Brillouin study of confined eigenvibrations of silver nanocubes

Solid State Communications 152 (2012) 501–503

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Brillouin study of confined eigenvibrations of silver nanocubes J.Y. Sun a , Z.K. Wang a , H.S. Lim a , V.L. Zhang a , S.C. Ng a , M.H. Kuok a,∗ , W. Zhang b , S. Firdoz b , X.M. Lu b,∗∗ a

Department of Physics, National University of Singapore, Singapore 117542, Singapore

b

Chemical and Biomolecular Engineering, National University of Singapore, Singapore 119260, Singapore

article

info

Article history: Received 1 November 2011 Received in revised form 13 December 2011 Accepted 29 December 2011 by R. Merlin Available online 5 January 2012 Keywords: A. Noble metals C. Nanocubes D. Confined eigenmodes E. Brillouin light scattering

abstract The particle-size dependence of the eigenfrequencies of confined acoustic modes of aggregates of highly monodisperse silver nanocubes has been measured by Brillouin light scattering. Four Brillouin peaks are observed whose frequencies scale with inverse cube edge length. They are assigned to the Dd1 , Dd2 , Ds1 (symmetric breathing vibration) and Ds2 dilatational modes, labeled using the Demarest classification scheme. The assignment, which is based on numerically simulated frequencies and displacement profiles of the modes, as well as their calculated scattering cross-sections, differs from those of earlier studies on nanocubes. The elastic moduli of the silver cubes, with mean edge lengths between 59–102 nm, are found to be comparable to those of the bulk material, indicating the absence of sample defects like pores and cracks. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction Investigations of the confined vibrational eigenmodes of an object, which arise from quantization due to spatial confinement, are of importance because they provide information on the mechanical and thermal properties of the object. For instance, the mechanical properties and the specific heat due to the phonons of an object can be extracted from data on its eigenmodes [1–3]. Knowledge of confined acoustic modes is also employed in areas such as the detection of gravitational waves by resonant-mass detectors [4]. Lamb has theoretically analyzed the confined acoustic modes of the sphere, classifying them into spheroidal and torsional vibrations [5]. Aside from the highly symmetric sphere, the cube is one of the few geometric shapes whose modes have been comprehensively characterized theoretically. The vibrations of the cube have been numerically calculated by Demarest using the Rayleigh–Ritz method [6]. He categorized them as dilatational (D), torsional (T ), shear (S), and flexural (F ) modes, and labeled them

∗ Correspondence to: Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore. Tel.: +65 65162609; fax: +65 67776126. ∗∗ Correspondence to: Department of Chemical and Biomolecular Engineering, Blk E5, 4 Engineering Drive 4, #03-02, Singapore 119260, Singapore. Tel.: +65 65161071; fax: +65 6779 1936. E-mail addresses: [email protected] (M.H. Kuok), [email protected] (X.M. Lu). 0038-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.12.043

using notations such as Ds1 . The capital letter of the notation refers to the initial letter of the fundamental mode, the lower case subscript refers to the respective subset s (denoting symmetric), a (antisymmetric) or d (doublet), while the numerical subscript indicates the energy ordering of the modes within a subset. Very little experimental information on the confined acoustic phonons of nanosize cubes is available. Using time-resolved spectroscopy, Petrova et al. [7] observed only two modes in silver nanocubes. They attributed them to the breathing mode and an asyet unclassified nontotally symmetric mode, induced respectively by uniform and nonuniform initial strains arising from ultrafast laser-induced heating. In their Brillouin study of GeO2 nanocubes, Li et al. could identify only one mode, namely the lowest-energy one as having torsional-like character [8]. Hence the resonance vibrations of the cube are not well explored experimentally. Also, Li et al. found that their GeO2 nanocubes, with mean edge lengths L in the 150–600 nm range, had elastic constants only one half those of bulk GeO2 , ascribing the huge reduction to sample defects [8]. On the other hand, Petrova et al. deduced that the elastic constants of their cubes (L = 36–86 nm) were the same as those of bulk silver. However, with standard size deviations as large as 20%, their measurement precision was limited by sample polydispersity. Metal nanoparticles have attracted intense interest due to their diverse applications ranging from optical sensing to drug delivery [9–11]. In particular, the interest in noble metal nanoparticles is driven by their applications in surface plasmonics. Plasmonic particles are able to enhance and confine electromagnetic fields, which has led to the development of surface-enhanced Raman scattering

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Fig. 1. FESEM image of silver nanocubes of mean edge length L = 67 nm.

