On the extinction multipole plasmons in gold nanorods

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Journal of Quantitative Spectroscopy & Radiative Transfer 107 (2007) 306–314 www.elsevier.com/locate/jqsrt

On the extinction multipole plasmons in gold nanorods Boris N. Khlebtsova, Andrei Melnikova, Nikolai G. Khlebtsova,b, a

Institute of Biochemistry and Physiology of Plants and Microorganisms, Russian Academy of Sciences, 13 Pr. Entuziastov, Saratov 410049, Russia b Saratov State University, 83 Astrakhanskaya Str., Saratov 410026, Russia Received 5 March 2007; accepted 6 March 2007

Abstract On the basis of the T-matrix formalism and numerical simulations, we derive an explicit rule for partial multipole contributions to the plasmon resonances of gold nanorods at a fixed or random orientation. The parity of a given spectral resonance number n coincides with the parity of their multipole contributions l, where l is equal to or greater than n, and the total resonance magnitude is determined by the lowest multipole contribution. We also investigate the dependence of multipole plasmons on the size, shape, and orientation of nanorods with respect to the polarized incident light. It is shown that the multipole resonance wavelengths as a function of the aspect ratio divided by the resonance number collapse onto one linear scaling curve. This scaling is explained by using the plasmon standing wave concept introduced by Schider et al. [Plasmon dispersion relation of Au and Ag nanowires. Phys Rev B 2003;68:155427]. r 2007 Elsevier Ltd. All rights reserved. Keywords: Gold nanorods; T-matrix method; Multipole plasmon resonance; Light scattering and extinction

1. Introduction Metal nanorods possess unique optical properties because of their tunable dipole plasmon resonances, as demonstrated in pioneering experiments [1,2]. Owing to the shape- and size-controlled scattering and absorption properties [3–9], the gold nanorods have found promising applications in diagnostic [10] and photothermal therapy [11] of cancer cells in vitro. New potentialities for biomedical application of metal nanorods are related to their enhanced plasmon resonance sensitivity to the dielectric environment [12–15] and orientation with respect to polarized incident light [16], including laser orientation alignment and trapping [17], strong light scattering oscillations induced by Brownian rotation [18,19], and unusual depolarization of scattered light [20,21]. Keeping in mind the recent advantages in synthesis [22,23] and functionalization [24] of gold nanorods, one can predict rapid progress in biomedical technologies using conjugates of gold nanorods with biomolecules. For a comprehensive discussion of chemistry, optics, and biomedical applications of metal nanorods, the readers are referred to recent reviews [15,16,25–28]. Corresponding author. Institute of Biochemistry and Physiology of Plants and Microorganisms, Russian Academy of Sciences, 13 Pr. Entuziastov, Saratov 410049, Russia. Fax: +7 8452 97 8303. E-mail address: [email protected] (N.G. Khlebtsov).

