Electron Energy-Loss Spectroscopy Calculation in Finite-Difference Time-Domain Package

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Electron Energy-Loss Spectroscopy Calculation in Finite-DifferenceTime-Domain Package

Yang Cao,, Alejandro Manjavacas,*,, Nicolas Large,*,,, and Peter Nordlander*,,

Department of Physics and Astronomy, Department of Electrical and Computer Engineering, and Laboratory for Nanophotonics,Rice University, 6100 Main Street, Houston, Texas 77005, United States

*S Supporting Information

ABSTRACT: Electron energy-loss spectroscopy (EELS) is a unique tool that isextensively used to investigate the plasmonic response of metallic nanostructures. Wepresent here a novel approach for EELS calculations using the nite-difference time-domain (FDTD) method (EELS-FDTD). We benchmark our approach by directcomparison with results from the well-established boundary element method (BEM)and published experimental results. In particular, we compute EELS spectra for

spherical nanoparticles, nanoparticle dimers, nanodisks supported by varioussubstrates, and a gold bowtie antenna on a silicon nitride substrate. Our EELS-FDTD method can be easily extended to more complex geometries and congurations. This implementation can also be directlyexported beyond the FDTD framework and implemented in other Maxwell s equation solvers.

KEYWORDS: numerical recipe, electron energy-loss spectroscopy (EELS), nite-difference time-domain (FDTD),plasmonic nanostructures, gold nanoparticle, silver dimer, gold nanodisk, bowtie antennas

E nergy-loss spectroscopy using fast electrons was used inthe rst experimental detection of surface plasmons inmetals.14 Since these pioneering studies, electron energy-lossspectroscopy (EELS) has become a unique tool for probingsurface plasmons of metallic nanostructures with unprece-dented spatial (


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vicinity of its trajectory (cf. Figure1a). This, in turn, induces an

electric eldEind

that acts back on the electron, exerting a forcethat produces the energy loss. This energy loss can be writtenas9

= =

E e t t tv E r[ ( ), ]d ( )deEELSind

0 EELS

wheree is the elementary charge, re(t) represents the electrontrajectory, and

=

+

e e t tv E r( ) Re{ [ ( ), ]}d

( )

i teEELS

ind

bulk (1)

is the energy loss probability per unit of frequency.9 The

second term of this expression,

bulk, represents the bulk lossprobability. This contribution can be calculated using thefollowing analytical expression9

=

e L

c

c q

cIm

1ln

/

/bulk

2

2

2

2

2

c

2 2

2 2

where L is the length of the electron trajectory inside themedium, is the medium dielectric permittivity, and qc 1[(meout)

2 + (/)2]1/2 is the cutoff momentumdetermined by the electron mass me, the electron velocity ,and the collection angleoutof the microscope. This expressionis valid within the local response approximation, in which onlylow enough momentum transfers belowqcare collected. In the

remainder of this paper, we do not consider the bulkcontribution. In addition, and without loss of generality, weassume the electron trajectory to be in the (x,z) plane, parallelto z-axis, and separated from the origin by the impactparameterb, so v= and re(t) = (b,0,t).

The electric eld created by an arbitrary currentdistribution(such as a beam of electrons) can be written as33

= t iE r G r r j r r[ ( ), ] 4 ( , , ) ( , )d (2)in terms of the Green tensor of Maxwells equationsG(r,r,)(note that we use Gaussian units). This quantity is dened asthe solution of

+ =

c cr G r r r r I

( , ) ( , , )

1

( )

2

2 2

where(r,) is the permittivity of the medium and Iis the unittensor. Using the electron current density relevant for EELS

j(r,) = e(x b)(y)eiz/ and eq2we can rewrite theloss probability (eq1) as9

=

e z z

G z z z z

( ) 4

cos ( )

Im[ ( , , )]d dzz

EELS

2

ind(3)

Here, Gzzind(z,z,) = [G(z,z,) G0(z,z,)] z, with

G0(r,r,) being the Green tensor for vacuum (to simplifythe notation we have omitted the lateral spatial coordinates x =

bandy= 0 in the arguments of the Green functions and relatedquantities). In the derivation of this expression, we have alsoused the reciprocity property of the Green tensor G(r,r,) =GT(r,r,). Interestingly,Gzz

ind(z,z,) can be obtained from thez-component of the electric eld induced at position z, by anelectric dipole of amplitude p(z,) placed at z and orientedalongz-axis

=

G z zE z

p z( , , )

1

4

( , )

( , )zz

zind2

ind

Using this expression, we can rewrite eq3 as

=

e z z

E z

p zz z

( ) cos ( )

Im ( , )

( , )d dz

EELS

2

2

ind

(4)

Therefore, all we need to do in order to obtain the lossprobability is to compute the induced electric eld generated byan electric dipole along the electron trajectory. Notice that eq4involves only the imaginary part of the induced eld, whichremains nite even at the position of the dipole. The use of theinduced eld, in place of the total eld, for nonpenetratingtrajectories is not required by the theoretical formalism, sincean electron cannot produce energy loss in absence of materialstructures. However, due to the nite accuracy of the FDTD

Figure 1. Description of the geometry employed to calculate the electron energy-loss spectrum. (a) Schematics showing the electric eld of anelectron moving with velocityvalong a straight-line trajectory separated from a metallic nanostructure by the impact parameter b. (b, c) Geometryused to calculate the induced electric eld Ez

ind(z,) =Ez(z,) Ez0(z,) for a nanostructure. Here Ezand Ez

0 represent the z-component of theelectric eld generated by an electric dipole p(z,) at position z in presence and in absence of the metallic nano-object, respectively. (d) FDTDsimulation setup showing (i) the FDTD simulation domain with PML boundary conditions, (ii) the nanostructure geometry, (iii) the electric dipolesource, (iv) the 1D monitor used to record the electric eld along the electron path, and (v) the override meshes used to improve the discretizationfor the nanostructure and the electron trajectory.

