Local field enhancement factor of spheroidal core–shell nanocomposites with passive and active dielectric cores

We studied the effects of depolarization factor (L), metal fraction (p), and dielectric function of host matrix (ε h ) on the local field enhancement factor (LFEF) of spheroidal core–shell nanocomposites (NCs) with passive and active dielectric cores. Solving Laplace’s equations in the quasi-static limit, we obtained expressions of electric potentials for spheroidal core–shell NCs. Then, by introducing L and the Drude-Sommerfeld model into these expressions, we derived the equation of LFEF in the core of spheroidal core–shell NCs. The results show that whether L, p, and/or ε h vary or kept constant, LFEF of the spheroidal core–shell NCs possesses two sets of peaks with passive dielectric core, whereas only a set of peak is observed with active dielectric core. In NCs with passive dielectric core, an increase in any of these parameters resulted in a more pronounced LFEF peaks in the first set of resonances. With both passive and active dielectric cores, increasing L increases the peaks of LFEF of spheroidal core–shell NCs, whereas increasing p shows decreasing tendency on the peaks of LFEF of the same material with active dielectric core. Moreover, the highest peak of LFEF is obtained by increasing L than p or ε h indicating that change in the geometry of spheroidal core–shell NCs has the highest effect on the LFEF than the metal concentration and host dielectric function. With the same increase in ε h , intensities of LFEF of the spheroidal core–shell NCs decrease when the dielectric core is passive and increase when the dielectric core is active. Briefly, the number and intensities of peaks of LFEF of spheroidal core–shell NCs vary greatly when its core is made either passive or active dielectric. Furthermore, by changing parameters like L, p, and ε h , adjustable LFEF could be obtained and used for applications in optical sensing, nonlinear optics, and quantum optics.


Introduction
Recently, the study of nanoparticles have attracted the attention of many scientists worldwide. When light interacts with such nanoparticles, their optical properties are considerably different from their corresponding bulk structures [1]. The optical responses of metal coated core-shell nanoparticles are governed by oscillations of the surface electrons in the electrical potential made by the positively charged ionic core [2]. It has been shown that metals have fast and strong nonlinear response [3,4], and if combined with dielectrics [5,6], they are good candidates for various applications in different fields of science and technology [7,8] such as nonlinear optics [9]. Furthermore, many innovative concepts and applications of such nanocomposite materials have been developed over the past few years [10][11][12][13]. To control light by light in the systems, combining metals with dielectrics has two main purposes [14]. One is allowing light to enter more deeply into metals, and this can be achieved by canceling reflections at interference. The other is achieving light localization which in turn leads to an enhanced nonlinear response [15][16][17].
During the last three decades, the optical properties of nanoparticles have been widely studied in various ways. Among these, core-shell NCs are found to have vast applications in different fields of science and technology [18]. It is also found that the optical properties of dielectric core-metallic shell NCs are strongly dependent on the size, metal fraction, spatial distribution of the core-shell structure, and the embedding medium [19][20][21][22]. Other research evidences further demonstrated that although size and embedding medium are important parameters, the surface plasmon resonances of the core-shell nanoparticles depend much more sensitively on the particle shapes [23][24][25]. Hence, changing the shape of core-shell NCs is an alternative way to tune the resonances of surface plasmons and hence their optical properties. Due to their geometric shape providing tunability of their optical properties, spheroidal core-shell NCs have attracted significant interest [26,27]. Since such geometry supports plasmon resonances, spheroidal core-shell NCs composed of a dielectric core coated by a metallic shell is one of the most useful structure for wider tunability range from the visible to infrared regions of electromagnetic spectrum [28,29].
However, many of the studies conducted so far focused on spherical and cylindrical core-shell NCs [30,31]. In other studies, researchers investigated plasmonic properties of spheroidal core-shell nanoparticles such as extinction, absorption, scattering cross sections, and field enhancement factor as a function of parameters angle, wavelength, and distance from the center [32]. For coated spheroidal NCs, the effect of depolarization factor and the surrounding dielectric constant on the optical bistability and absorption spectra has been studied [33,34]. To the best of our knowledge, the effect of passive and active dielectric cores on local field enhancement factor (LFEF) has not been largely investigated for spheroidal core-shell NCs. Thus, in the present study, we theoretically and numerically investigated the effects of passive and active dielectric cores, metal fraction, and host matrices on the LFEF of dielectric core-metallic shell spheroidal NCs.

