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Xe ion irradiation-induced polycrystallization of Ag nanoparticles embedded in SiO2 and related optical absorption property

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Abstract

The Xe ion irradiation-induced structural modifications of implantation-synthesized Ag NPs were explored in this paper. Our results clearly show that Xe ion irradiation could induce transformation of crystalline Ag NPs into polycrystalline ones with various shapes. The modified Ag NPs exhibit a notable shoulder absorption band around 550 nm besides the surface plasmon resonance (SPR) peak located at about 400 nm. Reduction of the electron mean-free-path and interaction between crystallites in the formed polycrystalline NPs have been proposed to interpret the appearance of the shoulder absorption band. Moreover, our results also show that subsequent thermal annealing at 400 °C for 1h can restore polycrystalline Ag NPs to crystalline ones with spherical shape, which are nearly aligned around the end of the Ag ion range, and contribute a much narrower SPR absorption peak at 400 nm.

© 2014 Optical Society of America

1. Introduction

Noble metal nanoparticles (NPs) have attracted much attention due to the fundamentally different properties and behaviors from those of their conventional bulk counterparts [1,2]. Especially, the NPs support the localized surface plasmon resonance (SPR), which is described as the collective coherent oscillations of the free electrons in the conduction band. As we know, the localized SPR can drastically enhance the optical electrical field close to the particle surface [13]. Therefore, NPs have been employed in diverse applications, such as biosensors [4], antibacterial materials [5], and surface enhanced Raman scattering [6], etc. Besides, the localized SPR effect is dependent on the size, morphology, spatial orientation, optical parameters of NPs and the background materials. The SPR absorption band in the optical absorption (OA) spectrum, an important fingerprint for materials, thereby can be engineered easily by tailoring the structural parameters of NPs.

Among different possible techniques to synthesize such NP-based materials, e.g., ion exchange [6], sol-gel method [7], ion implantation and/or irradiation [812], etc., ion implantation followed by a post heavy ion irradiation has been proved to be very versatile. For instance, ion implantation can introduce a desired amount of the guest phase into a host matrix; especially, the concentration and depth distribution of implants can be controlled to a certain degree by precisely selecting the implantation parameters [810]. Post heavy ion irradiation can be used to further tailor the size and spatial distribution of the embedded NPs as well as their shape, and, accordingly, the SPR properties can be engineered [1012]. In addition to the modulations in size, shape and spatial distribution of NPs, Wang et al. have also shown that post heavy ion irradiation may result in transformation of crystalline intermetallic compounds and/or ceramics into polycrystalline nanostructures [13]. In comparison with crystalline NPs, a polycrystalline particle generally contains multiple grain boundaries within individual crystallites [1315], which could possess intriguing optical properties. For example, Nagpal et al. have presented that polycrystalline particles fabricated by lithography can increase the attenuation and damping of surface plasmons due to electron scattering at grain boundaries [16]. Moreover, the polycrystalline nanostructures may display exceedingly bright luminescence [17] and strongly enhanced Raman scattering [17,18]. These findings indicate that the crystallinity of NPs could exert significant influences on their optical properties and thus the crystalline structures of NPs as well as the mechanism of their effects on optical behaviors deserve deep investigations. In the past decades, although the crystalline-to-amorphous transition has frequently been reported [1921], the irradiation-induced polycrystallization of noble metal NPs has not been systematically studied. Even the influence of polycrystalline nanostructures on SPR properties is not yet clearly understood.

In our recent work [22], we have synthesized quasi-two-dimensional Ag NPs in silica by Xe ion irradiation and subsequent Ag ion implantation, and observed the formation of polycrystalline Ag particles. However, for suppressing the heavy sputtering effect, the adopted Ag ion fluence was only 2 × 1016 cm−2 so that the maximum diameter of finally formed Ag NPs was only 7 nm. Moreover, a high Ag ion fluence could also cause a relatively high ion beam heating under practical conditions (e.g., operating a long-time implantation into a low thermal-conductive substrate by using a fixed flux density), which could lead to the crystallization of Ag NPs. Nevertheless, a high ion fluence generally favors relatively high volume fraction and large mean size for NPs, which can result in an enhanced SPR excitation [1,2]. Noticing the interaction between heavy ions and target particles, which could lead to the displacement cascades, defect accumulation, phase segregation and decomposition, etc. [13], heavy ion irradiation on preformed Ag NPs with large sizes could be more suitable to fabricate numerous and various polycrystalline NPs. In this paper, we focused on the procedure to transform the implantation-synthesized Ag NPs into various polycrystalline Ag NPs by post 500 keV Xe ion irradiation, and the SPR behaviors of polycrystalline Ag NPs as well as the underlying mechanisms were investigated in detail. Moreover, the thermal stability of polycrystalline Ag NPs was also studied. Our results show that as compared with crystalline Ag NPs, the polycrystalline ones present a pronounced shoulder band around 550 nm besides the SPR absorption peak at about 400 nm. After 400 °C annealing in a nitrogen atmosphere, various polycrystalline Ag NPs transform into spherical and crystalline ones, and a single layer of large Ag NPs is formed near the end of the Ag ion range, contributing a much narrower Ag SPR absorption peak at 400 nm. This work suggests a very promising method for fabrication of SPR-based nanocomposites with unique configurations of NPs.

