## 1. Introduction

Plasmonics and the recent birth of metamaterials (MMs) [1, 2, 3, 4] and transformation optics [5, 6] are currently opening a gateway to the development of a family of novel devices with unprecedented functionalities ranging from sub-wavelength plasmonic waveguides and optical nanoresonators [7] to superlenses, hyperlenses [8] and light concentrators [9]. Coupling between photons and surface plasmon polaritons (SPPs) [10, 11] is enabled by the periodically nanostructured metallic films allowing for exceptional and tunable optical properties determined by a combination of design geometry, the surrounding dielectric permittivity and the choice of metal [12, 13]. SPPs have promoted new applications in many fields such as microelectronics [14], photovoltaics [15], near-field sensing [16], laser technology [17, 18], photonics [19], meta-materials design [2], high-order harmonics generation [20] or charged particle acceleration [21]. Recently, it has been shown that by nanostructuring the metal surface, it is possible to modify the dispersion of SPPs or excite the SPPs in a prescribed manner [22, 23].

The process of replacement of the uniaxial medium by a biaxial crystal [24], an indefinite medium [25] and a structurally chiral material [26] may enforce the presence of hybrid surface waves with some parallel characteristics. In the latter case, a methodology developed by Tamm [27] was adopted seeking to find a new type of surface wave, called as Dyakonov-Tamm wave, as it combines the features of Dyakonov surface waves (DSWs) and Tamm states. The use of structured materials with extreme anisotropy provided a fertile background aiming to increase the range of directions of DSWs substantially, as it is compared with the rather narrow range observed with natural birefringent materials [28]. Especially, outstanding results take place if the metallic nanoelements are employed to the anisotropic structures, as it occurs, for example, with a simple metal-dielectric multilayer, a case where the angular range may surpass half of a right angle [29]. The propagation length of these DSWs is drastically limited by the penetration depth inside the lossy MM [30] as it is caused by the specific damping capacity of metals.

The examination of two different interfaces, i.e. MM/dielectric and MM/TCO, is of the particular importance. Surface waves of different kinds, including DSWs along with traditional-like SPPs, are examined. Contrarily, the introduction of MM/TCO interface leads to a transformation of the traditional-like SPPs. As a consequence, the new types of surface waves are found.

Moreover, hyperbolic metamaterials, being special kind of anisotropic metamaterial with dielectric tenor elements having the mixed signs, have attracted growing attention due to their ability to support very large wave vectors. Their exotic features give rise to many intriguing applications, such as sub-wavelength imaging [31, 32] and hyper-lens [33, 34] that are infeasible with natural materials. In this paper, we demonstrate the existence of a new kind of surface wave between two anisotropic metamaterials. In contrast to extensively studied surface waves such as SPPs and Dyakonov waves, whose in-plane wave vector is greater than that of the bulk modes, the surface waves supported by the nanostructured semiconductor metamaterial cross the light line, and a substantial portion at lower frequencies lies above the free-space light line, which typically separates non-radiative (bound) and radiative (leaky) regions.

## 2. Transparent conducting oxide (TCO)—dielectric composite heterostructure-based multilayer metamaterial

The structure of the metamaterial (MM) is shown in **Figure 1**, where *d _{TCO}* and

*d*represent the thicknesses of TCO and dielectric layers, respectively. All the involved media are nonmagnetic, so the magnetic permeability of every medium is the same as that of vacuum. In our numerical calculation, we use a semi-infinite TCO/PbS MM as an example to explore the dispersive features of hyperbolic surface plasmon polaritons (HSPPs), where

_{d}*ε*= 18.8 for PbS layers, and

_{d}*ε*is calculated using the parameters presented in [35].

_{TCO}Consequently, an example to estimate the limitation on the structure period under the effective-medium theory (EMT) is considered. Proposing model, when the wavelength of radiation is much larger than the thickness of any layer, one can apply the effective-medium approach based on averaging the structure parameters.

