Numerical Study of the Optical Response of $\text{ITO}$-${\text{In}_{{2}}{\text O}_{{3}}}$ Core-Shell Nanocrystals for Multispectral Electromagnetic Shielding

Nowadays, materials to protect equipment from unwanted multispectral electromagnetic waves are needed in a broad range of applications including electronics, medical, military and aerospace. However, the shielding materials currently in use are bulky and work effectively only in a limited frequency range. Therefore, nanostructured materials are under investigation by the relevant scientific community. In this framework, the design of multispectral shielding nanomaterials must be supplemented with proper numerical models that allow dealing with non-linearities and being effective in predicting their absorption spectra. In this study, the electromagnetic response of metal-oxide nanocrystals with multispectral electromagnetic shielding capability has been investigated. A numerical framework was developed to predict energy bands and electron density profiles of a core-shell nanocrystal and to evaluate its optical response at different wavelengths. To this aim, a finite element method software is used to solve a non-linear Poisson's equation. The numerical simulations allowed to model the optical response of $\mathbf {ITO}$-$\mathbf {In_{2}O_{3}}$ core-shell nanocrystals and can be effectively applied to different nanotopologies to support an enhanced design of nanomaterials with multispectral shielding capabilities.

T HE development of electronic devices operating on a wide range of frequency from one side opened a plethora of new applications while, on the other side, came along with a substantial and critical rise of electromagnetic interference, i.e., a set of unwanted collateral effects as device malfunctions and human exposure to electromagnetic radiation [1], [2]. To limit the impact of such unwanted effects, the interest and the demand for multispectral shielding materials grew exponentially in the last few years. Shielding materials that can successfully reduce the susceptibility to external radiation are needed in a broad set of applications as electronics medical, military and aerospace [3], [4], [5], [6], [7], [8]. For instance, smart electronics and telecommunication satellites operate in the dual-band spectrum of microwaves and infrared radiations [9], while medical equipment mostly work using microwaves and X-rays [10]. Additionally, the electromagnetic radiation emitted by the Sun covers a wide spectrum including microwave, infrared and ultraviolet wavelengths, which aerospace aircraft must cope with and endure [10]. Unfortunately, multispectral shielding requirements are not met by the shielding materials that are currently on the market, which are primarily synthesized for single-band electromagnetic waves. In fact, each band of the electromagnetic spectrum calls for unique attenuation mechanism [5], [11], [12], [13], [14]. Ultraviolet photons, for example, have enough energy to induce electron transitions between different energy levels (see Fig. 1). The wavelength of absorbed light has the energy required to promote an electron from a lower to a higher energy level resulting in the absorption of the ultraviolet photon. In this context, UV shielding materials are typically used in sunscreen applications since ultraviolet overexposure can harm living human cells [15], [16]. Considering  low infrared emissivity and low thermal conductivity [17]. The former, according to Kirchhoff's equation of thermal radiation for opaque materials, can be attained by high reflectivity, which in turn depends on electrical conductivity [18].
In this framework, the study of nanoscale radiators, scatterers and absorbers is a fundamental challenge to enable industrial and biomedical plasmonic applications [19]. Nonetheless, the design of nanomaterials with suitable properties must be supported with proper theoretical and numerical tools. To date, the literature focused on the analytical analysis of metallic, spherical nanoparticles under resonance conditions [20], especially considering scattering [21], [22], [23]. By solving the full dipole equation for the dominant mode radiation damping of the oscillating nanocrystal (NC), closed-form equations to study the near-or far-field characteristics can be derived. Several models have been developed to quantitatively describe the resonant behavior of NC (radii from 5 nm to 50 nm), through lumped-impedance representation based on the quasi-static approximation [24], [25]. The latter holds, depending on the scatterer size, the radiation wavelength and the field homogeneity in the nanoparticle [26]. The classic circuit concept, which is limited to the fundamental dipole mode, could be extended to predict the optical spectrum of nanoparticles independently on wavelength, size and material properties constraints [27]. Further efforts, such as the closed-form solution from [28], were spent in accounting for third order non-linear susceptibility in metallic nanostructures. However, when quasi-static approximation and circuital models do not apply, i.e., when high non-linearities are in place, numerical methods to investigate the interaction between the electromagnetic radiation and the NC are adopted [29]. For instance, recently, in [30], the surface enhanced Raman scattering of multiple nanostructures was studied by developing a wide-band nested equivalent source approximation accelerated method of moments. Finite element method (FEM) solvers, in particular, have been assessed to be effective numerical tools to investigate the performance of absorbing nanostructures [29], [31], [32]. So far, the studies about the numerical analysis of innovative nanomaterials mostly dealt with scattering, focusing with noble metals (e.g., gold nanosphere) [29] or dielectric nanoparticles [23], and core-shell systems have been understudied [22]. There is lack of numerical framework capable of handling extinction and scattering mechanisms of nanomaterials. Furthermore, any of the aforementioned models can predict the multispectral optical response of nanomaterials starting from their electronic properties.