2. Experimental

By using a metal vapor vacuum arc (MEVVA) implanter, optical-grade amorphous SiO2 slices of 1.0 mm in thickness were implanted with 45 keV Ag ions to a fluence of 5 × 1016 cm−2 to fabricate the embedded Ag NPs. For convenience, the Ag implanted SiO2 is labeled as the Ag sample. Ag ion implantation was carried out at a tilted incident angle of 45° from the normal of sample surface and with a beam current density of about 5 μA/cm2. For the purpose of uniformity, the target plate was rotated at a constant speed during the whole implantation. Some of the Ag implanted SiO2 slices were subsequently subjected to irradiation by 500 keV Xe ions at normal incidence to a fluence of 2 × 1016 cm−2 (labeled as the Ag + Xe sample). Xe ion irradiation was performed on an LC-4 high energy ion implanter with beam current density below 1 μA/cm2. SRIM 2013 code simulations show that the projected ranges of Ag and Xe ions are about 29 and 200 nm, respectively [23].

After implantation and irradiation, the Ag and Ag + Xe samples were furnace annealed in a flowing nitrogen atmosphere for 1 h. The annealing temperature was varied from 200 to 500 °C. Cross-sectional transmission electron microscope (XTEM) observations were made to evaluate structure, morphology, and size distribution of NPs by using a JEOL 2010 microscope operating at an acceleration voltage of 200 kV. Grazing incidence X-ray diffraction (GIXRD) measurements were performed at 0.3° incidence angle to identify the formation of Ag NPs along with their structures on a Philips X’pert pro X-ray diffractometer by using Cu Kα line (~0.154 nm) as the radiation source. Moreover, a UV-3100PC ultraviolet–visible–near infrared spectrophotometer was used to characterize the OA properties related to NPs.

3. Results and discussion

Figure 1 presents the typical XTEM micrographs taken of the Ag and Ag + Xe samples. One can see that Ag ion implantation gives rise to formation of NPs. The NPs mainly take on spherical shape and are distributed in a sample region from surface to 45 nm in depth, as shown in Fig. 1(a). Moreover, it can also be found that there are lots of large NPs to distribute in the shallow region close to the surface while many small ones are located around the end of the ion range. Based on the image-processing method, the mean diameters of NPs distributed in these two regions are 12.0 ± 2.8 nm and 3.6 ± 1.0 nm, respectively. High-resolution TEM (HRTEM) observations of NPs together with their corresponding fast Fourier transform (FFT) patterns confirm that the formed NPs are face-centered cubic (fcc) Ag ones, and have good-quality crystalline phases. As an example, Fig. 2(a) gives the HRTEM image of one particle as well as its corresponding FFT pattern.

 figure: Fig. 1

Fig. 1 XTEM images showing overall morphologies of NPs created in the (a) Ag and (b) Ag + Xe samples.

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 figure: Fig. 2

Fig. 2 (a) HRTEM image together with corresponding FFT pattern (inset) of one Ag NP in the Ag sample, (b) close view of the sheet-like nanostructure created in the Ag + Xe sample, and (c-e) HRTEM images showing various polycrystalline Ag NPs in the Ag + Xe sample.

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After Xe ion irradiation to a fluence of 2 × 1016 cm−2, both morphology and distribution of Ag NPs are remarkably changed, as shown in Fig. 1(b). From the image, it is clear that the quantity of spherical Ag NPs significantly decreases in the sample region close to the surface, while lots of nanostructures, which are possibly sheet-like due to their larger cross-sectional area and lower contrast than spherical NPs, are created. As for the sample region around the end of the Ag ion range, spherical Ag NPs still dominate. However, their size increases and volume fraction sharply decreases. Close views of the sheet-like nanostructures reveal that they are mainly composed of many small interconnected crystallites with the minimum grain size of ~2.0 nm (as an example, see Fig. 2(b)). Moreover, according to the atomic lattice fringes shown in Fig. 2(b), spacings of the crystallographic planes have been obtained, i.e., 0.235 and 0.205 nm, which are close to Ag (111) and (200) lattice planes, respectively.

