surface electromagnetic modes in prolate spheroids of gold, aluminum, and copper

2552 J. Opt. Soc. Am. B/Vol. 5, No. 12/December 1988 Bloemer et al.

Surface electromagnetic modes in prolate spheroids of gold,aluminum, and copper

M. J. Bloemer, M. C. Buncick, R. J. Warmack, and T. L. Ferrell

Health and Safety Research Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6123

Received December 22, 1987; accepted August 3, 1988Optical absorbance measurements are presented for discontinuous films of gold, aluminum, and copper as afunction of wavelength, polarization, and angle. Anomolous peaks in the absorbance spectra are observed for allthree metals. Two peaks are detected for gold, and one peak is detected for both aluminum and copper. Compari-son with electromagnetic theory is made on the basis that the microstructures can be modeled as prolate spheroids.The theoretical model is tested experimentally for various minor-to-major-axis ratios of the spheroid.

INTRODUCTION

The anomolous optical properties of discontinuous metalfilms are widely known.' Much of the structure in opticaltransmission and reflection spectra has been attributed tocollective electronic excitations (surface plasmons 2) local-ized on each particle. In the surface-plasmon model theenergy of the excitation depends on the shape of the particleas well as on the dielectric response of the metal. Thesamples are generally prepared by a normal-incidence evap-oration onto a smooth substrate. The resulting metal islandfilm consists of small, irregularly shaped particles. Anneal-ing the sample tends to reshape the particles uniformly intowhat can be modeled as oblate spheroids. 3 Other investiga-tors4 prepare the samples by an evaporation in an inert gas ata few-milliTorr pressure, yielding spherically shaped parti-cles.

Much interest has developed in recent years in surface-enhanced Raman scattering (SERS).5 Collections of smallmetallic particles, such as metal island films, have providedlarge enhancements to the Raman signal. Experimentaland theoretical studies with SERS indicate that an electro-magnetic (surface-plasmon) model is appropriate to explainpart of the enhancement. Researchers 6 7 have been corre-lating optical properties of the samples with SERS measure-ments. The largest enhancements occur when the surface-plasmon resonance is near the incident photon energy. Therequirement of a well-characterized sample is critical toidentifying the various elements involved in SERS.

For many materials, prolate spheroids used as SERS sub-strates have an advantage over oblate spheroids because ofthe lightning-rod effect.8 The small radius of curvature atthe tips of a prolate spheroid tends to concentrate the elec-tromagnetic fields leading to large SERS signals. The un-derstanding of the optical properties of prolate spheroids issimplified if the major axes of the spheroids are parallel.Liao9 has developed a method to produce ordered arrays ofprolate spheroids on the sides of silicon dioxide posts. Themethod of Liao will routinely produce structures with mi-nor-to-major-axis ratios of R = 0.3. The fabrication proce-dure is somewhat complicated and requires microlithograph-

ic techniques. Liao and Stern' 0 detected Raman signals byusing aluminum microstructures. The SERS signal wasattributed to the lightning-rod effect and not to resonantbehavior in the spheroid.

In the present paper we present absorbance spectra ofgold, aluminum, and copper microstructures. All three mi-crostructures display resonant behavior in the visible re-gime. The spectra are compared with an electrodynamictheory in which the particles are modeled as prolate spher-oids. The resonant energy is observed to depend on the axisratio of the spheroid.

The prolate spheroids are produced by an oblique evapo-ration onto a dielectric substrate with a microscopicallyrough surface. The simple technique provides macroscopicassemblies of prolate spheroids with parallel major axes. Asimilar technique was employed in Ref. 11 to produce silverprolate spheroids with axis ratios of R = 0.14. In Ref. 11 aconducting layer of tin oxide on quartz was chosen in orderto test the effect of an applied field on the growth process.In the experiment described below a layer of calcium fluo-ride provides the roughness. The uncomplicated procedureprovides adjustable aspect ratios and large eccentricities.Also, because the lightning-rod effect is a geometrical factorthat depends strongly on the eccentricity, the ability to varyparticle elongation should be ideally suited for SERS stud-ies.

