OSA Publishing > Optical Materials Express > Volume 9 > Issue 10 > Page 4090
Alexandra Boltasseva, Editor-in-Chief
Hot spot statistics and SERS performance of self-assembled silver nanoisland films
Ekaterina Babich, Sergey Scherbak, Fabrice Asonkeng, Thomas Maurer, and Andrey Lipovskii
Ekaterina Babich,1,* Sergey Scherbak,1,2 Fabrice Asonkeng,3 Thomas Maurer,3 and Andrey Lipovskii1,2
1Peter the Great St. Petersburg Polytechnic University, 29 Polytechnicheskaya, St. Petersburg 195251, Russia
2St. Petersburg Academic University, 8/3 Khlopina, St. Petersburg 194021, Russia
3Université de Technologie de Troyes, 12 rue Marie Curie, CS 42060, Troyes10004, France
*Corresponding author: babich_katherina@gmail.com
Ekaterina Babich https://orcid.org/0000-0003-4970-2591
E Babich
S Scherbak
F Asonkeng
T Maurer
A Lipovskii
•https://doi.org/10.1364/OME.9.004090
Ekaterina Babich, Sergey Scherbak, Fabrice Asonkeng, Thomas Maurer, and Andrey Lipovskii, "Hot spot statistics and SERS performance of self-assembled silver nanoisland films," Opt. Mater. Express 9, 4090-4096 (2019)
Wafer-scale silver nanoislands with ∼5 nm interstitial gaps for surface-enhanced Raman scattering (OME)
High-performance surface-enhanced Raman scattering substrate prepared by self-assembling of silver nanoparticles into the nanogaps of silver nanoislands (AO)
High-performance 3D flexible SERS substrate based on graphene oxide/silver nanoparticles/pyramid PMMA (OME)
Localized surface plasmon resonance
Nanopositioning equipment
Surface enhanced Raman scattering
Original Manuscript: July 12, 2019
Revised Manuscript: August 21, 2019
Manuscript Accepted: August 22, 2019
Discussion and modeling
The statistics of hot spots in the ensemble of self-assembled silver nanoislands grown on the surface of silver-to-sodium ion exchanged glass under hydrogen annealing was studied. The comparison of the surface enhanced Raman scattering (SERS) performance of the nanoisland films at different growth stages with the developed model of the hot spots formation revealed that at the nanoislands coalescence stage the enhancement is dominated by the increase of electric field in the hot spots, not by their surface density. At earlier growth stages, the linear relation between the SERS signal and the hot spots surface density is expected.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Nowadays, the development of highly efficient optical sensors is one of the urgent scientific and engineering problems, and sensors based on localized surface plasmon resonance (LSPR) are of the particular interest [1]. Optically excited LSPR in metal nanoparticles (NPs) increases the electric field (E-field) of incident lightwave in their vicinity. In the ensembles of metal NPs, small gaps between neighbor particles additionally enhance the local E-field and form so-called hot spots. This high E-field, in its turn, essentially enhances Raman scattering from biomolecules or other analytes absorbed on the metal surface, the maximal enhancement being up to 1010–1014 [2]. The ensembles of self-assembled metal NPs that are metal island films (MIF) integrated either onto solid or flexible dielectric substrates have attracted attention as prospective large-scale substrates for SERS [3,4]. MIF are conventionally fabricated via depositing metals onto dielectric substrates using thermal or electron beam evaporation [5], sputtering [6], and chemical or plasma-chemical metal deposition [7] followed by a high temperature annealing. Recently a novel technique to form silver nanoisland film (SNF) on the glass surface has been demonstrated [8], in which silver NPs were formed via self-assembly of out-diffused silver atoms in the course of annealing a silver-enriched glass in hydrogen. The enrichment of the glass with silver ions is employed by silver-to-sodium ion exchange [9,10]. The technique is notable for easy multiplication of the substrates, repeatability and stability of the SNF, and minimal fabrication costs. Moreover, in our experiments SNFs demonstrated SERS activity being the promising large-scale substrates for chemical sensing [11]. However, the impact of the SNF morphology, which changes in the course of the SNF growth, on their SERS activity is still unclear, while this issue is critical for applications. The objective of this study is to establish the relationship between the morphology of self-arranged metal nanoparticles, their hot spot statistics and Raman signal enhancement using modeling and experiments with SNF.
