RESEARCH

ARTIC

LE

Copyright copy 2010 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofComputational and Theoretical Nanoscience

Vol 7 1610ndash1615 2010

The Influence of Particle Shapes on theOptical Response of Nearly Touching

Plasmonic Nanoparticle Dimers

Xudong Cuilowast and Daniel ErniGeneral and Theoretical Electrical Engineering (ATE) Faculty of Engineering

University of Duisburg-Essen D-47048 Duisburg Germany

In this paper we provide a systematic analysis of the optical properties of different nanoscopicdimer structures with relatively small gap distances In particular we have focused on two differentaspects namely the feasibility of functionalizing optical nanodimers while analyzing the influenceof different particle shapes with respect to the proper contact regions as well as the impact ofpotential fabrication imperfections Both scenariosmdashfunctionalization and perturbationmdashare rootedin the significant variations of the nanodimerrsquos optical properties such as a dramatically altered fieldenhancement together with a significant shift in the resonance wavelength Referring to a state-of-the-art nanoprocessing technology we have forecasted in our outlook that emergent changes in thechemical composition especially of the metallic part will add a further dimension to the uncertaintiesthat have to be faced in functional nanoparticle desing Hence the proper determination of thenanoparticlersquos shape and the corresponding material properties may become constitutive in anydesign and optimization procedure of functional plasmonic nanostructures

Keywords Numerical Modelling Localized Surface Plasmons Nearly Touching PlasmonicDimers Photonic Molecules Nanowire Fabrication Process Variations

1 INTRODUCTION

Confining light at nanoscale dimensions and creating so-called lsquohot-spotsrsquo 12 by using metallic nanostructures isan ever-growing research field in state-of-the-art plas-monic research The underlying physics is governedby the excited (localized) surface plasmon resonancesbetween the metal and dielectric interface under properlight illumination conditions2 Such nanostructures at res-onance provide significantly enhanced electromagneticfield strengths around the structures hence light can beconfined within the nanoscale and potentially amplifylight-matter interactions Applications based on enhancedlight-matter interactions have been proposed such aseg surface-enhanced Raman spectroscopy (SERS)3 tip-enhanced Raman spectroscopy (TERS)4 single moleculedetection5 and higher-order optical nonlinearities6 Thehuge field enhancement that is indispensable for allabove-mentioned applications is however predominantlypresent in nanostructures where both the surface plas-mon resonance and the lightning-rod effect are taken intoaccount7 With this regard many interesting structures

lowastAuthor to whom correspondence should be addressed

have been numerically and experimentally investigatedsuch as eg coupled nanorods8 nano-bowtie structures9

and nanodimers10 Nanodimers with spherical shapes areof particular interest due to their simple shapes and thewell-established fabrication technology Nevertheless theproper shapes in the contacting regions are somehow cru-cial with respect to the resulting field enhancement sinceincreasing field enhancement factors typically arises fromdecreasing gap sizes in the contact region Meanwhilesmall sharp corners with tiny gaps are difficult to fab-ricate due to technological limitations This aspect alsoholds for spherical dimers as the shapes are not perfectlyspherical in reality In particular it would be interesting toquantify how the shapes at the nearly contacting regionsinfluence the resulting optical properties of the nanodimersuch as the field enhancement and the resonance shift Inthe remainder of the paper we will investigate those issuesand target the question how imperfections will hamper theextraction efficiency of information regarding the function-ality of the nanodimer especially for the case when theimperfectionsmdashnamely the shape deviationsmdashare provid-ing lowered field enhancement factors within the gapIt is worth mentioning that with the progress of nanofab-

rication and experimental techniques even small gap sizes

1610 J Comput Theor Nanosci 2010 Vol 7 No 8 1546-1955201071610006 doi101166jctn20101525

RESEARCH

ARTIC

LE

Cui and Erni The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers

in the order of sim1 nm have become practically feasible11

Although nanoparticle dimers have been intensively inves-tigated a systematic numerical analysis of nearly touchingdimers with different shapes are rarely seen in current liter-ature especially when fabrication imperfections are takeninto accountIn functional plasmonics numerical simulations have

now become indispensable for a better understanding ofthe experimental results not to mention the underlyingphysical mechanisms The major challenges for such kindof numerical analysis are the following for instanceto correctly address the field between such a tiny gapan extremely fine meshing is needed which inevitablyresults in large matrices and hence long computationtime In addition the numerical simulations may becomeunstable and the convergence gets degraded due to theextreme variations in mesh sizes In the presented work afrequency-domain Finite Element Method (FEM) is usedto investigate the optical response of the plasmonic dimerand the influence of the highly dispersive material prop-erties The choice of FEM is straightforward and basedon the fact that FEM supports conformal meshing pronefor arbitrary geometries without introducing large numeri-cal errors Our numerical experiments have shown that theoverall simulations are carried out within moderate com-putation time while showing good convergence Two sce-narios are considered in our analysis of the nanodimer(1) a gap distance between 2 nm and 10 nm where nocharge transfer occurs between the two nanoparticles and(2) the gap distance shrinks to zero which is tantamountto the case of two touching nanoparticles It should benoted for both cases classical material models as wellas classical electrodynamics are still valid12 as we donot analyze nanodimers with a gap size less than 1 nmwhere electron tunneling is supposed to be present Elec-tron tunneling would decrease the field in the gap as indi-cated by a recent quantum electronic analysis13 In sucha case it is generally assumed that the quantum effectsshould be taken into account because they may consid-erably affect the dispersion behavior of the underlyingmaterial response13 Besides the shape considerations inthe gap region we have addressed some tentative materialaspects in the outlook where we speculate on the influ-ence of material composition in the context of a recentlyfabricated nanowire structure Within our work we try tounderpin that the optical properties can significantly devi-ate from the desired specifications since the material prop-erties might have been varied during the processing of thenanostructures We believed that those findings are neces-sary to be included in any reliable design procedure forfunctional plasmonic nanodevices

2 SIMULATION PRELIMINARIES

Since extreme small meshes are indeed required to resolvethe fields in the gap the numerical simulations become

quite challenging In order to simplify the computationour conceptual ideas are presented along a two dimen-sional (2D) analysis Another convincing argument besidesthe computation time refers to convergence of the threedimensional (3D) FEM analysis which is degrading whenthe mesh sizes are becoming extremely small This is dueto larger matrix factorization and the increased iterationnumbers in the underlying solver as well as with respectto the different mesh generation schemes used in both rep-resentations However our preliminary 3D studies haveshown that reasonable convergence is obtainable whenthe commercialized FEM code COMSOL Multiphysics isemployed in conjunction with the direct solver PRASADOn a standard PC having a 2 GHz Intel dual-core pro-cessor with 32 GB RAM the elapsed time for computingthe optical response of a 3D nanodimer at one frequencypoint amounts to roughly 5 minutes For 2D calculationsdirect solvers like UMFPACK are much more efficient dueto less memory requirements even when the meshes in thegap region are extremely fine ie less than 01 nmNoble metals like gold and silver are widely used in

functional nanophotonics because of the relatively smalllosses at optical frequencies Despite its larger loss goldis mostly preferred as it is chemically inert against envi-ronmental influences In addition the plasmon frequencyfor bulk gold materials is smaller than that of bulk silvermaterials For a single gold spherical particle the surfaceplasmon resonance (SPR) is typically located in the greenregion of the visible spectrum whereas the SPR of a silvernanoparticle confines to the ultraviolet wavelength rangetherefore gold and silver are selectively used when differ-ent functionalities and applications are required Through-out our analysis we have used a material model that hasbeen based on the well-known experimental data for goldprovided by Johnson et al14

The simulated nanostructures are shown in Figure 1The gap size g is selected as the minimal distance between

(a)

(c)

(b)

(d)

Fig 1 (a)ndash(d) The different shapes of the simulated nanodimers Thegap size g coincides with the minimal distance between the two circu-lar particles The excitation consists of a y-polarized plane wave whichpropagates from below in the x-direction

J Comput Theor Nanosci 7 1610ndash1615 2010 1611

RESEARCH

ARTIC

LE

The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers Cui and Erni

the two 2D particles As an excitation we used a planewave that is polarized in the y-direction and propagatesfrom below along the x-direction having an electric fieldstrength of Ey = 1 Vm The choice of polarization is basedon the fact that the electric field in the gap is dramaticallyenhanced for a polarization along the axis of dimer8ndash10

Four configurations are considered in this work (cf Fig 1)(a) a circular dimer(b) a one-cut semi-circular dimer(c) a two-cut semi-circular dimer and(d) a square-shaped dimer

The four configurations stand for the typical shapes offabricated nanostructures The radii of the circular parti-cle regions are chosen to be r = 10 nm the cut is set ath= 4 nm and the apparent corners are rounded accordingto a radius of 2 nm for numerical reasons

3 RESULTS AND DISCUSSION

31 Circular Nanodimer

First we consider the circular structure shown inFigure 1(a) The nanodimer consists of two identical Aucircular particles Note that the optical properties andthe field enhancement of such nanodimers have alreadybeen intensively investigated Since nanodimers are oftenproposed as SERS substrate for sensing applications1ndash5

field enhancement and resonance wavelength are importantissues to be considered For spherical and circular singleparticles the resonance wavelength can be precisely pre-dicted by Mie theory For the proper dimer structure Mietheory is not directly applicable and the analysis has to bebased on numerical methods The resonance wavelengthof a single Au particle having a diameter of 20 nm islocated at 523 nm as depicted in Figures 2(a) and (b) The

Fig 2 The circular nanodimer with corresponding particle diameters of20 nm (a) the spectral response of the electric field strength in the centerof the gap as a function of the different gap sizes g (b) the intensityplot of the electric field strength for a single circular nanoparticle at res-onance (523 nm) (c) the corresponding electric field distribution for thecircular nanodimer with a gap size of g = 2 nm supporting a resonancewavelength of 528 nm

resonance wavelength of the corresponding nanodimer isconsiderably red-shifted (eg by 5 nm) for gap sizes gwell below 4 nm compared to the single particle resonance[cf Figs 2(a) and (c)] The resonance shift is partiallydue to mode coupling between the two particles10 whereboth the coupling and the redshift get inevitably smallerwith increasing gap size g A similar behavior is alsoobserved in bow-tie structures and other paired nanopar-ticle configurations9 The nearly touching dimer thereforeconstitutes an attractive nanoparticle configuration with apronounced wavelength tunability ranging from the visibleto the near infrared10 and hence is best suited for sens-ing and nano-antenna applications As already mentionedshrinking gap sizes g below 4 nm typically yield a con-tinuous redshift in the resonance wavelength15 Howevera recent analysis which has comprised all the emergingquantum effects has revealed that for gap sizes smallerthan 05 nm the resonances are getting blue-shifted due tothe emergent charge transfer13