In addition, HRTEM observations demonstrate various polycrystalline nanostructures. As examples, Figs. 2(c) and 2(d) present two typical multiply twinned particles, where the 5-fold axis can be found. Even the former could be a star-decahedron structure that is a remarkably stable structure that appears in the truncated decahedral NPs [24]. Twinning is one of the most common planar defects in nanocrystals, and it has been frequently observed in fcc structured metallic nanocrystals prepared by colloidal growth methods [2427]. However, it is seldom reported for the implantation-synthesized metallic NPs. Finally, it should be pointed out that polycrystalline NPs with spherical shape have also been revealed in the Ag + Xe sample through HRTEM observations. Figure 2(e), as an example, shows one spherical Ag NP that probably consists of three crystallites.

The transformation of crystalline Ag NPs to polycrystalline ones under Xe ion irradiation is closely associated with interactions of Ag NPs with Xe ions, which can be interpreted as follows. Xe ions into SiO2 embedded with Ag NPs could lose their energies via collisions with atoms of substrate. On the one hand, the interactions between Xe ions and Ag nuclei, i.e., nuclear collisions, could lead to formation of displaced Ag in the Ag NPs, and thus various types of damage could be produced in Ag NPs, such as dislocations, stacking faults, and polycrystalline structures, etc. [13,28]. Actually, the creation of damage in Ag NPs under Xe ion irradiation has been clearly revealed in our previous study, where the relatively low fluence of 1 × 1016 cm−2 was used [29]. However, possibly due to the relatively weak ion beam heating and/or insufficient collision processes, the NPs finally formed in that case were still mainly spherical in shape, and numerous lattice distortions were observed in the Ag NPs. For the present case, the applied fluence of Xe ions is sufficiently high, so some Ag NPs could be dissolved, and abundant dispersed Ag atoms in the implanted region would be produced. On the other hand, nuclear collisions occurring between Xe ions and the SiO2 substrate could create lots of defects in the substrate together with a build-up of a compressive in-plane stress there [30,31]. Owing to the ion beam heating from the ion kinetic energy deposition during Xe ion irradiation and the fact that the defects usually act as the nucleation sites for the NPs, dissolved Ag atoms could aggregate around defects and/or be absorbed by more stable, larger NPs. Simultaneously, the irradiation-induced stress may act on the NPs which are aggregating and growing, thus modifying their surface energies and surface tensions, and thus altering the energetic configurations [30]. Consequently, sheet-like and numerous polycrystalline Ag NPs could be produced.

Figure 3 shows the GIXRD patterns recorded for the Ag samples before and after Xe ion irradiation. It is clear that the Ag sample exhibits two intense diffraction peaks at 38.1° and 44.4°, corresponding to Ag (111) and Ag (200) planes, respectively. The result is in good agreement with that found in the XTEM measurements, indicating formation of Ag NPs. After Xe ion irradiation, intensities of both the diffraction peaks significantly reduce. Loss of long-range lattice periodicity in the formed polycrystalline Ag NPs, as demonstrated in Figs. 2(b-e), could explain such intensity reduction of the diffraction peaks.

 figure: Fig. 3

Fig. 3 GIXRD patterns of the Ag samples before and after Xe ion irradiation.

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Figure 4(a) presents the OA spectra recorded in the Ag and Ag + Xe samples. It can be seen that a pronounced absorption peak at about 400 nm attributed to the Ag SPR appears in the Ag sample, clearly indicating the formation of Ag NPs in SiO2. The result is well in line with those from XTEM and GIXRD measurements. After Xe ion irradiation, a broad absorption band occurs in the wavelength range of 350-700 nm. Previous studies have shown that various negatively charged and/or oxygen vacancy defects might be present after Xe ion irradiation into virgin amorphous silicas [32,33]; nevertheless, these intrinsic defect-induced maximum absorptions are produced only in the ultraviolet spectral region [11,33,34]. Thus this broad absorption band mainly originates from various Ag NPs. Furthermore, this absorption band could be approximately fitted by the sum of two Gaussian functions