Theoretical bases and calculations
We considered spheroidal core-shell nanocomposite in the quasi-static approximation. To investigate its effect on LFEF, the dielectric function of the core was taken to be either passive or active. The dielectric function, ε d of the core is called passive or active depending on the response of the material to the applied electric field. This dielectric function can be written as [35]: where d e ¢ and d e are the real and imaginary parts of the dielectric function of the core material, respectively. When 0 d e = , the core's dielectric function will be passive and when 0 d e < , it will be active. In this study, both components of the dielectric functions were separately considered.
2.1. Electric potential distribution in spheroidal core-shell nanocomposite Using spheroidal core-shell NCs, it is possible to investigate spherical and cylindrical core-shells by limiting cases [36]. However, our approach in the present work is the reverse of this. By applying boundary conditions and solving the Laplace's equations for spherical core-shell NCs, the electric potential distributions in the core, shell, and host matrix were obtained. Then, by employing the depolarization factors, we obtained the corresponding equations for a system of spheroidal nanocomposites.
Consider the core-shell nanocomposite shown in figure 1, where the dielectric core has a radius r 1 and dielectric permittivity ε d . The shell is characterized by the radius r 2 and dielectric permittivity ε m (where r 1 < r 2 ). The host material has an electric permittivity ε h . Then, the electric potential distribution in the dielectric core (Φ d ), metallic shell (Φ m ), and host matrix (Φ h ) are, respectively, given by [37]: where E 0 is the applied electric field, r and θ are the spherical coordinates of the observation point (the z-axis, chosen along the vector E 0 ), A, B, C, and D are unknown coefficients to be calculated using the continuity equations of the electric potential and displacement vector at the boundaries between core-shell and shell-host matrix interfaces. At the boundary between the dielectric core and the metallic shell, the potentials must satisfy the following boundary conditions [38]: (2) and (3) into equation (5), and simplifying, we obtain:

Substituting equations
Similarly, at the boundary between the metallic shell and the host matrix, the equation for the potentials can be written as: (3) and (4) into equation (7), we found the expression which relates the coefficients B, C, and D as follows: The continuity equations for the displacement vector at the interfaces are given by By substituting equations (2)-(4) into equations (9) and (10) and simplifying, we obtain By introducing the depolarization factor L into equations of spherical shape, i.e., equations (11) and (12), the equivalent equations for spheroidal shape can respectively be given by Now, by substituting equation (6) into equation (13), the expression for B is obtained to be  If we substitute the expression for B in equation (15) into equation (8), the equation that relates C and D can be written as . Furthermore, by substituting equation (15) into equation (14), we obtain e e e -+ + + -- The ratio of equations (16) and (17) leads to e e e e e e + -- Hence, the expressions for the unknown coefficients A, B, C, and D of equations (2)-(4) can be summarized as follows: Local field enhancement factor of spheroidal core-shell nanocomposite Since we are using the quasi-static approach, the electric field E 0 (assumed to be directed along the z-axis) is uniform. Then, by using the relation E = − ∇Φ, the local field in the dielectric core E d of the spheroidal nanocomposite can be written as [40]: where A is the complex function given by equation (19). The real quantity, |A| 2 , is known as the local field enhancement factor (LFEF). If we substitute equation (24) into equation (19), and then squaring and simplifying it, we find the LFEF in the dielectric core to be:

Results and discussion
We studied the effect of passive and active dielectric cores on LFEF of spheroidal core-shell NCs by varying the depolarization factor (L), metal fraction (p), and dielectric function of the host matrix (ε h ). The observed effects of those parameters and findings of the study were discussed accordingly and presented below.