0022-4073/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2007.03.006

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By now, the dipole plasmon resonances of metal nanorods have been studied in detail, both theoretically and experimentally [3–9,14,15]. In contrast, there are limited data on the multipole resonances in metal nanorods. As far as we are aware, the first reports on quadrupole modes were published by Barber and coworkers [29] for the local field enhancement factor of silver spheroids and the extinction and light scattering spectra of gold 2D lithographic nanostructures. The quadrupole modes also have been studied for silver [30] and gold spheroids, right circular cylinders, s-cylinders (cylinders with semispherical ends), and rectangular prisms [4,31]. DDA calculations coupled with experimental measurements allowed identification of two distinct quadrupole resonances for silver [32] and gold [33] nanoprisms. The past few years have seen the publication of the first observations of multipole plasmon resonances in gold and silver nanowires deposited onto a dielectric substrate (see, e.g., Refs. [34,35] and references therein). Quite recently, Mirkin and Schatz’s group reported on the first experimental observation and DDA simulation of multipole plasmon resonances in colloidal gold nanorods [36]. The assignment and interpretation of multipole resonances have been done on the basis of the standingwave picture [37,38] and DDA calculations [36]. For a review of recent works based on DDA simulations, the readers are referred to Refs. [3,5–9,39]. However, the DDA algorithm does not include explicit multipole representation. Instead, it treats the optical properties of nanorods in terms of dipole interactions of a huge dipole array that mimics the electrodynamic properties of a real particle. Here we report on the first, to the best of our knowledge, T-matrix analysis of multipole excitations in gold nanorods. We derive an explicit rule for partial multipole contributions to the observed plasmon resonances of randomly oriented nanorods. In addition, we investigate the relationships between the multipole plasmons of nanorods and their size, shape, and orientation with respect to polarized incident light, on the multipole basis and in more detail than is provided in previous publications [4,36,39]. Finally, we show, for the first time, that the multipole resonance wavelengths as a function of the aspect ratio divided by the resonance number collapse onto one linear curve. This scaling property is explained by using the plasmon standing wave concept [37,38]. 2. Models and methods The shape of gold nanorods can be modeled by a right circular cylinder of length L and diameter d with flat (the hard template fabrication method [36]) or semispherical (the seed-mediated growth method [22], scylinder model [31]) ends. From a computational point of view, the prolate spheroid has certain advantages because of its regular shape, which retains the principal optical properties of metal nanorods (see, however, discussion of Fig. 3 in Section 3). Here we use the above three models, which can be specified by the nanorod minor size d (nm), the length L, or the aspect ratio e ¼ L=d. The dielectric functions of the surrounding medium em ðlÞ (water) and gold nanorods eðl; Rev Þ were calculated according to Refs. [15,20], with the function eðl; Rev Þ accounting for the electron surface-scattering effects through the equivolume radius Rev and the size-limited particle damping constant gp ¼ gbulk þ vF =Rev , vF is the Fermi velocity. By using Rev for calculation of gp we follow, in a sense, the Coronado and Schtaz approach [40] in which the effective free path of conductive electrons is expressed in terms of the particle volume V and surface S, Leff 4V =S. Note that the need for inclusion of the size-limiting effects in nanoparticle optics through the eðl; Rev Þ function is debatable at present, see, e.g., the discussion in Refs. [15,41]. Previous single-particle experiments with gold nanorods [42] and silica/gold nanoshells [43] were explained in terms of the bulk dielectric function eðlÞ, whereas recent resonance light scattering [44] and absorption [45,46] single-particle measurements confirmed the size-dependent surface-scattering contribution to the bulk damping constant. Although our calculations included thin nanorods, here we mainly discuss the multipole resonances excited in large particles, where the size limiting effects can be neglected. The multipole contributions to the extinction, scattering, and absorption spectra can be expressed in terms of the corresponding cross sections, normalized to the geometrical cross-section pR2ev : Qext;sca;abs ¼

N X l¼1

qext;sca;abs . l

(1)

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For randomly oriented particles, the multipole partial contributions can be calculated with analyticalaveraging formulas [47,48]. Specifically, the multipole extinction contribution is given by 2 2 qext l ¼ ð2=k Rev Þ Spurl ðT ss Þ,

(2)