In current research it has been observed that smallgaps are helpful to provide large field enhancement fac-tors due to both the increased charge concentration andthe strong mode coupling in the gap region8ndash10 When thegap size shrinks below 1 nm charge transfer mechanismsare assumed to occur between the two particles The elec-tric field is then getting ldquoshort-circuitedrdquo by the emer-gent electron-tunneling channel leading to a considerabledecrease in the field strength13 It is worth mentioning thatstill no experimental results are available for such smallgap sizes because of lacking resolution within state-of-the-art fabrication technologies Besides the analysis and thepromotion of novel functionalities in optical nanodimersthis is good reason to further conduct numerical experi-ments in order to quantify the requirements that rendersfuture nanoprocessing feasible

32 One-Cut Semi-Circular Nanodimer

Because of the parameter variations in the fabrica-tion technology nanoparticle shapes usually deviate fromtheir intended geometry even though some self-assemblytechniques16 are applied during particle processing A typ-ical example regarding the deformation of spherical (andcircular) nanodimers refers to the fact that the poten-tial contact regions are not ideally curved The spe-cific area rather supports facet-like structures due toproximity and lag effects in the involved etching andgrowing processing steps11 which can be approximatedwhen executing a corresponding cut at the circularnanodimerThe electric field distribution around such semi-circular

nanoparticle respective such nanodimer is shown inFigure 3 Since the overall size of the nanodimer is wellin the sub-wavelength range it responds to the impinging

1612 J Comput Theor Nanosci 7 1610ndash1615 2010

RESEARCH

ARTIC

LE

Cui and Erni The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers

Fig 3 The one-cut semi-circular nanodimer with corresponding parti-cle diameters of 20 nm (a) the spectral response of the electric fieldstrength in the center of the gap as a function of the different gap sizes g(b) the intensity plot of the electric field strength for a single semi-circularnanoparticle at resonance (511 nm) (c) the corresponding electric fielddistribution for the semi-circular nanodimer with a gap size of g = 2 nmsupporting a resonance wavelength of 532 nm The apparent corners arerounded (r = 2 nm) for numerical reasons

field as a dipole showing a similar behavior as theperfectly circular particle [cf Fig 2(b)] except for anincreased resonance wavelength at 532 nm Comparingnow the resonance wavelength of the deformed single par-ticle at 511 nm [cf Fig 3(b)] with the perfectly circu-lar one at 523 nm reveals a blueshift of 12 nm Thisblue shift is not solely attributed to the smaller particlevolume but rather to the altered distribution of the sur-face charge density The blueshift increases with expand-ing cut depth h and even more resonances emerge inthe spectral range as a result of mode hybridization10

For the corresponding semi-circular nanodimer the situa-tion has changed when compared to the perfectly circu-lar nanodimer Contrary to the circular case reducing thegap size of the semi-circular nanodimer results now in adecrease of the electric field strength in the gap region[cf Fig 3(a)] Referring to a comparable gap size thisreduced field strength is attributed on one hand to theenlarged field volume that is bounded by the increasedsize of the very proximate particle surfaces in the propergap region [cf Fig 3(c)] and on the other hand thedecrease is governed by the correspondingly reduced sur-face charge density in this area15 However the spreadin the field distribution has some drawbacks since thelight-matter interaction would be weakened For instancein applications related to Raman spectroscopy the fieldenhancement factor is a very crucial figure for the signalgeneration because of its highly super-linear contributionto the Raman cross sections3ndash4 Similar signal degradationscould also emerge during ongoing experiments because offabrication imperfections

33 Two-Cut Semi-Circular Nanodimer

An alternative technology-related deviation from the per-fect circular dimer shape is represented by the two-cutsemi-circular nanodimer This structure is shown inFigure 1(c) where the cut parts are consecutively arrangednormal to the dimer axis Compared to the previousparticle shapes described in Sections 31 and 32 the res-onance wavelength at 507 nm of a single two-cut semi-circular nanoparticle is blue-shifted by 16 nm comparedto the single circular particle (at 523 nm) and blue-shiftedby 4 nm with respect to the single-cut semi-circular caseat 511 nm An intuitive explanation of the wavelengthshift would obviously refer to the reduced particle volumeNevertheless the situation is more complex as the modessupported by the single structures are hybridized lead-ing to significant changes in the surface charge densitiesAs displayed in Figures 4(b) and (c) the two-cut semi-circular particle structures yield the largest field enhance-ment not in gap region but in the rounded corners dueto the lightning-rod effect This is different compared tothe nanodimer described in Section 31 where the fieldmainly concentrates in the gap region Please note thatfor the nanodimers described in Sections 32 and 33 thefield strength in the gap depend very much on the cutdepth h (ie on the area of the flat proximate interfaces)namely smaller cut depths yield stronger electric fieldsThis also means that (circularly) curved particle interfacesare crucial for the field enhancement in nearly touchingnanodimers However small randomly distributed point-like defects on the particle interfaces can also help toachieve a large field enhancement1 but such defects arehard to control especially at these interfaces When the

Fig 4 The two-cut semi-circular nanodimer with corresponding par-ticle diameters of 20 nm (a) the spectral response of the electric fieldstrength in the center of the gap as a function of the different gap sizes g(b) the intensity plot of the electric field strength for a single two-cutsemi-circular nanoparticle at resonance (507 nm) (c) the correspondingelectric field distribution for the two-cut semi-circular nanodimer with agap size of g = 2 nm supporting a resonance wavelength of 535 nm Theapparent corners are rounded (r = 2 nm) for numerical reasons

J Comput Theor Nanosci 7 1610ndash1615 2010 1613

RESEARCH

ARTIC

LE

The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers Cui and Erni

gap size g is zero the two particles are supposed to toucheach other and the nanodimer becomes a single particleThe resonance wavelength is slightly red-shifted comparedto the single structure The largest electric field strengthwas then found in the wedge-like groove of the junctionregion showing that also ldquoinverted cornersrdquo may becomeindispensable for a large field enhancement This effect isattributed to local charge separation in the corner of thegroove17

34 Square-Shaped Nanodimer

As a final nanostructure we consider the square-shapednanodimer shown in Figure 1(d) where the particlersquoscross-section consists of two parallel aligned identicalsquares with a side length of 20 nm This is slightly dif-ferent from the nanostructures in the previous Section 33because no curved boundaries exist except for the roundedcorners that are introduced for numerical reasons The res-onance wavelength of the single square lies at 521 nmwhich is slightly blue-shifted by only 2 nm compared tothe perfect circular particle Interestingly the resonancewavelength of the square-shaped nanodimer turned out tobe insensitive (=+2 nm) against decreasing gap sizesdown to 2 nm [cf Fig 5(a)] whereas the three nanodimersof the previous Sections 31 32 and 33 provide signifi-cant redshifts already at gap widths around 4 nm whichincrease to 11 nm 20 nm and 28 nm respectively whenthe gap width reaches a value of 2 nm Generally speakingthe aforementioned insensitivity of the resonance shift mayprovide relaxed parameter tolerances to structure fabrica-tion The reduced resonance shift could also be beneficialto SERS experiments since the spatial mismatch between

Fig 5 The square-shaped nanodimer with a side length of 20 nm(a) the spectral response of the electric field strength in the center of thegap as a function of the different gap sizes g (b) the intensity plot of theelectric field strength for a single square-sized nanoparticle at resonance(521 nm) (c) the corresponding electric field distribution for the square-sized nanodimer with a gap size of g = 2 nm supporting a resonancewavelength of 523 nm The apparent corners are rounded (r = 2 nm) fornumerical reasons

the laser excitation and the resonant Raman excitationcan be easily minimized However due to the poor fieldenhancement in the proper gap region the square-shapednanodimer will never become the primary choice for appli-cations related to Raman spectroscopy As displayed inFigure 5(b) the largest electric field strengths is locatedat the rounded corners Optimizing functional nanodimersaccording to multiple objectives will inevitably lead to aholistic design scenario where simultaneous mechanismssuch as the lightning rod effect and the emerging surfaceplasmon resonances have to be taken into account likeeg in the various well-documented design examples ofbow-tie nanostructures

4 OUTLOOK AND CONCLUSION

Before concluding our numerical design study on the var-ious dimer shapes we want to reflect on the material influ-ence by adding a very preliminary and hence tentativeforecast on how the process related variations of materialcomposition may probably alter future design strategiesAn important issue regarding the design of functionalnanoparticles is the availability of a proper material modelused in the numerical simulation Usually bulk materialmodels based on experimental data14 are applied becausethe dimension of the nanoparticle is still within the scopeof classical theory if the particle size is supposed to belarger than 10 nm However due to different processingsteps in the underlying fabrication technology the chem-ical composition of the metallic nanoparticle might bealtered giving rise to significant changes in its opticalresponse Thus a reliable nanoparticle design has alwaysto account for the temporal history of material composi-tion that goes along with the various processing steps inthe corresponding fabrication technology To underpin thisstatement we have analyzed the compositions of charac-teristic sample namely an AundashIn nanowire that has beenfabricated using a processing technology based on catalyticgrowth The conpositional analysis is carried out in theframework of the Energy Filtered Transmission ElectronMicroscopy (EFTEM) The EFTEM is a typical TEM tech-nique in which only electrons of particular kinetic ener-gies are used for image formation18 and thus giving riseto a spatially resolved elemental mapping The chemicalcomposition of an AundashIn nanowire is shown in Figure 6as well as the shape of the nanowire whose metallic partresembles to one of the semi-cut particle shapes From theEFTEM data we can now conclude that the intentionallypure metallic part rather corresponds to mixture of goldwith an In and As alloy where the stoichiometric goldcomposition has been reduced during processing to a valueslightly below 71 In this case we have estimated theresulting redshift of a spherical nanoparticlersquos resonancewavelength due to the altered metal composition to be inthe order of 12 just by using the Froumlhlich condition in

1614 J Comput Theor Nanosci 7 1610ndash1615 2010

RESEARCH

ARTIC

LE

Cui and Erni The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers

Fig 6 (a) TEM image of the fabricated AundashIn nanowire The white part corresponds to metal and the gray part represents dielectric Red lineindicates the scanning line for the composition measurement The positions (1) and (2) indicate the points where the chemical composition is measured(b) cross line counts along the aforementioned red line (c) measured elements at the corresponding positions (1) and (2)

the framework of a simple Drude model Even if this isa very straightforward measure it provides a valuable esti-mate of the discussed material influence not to mentionthe existing lack of reliable optical material models forsuch specific alloys We therefore conclude that the mate-rial properties of functional nanostructures have always tobe retrieved according to the different processing stepsThe punch line of this story is that reliable design pro-cedures may now become extremely challenging becauseany of such designs will unconditionally rely on suitablytailored material models A detailed analysis of this spe-cific nanowire will be presented elsewhereTo summarize we have numerically investigated the

optical properties of nearly touching plasmonic nanopar-ticle dimers with focus on the field enhancementand resonance wavelengths when taking the fabricationimperfectionsndashnamely the shape deviationsndashinto accountWith this regard four typical nanostructures have beenconsidered the circular dimer the one-cut semi-circulardimer the two-cut semi-circular dimer and the square-shaped dimer We have observed that the optical prop-erties such as the field enhancement and the resonancewavelength strongly depend on the particle shape in theproper gap region and furthermore we have identifiedthe square-shaped dimer to be virtually insensitive to gapwidth variations The studies provide important detailsto nano engineering not to mention the issue that thediscrepancy between the practical engineering processand the underlying numerical analysis is suspected toremain a fundamental one This reasoning is also sup-ported when looking to the variation of the materialcomposition during the various processing steps As anexample we used TEM to characterize the compositionsof an AundashIn nanowire that has been grown by catalyticmethods We found that especially the important metallicparts are subject to large stoichiometric variations when

alloyed with different constituents This indicates that inany reliable design procedure the proper determinationand modeling of material properties at the nanoscale willbecome an essential part rendering nanoengineering apretty challenging venture

