## 2 Theoretical Approach  
In the frame of the present study, we investigate the plane interface separating a nanocomposite (NC) semi-infinite layer and a hypercrystal that is adjacent to it (Figure 1). It is of particular importance to mention that the surface waves propagate along \(z\) axis. It is possible to construct the system under investigation by means of molecular beam epitaxy [17], chemical vapor deposition, atomic layer deposition and sacrificial etching [18]. Nanocomposite is understood as a non-conductive transparent matrix with a permittivity \(\varepsilon_{n}\). Semiconductor nanoparticles with permittivity \(\varepsilon_{m}\) are regularly distributed in its host material. The dielectric function of the TCO based nanoparticles is of special importance to r the academic community. The topic has become one of significance due to the metal being opaque to light. The parameters of the Drude-Lorentz approach for AZO, GZO, and ITO are obtained from experimental data [2].  
We have made an assumption that the size of the inclusions suspended in a dielectric matrix is much smaller than the wavelength and the electromagnetic field penetration depth in the material. It is worthwhile to mention that an effective Maxwell Garnett model can be employed, aiming to characterize the optical parameters of the nanocomposite under consideration. The former approach is possible if the interference impacts of the inclusions are neglected. In this relation, the homogenization of the system should be performed, and the effective dielectric permittivity of the nanocomposite can be expressed as follows:  
\[\varepsilon_{nc}=\varepsilon_{n}\bigg{[}1+\frac{f}{(1-f)/3+\varepsilon_{n}/( \varepsilon_{m}-\varepsilon_{n})}\bigg{]} \tag{1}\]  
where \(f\) is the filling factor, i.e., the fraction of nanoparticles in the matrix.  
Based on the effective medium approximation, the effective permittivities of the nanowire metamaterial (hypercrystal) can be obtained according to [19]:  
\[\varepsilon_{\bot}=\varepsilon_{d}\bigg{[}\frac{\varepsilon_{m}(1+\rho)+ \varepsilon_{d}(1-\rho)}{\varepsilon_{m}(1-\rho)+\varepsilon_{d}(1+\rho)} \bigg{]} \tag{2}\]  
\[\varepsilon_{||}=\varepsilon_{m}\rho+\varepsilon_{d}(1-\rho) \tag{3}\]  
Here, \(\rho\) is the metal filling fraction ratio which is defined as:  
\[\rho=\frac{nanowire\ area}{unit\ cell\ area} \tag{4}\]  
The calculation of the metal filling fraction (\(\rho\)) taking into account the data sheet of the pore diameter (\(d\)) and spacing (\(h\)) (Figure 1b). The following equation [20,21] is applied, taking into consideration a perfect hexagonal structure:  
\[\rho=\frac{\pi d^{2}}{2\sqrt{3}h^{2}} \tag{5}\]  
One may attain a surface mode with the propagation constant [22] by evaluating the tangential components of the electric and magnetic fields at the interface.  
\[\beta=k\left(\frac{\Big{(}\varepsilon_{||}-\varepsilon_{nc}\Big{)}\varepsilon _{\bot}\varepsilon_{nc}}{\varepsilon_{\bot}\varepsilon_{||}-\varepsilon_{nc }^{2}}\right)^{1/2} \tag{6}\]  
Figure 1: Schematic design under consideration, comprising a semi-infinite hypercrystal (\(x>0\)) and a nanocomposite with semiconductor inclusions (\(x<0\)) (**a**), metamaterial (hypercrystal) unit cell (**b**).  
By substituting (1)-(3) in (6), the resulted dispersion relation is as follows:  
\[\begin{array}{l}\beta=k\left(-\frac{\epsilon_{n}ba(\epsilon_{n}a+\epsilon_{m} \rho-\epsilon_{d}(\rho-1))}{\left(\epsilon_{n}^{2}a^{2}+\frac{(\epsilon_{m}- \epsilon_{d}(\rho-1))}{\epsilon_{d}(\rho+1)-\epsilon_{m}(\rho-1)}\right)( \epsilon_{d}(\rho+1)-\epsilon_{m}(\rho-1))}\right)^{1/2}\\ a=\frac{f}{\frac{f}{f}+\frac{\epsilon_{m}}{\epsilon_{m}-1}-\frac{1}{3}}-1,b=( \rho-1)\epsilon_{d}^{2}-\epsilon_{m}\epsilon_{d}(\rho+1).\end{array} \tag{7}\]