 In this work we consider NPswhich are large enough to employ the classical EM the-ory However they are small enough to observe the depen-dence of the optical properties with its size and shapeThis means that the inhomogeneities of the particle aremuch smaller as compared to the wavelength of the inci-dent EM field such that each point of the nanoparticlecan be described in terms of its macroscopic dielectricfunction which depends on the frequency only Here werestrict ourselves to the elastic or coherent case where thefrequency of the absorbed and scattered light is the sameas the frequency of the incident light

When the size of a homogeneous particle is muchsmaller than the wavelength of the incident light the NPsfeel a field spatially constant but with a time dependentphase This is known as the quasi-static approximationwhich is characterized by keeping the time but not the spa-tial dependence of the EM field The electric field causesthe conduction electrons to oscillate coherently displacingthe electron cloud from the nuclei The positive chargesare assumed to be immobile while the negative chargesare allowed to move under the influence of the incidentfield The long range correlation of electrons caused byCoulomb forces and the attraction between positive andnegative charges results in a restoring force that changesthe oscillation frequency of the electron cloud with respectto the positive background In metallic particles the col-lective motion of electrons produces surface modes whosenumber position and width are determined by the parti-cle shape and variations of the dielectric function Sincethe main effect producing the restoring force is the sur-face polarization the proper resonances depend on theNP shape For example ellipsoids with three differentaxes have three different oscillation frequencies depend-ing on the relative length of the axes such that the optical

Fig 1 Schematic model of light extinction due to scattering andabsorption effects

response of the NP is sensitive to the choice of the lightpolarization

For nanoparticles of less than 10 nm radiation pro-cesses are negligible then the particle only absorbs energythrough the excitation of surface plasmon resonances4

Altering the surface by modifying the size shape andorenvironment of the conductor the properties of SPR canbe tailored SPR of small particles can be studied in termsof the strength of the coupling to the applied field of theoptically active electromagnetic surface modes of the sys-tem These modes only depend on the morphology of theparticle and not on its material properties10

The optical response of a NP characterized by a dielec-tric function can be obtained by finding the solutionof the Maxwellrsquos equations In 1908 Gustav Mie foundthe exact answer for a homogeneous spherical particle ofarbitrary size11 However exact solutions of the Maxwellrsquosequations for particles with other arbitrary shapes are notstraight forward and only approximations are possibleBecause of the complexity of the systems being studiedhere efficient computational methods capable of treatinglarge size systems are essential In the last few years sev-eral numerical methods have been developed to determinethe optical properties of non-spherical particles such as theDiscrete Dipole Approximation T-matrix Spectral Repre-sentation Finite Difference methods etc12 In this workwe employ the Discrete Dipole Approximation

21 Discrete Dipole Approximation

The Discrete Dipole Approximation (DDA) is a well-suited technique for studying scattering and absorption ofelectromagnetic radiation by particles with sizes of theorder or less of the wavelength of the incident light DDAhas been applied to a broad range of problems includ-ing interstellar dust grains ice crystals in the atmosphereinterplanetary dust human blood cells surface featuresof semiconductors metal nanoparticles and their aggre-gates and more DDA was first introduced by Purcelland Pennypacker13 and has been subjected to severalimprovements in particular those made by Draine andcollaborators14ndash16 Below we briefly describe the maincharacteristics of DDA and its numerical implementationthe DDSCAT code17 For a more complete description ofDDA and DDSCAT the reader can consult Refs [12ndash16]

DDSCAT builds up a solid object using an array ofN polarizable entities located in a periodic lattice whichresembles the shape and size of the particle under studysee Figure 2 These polarizable entities are located at thepositions ri with i= 12 N DDSCAT assigns to eachentity a dipole moment given as

pi = harri middot Ei loc (1)

where harri is the dipolar polarizability of the entity at ri and

Ei loc is the total electric field acting on the i-th dipole also



IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE










Ei ind =Nsumj=1




harrAij middot pj =

eikrij

r3ij


+ 1minus ikrij r2ij


](4)



pi = harri middot

(Eiinc +

Nsumj=1


(5)



Cext =4k

E02Nsumj=1


Cabs =4k

E02Nsumj=1





LDR = CM

1+CMb1 +b2+b3Sk2d

(8)



minus1 = 4nCM

1minus4nCM3(9)









IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE



intra= 1minus 2p

+ i (11)





a=bulkminusintra+

1minus 2p

+i+ia

(12)




Qext =Cext

A(13)




31 Cubic Morphology






IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE





300 350 400 4500

2

4

6

Wavelength (nm)

Ext

inct

ion

effic

ienc

y










IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE


(a) (b)

K

E

(c)



300 400 500 600

1

3

5

Wavelength (nm)

Ext

inct

ion

effic

ienc

y

parallel

parallel

perpendicular


300 400 500 600

1

2

3

Ext

inct

ion

effic

ienc

y

parallel

parallel

perpendicular

350 450 5500

1

2 average

(a)

(b)







IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE





0

2

4

Ext

inct

ion

effic

ienc

y


regularMarks

starrounded

0

2

4

Ext

inct

ion

effic

ienc

y


regularMarks

starrounded

300 400 500 600 7000

2

4

6

Wavelength (nm)