In this study, we adopted numerical tools to simulate the electromagnetic response of metal oxide nanocrystals (MO-NCs) with multispectral shielding capabilities. Based on different principles, effective shielding can be achieved by core-shell MO-NCs in a wide range of frequencies, from infrared to ultraviolet. Despite several plasmonic semiconductors can be used for shielding applications (Fig. S1, Supplementary Material) [33], [34], [35], Sn : NCs were considered as a case study due to their relative ease in the synthesis of core-shell structures. We first modeled an isolated core-shell NCs, pre-and post-photodoping, to account for their optical response by performing full-field optical simulations using steady state carrier concentration profiles. The latter are obtained by solving Poisson's equation over a spherical NC. Then, we predicted their far-field optical response and analyzed the near-field characteristics around the NCs.

II. ELECTROSTATIC SIMULATIONS
In this section, a three-dimensional model of a single NC is presented assuming an uniform dopant concentration from the center up to the interface between the core and shell regions. Charges in surface states are considered to be located outside the NC. To obtain the electric field at the surface, the Poisson's equation must be integrated from the center to the surface of the NC.

A. General Model
The Poisson's equation is defined as [36]: where ε ∞ (r) and ε 0 are the high-frequency dielectric permittivity of the NC and the permittivity of free space, respectively. To integrate Poisson's equation, an explicit expression for the charge density at any point, ρ(r), is needed, which is given by [36], [37]: where p h and n e are the carrier density contribution of mobile electrons and holes, respectively, ρ D (r) and ρ A (r) are the radially changing donor and acceptor dopant density, respectively, while e is the unit electron charge. In the following, energy and potential terms are indicated as follows: E F is the Fermi energy while E C and E V are the energies at the bottom of the conduction band (CB) and the top of the valence band (VB), respectively. For the sake of convenience, it has been defined an intrinsic energy E I , which is defined as E C +E V 2 . At any point r in the NC, it is possible to define a generic potential [38]: where E P and E Q are two generic energy levels. In particular, the dimensionless potential u between E F and E I is defined as: where k B is the Boltzmann constant, T is the temperature and ϕ is the potential difference: At any point inside the NC, the electron volume density, n e is given by [38]: where E CU is the upper edge of the CB, D C (E) is the density of states, which is the number of states per unit energy per unit volume [38]: where h is the Planck's constant, m * e is the effective electron mass in the CB which is equal to 0.4 times the electron mass in the case of ITO and In 2 O 3 [36], while f (E) is the Fermi-Dirac distribution function given by [38]: The use of the density of states expression for bulk structures is justified by the absence of quantum confinement effects (discussion in Supplementary Material). In (6), the factor two in front of the integral is due to the fact that each state in the CB can host two electrons of opposite spin. It must be highlighted that, usually, the integration limit E CU is replaced with +∞ to account for the fact that the integrating function vanishes exponentially at high energies. To compute this term, approximated expressions are used [37]. However, in this case, we selected this value iteratively by looking for a value large enough to ensure the convergence to a solution and to achieve an accuracy no larger than 1%. By replacing (7) and (8) into (6), we get: Rearranging (9), we obtain: where F j (η) is the integral Fermi-Dirac function defined as [39]: being j ∈ Z + , η and x the variable and the dummy term, respectively. Similarly, the hole volume density in the VB, p h , is [38]: where D V (E) is the number of quantum states in the VB per unit energy per unit volume, which is given by [38]: where m * h is the hole's effective mass in the VB equal to 0.6 times the electron mass in the case of ITO and In 2 O 3 [36]. In (12), the lower integration limit is taken to -∞ based on the assumption that E F is several k B T above the bottom of the VB. Finally, by inserting (8) and (13) in (12), we obtain: Once the electron and hole densities are obtained, to determine the charge density in any point (see (2)), information on the charge concentration due to ionized impurities is needed. We assumed that the impurity concentration is sufficiently low to consider the occupation of each site independently, i.e., the states are non-interacting. Under this assumption, a simple model for donors and acceptors can be used. Donors are considered neutral impurity atoms, each of which can release an unpaired electron thus becoming positively charged; while acceptors are atoms of impurities that can gain an electron (which goes into a coupled state) thus becoming negatively charged. Given the energy level and the concentration of the donors, E D and N D , respectively, the concentration of electrons in donor sites can be found as [40]: where the coefficient 2 in (15) is the ground-state degeneracy of the donor impurity due to the possibility for the electron occupying a given donor site to be of either spin [37], [41], which can be rewritten in terms of potentials: The expression for the occupation of the acceptor sites can be found similarly to (15), just considering that, in this case, the electron must be paired [37], [41]. The acceptor electron energy, E A , and the concentration of acceptor sites, N A , can be combined to obtain the concentration of electrons in acceptor sites, ρ A : Again, in terms of potentials, we have:  Finally, by inserting (2) in (1) and replacing (10), (14), (16), (18) in (2), then (1) reads as:

B. Pre-Photodoped NCs
When dealing with spherical NCs, we solved numerically the Poisson's equation (19) using the built-in Mathematics module of the Comsol Multiphysics v5.6 (Comsol Inc., Burlinghton, MA, USA) FEM software. The variable is discretized by using quadratic Lagrange element and a free triangular mesh is used in the NC analysis domain. The built-in Multifrontal Massively Parallel sparse direct Solve tool, with a non-linear Newton solver is adopted, with pivoting, row pre-ordering, an initial damping factor τ d = 10 −4 and a maximum number of iterations N iter = 10 3 . Numerical simulations are run setting the parameters according to [42], see Table I.
The non-equilibrium CB minimum (E C ), VB maximum (E V ) and intrinsic band (E I ) levels for ITO and In 2 O 3 used in the simulations are listed in Table II, where E I = 0 eV is set for ITO as a reference.
In the case of NCs consisting of a single material only (ITO NCs), energy levels, number and concentration of donors, relative dielectric permittivity and high-frequency dielectric permittivity are spatially constant. For a combination of different materials (ITO-In 2 O 3 core-shell NCs), we modeled the elemental composition of the materials by introducing a radial-dependent permittivity profile (ε ∞ (r)), non-equilibrium bandgap energy , and a radial-dependent donor distribution profile (N D = N D (r)) to take into account the materialdependence of the NCs. In our case, the diffusion effects on the ITO-In 2 O 3 core-shell NCs are investigated by introducing two different donor concentration levels for ITO and In 2 O 3 , N D,core and N D,shell , respectively. Therefore, we considered the donor distribution profile as follows: where N D,shell = 10 21 m −3 , N D,core = 10 26 m −3 are taken from experimental data in [36], and R core = 5.5 nm is the radius of the core. Different shape factors, i.e., k sh = 5, 25 and 100, are considered in solving the Poisson's equation. This analysis pointed out that differences in the electron density profiles are negligible for different k sh > 1 The spatially-dependent carrier density profile, i.e., n e (r), inside a NC is first solved for different parameters, see Fig. 2. As previously reported in [36], the effect of the Fermi level pinning on the equilibrium conduction band profile is modelled by considering different fixed NC surface states (E S ) and introducing additional electronic interfaces (core-shell system). We found that n e (r) is enough to completely suppress the metallic behavior of carriers close to the NC surface (Fig. 2). According to the surface level pinning E S , the carrier density profile at the surface determines a region depleted of carriers, which effectively suppresses the metallic behaviour at the surface, thus confining the free electrons in an inner region. As a result, this depletion region effectively acts as a dielectric. When E S increases from 0 eV to 2 eV, we observed an increase in the depletion layer width, which affects a larger fraction of the NC volume. The introduction of a shell layer with different thicknesses (t s ) which acts as an additional electronic interface beyond the surface of the nanoparticle, is another parameter that allows controlling the depletion layer and the carrier density profiles. Fig. 2 shows the tuning of the carrier density profile in the ITO-In 2 O 3 core-shell NC system. Here, E S = 2 eV is set [36]. It is important to underline that, in the core-shell system under analysis, two electronic interfaces occur: ITO-In 2 O 3 and In 2 O 3 -NC surface. While in uniform NCs, i.e., t s = 0 nm, the carrier density profile is determined by the radial depletion region near the NC surface, the addition of a shell layer with increasing thickness modifies the depletion layer resulting in a double-bent carrier density profile. With increasing t s , a more pronounced bending of the carrier density distribution is observed, showing a non-trivial profile. The double bending can be explained by a redistribution of free carriers into the shell, while the donor sites N D (r) remain fixed. The carrier density in large regions of the undoped shell (In 2 O 3 ) reaches values larger than 10 26 m −3 . These effects can be further extended by introducing different materials or by combining several materials together in other core-shell NCs structures and/or multiple materials sequentially in core-multishell NC architectures [36].