G1(λ) = 0.1593exp(((λ401.3)/22.82)2)           + 0.01059exp(((λ434.3)/13.5)2)+ 4.13exp(((λ453.1)/63.97)2)  4.051exp(((λ453.8)/63.39)2)           + 0.06499exp(((λ550.3)/516.1)2)                              
and
G2(λ) = 0.08275exp(((λ550.1)/131.4)2) + 0.0734exp(((λ278.7)/1273)2)
where λ is the wavelength of incident light. The calculated absorption peaks by G1(λ) and G2(λ) locate at 400 (peak 1) and 550 nm (peak 2), respectively, as shown in the inserted figure in Fig. 4(a). The absorption peak at 400 nm should result from the spherical Ag NPs as revealed by XTEM observations (Fig. 1(b)). Actually, based on Mie theory [1,2], simulations show that an individual spherical Ag NP with a radius of 6.0 nm embedded in SiO2 can give rise to an absorption peak at nearly the same wavelength position, as presented in Fig. 4(b). As for the absorption peak around 550 nm, we argue that it is most probably related to Xe ion irradiation-induced modifications in the Ag NPs, i.e., the formation of various polycrystalline Ag NPs, as demonstrated in Fig. 2. The formation of polycrystalline Ag NPs could reduce the electron mean-free-path (EMFP), l, in the NPs due to the loss of lattice periodicity. Earlier work by Amekura et al. [21] has indicated that the reduced l would indeed result in weakening and widening, or even redshift of the Cu SPR peak, even if the particle size remains virtually unchanged. By considering the reduced EMFP in polycrystalline Ag NPs, calculations have also been carried out for various l values in [22]. Nevertheless, it was found that even if the l value reduces to 1.00 nm, and the corresponding NP radius is about 1.02 nm, the Ag SPR peak could not red shift to 500 nm and above. Therefore, the appearance of the 550 nm absorption peak could not be simply attributed to the reduced EMFP in the modified Ag NPs, and other possibilities should be taken into account.

 figure: Fig. 4

Fig. 4 (a) OA spectra of the Ag and Ag + Xe samples. Inset gives the Gaussian fit band with two peaks (1, 2) for the Ag + Xe sample. (b) Calculated absorption cross section σ spectra of a crystalline spherical Ag NP by Mie theory and of polycrystalline Ag NPs by MG theory with different interaction parameters K. A refractive index of 1.46 was used for SiO2 matrix and all calculations are normalized to the height of the spectrum calculated by Mie theory.

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As known, a polycrystalline particle generally consists of several crystallites and each crystallite can be further assumed to be a small nanocluster; thus, the interaction between crystallites cannot be neglected due to the very short cluster distance [1,14,15]. Considering such interaction, the absorption spectrum of metallic NPs embedded in a medium with dielectric constant εm can be calculated according to the Maxwell-Garnett (MG) theory which takes into the interaction of NPs by introducing an interaction parameter K [35]. MG theory regards the composite as a uniform medium with an effective complex dielectric constant εeff = ε1,eff + iε2,eff, and the absorption spectrum is determined by the following expression

σabs=7.16×104ω[ε1,eff+(ε1,eff2+ε2,eff2)1/2]1/2
where

{ε1,eff=εm+AC+BDC2+D2ε2,eff=BCADC2+D2
A=f(ε1εm),        B=fε2C=εm+β(ε1εm)f(ε1εm)(13εm +K4πεm)D=βε2fε2(13εm +K4πεm)

In the above expressions, f denotes the metallic volume fraction in the matrix, ε = ε1 + iε2 is the complex dielectric function of the metal, and β is a parameter depending on the particle geometry. In the context of this model the crystallites in a single polycrystalline NP may be regarded as an aggregation of crystallites in the dielectric silica, and hence represent a kind of inhomogeneity in dipole distribution in the silica that can give rise to a nonzero K value [35]. Determining the f and β values of crystallites in the substrate could be difficult. However, we may tentatively set f = 0.2 (MG theory is limited to a low metallic volume fraction [2,36]) and β = 1/3 (assuming each crystallite to be a small spherical nanograin). The complex dielectric function of Ag NPs changes with EMFP and the l has a value of 1.0 nm according to the minimum grain size (~2.0 nm) on the basis of the analysis above on Fig. 2(b). On these bases, the absorption spectra of polycrystalline Ag NPs were calculated, as presented in Fig. 4(b). These calculations could not provide rigorously quantitative information. Nevertheless, it can be seen from Fig. 4 that polycrystalline Ag NPs, when taking into account both the effects of reduced EMFP and interaction between crystallites, can indeed cause a wide SPR absorption peak in a long wavelength range. In fact, a further calculation with K = 17.4 could well reproduce nearly the same absorption band as peak 2 shown in the inset of Fig. 4(a). For more clarity, the polycrystalline nanostructure as well as its influence on the EMFP is schematically illustrated in Fig. 5.The results further suggest that polycrystallization of metallic NPs could be an important factor to modify their SPR.

 figure: Fig. 5

Fig. 5 Schematic illustration of the crystalline and polycrystalline Ag NPs as well as their influences on the EMFP.