THEORY

A theory to describe the absorbance spectra of prolate spher-oids is given in Ref. 11. Briefly, Laplace's equation is solvedin prolate spheroidal coordinates. Implicit in the analysis isthe assumption of nonretarded electrodynamics of a particlethat does not interact with neighboring particles or the sub-strate. Applying the electrodynamic boundary conditionsto the solutions inside and outside the spheroid, we obtainthe dispersion relation for prolate spheroids:

E(W) = Em(W) = Pim(17) Qi m'(fl)Q1,.(710)Plf(1 ) | 7=,.0

(1)

0740-3224/88/122552-08$02.00 © 1988 Optical Society of America

Vol. 5, No. 12/December 1988/J. Opt. Soc. Am. B 2553

C04-'

C

L

4-H

Ua)'-4

0)

-0.

-2.

-4.

-6.

-8.2.0 3.0

Shape Parameter

4.0

Fig. 1. The dielectric function for the two dipole modes of surface-plasmon oscillations on prolate spheroids as a function of shapeparameter n. Large values of the shape parameter correspond tospheres, and small values correspond to needles.

where Pi~m(?) and Ql,m(t7) are associated Legendre functionsof the real argument. The prime denotes the derivativeswith respect to the argument. The prolate spheroidal coor-dinate 2 determines the shape of the spheroid. The coordi-nate -q is related to the minor-to-major-axis ratio to R = (q2

-

1)1

/2/n. The samples to be considered below have axis ratios

of R < 0.5. Because of the nonlinear relationship between Xj

and R, the corresponding values of in are those close to 1.The dispersion relation is plotted in Fig. 1. The 1 = 1, m = 1

and 1 = 1, m = 0 modes correspond to dipoles oriented alongthe minor and major axes of the spheroid, respectively. Theresonant energies of the modes are found from the experi-mentally determined values of the dielectric function c(w).Equation (1) is easily modified for the spheroid embedded ina dielectric medium.

The scattering and absorption cross sections are found byassuming that the prolate spheroid is immersed in a uniformbut time-varying field. From the cross sections, the absor-bance of the spheroid can be found for various polarizationsand angles of incident light. For a particular material, theaxis ratio of the spheroid determines the peak positions inthe absorbance spectra. The experimental absorbancepeaks tend to be wider than the theoretical curves. AGaussian distribution of shapes is included in the theoreticalspectra for better agreement. Electron micrographs of thesamples show this to be a reasonable approach.

EXPERIMENT

The microstructures were formed by a simple procedure ofoblique evaporation onto a rough surface. First, cleanquartz slides (1.2 cm X 2.5 cm) were mounted in a cryo-pumped, electron-beam evaporator for a normal-incidence

evaporation of calcium flouride. A mass thickness of 210nm, as measured by a crystal thickness monitor, was deposit-ed at a pressure of 10-6 Torr and a rate of 1 nm/sec. Elec-tron micrographs of various thicknesses of calcium fluorideare presented in Ref. 12. The hills and canyons of thecalcium fluoride surface produce a unique mask for highlyoblique evaporations.

Next, the sample was mounted upon a large-area brassplate, which is held 30 cm above the evaporation crucible.The sample is positioned at near grazing incidence withrespect to the crucible, -88°. The angle was determined bysighting along the surface of the plate to a known distancefrom the center of the crucible. For each metal three sam-ples were made, with increasing amounts of material depos-ited. The crystal in the thickness monitor was positionedwith its surface perpendicular to the path of the metal vapor.Mass thicknesses measured by the thickness monitor were100, 150, and 200 nm, deposited at a rate of 1-1.5 nm/sec.

Because the samples are nonconductors, a conducting lay-er of gold was sputtered onto the surface so that the micro-structures could be viewed in a scanning electron micro-graph. A minimum mass thickness of approximately 7 nmof gold was required to prevent severe charging problems.The absorbance spectra were recorded without the goldoverlayer. Figure 2 displays electron micrographs of themicrostructures formed by a 200-nm evaporation of gold,aluminum, and copper. The angle of observation is alongthe sample normal. The aluminum particles are shorter andwider than are the gold or copper particles. It was notedthat gold and copper tend to form a bead in the cruciblewhen heated. Aluminum was observed to wet the surface ofthe crucible, thereby presenting a larger solid angle to apoint on the substrate. The effect of the charge dimensionson the microstructure shape is unclear, but it may contributeto the variation of the aluminum microstructures from cop-per and gold. Surface tension and surface mobility 0 ofaluminum on the calcium fluoride could also be importantfactors.