C. Post-Photodoped NCs
In this subsection, the electronic structure of core-shell NCs is investigated out of the equilibrium conditions by introducing multiple free-charge carriers with photodoping [16], [43], [44], [45]. This analysis aims to determine the effect of photodoping on the carrier density distribution of the system. In this way, we go beyond the results shown in Fig. 2 while assessing the dynamic, post-synthetic variation of the carrier density profiles via light-induced charge injection (photodoping). Hence, taking into account the ITO-In 2 O 3 system, we solved Poisson's equation with the numerical methods introduced before. The effect of the photodoping is intended to be an "on-off" effect that does not depend on the frequency of the impinging source. For this reason, we simply included additional generation (G(r)) and recombination (R(r)) wavelength-independent terms into the Poisson's equation [46]: Notably, the recombination effects are neglected, i.e., R(r) = 0, since the media surrounding the NC acts as an hole scavenger suppressing carrier recombination [36]. On the other hand, the generation term G(r) = 0 extends the Poisson's equation representing the spatial distribution of the extra free carriers introduced into the system via photodoping. The generation term can be modelled by the following Gaussian distribution [46]: where N max is the peak number of photoelectrons introduced, V is the volume of the NC, σ f is the shape factor that rules the spatial distribution of the photoelectrons into the NC, whose radius is R NC . Here, the terms N max and σ f have been derived from experimental results in [36], [42]. In particular, the σ f term is numerically tuned to reduce the discrepancy with the experimental findings reported in [36]. The comparison of the carrier density profiles both pre-and post-photodoping is shown in Fig. 3. Similarly to the nonphotodoped case discussed in section II-B, when increasing t s a bending of the carrier density distribution occurs and the observed double bending becomes more pronounced upon photodoping in all samples, thus affecting the carrier density distribution. Indeed, post-photodoping, for samples calling for t s > 2.7 nm, the carrier density profile approaches a step function with two distinct density levels in a two-step profile. The maximum n e achieved in the NC shell differs from the one in the core, reaching values of around 4 · 10 26 m −3 , while the carrier density in the core remains almost constant.

III. OPTICAL SIMULATION
We assume a time-harmonic plane wave excitation, linearly polarized along the x-direction while propagating along the z-axis, with intensity of 1 Vm −1 , that impinges a penetrable object placed in a bounded 3D domain Ω. In the latter, both outside and inside regions Maxwell's equations are satisfied (Fig. 1). The dielectric properties, i.e., the space-dependent complex permittivity of the space, are defined as: where ε m = 2.27 is the relative permittivity of the surrounding medium (toluene) [42] and ε r (r) is the space-dependent complex permittivity: where ε r (r) = {ε r (r)} and ε r (r) = {−ε r (r)} are the real and the imaginary part of the complex permittivity, respectively. By considering the analytical solution of Maxwell's equations in the case of an electromagnetic field interacting with a spherical particle, it is possible to derive a set of field expressions expanded in the spherical coordinate system [23]. From the Lorenz-Mie wavelength-dependent complex scattering electric (a q ) and magnetic (b q ) coefficients, it is possible to rigorously quantify the NC electric properties and size contributions to the overall scattering behavior. The determination of a q and b q coefficients requires the solution of the boundary value problem, i.e., the continuity of all the components of the tangential field at the interface, assuming that the solutions consists of functions that vanish at infinity. The a q and b q coefficients are described by a set of functions that consists of a combination of spherical Riccati-Bessel (ψ q = xj q (x)) and Riccati-Hankel (ξ q = xh (1) q (x)) functions of the first kind: where the superscript means derivative with respect to the dummy variable x, q is the order of the Bessel function, k is the wavenumber of the impinging light and m is the relative complex refractive index, defined as: where ν is the refractive index and κ is the extinction coefficient.