spectroscopy (SERS) [12], and they show promise in photonic applications such as nonlinear optics [13]. In this work, we used highly monodisperse silver nanocubes, and the nondestructive, noncontact technique of Brillouin light scattering, to investigate their resonance modes. With highquality samples, the eigenfrequencies and their dependence on particle size were measured with higher precision, thus yielding more accurate elastic modulus values for the nanocubes. A thorough characterization of the observed vibrations, based on calculated mode frequencies and intensities as well as numerically simulated displacement profiles, was carried out, with the modes labeled according to the Demarest classification scheme [6]. The experimental data allowed us to ascertain whether there were any defects in the nanosize particles studied. 2. Experimental details Four aggregates of Ag cubes with respective mean edge lengths L = 59, 67, 78 and 102 nm (size polydispersity ≈4%), were prepared in ethylene glycol and stabilized with poly(vinylpyrrolidone) (PVP), following the synthetic method of Im et al. [14]. The particle sizes were determined by field-emission scanning electron microscopy (FESEM). A typical FESEM image, that of the 67 nm cubes, shown in Fig. 1 reveals that the synthesized particles are monodisperse and their shapes are well-defined. Prior to the Brillouin measurements, various samples of the synthesized aggregates of loose silver nanocubes were prepared by using an eye dropper to drop colloidal water solutions of the nanocubes onto clean pieces of silicon wafer, followed by vacuum drying at room temperature for several hours. All the Brillouin measurements were carried out in the 180°-backscattering geometry at room temperature. The 514.5 nm radiation of an argon-ion laser was used to excite the spectra, with the incident laser power kept low at a few milliwatts, to avoid overheating of the samples. The scattered light was frequency analyzed with a (3 + 3)-pass tandem Fabry–Pérot interferometer, which was equipped with a silicon avalanche diode detector. 3. Results and discussion Fig. 2(a) shows a representative spectrum, that of the 67 nm silver cubes, which contains four peaks due to the vibrational eigenmodes of the particles. An earlier study [15] has shown that the vibrations of individual nanoparticles are not significantly affected by contacting neighboring nanoparticles, and, in our case, the Si wafer substrates as well, since our samples are opaque and thick. All spectra recorded were fitted with Lorentzian functions, and the resulting eigenfrequencies of the four samples studied are plotted as a function of their inverse cube edge length in Fig. 3. The mode displacement profiles and frequencies ( f ) were calculated numerically, based on the finite element method, using COMSOL Multiphysics [16]. The values of the Young modulus,

Fig. 2. Brillouin spectra of Ag nanocubes with edge length L = 67 nm. (a) Measured spectrum. The experimental data are denoted by dots. The spectrum is fitted with Lorentzian functions (dotted curves) and a background (dashed curve), while the resultant fitted spectrum is shown as a solid curve. (b) Measured spectrum with the background subtracted. (c) Calculated spectrum broadened with a Lorentzian line shape.

Poisson ratio, and density used in the computation were those for bulk silver, namely 74 GPa, 0.37, and 10.5 g/cm3 respectively. Fig. 3 reveals that the experimental size dependence of the eigenfrequencies is well reproduced by the numerical calculations (see the discussion below on mode assignment). It is to be noted that the elastic parameters were not fitted to the Brillouin data, but were those of bulk silver. The good agreement means that the elastic modulus values of the Ag nanocubes studied were the same as those of bulk silver, thus suggesting that the samples were free from defects like pores and cracks [8]. Also, the measured scaling of the eigenfrequencies of the silver cubes with their inverse edge lengths ( f ∝ 1/L) was consistent with the results of Li et al. in their work on GeO2 nanocubes [8]. This lends support to the finding of Pan et al. [17], who observed that the eigenfrequencies of any free regular-shaped homogeneous object always scale with its inverse linear dimension, and that this universal relationship holds for such objects of any classical size and material. The mode assignment of the observed eigenvibrations was made as follows. The Brillouin intensities of the modes of the Ag cubes were estimated using a macroscopic approach, in which the intensity of the mode p of frequency fp is approximately given by Montagna [18] Ip ∝

 2   e−iq•r q • ep dV  ,   2 fp 1 

(1)

where ep is the normalized eigenvector, and q the exchanged wavevector. The integration was taken over the skin depth of the metallic cube rather than over its entire volume, and Ip was averaged over various qs ranging from 0 to 4π n/λ, where n is the refractive index of silver, and λ the laser wavelength. Calculations

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ν17 , as an as-yet unclassified nontotally symmetric mode and the breathing mode respectively. A comparison of their experimental data with ours reveals that their ν17 vibration corresponds to our Ds1 symmetric breathing mode, while their ν8 one corresponds to our Dd2 dilatational mode. It is likely that Petrova et al. were not able to measure the Dd1 and Ds2 vibrations we observed as they were not induced by strains arising from ultrafast laserinduced heating. However, our assignment differs from that, based on only calculated mode frequencies, for GeO2 nanocubes by Li et al. [8], who deduced that their lowest-energy mode observed had torsional-like character. This is not unexpected, as the energy ordering of the acoustic modes is material dependent. 4. Conclusions

Fig. 3. Dependence of measured and calculated mode frequencies on the inverse edge length of the Ag nanocubes. The experimental data are denoted by dots with error bars, while the lines represent the theoretical frequencies of the Dd1 , Dd2 , Ds1 and Ds2 dilatational modes.

