2.1. Optical Characteristics in the X-Ray Regime

X-rays interact rather weakly with matter (e.g., absorption lengths are on the order of millimeters). Hence, X-ray refractive indices are extremely close to or below 1.0 (that of vacuum). Figure 1 gives the refractive index of silicon crystal as a function of wavelength from the infrared to extreme UV at 30 nm [29, 31]. It exhibits three peaks at 3.5, 4.6, and 5.5 eV which are referred to as E1, E2, and E3 [30, 32, 33]. The E1 (3.4 eV) generally corresponds to transitions between the valence band maximum and the edge of the conduction band minimum at the gamma point (the center of the Brillouin zone), while the E2 (4.4 eV) and E3 (5.3 eV) involve transitions between the valence band and higher complex band extrema. These exhibit strong optical responses and contribute significantly to the absorption and reflectance spectra of silicon. The figure shows that, for shorter and shorter wavelengths, the real refractive index approaches 1.0 (characteristic of vacuum) and begins to dip below it at 197 nm incident wavelength, reaching as deep as 0.24 at 100 nm, then rising back asymptotically to 1.0 (reaching 0.92 at 30 nm). Having a maximum dip of 0.76 below the vacuum makes silicon one of the most responsive materials. In the region below 1.0, one can write the complex refractive index as n = 1 − δ + iβ, where δ is the refractive index decrement and β is the absorption index [34, 35]. The real part of the refractive index (1 − δ) describes the phase shift of the X-rays, while the imaginary part (β) describes the attenuation (or absorption). Theoretical analysis shows that the decrement can be written as δ = (ρ_er_eλ²)/(2π), where ρ_e is the total electron density of the material and λ is the wavelength of te incident radiation and inversely proportional to r_e the classical electron radius, a fundamental constant, equal ∼ 2.818 × 10⁻¹⁵ m. Below 30 nm in the soft to hard x-ray regime (100 eV to 10 keV), the decrement continues to rise asymptotically but may not be fully smooth to 1.0 (δ decreases) as shown in the inset (lower right).

For even higher photon energies in the gamma regime [36], the expression for δ is the same as in the X-ray or EUV. The newly measured index of refraction (|δ|) in the γ regime for γ energies in the range 0.1 to 10 MeV (shown in the inset upper tight) exhibits the same dependencies as X-rays. The decrement dip δ continues to drop and asymptotically reaches 1.0, but it also runs into a local resonance at ∼ 0.9 MeV, and it reverses the sign by crossing 1.0. In this MeV range, additional quantum mechanical effects are exhibited above and beyond the X-ray response. The blue dashed curve shows the negative δ from the virtual photo effect, which is confirmed by the measured values for lower γ energies [36]. For the positive δ, virtual pair creation occurs. Also, the superposition of the two δ contributions is shown in the inset.

The dependence of the refractive index on the material can be easily seen in Figures 2(a), 2(b), 2(c), 2(d), which give the refractive index n for gold [39, 40] (two figures) and silica [37, 38, 41], in addition to that of silicon [29, 31] in the same photoenergy, respectively. It is to be noted that there is a lack of accurate measurements, and data are lacking over the entire X-ray range. Although at wavelengths < 50 nm, all materials have indices of refraction ≈ 1.0, the approach and the wavelength where it dips below 1.0 depend on the material composition. The X-ray refractive index is influenced by the atomic number (Z) of the material. Generally, materials with higher Z values exhibit stronger X-ray scattering, leading to a greater change in the refractive index. However, the relationship is not strictly monotonic due to resonant absorption edges. In plasmonic materials (e.g., gold), the refractive index is not as smooth as dielectric or semiconductor dielectric material. Figures 2(b), 2(c) show that the refractive index n of gold dips below 1.0 more than once, perhaps due to its oscillatory nature resulting from plasma electronic oscillations in the metal. It dips below 1.0 in the infrared/visible region (0.7–2), in addition to the dip in the region of interest, namely, EUV and X-ray regions. It effectively dips below 1.0 at wavelength of 60 nm, but briefly rises above 1.0 at and then stays down through the hard X-ray regime. In plasmonic materials (e.g., gold), with surface plasmon resonance exhibiting strong optical resonances that can be tuned by changing the material’s shape or surrounding environment, especially dielectric materials of high refractive index (high-k bulk material on the visible like silicon for example) can exhibit Mie resonances, leading to strong enhancement of electromagnetic fields within the material. Figure 2(d) on the other hand is the refractive index of silica whose refractive index dips late below 1.0 at 0.6 nm due to its high band gap (9.1 eV).