To describe the energy balance between the scattering and extinction mechanisms within Ω, the far-field scattering and extinction cross-section efficiencies are typically considered. The extinction (Q ext ), scattering (Q sca ) and absorption (Q abs ) cross-sections can be expressed as a combination of the Lorenz-Mie coefficients as follows: First, we evaluated the ratio between the absorption and scattering coefficients (Q abs /Q sca ) for uniform ITO NCs with n e = 10 27 m −3 having a radius varying from 1 nm to 20 nm and considering a wavelength dependent response between 290 nm and 2000 nm. For every point of the NC, the resonant condition for the plasmonic peak can be obtained from the complex-valued dielectric function of the NC, which is given by a Drude model: where ω p = n e (r)e 2 ε 0 m * e is the bulk plasma frequency and γ(r) is a damping term that provides the free-electron description inside the NC. The frequency-radial distribution of the complex electric permittivity simulated for the considered NCs is shown in Fig. S2 (Supplementary Material), where panels a, b, e and f (c, d, g and h) refer to pre-photodoped and post-photodoped NCs, respectively. In the range of values spanning between -135 < ε r < 5 and -370 < ε r < 0 we found that by setting q = 10 in (27), Q sca << Q abs applies within our range of interest. This means that the scattering contribution to the optical response can be neglected (i.e., extinction and absorption components are equivalent). In addition, for the range of r and k values considered in this work, i.e., r · k ≤ 1, the extinction and scattering cross-sections reduces to the first term of the series (27). The latter is a sufficient condition to describe the dissipation of light by absorption and scattering, which in the case of linear regime of r · k limits to the well-known relationship for Rayleigh scattering [47]. Hence, it can be stated that scattering is negligible and the extinction can be approximated with the absorption. Moreover, it is possible to highlight that, although in the range of ε r values obtained for the NCs under investigation (see the area bounded by the red dashed box in Fig. 4) the absorption process dominates over the scattering one. Prospectively, it would be interesting to study materials calling for ε r > 0 as they present resonances where absorption is even more efficient. For this reason, it has been evaluated Q abs /Q sca by varying real and imaginary parts of the complex electric permittivity in the range [-400, 400] while setting the NC radius and the wavelength of the incident light r = 10 nm and λ = 290 nm (λ = 1 k ), see Fig. 4. From the latter analysis, it is straightforward to observe that by varying ε r , the absorption-to-scattering ratio is dominated by the absorption, even in the limit case for which r λ = 0.03.

A. Power Balance Model
In this subsection, we evaluated the near-and far-field responses of the NCs under analysis with the purpose of correlating their carrier concentration profiles and their optical properties. Indeed, owing to the proportionality between ω p and n e , the plasmonic resonances represent a source of information for the doping processes occurring in NCs [48], [49], [50]. In fact, at the interface between the NCs and the medium, the real part of the dielectric function changes sign across the interface, thus showing coherent delocalized electron oscillations (i.e., surface plasmons). In NCs of size comparable to or smaller than the wavelength of light used to excite the plasmon, a localized surface plasmon (LSP) occurs, as a result of the confinement of a surface plasmon. In resonance conditions, the coupling of the incident radiation with LSP leads to a loss of energy when Fig. 5. Simulated absorption spectra (a and b) and electric field distribution (c-j) of the NC under analysis. Panels a and b refer to pre-photodoped (black line) and post-photodoped (red line) ITO NC consisting of the core only (t s = 0 nm) and to pre-photodoped (black line) and post-photodoped (red line) ITO-In 2 O 3 core-shell NC with t s = 4.25 nm. Panels from c and d refer to the electric field distribution (in Vm −1 ) at 4900 cm −1 for the only core NC and the core-shell NC, respectively. Panels e, g and i refer to the electric field patterns of the ITO NC at ω = 3700 cm −1 , 5300 cm −1 and 5700 cm −1 , respectively, while panels f, h and j refer to the electric field patterns of the ITO-In 2 O 3 (t s = 4.25 nm) core-shell NC at ω = 3700 cm −1 , 5300 cm −1 and 5700 cm −1 .