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Finally, thermal stability of the formed polycrystalline Ag NPs due to Xe ion irradiation together with their OA properties has been also investigated. Our results reveal that thermal annealing at elevated temperatures could give rise to restoration of polycrystalline Ag NPs to crystalline ones with approximately spherical shape. Moreover, the broad absorption peak at about 550 nm gradually reduces and disappears when the annealing temperature reaches 400 °C. As an example, Fig. 6(a) presents the typical XTEM image taken of the Ag + Xe sample after 400 °C annealing for 1h in a flowing nitrogen atmosphere. The corresponding OA spectrum is given in Fig. 6(b). Moreover, for the purpose of comparison, the OA spectrum of the Ag sample after 400 °C annealing is also shown in Fig. 6(b). From Fig. 6(a), it can be seen that the sheet-like Ag nanostructures created in the Ag + Xe sample completely disappear, and spherical ones with smaller particle size and lower density become dominant in the shallow region close to the substrate surface. Interestingly, nearly aligned spherical Ag NPs with a mean diameter of about 12.2 nm are formed around the end of the Ag ion range. HRTEM inspection of one large particle (see the inset in Fig. 6(a)) shows that the particle is crystalline and the estimated interplanar distance of ~0.235 nm corresponds closely to the spacing of Ag(111) plane. The result clearly demonstrates that subsequent annealing at an appropriate temperature could result in restoration of polycrystalline Ag NPs to crystalline ones with spherical shape and well-aligned spatial distribution.

 figure: Fig. 6

Fig. 6 (a) XTEM image of the Ag + Xe sample after 400 °C annealing. The inset gives the HRTEM image of one particle together with the measured spacings of the crystallographic plane, (b) OA spectra of the Ag and Ag + Xe samples after 400 °C annealing.

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Recently, to satisfy the needs for plasmonic applications exploiting optical near-field effects of NP assemblies, ultra low energy ion implantation (ULEII) [37] and a two-step method, i.e., Xe ion irradiation followed by Ag ion implantation [22], have been proposed to optimize the fabrication of well-aligned NPs in host matrices. Unfortunately, owing to the heavy sputtering effect associated with the synthesis of NPs using ion implantation, the proposed methods are often performed with very low ion fluence so that the particle sizes of finally formed NPs are often very small, negatively affecting the optical properties of the nanocomposites. For the present case, since large Ag NPs could be preformed in SiO2 by Ag ion implantation at a high ion fluence, post 500 keV Xe ion irradiation followed with subsequent annealing could be more suitable for the fabrication of well-aligned NPs with large sizes. As for the transformation from polycrystalline Ag NPs to crystalline ones, it could be qualitatively interpreted as follows. Owing to extremely large surface area and defective structure, the polycrystalline NPs have high surface energy and substantial lattice strain so that atoms on them are generally very unstable [38]. Although bulk Ag has a high melting point Tm ~960°C, if the Ag particle size is reduced to nanoscale, the Tm can be significantly lowered [39]. During subsequent annealing, the crystal structure of polycrystalline NPs gradually reaches higher degrees of periodicity, develops a more ordered pattern, and becomes more stabilized in terms of the free energy. Especially, each sheet-like Ag nanostructure could evolve to be as nearly spherical in shape as possible in order to decrease the surface tension and attain the most thermodynamically stable configuration. This would be accomplished initially by surface diffusion and finally by surface melting [38,40]. In addition, the nucleation and thermal growth of large NPs at the expense of the melted Ag atoms, i.e., the Ostwald ripening [41], might also occur. As a result, small Ag NPs with a low particle density would form in the shallow region while large and crystalline ones would be nearly aligned around the end of the Ag ion range due to the fact that spherical Ag NPs with large sizes are dominant even after Xe ion irradiation. As shown in Fig. 6(b), it is clear that accompanying the disappearance of sheet-like nanostructures and the formation of aligned spherical Ag NPs around the end of the Ag ion range, the absorption band at about 550 nm disappears, while a much narrower Ag SPR absorption peak with the full width at half maximum (FWHM) of ~47 nm occurs at 400 nm as compared with that (the FWHM is ~72 nm) in the Ag sample after thermal treatment under the same conditions. The above results suggest that post Xe ion irradiation in the keV energy range together with thermal annealing could be an effective method to fabricate planar NP arrays embedded in a SiO2 matrix and to modify the OA property of Ag NPs.