ACS Photonics Article

DOI: 10.1021/ph500408eACS Photonics2015, 2, 369375

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calculations we choose to work with the induced eld in orderto minimize any numerical instability originating from thecalculation of the elds. In order to calculate the eld, we canemploy any Maxwells equation solver. Here, we choose to

work withthe commercial software package Lumerical FDTDSolutions29 due to its convenient user environment. Thecomputation procedure starts by setting a 3D FDTD simulation

domain with perfectly matched layers (PMLs) to preventspurious reections from outer boundaries (Figure1b,d).34Wethen insert the nanostructure and dene an override mesh thatallows us to manually adjust the mesh grid size in a particularregion. This allows us to optimize the discretization of thephysical object and to improve the convergence. After that, weplace a 1D (linear) monitor along the electron trajectory thatallows us to calculate the electric eld at specic points. Themonitor is extended across the entire simulation domainthrough the PMLs. Again, to improve the convergence and toensure a proper spatial discretization along the electrontrajectory, we place a second override mesh on top of themonitor. Next, we position an electric point dipole p(z,) onthe mesh grid points, aligned with the electron path, and

successively displace it from mesh point to mesh point alongthe electron trajectory fromz=zmax(upper PML) toz= zmin(lower PML). This electric dipole acts as a source in Maxwell sequations. For each position, we record thez-component of thetotal electric eldEz(z,) along the entire monitor (Figure1b).To obtain the induced electric eldEz

ind(z,), we subtract thebackground electric eld Ez

0(z,) from Ez(z,). The formerquantity is calculated using the previously described protocol

but removing all the physical objects (e.g., nano-object andsubstrate) from the simulation domain (Figure 1c). Forsituations where the electron trajectory penetrates into theabsorbing medium (e.g., the metal), one needs to calculate thecorresponding background electric eld generated by an electricdipole placed in an innite space lled with the correspondingmaterial. In practice, this can be accomplished by placing thedipole at the center of a sphere composed of this material,

whose diameter must be chosen large enough to minimize theeld spill out into the surrounding medium. Here we choosethis diameter equal to the monitor length. In metals, as the eldof the electric dipole decays to zero after a few tens ofnanometers, no eld exits the micron-sized metallic domain

which, thus, can be considered as innite from the dipolespoint of view. Incidentally, since FDTD simulations are notstable when an absorbing medium is extended through thePMLs, this forces us to enlarge the simulation domain and themonitor to prevent the absorbing medium to reach into thePMLs. Once the induced electric eld is calculated, the EELS

spectrum is readily computed using eq4. Lumerical and Matlabscripts used for the postprocessing (i.e., calculation of theinduced electric eld and calculation of the integral using eq4)are presented in the Supporting Information, S1. Finally, wenote that the velocity of the electron only enters in eq 4through the cosine function. This allows us to compute EELSspectra for any electron velocity from a single FDTDcalculation. It is important to notice that one can eithercalculate all the dipole positions in one single FDTDcalculation or split each dipole position into smallersubcalculations to increase the parallelization and optimiza-tion.29 The results presented in this paper are performed usingthe later procedure.

ISOLATED NANOSTRUCTURE

We rst illustrate the EELS-FDTD implementation for the caseof an isolated gold nanosphere of diameter a = 160 nm placedin vacuum. To benchmark our method, BEM calculations areperformed using an electron source implemented in the axial-symmetry version of this semianalytical method following theformalism established in ref15. In addition, the simple spherical

geometry allows us to perform analytical (Mie theory).31

Weuse the dielectric function of gold tabulated by Johnson andChristy.35 While the experimental values are used as is in theBEM and Mie calculations, analytical multicoefficient models(MCMs) are used in FDTD to t these experimental data andovercome the difficulty of adapting spectrally tabulateddielectric permittivities into time-domain methods (cf.Supporting Information, S2).29 The nanoparticle center isplaced at the origin of the coordinate system. To ensure a goodconvergence (cf.Supporting Information, S3), we set a monitorlength of 1500 nm (i.e., zmax(min) = 750 nm). The otherparameters used in the Lumerical FDTD Solutions simulationare set as follows: a simulation time of 100 fs with an auto-shutoffparameter of 105, a mesh accuracy of 5 (i.e., 22 mesh

points per wavelength), and mesh renement algorithm set toconformal variant 0allowing for a nonuniform mesh over theFDTD domain. The physical meaning of these proprietaryparameters is provided inSupporting Information, S3. An initialsimulation is performed to calculate the total electric eldEzateach point along the electron trajectory in the frequency range13 eV. A second calculation is then performed for the sameelectric dipole positions in absence of the nanoparticle tocalculate the background electric eld generated by the dipolein vacuumEz

