# The Study of the Surface Plasmon Polaritons at the Interface Separating Nanocomposite and Hypercrystal

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{n}, in the volume of which regularly distributed semiconductor nanoparticles with permittivity ε

_{m}. The frequency dependent dielectric function of the TCO based nanoparticles is of particular interest. An emergence of high-conducting metal being transparent has opened wide avenues recently. The issue has attracted lots of interest within the scientific community because of the metal being opaque for light. From the perspectives of the potential applications, transparent conducting metals described by high DC conductivity (σ

_{DC}) are anticipated for optoelectronic devices, ranging from solar cells to electronic paper, touch screens and displays. Though, since ${\sigma}_{DC}={n}_{e}{e}^{2}\tau /m$ (with τ being relaxation time of the electron and m—electron mass) of a metal is associated with plasmon frequency ${\omega}_{p}^{2}={n}_{e}{e}^{2}/{\epsilon}_{0}m$ through the free-electron density n

_{e}, a high-conducting metal (with a high n

_{e}) is certainly opaque for light because of its permittivity ε being typically very negatively affected by its high ω

_{p}. Conventional techniques to produce transparent conducting metals include the decrease of the n

_{e}, by utilizing transparent conducting oxides (TCOs). The parameters of the Drude–Lorentz approach for aluminum-doped zinc oxide (AZO), Ga-doped ZnO (GZO) and indium tin oxide (ITO) gained from experimental data [12] are presented in Table 1.

_{d}is the permittivity of the host material. By making a step forward towards complex nanostructures, we made an assumption that the wavelength and the electromagnetic field penetration depth in the material are much larger that the size of inclusions suspended in a dielectric matrix. It is worthwhile mentioning that effective Maxwell Garnett model can be employed aiming to characterize the optical properties of the nanocomposite under consideration. The former approach is possible, if the interference effects of the inclusions are neglected and their volume fraction is as small as 1/3. Thus, one may apply the homogenization procedure and the effective complex permittivity of the nanocomposite can be expressed as follows

_{n}is the permittivity of the host material of the nanocomposite and f is the number of nanoparticles in the matrix.

_{d}is the permittivity of the host material, ε

_{m}is the permittivity of the inclusions embedded into the host material and ρ is the metal filling fraction ratio, which is calculated as:

## 3. Results

^{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 ω

_{||0}[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 ${\epsilon}_{n}=2.25$, ${\epsilon}_{d}=11.8$. It is worthwhile noting that the regime of DSW propagation takes place if ${\epsilon}_{\left|\right|}\left(\omega \right),\text{\hspace{0.17em}}{\epsilon}_{nc}\left(\omega \right)>0$. To have a deeper insight into the problem, we investigated permittivity components versus conductivity. Doing so, in Figure 3 permittivity function is plotted if ω = 0.3 × 10

^{14}Hz (Figure 3a) and ω = 3 × 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 Figure 3a and Figure 4a, one may conclude that ${\epsilon}_{nc}\left(\sigma \right)>{\epsilon}_{\perp}\left(\sigma \right)$ in case of the hyperbolic regime and ${\epsilon}_{nc}\left(\sigma \right)<{\epsilon}_{\perp}\left(\sigma \right)$ 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 ${\epsilon}_{nc}\left(\omega \right),\text{\hspace{0.17em}}{\epsilon}_{\perp}\left(\omega \right)>0$ and ${\epsilon}_{\left|\right|}\left(\omega \right)<0$. 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 ${\epsilon}_{nc}\left(\omega \right),\text{\hspace{0.17em}}{\epsilon}_{\perp}\left(\omega \right),\text{\hspace{0.17em}}{\epsilon}_{\left|\right|}\left(\omega \right)>0$ and if ${\epsilon}_{nc}\left(\sigma \right),\text{\hspace{0.17em}}{\epsilon}_{\perp}\left(\sigma \right)>0$, ${\epsilon}_{\left|\right|}\left(\sigma \right)<0$.

_{0}then SPPs propagate at the interface and if Re(β) < k

_{0}then SPPs cannot propagate at the interface of two media. k

_{0}= 2πω/c is the wave vector of the electromagnetic wave in free space. Here ω

_{1}= 0.3 × 10

^{14}Hz, ω

_{2}= 0.3 × 10

^{14}Hz and c = 3 × 10

^{8}m/s, which gives k

_{01}= 6.28 × 10

^{5}1/m and k

_{02}= 6.28 × 10

^{6}1/m. The value of Re(β) varies from 0 to 15 × 10

^{6}1/m versus conductivity. The variations of Re(β) and Im(β) versus conductivity are shown in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10. The absolute value of complex conductivity strongly affects the SPPs propagation at the interface of two media. The real part of dispersion relation of SPPs increases with the absolute value of complex conductivity.