Acknowledgments We thank Dr Zi-An Li for theTEM characterization of the nanowire

References

1 W H Zhang X D Cui B S Yeo T Schmid Ch Hafner andR Zenobi Nano Lett 7 1401 (2007)

2 W L Barnes A Dereux and T W Ebbesen Nature 424 824(2003)

3 A Campion and P Kambhampati Chem Soc Rev 27 241 (1998)4 N Hayazawa Y Inouye Z Sekkat and S Kawata Opt Commun

183 333 (2006)5 S Nie and S R Emory Science 275 1102 (1997)6 M Danckwerts and L Novotny Phys Rev Lett 98 026104 (2007)7 H X Xu J Aizpurua M Kall and P Apell Phys Rev E 62 4318

(2000)8 J Aizpurua G W Bryant L J Richter and F J Garcia de Abajo

Phys Rev B 71 235420 (2005)9 D P Fromm A Sundaramurthy P James G Kino and W E

Moerner Nano Lett 4 957 (2004)10 P Nordlander C Oubre E Pordan K Li and M I Stockman

Nano Lett 4 899 (2004)11 W Y Li P H C Camargo X M Lu and Y N Xia Nano Lett

9 485 (2009)12 E Moreno D Erni Ch Hafner and R Vahldieck J Opt Soc

Am A 19 101 (2002)13 J Zuloaga E Prodan and P Nordlander Nano Lett 9 887

(2009)14 P B Johnson and R W Christy Phys Rev B 6 4370 (1972)15 I Romero J Aizpurua G W Bryant and F Javier Garcia de Abajo

Opt Express 14 9988 (2006)16 M Li H Schnablegger and S Mann Nature 402 393 (1999)17 R Kappeler D Erni X Cui and L Novotny J Comput Theor

Nanosci 4 686 (2007)18 L Reimer Advances in Electronics and Electron Physics edited by

P W Hawkes Academic Press London (1991) Vol 81

Received 25 September 2009 Accepted 27 October 2009

J Comput Theor Nanosci 7 1610ndash1615 2010 1615

RESEARCH

ARTIC

LE

Cui and Erni The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers

in the order of sim1 nm have become practically feasible11

Although nanoparticle dimers have been intensively inves-tigated a systematic numerical analysis of nearly touchingdimers with different shapes are rarely seen in current liter-ature especially when fabrication imperfections are takeninto accountIn functional plasmonics numerical simulations have

now become indispensable for a better understanding ofthe experimental results not to mention the underlyingphysical mechanisms The major challenges for such kindof numerical analysis are the following for instanceto correctly address the field between such a tiny gapan extremely fine meshing is needed which inevitablyresults in large matrices and hence long computationtime In addition the numerical simulations may becomeunstable and the convergence gets degraded due to theextreme variations in mesh sizes In the presented work afrequency-domain Finite Element Method (FEM) is usedto investigate the optical response of the plasmonic dimerand the influence of the highly dispersive material prop-erties The choice of FEM is straightforward and basedon the fact that FEM supports conformal meshing pronefor arbitrary geometries without introducing large numeri-cal errors Our numerical experiments have shown that theoverall simulations are carried out within moderate com-putation time while showing good convergence Two sce-narios are considered in our analysis of the nanodimer(1) a gap distance between 2 nm and 10 nm where nocharge transfer occurs between the two nanoparticles and(2) the gap distance shrinks to zero which is tantamountto the case of two touching nanoparticles It should benoted for both cases classical material models as wellas classical electrodynamics are still valid12 as we donot analyze nanodimers with a gap size less than 1 nmwhere electron tunneling is supposed to be present Elec-tron tunneling would decrease the field in the gap as indi-cated by a recent quantum electronic analysis13 In sucha case it is generally assumed that the quantum effectsshould be taken into account because they may consid-erably affect the dispersion behavior of the underlyingmaterial response13 Besides the shape considerations inthe gap region we have addressed some tentative materialaspects in the outlook where we speculate on the influ-ence of material composition in the context of a recentlyfabricated nanowire structure Within our work we try tounderpin that the optical properties can significantly devi-ate from the desired specifications since the material prop-erties might have been varied during the processing of thenanostructures We believed that those findings are neces-sary to be included in any reliable design procedure forfunctional plasmonic nanodevices

2 SIMULATION PRELIMINARIES

Since extreme small meshes are indeed required to resolvethe fields in the gap the numerical simulations become

quite challenging In order to simplify the computationour conceptual ideas are presented along a two dimen-sional (2D) analysis Another convincing argument besidesthe computation time refers to convergence of the threedimensional (3D) FEM analysis which is degrading whenthe mesh sizes are becoming extremely small This is dueto larger matrix factorization and the increased iterationnumbers in the underlying solver as well as with respectto the different mesh generation schemes used in both rep-resentations However our preliminary 3D studies haveshown that reasonable convergence is obtainable whenthe commercialized FEM code COMSOL Multiphysics isemployed in conjunction with the direct solver PRASADOn a standard PC having a 2 GHz Intel dual-core pro-cessor with 32 GB RAM the elapsed time for computingthe optical response of a 3D nanodimer at one frequencypoint amounts to roughly 5 minutes For 2D calculationsdirect solvers like UMFPACK are much more efficient dueto less memory requirements even when the meshes in thegap region are extremely fine ie less than 01 nmNoble metals like gold and silver are widely used in

functional nanophotonics because of the relatively smalllosses at optical frequencies Despite its larger loss goldis mostly preferred as it is chemically inert against envi-ronmental influences In addition the plasmon frequencyfor bulk gold materials is smaller than that of bulk silvermaterials For a single gold spherical particle the surfaceplasmon resonance (SPR) is typically located in the greenregion of the visible spectrum whereas the SPR of a silvernanoparticle confines to the ultraviolet wavelength rangetherefore gold and silver are selectively used when differ-ent functionalities and applications are required Through-out our analysis we have used a material model that hasbeen based on the well-known experimental data for goldprovided by Johnson et al14

The simulated nanostructures are shown in Figure 1The gap size g is selected as the minimal distance between

(a)

(c)

(b)

(d)

Fig 1 (a)ndash(d) The different shapes of the simulated nanodimers Thegap size g coincides with the minimal distance between the two circu-lar particles The excitation consists of a y-polarized plane wave whichpropagates from below in the x-direction

J Comput Theor Nanosci 7 1610ndash1615 2010 1611

RESEARCH

ARTIC

LE

The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers Cui and Erni

the two 2D particles As an excitation we used a planewave that is polarized in the y-direction and propagatesfrom below along the x-direction having an electric fieldstrength of Ey = 1 Vm The choice of polarization is basedon the fact that the electric field in the gap is dramaticallyenhanced for a polarization along the axis of dimer8ndash10

Four configurations are considered in this work (cf Fig 1)(a) a circular dimer(b) a one-cut semi-circular dimer(c) a two-cut semi-circular dimer and(d) a square-shaped dimer

The four configurations stand for the typical shapes offabricated nanostructures The radii of the circular parti-cle regions are chosen to be r = 10 nm the cut is set ath= 4 nm and the apparent corners are rounded accordingto a radius of 2 nm for numerical reasons

3 RESULTS AND DISCUSSION

31 Circular Nanodimer

First we consider the circular structure shown inFigure 1(a) The nanodimer consists of two identical Aucircular particles Note that the optical properties andthe field enhancement of such nanodimers have alreadybeen intensively investigated Since nanodimers are oftenproposed as SERS substrate for sensing applications1ndash5

field enhancement and resonance wavelength are importantissues to be considered For spherical and circular singleparticles the resonance wavelength can be precisely pre-dicted by Mie theory For the proper dimer structure Mietheory is not directly applicable and the analysis has to bebased on numerical methods The resonance wavelengthof a single Au particle having a diameter of 20 nm islocated at 523 nm as depicted in Figures 2(a) and (b) The

Fig 2 The circular nanodimer with corresponding particle diameters of20 nm (a) the spectral response of the electric field strength in the centerof the gap as a function of the different gap sizes g (b) the intensityplot of the electric field strength for a single circular nanoparticle at res-onance (523 nm) (c) the corresponding electric field distribution for thecircular nanodimer with a gap size of g = 2 nm supporting a resonancewavelength of 528 nm

resonance wavelength of the corresponding nanodimer isconsiderably red-shifted (eg by 5 nm) for gap sizes gwell below 4 nm compared to the single particle resonance[cf Figs 2(a) and (c)] The resonance shift is partiallydue to mode coupling between the two particles10 whereboth the coupling and the redshift get inevitably smallerwith increasing gap size g A similar behavior is alsoobserved in bow-tie structures and other paired nanopar-ticle configurations9 The nearly touching dimer thereforeconstitutes an attractive nanoparticle configuration with apronounced wavelength tunability ranging from the visibleto the near infrared10 and hence is best suited for sens-ing and nano-antenna applications As already mentionedshrinking gap sizes g below 4 nm typically yield a con-tinuous redshift in the resonance wavelength15 Howevera recent analysis which has comprised all the emergingquantum effects has revealed that for gap sizes smallerthan 05 nm the resonances are getting blue-shifted due tothe emergent charge transfer13

In current research it has been observed that smallgaps are helpful to provide large field enhancement fac-tors due to both the increased charge concentration andthe strong mode coupling in the gap region8ndash10 When thegap size shrinks below 1 nm charge transfer mechanismsare assumed to occur between the two particles The elec-tric field is then getting ldquoshort-circuitedrdquo by the emer-gent electron-tunneling channel leading to a considerabledecrease in the field strength13 It is worth mentioning thatstill no experimental results are available for such smallgap sizes because of lacking resolution within state-of-the-art fabrication technologies Besides the analysis and thepromotion of novel functionalities in optical nanodimersthis is good reason to further conduct numerical experi-ments in order to quantify the requirements that rendersfuture nanoprocessing feasible

