1. Introduction

In the last decades, hyperbolic metamaterials (HMMs) gained a central role among other metamaterials [1–5] due to their unique ability to manipulate electromagnetic fields [6–15]. As a rule, for generation of the hyperbolic dispersion relation, two structures are generally used: a stack of sub-wavelength alternating metallic and dielectric layers, or a lattice of metallic nanowires embedded in a dielectric matrix, called a nanowire array. The particular properties of HMMs are based on the excitation of coupled surface plasmons in the metal layers or nanowires inside of HMMs [16–22]. The electric field of surface plasmons decays very fast with distance from the interface between the dielectric and metal, and is defined by distances shorter than the wavelength. When the metallic/dielectric layers or metallic nanowires in a dielectric matrix are periodically arranged with the sub-wavelength separation, the electromagnetic fields of the individual plasmonic interfaces overlap and couple, giving rise to a collective response. Such an optical response of these media can be homogenized via an effective medium theory [23–27] and interpreted as arising from the bulk effective media with a hyperbolic effective permittivity tensor.

There is a growing interest in introducing a possibility of reconfiguration or tunability of HMM structures, allowing a tuning of their properties and functions. In this respect, liquid crystals (LCs) stand out as the preferred means of achieving tunability, as they possess the broadband transparency, large optical birefringence, and easy susceptibilities to applied fields. An analysis of the aligned nematic LC cells containing the nanospheres was made in papers [28,29]. Authors of the papers have shown that these media provided a new type of metamaterial whose index of refraction is tunable from negative, through zero, to positive values. In paper [30] it was shown that LCs containing randomly dispersed metallic spherical nanoparticles exhibit hyperbolic properties in the visible spectrum. It also demonstrated the possibility of tunable wavelength properties by choosing the dielectric-metal core-shell nanoparticles. The nanoparticle shell anisotropy was observed in phospholipid vesicles (∼50 nm radius) [31] and the SiO₂@Au core–shell nanoparticles with 50 nm core diameter and 10 nm shell thickness [32]. Tuning of the plasmon resonance in the nanoribbon arrays with the adjacent LC layer was studied in papers [33,34]. The nanosphere dispersed LCs as a hyperbolic metamaterial for wide-angle negative–positive refraction and reflection were proposed in paper [35], where it was shown that the real parts of effective ordinary and extraordinary permittivities of such media can have opposite signs in the near IR spectral region. A voltage-tunable hyperbolic dispersion metamaterial consisting of silver nanowires and nematic LCs was studied in paper [36]. The metamaterial showed an electro-optic effect, in which the dispersion relation switched between elliptic and hyperbolic dispersions in the visible waveband with an external electric field. It was analyzed the optical refraction in a fluid system, which contained silica-coated gold nanorods dispersed in silicone oil under an external electric field [37]. Because of the formation of a chain-like or lattice-like structure of dispersed nanorods along the electric field, the fluid showed hyperbolic properties resulting in optical negative refraction for transverse magnetic waves.

In our paper, we study a system of the core-shell nanowire array in the LC supposing that the nanowire shell permittivity is anisotropic with radially power-law dependent components. The paper is organized as follows. In Section 2 we derive the analytical expressions for the effective permittivity components of the system under consideration. In Section 3 we find the frequency areas of the HMM when the nanowires core is either Ag or Au and the nanowires are embedded in the LC 5CB. Dependence of the frequency position of these areas on the core-shell nanowire parameters are also analyzed. Section 4 contains results showing influence of the LC orientational state on the HMM areas position. Some conclusions of our study are presented in Section 5.

2. Effective permittivity tensor of the LC matrix with the cylindrical core-shell nanowires

Consider a system of cylindrical nanowires embedded in a nematic liquid crystal (LC). The nanowires are periodically arranged in the LC matrix with the sub-wavelength separation.

Frequency $\omega$ and wave vector $k=(k_x,k_y,k_z)$ of the extraordinary electromagnetic wave propagating in HMMs must satisfy to the dispersion relation (see, for example [6],)

To provide the cylindrical symmetry of the LC matrix containing the nanowires, we suppose that there are strong forces aligning the LC director along the nanowires. It can be an external field orienting the LC director along the nanowires or the planar director anchoring with the nanowire surface.