transmitted to the plasmons, resulting in a maximum absorption. Frequencies of light below the resonance are reflected by a material because the electrons in the material screen the electric field of the light. This principle can be used for shielding purposes. Hence, we coupled the radial carrier distribution already obtained by solving numerically the non-linear Poisson's equation with the high-frequency formulation of Maxwell's equation relating the carrier density of the NCs to the plasmonic resonance and simulating the absorption spectra. The frequency-domain equations are given by the lossy Helmholtz's wave equations: where σ is the electric conductivity and μ is the magnetic permeability. The power associated to the electromagnetic wave traveling into (or radiated from) the NC (inside the Ω domain) can be expressed as a flux integral of the inward-directed (or outward-directed) normal component of the Poynting vector over the surface ∂Ω [51], [52]. In addition, similar expressions can also be derived for scattered and extinguished power by integrating the associated components of the Poynting vector. First, the time average Poynting vector can be decomposed into its total time-harmonic field (E, H) components [51], [52]: where S inc , S sca and S ext represent incident, scattered and extinguished components, respectively, where Please note that, here, the incident fields (E inc , H inc ) are the fields generated by the sources in the absence of the scatterer. Then, it is possible to derive the relations needed to evaluate the absorbed, extinguished, scattered and radiated powers. The absorbed power is defined as the power traveling into the NC and is given by the integral of the inward-directed normal component of the Poynting vector. By applying Poynting's theorem and the divergence theorem, the absorbed power in terms of volumetric quantities is found as follows: wheren is the outward-directed surface normal. The power associated to the extinction process is the total power removed from the incident field (i.e., the sum of the absorbed and the scattered powers) due to the presence of the NC. As for the extinction power, it can be obtained as: Finally, the power scattered from the NC is given by the integral of the outward-directed normal component of S sca over ∂Ω: Hence, according to (34)- (36), it is straightforward to express the scattered power as the difference between the extinction power and the absorbed one:

B. Numerical Simulations
In this subsection, we numerically simulated the NCs optical response by using the Electromagnetic Waves, Frequency Domain module of COMSOL Multiphysics software. The electric field vector was discretized with quadratic Lagrange elements and a frequency domain direct Newton solver (τ d = 1, N iter = 10 3 ) was used. Free triangular mesh elements for the NC and the surrounding medium were used, while a swept mesh was used for the perfectly matched layer. We focused on ω ∈ [500, 10000] cm −1 , with a 100 cm −1 step. Numerical simulations are performed on two showcase NCs calling for t s = 0 nm (only core) and t s = 4.25 nm (core-shell structure).
The real and the imaginary part of ε(ω, r) obtained from the Drude model by varying ω and the point in the NC are shown in Fig. S2a-d and S2e-h for only core and core-shell NCs, respectively (Supplementary Material). First, we simulated the far-field optical spectra of the NCs from n e (r) by varying t s and photodoping conditions, see Figs. 5 a,b and S3 (Supplementary Material). Fig. 5 a shows the corresponding absorption of a single NC calculated using the effective dielectric function of the t s = 0 nm NC, with a peak at 4900 cm −1 (2041 nm, in the NIR wavelength region). In the pre-photodoping case (black plot), the spectrum is ruled by strong resonance in the nearinfrared region which is likely due to localized surface plasmon resonances resulting from free electrons in the highly doped semiconductor (10 27 m −3 , Fig. 3). After the introduction of charge carriers (post-photodoping, red plot), a stronger localized surface plasmon resonance peak is observed. In the case of the pre-photodoped t s = 4.25 nm NC depicted in Fig. 5 b (black plot) and Fig. S3a (Supplementary Material), we observe a shift of the localized surface plasmon resonance upon shell growth, with a peak at 5300 cm −1 (1887 nm -NIR), which is due to the presence of the shell that modifies the dielectric properties of the space surrounding the NC. After photodoping (red plot), two peaks appear in the spectrum (3700 and 5700 cm −1 corresponding to 2703 and 1754 nm), indicating a more complex carrier density profile within the core-shell NC which induces an independent resonant mode, generated by a sufficiently high carrier density in the NC shell. These results show that shielding properties can be controlled by changing shell thickness and/or photodoping conditions (Fig. S3, Supplementary Material).