The samples with less material deposited have spheroidswith approximately the same minor-axis length as in Fig. 2but with shorter major-axis lengths. Micrographs of thesamples at other angles seem to indicate that the major axisof the spheroids is oriented -70° from the surface normal.

Absorbance spectra were obtained within 4 h of samplefabrication by using a double-beam spectrophotometer. 13

Because spectrophotometers are often used to measure theabsorbance (A = -logi0T) of liquids, T refers to the internaltransmittance. For our samples, T is taken to be the trans-mittance.

The reference and sample mounts could be rotated toobtain angular dependence. The two light beams were po-larized by Glan prisms. Samples were oriented in the spec-trophotometer with the major axes of the spheroids in theplane of incidence. Thus, for s-polarized light (electric fieldvector perpendicular to the plane of incidence) incidentupon the sample, the electric field vector is always orientedalong the minor axis of the spheroid.

In the path of one beam we placed a reference of 210 nm ofcalcium fluoride on quartz. The intensity of the light thatpasses through the reference is considered to be the intensityincident upon the sample with the spheroids. The reference

Bloemer et al.

2554 J. Opt. Soc. Am. B/Vol. 5, No. 12/December 1988

a I helps to account for effects of the substrate on the measuredspectra. It will not account for depolarizing effects of thesubstrate, which tend to decrease the energy of the resonantmodes in the particle.'4 The reflectivity at the microstruc-ture-calcium fluoride interface could also influence themeasured absorbance.

RESULTS

GoldFigure 3 is the absorbance of s-polarized light for the samplewith gold spheroids illustrated in Fig. 2a. Incident anglesnoted in the spectra are with respect to the major axes of theprolate spheroids. All three absorbance curves have a peaknear 500 nm, corresponding to the excitation of the = 1, m= 1 surface-plasmon mode.

Figure 4 is the absorbance of p-polarized light for the goldsample displayed in Fig. 2a. At a 90° angle of incidence theelectric field vector is parallel to the major axis of the gold

b ~~~~73! F J ~~~~~~~spheroid, and only the 1= 1, m = 0 mode is stimulated. Forsmaller angles there is a decrease in the electric field compo-nent along the major axis of the spheroid with a correspond-ing decrease in peak height. For a 0' angle of incidence withp-polarized light, the incident electric field vector is orientedparallel to the minor axis of the spheroid. Therefore at 06only the short wavelength'( = 1, m = 1) mode is excited.

Figure 5 shows the theoretical absorbance of p-polarizedlight for gold prolate spheroids when the tabulated dielectricfunctions of Hagemann et a.15 are used. (This reference isused as a source for all the dielectric functions mentionedbelow.) To match the peak positions of the experimentalabsorbance, the gold spheroids are modeled as having aminor-to-major-axis ratio of R = 0.17. The peak heightshave been normalized to the experimental spectra.

Is]=_ _ _ _ 0_ f 90C

0 . 8~~~~~~~~~2

a) 0.6_C.)

L0~0.4-0

Fig. 2. Micrographs of structures formed by a 200-nm oblique 0. 0 .evaporation onto a calcium fluoride layer: a, gold; b, aluminum; c, 300 500 700 900copper. Samples were coated with a 7-nm layer of gold to prevent Wavelength (nm)charging in the scanning electron micrograph. The observation Fig. 3. Absorbance of s-polarized light for sample with gold prolateangle is along the sample normal. The length of the bar corre- spheroids. Sample is that of Fig. 2a. Inset indicates incidentsponds to 199 nm. angles.

Bloerner et al.


i

0

ajvC)

MC_Co

in.0

0.

0.

0

0.0L300 500 700

Wavelength (nm)900

Fig. 4. Experimental absorbance of p-polarized light for the goldprolate spheroids of Fig. 2a.

I.

0.

a.)

C.0L0U,.0

0

0.

0.