where k is the wave number in the medium, s  ðl; m; pÞ stands for T-matrix multi-index [48], and the trace (Spur) is taken over all T-matrix indices except for the multipole order l. For particles at a particular fixed orientation (specified by the angle a between the vector k and the nanorod axis a), we consider two fundamental cross sections corresponding to the TM and TE plane wave configurations, where E 2 ðk; aÞ plane and E ? ðk; aÞ plane, respectively. Modification of the T-matrix method surface integrals for s-cylinders can be found in Ref. [20]. For calculations, we used our extended-precision T-matrix code, which was tested by comparison with the separation of variables method [49]. Additionally, all numerical T-matrix data were checked by convergence tests as described in Ref. [50], so the accuracy of all presented data is better than 1%. Currently, there are various numerical techniques for calculating the optical properties of arbitrarily shaped particles (see, e.g., book [50, Ch. 6]). The main advantage of the T-matrix approach, compared with other numerical approaches (including those using multipole expansions), is related to analytical orientationaveraging solutions [47,50], which make such calculations very efficient and accurate. 3. Results and discussion Consider first the optimal geometrical parameters for excitation of high multipole plasmons in randomly oriented gold nanorods. Fig. 1 shows the extinction spectra Qext ðlÞ of randomly oriented gold s-cylinders with minor axes d ¼ 40, and 80 nm and aspect ratios e ¼ 2, 4, 6, 8, and 10. The spectral resonances are numbered according to the designation Qn  Qext ðln Þ, n ¼ 1; 2; . . . from far infrared to visible, while the symbol ‘‘0’’ in Fig. 1 designates the shortest wavelength resonance located at the transversal dipole TE mode. At a fixed nanorod thickness, an increase in the aspect ratio results in the red shifting of dipole resonance ðn ¼ 1Þ and in appearance of high-multipole resonances. The greater the aspect ratio, the more the number of multipole resonances can be observed. On the other hand, an increase in the particle diameter at a fixed aspect ratio also results in excitation of high multipoles and moves the spectrum to the red. Keeping in mind the logarithmic ordinate scale in all plots (from 0.2 to 50), we conclude that the excitation of high multipoles is not effective for very small particle diameters dp10 nm for all aspect ratios considered. The above dependences of multipole d = 80 nm

d = 40 nm 0

e=2

1

1 10

1 0

Qext

1 10 1 10

0

1 10 1 400

e=4

0

1

2

e=4

2 e=6

0

e=2

1

0

d = 20 nm

1

2

3

2

Qext

10

0

1

3

0

e=6

2

3

e=8

3

4

2

e=8 2

0 4

600

3

0

1 e = 10

800 1000 1200 1400 1600 Wavelength [nm]

400

600

e = 10 5

4

3 2

800 1000 1200 1400 1600 Wavelength [nm]

Fig. 1. Extinction spectra of randomly oriented gold s-cylinders with diameters d ¼80 (right), 40 (left), and 20 nm (left, dashed curves), and aspect ratios e ¼ L=d ¼ 2; 4; 6; 8, and 10. The numbers n ¼ 125 designate the total resonances Qn ¼ Qext ðln Þ; the number ‘‘0’’ stands for the short wavelength multipole resonance ð522 nmÞ, determined by the TE dipole excitation.

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plasmons on the particle size and shape were observed for the all three particle shapes considered. Thus, in the remaining of this work, we shall discus the multipole plasmons in thick nanorods (d ¼ 40 and 80 nm). Note that we encountered some convergence problems for these particles with aspect ratios greater than 9. Fig. 2a shows the extinction spectra Qext ðlÞ of randomly oriented gold spheroids with a minor axis d ¼ 80 nm and an aspect ratio e ¼ 10. The first resonance, corresponding to the longitudinal dipole excitation, is located in the far infrared and is not shown here. By contrast to the transversal and longitudinal dipole resonances of small nanorods [1], five additional multipole resonances can be identified in Fig. 2a (the 6th resonance looks like a weak shoulder, but it is clearly seen on the multipole contribution curve, designated q6 ). For right circular and s-cylinders, the multipole extinction spectra are analogous to plots in Fig. 2a, so we do not discuss these data here. Fig. 2a also shows six extinction spectra for partial multipole contributions qext l ðlÞ (for simplicity, we shall omit the symbol ‘‘ext’’ hereafter). First of all, we note that all multipole excitations contribute to the short wavelength ‘‘zero’’ resonance, although the major contribution is due to the dipole term. A close inspection of the spectra in Fig. 2a leads to the following conclusions: (1) the parity of multipole contributions coincides with the parity of total resonance; (2) for a given spectral resonance number n, the number of partial multipole contributions l are equal to or greater than n: X X Q2n ¼ q2l ðl2n Þ; Q2nþ1 ¼ q2lþ1 ðl2nþ1 Þ. (3) lXn

lXn

Total resonances Qn 0 65 4 3

2

Qext q1

0

q2 q3

-2

q4 q5

-4

q6 400

800

1200

Wavelength [nm]