2. Experimental
Four glass substrates with SNFs on their surface and silver NPs in bulk were fabricated by the out-diffusion technique. The silver NPs in the bulk of the glass grew due to the penetration of hydrogen and its interaction with the diffused silver ions [12]. The Menzel glass slides [13] were immersed in (AgNO3)5wt%(NaNO3)95wt% melt heated up to 325°C for 20 min and annealed at 250°C in hydrogen atmosphere for 5, 10, 15 and 20 min. The SNFs formed on these slides were marked as S5, S10, S15 and S20, respectively.
The surface of each SNF was characterized by Atomic Force Microscope (AFM) measurements (Bruker's DIMENSION-ICON AFM). The NPs size distribution, surface density and the fraction of substrate surface occupied by the NPs (fill factor) were revealed from the AFM data processed with ImageJ and Gwyddion software.
We found the extinction band position of SNF associated with LSPR as residual of the optical extinction spectra measured with SNF on the glass surface and after the SNF removal. This procedure allowed excluding the impact of the bulk silver NPs and glass matrix in the samples' extinction. The measurements were performed with Specord 50 spectrophotometer in the spectral range 300-900 nm.
The Raman measurements were performed at 632.8 nm (LabRAM HR800, Horiba) excitation wavelength. In the experiments, we deposited 1 µl droplets of BPE (1,2-Di(4-pyridyl)ethylene 97%) water solution at 10−3 M concentration on the surfaces of S5 – S20 samples and virgin glass without SNF, dried them and measured spectra in 5 different points in the centers of the droplets. The diameter of the dried droplet was ∼3 mm, the same for all samples, that allowed comparing SERS by the SNFs of different morphology. The experiments were performed with 10x/0.3 objective, maximal laser power of 7 mW and the acquisition time of a single spectrum of 30 s.
The AFM images of S5 – S20 SNFs and the corresponding histograms of the nanoislands' radius distribution are presented in Fig. 1(a) and (b), respectively.
Fig. 1. (а) AFM images of S5 – S20 SNFs and (b) corresponding nanoislands' radius distribution. The SNFs fill factor FF, NPs surface density N, average radius $\bar{R}$ and total volume per 1 µm2 $V = {2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}N\pi {\bar{R}^3}$ are denoted.
As reported earlier, the nanoislands are shaped as semispheres/semispheroids [14]. As seen, the nanoislands are randomly distributed on the substrates, their typical radii are in the range of 5–20 nm, the heights are below 30 nm, and very few NPs are more than 40 nm in size. The accuracy of the lateral size measurements is ∼2 nm in correspondence with the AFM probe used (SCANASYST-AIR tip). The radius distributions were approximated with log-normal ones, from which the average radii, $\bar{R}$, were determined. The SNFs fill factor, FF, and surface density of NPs, N, were obtained from AFM measurements and denoted in Fig. 1 with the estimation of the total volume of the NPs per unit area, V.
These data allow following the dynamics of the nanoislands formation via out-diffusion. During the first 10 min (S5, S10) of the hydrogen annealing, silver ions in the glass bulk are reduced, silver atoms go to the glass surface and the NPs are formed [12,15]. This results in the increase of the surface density and the total volume of silver NPs. In S15 the total volume of the NPs is still slightly increased, however their surface density dropped. This allows supposing that, in parallel with silver out-diffusion, the coalescence of the NPs (the redistribution of silver between larger and smaller nanoparticles through surface diffusion in the favor of the larger ones) has started [16]. For longer annealing times the out-flow of silver stops because of the depletion of silver in the subsurface glass layer [17], and coalescence occurs under the absence of the inflow of silver atoms from the glass bulk. The latter is evidenced by the nearly same total volume of the NPs in S15 and S20, and by the drop of the surface density of the NPs in parallel with the increase of the average radius.