In the photoenergy or wavelength regime and higher intensities we consider in this paper (5–100 nm), we believe the above quantum effects would not play significant roles. At these wavelengths, the photoenergies will not be sufficient to produce virtual electron–positron pairs, nor virtual photoelectrons as would be in the few MeV gamma-ray regimes [36]. Other effects like stochastic effects would be minimized at higher incident intensity. Compton scattering, where photons interact with electrons, and Delbrück scattering, where photons interact with the nucleus, potentially produce electron–positron pairs. The effect was observed at the photon energy of 2.754 MeV [42], which in the soft/EUV conditions would be insignificant.

Other quantum effects that must be considered are due to quantum confinement in nanoparticles. In those particles, electrons are subject to quantum confinement and quantum wave reflection/interference off their boundaries, which create discrete energy levels, and hence, electron oscillations are restricted to those electrons within these discrete energy states. Moreover, such confinement leads to enhanced e-e correlation with diminishment of e-phonon interaction which leads to enhanced multielectron excitation by a single photon [43, 44]. Those effects are expected to be stronger under very strong confinement in ultrasmall nanoparticles ≤ 1.0 nm across. We neglect those effects in the present studies as we consider particles or films ≥ 3.0 nm. Moreover, on the ultrasmall limit with a small number of atoms, polarizability is a more appropriate term; nevertheless, the dielectric constant for ultrasmall sizes has been used in the literature. The dielectric constants of ultrasmall Si particles were examined by placing the particles on a silicon substrate, and a scanning tunneling microscope (STM) was used to measure the charging energy of different-sized particles (from I-V spectrum). The measured dielectric constant k and the results from Penn Model and pseudopotential calculations [45] are presented in Figure 3(a), while Figure 3(b) shows the layout of the measurements of a single particle (with allowance of spectroscopy using external light irradiation). In the case of a closely packed thin film (Figure 3(c)), corrections due to the silicon interface as well as the vacuum interface were estimated. The Si substrate typically increases the capacitance by less than 10 percent, only serving to lower the charging energy and therefore will not increase the largest charging energy barrier. For particles protruding above the vacuum interface, the capacitance decreases, hence increasing the largest charging energy. Since we get consistent values from sample-to-sample for each size and a consistent trend with size, the correction is negligible if any. Measured k as function of particle diameter shows that all particles with the exception of the 2.9 nm have values below the bulk value of 11.9.

The trend of the measurement shows that for the ultrasmall particle regime, the refractive index drops toward the vacuum value as the size of the nanostructure drops; thus, it is most probably that for X-rays, materials of all sizes will have a refractive index near 1.0. It is generally true that at wavelengths less than 50 nm (EUV), the refractive indices of materials are close to 1.0 regardless of their dimension. To avoid any prospectus effects, we will not use a thickness of 1 nm only in this paper. We will use 1, 3, 12, and 24 nm to allow comparison and evaluation.