Total resonances Qn 0 65 4 3

8 2

1600

8

6

4

Cross sections

Ext. cross section, log 10Qext

For example, the first and second resonances are determined by the dipole q1 ðl1 Þ and quadrupole terms q2 ðl2 Þ, respectively, whereas the high multipole contributions q3 ðl1 Þ, q5 ðl1 Þ; . . . or q4 ðl2 Þ, q6 ðl2 Þ . . . decrease up to one or two orders per multipole number. Sosa et al. [3] pointed out that the absorption resonances of large particles (to say, larger than 50 nm) may be masked by strong scattering located at the positions of absorption plasmons. Indeed, the authors [3] showed such masked absorption spectra for silver spheroids at fixed orientations. Fig. 2b shows the extinction, scattering, and absorption spectra of randomly oriented gold spheroids in water. First, we note that all the extinction, scattering, and absorption resonances are located near the same wavelengths. Further, only the dipole (n ¼ 1, not shown) and quadrupole ðn ¼ 2Þ scattering resonances give the dominant contribution to the corresponding extinction resonances. However, for resonances of numbers n ¼ 3 and 4, the absorption contributions dominate. Finally, the extinction resonance of number 5 is due to absorption only (see insert in Fig. 2b), whereas the corresponding scattering contribution is negligible. Thus, we conclude that the locations of extinction plasmon resonances coincide with the dominant scattering or absorption plasmon resonances and the scattering does not mask the absorption plasmon resonances. The same conclusion is valid for gold right circular and s-cylinders.

2

6 5

4

Qext 2

Qsca

0

Qabs

2 400

800 1200 Wavelength [nm]

1600

Fig. 2. (a) Extinction spectrum Qext ðlÞ of randomly oriented gold spheroids (d ¼ 80 nm; L ¼ 800 nmÞ in water. Curves q1 2q6 show the spectra of separate multipole contributions. The upper row of numbers n ¼ 226 designate the total resonances Qn ¼ Qext ðln Þ; the number ‘‘0’’ stands for the short wavelength multipole resonance ð522 nmÞ, determined by the TE dipole excitation. The resonance Q1 is located in the far infrared and is not shown. (b) Comparison of the absorption and scattering contributions to the total extinction cross section. The insert illustrates negligible contribution of scattering to the extinction resonance of number 5.

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Qext

k TM

α=0

α=53

73

1 400

Q0(TE) Q0

5

Random

4

800 1200 Wavelength [nm]

TE

10

2 3

5

E

15

90

0

k

20

38

E

10

25

α

α

Q2

ETM

Q3

Q5 Q5(TE) Q4

0 1600

0

20 40 60 80 Orientation angle α, degrees

Fig. 3. (a) Extinction spectra calculated for TM excitation of spheroids at optimal orientation angles a, which ensure the maximal contribution to the multipole spectral resonances n ¼ 225 randomly oriented particles; (b) orientation dependence of the total resonances Qn ; n ¼ 0; 225 at TM excitation (l0 ¼ 522; l25 ’ 1428, 1030, 832, and 722 nm, respectively). The dashed curves Q0 ðTEÞ and Q5 ðTEÞ correspond to the TE-exited resonances at 522 and 722 nm, respectively.

