Abstract
The presence of electromagnetic waves on two-dimensional interfaces has been extensively studied over the last several decades. Surface plasmonic polariton (SPP), which normally exists at the interface between a noble metal and a dielectric, is treated as the most widely investigated surface wave. SPPs have promoted new applications in many fields such as microelectronics, photovoltaics, etc. Recently, it has been shown that by nanostructuring the metal surface, it is possible to modify the dispersion of SPPs in a prescribed manner. Herein, we demonstrate the existence of a new kind of surface wave between two anisotropic meta-materials. In contrast to extensively studied surface waves such as SPPs and Dyakonov waves, 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. Consequently, the proposed structure will interact with the material via leaky waves.
Keywords
- metamaterials
- semiconductor
- surface plasmon polaritons
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
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
It is interesting to notice that in the case of the MM interface, the obtained result for the dispersion is as follows:
where
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 (
The curve colors correspond to different frequency regions. For
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
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
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 ds = 0.35 nm. PbS with relative permittivity εd = 18.8 and slab thickness td = 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
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
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
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 × 1025 m−3, N = 3 × 1025 m−3, N = 4 × 1025 m−3, N = 5 × 1025 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 × 1025 m−3 possesses the longer propagation length than the mode corresponding to the case N = 5 × 1025 m−3.
The existence of the boundary modes associated with the second case under consideration, i.e.
In contrast to the previous case, we now discuss the instance denoted as
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.
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