First, the dispersion features of HSPPs are investigated. On the contrary to the approach presented in [36, 37, 38], the damping term in the TCO is not ignored in the process of analysing and calculating their dispersion properties. It is worth to mention that this particular MM is equivalent to a uniaxial-anisotropy effective medium, with its anisotropy axis (the optical axis) along the structure periodicity in the long-wavelength limit. Its effective permittivity tensor is written as

in the principal-axis coordinate system. The principal values of the tensor are expressed with [39, 40]

where

Electric and magnetic fields’ tangential components need to be evaluated at the interface in order to get metamaterial interface confined surface mode unique dispersion [42]:

where *k* is the absolute value of wave vector in vacuum and *β* is the component of the wave vector parallel to the interface.

It is interesting to notice that in the case of the MM interface, the obtained result for the dispersion is as follows:

where *ε _{R}* is the permittivity of the material at the right-hand side of the interface.

It is of particular interest to obtain the dispersion relation for the interface states in the effective media approach corresponding to the MM interface, having in mind that material at the right-hand side is the same as employed in the MM. This dispersion relation reads

As a matter of fact, we obtain a surprising result: the dispersion of a (single) interface mode does not depend on the thicknesses of the layers, and it coincides with the dispersion of a conventional surface plasmon at metal-dielectric interface.

### 2.1. The mode structure

In the case of a spatially infinite anisotropic material, invariant in two directions, the electromagnetic wave dispersion can be plotted for both MM/dielectric and MM/TCO cases. Thus, herein, we present analysis performed after the homogenization of the MM corresponding to the MM/dielectric and MM/TCO interfaces. Doing so, in the numerical calculations, the semi-infinite AZO/PbS MM is taken as an example. We will first review the optical properties by depicting the curves of the principal values (**Figures 2** and **3**. **Figures 2** and **3** has an effect on the dispersion curves of the HSPPs. As shown in **Figures 2** and **3**, the parameter that principally defines the tunability of the effective optical parameters of our metamaterial is the TCO-filling ratio, *f _{TCO}*. The dispersion curves are illustrated in

**Figures 4**and

**5**for various TCO-filling ratios.

The curve colors correspond to different frequency regions. For *f _{TCO}* = 0.3, there are three HSPPs belonging to three different kinds, respectively, in the case of MM/dielectric interface and two kinds of HSPPs in the case of MM/TCO interface. The upper waves exist in the gray region for MM/TCO case. Three exceptional cases related to

*f*= 0.5, 0.7 and 0.9. In the case of

_{TCO}*f*= 0.5, two kinds of HSPPs exist in two color regions, i.e. cyan and grey for MM/dielectric and MM/TCO interfaces. The upper short curves in the case of

_{TCO}*f*= 0.7 lie in the gray region, as illustrated in

_{TCO}**Figures 4**and

**5**. The other HSPPs lie in the orange region. The additional HSSPs lying in the magenta region correspond to the case of MM/dielectric interface. The extension of the magenta range for the case

*f*= 0.9, two kinds of HSPPs are found lying in the orange and cyan regions for MM/TCO case and three kinds of HSPPS lying in the cyan, orange and gray regions for MM/dielectric case.

_{TCO}**Figure 4**also demonstrates that there always is one HSPP in the cyan region for various TCO-filling ratios in MM/dielectric case. It is worthwhile mentioning that the case

**Figure 5**).

Based on the necessary condition for the existence of the DSW [39], the HSPP in the green region (**Figure 4**) is similar to the DSW so that it should be called Dyakonov-like SPP [2] or the Dyakonov defined in [39]. Moreover, the frequency range of the DSW existence can be extended by replacing the material at the right-hand side of the interface with **Figures 2** and **4**). The former extension is demonstrated by the dark-green color in **Figure 2**. It is worthwhile mentioning that extension of the gray region in **Figure 3** is possible due to the employment of ITO instead of GZO at the right-hand side of the interface (the extension of the gray range is depicted by the light-gray color). Due to the negative principal values of the effective permittivity in the orange region, the HSPP in this region is similar to the traditional SPP, so we should name it the traditional-like SPP. The others are new types of HSPPs. Thus, all the HSPPs are divided into five kinds, situated in the five color regions in **Figures 4** and **5**, respectively. In the case of AZO/PbS MM and air/dielectric interface, five types of the HSPPs exist, and only three are seen in the case of AZO/PbS MM and TCO interface.