The effect of dielectric function of the core on LFEF of core-shell nanocomposite
To investigate the effect of passive and active dielectric core on the LFEF of spheroidal core-shell NCs, all other parameters such as depolarization factor, metal fraction, dielectric constants of the metallic shell, and that of the host matrix were kept constant. Then, to see the effect, the dielectric function of the core was made passive and active alternately. Under these conditions, the study shows that the LFEF of the spheroidal core-shell NCs posses two peaks with the passive dielectric core whereas only one peak is observed with active dielectric core (figures 2(a) and (b)). As the depolarization factor increases from 0.34 to 0.42 equally in both passive and active core, the first set of peaks (counted from left to right) of LFEF of the spheroidal core-shell NCs show similar patterns both in passive and active dielectric core. However, all peaks of LFEF were more pronounced in the passive dielectric core than its corresponding active core. This may indicate that passive dielectric core is preferable to the active one when higher and two sets of peaks of local filed enhancement factors are required. The nanocomposite with passive dielectric core, although the first and the second sets of the corresponding peaks of LFEF increase with increase in depolarization factor, the intensities of the first peaks are comparatively larger ( figure 2(a)). This might be attributed to the case that the core holds the applied field as the field passes from the metallic shell to the dielectric core. From this figure, it is observed that with an increase in the depolarization factor, the first and the second sets of peaks are blue shifted (towards larger z or shorter λ) and red shifted (towards smaller z or longer λ), respectively. This caused the horizontal positions between the corresponding peaks of the LFEF to become closer to each other. It is worth noting that our findings on figure 2(a) is similar to the previous study showing that LFEF possesses two peaks when the dielectric core with real dielectric function (passive) is considered in metal coated spheroidal core-shell NCs [36].
Hence, the number and the intensities of peaks of LFEF of spheroidal core-shell NCs vary greatly in the same material when its core is made passive and active dielectric.

The effect of metal fraction on LFEF of spheroidal core-shell NCs
Next, we considered the effect of metal fraction on the LFEF core-shell NCs. For the same parameters, the LFEF of spheroidal core-shell NCs vary with the metal concentration both with passive and active dielectric cores, as shown in figures 3(a) and (b). It shows that when the metal fraction of spheroidal core-shell NCs with passive dielectric core increases, the intensities of peaks of the first set of LFEF increases and are blue shifted. Similarly, the peaks of the second set also increases in intensity, however, they are slightly red shifted ( figure 3(a)). We have seen that for a constant metal fraction (p=0.99) and the same change in depolarization factor, peaks of LFEF varies in number and intensities in NCs with passive dielectric core ( figure 2(a)). On the other hand, when the metal fraction p is changed uniformly from 0.91 to 0.99 for the same depolarization factor (L = 0.40), the patterns of the LFEF observed in figure 3(a) is similar to the pattern observed in figure 2(a), but their intensities were about ten times larger in figure 2(a). Similarly, when figures 2(b) and 3(b) are compared, the patterns of the LFEF observed seem to be similar. Yet, larger intensities were noticed when the depolarization factor increases than does the metal fraction. When the depolarization factor changes, obviously the geometry of the spheroidal core-shell NCs also changes. This might indicate that the change in the geometry of the spheroidal core-shell NCs has larger effect in increasing the LFEF than the metal concentration. These results are in agreement with that reported in [44].
Furthermore, when the dielectric core becomes active (perturbed by an incident field), the peak values of the LFEF in the core-shell structure increase with increase in the depolarization factor provided that the metal fraction is kept constant ( figure 2(b)). However, when the metal fraction increases at constant depolarization factor, the peak values tend to decrease ( figure 3(b)). Hence, it is observed that the depolarization factor and the metal fraction show different effects on the LFEF of spheroidal core-shell NCs with active dielectric core. Moreover, for the system of nanocomposites considered, whether the depolarization or metal fraction varies, only a single peak is observed when the core is active dielectric.
Hence, we found that when the core of the metallic shell of the spheroidal core-shell NCs is made active, it seems that the whole system behaves like a metal nanoparticle. However, when the dielectric core is passive, two peaks of the LFEF were observed for spheroidal core-shell NCs.
3.3. The effect of the host matrix on LFEF of spheroidal core-shell NCs Finally, we investigated the effect of the dielectric function of the host matrix on the LFEF of the spheroidal coreshell NCs with passive and active dielectric cores. For, the NCs with passive dielectric core, it is observed that the intensities of the LFEF decrease with increase in the dielectric function of the host matrix ( figure 4(a)).
Moreover, for the same increase in the dielectric function of the host material, the first set of peaks (counted from left to right) of LFEF were red shifted, while the second peaks show no shifts ( figure 4(a)). When the dielectric function of the host matrix increases while L and p are kept constant, the peaks of the LFEF are red shifted ( figure 4(b)). The result also show that the effect of dielectric function of the host material on the LFEF is different in passive and active dielectric core of the spheroidal core-shell NCs. That is, for the same increase in the dielectric function of the host matrix, the intensity of LFEF, respectively, decreases and increases when the dielectric core is made to be passive and active (figures 4(a) and (b)). Comparing figures 3(a) with 4(a), and 3(b) with 4(b), it is seen that the metal fraction and dielectric function of the host matrix have reverse effects on the LFEF of spheroidal core-shell NCs. Generally, the results agree with other research findings in that varying the