The nanowire array in the LC matrix creates a heterogeneous medium. Using an effective medium theory we can describe the electric properties of such a system by the effective permittivity. To obtain the effective permittivity components of the LC with the cylindrical nanowires, ${\epsilon }_{\perp },{\epsilon }_{\parallel }$, we can use, for example, the results obtained in paper [38] for the LC filled with ellipsoidal particles. In our case the particle long axis length is set to be infinite, and we denote the nanowire permittivity components perpendicular and parallel to the nanowire axis by${\epsilon }_{cyl,\perp }$, and ${\epsilon }_{cyl,\parallel }$, respectively; we arrive at the following expressions for the effective permittivity components of the LC with the cylindrical nanowires:

Now we suppose that each nanowire contains a metallic core with a radius $b$ and a permittivity ${\epsilon }_c$covered by the anisotropic dielectric shell with an external radius $a$(Fig. 1). We consider the case of the anisotropic nanowire shell with the permittivity tensor main components ${\epsilon }_r,{\epsilon }_{\phi },{\epsilon }_z$ (in the cylindrical coordinate system), supposing the z-axis to be directed along the nanowire. We also assume that the components ${\epsilon }_r,{\epsilon }_{\phi }$ of the shell permittivity are radially power-law dependent,

 

Fig. 1 Cross-section of cylindrical nanowire with radially inhomogeneous anisotropic permittivity of the shell, which covers the metallic core with permittivity${\epsilon }_c$;${\epsilon }_{r,\phi }$are radially power-law dependent components of the shell permittivity;$a$and $b$are outer and inner radiuses of the shell.

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Since the core-shell nanowires are optically inhomogeneous we need to perform the so-called “internal homogenization” [39–41]. In the case of the core-shell nanowires with radially anisotropic and power-law dependent components ${\epsilon }_r,{\epsilon }_{\phi }$of the shell permittivity, the problem is analytically tractable if the external media (the LC matrix) also has the cylindrical symmetry.

In the cylindrical coordinates the electric displacement vector takes the form $\begin{array}{l}{H}_1=1,H_2=\rho ,H_3=1\\ \text{grad}f=\frac{\partial f}{\partial \rho }{e}_{\rho }+\frac{1}{\rho }\frac{\partial f}{\partial \phi }{e}_{\phi }+\frac{\partial f}{\partial z}{e}_z\end{array}$. Setting the external electric field $E_0$ directed perpendicular to the cylindrical axis (the z-axis), the solution to the equation $divD=0$for the potential function$\varphi$ ($E=-\nabla \varphi$) yields:

The boundary conditions at $r=a$ and $r=b$ for the nanowire embedded in the LC with the director parallel to the nanowire reduce to the following equations:

The electric potential created by a cylinder of radius $a$ polarized by an external electric field $E_0$ is $\varphi ={\alpha }_{\perp }(a^2\cos \phi /2r)E_0$, where ${\alpha }_{\perp }$ is a polarizability of the cylinder in direction perpendicular to the cylinder long axis. This expression can be compared to the expression for$\varphi$ in Eq. (5) for $r>a$, yielding an expression for ${\alpha }_{\perp }=(2F/a^2)E_0$, where ${\alpha }_{\perp }$ is now regarded as the effective polarizability of the core-shell cylinder.

The two dimensional generalization of the Clausius-Mossotti formula (see [42]), relating the polarizability of a cylinder and its permittivity ${\epsilon }_{cyl,\perp }$, is ${\alpha }_{\perp }=2({\epsilon }_{cyl,\perp }-{\epsilon }_{m\perp })/({\epsilon }_{cyl,\perp }+{\epsilon }_{m\perp })$, where ${\epsilon }_{m\perp }$ is a permittivity of the surrounding medium in direction perpendicular to the cylinder and in our case ${\epsilon }_{m\perp }={\epsilon }_{\perp }^{LC}$. This permits the derivation of an expression for the effective permittivity ${\epsilon }_{cyl,\perp }$ of the core-shell cylinder in terms of ${\alpha }_{\perp }=(2F/a^2)E_0$. The constant $F$is found by solving Eqs. (7). Omitting the details of the calculation, we obtain the following expressions for the effective permittivity component of the cylindrical core-shell nanowire ${\epsilon }_{cyl,\perp }$:

We consider the cases when the nanowire core material is Ag or Au. For these materials the frequency dependence of the permittivity ${\epsilon }_c$ is very well described by the function, which accounts for the interband transitions [43]

Table 1. ^a

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As the LC matrix, we choose the typical nematic LC 5CB. The permittivity frequency dispersion of the LC 5CB was measured by the authors of paper [44] and fit by the following analytical expressions:

Table 2. ^a

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Furthermore, we calculate the real parts of the effective permittivity components (2), (3), which determine the frequency regions of HMM characterized by the sign of the inequality$\mathrm{Re}{\epsilon }_{\perp }(\omega )\cdot \mathrm{Re}{\epsilon }_{\parallel }(\omega )<0$. For this, we use Eqs. (8), (9) for the core-shell nanowire effective permittivity, Eqs. (11a), (11b) for the LC 5CB permittivity, and Eqs. (10) for the permittivity of the Ag or Au cores.

