## 3 Results  
The propagation of SPPs at the boundary of nanocomposite and hypercrystal is investigated. The absolute value of complex conductivity \(|\sigma|\) varies from 0 to 6 \(\times\) 10\({}^{7}\) S/m. Herein, the permittivity components of a hypercrystal and nanocomposite versus frequency are studied numerically aiming to identify the frequency ranges of Dyakonov surface waves (DSWs) and SPP waves existence (Figure 2). In the frequency ranges below the frequency \(\omega\)\(|\)\(|\)\(|\) [25] the semiconductor-dielectric metamaterial possesses hyperbolic properties. It is worthwhile noting that in this frequency range the presence of conventional surface plasmon polaritons waves with propagation parallel to the optical axis is feasible under specific conditions. One may conclude from Figure 2 that propagation of DSW is possible in case of \(\varepsilon_{n}=2.25\), \(\varepsilon_{d}=11.8\). It is worthwhile noting that the regime of DSW propagation takes place if \(\varepsilon_{||}(\omega),\ \varepsilon_{nc}(\omega)>0\). To have a deeper insight into the problem, we investigated permittivity components versus conductivity. Doing so, in Figure 3 permittivity function is plotted if \(\omega\) = 0.3 \(\times\) 10\({}^{14}\) Hz (Figure 3a) and \(\omega\) = 3 \(\times\) 10\({}^{14}\) Hz (Figure 3b). The former allows us to investigate conductivity dependent permittivity functions for both regimes, i.e., hyperbolic and conventional. Moreover, we studied the phenomenon of conductivity dependent functions for the DSW regime (Figure 4a). Comparing Figures 3a and 4a, one may conclude that \(\varepsilon_{nc}(\sigma)>\varepsilon_{\perp}(\sigma)\) in case of the hyperbolic regime and \(\varepsilon_{nc}(\sigma)<\varepsilon_{\perp}(\sigma)\) for DSW waves. Moreover, it is interesting to compare the conditions that are valid in case of hyperbolic and DSW regimes for both, i.e., frequency and conductivity dependent functions. Thus, it is seen in Figure 2a that hyperbolic properties of metamaterial are possible if \(\varepsilon_{nc}(\omega)\), \(\varepsilon_{\perp}(\omega)>0\) and \(\varepsilon_{||}(\omega)<0\). Dealing with the conductivity dependent functions, one may conclude that the same conditions are needed in other to obtain hyperbolic regime. On the contrary to the described case conditions for existence of DSW regime are different in two different planes, i.e., DSW is obtainable if \(\varepsilon_{nc}(\omega)\), \(\varepsilon_{\perp}(\omega)\), \(\varepsilon_{||}(\omega)>0\) and if \(\varepsilon_{nc}(\sigma)\), \(\varepsilon_{\perp}(\sigma)>0\), \(\varepsilon_{||}(\sigma)<0\). Figure 2: Relative permittivity components of the nanocomposite and hypercrystal versus frequency. Herein, \(f=0.3\). (**a**), \(\varepsilon_{n}=11.8\), \(\varepsilon_{d}=2.25\); (**b**) \(\varepsilon_{n}=2.25\), \(\varepsilon_{d}=11.8\). Herein ITO inclusions are employed in nanocomposite and hypercrystal. Figure 3: Relative permittivity components of the nanocomposite and hypercrystal versus conductivity. Herein, \(f=0.3\), \(\varepsilon_{n}=11.8\), \(\varepsilon_{d}=2.25\). Herein ITO inclusions are employed in nanocomposite and hypercrystal. (**a**) \(\omega=0.3\times 10^{14}\) Hz; (**b**) \(\omega=3\times 10^{14}\) Hz. In Figures 5-10 the plots are obtained for real and imaginary parts of the dispersion relation along with the transmittance characteristics. The real part of propagation constant is related to group velocity and the imaginary part is related to damping of SPPs. If \(\text{Re}(\beta)>k_{0}\) then SPPs propagate at the interface and if \(\text{Re}(\beta)<k_{0}\) then SPPs cannot propagate at the interface of two media. \(k_{0}=2\pi\omega/c\) is the wave vector of the electromagnetic wave in free space. Here \(\omega_{1}=0.3\times 10^{14}\) Hz, \(\omega_{2}=0.3\times 10^{14}\) Hz and \(c=3\times 10^{8}\) m/s, which gives \(k_{01}=6.28\times 10^{5}\) 1/m and \(k_{02}=6.28\times 10^{5}\) 1/m. The value of \(\text{Re}(\beta)\) varies from 0 to \(15\times 10^{6}\) 1/m versus conductivity.