2.2. Mie Scattering

We examine the effect of a dielectric semiconductor coating layer on the X-ray/EUV scattering response of a plasmonic AuNP (core–shell architecture). We theoretically analyze the plasmonic and polarizmon dielectric effects involved by conducting Mie-type scattering [22–25] studies as well as FDTD analysis [26–28] of the field distribution of X-ray scattering. We examined individual bare AuNPs and compared it to the response of coated AuNP with a thin dielectric/semiconductor material including closely packed clusters of luminescent 3-nm silicon particles. Mie studies involve the direct analytical solutions of Maxwell’s equations of particles in the presence of electromagnetic fields E and B using series expansions of the involved fields into partial waves of different spherical symmetries to model the interaction of metal–semiconductor nanoparticles with electromagnetic radiation.

We focus on the scattering of X-ray/EUV in the wavelength range 5–200 nm of a bare gold antiparticle as well as concentric core–shell structure of gold and silicon material. Table 1 gives the results for a bare gold particle of 10 nm diameter and 10-nm AuNPs with a dielectric coating consisting of an ultrathin layer of 3-nm-thick silicon. The table gives the real refractive index for both materials (silicon and gold), extracted from Figure 1 at selected wavelengths in X-ray/EUV range. The figure and table show that the materials begin to dip below 1.0 at 197 for silicon and 70 at nm for gold. Note that silicon exhibits one of the largest dips below vacuum (0.76) among materials, whereas gold, for example, dips only by a value of 0.22. The refractive indices exhibit variation with a wavelength that is not monotonic, exhibiting gold local global dips with oscillatory behavior. One notices that the dips in the two materials are not synchronous resulting in large refractive differences that change in sign. Since one expects the scattering of core–shell architecture that depends on the relative refractive index, the effect may not be straightforward to predict, making it more difficult to discern, with dependence on density, atomic weight, atomic number, and anomalous scattering factors of the materials. All materials have a refractive index near one in the X-ray regime. However, the effect of differences caused by those dependencies becomes significant in the approach of 1.0. The results show that the cross section of the Mie scattering for both structures is less than the geometrical cross section (78.5 and 201 nm² for the bare and the core–shell structures), which indicates as expected that the Mie scattering is weak in this range. Second, the coated AuNP, having an ultrathin coating of 3 nm silicon, shows an enhancement over the bare case, with σ core/σ bare showing an enhancement for most wavelengths. The enhancement reaches as large as an order of magnitude, especially in the soft X-ray/EUV case.

We examined the effect of the size of the gold core on the scattering cross section keeping the thickness of the silicon cap constant at 3 nm. Table 2 gives the Mie scattering of 10- and 5-nm-diameter AuNPs with 3-nm-thick silicon caps in the spectral range of 5–200 nm wavelength.

As expected, the scattering grows with the size of the scattering structure. The table shows that increasing the size of the core size increases the scattering cross section. However, the increase in the core size is bound to increase the size of the silicon shell which also contributes to the increase.

Table 3 gives the effect of the thickness of the silicon capping of AuNPs, keeping the gold core at a diameter of 5 nm. The table details the X-ray scattering cross section for the thickness cases of 3, 12, and 24 nm, respectively. As the thickness of the silicon cap rises from 3 to 12 to 24 nm, the scattering rises in the cross section. For 40 nm wavelength, it rises from 1.3 to 139, to 1620 nm².

Finally, we examine the effect of the gold core on the Mie scattering. We replace the AuNP with a vacuum core, which results in a silicon bubble. We calculated the scattering results for a vacuum core for a silicon thickness of 12 nm. The results show essentially very little effects of the gold part. Table 4 gives the case of 12 nm thickness as an example, showing that the gold nanostructure is essentially a “passive” substrate that holds the thin dielectric silicon layer. The trend indicates that dielectric layer coatings dominate the scattering of X-rays. The gold nanostructure essentially acts as a “passive” substrate to hold the thin dielectric silicon layer.

