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12TWO-COMPONENT DIELECTRIC FUNCTION OF GOLD NANOSTARS: NOVEL CONCEPT FOR THEORETICAL MODELING AND ITS EXPERIMENTAL VERIFICATION
NIKOLAI G. KHLEBTSOV1'2, SERGEY V. ZARKOV3, VITALY A. KHANADEEV1 AND YURI A. AVETISYAN3
1Institute of Biochemistry and Physiology of Plants and Microorganisms, RussianAcademy of Sciences,
2Saratov State University, Russia 3Institute of Precision Mechanics and Control, Russian Academy of Sciences, Russia
Abstract
Plasmon resonances of gold nanostars can be tuned across 600-2000 nm, which makes them an attractive platform for applications. Rational design of nanostar morphology requires adequate computational models. The common approach, based on electromagnetic simulations with the bulk dielectric function, is not applicable to sharp nanostar spikes, typical of plasmon resonances above 800 nm. Here we suggest a two-component dielectric function in which the nanostar core is treated as a bulk material, whereas the size-corrected dielectric function of the spikes is treated in terms of a modified Coronado-Schatz model. In contrast to common simulations with bulk gold constants and in agreement with experimental observations, the simulated nanostar spectra show a strong reduction in the Q factor of the plasmonic peak. The effect of NIR water absorption on the calculated cross sections is negligible, and the simulated and measured two-peaked UV-vis-NIR extinction spectra of water colloids and AuNST monolayers on glass in air are in qualitative agreement.
Gold nanostars (AuNSTs) have attracted much research attention because they are uniquely tunable across the vis-NIR-I-II spectral bands, are easy to fabricate, can generate strong local fields near the spikes, and are low cytotoxic. These properties make AuNSTs promising for bioimaging, photothermal, and sensing applications [1,2] The main obstacles to applications and fundamental studies of AuNSTs are their size polydispersity and multibranched random morphology, which result from poorly controlled seed-mediated synthesis [3] On the other hand, it is owing to their multibranched morphology that AuNSTs are much less sensitive to their orientation with respect the incident polarized light than are their well-developed and precisely controlled alternative—Au nanorods.
Rational design of AuNST morphology for single-particle photothermal and imaging platforms [4] and 2D SERS arrays requires adequate computational models including the frequency-dependent dielectric function of gold. Until now, to the best of our knowledge, the common practice—without any exceptions—has been the use of the bulk Au optical constants tabulated by Johnson and Christy (J-Ch) [5]. However, AuNSTs consist of two morphological components: (1) a quasispherical core with a typical diameter of above 20 nm; (2) thin anisotropic spikes with aspect ratios(length/base) in the range 3-8. Because of the strong size-dependent correction, the dielectric function of the thin anisotropic spikes may be very different from the tabulated bulk values. It is well known that simulations with a size-corrected dielectric function strongly increase the width of the plasmonic peak and decrease its amplitude. What is more, the recent developmentОшибка! Закладка не определена. of 6-tip AuNSTs with the main plasmonic peak in the range 1000-2000 nm makes the use of the J-Ch data questionable. A very thorough reexamination of the bulk Au optical constants by Olmon et al. [6] for evaporated, template-stripped, and single-crystal gold samples clearly shows a notable deviation from the J-Ch data. Evidently, the input dielectric function may strongly affect the peak position. As a result, the fitting of simulated spectra to experimental measurements by varying AuNST morphology may produce biased parameters.