To reveal the relations between the morphology of the SNFs and their SERS activity we compared Raman spectra of BPE measured using S5 – S20 and virgin glass (see Fig. 2(a)). The excitation at 632.8 nm was chosen far from the LSPRs of the SNFs to eliminate the impact of the resonance in NPs on the E-field enhancement averaged over the ensembles of NPs. Thus, we used non-resonant SERS excitation, and to show this we plotted the extinction spectrum of S15 in Fig. 2(a). The LSPRs in other S5, S10 and S20 SNFs also lay in the spectral range of 435–455 nm, the heights of the extinction peaks being comparable.
Fig. 2. (а) BPE SERS spectra measured using S5 – S20 SNFs and (b) 1607 cm−1 BPE Raman line intensity vs SNFs fill factor. Inset: Extinction spectrum of S15. The LSPR of the SNF and the wavelength of the SERS excitation are marked.
As seen in Fig. 2(a), there is no signal from the virgin glass and the strongest enhancement is provided by S15, the spectra from S5 and S20 being about the same intensity in spite of essential difference in their morphology. In Fig. 2(b) the dependence of the intensity of 1607 cm−1 BPE Raman line vs the SNFs fill factor is plotted, the dispersion of the SERS intensity is indicated with error-bars. We chose this parameter because it has clear relation with the average radius of nanoislands and their surface density:
(1)$$FF = (\frac{1}{S}\sum\limits_i {\pi R_i^2} ) \cdot 100\%\le N\pi {\bar{R}^2} \cdot 100\%$$
One can see that the maximal enhancement in S15 corresponds to the maximal fill factor.
4. Discussion and modeling
The SERS signal is known to be proportional to the fourth power of the local E-field, ISERS∼E4 [18], and in a multi-particle system the enhancement is mainly provided by the hot spots because of high E-field in the gaps between neighbor NPs [19]. Thus, the increase in SERS efficiency can be provided by both the increase in the surface density of the hot spots, NHS, and the growth of the E-field in the hot spots under increasing the fill factor. The first results in a linear dependence and the second in a power relation of SERS signal to the surface density of the hot spots. To clarify the connection of Raman enhancement and SNFs fill factor, we calculated the maximum of E-field in the gap between two neighbor NPs as the function of the gap width and modeled the statistics of the hot spots concentration.
Since the SNF is the ensemble of randomly placed NPs, we have developed the algorithm of random filling an area by non-overlapping circles, which have log-normal radius distribution. Basically, the algorithm can be described as a following repeatable cycle: 1) to define a pull of NPs with given radius distribution; 2) to place a NP randomly in a given area; 3) to check for overlapping it with already placed NPs: if it does overlap to go to step 2 again, if it does not – to remember the NPs configuration and repeat the cycle till all the NPs from the pull are placed. We varied the total number of NPs in 1 µm2 and calculated the quantity of small gaps (hot spots) between the NPs vs. fill factor defined by the particle number and average radius - see Eq. (1). We considered as a hot spot any gap between the edges of two neighbor NPs, g, which did not exceed 0.5r, where r is the radius of the smaller NP in the couple. As discussed below, this provides strong enough interaction of the NPs.
The E-field modeling was performed numerically in COMSOL Multiphysics environment (wave-optics module, frequency domain) for the pair of the same semispherical NPs 5 nm in radius [20]. Two NPs were placed on a glass with dielectric permittivity 2.25 in air (upper environment) and normally illuminated by a plane lightwave of a unit magnitude. Generally, the dependence of the E-field in the gap on the angle between the particle-particle axis and the incident wave polarization, φ, is described as ∼|cos(φ)|, and we considered the case with φ=0 as an upper estimation, all the gap dependences for different angles being proportional. The E-field was then calculated everywhere, including the gap between the NPs, using a finite elements method. Essentially, for NPs, the size of which is below 50 nm, the electrodipole approximation is valid [21], and their plasmonic properties do not directly depend on the size. Therefore, the dependences in Fig. 3(a) are general for the NPs with size below 50 nm and qualitatively general for all positions of the particle-particle axis relatively to the E-field direction. The E-field has maximum near the NPs surfaces and decay with the distance from the surfaces being minimal at the center between NPs (see the inset in Fig. 3(a)). The maximum of E-field decrease when the g increases. As follows from Fig. 3(a), the hot spots with g < 0.1r provide at least 14-fold enhancement of the incident field, whereas the hot spots with g = 0.5r only about 6-fold and with g = 0.3r about 8-fold. This means that input of the hot spots with g < 0.1r in SERS is higher than the one of spots with g∼0.5r or with g∼0.3r by at least an order of magnitude, for SERS signal is proportional to the 4th power of local E-field: (14/6)4∼28 and (14/8)4∼9.3. The number of the hot spots with g < 0.1r, g < 0.3r and g < 0.5r as the function of the film fill factor (a plain substrate) is presented in Fig. 3(b). All dependences correspond to the cube law and are similar to each other, i.e. statistically equivalent. This means that the fractions of the hot spots with g < 0.1r, 0.1r < g < 0.3r and 0.3r < g < 0.5r do not depend on the fill factor and quantities of these hot spots are proportional with some weights to the quantity of the hot spots with g < 0.5r as well as the corresponding inputs in SERS enhancement. Therefore, one can expect the linear increase in SERS signal with the increase in the number of the hot spots with g < 0.5r. Note that the presented dependences were calculated for a given log-normal distribution with $\bar{R} = 12$ nm and dispersion = 0.3 but they are qualitatively the same for normal distribution as well and all considered radii and dispersions presented in Fig. 1(b).