32 One-Cut Semi-Circular Nanodimer

Because of the parameter variations in the fabrica-tion technology nanoparticle shapes usually deviate fromtheir intended geometry even though some self-assemblytechniques16 are applied during particle processing A typ-ical example regarding the deformation of spherical (andcircular) nanodimers refers to the fact that the poten-tial contact regions are not ideally curved The spe-cific area rather supports facet-like structures due toproximity and lag effects in the involved etching andgrowing processing steps11 which can be approximatedwhen executing a corresponding cut at the circularnanodimerThe electric field distribution around such semi-circular

nanoparticle respective such nanodimer is shown inFigure 3 Since the overall size of the nanodimer is wellin the sub-wavelength range it responds to the impinging

1612 J Comput Theor Nanosci 7 1610ndash1615 2010

RESEARCH

ARTIC

LE

Cui and Erni The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers

Fig 3 The one-cut semi-circular nanodimer with corresponding parti-cle diameters of 20 nm (a) the spectral response of the electric fieldstrength in the center of the gap as a function of the different gap sizes g(b) the intensity plot of the electric field strength for a single semi-circularnanoparticle at resonance (511 nm) (c) the corresponding electric fielddistribution for the semi-circular nanodimer with a gap size of g = 2 nmsupporting a resonance wavelength of 532 nm The apparent corners arerounded (r = 2 nm) for numerical reasons

field as a dipole showing a similar behavior as theperfectly circular particle [cf Fig 2(b)] except for anincreased resonance wavelength at 532 nm Comparingnow the resonance wavelength of the deformed single par-ticle at 511 nm [cf Fig 3(b)] with the perfectly circu-lar one at 523 nm reveals a blueshift of 12 nm Thisblue shift is not solely attributed to the smaller particlevolume but rather to the altered distribution of the sur-face charge density The blueshift increases with expand-ing cut depth h and even more resonances emerge inthe spectral range as a result of mode hybridization10

For the corresponding semi-circular nanodimer the situa-tion has changed when compared to the perfectly circu-lar nanodimer Contrary to the circular case reducing thegap size of the semi-circular nanodimer results now in adecrease of the electric field strength in the gap region[cf Fig 3(a)] Referring to a comparable gap size thisreduced field strength is attributed on one hand to theenlarged field volume that is bounded by the increasedsize of the very proximate particle surfaces in the propergap region [cf Fig 3(c)] and on the other hand thedecrease is governed by the correspondingly reduced sur-face charge density in this area15 However the spreadin the field distribution has some drawbacks since thelight-matter interaction would be weakened For instancein applications related to Raman spectroscopy the fieldenhancement factor is a very crucial figure for the signalgeneration because of its highly super-linear contributionto the Raman cross sections3ndash4 Similar signal degradationscould also emerge during ongoing experiments because offabrication imperfections

33 Two-Cut Semi-Circular Nanodimer

An alternative technology-related deviation from the per-fect circular dimer shape is represented by the two-cutsemi-circular nanodimer This structure is shown inFigure 1(c) where the cut parts are consecutively arrangednormal to the dimer axis Compared to the previousparticle shapes described in Sections 31 and 32 the res-onance wavelength at 507 nm of a single two-cut semi-circular nanoparticle is blue-shifted by 16 nm comparedto the single circular particle (at 523 nm) and blue-shiftedby 4 nm with respect to the single-cut semi-circular caseat 511 nm An intuitive explanation of the wavelengthshift would obviously refer to the reduced particle volumeNevertheless the situation is more complex as the modessupported by the single structures are hybridized lead-ing to significant changes in the surface charge densitiesAs displayed in Figures 4(b) and (c) the two-cut semi-circular particle structures yield the largest field enhance-ment not in gap region but in the rounded corners dueto the lightning-rod effect This is different compared tothe nanodimer described in Section 31 where the fieldmainly concentrates in the gap region Please note thatfor the nanodimers described in Sections 32 and 33 thefield strength in the gap depend very much on the cutdepth h (ie on the area of the flat proximate interfaces)namely smaller cut depths yield stronger electric fieldsThis also means that (circularly) curved particle interfacesare crucial for the field enhancement in nearly touchingnanodimers However small randomly distributed point-like defects on the particle interfaces can also help toachieve a large field enhancement1 but such defects arehard to control especially at these interfaces When the

Fig 4 The two-cut semi-circular nanodimer with corresponding par-ticle diameters of 20 nm (a) the spectral response of the electric fieldstrength in the center of the gap as a function of the different gap sizes g(b) the intensity plot of the electric field strength for a single two-cutsemi-circular nanoparticle at resonance (507 nm) (c) the correspondingelectric field distribution for the two-cut semi-circular nanodimer with agap size of g = 2 nm supporting a resonance wavelength of 535 nm Theapparent corners are rounded (r = 2 nm) for numerical reasons

J Comput Theor Nanosci 7 1610ndash1615 2010 1613

RESEARCH

ARTIC

LE

The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers Cui and Erni

gap size g is zero the two particles are supposed to toucheach other and the nanodimer becomes a single particleThe resonance wavelength is slightly red-shifted comparedto the single structure The largest electric field strengthwas then found in the wedge-like groove of the junctionregion showing that also ldquoinverted cornersrdquo may becomeindispensable for a large field enhancement This effect isattributed to local charge separation in the corner of thegroove17

34 Square-Shaped Nanodimer

As a final nanostructure we consider the square-shapednanodimer shown in Figure 1(d) where the particlersquoscross-section consists of two parallel aligned identicalsquares with a side length of 20 nm This is slightly dif-ferent from the nanostructures in the previous Section 33because no curved boundaries exist except for the roundedcorners that are introduced for numerical reasons The res-onance wavelength of the single square lies at 521 nmwhich is slightly blue-shifted by only 2 nm compared tothe perfect circular particle Interestingly the resonancewavelength of the square-shaped nanodimer turned out tobe insensitive (=+2 nm) against decreasing gap sizesdown to 2 nm [cf Fig 5(a)] whereas the three nanodimersof the previous Sections 31 32 and 33 provide signifi-cant redshifts already at gap widths around 4 nm whichincrease to 11 nm 20 nm and 28 nm respectively whenthe gap width reaches a value of 2 nm Generally speakingthe aforementioned insensitivity of the resonance shift mayprovide relaxed parameter tolerances to structure fabrica-tion The reduced resonance shift could also be beneficialto SERS experiments since the spatial mismatch between

Fig 5 The square-shaped nanodimer with a side length of 20 nm(a) the spectral response of the electric field strength in the center of thegap as a function of the different gap sizes g (b) the intensity plot of theelectric field strength for a single square-sized nanoparticle at resonance(521 nm) (c) the corresponding electric field distribution for the square-sized nanodimer with a gap size of g = 2 nm supporting a resonancewavelength of 523 nm The apparent corners are rounded (r = 2 nm) fornumerical reasons

the laser excitation and the resonant Raman excitationcan be easily minimized However due to the poor fieldenhancement in the proper gap region the square-shapednanodimer will never become the primary choice for appli-cations related to Raman spectroscopy As displayed inFigure 5(b) the largest electric field strengths is locatedat the rounded corners Optimizing functional nanodimersaccording to multiple objectives will inevitably lead to aholistic design scenario where simultaneous mechanismssuch as the lightning rod effect and the emerging surfaceplasmon resonances have to be taken into account likeeg in the various well-documented design examples ofbow-tie nanostructures

4 OUTLOOK AND CONCLUSION

Before concluding our numerical design study on the var-ious dimer shapes we want to reflect on the material influ-ence by adding a very preliminary and hence tentativeforecast on how the process related variations of materialcomposition may probably alter future design strategiesAn important issue regarding the design of functionalnanoparticles is the availability of a proper material modelused in the numerical simulation Usually bulk materialmodels based on experimental data14 are applied becausethe dimension of the nanoparticle is still within the scopeof classical theory if the particle size is supposed to belarger than 10 nm However due to different processingsteps in the underlying fabrication technology the chem-ical composition of the metallic nanoparticle might bealtered giving rise to significant changes in its opticalresponse Thus a reliable nanoparticle design has alwaysto account for the temporal history of material composi-tion that goes along with the various processing steps inthe corresponding fabrication technology To underpin thisstatement we have analyzed the compositions of charac-teristic sample namely an AundashIn nanowire that has beenfabricated using a processing technology based on catalyticgrowth The conpositional analysis is carried out in theframework of the Energy Filtered Transmission ElectronMicroscopy (EFTEM) The EFTEM is a typical TEM tech-nique in which only electrons of particular kinetic ener-gies are used for image formation18 and thus giving riseto a spatially resolved elemental mapping The chemicalcomposition of an AundashIn nanowire is shown in Figure 6as well as the shape of the nanowire whose metallic partresembles to one of the semi-cut particle shapes From theEFTEM data we can now conclude that the intentionallypure metallic part rather corresponds to mixture of goldwith an In and As alloy where the stoichiometric goldcomposition has been reduced during processing to a valueslightly below 71 In this case we have estimated theresulting redshift of a spherical nanoparticlersquos resonancewavelength due to the altered metal composition to be inthe order of 12 just by using the Froumlhlich condition in

1614 J Comput Theor Nanosci 7 1610ndash1615 2010

RESEARCH

ARTIC

LE

Cui and Erni The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers

Fig 6 (a) TEM image of the fabricated AundashIn nanowire The white part corresponds to metal and the gray part represents dielectric Red lineindicates the scanning line for the composition measurement The positions (1) and (2) indicate the points where the chemical composition is measured(b) cross line counts along the aforementioned red line (c) measured elements at the corresponding positions (1) and (2)

the framework of a simple Drude model Even if this isa very straightforward measure it provides a valuable esti-mate of the discussed material influence not to mentionthe existing lack of reliable optical material models forsuch specific alloys We therefore conclude that the mate-rial properties of functional nanostructures have always tobe retrieved according to the different processing stepsThe punch line of this story is that reliable design pro-cedures may now become extremely challenging becauseany of such designs will unconditionally rely on suitablytailored material models A detailed analysis of this spe-cific nanowire will be presented elsewhereTo summarize we have numerically investigated the

optical properties of nearly touching plasmonic nanopar-ticle dimers with focus on the field enhancementand resonance wavelengths when taking the fabricationimperfectionsndashnamely the shape deviationsndashinto accountWith this regard four typical nanostructures have beenconsidered the circular dimer the one-cut semi-circulardimer the two-cut semi-circular dimer and the square-shaped dimer We have observed that the optical prop-erties such as the field enhancement and the resonancewavelength strongly depend on the particle shape in theproper gap region and furthermore we have identifiedthe square-shaped dimer to be virtually insensitive to gapwidth variations The studies provide important detailsto nano engineering not to mention the issue that thediscrepancy between the practical engineering processand the underlying numerical analysis is suspected toremain a fundamental one This reasoning is also sup-ported when looking to the variation of the materialcomposition during the various processing steps As anexample we used TEM to characterize the compositionsof an AundashIn nanowire that has been grown by catalyticmethods We found that especially the important metallicparts are subject to large stoichiometric variations when

alloyed with different constituents This indicates that inany reliable design procedure the proper determinationand modeling of material properties at the nanoscale willbecome an essential part rendering nanoengineering apretty challenging venture