Quite recently, we suggested [7] a two-component dielectric function in which the nanostar core and spikes are treated in terms of different dielectric functions:
k —, rX r £ ^e , n.
e—, r) = i (1)
k (— r X r £Vspke ,
where Vcore and V^ are the volumes of the core and spike, respectively. The core and spike components are calculated as ecs (— r) = eb —) + As (a, r,lcs), with the Lorenz-Drude correction to the tabulated bulk value sb (—:
2 2
As—, r,lc s) = As—,) = -—p.---^-- . (2)
— +—Yb - +—(Yb + Yc,s)
where —p is the plasma frequency; yb is the damping constant of bulk gold; lc s is the effective path length of the electrons in the core and spikes, respectively; and yc s takes into account three possible contributions from radiation
damping, surface-electron scattering, and chemical interface damping (CID) [8] yc s = y^ + yTI + yC'D • We have
shown [Ошибка! Закладка не определена.] that for typical AuNST cores, the size correction is small and the core dielectric function can be taken as the bulk value sb (rn). The radiation damping is proportional to the particle volume, and for thin spikes, we can neglect this contribution. Further, for spherical particles of radius R , the surface-scattering term scales like 1/ R . The same scaling is expected for CID damping, because the number of s-electrons interacting with the surface adsorbate is proportional to the particle volume, whereas the number of adsorbate molecules scales like R2. Thus, the surface and CID correction terms can be combined into one relation,
Ys = (AT" + A™) Vf = A/f, (3)
s s
where As takes into account the surface-electron scattering and CID contributions, vF is the Fermi velocity, and ls is the effective path length. In Eq. (3), the effective length of the size-corrected dielectric function can be calculated in terms of a modified Coronado-Schatz model [9] for a conical spike of base radius Rs and height (length) h [Ошибка! Закладка не определена.].
The next crucial point concerns appropriate values for the damping parameter As. Recent comparisons of simulated and experimental spectra on the basis of extinction measurements of monodisperse Au nanorod colloids, absorption experiments with single Au nanospheres and scattering experiments with single Au nanorods have shown that the best fitting value is about As = 1/3 . This value should be considered a low limit for cetyltrimethylammonium bromide-coated Au nanorods. For other ligands, the CID contribution can be different. For example, in the case of dodecanethiol adsorption on Au nanorods, Foerster et al. [Ошибка! Закладка не определена.] reported Amrf = 0.12 and Acd = 0.34 , thus giving the total A = 0.46. On the other hand, for small Au clusters, the maximal A
value approaches 1. In summary, the possible range of As values is 0.3 -1.
In addition to the conventional NIR-I biotissue window (700-1000 nm), two other optical windows, namely NIR-II (1000-1350 nm) and NIR-III or shortwave IR (SWIR) (1550-1870 nm), have been identified recently [10]. In the NIR-III spectral band, water absorbs the incident light strongly. Therefore, an important question arises: what is the physical meaning, if any, of the commonly used extinction, absorption, and scattering cross sections for particles hosted in an absorbing medium [11]? This question has been extensively dealt with in recent years (for details, the readers are referred to paper [12] and references therein). The main conclusion is that for an absorbing host medium, the electromagnetic energy budget depends on the geometry of the measuring volume, the properties of the scattering particle, and the particle's position with respect to the incident light and detector. The only observable integral quantity is the extinction cross section, which cannot be written as the sum of the absorption and scattering cross sections in the same way as for a nonabsorbing host medium. In fact, being operationally defined in the usual way, the scattering and absorption cross sections become dependent on the particle position. By contrast, the extinction cross section is independent of the particle position and characterizes the particle itself.
To make rough estimates of water-absorption effects, we consider small gold spheres and spheroids in an absorbing host medium. For small spheroidal particles with semiaxes (a,b,b), we suggested a new analytical expression [Ошибка! Закладка не определена.]:
= nRlQxt^b = П ^ Im3 gg (-1) 1) (4)
ф2 3 + 3La,b (gr -1)
where k0 = 2п/2 is the wave number in vacuum, n2 = n'2 + iri2=y[g is the medium's refractive index, and gr = g / g2 is the particle relative permeability, Rev is the equivolume sphere radius, and La b are the geometrical depolarization factors of the spheroids (La + 2Lb = 1). For randomly oriented particles (Cxt) = (Cext a + 2Cext b) / 3. We
have found that the effect of host absorption on the calculated cross sections is moderate for spheres and is quite negligible for spheroids. This validates our computations of the AuNST cross sections by using COMSOL Multiphysics v. 5.1 with a nonabsorbing refractive index of water.