0. Schematics of these two congurations is shownin Figure1b,c. Then, the induced electric eld is computed as

Ezind(z) = Ez(z) Ez

0(z) and inserted into eq4to obtain theEELS spectra. The results of this calculation are shown with

blue lines in Figure2for three different impact parameters:b =

120 nm (away from the nanoparticle, bottom), b = 82 nm (inclose proximity to the nanoparticle, center), and b = 0 nm(through the center of the nanoparticle, top). In all the cases,

we use an electron velocity equal to half of the speed of light invacuumc (i.e., = 0.5c), which corresponds to a kinetic energyof 80 keV.

The results obtained with the EELS-FDTD implementation(blue lines) show a strong peak at 2.4 eV, in very goodagreement with BEM (red lines) and Mie theory (blacktriangles) calculations. This peak corresponds to the quad-rupolar mode of the nanoparticle. The dipolar mode of thenanoparticle only appears as a shoulder in the spectrum atsmaller energies. Interestingly, the position of this peakdepends on the impact parameter due to retardation effects,

originating from the frequency dependence of the eld fromthe electron. In contrast to several published methods,1820,22

the EELS-FDTD implementation can also handle penetratingtrajectories (Figure2, top). However, for such cases one has to

be careful when performing the FDTD simulation.The relative position of the electric dipoles with respect to

the nanoparticle surface can articially introduce numericalerrors. We show inSupporting Information, S5, that when anelectric dipole is placed exactly at the nanoparticle surface, itproduces an overestimation of the EELS signal. This is easilysolved by slightly displacing the entire nanostructure along thez-axis with respect to the monitor mesh grid (typically 1/3mesh step). The discrepancies observed between FDTD and



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BEM results (Figure2, top) can be minimized by increasing thenumber of dipoles used in the calculation, in particular in thepart of the trajectory close to the nanostructure, as shown inSupporting Information, S6.

Although our EELS-FDTD implementation requires per-

forming of a large number of short subcalculations (i.e., one perdipole position), it allows for reaching a better convergencelevel at lower computational cost than DDA18 in speciccongurations (cf. Figure S4 and Table S2 in the SupportingInformation). BEM, due to the axial-symmetry nature of thisparticular problem, allows the user to perform the samecalculation much faster. A comparison of the computationalresource used by FDTD, DDA, and BEM to calculate the EELSspectrum for an impact parameterb= 82 nm (Figure2, center)is provided inSupporting Information, S4. A more general anddetailed comparison between the three methods can be foundin ref36.

INTERACTING NANO-OBJECTS

Nanoparticle dimers have been extensively studied usingEELS.11,13,2224,37,38 For this reason, they constitute anotherideal system to benchmark our EELS-FDTD method. Here, westudy a dimer of closely spaced (i.e., strongly interacting) silvernanospheres of diameter a = 160 nm placed in vacuum. Thegap size is xed to g= 5 nm,and we use the dielectric functionfor silver tabulated by Palik39 in the BEM calculations andMCMs t of the later in the FDTD calculations (SupportingInformation, S2). The other simulation parameters are thesame as for the gold nanosphere (cf. Supporting Information,S3).

Figure3 shows the results obtained with our EELS-FDTDmethod (blue lines) for three different impact parameters:b =

164.5 nm (dimer end, top),b= 82.5 nm (through the center ofone of the NP, center), and b = 0 nm (in the gap, bottom). Inall the cases, we use an electron velocity of= 0.5c(80 keV). Itis well-known that for large gaps the nanoparticles interact only

weakly and the resulting dimer plasmons are essentiallybonding and antibonding combinations of the nanoparticleplasmons of the same multipole order (e.g., l = 1, dipole).40

Here, due to the very small gap-to-diameter ratio (g/a= 0.03),the plasmon modes of the dimer contain contributions from allmultipole orders. When the electrons pass in close proximity tothe dimer end (upper spectra), both FDTD and BEM show theappearance of two weak localized surface plasmon resonances(LSPRs) at 1.5 and 2.3 eV and stronger features at 3.25 and 3.5eV (i, ii, iii, and iv, respectively). Though mixed with higher

values ofl, the 3.5 eV is dominated by the antibonding dipole,while the 3.25 eV feature results from the strong hybridizationof high order modes. The low energy and weak features

observed at 1.5 and 2.3 eV are the longitudinal and transversebonding dipole modes, respectively. The EELS maps associatedwith the four LSPRs (iiv) are shown in Figure3b with a 4 nmspatial resolution. For the sake of simplicity and forcomputational considerations, we here excluded penetratingtrajectories. Although the present dimer is much larger (i.e.,more retardation effects) and has a smaller gap-to-diameterratio (i.e., more hybridization), the results are in goodagreement with EELS measurements reported in litera-ture.11,23,24,37,38 The large energy splitting between the bondingand antibonding dipolar LSPRs is the signature of a strongcoupling.37,38,40 When the electron beam follows a penetratingtrajectory (center spectra) the high energy antibonding dipole

Figure 2.EELS spectra for a gold nanosphere of diameter a = 160 nmcalculated with the EELS-FDTD implementation (blue lines) for threeimpact parameters. From top to bottom: b= 0 nm,b= 82 nm, andb=120 nm. The EELS-FDTD calculations are compared with BEMcalculations (red lines), and with Mie theory (black triangles, only forb > a/2). The spectra for b = 120 nm are multiplied by 10 toimprove the clarity of the gure. The electron velocity is taken equal to0.5c (i.e., 80 keV) in all cases.