O.300 500 700

Wavelength (nm)

in the calculation. The effect of the substrate can be seen bycomparing the 00 absorbance of Figs. 3 and 4. According tothe theory outlined above, these spectra should be identical.Two discrepancies are evident. The first is that the overallabsorbance is higher for s-polarized light. We believe this isdue to variation in reflectivity at the microstructure-sub-strate interface for s-polarized and p-polarized light. Thesecond feature noted is the slight red shift in peak positionfor s-polarized light compared with p-polarized light. Thetheoretical peak position for the I = 1, m = 1 mode agreesfavorably with the p-polarized data of Fig. 4. Deviationsfrom a perfect spheroidal shape will result in variations inthe peak positions. In such a case the geometrical crosssection, when viewed along the major axis of the particle,would be elliptical and not circular. Another possibility is adepolarizing substrate interaction,' 4 which is stronger fordipoles oriented parallel to the surface than for those orient-ed perpendicular to the surface. A substrate interactionwould also help to account for the observed red shift (i.e., thediscrepancy in experimental and theoretical values of axisratio R) in the = 1, m = 0 mode for a given particle shape.

Also in the experimental spectra, the background absor-bance is larger than theory indicates. The theory considersonly surface effects. The higher experimental backgroundcould be due to scattering from the bulk or to a higherreflectivity at the sample's microstructure-substrate inter-face compared with the reference.

Figure 6 is a plot of Eq. (1), made by using the dielectricfunction of gold. The shape parameter al has been related tothe minor-to-major-axis ratio of the prolate spheroid. Theshort-wavelength mode is nearly independent of the axisratio, in contrast to the long-wavelength mode. By adjust-ing the amount of gold that is evaporated, is is possible to

I.

0.

0~1-ICO.

U,

x-H

900

Fig. 5. Theoretical absorbance of p-polarized light for gold prolatespheroids with a 0.17 mean axis ratio.

From the micrograph of Fig. 2a, an average value of R 0.19 appears to be appropriate, but it is difficult to deter-mine the exact ratio from the micrograph. The sampleswere coated with a thin layer of gold to prevent charging inthe microscope. The coating could lead to distortions in theactual shape of the spheroid. For these elongated particles,an uncertainty in the minor-axis length will cause R to fluc-tuate more strongly than uncertainties in the major-axislength.

In addition, the effect of the substrate was not considered

0.

0.

0.

0. 0L300 500 700 900

Wavelength (nm)

Fig. 6. Dispersion relation for the resonant dipole modes for a goldprolate spheroid as a function of the minor-to-major-axis ratio.The short-wavelength I = 1, m = 1 mode and the long-wavelength 1= 1, m = 0 mode merge for a spherical particle.

Bloemer et al.


.

0 .

a)

U

CoID.00(I).0

0 .

0 .

0

0 .300 500 700

Wavelength (nm)

ratio. The red-shifted curves are for an aluminum spheroidembedded in aluminum oxide. For the alumina, a value of 3for the real part of the dielectric function and a value of 0 forthe imaginary part were used in the calculation.

Figure 10 is the absorbance spectra of the three aluminumsamples formed by mass thickness evaporations of 100, 150,and 200 nm. The spectra with peaks are for p-polarized

I

0.

a)cC.0L0U).0900

Fig. 7. Experimental absorbance of three samples for p-polarizedlight incident at 900 with respect to the major axis of the spheroid.The samples were fabricated by evaporating a, 200-; b, 150-; and c,100-nm mass thicknesses of gold.

vary the major-axis length of the spheroid and hence theresonant energies of the dipole modes.

Figure 7 is the experimental absorbance of p-polarizedlight for the three samples with gold spheroids. The angle ofincidence is 90° with respect to the major axis of the spher-oid. The major-axis length of the spheroid increases withthe mass thickness of the film as measured by the thicknessmonitor. As the major axes of the spheroids lengthen, theresonant energy of the = 1, m = 0 mode decreases.

Figure 8 is the theoretical absorbance corresponding tothe experimental spectra in Fig. 7. The increase in absor-bance for spheroids of longer major axes is due to an increasein the size of the spheroid. For the spheroids formed byevaporations of 100 and 150-nm mass thicknesses, the mi-crographs indicate slightly larger values of R (0.04 larger)than those predicted by theory.

AluminumAluminum has a much higher surface-plasmon energy thangold or copper. A small aluminum sphere has a resonancenear 8.75 eV. Small here means that the dipole approxima-tion is appropriate and that retardation effects are negligi-ble. An aluminum prolate spheroid must have a small mi-nor-to-major-axis ratio in order to have a surface-plasmonresonance in the visible.