A simplified 6-spike geometrical model of AuNSTs was obtained from the TEM data of Tsoulos et al. [Ошибка! Закладка не определена.] by fitting the calculated absorption peak position (with J-Ch Au constants) to the experimental range 1800-1900 nm. A similar model was also derived from TEM images of the AuNSTs synthesized in this work. We use the scattering geometry in which the incident electric field is polarized along one of the side spikes, because the optical properties of AuNSTs are determined mainly by the in-plane dipole excitation of spikes.
Figures 1A,B show the AuNST absorption and scattering spectra calculated with the bulk Au optical constants of J-Ch and Olmon et al. In addition to the main plasmonic peak, associated with the in-plane dipolar excitation of the collinear spike, there are two minor peaks (near 710 and 930 nm),associated with the hybridization of the core and spike multipolar plasmons. The spike peaks are increased by bonding with the gold core, which serves as a plasmonic antenna. The integral absorption cross section dominates the scattering one by an almost one order of magnitude. The positions and magnitudes of the main peaks calculated with the two sets of bulk Au optical constants differ noticeably. Thus, any fitting procedure based on the comparison of experimental and calculated NIR-II-III spectra with the J-Ch input optical constants may produce biased output data.
Figures 1C,D show the spike size-correction effect on the absorption and scattering spectra. As expected, the size correction results in noticeable decrease and broadening of the main peaks. The minor peaks retain their magnitudes but also become broadened. The plasmonic dumping in the spikes affects the scattering peaks of the AuNSTs much more strongly than it affects the corresponding absorption peaks (see the insets in panels C and D).
Figure 2A shows three types of nanostars, as found by SEM. The NST-1 type particles (~74%) were typical surfactant-free AuNSTs with an average of 6 asymmetrical spikes. The type 2 particles (NST-2; 12%) were highly symmetrical multispiked "sea urchins" (average spike number, 20). Finally, the type 3 particles (NST-3; 14%) mostly had small protrusions and sometimes also one or two small spikes. Typically, the reported extinction spectra of the AuNSTs synthesized with the Vo-Dinh protocol display a single major peak between 650-1100 nm. Except for rare cases, there seem to be no studies of colloids in the NIR-SWIR region of 1200-2300 nm, because such spectrophotometers are uncommon. Close inspection of the reported UV-vis-NIR spectra of AuNSTs suggests that possible SWIR peaks were simply missed because of the limited spectral ranges of the spectrophotometers used.
Figure 1: Absorption (A, C) and scattering (B, D) spectra of AuNSTs, as calculated with the bulk Au optical constants of Olmon et
al. and J-Ch. Panels (C) and (D) illustrate the plasmonic dumping effect in the spikes with different dumping parameters. For cores, the bulk constants are used. The insets show the geometrical model of AuNSTs (A) and TEM image (B, scale bar is 50 nm).
Figure 2: (A) Three types of AuNSTs, as found by SEM: NST-1 (cyan circles), NST-2 (magenta), and NST-3 (red). Panels (B) and (C) compare experimental (black) and simulated (blue) extinction spectra for water colloids (B) and bilayer on glass in air (C).
It is generally believed that the long spikes of surfactant-free AuNSTs correspond to typical broad plasmonic peaks observed experimentally between 800-1200 nm. In fact, however, such NIR peaks should be attributed to the short spikes of NST-1A subensemble, whereas the long-spike NST-1B subensemble produces the second plasmonic peak located between 1800-2300 nm [Ошибка! Закладка не определена.]. To the best of our knowledge, this plasmonic feature of surfactant-free AuNSTs has never been noted previously. By using statistical data and derived electromagnetic model we were able to reproduce the main spectral features of water AuNST colloids and bilayer on glass in air (Figs. 2B,C, blue curves).