## 4. Conclusions

_{0}. It is worthwhile mentioning that propagation of SPPs was achieved at the interface with variation of complex conductivity. The conducted study allows one to conclude on the conditions of surface waves propagation in the complex conductivity plane. Thus, ${\epsilon}_{nc}\left(\sigma \right),\text{\hspace{0.17em}}{\epsilon}_{\perp}\left(\sigma \right)>0$ and ${\epsilon}_{\left|\right|}\left(\sigma \right)<0$ in the case of hyperbolic regime and ${\epsilon}_{nc}\left(\sigma \right),\text{\hspace{0.17em}}{\epsilon}_{\perp}\left(\sigma \right)>0$, ${\epsilon}_{\left|\right|}\left(\sigma \right)<0$ for Dyakonov surface waves. The potential applications of this works are in the fields of the development of waveguides sources, near-field optics, surface-enhanced Raman spectroscopy, data storage, solar cells, chemical sensors and biosensors.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- Kneipp, K. Surface-enhanced Raman scattering. Phys. Today
**2007**, 60, 40–45. [Google Scholar] [CrossRef] [Green Version] - Juan, M.L.; Righini, M.; Quidant, R. Plasmon nanooptical tweezers. Nat. Photonics
**2011**, 5, 349. [Google Scholar] [CrossRef] - Pendry, J.B. Negative refraction makes a perfect lens. Phys. Rev. Lett.
**2000**, 85, 3966–3969. [Google Scholar] [CrossRef] [PubMed] - Luo, Y.; Aubry, A.; Pendry, J.B. Electromagnetic contribution to surface-enhanced Raman scattering from rough metal surfaces: A transformation optics approach. Phys. Rev. B Condens. Matter Mater. Phys.
**2011**, 83, 155422. [Google Scholar] [CrossRef] [Green Version] - Li, J.; Ye, J.; Chen, C.; Hermans, L.; Verellen, N.; Ryken, J.; Jans, H.; Van Roy, W.; Moshchalkov, V.V.; Lagae, L.; et al. Biosensing using diffractively coupled plasmonic crystals: The figure of merit revisited. Adv. Opt. Mater.
**2015**, 3, 176–181. [Google Scholar] [CrossRef] - Li, X.; Ren, X.; Xie, F.; Zhang, Y.; Xu, T.; Wei, B.; Choy, W.C.H. High-performance organic solar cells with broadband absorption enhancement and reliable reproducibility enabled by collective plasmonic effects. Adv. Opt. Mater.
**2015**, 3, 1220–1231. [Google Scholar] [CrossRef] - Giannini, V.; Fernández-Domínguez, A.I.; Heck, S.C.; Maier, S.A. Plasmonic nanoantennas: Fundamentals and their use in controlling the radiative properties of nanoemitters. Chem. Rev.
**2011**, 111, 3888–3912. [Google Scholar] [CrossRef] - Stiens, J.; Vounckx, R.; Veretennicoff, I. Slab plasmon polaritons and waveguide modes in four-layer resonant semiconductor waveguides. J. Appl. Phys.
**1997**, 81, 1–4. [Google Scholar] [CrossRef] - Singh, M.R.; Racknor, C. Nonlinear energy transfer in quantum dot and metallic nanorod nanocomposites. J. Opt. Soc. Am. B
**2015**, 32, 2216–2222. [Google Scholar] [CrossRef] - Singh, M.R.; Brassem, G.; Yastrebov, S. Enhancement of Radiative and Nonradiative Emission in Random Lasing Plasmonic Nanofibers. Annalen Physik
**2021**, 533, 2000558. [Google Scholar] [CrossRef] - Singh, M.R. A Review of Many-Body Interactions in Linear and Nonlinear Plasmonic Nanohybrids. Symmetry
**2021**, 13, 445. [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] - Feigenbaum, E.; Diest, K.; Atwater, H.A. Unity-Order Index Change in Transparent Conducting Oxides at Visible Frequencies. Nano Lett.
**2010**, 10, 2111. [Google Scholar] [CrossRef] - Das, S.; Salandrino, A.; Wu, J.Z.; Hui, R. Near-infrared electro-optic modulator based on plasmonic graphene. Opt. Lett.
**2015**, 40, 1516. [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. [Google Scholar] [CrossRef] [PubMed] - 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] - Peragut, F.; Cerutti, 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] - Ali, K.; Ullah, M.; Bacha, B.A.; Jabar, M.S.A. Complex conductivity-dependent two-dimensional atom microscopy. Eur. Phys. J. Plus
**2019**, 134, 618. [Google Scholar] [CrossRef] - Khan, N.; Bacha, N.B.A.; Iqba, A.; Rahman, A.U.; Afaq, A. Gain-assisted superluminal propagation and rotary drag of photon and surface plasmon polaritons. Phys. Rev. A.
**2017**, 96, 013848. [Google Scholar] [CrossRef] - Shekhar, P.; Atkinson, J.; Jacob, Z. Hyperbolic metamaterials: Fundamentals and applications. Nano Converg.
**2014**, 1, 14. [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] - Ioannidis, T.; Gric, T.; Rafailov, E. Controlling surface plasmon polaritons propagating at the interface of low-dimensional acoustic metamaterials. Waves Random Complex Media
**2021**, submitted. [Google Scholar]

**Figure 1.**Schematic system under consideration, involving a semi-infinite hypercrystal (x > 0) and a nanocomposite with semiconductor inclusions (x < 0) (

**a**) and metamaterial (hypercrystal) unit cell (

**b**).

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

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

**b**) ${\epsilon}_{n}=2.25$, ${\epsilon}_{d}=11.8$. 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, ${\epsilon}_{n}=11.8$, ${\epsilon}_{d}=2.25$. Herein ITO inclusions are employed in nanocomposite and hypercrystal. (

**a**) ω = 0.3 × 10

^{14}Hz; (

**b**) ω = 3 × 10

^{14}Hz.