Fig. 3. (a) Calculated dependence of maximum of E-field on the gap g between two NPs of radius r; inset: the dependence of E-field on the position in the gap (x/r) for g = 0.5r. (b) Calculated dependences of the number of hot spots with different g on fill factor; inset: experimental points for S5 – S20 SNFs. (c) Measured 1607 cm−1 BPE Raman line intensity vs surface density of hot spots with g < 0.5r, NHS, and the diagrams of the interparticle gaps distribution in the SNFs. The legend is in the upper left corner of the graph.
We compared the modeled statistics with the experimental data processed the similar way: the distances between NPs in randomly chosen 1 µm2 area of AFM images were sorted by the criteria of g < 0.1r, g < 0.3r and g < 0.5r. The results presented in Fig. 3(b) and (c) clearly show that the developed algorithm is valid only for random distribution of NPs on the surface in presence of a source "to feed" the particles: the discrepancy between the modeling and the experiments is minimal for S5 and S10 and fewer than 5%. It almost does not describe phase transition processes like coalescence observed in S15 and S20 SNFs. This is evidenced by the essentially higher, over 20%, discrepancy for these samples. Coalescence results in the convergence of the NPs to each other and the increase of the fraction of the hot spots with smaller gaps as seen in the presented experimental statistics.
In Fig. 3(c), we plotted the SERS signal as the function of the surface density of the hot spots in the studied samples S5 – S20. Here one can see that, as expected, the SERS signal is maximal for the maximal surface density of the hot spots. However, there is no linear relation between the Raman enhancement provided by the SNFs and the surface density of the hot spots. This allows concluding that in the growth of SERS signal the local E-field increase dominates over the increase in the number of the hot spots, i.e. the hot spots with closer placed NPs give more significant impact in SERS. For example, S5 and S20 provide approximately the same level of SERS signal despite almost twice difference in the surface density of the hot spots with g < 0.5r. However, S20 contains more "hottest" spots with g < 0.1r than S5. In the same manner, S10 and S15 can be compared. These samples have approximately the same hot spots surface density, but S15 provides twice-stronger SERS signal. This is probably because S15 contains more "hotter" spots with 0.1r < g < 0.3r than S10. Nevertheless, in the studied range of the SNFs fill factors, the highest fill factor corresponds to both the highest SERS signal and the highest surface density of the hot spots. This allows using MIF fill factor as a universal parameter defining the SERS efficiency in the ensembles of NPs.
The study shows that the optimal for SERS applications out-diffused silver nanoisland film corresponds to early beginning of the coalescence stage. This also corresponds to the maximal fill factor ∼40%, which, as shown by the experiments and modeling performed, reflects both the smallest gaps between the particles (hot spots) and the highest E-field in these hot spots. The change in SERS signal is dominated by the local E-field enhancement when coalescence of metal films begins and by the hot spots surface density for films at earlier stages of their growth. In the latter case, SERS signal linearly increases with hot spots surface density. The developed model provides a general relation between the hot spots surface density and the fill factor of the ensemble of NPs. These results are of general character and can be applied to other ensembles of self-arranged NPs formed in the presence of a source of metal, e.g. by sputtering.
