# Controlling Surface Plasmon Polaritons Propagating at the Boundary of Low-Dimensional Acoustic Metamaterials

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Approach

_{n}. Semiconductor nanoparticles with permittivity ε

_{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].

## 3. Results and Discussions

_{||0}the semiconductor-dielectric metamaterial possesses hyperbolic properties. It can be seen from Figure 2, that the propagation of DSW is possible in the case of ${\epsilon}_{n}=2.25$, ${\epsilon}_{d}=11.8$. It is worthwhile to note that the regime of DSW propagation is possible in the case of ${\epsilon}_{\left|\right|},\text{\hspace{0.17em}}{\epsilon}_{nc}>0$. Moreover, it is possible to increase the frequency range of DSW existence by changing the nature of inclusions, i.e., by replacing AZO with ITO.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Halperin, W.P. Quantum size effects in metal particles. Rev. Mod. Phys.
**1986**, 58, 533–606. [Google Scholar] [CrossRef] - Naik, G.V.; Shalaev, V.M.; Boltasseva, A. Alternative plasmonic materials: Beyond gold and silver. Adv. Mater.
**2013**, 25, 3264–3294. [Google Scholar] [CrossRef] [PubMed] - Feigenbaum, E.; Diest, K.; Atwater, H.A. Unity-Order Index Change in Transparent Conducting Oxides at Visible Frequencies. Nano Lett.
**2010**, 10, 2111–2116. [Google Scholar] [CrossRef] - Sorger, V.J.; Lanzillotti-Kimura, N.D.; Ma, R.-M.; Zhang, X. Ultracompacy silicon nanophotonic modulator with broadband response. Nanophotonics
**2012**, 1, 17. [Google Scholar] [CrossRef][Green Version] - Cai, W.; White, J.S.; Brongersma, M.L. Compact, High-Speed and Power-Efficient Electrooptic Plasmonic Modulators. Nano Lett.
**2009**, 9, 4403–4411. [Google Scholar] [CrossRef] [PubMed] - Das, S.; Salandrino, A.; Wu, J.Z.; Hui, R. Near-infrared electro-optic modulator based on plasmonic graphene. Opt. Lett.
**2015**, 40, 1516–1519. [Google Scholar] [CrossRef] - Ye, C.; Khan, S.; Li, Z.R.; Simsek, E.; Sorger, V.J. λ-size ITO and graphene-based electro-optic modulators on SOI. IEEE J. Sel. Top. Quantum Electron.
**2014**, 20, 40. [Google Scholar] - Vasudev, A.P.; Kang, J.-H.; Park, J.; Liu, X.; Brongersma, M.L. Electro-optical modulation of a silicon waveguide with an “epsilon-near-zero” material. Opt. Express
**2013**, 21, 26387. [Google Scholar] [CrossRef] - Das, S.; Fardad, S.; Kim, I.; Rho, J.; Hui, R.; Salandrino, A. Nanophotonic modal dichroism: Mode-multiplexed modulators. Opt. Lett.
**2016**, 41, 4394–4397. [Google Scholar] [CrossRef] - Barnes, W.L.; Dereux, A.; Ebbesen, T.W. Surface plasmon subwavelength optics. Nature
**2003**, 424, 824–830. [Google Scholar] [CrossRef] [PubMed] - Zhang, J.; Zhang, L.; Xu, W. Surface plasmon polaritons: Physics and applications. J. Phys. D Appl. Phys.
**2012**, 45, 113001. [Google Scholar] [CrossRef] - Peragut, F.; Cerruti, L.; Baranov, A.; Hugonin, J.P.; Taliercio, T.; De Wilde, Y.; Greffet, J.J. Hyperbolic metamaterials and surface plasmon polaritons. Optica
**2017**, 4, 1409–1415. [Google Scholar] [CrossRef] - Zhukovsky, S.V.; Andryieuski, A.; Sipe, J.E.; Lavrinenko, A.V. From surface to volume plasmons in hyperbolic metamaterials: General existence conditions for bulk high-k waves in metal-dielectric and graphene-dielectric multilayers. Phys. Rev. B
**2014**, 90, 155429. [Google Scholar] [CrossRef][Green Version] - Mahmoodi, M.; Tavassoli, S.H.; Takayama, O.; Sukham, J.; Malureanu, R.; Lavrinenko, A.V. Existence Conditions of High-k Modes in Finite Hyperbolic Metamaterials. Laser Photon Rev.
**2019**, 13, 1800253. [Google Scholar] [CrossRef][Green Version] - Takayama, O.; Lavrinenko, A. Optics with hyperbolic materials. J. Opt. Soc. Am. B
**2019**, 36, F38–F48. [Google Scholar] [CrossRef][Green Version] - Ferrari, L.; Lu, D.; Lepage, D.; Liu, Z. Enhanced spontaneous emission inside hyperbolic metamaterials. Opt. Express
**2014**, 22, 4301–4306. [Google Scholar] [CrossRef] - Hoffman, A.; Alekseyev, L.; Howard, S.; Franz, K.; Wasserman, D.; Podolskiy, V.; Narimanov, E.; Sivco, D.; Gmachl, C. Negative refraction in semiconductor metamaterials. Nat. Mater.
**2007**, 6, 946–950. [Google Scholar] [CrossRef] - Feng, J.; Chen, Y.; Blair, J.; Kurt, H.; Hao, R.; Citrin, D.S.; Summers, C.J.; Zhou, Z. Fabrication of annular photonic crystals by atomic layer deposition and sacrificial etching. J. Vac. Sci. Technol. B Microelectron. Nanometer Struct.
**2009**, 27, 568. [Google Scholar] [CrossRef] - Shekhar, P.; Atkinson, J.; Jacob, Z. Hyperbolic metamaterials: Fundamentals and applications. Nano Converg.
**2014**, 1, 1–17. [Google Scholar] [CrossRef] [PubMed][Green Version] - Starko-Bowes, R.; Atkinson, J.; Newman, W.; Hu, H.; Kallos, T.; Palikaras, G.; Fedosejevs, R.; Pramanik, S.; Jacob, Z. Optical characterization of epsilon-near-zero, epsilon-near-pole, and hyperbolic response in nanowire metamaterials. J. Opt. Soc. Am. B
**2015**, 32, 2074–2080. [Google Scholar] [CrossRef][Green Version] - Gric, T.; Hess, O. Surface plasmon polaritons at the interface of two nanowire metamaterials. J. Opt.
**2017**, 19, 085101. [Google Scholar] [CrossRef] - Iorsh, I.; Orlov, A.; Belov, P.; Kivshar, Y. Interface modes in nanostructured metal-dielectric metamaterials. Appl. Phys. Lett.
**2011**, 99, 151914. [Google Scholar] [CrossRef][Green Version] - Maier, S. Surface Plasmon Polaritons at Metal/Insulator Interfaces. In Plasmonics: Fundamentals and Applications; Springer: New York, NY, USA, 2007. [Google Scholar]
- Gric, T.; Gorodetsky, A.; Trofimov, A.; Rafailov, E.U. Tunable Plasmonic Properties and Absorption Enhancement in Terahertz Photoconductive Antenna Based on Optimized Plasmonic Nanostructures. J. Infrared Millim. Terahertz Waves
**2018**, 39, 1028–1038. [Google Scholar] [CrossRef]