To gain insight into the physics of multipole excitations, it is instructive to compare the extinction spectra calculated for random orientations and for a particular orientation of a nanorod (Fig. 3a) [31,36]. First, we note that TM configurations make contributions only to the multipole plasmons, whereas TE configurations contribute to the single spectral resonance located near 520 nm, by close analogy with the usual dipole resonance of gold spheres in water. That is why we investigated the orientation dependence of the extinction spectra for a TM incident wave, to find the specific orientations that make the most contribution to the total spectra of randomly oriented nanorods. It is clear that the first longitudinal dipole-like resonance Q1ðTMÞ is effectively excited when a rod is oriented along the electric field. For the quadrupole resonance, the optimal excitation angle is 531, in full agreement with our previous finding [31]. The resonances of numbers n ¼ 3; 4; 5 are excited most effectively by particles that are in TM orientation configurations near the orientation angles a ¼ 38 , 731, and 901, respectively. Fig. 3b shows the orientation dependence of the first five resonance magnitudes Qn ðaÞ for a TM incident wave. Because of the symmetry restrictions [36], there exists an evident difference between the angular behaviors of even and odd TM resonances. The even resonances are effectively excited at intermediate orientations between the longitudinal and the transverse ones, whereas the odd resonances reach their maxima at perpendicular ða ¼ 90 ) orientation. However, we have found an exception from this rule for spheroids: the maximal resonance magnitude Q3 ðl3 ’ 1005 nmÞ is observed at a ¼ 38 , whereas at a ¼ 90 the resonance magnitude Q3 is a bit lower. In contrast, for s-cylinders, the maximal resonance magnitude Q3 ðl3 ’ 1030 nmÞ is observed at perpendicular orientation ða ¼ 90 Þ, whereas the second orientation angle for optimal (but not maximal!) excitation equals a ¼ 42 . This example clearly shows that both the spectral resonance position and its orientation dependence are shape-dependent. For instance, the spectral position of four resonances 5, 4, 3, and 2 are l52 ¼ 675, 785, 1005, and 1460 nm for spheroids and l52 ¼ 722, 832, 1030, and 1428 nm for scylinders. In addition to the TM resonance curves, we also show the orientation dependence of the Q0ðTEÞ and Q5ðTEÞ resonances that corresponds to the TE-excitations at 522 and 722 nm. For small nanorods, the transversal plasmon does not depend on the particle orientation. In our case, the evolution of Q0ðTEÞ reveals an evident variation with a because of finite particle size, whereas the Q5ðTEÞ resonance is still not orientation dependent. In a recent publication [36], the experimental extinction spectra of randomly oriented gold nanorods were compared with DDA simulations based on the TEM geometrical parameters of particles. Fig. 4 reproduces the measured (circles) and DDA-calculated (dashed curve 3) spectra for gold nanorods with a thickness of 85 nm and a length of 735 nm (adapted from Ref. [36]). We also show calculated spectra for right circular cylinders and s-cylinders possessing the same thickness and length (85 and 735 nm). The ordinates of the spectra were normalized so that the measured and calculated (curve 1, right circular cylinders) quadrupole resonances were equal. The best agreement between measurements and simulations was obtained for the right circular cylinders, i.e. for the shape that is the best approximation to the TEM images of hard-template

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Exp.[36] 6 2

Extinction

1 3

4

2

400

800 1200 Wavelength [nm]

1600

1600 1400 1200 1000

n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8

λn=252+235*e/n

800 600 2 4 6 Normalized axis ratio, e/n

Resonance wavelength, λn [nm]

Resonance wavelength, λn [nm]

Fig. 4. Extinction spectra of randomly oriented gold nanorods with d ¼ 85 nm and L ¼ 735 nm. Calculations for right circular (1) and scylinders (2). The experimental spectrum (circles) and DDA simulation (3) were adapted from Ref. [36].

1600 1400 1200 1000

n=1 n=2 n=3 n=4 n=5

λn=143.6+272.4*e/n

800 600 2 4 6 Normalized axis ratio, e/n

Fig. 5. The linear scaling of multipole resonance wavelengths ln vs normalized aspect ratio e=n. Calculations for randomly oriented gold spheroids (a) and s-cylinders (b) in water. The particle diameter d ¼ 80 nm, the aspect ratio e ¼ L=d ¼ 2220 (a), 2–12 (b), and the resonance number n ¼ 128.

nanorods. Note also the significant improvements between calculations and measurements in the shortwavelength part of the spectrum, as compared with DDA simulations [36]. Finally, we discus the size- and shape-dependent scaling of multipole resonances. It is well known [51] that the dipole resonance depends almost linearly on the rod aspect ratio, with the angular slope being a weak function of the rod thickness and shape [15,20,52,53]. On the other hand, the angular slope of the multipole resonances varies significantly with the resonance number [36]. Fig. 1 also illustrates a complex behavior of resonances as a function of the particle size and shape. We found, however, that the multipole resonance wavelengths ln obey the following universal scaling: ln ¼ f ðe=nÞ ’ a þ bðe=nÞ,

(4)

where the constants a and b do not depend on the resonance number. Fig. 5a illustrates the scaling law (4) for a particular example of randomly oriented gold spheroids in water (d ¼ 80 nm, e ¼ 2  20). It is evident that for resonance numbers n ¼ 1  8, all data collapse into single linear function (4) with a ¼ 252 and b ¼ 235.