Acknowledgments We thank Dr Zi-An Li for theTEM characterization of the nanowire

References

1 W H Zhang X D Cui B S Yeo T Schmid Ch Hafner andR Zenobi Nano Lett 7 1401 (2007)

2 W L Barnes A Dereux and T W Ebbesen Nature 424 824(2003)

3 A Campion and P Kambhampati Chem Soc Rev 27 241 (1998)4 N Hayazawa Y Inouye Z Sekkat and S Kawata Opt Commun

183 333 (2006)5 S Nie and S R Emory Science 275 1102 (1997)6 M Danckwerts and L Novotny Phys Rev Lett 98 026104 (2007)7 H X Xu J Aizpurua M Kall and P Apell Phys Rev E 62 4318

(2000)8 J Aizpurua G W Bryant L J Richter and F J Garcia de Abajo

Phys Rev B 71 235420 (2005)9 D P Fromm A Sundaramurthy P James G Kino and W E

Moerner Nano Lett 4 957 (2004)10 P Nordlander C Oubre E Pordan K Li and M I Stockman

Nano Lett 4 899 (2004)11 W Y Li P H C Camargo X M Lu and Y N Xia Nano Lett

9 485 (2009)12 E Moreno D Erni Ch Hafner and R Vahldieck J Opt Soc

Am A 19 101 (2002)13 J Zuloaga E Prodan and P Nordlander Nano Lett 9 887

(2009)14 P B Johnson and R W Christy Phys Rev B 6 4370 (1972)15 I Romero J Aizpurua G W Bryant and F Javier Garcia de Abajo

Opt Express 14 9988 (2006)16 M Li H Schnablegger and S Mann Nature 402 393 (1999)17 R Kappeler D Erni X Cui and L Novotny J Comput Theor

Nanosci 4 686 (2007)18 L Reimer Advances in Electronics and Electron Physics edited by

P W Hawkes Academic Press London (1991) Vol 81

Received 25 September 2009 Accepted 27 October 2009

J Comput Theor Nanosci 7 1610ndash1615 2010 1615

RESEARCH

ARTIC

LE

The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers Cui and Erni

the two 2D particles As an excitation we used a planewave that is polarized in the y-direction and propagatesfrom below along the x-direction having an electric fieldstrength of Ey = 1 Vm The choice of polarization is basedon the fact that the electric field in the gap is dramaticallyenhanced for a polarization along the axis of dimer8ndash10

Four configurations are considered in this work (cf Fig 1)(a) a circular dimer(b) a one-cut semi-circular dimer(c) a two-cut semi-circular dimer and(d) a square-shaped dimer

The four configurations stand for the typical shapes offabricated nanostructures The radii of the circular parti-cle regions are chosen to be r = 10 nm the cut is set ath= 4 nm and the apparent corners are rounded accordingto a radius of 2 nm for numerical reasons

3 RESULTS AND DISCUSSION

31 Circular Nanodimer

First we consider the circular structure shown inFigure 1(a) The nanodimer consists of two identical Aucircular particles Note that the optical properties andthe field enhancement of such nanodimers have alreadybeen intensively investigated Since nanodimers are oftenproposed as SERS substrate for sensing applications1ndash5

field enhancement and resonance wavelength are importantissues to be considered For spherical and circular singleparticles the resonance wavelength can be precisely pre-dicted by Mie theory For the proper dimer structure Mietheory is not directly applicable and the analysis has to bebased on numerical methods The resonance wavelengthof a single Au particle having a diameter of 20 nm islocated at 523 nm as depicted in Figures 2(a) and (b) The

Fig 2 The circular nanodimer with corresponding particle diameters of20 nm (a) the spectral response of the electric field strength in the centerof the gap as a function of the different gap sizes g (b) the intensityplot of the electric field strength for a single circular nanoparticle at res-onance (523 nm) (c) the corresponding electric field distribution for thecircular nanodimer with a gap size of g = 2 nm supporting a resonancewavelength of 528 nm

resonance wavelength of the corresponding nanodimer isconsiderably red-shifted (eg by 5 nm) for gap sizes gwell below 4 nm compared to the single particle resonance[cf Figs 2(a) and (c)] The resonance shift is partiallydue to mode coupling between the two particles10 whereboth the coupling and the redshift get inevitably smallerwith increasing gap size g A similar behavior is alsoobserved in bow-tie structures and other paired nanopar-ticle configurations9 The nearly touching dimer thereforeconstitutes an attractive nanoparticle configuration with apronounced wavelength tunability ranging from the visibleto the near infrared10 and hence is best suited for sens-ing and nano-antenna applications As already mentionedshrinking gap sizes g below 4 nm typically yield a con-tinuous redshift in the resonance wavelength15 Howevera recent analysis which has comprised all the emergingquantum effects has revealed that for gap sizes smallerthan 05 nm the resonances are getting blue-shifted due tothe emergent charge transfer13

In current research it has been observed that smallgaps are helpful to provide large field enhancement fac-tors due to both the increased charge concentration andthe strong mode coupling in the gap region8ndash10 When thegap size shrinks below 1 nm charge transfer mechanismsare assumed to occur between the two particles The elec-tric field is then getting ldquoshort-circuitedrdquo by the emer-gent electron-tunneling channel leading to a considerabledecrease in the field strength13 It is worth mentioning thatstill no experimental results are available for such smallgap sizes because of lacking resolution within state-of-the-art fabrication technologies Besides the analysis and thepromotion of novel functionalities in optical nanodimersthis is good reason to further conduct numerical experi-ments in order to quantify the requirements that rendersfuture nanoprocessing feasible

32 One-Cut Semi-Circular Nanodimer

Because of the parameter variations in the fabrica-tion technology nanoparticle shapes usually deviate fromtheir intended geometry even though some self-assemblytechniques16 are applied during particle processing A typ-ical example regarding the deformation of spherical (andcircular) nanodimers refers to the fact that the poten-tial contact regions are not ideally curved The spe-cific area rather supports facet-like structures due toproximity and lag effects in the involved etching andgrowing processing steps11 which can be approximatedwhen executing a corresponding cut at the circularnanodimerThe electric field distribution around such semi-circular

nanoparticle respective such nanodimer is shown inFigure 3 Since the overall size of the nanodimer is wellin the sub-wavelength range it responds to the impinging

1612 J Comput Theor Nanosci 7 1610ndash1615 2010

RESEARCH

ARTIC

LE

Cui and Erni The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers

Fig 3 The one-cut semi-circular nanodimer with corresponding parti-cle diameters of 20 nm (a) the spectral response of the electric fieldstrength in the center of the gap as a function of the different gap sizes g(b) the intensity plot of the electric field strength for a single semi-circularnanoparticle at resonance (511 nm) (c) the corresponding electric fielddistribution for the semi-circular nanodimer with a gap size of g = 2 nmsupporting a resonance wavelength of 532 nm The apparent corners arerounded (r = 2 nm) for numerical reasons

field as a dipole showing a similar behavior as theperfectly circular particle [cf Fig 2(b)] except for anincreased resonance wavelength at 532 nm Comparingnow the resonance wavelength of the deformed single par-ticle at 511 nm [cf Fig 3(b)] with the perfectly circu-lar one at 523 nm reveals a blueshift of 12 nm Thisblue shift is not solely attributed to the smaller particlevolume but rather to the altered distribution of the sur-face charge density The blueshift increases with expand-ing cut depth h and even more resonances emerge inthe spectral range as a result of mode hybridization10

For the corresponding semi-circular nanodimer the situa-tion has changed when compared to the perfectly circu-lar nanodimer Contrary to the circular case reducing thegap size of the semi-circular nanodimer results now in adecrease of the electric field strength in the gap region[cf Fig 3(a)] Referring to a comparable gap size thisreduced field strength is attributed on one hand to theenlarged field volume that is bounded by the increasedsize of the very proximate particle surfaces in the propergap region [cf Fig 3(c)] and on the other hand thedecrease is governed by the correspondingly reduced sur-face charge density in this area15 However the spreadin the field distribution has some drawbacks since thelight-matter interaction would be weakened For instancein applications related to Raman spectroscopy the fieldenhancement factor is a very crucial figure for the signalgeneration because of its highly super-linear contributionto the Raman cross sections3ndash4 Similar signal degradationscould also emerge during ongoing experiments because offabrication imperfections

33 Two-Cut Semi-Circular Nanodimer

An alternative technology-related deviation from the per-fect circular dimer shape is represented by the two-cutsemi-circular nanodimer This structure is shown inFigure 1(c) where the cut parts are consecutively arrangednormal to the dimer axis Compared to the previousparticle shapes described in Sections 31 and 32 the res-onance wavelength at 507 nm of a single two-cut semi-circular nanoparticle is blue-shifted by 16 nm comparedto the single circular particle (at 523 nm) and blue-shiftedby 4 nm with respect to the single-cut semi-circular caseat 511 nm An intuitive explanation of the wavelengthshift would obviously refer to the reduced particle volumeNevertheless the situation is more complex as the modessupported by the single structures are hybridized lead-ing to significant changes in the surface charge densitiesAs displayed in Figures 4(b) and (c) the two-cut semi-circular particle structures yield the largest field enhance-ment not in gap region but in the rounded corners dueto the lightning-rod effect This is different compared tothe nanodimer described in Section 31 where the fieldmainly concentrates in the gap region Please note thatfor the nanodimers described in Sections 32 and 33 thefield strength in the gap depend very much on the cutdepth h (ie on the area of the flat proximate interfaces)namely smaller cut depths yield stronger electric fieldsThis also means that (circularly) curved particle interfacesare crucial for the field enhancement in nearly touchingnanodimers However small randomly distributed point-like defects on the particle interfaces can also help toachieve a large field enhancement1 but such defects arehard to control especially at these interfaces When the

Fig 4 The two-cut semi-circular nanodimer with corresponding par-ticle diameters of 20 nm (a) the spectral response of the electric fieldstrength in the center of the gap as a function of the different gap sizes g(b) the intensity plot of the electric field strength for a single two-cutsemi-circular nanoparticle at resonance (507 nm) (c) the correspondingelectric field distribution for the two-cut semi-circular nanodimer with agap size of g = 2 nm supporting a resonance wavelength of 535 nm Theapparent corners are rounded (r = 2 nm) for numerical reasons

J Comput Theor Nanosci 7 1610ndash1615 2010 1613

RESEARCH

ARTIC

LE

The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers Cui and Erni

gap size g is zero the two particles are supposed to toucheach other and the nanodimer becomes a single particleThe resonance wavelength is slightly red-shifted comparedto the single structure The largest electric field strengthwas then found in the wedge-like groove of the junctionregion showing that also ldquoinverted cornersrdquo may becomeindispensable for a large field enhancement This effect isattributed to local charge separation in the corner of thegroove17