Conclusions
We investigated the effect of passive and active dielectric cores on the LFEF of spheroidal core-shell NCs by varying L, p, and ε h . The results show that whether L, p, or ε h vary or kept constant, the LFEF of the spheroidal core-shell NCs possesses two sets of peaks in NCs with passive dielectric core whereas only one set of peak is observed in the NCs with active dielectric core. In addition, the two peaks observed in spheroidal core-shell with passive dielectric core, the first set of peaks are more pronounced than the second, indicating that passive dielectric core is preferable to the active one when higher and two sets of peaks of LFEF are required. We also found that as L increases, the peaks of LFEF are increased and blue shifted in both passive and active dielectric cores. Moreover, when p in the passive dielectric core increases, the intensities of the peaks of the first set of LFEF increases and are blue shifted. Similarly, the second set of peaks also increases in intensity, however, they are slightly red shifted. Yet, larger intensities were noticed when L increases than does p, indicating that change in the geometry of spheroidal core-shell NCs has larger effect in increasing the LFEF than the metal concentration.
Furthermore, with an increase in ε h , the intensity of the LFEF decreases when the dielectric core is passive and increases when the core is active. Moreover, when ε h increases while keeping L and p constant, the peaks of the LFEF are red shifted in the spheroidal NCs with both types of dielectric cores. For the same increase in ε h , the intensity of the LFEF decreases and increases when the dielectric core is passive and active, respectively. For spheroidal core-shell NCs with active dielectric core, increasing L and ε h increases the peak intensities of the LFEF, however, the peak positions are reversed. Moreover, p and ε h have reverse effects on the LFEF of the spheroidal core-shell NCs. We found that the peak values of the LFEF of the spheroidal core-shell NCs can be tuned by changing the depolarization factor, metal fraction, and dielectric function of the host medium of the NCs. Moreover, the number and the intensities of these peaks vary significantly when its dielectric core is made passive than active. Hence, by changing these parameters and types of dielectric cores, adjustable LFEF of spheroidal core-shell NCs could be obtained and used for applications in optical sensing, nonlinear optics, and quantum optics.

Acknowledgments
This work is supported financially by Addis Ababa University and Bonga University.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).