Figure 3. (a) EELS spectra for a dimer of silver nanospheres ofdiametera = 160 nm separated by a gap g= 5 nm. Results obtainedwith the EELS-FDTD implementation (blue lines) are compared withBEM calculations (red lines) for three different impact parameters,from top to bottom:b = 164.5 nm, b = 82.5 nm, and b = 0 nm. In allcases, the electron velocity is xed to 0.5c(i.e., 80 keV). Inset: Zoom-in view of the low energy part of the upper spectrum. (b) EELS mapscalculated at (i) 1.5, (ii) 2.3, (iii) 3.25, and (iv) 3.5 eV.



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(3.5 eV) remains strong and the transverse bonding dipole (2.3eV) is strengthened. For a penetrating electron trajectory, alsothe bulk plasmon mode at 3.8 eV can be excited.11,24 However,as we mentioned in theMethodsection we have chosen not toinclude the bulk contribution. For the gap center trajectory(lower spectra), the bonding dipolar dimer mode cannot beexcited and the spectrum is dominated by the antibondingdipolar LSPR.11,23,24 Incidentally, due to retardation effects theposition of this mode is slightly different for the differentexcitation congurations. This spectral shift hasbeen observedexperimentally for dimers of large nanoparticles.24

SUPPORTED NANOSTRUCTURES

EELS experiments require very thin nonabsorbing substrates tominimize energy losses. Typical EELS substrates are made ofmica, silica (SiO2), silicon nitride (Si3N4, SiNx), or carbon (C),and their thickness generally ranges from 5 to 50 nm. Eventhough such thin substrates introduce a negligible EELS

background and are often considered to have a negligible effecton the optical properties of the supported nanostructure (i.e.,small spectral shift with respect to the free-standing

nanostructure), they can be crucial in some situations.41

Forthis reason, we calculate the EELS spectra for gold nanodisks ofdiametera = 50 nm and height h = 15 nm placed on 30 nmthick substrates of different dielectric permittivities: = 1 (free-standing);= 2 (SiO2);

42,43 and = 4 (Si3N4).42,44 As in the

case of the isolated sphere, we use the gold dielectric functiontabulated by Johnson and Christy35 in the BEM calculationsand MCMs t of the later in the FDTD calculations(Supporting Information, S2). The FDTD simulation param-eters are the same as in previous cases (cf. SupportingInformation, S3), and the electron velocity is set to = 0.5c(i.e., 80 keV). The results are shown in Figure 4 for FDTD(blue lines) and BEM (red lines) for an impact parameter b =27 nm (in close proximity to the nanodisk). As expected, the

dipolar LSPR (2.3 eV for = 1) red-shifts with increasingpermittivity of the substrate (1.97 eV for = 4). Interestingly, ithas to be noticed a change in the LSPR line shape for = 4.This effect, along with the spectral shift clearly shows that even

very thin substrates can have a signicant impact on the EELSspectrum and may need to be included in the simulations.

Finally, to highlight the power and the exibility of ourEELS-FDTD implementation, we perform EELS calculationsfor a supported bowtie antenna. This complex structure iscomposed of two gold equilateral triangles with a lateral lengtha = 80 nm, a heighth = 15 nm, separated by a gap g= 4 nm.The gold bowtie structure is placed on top of a 30 nm thick SiNsubstrate (calculations for other substrates are shown inSupporting Information, S7). We also include in the simulationa 2.5 nm chromium (Cr) adhesion layer. The dielectricpermittivity of SiN is assumed to be constant and equal to5.5,45 while the corresponding one for Cr is described byMCMs t of tabulated data (Supporting Information, S2).39

The FDTD simulation parameters are given in SupportingInformation, S3.

The results of this simulation are shown in Figure 5. There,we observe that for edge excitations (Figure 5a) the spectra

display two LSPRs (i, iv) located at 1.27 and 2.39 eV,respectively. In this case, different impact parameters produce

very different intensity ratios for these LSPRs. On the otherhand when electron trajectory crosses the center of the gap(Figure5b) the EELS spectrum displays two distinct LSPRs (ii,iii) at 1.68 and 2.17 eV, respectively, with relative intensitiesthat are much less dependent on the impact parameter. Whendisplacing the electron trajectory off-axis, the LSPR at 2.17 eV(iii) becomes weaker progressively, while the one at 1.68 eV(ii) remains unchanged. All these results, along with the resultsfor a single triangular prism (Supporting Information, S8), arein good agreementwith the experimental observations by Yangand co-workers.10,12 The small discrepancies, mainly in the line

widths, are related to the presence of a thicker Cr adhesionlayer in our calculations, which is known to introduce a

Figure 4.EELS spectra for a gold nanodisk of diameter a= 50 nm andheight h = 15 nm placed on top of a 30 nm thick substrate withdielectric permittivity = 1, 2, and 4 calculated with the EELS-FDTDimplementation (blue lines) and BEM (red lines). The impactparameter is xed tob = 27 nm and the electron velocity is 0.5 c(i.e.,80 keV).