The real refractive index n of silica (SiO2) at longer wavelengths (e.g., 500 nm), is around 1.464, but at short wavelengths, it is important for understanding its behavior in the X-ray region. Theoretical models using atomic scattering factors of silicon and oxygen provide good agreement with experimental data. We now examine silica as another silicon-based material for coating AuNP. Table 5 gives the EUV/X-ray refractive index of SiO₂ and index of gold in the range 210–25 nm wavelength. Note that the interesting range in which the refractive index approaches 1.0 is 40–25 nm. Table 6 gives the Mie scattering of 10-nm-diameter bare AuNP, as well as with a silica cap of 3 nm thickness in the spectral range of 25–40 nm wavelength, showing more than fourfold enhancement.

It should be noted that the coating may also be produced by depositing closely packed H-terminated 1 nm particles [46–53]. Hydrophobic forces in an aqueous environment can be used to form this kind of assembly by driving the particles toward and nucleate on the gold surface. Hydrophobic forces have been used to form colloidal crystals with closely packed particles, The particle distribution of the nanoparticles is expected to be random but reasonably uniform. This provides an interparticle atomic scale gap of 3 Å, with a density of the order 10¹²–10¹³/cm². For a AuNP of 10 nm diameter, a total of ∼360 particles of 1 nm diameter are expected for a single monolayer.

2.3. FDTD Analysis (Scattered Field Distribution)

It is interesting to determine the field distribution, angular distribution, spatial range, and focusing effects of the core–shell nanostructures. We calculate these field distribution effects using the FDTD computations in which a discrete solution to Maxwell’s equations is obtained [22–28]. The method is based on central difference approximations of the spatial and temporal derivatives of the curl equations. We use the software of COMSOL Multi-Physics Software 5.3a AC/DC module electromagnetic waves. In the software, the simulation parameters, e.g., the dimensions of perfectly matched layer (PML) and grid size, are a physics-based mesh, with a max element size 0.35 nm, a min element size 0.000 05 nm, maximum growth rate 1.3, curvature factor 0.3, and resolution of narrow regions 1. For simulating unbounded problems, the method uses a reflectionless absorbing boundary (PML method). Moreover, the procedure can absorb evanescent waves and near fields. The procedure can handle the modeling of very complex geometries by using subcell modeling techniques and local subgrids. Subgrids are embedded within the global grid to locally resolve fine geometric structures without sacrificing the global space/time scale. The applications of subcell models and subgridding methods enhance the efficiency and the accuracy of the FDTD method. We apply FDTD to the core–shell architecture. As to the optical constants used in the calculations, and the wavelength dependence of the refractive index for silicon, silicon and gold are taken from previous data and the associated Figures 1 and 2 [29–34]. Those cover the region of X-ray–EUV region 4–20 nm. For the core metal (Au) nanoparticles, we used the values embedded in the COMSOL software package [40]. It is to be noted that there is lack of accurate measurements of the optical constants in the X-ray/EUV regime, and data are lacking over the entire X-ray range. Although at wavelengths < 50 nm, all materials have indices of refraction ≈ 1.0, the approach and the wavelength where it dips below 1.0 depend on some atonic/electronic characteristics of the material composition.

In the optical configuration, we use a linearly polarized incident beam of amplitude E_inc (or background field E_b) with an amplitude of 1.0 V/m. A plane wave source is used with the incident beam propagating along the x direction, and the polarization is along the z direction (see the coordinate system and configuration in Figures 4(a), 4(d), 4(g)). The scattered field is defined as E_sca = E_total − E_inc, where E_total is the total field and E_inc is the incident field. This definition is enforced at the boundary between the total and scattered fields by adding in or subtracting out a correction term for the update equations on and next to this boundary.