34 Square-Shaped Nanodimer

As a final nanostructure we consider the square-shapednanodimer shown in Figure 1(d) where the particlersquoscross-section consists of two parallel aligned identicalsquares with a side length of 20 nm This is slightly dif-ferent from the nanostructures in the previous Section 33because no curved boundaries exist except for the roundedcorners that are introduced for numerical reasons The res-onance wavelength of the single square lies at 521 nmwhich is slightly blue-shifted by only 2 nm compared tothe perfect circular particle Interestingly the resonancewavelength of the square-shaped nanodimer turned out tobe insensitive (=+2 nm) against decreasing gap sizesdown to 2 nm [cf Fig 5(a)] whereas the three nanodimersof the previous Sections 31 32 and 33 provide signifi-cant redshifts already at gap widths around 4 nm whichincrease to 11 nm 20 nm and 28 nm respectively whenthe gap width reaches a value of 2 nm Generally speakingthe aforementioned insensitivity of the resonance shift mayprovide relaxed parameter tolerances to structure fabrica-tion The reduced resonance shift could also be beneficialto SERS experiments since the spatial mismatch between

Fig 5 The square-shaped nanodimer with a side length of 20 nm(a) the spectral response of the electric field strength in the center of thegap as a function of the different gap sizes g (b) the intensity plot of theelectric field strength for a single square-sized nanoparticle at resonance(521 nm) (c) the corresponding electric field distribution for the square-sized nanodimer with a gap size of g = 2 nm supporting a resonancewavelength of 523 nm The apparent corners are rounded (r = 2 nm) fornumerical reasons

the laser excitation and the resonant Raman excitationcan be easily minimized However due to the poor fieldenhancement in the proper gap region the square-shapednanodimer will never become the primary choice for appli-cations related to Raman spectroscopy As displayed inFigure 5(b) the largest electric field strengths is locatedat the rounded corners Optimizing functional nanodimersaccording to multiple objectives will inevitably lead to aholistic design scenario where simultaneous mechanismssuch as the lightning rod effect and the emerging surfaceplasmon resonances have to be taken into account likeeg in the various well-documented design examples ofbow-tie nanostructures

4 OUTLOOK AND CONCLUSION

Before concluding our numerical design study on the var-ious dimer shapes we want to reflect on the material influ-ence by adding a very preliminary and hence tentativeforecast on how the process related variations of materialcomposition may probably alter future design strategiesAn important issue regarding the design of functionalnanoparticles is the availability of a proper material modelused in the numerical simulation Usually bulk materialmodels based on experimental data14 are applied becausethe dimension of the nanoparticle is still within the scopeof classical theory if the particle size is supposed to belarger than 10 nm However due to different processingsteps in the underlying fabrication technology the chem-ical composition of the metallic nanoparticle might bealtered giving rise to significant changes in its opticalresponse Thus a reliable nanoparticle design has alwaysto account for the temporal history of material composi-tion that goes along with the various processing steps inthe corresponding fabrication technology To underpin thisstatement we have analyzed the compositions of charac-teristic sample namely an AundashIn nanowire that has beenfabricated using a processing technology based on catalyticgrowth The conpositional analysis is carried out in theframework of the Energy Filtered Transmission ElectronMicroscopy (EFTEM) The EFTEM is a typical TEM tech-nique in which only electrons of particular kinetic ener-gies are used for image formation18 and thus giving riseto a spatially resolved elemental mapping The chemicalcomposition of an AundashIn nanowire is shown in Figure 6as well as the shape of the nanowire whose metallic partresembles to one of the semi-cut particle shapes From theEFTEM data we can now conclude that the intentionallypure metallic part rather corresponds to mixture of goldwith an In and As alloy where the stoichiometric goldcomposition has been reduced during processing to a valueslightly below 71 In this case we have estimated theresulting redshift of a spherical nanoparticlersquos resonancewavelength due to the altered metal composition to be inthe order of 12 just by using the Froumlhlich condition in

1614 J Comput Theor Nanosci 7 1610ndash1615 2010

RESEARCH

ARTIC

LE

Cui and Erni The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers

Fig 6 (a) TEM image of the fabricated AundashIn nanowire The white part corresponds to metal and the gray part represents dielectric Red lineindicates the scanning line for the composition measurement The positions (1) and (2) indicate the points where the chemical composition is measured(b) cross line counts along the aforementioned red line (c) measured elements at the corresponding positions (1) and (2)

the framework of a simple Drude model Even if this isa very straightforward measure it provides a valuable esti-mate of the discussed material influence not to mentionthe existing lack of reliable optical material models forsuch specific alloys We therefore conclude that the mate-rial properties of functional nanostructures have always tobe retrieved according to the different processing stepsThe punch line of this story is that reliable design pro-cedures may now become extremely challenging becauseany of such designs will unconditionally rely on suitablytailored material models A detailed analysis of this spe-cific nanowire will be presented elsewhereTo summarize we have numerically investigated the

optical properties of nearly touching plasmonic nanopar-ticle dimers with focus on the field enhancementand resonance wavelengths when taking the fabricationimperfectionsndashnamely the shape deviationsndashinto accountWith this regard four typical nanostructures have beenconsidered the circular dimer the one-cut semi-circulardimer the two-cut semi-circular dimer and the square-shaped dimer We have observed that the optical prop-erties such as the field enhancement and the resonancewavelength strongly depend on the particle shape in theproper gap region and furthermore we have identifiedthe square-shaped dimer to be virtually insensitive to gapwidth variations The studies provide important detailsto nano engineering not to mention the issue that thediscrepancy between the practical engineering processand the underlying numerical analysis is suspected toremain a fundamental one This reasoning is also sup-ported when looking to the variation of the materialcomposition during the various processing steps As anexample we used TEM to characterize the compositionsof an AundashIn nanowire that has been grown by catalyticmethods We found that especially the important metallicparts are subject to large stoichiometric variations when

alloyed with different constituents This indicates that inany reliable design procedure the proper determinationand modeling of material properties at the nanoscale willbecome an essential part rendering nanoengineering apretty challenging venture

Acknowledgments We thank Dr Zi-An Li for theTEM characterization of the nanowire

References

1 W H Zhang X D Cui B S Yeo T Schmid Ch Hafner andR Zenobi Nano Lett 7 1401 (2007)

2 W L Barnes A Dereux and T W Ebbesen Nature 424 824(2003)

3 A Campion and P Kambhampati Chem Soc Rev 27 241 (1998)4 N Hayazawa Y Inouye Z Sekkat and S Kawata Opt Commun

183 333 (2006)5 S Nie and S R Emory Science 275 1102 (1997)6 M Danckwerts and L Novotny Phys Rev Lett 98 026104 (2007)7 H X Xu J Aizpurua M Kall and P Apell Phys Rev E 62 4318

(2000)8 J Aizpurua G W Bryant L J Richter and F J Garcia de Abajo

Phys Rev B 71 235420 (2005)9 D P Fromm A Sundaramurthy P James G Kino and W E

Moerner Nano Lett 4 957 (2004)10 P Nordlander C Oubre E Pordan K Li and M I Stockman

Nano Lett 4 899 (2004)11 W Y Li P H C Camargo X M Lu and Y N Xia Nano Lett

9 485 (2009)12 E Moreno D Erni Ch Hafner and R Vahldieck J Opt Soc

Am A 19 101 (2002)13 J Zuloaga E Prodan and P Nordlander Nano Lett 9 887

(2009)14 P B Johnson and R W Christy Phys Rev B 6 4370 (1972)15 I Romero J Aizpurua G W Bryant and F Javier Garcia de Abajo

Opt Express 14 9988 (2006)16 M Li H Schnablegger and S Mann Nature 402 393 (1999)17 R Kappeler D Erni X Cui and L Novotny J Comput Theor

Nanosci 4 686 (2007)18 L Reimer Advances in Electronics and Electron Physics edited by

P W Hawkes Academic Press London (1991) Vol 81

Received 25 September 2009 Accepted 27 October 2009

J Comput Theor Nanosci 7 1610ndash1615 2010 1615

RESEARCH

ARTIC

LE

Cui and Erni The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers

Fig 3 The one-cut semi-circular nanodimer with corresponding parti-cle diameters of 20 nm (a) the spectral response of the electric fieldstrength in the center of the gap as a function of the different gap sizes g(b) the intensity plot of the electric field strength for a single semi-circularnanoparticle at resonance (511 nm) (c) the corresponding electric fielddistribution for the semi-circular nanodimer with a gap size of g = 2 nmsupporting a resonance wavelength of 532 nm The apparent corners arerounded (r = 2 nm) for numerical reasons

field as a dipole showing a similar behavior as theperfectly circular particle [cf Fig 2(b)] except for anincreased resonance wavelength at 532 nm Comparingnow the resonance wavelength of the deformed single par-ticle at 511 nm [cf Fig 3(b)] with the perfectly circu-lar one at 523 nm reveals a blueshift of 12 nm Thisblue shift is not solely attributed to the smaller particlevolume but rather to the altered distribution of the sur-face charge density The blueshift increases with expand-ing cut depth h and even more resonances emerge inthe spectral range as a result of mode hybridization10

For the corresponding semi-circular nanodimer the situa-tion has changed when compared to the perfectly circu-lar nanodimer Contrary to the circular case reducing thegap size of the semi-circular nanodimer results now in adecrease of the electric field strength in the gap region[cf Fig 3(a)] Referring to a comparable gap size thisreduced field strength is attributed on one hand to theenlarged field volume that is bounded by the increasedsize of the very proximate particle surfaces in the propergap region [cf Fig 3(c)] and on the other hand thedecrease is governed by the correspondingly reduced sur-face charge density in this area15 However the spreadin the field distribution has some drawbacks since thelight-matter interaction would be weakened For instancein applications related to Raman spectroscopy the fieldenhancement factor is a very crucial figure for the signalgeneration because of its highly super-linear contributionto the Raman cross sections3ndash4 Similar signal degradationscould also emerge during ongoing experiments because offabrication imperfections

33 Two-Cut Semi-Circular Nanodimer

An alternative technology-related deviation from the per-fect circular dimer shape is represented by the two-cutsemi-circular nanodimer This structure is shown inFigure 1(c) where the cut parts are consecutively arrangednormal to the dimer axis Compared to the previousparticle shapes described in Sections 31 and 32 the res-onance wavelength at 507 nm of a single two-cut semi-circular nanoparticle is blue-shifted by 16 nm comparedto the single circular particle (at 523 nm) and blue-shiftedby 4 nm with respect to the single-cut semi-circular caseat 511 nm An intuitive explanation of the wavelengthshift would obviously refer to the reduced particle volumeNevertheless the situation is more complex as the modessupported by the single structures are hybridized lead-ing to significant changes in the surface charge densitiesAs displayed in Figures 4(b) and (c) the two-cut semi-circular particle structures yield the largest field enhance-ment not in gap region but in the rounded corners dueto the lightning-rod effect This is different compared tothe nanodimer described in Section 31 where the fieldmainly concentrates in the gap region Please note thatfor the nanodimers described in Sections 32 and 33 thefield strength in the gap depend very much on the cutdepth h (ie on the area of the flat proximate interfaces)namely smaller cut depths yield stronger electric fieldsThis also means that (circularly) curved particle interfacesare crucial for the field enhancement in nearly touchingnanodimers However small randomly distributed point-like defects on the particle interfaces can also help toachieve a large field enhancement1 but such defects arehard to control especially at these interfaces When the