Figure 5.EELS spectra for a gold bowtie antenna calculated with theEELS-FDTD implementation. Each triangle has a lateral lengtha = 80nm, height h = 15 nm, and gap g= 4 nm. The bowtie antenna issupported by a 30 nm thick SiN substrate. A 2.5 nm chromiumadhesion layer is included. The electron velocity is taken equal to 0.5 c(i.e., 80 keV) in all cases. (a) EELS spectra for edge excitation withfour different impact parameters: 2 (blue), 5 (purple), 10 (red), and15 nm (black; cf. inset). (b) EELS spectra for gap excitation with vedifferent impact parameters: 0 (blue), 2 (purple), 5 (red), 10 (brown),and 15 nm (black; cf. inset). (c) EELS maps calculated at (i) 1.27, (ii)1.68, (iii) 2.17, and (iv) 2.39 eV.



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broadening and a red-shift of the LSPRs (cf. SupportingInformation, S8).46,47

Figure5c shows the EELS maps corresponding to the fourLSPRs (iiv) calculated using our EELS-FDTD implementa-tion with a 2 nm spatial resolution. Similarly to the dimer, wehere choose to exclude penetrating trajectories. The nature ofthe LSP modes can straightforwardly be determined from thesemaps. Modes (i) and (ii) correspond to the dipolar bonding(bright) and antibonding (dark) modes, respectively. The mapfor 2.39 eV shows strong EELS signal from each bowtie edgeunderlining the high-order nature of mode (iv). These resultsare also in excellent quantitative agreement with the recentstudies by Yang and co-workers.10,12 Finally, mode (iii) displaysa EELS signal which is strongly localized at the bowtie gap. Thisspatial connement directly correlates with the rapid vanishingof mode (iii) when the electron trajectory is displaced off-gap(Figure5b). Interestingly, this mode was not imaged by Duanet al. because their excitation geometry was off-gap excitation.10

CONCLUSIONS

We have presented a simple procedure to calculate the energy

loss probability of fast electrons interacting with metallicnanostructures. Although this method can be implementedwith any Maxwells equation solver we have chosen here towork with the commercial package Lumerical FDTDSolutions29 due to the exibility of the FDTD method andits user-friendly environment. Contrary to most of the well-established methods, we have shown that this implementationcan deal with both penetrating and nonpenetrating trajectoriesand nanostructures of arbitrary geometries and morphologies,including substrates and adhesion layers. We have bench-marked our EELS-FDTD implementation by comparing theresults with the well-established BEM method for differentrepresentative nanostructures, such as nanospheres, nano-particle dimers, and a nanodisk supported by a substrate.

Furthermore, we have applied this method to study the EELSspectrum of a complex system consisting of a supported bowtieantenna, and we have mapped the LSPR modes of the latter.Our EELS-FDTD method provides a simple and convenientapproach for the calculation of EELS spectra and maps fromcomplex nanostructures of arbitrary shape and composition.

ASSOCIATED CONTENT

*S Supporting Information

(S1) Lumerical and Matlab scripts for postprocessing and EELSspectrum computation. (S2) Dielectric function of metals:Multicoefficient models in FDTD. (S3) Convergence of theEELS spectra: FDTD Lumerical parameters. (S4) Computa-tional resource: EELS-FDTD versus BEM versus e-DDA. (S5)Penetrating trajectory: Numerical error and workaround. (S6)Convergence of the EELS spectra: Electron path meshing. (S7)EELS spectra of a bowtie antenna: Substrate effect. (S8) EELSspectra of a single gold triangular prism: Effect of the Cr layer.This material is available free of charge via the Internet athttp://pubs.acs.org.

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected].*E-mail: [email protected].*E-mail: [email protected].

Present AddressDepartment of Chemistry, Northwestern University, Evan-ston, IL, United States (N.L.).

Notes

The authors declare no competing nancial interest.

ACKNOWLEDGMENTS

The authors thank F. J. Garca de Abajo for his insight andstimulating discussions. This work was supported by the Robert

A. Welch Foundation under Grant C-1222, the Cyberinfras-tructure for Computational Research funded by NSF underGrant CNS-0821727, and by the Data Analysis and Visual-ization Cyberinfrastructure under NSF Grant OCI-0959097.

A.M. acknowledges support from the Welch Foundation underthe J. Evans Attwell-Welch Fellowship for Nanoscale Research,administrated by the Richard E. Smalley Institute for NanoscaleScience and Technology (Grant L-C-004).

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(17) Geuquet, N.; Henrard, L. EELS and optical response of a noblemetal nanoparticle in the frame of a discrete dipole approximation.Ultramicroscopy 2010, 110, 1075.