Figure 4 gives the scattered field distributions in the incident plane (plane of propagation and polarization) for the case of 5-nm-diameter AuNPs with different coatings and incident wavelengths. The figure uses a color-coded metric given on the right side of each frame. Note that the color code is different for each but, in all, the dark red is the strongest and the dark blue is the weakest. In all frames, the incident X-ray is a plane wave, incident from the left. The scattered field is normalized to the incident field amplitude of 1 V/cm. Figures 4(a), 4(b), 4(c) show a bare AuNP and 20, 10, and 5 nm incident wavelength, respectively. Figures 4(d), 4(e), 4(f) show a AuNP with 1-nm coating (e.g., silica) and incident wavelength of 20, 10, and 5 nm, respectively. Figures 4(g), 4(h), 4(i) show a AuNP with 1-nm silicon film coating shell and incident wavelength of 20, 10, and 5 nm, respectively. The trend in the frames shows that the scattering is restricted to the forward direction, getting more directional as the wavelength drops. The figure also shows that the scattered polarization/plasmon field is stronger and more focused than the pure plasmonic field, i.e., has a shorter range (shorter effective focal length) with higher intensity. The strongest inner field is more conversing for silicon material coating (a high k-material precursor in the visible).

We now compare the scattered intensity at the center and surface of the structure. Figures 5(a), 5(b), 5(c) give as a function of incident wavelength the scattered FDTD electric field for the case of bare 5-nm-diameter AuNPs, and with 1 nm dielectric (silica) with refractive index in the visible of 1.45, but near 1.0 in the X-ray region and with 1-nm silicon-coating shell, respectively (in the visible, the refractive index is 3.5–3.6 but neat 1.0 in the X-ray region). The fields are plotted at points A, B, and C, at the center of the structure and the inner and outer edges of the shell as labeled in the figure. For the pure gold structure, it shows that the scattered field at the center of the nanoparticle is excluded especially at the longer wavelength. The addition of the dielectric coating increases the penetration of the scattered fields inside the structure. Also, it shows the oscillatory as well as the steadiness of the scattered field with wavelength.

Finally, we examine the total scattering cross section as a function of wavelength. Figures 6(a), 6(b), 6(c) give as a function of incident wavelength the scattered FDTD cross section for the case of 5-nm-diameter AuNPs, for the three cases: bare, with 1 nm silica, and with 1 nm silicon-coating shells, respectively. Figure 6(a) shows that for a bare AuNP, the scattering cross section is structureless over a wavelength range of 5–20 nm, falling from a maximum of 5 nm² when the incident wavelength matches the diameter of the particle, effectively falling to a level of 0.5 nm². Wrapping the particles by a silica shell enhances the cross section to 50 nm², as shown in Figure 6(b), but it remains structureless in this wavelength range while dropping faster to a level of 2 nm². On the other hand, for a silicon shell (high k material with a refractive index of 3.65 when under visible/UV), the response shows several interesting features (Figure 6(c)). First, it is enhanced to a level of 93 nm². Second, it remains nearly level over the range 5–13 nm with some residual oscillation with wavelength before it drops sharply at 15 nm to a level of ∼ 80 nm². The residual ripple is about 3 nm² peak to peak, with visibility of ∼ 1.5%. However, we should guard that FDTD calculations are sensitive to the input data in the X-ray regime as there is lack or inaccurate optical constants. We should note that lack of accurate measurements and data of the optical constants over the entire X-ray range and that the approach and the wavelength where the refractive index dips below 1.0 depend on the material composition. Moreover, the scattering results for using a different software (e.g., ANSYS Lumerical photonic solver) may give different scattering intensities even for the same input date, let alone with input refractive index from a different source. For example, bare AuNP ANSYS may give results like the COMSOL package (except scattering intensity). Moreover, for gold-coated silicon, both may exhibit Mie-type resonances; however, the spectral responses of shell-coated nanoparticles may differ in shape, position, and magnitude. This strong dependence on the used material data/models does not allow unconditionally trusting numerical simulation results without appropriate experimental confirmation. Therefore, exact numerical results might be suspect, but several trends and features can be less suspect for a given software/input data, such as the prospect of contrast enhancement, multiresonance response, stronger focusing, and near-field range, which can be useful for the development of novel testing.