**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**).

**Figure 2.**Relative permittivity components of the nanocomposite and hypercrystal versus frequency. Herein, f = 0.3. (

**a**,

**b**) ${\epsilon}_{n}=11.8$, ${\epsilon}_{d}=2.25$; (

**c**,

**d**) ${\epsilon}_{n}=2.25$, ${\epsilon}_{d}=11.8$. Herein AZO (

**a**,

**c**) and ITO (

**b**,

**d**) inclusions are employed in nanocomposite and hypercrystal.

**Figure 3.**Solution of the wave equation for different filling ratio f: (

**a**,

**b**)—${\epsilon}_{n}=2.25$, ${\epsilon}_{d}=11.8$; (

**c**,

**d**)—${\epsilon}_{n}=11.8$, ${\epsilon}_{d}=2.25$. Herein AZO (

**a**,

**c**) and ITO (

**b**,

**d**) inclusions are employed in nanocomposite and hypercrystal.

**Figure 4.**Dependence of transmission characteristics versus frequency for different filling factors: (

**a**,

**c**) the imaginary part of β; (

**b**,

**d**) the propagation length L

_{p}. The case ${\epsilon}_{n}=2.25$, ${\epsilon}_{d}=11.8$.is presented in (

**a**,

**b**); ${\epsilon}_{n}=11.8$, ${\epsilon}_{d}=2.25$—(

**c**,

**d**). All of the presented results were obtained for the AZO inclusions.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ioannidis, T.; Gric, T.; Rafailov, E. Controlling Surface Plasmon Polaritons Propagating at the Boundary of Low-Dimensional Acoustic Metamaterials. *Appl. Sci.* **2021**, *11*, 6302.
https://doi.org/10.3390/app11146302

**AMA Style**

Ioannidis T, Gric T, Rafailov E. Controlling Surface Plasmon Polaritons Propagating at the Boundary of Low-Dimensional Acoustic Metamaterials. *Applied Sciences*. 2021; 11(14):6302.
https://doi.org/10.3390/app11146302

**Chicago/Turabian Style**

Ioannidis, Thanos, Tatjana Gric, and Edik Rafailov. 2021. "Controlling Surface Plasmon Polaritons Propagating at the Boundary of Low-Dimensional Acoustic Metamaterials" *Applied Sciences* 11, no. 14: 6302.
https://doi.org/10.3390/app11146302