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Kneipp, H.; Kneipp, K. Electron energy loss and one- and two-photonexcited SERS probing of hot plasmonic silver nanoaggregates.Plasmonics2013, 2, 763.

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(27) Das, P.; Chini, T. K.; Pond, J. Probing higher order surfaceplasmon modes on individual truncated tetrahedral gold nanoparticleusing cathodoluminescence imaging and spectroscopy combined withFDTD simulations. J. Phys. Chem. C2012, 116, 15610.

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response of single gold nanostars. Nanotechnology 2013,24, 405704.(29) Lumerical Solutions, Inc.; http://www.lumerical.com/tcad-products/fdtd/.

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(36) Myroshnychenko, V.; Rodrguez-Fernandez, J.; Pastoriza-Santos,I.; Funston, A. M.; Novo, C.; Mulvaney, P.; Liz-Marzan, L. M.; Garcade Abajo, F. J. Modelling the optical response of gold nanoparticles.Chem. Soc. Rev. 2008, 37, 17921805.

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(38) Kadkhodazadeh, S.; de Lasson, J. R.; Beleggia, M.; Kneipp, H.;Wagner, J. B.; Kneipp, K. Scaling of the surface plasmon resonance in

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Opt. 1982, 21, 10691072.(43) Gao, L.; Lemarchand, F.; Lequime, M. Refractive index

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Supporting information for:

Electron Energy-Loss Spectroscopy Calculation in

Finite-Difference Time-Domain Package

Yang Cao,, Alejandro Manjavacas,,, Nicolas Large,,,, and Peter

Nordlander,,,

Department of Physics and Astronomy, Department of Electrical and Computer

Engineering, and Laboratory for Nanophotonics, Rice University, 6100 Main Street,

Houston, Texas 77005, United States

E-mail: [email protected]; [email protected]; [email protected]

S1. Lumerical and Matlab scripts for postprocessing and EELS spectrumcomputation

S2. Dielectric function of metals: Multicoefficient models in FDTD

S3. Convergence of the EELS spectra: FDTD Lumerical parameters

S4. Computational resource: EELS-FDTD vs BEM vs e-DDA

S5. Penetrating trajectory: Numerical error and workaround

S6. Convergence of the EELS spectra: Electron path meshing

S7. EELS spectra of a bowtie antenna: Substrate effect

S8. EELS spectra of a single gold triangular prism: Effect of the Cr layer

To whom correspondence should be addressedDepartment of Physics and AstronomyECE DepartmentLANP, Rice UniversityCurrent address: Department of Chemistry, Northwestern University, Evanston, IL, United States

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S2. Dielectric function of metals: Multicoefficient models in FDTD

To deal with the dispersive nature of optical materials, time-domain based methods typi-cally employ analytical models (e.g.,, Drude, Debye, and Lorentz models) to approximatethe dielectric permittivity. Here, we use multicoefficient models (MCMs), implemented inLumerical FDTD solutions, that rely on a more extensive set of basis functions to better fitdispersion profiles that are not easily described by Drude, Debye, and Lorentz models.

Figure S2: Real and imaginary parts of the dielectric permittivity of (a) gold, (b) silver, and(c) chromium. The squares are the experimental data tabulated by Johnson and ChristyS1

and by PalikS2 and the full lines are the MCM fits used in the FDTD simulations.

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S3. Convergence of the EELS spectra: FDTD Lumerical parame-ters

The override mesh is an object that enables the user to indicate where the interfaces arephysically significant to the problem. It allows for manually setting a finer mesh on particular

regions than the automatic meshing would have generated.The mesh accuracy is an integer from 1 to 8 defining the fineness of the mesh in theFDTD domain. The number of mesh points per wavelength (ppw), where the wavelengthis the shortest wavelength of the simulation bandwidth, is a major consideration for themeshing algorithm. Accuracy 1 to 8 corresponds to a target of 6 to 34 ppw by increment of4 ppw.

The auto-shutoffparameter is a built-in convergence criterion of Lumerical FDTD So-lutions, associated to the total amount of energy remaining in the simulation domain. Thelower the auto-shutoff, the less energy remains, the better the convergence.

More detailed information about the Lumerical FDTD specific parameters (shown inFigure S3 and Table S1) can be found in ref S3.

Figure S3: Convergence of the EELS-FDTD spectrum of a goldS1 nanosphere of diametera= 160 nm. The impact parameter is fixed to b= 82 nm and the electron velocity is takenequal to 0.5c (i.e., 80 keV) in all the cases. (a) FDTD domain size and monitor length; thesimulation domain is taken as cubic. (b) FDTD Lumerical mesh accuracy, ma. (c) FDTDLumerical auto-shutoff parameter. (d) Nanostructure override mesh (dx= dy = dz).

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Table S1: FDTD parameters for: (i) isolated goldS1 nanosphere of diameter a = 160 nm,(ii) dimer of silverS2 nanoparticles of diameter a = 160nm and gap g = 5 nm, (iii) goldS1

nanodisk of diameter a = 50 nm, and thickness h = 15nm on a 30nm thick Si3N4S4,S5

substrate, and (iv) goldS1 bowtie antenna of edge length a = 80 nm, thickness h = 15 nm,and gap g= 4 nm on a 30 nm thick SiNS6 substrate and 2.5 nm chromiumS2 adhesion layer.