4. Conclusions

In summary, polycrystallization of ion-beam-synthesized Ag NPs in SiO2 has been clearly demonstrated after irradiation by 500 keV Xe ions to a fluence of 2 × 1016 cm−2. Owing to reduction of the EMFP and interaction between crystallites in the formed polycrystalline NPs, the Xe ion irradiation-induced modifications in structure and morphology of Ag NPs have been shown to result in a strong absorption band around 550 nm besides the SPR peak located at about 400 nm. Moreover, subsequent annealing at 400 °C for 1h is sufficient to restore various polycrystalline Ag NPs to spherical and crystalline ones, and thus nearly-aligned large Ag NPs are formed near the end of the Ag ion range, which contribute a much narrower Ag SPR absorption peak at 400 nm. This work indicates that the polycrystallization can be a significant parameter to control the SPR of noble metal NPs. In addition, the above findings may also provide an effective ion-engineering technique to tailor both the configurations and SPR property of embedded NPs in a SiO2 substrate.

Acknowledgments

Authors acknowledge the financial supports from Natural Science Foundation of China (NSFC) (Nos. 11175129 and 11175235) and Natural Science Foundation of Tianjin (No. 12JCZDJC 26900).

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Figures (6)

Fig. 1
Fig. 1 XTEM images showing overall morphologies of NPs created in the (a) Ag and (b) Ag + Xe samples.
Fig. 2
Fig. 2 (a) HRTEM image together with corresponding FFT pattern (inset) of one Ag NP in the Ag sample, (b) close view of the sheet-like nanostructure created in the Ag + Xe sample, and (c-e) HRTEM images showing various polycrystalline Ag NPs in the Ag + Xe sample.
Fig. 3
Fig. 3 GIXRD patterns of the Ag samples before and after Xe ion irradiation.
Fig. 4
Fig. 4 (a) OA spectra of the Ag and Ag + Xe samples. Inset gives the Gaussian fit band with two peaks (1, 2) for the Ag + Xe sample. (b) Calculated absorption cross section σ spectra of a crystalline spherical Ag NP by Mie theory and of polycrystalline Ag NPs by MG theory with different interaction parameters K. A refractive index of 1.46 was used for SiO2 matrix and all calculations are normalized to the height of the spectrum calculated by Mie theory.
Fig. 5
Fig. 5 Schematic illustration of the crystalline and polycrystalline Ag NPs as well as their influences on the EMFP.
Fig. 6
Fig. 6 (a) XTEM image of the Ag + Xe sample after 400 °C annealing. The inset gives the HRTEM image of one particle together with the measured spacings of the crystallographic plane, (b) OA spectra of the Ag and Ag + Xe samples after 400 °C annealing.

Equations (5)

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G 1 ( λ )   =   0. 1593exp ( ( ( λ 4 0 1 . 3 ) / 22 . 82 ) 2 )             +   0.0 1 0 59exp ( ( ( λ 434 . 3 ) / 13 . 5 ) 2 ) +  4 . 13exp ( ( ( λ 453 . 1 ) / 63 . 97 ) 2 )    4 .0 51exp ( ( ( λ 453 . 8 ) / 63 . 39 ) 2 )             +   0.0 6499exp ( ( ( λ 55 0. 3 ) / 516 . 1 ) 2 )                                                            
G 2 ( λ )   =   0.0 8275exp ( ( ( λ 55 0. 1 ) / 131 . 4 ) 2 )   +   0.0 734exp ( ( ( λ 278 . 7 ) / 1273 ) 2 )
σ abs =7.16× 10 4 ω [ ε 1,eff + ( ε 1,eff 2 + ε 2,eff 2 ) 1/2 ] 1/2
{ ε 1,eff = ε m + AC+BD C 2 + D 2 ε 2,eff = BCAD C 2 + D 2
A=f( ε 1 ε m ),        B=f ε 2 C= ε m +β( ε 1 ε m )f( ε 1 ε m )( 1 3 ε m  + K 4π ε m ) D=β ε 2 f ε 2 ( 1 3 ε m  + K 4π ε m )
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