Fig 4 The two-cut semi-circular nanodimer with corresponding par-ticle diameters of 20 nm (a) the spectral response of the electric fieldstrength in the center of the gap as a function of the different gap sizes g(b) the intensity plot of the electric field strength for a single two-cutsemi-circular nanoparticle at resonance (507 nm) (c) the correspondingelectric field distribution for the two-cut semi-circular nanodimer with agap size of g = 2 nm supporting a resonance wavelength of 535 nm Theapparent corners are rounded (r = 2 nm) for numerical reasons

J Comput Theor Nanosci 7 1610ndash1615 2010 1613

RESEARCH

ARTIC

LE

The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers Cui and Erni

gap size g is zero the two particles are supposed to toucheach other and the nanodimer becomes a single particleThe resonance wavelength is slightly red-shifted comparedto the single structure The largest electric field strengthwas then found in the wedge-like groove of the junctionregion showing that also ldquoinverted cornersrdquo may becomeindispensable for a large field enhancement This effect isattributed to local charge separation in the corner of thegroove17

34 Square-Shaped Nanodimer

As a final nanostructure we consider the square-shapednanodimer shown in Figure 1(d) where the particlersquoscross-section consists of two parallel aligned identicalsquares with a side length of 20 nm This is slightly dif-ferent from the nanostructures in the previous Section 33because no curved boundaries exist except for the roundedcorners that are introduced for numerical reasons The res-onance wavelength of the single square lies at 521 nmwhich is slightly blue-shifted by only 2 nm compared tothe perfect circular particle Interestingly the resonancewavelength of the square-shaped nanodimer turned out tobe insensitive (=+2 nm) against decreasing gap sizesdown to 2 nm [cf Fig 5(a)] whereas the three nanodimersof the previous Sections 31 32 and 33 provide signifi-cant redshifts already at gap widths around 4 nm whichincrease to 11 nm 20 nm and 28 nm respectively whenthe gap width reaches a value of 2 nm Generally speakingthe aforementioned insensitivity of the resonance shift mayprovide relaxed parameter tolerances to structure fabrica-tion The reduced resonance shift could also be beneficialto SERS experiments since the spatial mismatch between

Fig 5 The square-shaped nanodimer with a side length of 20 nm(a) the spectral response of the electric field strength in the center of thegap as a function of the different gap sizes g (b) the intensity plot of theelectric field strength for a single square-sized nanoparticle at resonance(521 nm) (c) the corresponding electric field distribution for the square-sized nanodimer with a gap size of g = 2 nm supporting a resonancewavelength of 523 nm The apparent corners are rounded (r = 2 nm) fornumerical reasons

the laser excitation and the resonant Raman excitationcan be easily minimized However due to the poor fieldenhancement in the proper gap region the square-shapednanodimer will never become the primary choice for appli-cations related to Raman spectroscopy As displayed inFigure 5(b) the largest electric field strengths is locatedat the rounded corners Optimizing functional nanodimersaccording to multiple objectives will inevitably lead to aholistic design scenario where simultaneous mechanismssuch as the lightning rod effect and the emerging surfaceplasmon resonances have to be taken into account likeeg in the various well-documented design examples ofbow-tie nanostructures

4 OUTLOOK AND CONCLUSION

Before concluding our numerical design study on the var-ious dimer shapes we want to reflect on the material influ-ence by adding a very preliminary and hence tentativeforecast on how the process related variations of materialcomposition may probably alter future design strategiesAn important issue regarding the design of functionalnanoparticles is the availability of a proper material modelused in the numerical simulation Usually bulk materialmodels based on experimental data14 are applied becausethe dimension of the nanoparticle is still within the scopeof classical theory if the particle size is supposed to belarger than 10 nm However due to different processingsteps in the underlying fabrication technology the chem-ical composition of the metallic nanoparticle might bealtered giving rise to significant changes in its opticalresponse Thus a reliable nanoparticle design has alwaysto account for the temporal history of material composi-tion that goes along with the various processing steps inthe corresponding fabrication technology To underpin thisstatement we have analyzed the compositions of charac-teristic sample namely an AundashIn nanowire that has beenfabricated using a processing technology based on catalyticgrowth The conpositional analysis is carried out in theframework of the Energy Filtered Transmission ElectronMicroscopy (EFTEM) The EFTEM is a typical TEM tech-nique in which only electrons of particular kinetic ener-gies are used for image formation18 and thus giving riseto a spatially resolved elemental mapping The chemicalcomposition of an AundashIn nanowire is shown in Figure 6as well as the shape of the nanowire whose metallic partresembles to one of the semi-cut particle shapes From theEFTEM data we can now conclude that the intentionallypure metallic part rather corresponds to mixture of goldwith an In and As alloy where the stoichiometric goldcomposition has been reduced during processing to a valueslightly below 71 In this case we have estimated theresulting redshift of a spherical nanoparticlersquos resonancewavelength due to the altered metal composition to be inthe order of 12 just by using the Froumlhlich condition in

1614 J Comput Theor Nanosci 7 1610ndash1615 2010

RESEARCH

ARTIC

LE

Cui and Erni The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers

Fig 6 (a) TEM image of the fabricated AundashIn nanowire The white part corresponds to metal and the gray part represents dielectric Red lineindicates the scanning line for the composition measurement The positions (1) and (2) indicate the points where the chemical composition is measured(b) cross line counts along the aforementioned red line (c) measured elements at the corresponding positions (1) and (2)

the framework of a simple Drude model Even if this isa very straightforward measure it provides a valuable esti-mate of the discussed material influence not to mentionthe existing lack of reliable optical material models forsuch specific alloys We therefore conclude that the mate-rial properties of functional nanostructures have always tobe retrieved according to the different processing stepsThe punch line of this story is that reliable design pro-cedures may now become extremely challenging becauseany of such designs will unconditionally rely on suitablytailored material models A detailed analysis of this spe-cific nanowire will be presented elsewhereTo summarize we have numerically investigated the

optical properties of nearly touching plasmonic nanopar-ticle dimers with focus on the field enhancementand resonance wavelengths when taking the fabricationimperfectionsndashnamely the shape deviationsndashinto accountWith this regard four typical nanostructures have beenconsidered the circular dimer the one-cut semi-circulardimer the two-cut semi-circular dimer and the square-shaped dimer We have observed that the optical prop-erties such as the field enhancement and the resonancewavelength strongly depend on the particle shape in theproper gap region and furthermore we have identifiedthe square-shaped dimer to be virtually insensitive to gapwidth variations The studies provide important detailsto nano engineering not to mention the issue that thediscrepancy between the practical engineering processand the underlying numerical analysis is suspected toremain a fundamental one This reasoning is also sup-ported when looking to the variation of the materialcomposition during the various processing steps As anexample we used TEM to characterize the compositionsof an AundashIn nanowire that has been grown by catalyticmethods We found that especially the important metallicparts are subject to large stoichiometric variations when

alloyed with different constituents This indicates that inany reliable design procedure the proper determinationand modeling of material properties at the nanoscale willbecome an essential part rendering nanoengineering apretty challenging venture

Acknowledgments We thank Dr Zi-An Li for theTEM characterization of the nanowire

References

1 W H Zhang X D Cui B S Yeo T Schmid Ch Hafner andR Zenobi Nano Lett 7 1401 (2007)

2 W L Barnes A Dereux and T W Ebbesen Nature 424 824(2003)

3 A Campion and P Kambhampati Chem Soc Rev 27 241 (1998)4 N Hayazawa Y Inouye Z Sekkat and S Kawata Opt Commun

183 333 (2006)5 S Nie and S R Emory Science 275 1102 (1997)6 M Danckwerts and L Novotny Phys Rev Lett 98 026104 (2007)7 H X Xu J Aizpurua M Kall and P Apell Phys Rev E 62 4318

(2000)8 J Aizpurua G W Bryant L J Richter and F J Garcia de Abajo

Phys Rev B 71 235420 (2005)9 D P Fromm A Sundaramurthy P James G Kino and W E

Moerner Nano Lett 4 957 (2004)10 P Nordlander C Oubre E Pordan K Li and M I Stockman

Nano Lett 4 899 (2004)11 W Y Li P H C Camargo X M Lu and Y N Xia Nano Lett

9 485 (2009)12 E Moreno D Erni Ch Hafner and R Vahldieck J Opt Soc

Am A 19 101 (2002)13 J Zuloaga E Prodan and P Nordlander Nano Lett 9 887

(2009)14 P B Johnson and R W Christy Phys Rev B 6 4370 (1972)15 I Romero J Aizpurua G W Bryant and F Javier Garcia de Abajo

Opt Express 14 9988 (2006)16 M Li H Schnablegger and S Mann Nature 402 393 (1999)17 R Kappeler D Erni X Cui and L Novotny J Comput Theor

Nanosci 4 686 (2007)18 L Reimer Advances in Electronics and Electron Physics edited by

P W Hawkes Academic Press London (1991) Vol 81

Received 25 September 2009 Accepted 27 October 2009

J Comput Theor Nanosci 7 1610ndash1615 2010 1615

RESEARCH

ARTIC

LE

The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers Cui and Erni

gap size g is zero the two particles are supposed to toucheach other and the nanodimer becomes a single particleThe resonance wavelength is slightly red-shifted comparedto the single structure The largest electric field strengthwas then found in the wedge-like groove of the junctionregion showing that also ldquoinverted cornersrdquo may becomeindispensable for a large field enhancement This effect isattributed to local charge separation in the corner of thegroove17

34 Square-Shaped Nanodimer

As a final nanostructure we consider the square-shapednanodimer shown in Figure 1(d) where the particlersquoscross-section consists of two parallel aligned identicalsquares with a side length of 20 nm This is slightly dif-ferent from the nanostructures in the previous Section 33because no curved boundaries exist except for the roundedcorners that are introduced for numerical reasons The res-onance wavelength of the single square lies at 521 nmwhich is slightly blue-shifted by only 2 nm compared tothe perfect circular particle Interestingly the resonancewavelength of the square-shaped nanodimer turned out tobe insensitive (=+2 nm) against decreasing gap sizesdown to 2 nm [cf Fig 5(a)] whereas the three nanodimersof the previous Sections 31 32 and 33 provide signifi-cant redshifts already at gap widths around 4 nm whichincrease to 11 nm 20 nm and 28 nm respectively whenthe gap width reaches a value of 2 nm Generally speakingthe aforementioned insensitivity of the resonance shift mayprovide relaxed parameter tolerances to structure fabrica-tion The reduced resonance shift could also be beneficialto SERS experiments since the spatial mismatch between