FDTD simulation domain Override meshStructure e-path

Sim. time Size Mesh type Mesh Auto- dx, dy, dz dz [fs] [m] (accuracy) refinement shutoff [nm] [nm]

Sphere 100 1 auto non-uniform (5) conformal 0 105 2,2,2 2Dimer 100 1 auto non-uniform (7) conformal 0 106 1.5,2,2 2Disk 100 1 auto non-uniform (5) conformal 0 105 2,2,2 2Bowtie 100 1 auto non-uniform (5) conformal 0 106 1.5,2,2 2

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S4. Computational resource: EELS-FDTD vs BEM vs e-DDA

We calculate the EELS spectrum of an isolated goldS1 nanosphere of diameter a = 160nmwith an impact parameter b = 82 nm, for an energy range from 1 to 4 eV, and an energyresolution of 15 meV. The electron velocity is taken equal to 0.5c (i.e., 80 keV). The EELS-FDTD calculation is performed for a cubic simulation domain size (and monitor length)of 1.5 m, a mesh size dx = dy = dz = 1 nm for the nanoparticle with a mesh accuracyof 5 (i.e., 22 ppw), and an auto-shutoff parameter set to 106. The dipole position stepis fixed at dz = 2 nm. Figure S4a shows the computational time for each dipole positionin 0 z 750 nm (red dots and blue line). The average calculation time obtained forthis calculation is 8.61 min per dipole position (green dashed line), and the median valueis 8.80 min (orange dashed line). However, considerable time reduction can be achieved byreducing the mesh size, the mesh accuracy, and the auto-shutoff parameter while preservingthe convergence of the results (cf. Figure S3). Figure S4b shows the computational time foreach dipole position, for the same calculation, when the FDTD simulation domain size isset to 1 m, the mesh accuracy set to 2 (i.e., 10 ppw), and the auto-shutoff parameter set to

105. The average time is decreased to 4.62 min.

Figure S4: (a) EELS-FDTD computational time for each dipole position in 0 z 750 nm

(red dots and blue line). The mean value is 8.61 min (green dashed line), and the medianvalue is 8.80 min (orange dashed line). (b) Computational time in 0 z 500 nm (reddots and blue line). The mean value is 4.62 min (green dashed line), and the median valueis 4.53 min (orange dashed line). (c) FDTD, BEM, DDA, and Mie EELS spectra of anisolated gold nanosphere of diameter a= 160 nm with an impact parameter b= 82 nm andan electron velocity of 0.5c (i.e., 80 keV). The spectra are calculated with the parametersgiven in Table S2.

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Table S2 summarizes the statistics of the two FDTD calculations and compares themto the results obtained using BEMS7 and DDA.S8 BEM uses 100 boundary elements (i.e.,mesh size of about 2.5 nm) to descretize the profile of the nanoparticle. In DDA, we setan interdipole distance of d = 0.8 nm, corresponding to about 4.2 million dipoles in thenanoparticle. The FDTD, BEM, and DDA calculations are performed on 10 Intel Sandy-

bridge E2670 processors (64-bit, 20 MB Cache, 2.6 Ghz) with 40 GB of available memory(DDR3 1600 MHz).

Table S2: Comparison of the computational resource (time and memory) between FDTD(two calculations), DDA, and BEM. The EELS spectra are calculated for an isolated goldnanosphere of diameter a = 160 nm with an impact parameter b = 82 nm, for an energyrange from 1 to 4 eV, and an energy resolution of 15 meV (Figure S4c). The discretizationparameterdcorresponds to the minimum volume mesh size (FDTD), the boundary elementsize (BEM), and the interdipole distance (DDA). The total time is calculated for each EELScalculation when the sequences are run in serial. FDTD (a) and (b) refers to the panels (a)and (b) in Figure S4.

d Number of Number of Average time Total Average[nm] elements sequences per sequence comput. time memory

[min] [h] [MB]

FDTD (a) 1 6, 011, 499 750 dipole

8.62.1 105 2, 874Yees cells positions

FDTD (b) 1 1, 642, 560 500 dipole

4.60.6 35 1, 659Yees cells positions

DDA 0.8 4, 188, 896 200 energy

6220 >210 15, 135dipoles values

BEM 2.5 100 boundary 200 energy 6.8 s 25 min 14.50elements values

Due to its partitioning in subcalculations, the total computational time of the EELS-FDTD simulation can be drastically reduced by running the subcalculations in parallel (e.g.,using job arrays). Also, as the nanosphere possesses a symmetry with respect to the z-axis,the number of calculations, and hence the total computational time, can be reduced by afactor of 2 (i.e., 375 dipole positions) by using the reciprocity property of the Green tensor.As it can be seen from Figure S4c and Table S2, EELS-FDTD allows for reaching a betterconvergence level at lower computational cost than DDA. Furthermore, it is important tonotice that, in the current conditions, a DDA calculation with a smaller interdipole distance

results in errors and cannot be performed due to the extremely large number of dipoles (over17 millions ford= 0.5 nm). Also, the number of iterations increases with the magnitude ofthe permittivity in DDA, and becomes extremely large for gold in the NIR. This results incomputation time of several hours for each frequency when reaching energies below2.25eV.