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Physically, this scaling can be explained in terms of the standing plasmon wave concept and the plasmon wave dispersion law, as introduced for periodic nanowire arrays [37] and single nanowires [54,55]. The simplest estimation of the resonance position is given by [37] n

p 2p ¼ ¼ qeff ðon Þ, L leff n

(5)

where qeff ðon Þ is an effective wave number corresponding to the resonance frequency on , and the superscript ‘‘eff’’ accounts for the substrate, finite-size, and other effects. At a constant particle thickness, d, Eq. (5) predicts a linear scaling leff n L=nde=ne=n, in full accord with our finding given by Eq. (4). It should be emphasized that the scaling coefficients a and b in Eq. (4) depend on the particle size and morphology and that standing wave estimation (5) should be considered as a good approximation rather than as an exact quantitative description. Indeed, Fig. 5b shows the same plotting as in Fig. 5a but for s-cylinder model. In this case, the scaling law coefficients differ from those in Fig. 5a. To make this point more evident, consider two sets of rods with the same aspect ratios but with different diameters d 1 ¼ 80 nm and d 2 ¼ 40 nm. If estimation (5) were correct, one would expect that the scaling coefficient would satisfy a1 =a2 ¼ 2. However, from computer simulations we found the following ratios: a1 =a2 ’ 1:70 for s-cylinders and a1 =a2 ’ 1:59 for spheroids. A more detailed consideration, including the surrounding medium effects, will be presented in a separate work.

4. Conclusions In this work, we have shown that the parity of a nanorod multipole spectral resonance coincides with the parity of its multipole contributions, whose numbers are equal to or greater than the resonance number. Tmatrix calculations for the right circular cylinders and s-cylinders are in good agreement with the experimental spectra obtained by Mirkin, Schatz and coworkers [36]. It would also be interesting to investigate the dependence of multipole plasmons on the particle size, shape, orientation, material type (gold vs silver), and composition. As a preliminary part of this project, we have found that the multipole resonance wavelengths obey a universal linear scaling being plotted vs the particle aspect ratio divided by the resonance number. This remarkable property of multipole resonances can be understood in terms of a simple concept based on plasmon standing waves excited in metal nanowires by an electric filed of incident light [37,54]. The present study confirmed our previous finding [31] on optimal orientation angle for excitation of quadrupole resonance and extended the results of Refs. [4,31,39] to higher multipole plasmons. Finally, we note that the electrodynamic coupling between gold nanorods may affect their extinction spectra, as demonstrated by recent DDA [56–58] and recursive Green function [59] simulations. However, such effects have been observed only in experimental conditions in which the electrodynamic interaction of nanorods in a solution was due to aggregation induced by addition of salt (sodium citrate) [57] or bimolecular linkers [60]. For dilute nanorod solutions [36], the single-scattering approach allows one to describe all spectral peculiarities caused by multipole single-particle properties. It should also be noted that the anisotropic properties of particulate composite structures present significant interest because of potential applications (see, e.g., [56,57,59]).

Acknowledgments This research was partially supported by grants Nos. 05-02-16776a, 07-04-00302a (RFBR), RNP.2.1.1.4473, contracts No. 02.513.11.3043, and 02.512.11.2034 (the Ministry of Science and Education of RF). BK was supported by grants from the President of the Russian Federation (No. MK 2637.2007.2), 07-04-00301a, 0702-01434-a (RFBR), and INTAS YS Fellowship (No. 06-1000014-6421). We thank D.N. Tychinin for help in preparation of the manuscript and N.V. Voshchinnikov for SVM codes, which were used for benchmark testing of our extended precision T-matrix codes.

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