If the particles are embedded in a dielectric medium, thiswill tend to red shift the resonant energies of the dipolemodes. A fresh aluminum surface readily forms an oxidelayer even in vacuum. The oxidation process advances rap-idly at first and then slows as the oxide thickness increases,forming a diffusion barrier. 16

Figure 9 is a plot of the dispersion relation for the dipolemodes of an aluminum prolate spheroid in vacuum and in analumina medium as a function of the minor-to-major-axis

0.

0.41

0.21

0.0_300 500 700 900

Wavelength (nm)

Fig. 8. Theoretical absorbance of p-polarized light for gold prolatespheroids with mean axis ratios of a, 0.17; b, 0.20; and c, 0.28. Angleof incidence is 90° with respect to the major axis of the spheroid.

I.

0.

0-H4J

U)-Hl

x

0.

0 .

0 .

0.100 300 500 700 900

Wavelength (nm)

Fig. 9. Plot of the dispersion relation for the dipole modes of analuminum prolate spheroid as a function of the minor-to-major-axisratio: solid curves, spheroid in vacuum; dashed curves, spheroid inaluminum oxide.

Bloemer et al.


O.

0 .

CDU.C:(a.0C_0)En.0

0.

0

500 700Wavelength (nm)

900

Fig. 10. Experimental absorbance of the three aluminum samplesfor p-polarized light incident at 900: a, 200-; b, 150-; c, 100-nm massthickness evaporation. The dotted curve shows the typical absor-bance of s-polarized light for all three samples.

light incident at 90° with respect to the major axis of thespheroid and the electric field vector parallel to the majoraxis. The spectrum not displaying a peak (dotted curve)shows the typical absorbance of s-polarized light at anyincident angle.

The micrographs indicate an average minor-to-major-axisratio of R 0.28 for the sample formed by evaporating 200nm of aluminum and an axis ratio of R 0.45 for the 100-nmevaporation. The experimental variation in the mean axisratio and absorbance resonance of the three aluminum sam-ples tends to follow the theory for an aluminum spheroidembedded in alumina rather-than an aluminum spheroid invacuum. This result is somewhat surprising consideringthat the oxide layer is only 3-4 nm thick. Batson 1 7 hasexamined the shift in the surface-plasmon energy for alumi-num spheres as a function of the oxide thickness. Theextent of the shift depends on the curvature of the sphere aswell as on the oxide thickness. The lightning-rod effect ofthe = 1, m = 0 mode in prolate spheroids tends to concen-trate the fields at high-curvature areas on the spheroid. Anoxide layer at areas of high curvature could cause significantrelaxation of the = 1, m = 0 plasmon mode. AlthoughBatson's experimental data concern plasmon modes of > 1,it is interesting to note that his experimental data for singlespheres display a red shift larger than might be expected.Although no quantitative details are available to support thearguments given above, it is reasonable to expect that anoxide layer on a particle might have a different effect froman oxide layer on a flat film.

In Fig. 10 the red shift in the peak position for largeraspect ratios clearly indicates the excitation of the = 1, m =0 surface-plasmon mode. The = 1, m = 1 mode is notpresent in the s-polarized spectra because the resonance is ata much shorter wavelength.

CopperFigure 11 is the plot of the dispersion relation for the = 1modes for a copper prolate spheroid in vaccum. The shoul-der near 525 nm in Fig. 11 is due to a small variation in thereal part of the dielectric response of copper as a function ofenergy.

Figure 12 is the absorbance of p-polarized light incident at900 for the three copper samples. As in gold and aluminum,the resonant energy of the = 1, m = 0 mode of copperdecreases as the major-axis length increases. To match theexperimental and theoretical peak positions in the absor-bance spectra of copper spheroids, the theory requiresslightly smaller values of R (approximately 0.03) than themicrographs indicate. Although the absorbance measure-ments were made within 4 h of the copper evaporation, it ispossible that some oxide had built up on the copper parti-cles, and consequently the absorbance spectra would beslightly shifted from that of uncoated particles.

Also, Fig. 12 shows a typical absorbance spectrum for s-polarized light. No peak was observed for the 1 = 1, m = 1mode in any of the spectra for the three samples of copper.Examination of the dielectric function of copper shows thatin the energy range of the = 1, m = 1 mode, the value of theimaginary part is large. The I = 1, m = 1 mode is discernablein the gold absorbance spectra, but the mode is highlydamped because of the onset of an interband transition.The imaginary part of the dielectric function for copper islarger than that of gold at their corresponding energies forthe = 1, m = 1 mode.