**Figure 4.**Relative permittivity components of the nanocomposite and hypercrystal versus conductivity. Herein, f = 0.3, ${\epsilon}_{n}=2.25$, ${\epsilon}_{d}=11.8$. Herein ITO inclusions are employed in nanocomposite and hypercrystal. (

**a**) ω = 0.3 × 10

^{14}Hz; (

**b**) ω = 3 × 10

^{14}Hz.

**Figure 5.**Solution of the dispersion equation versus frequency (

**a**) and versus conductivity (

**b**). ${\epsilon}_{n}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2.25$, ${\epsilon}_{d}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}11.8$. Herein ITO inclusions are employed in nanocomposite and hypercrystal, f = 0.3 in (

**b**).

**Figure 6.**Solution of the dispersion equation versus frequency (

**a**) and versus conductivity (

**b**). ${\epsilon}_{n}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}11.8$, ${\epsilon}_{d}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2.25$. Herein ITO inclusions are employed in nanocomposite and hypercrystal, f = 0.3 in (

**b**).

**Figure 7.**Dependence of imaginary part of propagation constant versus frequency for different filling factors (

**a**) and versus conductivity for f = 0.3 (

**b**). ${\epsilon}_{n}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2.25$, ${\epsilon}_{d}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}11.8$. All the presented results are obtained for the ITO inclusions.

**Figure 8.**Dependence of imaginary part of propagation constant versus frequency for different filling factors (

**a**) and versus conductivity for f = 0.3 (

**b**). ${\epsilon}_{n}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}11.8$, ${\epsilon}_{d}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2.25$. All the presented results are obtained for the ITO inclusions.

**Figure 9.**Dependence of propagation length versus frequency for different filling factors (

**a**) and versus conductivity for f = 0.3 (

**b**). ${\epsilon}_{n}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2.25$, ${\epsilon}_{d}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}11.8$. All the presented results are obtained for the ITO inclusions.

**Figure 10.**Dependence of propagation length versus frequency for different filling factors (

**a**) and versus conductivity for f = 0.3 (

**b**). ${\epsilon}_{n}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}11.8,\text{\hspace{1em}}{\epsilon}_{d}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2.25$. All the presented results are obtained for the ITO inclusions.

**Table 1.**Drude–Lorentz parameters of plasmonic materials obtained from experimental data. One may approximate the materials dielectric function by the complex dielectric function: ${\epsilon}_{TCO}={\epsilon}_{b}-\frac{{\omega}_{p}^{2}}{\omega \left(\omega +i{\gamma}_{p}\right)}+\frac{{f}_{1}{\omega}_{1}^{2}}{\left({\omega}_{1}^{2}-{\omega}^{2}-i\omega {\gamma}_{1}\right)}$, with the values of the parameters outlined in the table [18]. Here ε

_{b}is the polarization response from the core electrons (background permittivity), ω

_{p}is the plasma frequency, γ

_{p}is the Drude relaxation rate.

AZO | GZO | ITO | TiN (Deposited at 800 °C) | TiN (Deposited at 500 °C) | ZrN | |
---|---|---|---|---|---|---|

ε_{b} | 3.54 | 3.23 | 3.53 | 4.86 | 2.49 | 3.47 |

ω_{p} (eV) | 1.75 | 1.99 | 1.78 | 7.93 | 5.95 | 8.02 |

γ_{p} (eV) | 0.04 | 0.12 | 0.16 | 0.18 | 0.51 | 0.52 |

f_{1} | 0.51 | 0.39 | 0.39 | 3.29 | 2.04 | 2.45 |

ω_{1} (eV) | 4.29 | 4.05 | 4.21 | 4.22 | 3.95 | 5.48 |

γ_{1} (eV) | 0.10 | 0.09 | 0.09 | 2.03 | 2.49 | 1.74 |

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.
The Study of the Surface Plasmon Polaritons at the Interface Separating Nanocomposite and Hypercrystal. *Appl. Sci.* **2021**, *11*, 5255.
https://doi.org/10.3390/app11115255

**AMA Style**

Ioannidis T, Gric T, Rafailov E.
The Study of the Surface Plasmon Polaritons at the Interface Separating Nanocomposite and Hypercrystal. *Applied Sciences*. 2021; 11(11):5255.
https://doi.org/10.3390/app11115255

**Chicago/Turabian Style**

Ioannidis, Thanos, Tatjana Gric, and Edik Rafailov.
2021. "The Study of the Surface Plasmon Polaritons at the Interface Separating Nanocomposite and Hypercrystal" *Applied Sciences* 11, no. 11: 5255.
https://doi.org/10.3390/app11115255