As expected, the traditional-like SPP has a typical dispersion in the case of the MM/dielectric interface, lying to the right of the light line. At the same time, the degradation of the dispersion in the orange region takes place in the case of the MM/TCO interface. The dispersion properties can be tuned with the TCO-filling ratio of the MM realization. While for low TCO-filling ratio in the case of the MM/TCO interface, two types of the modes are always present, for higher TCO-filling ratio, the disappearance of modes in the gray region takes place (**Figure 5(d)**).

The complex mode structure (**Figures 4** and **5**) corresponding to either MM/dielectric or MM/TCO interface emerges as a consequence of the confinement of plasmon polaritons in the direction perpendicular to the wave propagation. These electromagnetic surface waves arise via the coupling of the electromagnetic fields to oscillations of the conductor’s electron plasma.

The imaginary parts of the wave vector (i.e. absorption) are plotted in **Figures 6** and **7**. It should be mentioned that negative values of the absorption in **Figures 6** and **7** result from non-physical solutions of the dispersion equation and have been omitted in line with [43]. Taking advantage of the absorption resonances, one can show that the simple multilayer structures without possessing any periodic corrugation have the prospective to act as directive and monochromatic thermal sources [44].

## 3. Nanostructured semiconductor metamaterial

Another interesting MM structure depicted in **Figure 8** is periodic stack of semiconductor-dielectric layers called hyperbolic metamaterial heterostructure.

The effective permittivity of the semiconductor (Si) can be calculated as follows:

where *ε*_{∞} is the background permittivity and *ω _{p}* is the plasma frequency. The effective-medium approach [45] which is justified if the wavelength of the radiation considered is much larger than the thickness of any layer is applied aiming to describe the optical response of such a system. The dispersion relation for the surface modes localized at the boundary separating two anisotropic media [42] is found by applying the appropriate boundary conditions, i.e. matching the tangential components of the electrical and magnetic fields at the interface. It is interesting to note that heavily doped silicon (

*n*> 2.2 × 10

^{19}cm

^{−3}) has been shown to exhibit metallic properties at terahertz frequencies [46, 47] and has the potential to replace metals in such applications [48]. The case of a heavy-doped Si is considered, assuming that the doping level is

*N*= 5 × 10

^{19}cm

^{−3}[49].

It should be mentioned that dramatic control of the frequency range of the surface wave existence is mostly concerned with the modifications of the permittivities and thicknesses of the layers [50] employed in the HMMs. To further study the surface waves, the tangential components of the electric and magnetic fields at the interface should be evaluated, and a single surface mode with the propagation constant should be obtained aiming to get the unique dispersion relation for the surface modes confined at the interface between two metamaterials [42].

Using the (4) formula, we can describe the case

The dispersion for the case of

where

In the case of

The permittivity spectra for the perpendicular components of the considered multilayer heterostructure are demonstrated in **Figures 2(c)**, **3(a)**–**(c)** and **9(a)**. Tuning the doping level of the semiconductor may open a gateway to the frequency control of the hyperbolic dispersion curve as shown in **Figures 3(a)**, **4(a)**, **5(a)** and **9(a)**. It is assumed that d_{s} = 0.35 nm. PbS with relative permittivity ε_{d} = 18.8 and slab thickness t_{d} = 10 nm is chosen as the dielectric layer. It is clear that one has the potential to achieve the resonant behaviour of

Other than the doping level, the resonant behaviour of **Figures 9(b)** and **3(b)**. From **Figure 9(b)**, the shift of the resonant frequency of _{d} is increased can be clearly distinguished.