The spectra of gold, aluminum, and copper indicate slight-ly smaller values of minor-to-major-axis ratios than are pre-dicted by theory. These discrepancies may be accounted forin several ways. Coating the samples with a conductinglayer for viewing in a scanning electron micrograph can alterthe apparent shapes of the particles. Also, a depolarizing

I

0.

0.- ,

fu

-4

0

0.

0.

0.500 700

Wavelength (nm)

Fig. 11. Plot of the dispersion relation for a copper prolate spher-oid as a function of the minor-to-major-axis ratio.

Bloemer et al.

0.2--.

0. 0-300


1.

0.

wLC.C(U.0L0(I,.0

0.

0.

0.2- c

300 500 700 900Wavelength (nm)

Fig. 12. Experimental absorbance of p-polarized light incident at900 with respect to the major axes of the spheroids for the threecopper samples formed by a, 200-; b, 150-; and c, 100-nm massthickness evaporation. The dotted curve is the typical absorbanceof s-polarized light at any incident angle.

substrate interaction can lead to a red shift of the modes.Another important consideration is the significance of retar-dation. 8"19 Possible retardation effects on absorbance spec-tra include a red shift of the resonance, a broadening 'of theresonance, and excitation of higher-order modes. The ex-perimental spectra do not indicate an excitation of higher-order modes. For aluminum, higher-order modes are in theUV; for gold and copper, the modes can be strongly damped.There is a definite trend for the experimental absorbancewidth to broaden as the spheroids' major-axis length in-creases. Nonretarded electrodynamic theory also predicts abroadening for the samples studied in the present paper.For small values of the axis ratio, the resonant energy of the 1= 1, m = 0 mode is sensitive to small changes in shape (Figs.6, 9, and 11). Thus the same Gaussian distribution ofshapes applied to a sample with small minor-to-major-axisratios has a larger width compared with a sample with largeratios. This is illustrated in Fig. 8, in which all three spectrahave been generated by the same Gaussian distribution butare centered at different values of minor-to-major-axis ra-tios. The spectral broadening noted in Fig. 8 will not ac-count for the experimentally observed broadening of Fig. 7.Electron micrographs indicate that the particle shape distri-bution broadens in proportion to the amount of materialdeposited upon the sample. Closer agreement of theory andexperiment would result if the width of the Gaussian distri-bution were increased for the longer spheroids. We have notattempted to perform a detailed analysis of shape distribu-tions for each sample. The slight red shift of the absorbancepeaks might well be due to retardation effects, especially forthe long spheroids of Fig. 2. For the samples of the presentpaper, the difficulty in determining conclusively the effectsof retardation is the fact that the spheroid shape, as well asthe size, changes with the amount of material deposited.

CONCLUSION

Discontinuous films of gold, aluminum, and copper havebeen fabricated by an oblique evaporation onto a roughsubstrate. Electron micrographs indicate that the particlescan be modeled as prolate spheroids with the major axesoriented at -70' with respect to the sample normal. Anoma-lous peaks are observed in the absorbance spectra of all threemetals. Comparison with electromagnetic scattering theoryshows that the two absorbance peaks in gold are due to theexcitation of the I = 1, m = 0 and I = 1, m = 1 dipole surface-plasmon modes. The = 1, m = 0 mode is observed foraluminum and copper microstructures. The = 1, m = 1mode for aluminum is in the UV, and for copper it is stronglydamped. The resonant energy of the = 1, m = 0 mode isobserved to depend on the aspect ratio of the spheroid for allthree metals. Reasonable agreement of experimental parti-cle shapes from electron micrographs and theory is obtainedfor copper and gold, but agreement for aluminum requiresthe spheroid to be embedded in an oxide.

ACKNOWLEDGMENTS

This research was sponsored by the Office of Energy Storageand Distribution, Electric Energy Systems Program, and theOffice of Health and Environmental Research, U.S. Depart-ment of Energy, under contract DE-AC05-840R21400 withMartin Marietta Energy Systems, Inc. This research wassupported in part by an appointment to the PostgraduateResearch Training Program under contract DE-AC05-760R00033 between the U.S. Department of Energy andOak Ridge Associated Universities.

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surface electromagnetic modes in prolate spheroids of gold, aluminum, and copper

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