Ministry of Education and Science of the Russian Federation (3.2869.2017/PCh); Agence Nationale de la Recherche (ANR-18-CE09-0003); EIPHI Graduate School (ANR-17-EURE-0002).
F. A. and T. M. acknowledge financial support of NanoMat by the "Ministère de l'enseignement supérieur et de la recherche", the "Conseil régional Champagne-Ardenne", the "Fonds Européen de Développement Régional fund" and the "Conseil général de l'Aube".
1. Y. Oh, K. Kim, S. Hwang, H. Ahn, J. W. Oh, and J. R. Choi, "Recent advances of nanostructure implemented spectroscopic sensors - A brief overview," Appl. Spectrosc. Rev. 51(7-9), 656–668 (2016). [CrossRef]
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4. L. Bin Zhong, J. Yin, Y. M. Zheng, Q. Liu, X. X. Cheng, and F. H. Luo, "Self-assembly of Au nanoparticles on PMMA template as flexible, transparent, and highly active SERS substrates," Anal. Chem. 86(13), 6262–6267 (2014). [CrossRef]
5. A. Schaub, P. Slepička, I. Kašpárková, P. Malinský, A. Macková, and V. Švorčík, "Gold nanolayer and nanocluster coatings induced by heat treatment and evaporation technique," Nanoscale Res. Lett. 8(1), 249 (2013). [CrossRef]
6. Y. Hatakeyama, K. Onishi, and K. Nishikawa, "Effects of sputtering conditions on formation of gold nanoparticles in sputter deposition technique," RSC Adv. 1(9), 1815 (2011). [CrossRef]
7. M. Maicu, R. Schmittgens, D. Hecker, D. Glöß, P. Frach, and G. Gerlach, "Synthesis and deposition of metal nanoparticles by gas condensation process," J. Vac. Sci. Technol., A 32(2), 02B113 (2014). [CrossRef]
8. V. V. Zhurikhina, P. N. Brunkov, V. G. Melehin, T. Kaplas, Y. Svirko, V. V. Rutckaia, and A. A. Lipovskii, "Self-assembled silver nanoislands formed on glass surface via out-diffusion for multiple usages in SERS applications," Nanoscale Res. Lett. 7(1), 676 (2012). [CrossRef]
9. R. H. Doremus, "Electrical potentials of glass electrodes in molten salts," J. Phys. Chem. 72(8), 2877–2882 (1968). [CrossRef]
10. V. V. Zhurikhina, M. I. Petrov, K. S. Sokolov, and O. V. Shustova, "Ion-exchange characteristics of sodium-calcium-silicate glass: Calculation from mode spectra," Tech. Phys. 55(10), 1447–1452 (2010). [CrossRef]
11. E. Babich, A. Redkov, I. Reduto, and A. Lipovskii, "Self-Assembled Silver–Gold Nanoisland Films on Glass for SERS Applications," Phys. Status Solidi RRL 12(1), 1700226 (2018). [CrossRef]
12. M. Suszynska, L. Krajczyk, R. Capelletti, A. Baraldi, and K. J. Berg, "Microstructure and silver nanoparticles in ion-exchanged and deformed soda-lime silicate glasses," J. Non-Cryst. Solids 315(1-2), 114–123 (2003). [CrossRef]
13. ThermoScientific®, "Microscope Cover Slips," https://www.agarscientific.com/microscope-slides.
14. S. Chervinskii, V. Sevriuk, I. Reduto, and A. Lipovskii, "Formation and 2D-patterning of silver nanoisland film using thermal poling and out-diffusion from glass," J. Appl. Phys. 114(22), 224301 (2013). [CrossRef]
15. A. Miotello, G. De Marchi, G. Mattei, P. Mazzoldi, and A. Quaranta, "Clustering of silver atoms in hydrogenated silver-sodium exchanged glasses," Appl. Phys. A: Mater. Sci. Process. 70(4), 415–419 (2000). [CrossRef]
16. M. Smoluchowski, "Mathematical theory of the kinetics of the coagulation of colloidal solutions," Z Phys Chem 129, 129–168 (1917).