Ext

inct

ion

effic

ienc

y

(c) perpendicular

regularMarks

starrounded





IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE


4 CONCLUSIONS



References




Chem B 107 668 (2003)

















45 4597 (2006)




IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE










Ei ind =Nsumj=1




harrAij middot pj =

eikrij

r3ij


+ 1minus ikrij r2ij


](4)



pi = harri middot

(Eiinc +

Nsumj=1


(5)



Cext =4k

E02Nsumj=1


Cabs =4k

E02Nsumj=1





LDR = CM

1+CMb1 +b2+b3Sk2d

(8)



minus1 = 4nCM

1minus4nCM3(9)









IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE



intra= 1minus 2p

+ i (11)





a=bulkminusintra+

1minus 2p

+i+ia

(12)




Qext =Cext

A(13)




31 Cubic Morphology






IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE





300 350 400 4500

2

4

6

Wavelength (nm)

Ext

inct

ion

effic

ienc

y










IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE


(a) (b)

K

E

(c)



300 400 500 600

1

3

5

Wavelength (nm)

Ext

inct

ion

effic

ienc

y

parallel

parallel

perpendicular


300 400 500 600

1

2

3

Ext

inct

ion

effic

ienc

y

parallel

parallel

perpendicular

350 450 5500

1

2 average

(a)

(b)







IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE





0

2

4

Ext

inct

ion

effic

ienc

y


regularMarks

starrounded

0

2

4

Ext

inct

ion

effic

ienc

y


regularMarks

starrounded

300 400 500 600 7000

2

4

6

Wavelength (nm)

Ext

inct

ion

effic

ienc

y

(c) perpendicular

regularMarks

starrounded





IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE


4 CONCLUSIONS



References




Chem B 107 668 (2003)

















45 4597 (2006)




IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE



intra= 1minus 2p

+ i (11)





a=bulkminusintra+

1minus 2p

+i+ia

(12)




Qext =Cext

A(13)




31 Cubic Morphology






IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE





300 350 400 4500

2

4

6

Wavelength (nm)

Ext

inct

ion

effic

ienc

y










IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE


(a) (b)

K

E

(c)



300 400 500 600

1

3

5

Wavelength (nm)

Ext

inct

ion

effic

ienc

y

parallel

parallel

perpendicular


300 400 500 600

1

2

3

Ext

inct

ion

effic

ienc

y

parallel

parallel

perpendicular

350 450 5500

1

2 average

(a)

(b)







IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE





0

2

4

Ext

inct

ion

effic

ienc

y


regularMarks

starrounded

0

2

4

Ext

inct

ion

effic

ienc

y


regularMarks

starrounded

300 400 500 600 7000

2

4

6

Wavelength (nm)

Ext

inct

ion

effic

ienc

y

(c) perpendicular

regularMarks

starrounded





IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE


4 CONCLUSIONS



References




Chem B 107 668 (2003)

















45 4597 (2006)




IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE





300 350 400 4500

2

4

6

Wavelength (nm)

Ext

inct

ion

effic

ienc

y










IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE


(a) (b)

K

E

(c)



300 400 500 600

1

3

5

Wavelength (nm)

Ext

inct

ion

effic

ienc

y

parallel

parallel

perpendicular


300 400 500 600

1

2

3

Ext

inct

ion

effic

ienc

y

parallel

parallel

perpendicular

350 450 5500

1

2 average

(a)

(b)







IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE





0

2

4

Ext

inct

ion

effic

ienc

y


regularMarks

starrounded

0

2

4

Ext

inct

ion

effic

ienc

y


regularMarks

starrounded

300 400 500 600 7000

2

4

6

Wavelength (nm)

Ext

inct

ion

effic

ienc

y

(c) perpendicular

regularMarks

starrounded





IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE


4 CONCLUSIONS



References




Chem B 107 668 (2003)

















45 4597 (2006)




IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE


(a) (b)

K

E

(c)



300 400 500 600

1

3

5

Wavelength (nm)

Ext

inct

ion

effic

ienc

y

parallel

parallel

perpendicular


300 400 500 600

1

2

3

Ext

inct

ion

effic

ienc

y

parallel

parallel

perpendicular

350 450 5500

1

2 average

(a)

(b)







IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE





0

2

4

Ext

inct

ion

effic

ienc

y


regularMarks

starrounded

0

2

4

Ext

inct

ion

effic

ienc

y


regularMarks

starrounded

300 400 500 600 7000

2

4

6

Wavelength (nm)

Ext

inct

ion

effic

ienc

y

(c) perpendicular

regularMarks

starrounded





IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE


4 CONCLUSIONS



References




Chem B 107 668 (2003)

















45 4597 (2006)




IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE





0

2

4

Ext

inct

ion

effic

ienc

y


regularMarks

starrounded

0

2

4

Ext

inct

ion

effic

ienc

y


regularMarks

starrounded

300 400 500 600 7000

2

4

6

Wavelength (nm)

Ext

inct

ion

effic

ienc

y

(c) perpendicular

regularMarks

starrounded





IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE


4 CONCLUSIONS



References




Chem B 107 668 (2003)

















45 4597 (2006)




IP 1296784229Tue 13 Mar 2007 082815

RESEARCHARTICLE


4 CONCLUSIONS



References




Chem B 107 668 (2003)

















45 4597 (2006)



Documents

Influence of Morphology on the Optical Properties of Metal