In summary, Brillouin light scattering from a series of aggregates of highly monodisperse silver cubes, with mean edge lengths in the 59–102 nm range, has been measured. Based on calculated mode frequencies, intensities and displacement profiles, the four observed spectral peaks were assigned to the Dd1 , Dd2 , Ds1 and Ds2 dilatational vibrations of silver nanocubes, with Ds1 being the symmetric breathing vibration. The modes were labeled following the Demarest classification scheme. Our mode assignment differs from those reported previously for nanocubes. Our findings lend support to the observation that the eigenfrequencies of any free regularshaped homogeneous object always scale with its inverse linear dimension, and that this universal relationship holds for such objects of any classical size and material. The good agreement between the calculated and measured size-dependence of the eigenfrequencies suggests that the elastic moduli of the silver nanoparticles studied are the same as those of the bulk material and also that the particles are free from defects such as pores and cracks. Acknowledgment

Fig. 4. Simulated displacement profiles of the four observed dilatational modes of a silver cube. The displacement magnitudes are color-coded, with red denoting the maximal value. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

reveal that the dilatational Dd1 , Dd2 , Ds1 and Ds2 motions (Ds1 is the symmetric breathing mode) are the most intense. An exemplary calculated spectrum for these four modes of the L = 67 nm Ag cubes is presented in Fig. 2(c). As there is qualitative agreement between the experimental spectrum (with background subtracted) and the calculated one (see Fig. 2(b) and (c)), the observed Brillouin peaks are ascribed to these dilatational modes. Their numerically simulated displacement profiles are illustrated in Fig. 4. Petrova et al. [7] found that silver cubes smaller than 50 nm produce a single modulation in the transient absorption traces (i.e., one excited mode) due to uniform ultrafast laser-induced heating, while larger cubes produce two modulations (i.e., two excited modes) due to nonuniform heating. While time-resolved spectroscopy measures vibrational modes in the time domain, Brillouin light scattering measures them directly in the frequency domain. Also the latter technique detects the more intense modes that are thermally excited at room temperature, which generally number more than those observed by time-resolved spectroscopy. Interestingly, Petrova et al. identified the two Ag nanocube modes they observed, which they labeled as ν8 and

Funding by the Ministry of Education Singapore under research grants R-144-000-282-112 and R-279-000-273-133/298-112 is gratefully acknowledged. References [1] I.-C. Chen, C.-L. Weng, Y.-C. Tsai, J. Appl. Phys. 103 (2008) 064310. [2] M.H. Kuok, H.S. Lim, S.C. Ng, N.N. Liu, Z.K. Wang, Phys. Rev. Lett. 90 (2003) 255502. [3] H.S. Lim, M.H. Kuok, S.C. Ng, Z.K. Wang, Appl. Phys. Lett. 84 (2004) 4182–4184. [4] O.D. Aguiar, Research Astron. Astrophys. 11 (2011) 1–42. [5] H. Lamb, Proc. Lond. Math. Soc. 13 (1882) 189–212. [6] H.H. Demarest, J. Acoust. Soc. Am. 49 (1971) 768–775. [7] H. Petrova, C.-H. Lin, S. Liejer, M. Hu, J.M. McLellan, A.R. Siekkinen, B.J. Wiley, M. Marquez, Y. Xia, J.E. Sader, G.V. Hartland, J. Chem. Phys. 126 (2007) 094709. [8] Y. Li, H.S. Lim, S.C. Ng, M.H. Kuok, M.Y. Ge, J.Z. Jiang, Appl. Phys. Lett. 91 (2007) 093116. [9] X. Lu, M. Rycenga, S.E. Skrabalak, B. Wiley, Y. Xia, Annu. Rev. Phys. Chem. 60 (2009) 167–192. [10] Y.-H. Ye, Y.-W. Jiang, M.-W. Tsai, Y.-T. Chang, C.-Y. Chen, D.-C. Tzuang, Y.-T. Wu, S.-C. Leeb, Appl. Phys. Lett. 93 (2008) 263106. [11] W. Akahata, Z.-Y. Yang, H. Andersen, S. Sun, H.A. Holdaway, W.-P. Kong, M.G. Lewis, S. Higgs, M.G. Rossmann, S. Rao, G.J. Nabel, Nat. Med. 16 (2010) 334–338. [12] J.F. Li, Y.F. Huang, Y. Ding, Z.L. Yang, S.B. Li, X.S. Zhou, F.R. Fan, W. Zhang, Z.Y. Zhou, D.Y. Wu, B. Ren, Z.L. Wang, Z.Q. Tian, Nature 464 (2010) 392–395. [13] D.A. Genov, A.K. Sarychev, V.M. Shalaev, A. Wei, Nano Lett. 4 (2004) 153–158. [14] S.H. Im, Y.T. Lee, B. Wiley, Y. Xia, Angew. Chem. Int. Ed. 44 (2005) 2154–2157. [15] Y. Li, H.S. Lim, S.C. Ng, Z.K. Wang, M.H. Kuok, E. Vekris, V. Kitaev, F.C. Peiris, G.A. Ozin, Appl. Phys. Lett. 88 (2006) 023112. [16] COMSOL Multiphysics, Manual, Comsol, AB, Stockholm, Sweden. [17] H.H. Pan, Z.K. Wang, H.S. Lim, S.C. Ng, V.L. Zhang, M.H. Kuok, Appl. Phys. Lett. 98 (2011) 133123. [18] M. Montagna, Phys. Rev. B 77 (2008) 045418.