We also simulated the near-field optical properties inside and surrounding the NCs, see We depicted the near-field intensity pattern around the NCs for eight different cases: the electric field evaluated for a NC calling for t s = 0 nm pre-and post-photodoping at the resonance frequency (4900 cm −1 ) as well as the electric field evaluated for a NC calling for t s = 4.25 nm pre-and post-photodoping at 3700 cm −1 , 5300 cm −1 and 5700 cm −1 , all excited at the maxima localized surface plasmon resonances.
Numerical simulations suggested that the maximum intensity of the electric field is lower for thicker shells and that an increase in the depletion layer leads to a decrease in the maximum near-field intensity when exciting at peak frequencies. It is important to point out that the increased electric field is largely confined within the upper dielectric shell. This loss in near-field enhancement is the result of the stronger contribution of the high refractive index shell. However, near-field excitation at longer wavelengths than the localized surface plasmon resonances peak allows the field confinement to extend to larger sizes and into the surrounding dielectric medium. Although in this case the increase of the maximum field is slightly lower, it provides a solution to overcome the confinement of the field within the NC volume. These effects are further enhanced post-photodoping the NCs, also showing that in the case of a core-shell NC, a splitting of the plasmon resonance occurs due to the presence of two distinct levels of n e , as shown in Fig. 3, which lead to two distinct frequencies of plasma oscillation.

IV. CONCLUSION
In this study, we investigated the optical response of ITO-In 2 O 3 core-shell nanocrystals in static and dynamic (i.e., photodoping) conditions and for different core-shell architectures, illustrating the evolution of the dielectric function and the absorption peaks in the near-infrared region. Metal oxide nanocrystals are promising candidates to be used for multispectral electromagnetic shielding purposes due to their ability to block infrared radiation together with ultraviolet photons, combined with high transparency in the visible range. The experimental results obtained by means of numerical simulations show that the absorption spectrum of doped metal oxide nanoparticles can be engineered both during and after synthesis to better match the targeted electromagnetic interference.
We foresee the development of thin film fabricated by spincoating ITO-In 2 O 3 core-shell nanocrystals with thick shells (i.e., t s > 3 nm) and subsequently photodoped with ultraviolet treatments, as effective shielding coating for transparent screens for vehicles, aircraft and for the next-generation electronic devices. Moreover, similar colloidally-stable doped metal oxide nanocrystals, adapted to be environmentally friendly and skin-compatible, could be employed as photoprotective topical products (e.g., sunscreens) to shield ultraviolet radiation via absorption, preventing sunburns and several kinds of skin cancer.

IV. SUPPLEMENTARY DATA
All the data generated in this study has been deposited in the Zenodo database under accession code https://doi.org/10.5281/ zenodo.7193366. Giuseppe Mazzarella (Senior Member, IEEE) received the Graduation degree (Summa with Laude) in electronic engineering and the Ph.D. degree in electronic engineering and computer science from the Università Federico II of Naples, Naples, Italy, in 1984 and 1989, respectively. In 1990, he became an Assistant Professor with the Dipartimento di Ingegneria Elettronica, Università Federico II of Naples. He has been with the Dipartimento di Ingegneria Elettrica ed Elettronica, Università di Cagliari, first as an Associate Professor since 1990, and then a Full Professor, teaching courses in electromagnetics, microwave, antennas and remote sensing since 2000. His research interests include efficient design of large arrays of slots, power synthesis of array factor, with emphasis on inclusion of constraints, microwave holography techniques for the diagnosis of large reflector antennas, use of evolutionary programming for the solution of inverse problems, and in particular problems of synthesis of antennas and periodic structures. He is the author or coauthor of more than 100 papers in international journals. He is a Reviewer of many EM journals.
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