17. Y. Kaganovskii, A. Lipovskii, M. Rosenbluh, and V. Zhurikhina, "Formation of nanoclusters through silver reduction in glasses: The model," J. Non-Cryst. Solids 353(22-23), 2263–2271 (2007). [CrossRef]
18. P. L. Stiles, J. A. Dieringer, N. C. Shah, and R. P. Van Duyne, "Surface-Enhanced Raman Spectroscopy," Annu. Rev. Anal. Chem. 1(1), 601–626 (2008). [CrossRef]
19. A. Gopinath, S. V. Boriskina, W. R. Premasiri, L. Ziegler, B. M. Reinhard, and L. Dal Negro, "Plasmonic Nanogalaxies: Multiscale Aperiodic Arrays for Surface-Enhanced Raman Sensing," Nano Lett. 9(11), 3922–3929 (2009). [CrossRef]
20. F. Heisler, E. Babich, S. Scherbak, S. Chervinskii, M. Hasan, A. Samusev, and A. A. Lipovskii, "Resonant Optical Properties of Single Out-Diffused Silver Nanoislands," J. Phys. Chem. C 119(47), 26692–26697 (2015). [CrossRef]
21. S. A. Maier, Plasmonics: Fundamentals and Applications, 1st ed. (Springer, US, 2007).
Y. Oh, K. Kim, S. Hwang, H. Ahn, J. W. Oh, and J. R. Choi, "Recent advances of nanostructure implemented spectroscopic sensors - A brief overview," Appl. Spectrosc. Rev. 51(7-9), 656–668 (2016).
K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, "Single Molecule Detection Using Surface-Enhanced Raman Scattering (SERS)," Phys. Rev. Lett. 78(9), 1667–1670 (1997).
P. Mosier-Boss, "Review of SERS Substrates for Chemical Sensing," Nanomaterials 7(6), 142 (2017).
L. Bin Zhong, J. Yin, Y. M. Zheng, Q. Liu, X. X. Cheng, and F. H. Luo, "Self-assembly of Au nanoparticles on PMMA template as flexible, transparent, and highly active SERS substrates," Anal. Chem. 86(13), 6262–6267 (2014).
A. Schaub, P. Slepička, I. Kašpárková, P. Malinský, A. Macková, and V. Švorčík, "Gold nanolayer and nanocluster coatings induced by heat treatment and evaporation technique," Nanoscale Res. Lett. 8(1), 249 (2013).
Y. Hatakeyama, K. Onishi, and K. Nishikawa, "Effects of sputtering conditions on formation of gold nanoparticles in sputter deposition technique," RSC Adv. 1(9), 1815 (2011).
M. Maicu, R. Schmittgens, D. Hecker, D. Glöß, P. Frach, and G. Gerlach, "Synthesis and deposition of metal nanoparticles by gas condensation process," J. Vac. Sci. Technol., A 32(2), 02B113 (2014).
V. V. Zhurikhina, P. N. Brunkov, V. G. Melehin, T. Kaplas, Y. Svirko, V. V. Rutckaia, and A. A. Lipovskii, "Self-assembled silver nanoislands formed on glass surface via out-diffusion for multiple usages in SERS applications," Nanoscale Res. Lett. 7(1), 676 (2012).
R. H. Doremus, "Electrical potentials of glass electrodes in molten salts," J. Phys. Chem. 72(8), 2877–2882 (1968).
V. V. Zhurikhina, M. I. Petrov, K. S. Sokolov, and O. V. Shustova, "Ion-exchange characteristics of sodium-calcium-silicate glass: Calculation from mode spectra," Tech. Phys. 55(10), 1447–1452 (2010).
E. Babich, A. Redkov, I. Reduto, and A. Lipovskii, "Self-Assembled Silver–Gold Nanoisland Films on Glass for SERS Applications," Phys. Status Solidi RRL 12(1), 1700226 (2018).
M. Suszynska, L. Krajczyk, R. Capelletti, A. Baraldi, and K. J. Berg, "Microstructure and silver nanoparticles in ion-exchanged and deformed soda-lime silicate glasses," J. Non-Cryst. Solids 315(1-2), 114–123 (2003).