Fig 5 The square-shaped nanodimer with a side length of 20 nm(a) the spectral response of the electric field strength in the center of thegap as a function of the different gap sizes g (b) the intensity plot of theelectric field strength for a single square-sized nanoparticle at resonance(521 nm) (c) the corresponding electric field distribution for the square-sized nanodimer with a gap size of g = 2 nm supporting a resonancewavelength of 523 nm The apparent corners are rounded (r = 2 nm) fornumerical reasons

the laser excitation and the resonant Raman excitationcan be easily minimized However due to the poor fieldenhancement in the proper gap region the square-shapednanodimer will never become the primary choice for appli-cations related to Raman spectroscopy As displayed inFigure 5(b) the largest electric field strengths is locatedat the rounded corners Optimizing functional nanodimersaccording to multiple objectives will inevitably lead to aholistic design scenario where simultaneous mechanismssuch as the lightning rod effect and the emerging surfaceplasmon resonances have to be taken into account likeeg in the various well-documented design examples ofbow-tie nanostructures

4 OUTLOOK AND CONCLUSION

Before concluding our numerical design study on the var-ious dimer shapes we want to reflect on the material influ-ence by adding a very preliminary and hence tentativeforecast on how the process related variations of materialcomposition may probably alter future design strategiesAn important issue regarding the design of functionalnanoparticles is the availability of a proper material modelused in the numerical simulation Usually bulk materialmodels based on experimental data14 are applied becausethe dimension of the nanoparticle is still within the scopeof classical theory if the particle size is supposed to belarger than 10 nm However due to different processingsteps in the underlying fabrication technology the chem-ical composition of the metallic nanoparticle might bealtered giving rise to significant changes in its opticalresponse Thus a reliable nanoparticle design has alwaysto account for the temporal history of material composi-tion that goes along with the various processing steps inthe corresponding fabrication technology To underpin thisstatement we have analyzed the compositions of charac-teristic sample namely an AundashIn nanowire that has beenfabricated using a processing technology based on catalyticgrowth The conpositional analysis is carried out in theframework of the Energy Filtered Transmission ElectronMicroscopy (EFTEM) The EFTEM is a typical TEM tech-nique in which only electrons of particular kinetic ener-gies are used for image formation18 and thus giving riseto a spatially resolved elemental mapping The chemicalcomposition of an AundashIn nanowire is shown in Figure 6as well as the shape of the nanowire whose metallic partresembles to one of the semi-cut particle shapes From theEFTEM data we can now conclude that the intentionallypure metallic part rather corresponds to mixture of goldwith an In and As alloy where the stoichiometric goldcomposition has been reduced during processing to a valueslightly below 71 In this case we have estimated theresulting redshift of a spherical nanoparticlersquos resonancewavelength due to the altered metal composition to be inthe order of 12 just by using the Froumlhlich condition in

1614 J Comput Theor Nanosci 7 1610ndash1615 2010

RESEARCH

ARTIC

LE

Cui and Erni The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers

Fig 6 (a) TEM image of the fabricated AundashIn nanowire The white part corresponds to metal and the gray part represents dielectric Red lineindicates the scanning line for the composition measurement The positions (1) and (2) indicate the points where the chemical composition is measured(b) cross line counts along the aforementioned red line (c) measured elements at the corresponding positions (1) and (2)

the framework of a simple Drude model Even if this isa very straightforward measure it provides a valuable esti-mate of the discussed material influence not to mentionthe existing lack of reliable optical material models forsuch specific alloys We therefore conclude that the mate-rial properties of functional nanostructures have always tobe retrieved according to the different processing stepsThe punch line of this story is that reliable design pro-cedures may now become extremely challenging becauseany of such designs will unconditionally rely on suitablytailored material models A detailed analysis of this spe-cific nanowire will be presented elsewhereTo summarize we have numerically investigated the

optical properties of nearly touching plasmonic nanopar-ticle dimers with focus on the field enhancementand resonance wavelengths when taking the fabricationimperfectionsndashnamely the shape deviationsndashinto accountWith this regard four typical nanostructures have beenconsidered the circular dimer the one-cut semi-circulardimer the two-cut semi-circular dimer and the square-shaped dimer We have observed that the optical prop-erties such as the field enhancement and the resonancewavelength strongly depend on the particle shape in theproper gap region and furthermore we have identifiedthe square-shaped dimer to be virtually insensitive to gapwidth variations The studies provide important detailsto nano engineering not to mention the issue that thediscrepancy between the practical engineering processand the underlying numerical analysis is suspected toremain a fundamental one This reasoning is also sup-ported when looking to the variation of the materialcomposition during the various processing steps As anexample we used TEM to characterize the compositionsof an AundashIn nanowire that has been grown by catalyticmethods We found that especially the important metallicparts are subject to large stoichiometric variations when

alloyed with different constituents This indicates that inany reliable design procedure the proper determinationand modeling of material properties at the nanoscale willbecome an essential part rendering nanoengineering apretty challenging venture

Acknowledgments We thank Dr Zi-An Li for theTEM characterization of the nanowire

References

1 W H Zhang X D Cui B S Yeo T Schmid Ch Hafner andR Zenobi Nano Lett 7 1401 (2007)

2 W L Barnes A Dereux and T W Ebbesen Nature 424 824(2003)

3 A Campion and P Kambhampati Chem Soc Rev 27 241 (1998)4 N Hayazawa Y Inouye Z Sekkat and S Kawata Opt Commun

183 333 (2006)5 S Nie and S R Emory Science 275 1102 (1997)6 M Danckwerts and L Novotny Phys Rev Lett 98 026104 (2007)7 H X Xu J Aizpurua M Kall and P Apell Phys Rev E 62 4318

(2000)8 J Aizpurua G W Bryant L J Richter and F J Garcia de Abajo

Phys Rev B 71 235420 (2005)9 D P Fromm A Sundaramurthy P James G Kino and W E

Moerner Nano Lett 4 957 (2004)10 P Nordlander C Oubre E Pordan K Li and M I Stockman

Nano Lett 4 899 (2004)11 W Y Li P H C Camargo X M Lu and Y N Xia Nano Lett

9 485 (2009)12 E Moreno D Erni Ch Hafner and R Vahldieck J Opt Soc

Am A 19 101 (2002)13 J Zuloaga E Prodan and P Nordlander Nano Lett 9 887

(2009)14 P B Johnson and R W Christy Phys Rev B 6 4370 (1972)15 I Romero J Aizpurua G W Bryant and F Javier Garcia de Abajo

Opt Express 14 9988 (2006)16 M Li H Schnablegger and S Mann Nature 402 393 (1999)17 R Kappeler D Erni X Cui and L Novotny J Comput Theor

Nanosci 4 686 (2007)18 L Reimer Advances in Electronics and Electron Physics edited by

P W Hawkes Academic Press London (1991) Vol 81

Received 25 September 2009 Accepted 27 October 2009

J Comput Theor Nanosci 7 1610ndash1615 2010 1615

RESEARCH

ARTIC

LE

Cui and Erni The Influence of Particle Shapes on the Optical Response of Nearly Touching Plasmonic Nanoparticle Dimers

Fig 6 (a) TEM image of the fabricated AundashIn nanowire The white part corresponds to metal and the gray part represents dielectric Red lineindicates the scanning line for the composition measurement The positions (1) and (2) indicate the points where the chemical composition is measured(b) cross line counts along the aforementioned red line (c) measured elements at the corresponding positions (1) and (2)

the framework of a simple Drude model Even if this isa very straightforward measure it provides a valuable esti-mate of the discussed material influence not to mentionthe existing lack of reliable optical material models forsuch specific alloys We therefore conclude that the mate-rial properties of functional nanostructures have always tobe retrieved according to the different processing stepsThe punch line of this story is that reliable design pro-cedures may now become extremely challenging becauseany of such designs will unconditionally rely on suitablytailored material models A detailed analysis of this spe-cific nanowire will be presented elsewhereTo summarize we have numerically investigated the

optical properties of nearly touching plasmonic nanopar-ticle dimers with focus on the field enhancementand resonance wavelengths when taking the fabricationimperfectionsndashnamely the shape deviationsndashinto accountWith this regard four typical nanostructures have beenconsidered the circular dimer the one-cut semi-circulardimer the two-cut semi-circular dimer and the square-shaped dimer We have observed that the optical prop-erties such as the field enhancement and the resonancewavelength strongly depend on the particle shape in theproper gap region and furthermore we have identifiedthe square-shaped dimer to be virtually insensitive to gapwidth variations The studies provide important detailsto nano engineering not to mention the issue that thediscrepancy between the practical engineering processand the underlying numerical analysis is suspected toremain a fundamental one This reasoning is also sup-ported when looking to the variation of the materialcomposition during the various processing steps As anexample we used TEM to characterize the compositionsof an AundashIn nanowire that has been grown by catalyticmethods We found that especially the important metallicparts are subject to large stoichiometric variations when

alloyed with different constituents This indicates that inany reliable design procedure the proper determinationand modeling of material properties at the nanoscale willbecome an essential part rendering nanoengineering apretty challenging venture

Acknowledgments We thank Dr Zi-An Li for theTEM characterization of the nanowire

References

1 W H Zhang X D Cui B S Yeo T Schmid Ch Hafner andR Zenobi Nano Lett 7 1401 (2007)

2 W L Barnes A Dereux and T W Ebbesen Nature 424 824(2003)

3 A Campion and P Kambhampati Chem Soc Rev 27 241 (1998)4 N Hayazawa Y Inouye Z Sekkat and S Kawata Opt Commun

183 333 (2006)5 S Nie and S R Emory Science 275 1102 (1997)6 M Danckwerts and L Novotny Phys Rev Lett 98 026104 (2007)7 H X Xu J Aizpurua M Kall and P Apell Phys Rev E 62 4318

(2000)8 J Aizpurua G W Bryant L J Richter and F J Garcia de Abajo

Phys Rev B 71 235420 (2005)9 D P Fromm A Sundaramurthy P James G Kino and W E

Moerner Nano Lett 4 957 (2004)10 P Nordlander C Oubre E Pordan K Li and M I Stockman

Nano Lett 4 899 (2004)11 W Y Li P H C Camargo X M Lu and Y N Xia Nano Lett

9 485 (2009)12 E Moreno D Erni Ch Hafner and R Vahldieck J Opt Soc

Am A 19 101 (2002)13 J Zuloaga E Prodan and P Nordlander Nano Lett 9 887

(2009)14 P B Johnson and R W Christy Phys Rev B 6 4370 (1972)15 I Romero J Aizpurua G W Bryant and F Javier Garcia de Abajo

Opt Express 14 9988 (2006)16 M Li H Schnablegger and S Mann Nature 402 393 (1999)17 R Kappeler D Erni X Cui and L Novotny J Comput Theor

Nanosci 4 686 (2007)18 L Reimer Advances in Electronics and Electron Physics edited by

P W Hawkes Academic Press London (1991) Vol 81

Received 25 September 2009 Accepted 27 October 2009

J Comput Theor Nanosci 7 1610ndash1615 2010 1615