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S5. Penetrating trajectory: Numerical error and workaround

Figure S5: EELS spectra for a spherical gold S1 nanoparticle of diameter a= 160 nm calcu-lated with the EELS-FDTD implementation at b= 0 nm (i.e., penetrating trajectory). Theelectron velocity is taken equal to 0.5c (i.e., 80 keV). The override mesh placed along theelectron trajectory discretizes space for the placement of the electric dipoles. Each electricdipole are placed at one of these mesh points. (i) When the physical surface of the nanos-tructure coincides with the override mesh (i.e., when an electric dipole is placed at the NPsurface) the EELS signal is overestimated. This overestimation, introduced by numerical/computational errors, can be corrected by introducing a slight offset between the surface andthe mesh grid as shown in panels (ii) and (iii).

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S6. Convergence of the EELS spectra: Electron path meshing

Figure S6: EELS-FDTD spectra for a goldS1 nanodisk of diametera= 50 nm and thicknessh= 15 nm placed over a 30 nm thick SiO2 substrate with dielectric permittivity = 2.

S5,S9

The impact parameter is fixed to b= 27 nm and the electron velocity is 0.5c (i.e., 80 keV).Convergence of the calculated EELS spectrum can be achieved by increasing the number ofelectric dipoles. This is done by changing the stepdzof the override mesh used along theelectron trajectory. (i) Uniform distribution of electric dipoles along the z-axis (dz= 2 nm).

(ii) Uniform distribution of electric dipoles along the z-axis (dz= 1 nm).

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S7. EELS spectra of a bowtie antenna: Substrate effect

Figure S7: EELS spectra for a goldS1 bowtie antenna of edge length a = 80 nm, thicknessh = 15nm, and gap g = 5 nm on top of a 30 nm thick substrate. We vary the substratedielectric permittivity from 1 to 4. The electron velocity is fixed to 0.5c (i.e., 80 keV) inall the cases. The impact parameter is b = 0 nm (gap center). The bowtie nanostructureappears to be more sensitive to the presence of the substrate than a single triangular prism

(cf. Figure 4). In addition to a significant red-shift of the LSPRs, an increase of also givesrise to the appearance of high order modes.

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S8. EELS spectra of a single gold triangular prism: Effect of theCr layer

Figure S8: EELS spectra for a single goldS1 triangular prism of edge length a= 80 nm and

thickness h= 15 nm, on top of a 30 nm thick SiN

S6

substrate. The nanostructures include a2.5 nm (blue lines) and 1 nm (red lines) chromiumS2 adhesion layer. The electron velocity isfixed to 0.5c(i.e., 80 keV) in all the cases. (a) Edge excitationb = 15 nm. (b) Apex excitationb = 15 nm. The impact parameter b is taken as the distance from the prism end in panel(a) and prism apex in panel (b). The results are in good agreement with the experimentalobservations by Yang and co-workers.S10,S11

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References

(S1) Johnson, P. B.; Christy, R. Optical constants of the noble metals.Phys. Rev. B1972,6, 4370.

(S2) Palik, E. D., Ed.Handbook of Optical Constants of Solids; Academic Press: New York,

1985; Vol. 1.

(S3) Lumerical Solutions, Inc. http://www.lumerical.com/tcad-products/fdtd/.

(S4) Philipp, H. R. Optical properties of silicon nitride. J. Electrochem. Soc. 1973, 120,295300.

(S5) Baak, T. Silicon oxynitride; a material for GRIN optics. Appl. Opt. 1982, 21, 10691072.

(S6) Tanaka, M.; Saida, S.; Tsunashima, Y. Film properties of lowk silicon nitride filmsformed by hexachlorodisilane and ammonia. J. Electrochem. Soc.2000, 147, 22842289.

(S7) Garca de Abajo, F. J.; Howie, A. Retarded field calculation of electron energy loss ininhomogeneous dielectrics. Phys. Rev. B2002, 65, 115418.

(S8) Bigelow, N.; Vaschillo, A.; Iberi, V.; Camden, J. P.; Masiello, D. Characterization ofthe electron- and photon-driven plasmonic excitations of metal nanorods.ACS Nano2012, 6, 7497.

(S9) Gao, L.; Lemarchand, F.; Lequime, M. Refractive index determination of SiO2 layer inthe UV/Vis/NIR range: Spectrophotometric reverse engineering on single and bi-layer

designs. J. Europ. Opt. Soc. Rap. Public. 2013, 8, 13010.

(S10) Koh, A. L.; Fernandez-Domnguez, A. I.; McComb, D. W.; Maier, S. A.; Yang, J. K. W.High-resolution mapping of electron-beam-excited plasmon modes in lithographicallydefined gold nanostructures. Nano Lett. 2011, 11, 13231330.

(S11) Duan, H.; Fernandez-Domnguez, A. I.; Bosman, M.; Maier, S. A.; Yang, J. K. W.Nanoplasmonics: Classical down to the nanometer scale.Nano Lett. 2012, 12, 1683.

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Electron Energy-Loss Spectroscopy Calculation in Finite-Difference Time-Domain Package