ThermoScientific®, "Microscope Cover Slips," https://www.agarscientific.com/microscope-slides .
S. Chervinskii, V. Sevriuk, I. Reduto, and A. Lipovskii, "Formation and 2D-patterning of silver nanoisland film using thermal poling and out-diffusion from glass," J. Appl. Phys. 114(22), 224301 (2013).
A. Miotello, G. De Marchi, G. Mattei, P. Mazzoldi, and A. Quaranta, "Clustering of silver atoms in hydrogenated silver-sodium exchanged glasses," Appl. Phys. A: Mater. Sci. Process. 70(4), 415–419 (2000).
M. Smoluchowski, "Mathematical theory of the kinetics of the coagulation of colloidal solutions," Z Phys Chem 129, 129–168 (1917).
Y. Kaganovskii, A. Lipovskii, M. Rosenbluh, and V. Zhurikhina, "Formation of nanoclusters through silver reduction in glasses: The model," J. Non-Cryst. Solids 353(22-23), 2263–2271 (2007).
P. L. Stiles, J. A. Dieringer, N. C. Shah, and R. P. Van Duyne, "Surface-Enhanced Raman Spectroscopy," Annu. Rev. Anal. Chem. 1(1), 601–626 (2008).
A. Gopinath, S. V. Boriskina, W. R. Premasiri, L. Ziegler, B. M. Reinhard, and L. Dal Negro, "Plasmonic Nanogalaxies: Multiscale Aperiodic Arrays for Surface-Enhanced Raman Sensing," Nano Lett. 9(11), 3922–3929 (2009).
F. Heisler, E. Babich, S. Scherbak, S. Chervinskii, M. Hasan, A. Samusev, and A. A. Lipovskii, "Resonant Optical Properties of Single Out-Diffused Silver Nanoislands," J. Phys. Chem. C 119(47), 26692–26697 (2015).
S. A. Maier, Plasmonics: Fundamentals and Applications, 1st ed. (Springer, US, 2007).
Ahn, H.
Babich, E.
Baraldi, A.
Berg, K. J.
Bin Zhong, L.
Boriskina, S. V.
Brunkov, P. N.
Capelletti, R.
Cheng, X. X.
Chervinskii, S.
Choi, J. R.
Dal Negro, L.
Dasari, R. R.
De Marchi, G.
Dieringer, J. A.
Doremus, R. H.
Feld, M. S.
Frach, P.
Gerlach, G.
Glöß, D.
Gopinath, A.
Hasan, M.
Hatakeyama, Y.
Hecker, D.
Heisler, F.
Hwang, S.
Itzkan, I.
Kaganovskii, Y.
Kaplas, T.
Kašpárková, I.
Kim, K.
Kneipp, H.
Kneipp, K.
Krajczyk, L.
Lipovskii, A.
Lipovskii, A. A.
Liu, Q.
Luo, F. H.
Macková, A.
Maicu, M.
Maier, S. A.
Malinský, P.
Mattei, G.
Mazzoldi, P.
Melehin, V. G.
Miotello, A.
Mosier-Boss, P.
Nishikawa, K.
Oh, J. W.
Oh, Y.
Onishi, K.
Perelman, L. T.
Petrov, M. I.
Premasiri, W. R.
Quaranta, A.
Redkov, A.
Reduto, I.
Reinhard, B. M.
Rosenbluh, M.
Rutckaia, V. V.
Samusev, A.
Schaub, A.
Scherbak, S.
Schmittgens, R.
Sevriuk, V.
Shah, N. C.
Shustova, O. V.
Slepicka, P.
Smoluchowski, M.
Sokolov, K. S.
Stiles, P. L.
Suszynska, M.
Svirko, Y.
Švorcík, V.
Van Duyne, R. P.
Yin, J.
Zheng, Y. M.
Zhurikhina, V.
Zhurikhina, V. V.
Ziegler, L.
Anal. Chem. (1)
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Appl. Phys. A: Mater. Sci. Process. (1)
Appl. Spectrosc. Rev. (1)
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(1) F F = ( 1 S ∑ i π R i 2 ) ⋅ 100 % ≤ N π R ¯ 2 ⋅ 100 %