In summary, we have introduced a new two-component dielectric function for modeling the plasmonic properties of AuNSTs. The model takes into account the size-dependent plasmonic dumping mechanisms for thin spikes, whereas the core dielectric function is treated as a bulk material parameter. We have provided, for the first time, an analytical expression for the extinction cross sections of small spheroids embedded in an absorbing host medium. With this solution and COMSOL simulations, we have demonstrated negligible effects of water absorption on the calculated cross sections for a broad spectral range (300-2300 nm). By contrast, the size-dependent correction of plasmonic dumping in the spikes causes the dipolar and multipolar plasmonic peaks to broaden and decrease strongly.
Our extensive numerical simulations with various structural models have revealed several new features, which have not been reported previously:
(1) The core + two spikes nanostructure is a minimal structure that reproduces the basic spectral feature of 6-spike AuNAs and the more complex 20-spike AuNSTs.
(2) The addition of a third, 45o spike to a two-collinear spike configuration does not affect the two-spike spectra under collinear excitation; instead, it produces only a small shoulder under 45o excitation. When the incident field is directed along the 45o spike, it excites an intense dipolar plasmon and a strong local field around the neighbouring collinear spike, rather than around the exciting 45o spike. These observations contradict Zhu et al.'s [13] explanation of two peaks in AuNST spectra.
(3) The absorption spectra of spheroids and AuNAs in strongly absorbing media are close to those calculated for similar dielectric media. However, we have found unexpected behaviour of the scattering plasmonic peak, when water absorption increases it by almost 30%. Further work is needed to explain this effect.
(4) We have demonstrated a typical decrease in the quality factor of the absorption and scattering spectra with the damping parameter as increasing from 0 to 1. However, in some cases, we have found an unexpected increase in the main absorption peaks. This effect has been explained by a small increase in spike absorption, enhanced by plasmonic coupling with the core.
(5) For symmetrical multispiked AuNSTs, we have demonstrated the polarization invariance of absorption spectra and have also shown the invariance of scattering spectra. Although the absorption and scattering spectra of 6-spike AuNSts are not polarization invariant, we have shown (both numerically and analytically) the polarization invariance for a specific case when the electric field lies in the spike plane.
(6) We have explicitly demonstrated the broadening of spectra with an increase in the damping parameter and the absence of a broadening for polarization-dependent normalized spectra.
(7) For common surfactant-free AuNSTs, we have reported, for the first time, very intense SWIR plasmonic peaks around 1600-1900 nm. By thoroughly inspecting the SEM images of surfactant-free AuNSTs, we have found two clusters in the spike length-spike base diagrams and in the spike length and aspect ratio histograms. This morphological feature has been correlated with the experimentally observed two-peaked spectra. In contrast to general belief, we have shown that the common UV-vis-NIR plasmonic peak of the surfactant-free AuNSTs is related to the multiple short spikes, rather than to the long ones. The long spike produces an intense SWIR plasmonic mode, which has not been reported before for such nanostars.
(8) To simulate the experimental extinction spectra of colloids and bilayers on glass in air, we have developed a simplified three-fraction model consisting of typical Vo-Dinh AuNSTs (major NST-1 fraction, about 75%), multiple branched symmetrical sea urchins (NST-2), and particles with protrusions (NST-3). The spike length of the major fraction consists of two sub-ensembles of short (NST-1A) and long (NST-1B) spikes, which are presented on the same core and generate UV-vis-NIR (700-1100 nm) and SWIR (1600-1900 nm) plasmonic bands, respectively. With our model, we have demonstrated good agreement between simulated and measured spectra. The suggested model of the dielectric function and the reported results could be useful for a deeper understanding of the optical properties of morphologically complex AuNSTs.
Acknowledgment: This research was supported by the Russian Scientific Foundation (Project No. 18-14-00016). The work by VK on AuNST synthesis and characterization was supported by the Russian Scientific Foundation (Project No. 19-72-00120). The work by SZ and YA on electromagnetic simulations was supported by the Russian Foundation for Basic Research (Project No. 19-07-00378)
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