### 3.1. The mode structure

Guided by the homogenization of two HMs, the computed dispersion curves are demonstrated. Thus, in **Figure 13**, the dispersion curves for the case _{1} = 0.35 nm, d_{2} = 10 nm, d_{3} = 0.25 nm and d_{4} = 10.1 nm. Furthermore, due to the great interest in this case, **Figure 13** refers to the dispersion of surface waves with the calculated effective parameters shown in **Figures 9(a)**, **10(a)**, **11(a)** and **12(a)**; the blue line is the free-space light line.

The frequency ranges of surface wave can be tuned by changing the doping level of silicon. As it is shown in **Figure 9(a)**, doping level is correlated to the permittivity orthogonal component **Figure 13**. Furthermore, this correlation can be used to engineer the metamaterial surface wave just by controlling silicon sheet doping level.

As the silicon is not modeled as lossless, β is complex, leading to a finite propagation length (Eq. (2)), drawn in **Figure 13(b)**. In **Figure 13** the four modes (N = 2 × 10^{25} m^{−3}, N = 3 × 10^{25} m^{−3}, N = 4 × 10^{25} m^{−3}, N = 5 × 10^{25} m^{−3}) always lie to the right side of the light line and remain non-radiative (bound) SP modes throughout the certain frequency range. All the considered cases are of particular interest due to the fact that their dispersion relations cross the light line and a significant portion at lower frequencies lies above the free-space light line, which usually splits up non-radiative (bound) and radiative (leaky) regions. For the bound modes, longer propagation lengths take place at lower frequencies owning the dispersion that is close to linear. Mode corresponding to the case N = 4 × 10^{25} m^{−3} possesses the longer propagation length than the mode corresponding to the case N = 5 × 10^{25} m^{−3}.

The existence of the boundary modes associated with the second case under consideration, i.e. **Figure 14** tackles this problem by displaying four modes at the boundary of two different metamaterials with ε_{4} = 2.25. As seen from **Figure 14**, the smallest asymptotic frequency corresponds to the case N = 2 × 10^{25} m^{−3}.

In contrast to the previous case, we now discuss the instance denoted as **Figure 15** shows the dispersion curves of four different modes. The assessment and control of variation of the dielectric and semiconductor sheets’ fill factors is of critical importance (**Figure 15**). First, the impact of the thickness of the dielectric d_{d} on the dispersion curve (see **Figure 15(a)**) is considered. It is found that the upper limit moves to the higher frequencies as d_{d} is increased. The former is consistent with the effect of d_{d} on the frequency range of **Figures 14(a)** and **15(a)**, it is interesting to observe the Ferrell-Berreman modes which exist at energies near the ENZ of the hyperbolic metamaterial to the left of the light line [51, 52, 53].

## 4. Conclusion

During a study of the HSPPs in a one-dimensional TCO-dielectric MM, we can see that similar to graphene-dielectric MM [45], TCO-dielectric MM supports traditional-like SPPs having different patterns corresponding to two different interfaces. The dispersion equations of HSPPs are obtained based on the theoretical approach [42, 45]. Five kinds of HSPP, among which three kinds are new types of HSPPs and one is the Dyakonov-like SPP and another is the traditional-like SPP have been predicted. The existence of these HSPPs is dramatically influenced by the properties of and the relation among the principal values of the effective permittivity and the dielectric constant of the covering medium. It is worthwhile mentioning that the new types of the HSPPs arise because the principal values of the effective permittivity used in this chapter are functions of frequency and can be negative or positive. Moreover, it was demonstrated that used approach allows to predict surface mode with the dispersion that coincides with the dispersion of a surface plasmon at the boundary of two isotropic media corresponding to the MM interface if the material at the right-hand side is the same as employed in the MM.

Moreover, a new kind of surface wave between two nanostructured semiconductor metamaterials was demonstrated. It is shown that the dispersion diagrams are sensitively dependent on the semiconductor parameters. These findings open the gateway towards potential applications in both classical and quantum optical signal communication and processing.