3. Frequency position of the HMM areas

In Fig. 1 we present real parts of the effective permittivity components,$\mathrm{Re}{\epsilon }_{\perp }$and$\mathrm{Re}{\epsilon }_{\parallel }$, versus the wavelength for the LC 5CB matrix contained the core-shell nanowires with the Ag core (Figs. 2(a), 2(b)) and the Au core (Figs. 2(c),2(d)). As an example, we show results obtained when the nanowire shell possesses: i) isotropic permittivity ($m=0$,${\epsilon }_r={\epsilon }_{\phi }={\epsilon }_z$), ii) anisotropic permittivity with radially power-law dependent components ${\epsilon }_r,{\epsilon }_{\phi }$($m\ne 0$).

 

Fig. 2 Real parts of the effective permittivity components, $\mathrm{Re}{\epsilon }_{\perp }$(solid lines) and $\mathrm{Re}{\epsilon }_{\parallel }$(dashed lines), versus the wavelength for the system of the shell-core nanowires with the Ag core (a), (b) and Au core (c), (d) in the LC 5CB matrix for two values of the exponent $m$;${\epsilon }_{r0}=5,{\epsilon }_{\phi 0}=5,{\epsilon }_z=5$,$b/a=0.5,f=0.4$.

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It is seen from Fig. 2 that the system under consideration can possess two HMM areas: an area where $\mathrm{Re}{\epsilon }_{\parallel }<0$(and $\mathrm{Re}{\epsilon }_{\perp }>0$) and an area where $\mathrm{Re}{\epsilon }_{\perp }<0$(and$\mathrm{Re}{\epsilon }_{\parallel }>0$). The area $\mathrm{Re}{\epsilon }_{\parallel }<0$appears at $\lambda >860nm$in the case of the Ag core and at$\lambda >870nm$ in the case of the Au core. As we can see from Eqs. (3), (9) a position of the area$\mathrm{Re}{\epsilon }_{\parallel }<0$ depends on the metallic fill fraction $\rho =b^2/a^2$, the volume fill fraction of the nanowires $f$, the core metal permittivity ${\epsilon }_c$, and the ${\epsilon }_z$ component of the shell permittivity. It does not depend on the shell permittivity components ${\epsilon }_r,{\epsilon }_{\phi }$ and the exponent $m$value

The HMM area $\mathrm{Re}{\epsilon }_{\perp }<0$ is observed at smaller wavelengths and is more narrow than the area $\mathrm{Re}{\epsilon }_{\parallel }<0$. Its position depends on the ratio $b/a$, the fill fraction $f$, the shell permittivity components ${\epsilon }_r,{\epsilon }_{\phi }$ and, therefore, on the exponent$m$value, but it does not depend on the ${\epsilon }_z$ component of the shell permittivity (see Eqs. (2), (8)). It also depends on the type of the core metal as well (see also Fig. 2).

The frequency dependences of the Au and Ag permittivity, ${\epsilon }_c(\omega )$, significantly differ (see Fig. 3). As a result, the position of the HMM areas in the cases of the core-shell nanowires with the Ag core and the Au core are different. In particular, in the case of the Au core the area $\mathrm{Re}{\epsilon }_{\perp }<0$does not appear when the nanowire shell is isotropic (Fig. 2(c), $m=0$).

 

Fig. 3 Real (a) and imaginary (b) parts of the nanowire core permittivity versus the wavelength: Ag core - solid lines, Au core - dashed lines. For plotting, we used Eq. (10) and Table 1 obtained in paper [43].

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In Fig. 4 we show the dependence of the width of the HMM area $\mathrm{Re}{\epsilon }_{\perp }<0$ (${\lambda }_{\min },{\lambda }_{\max }$) on the exponent $m$value for nanowires with the Ag or Au cores in the LC 5CB matrix. It is seen that the area $\mathrm{Re}{\epsilon }_{\perp }<0$ exists only in the limited interval of the exponent $m$value. The width of this interval depends on the values of the shell permittivity parameters${\epsilon }_{r0}$ and ${\epsilon }_{\phi 0}$,$f$and ratio $b/a$.

 

Fig. 4 The width (${\lambda }_{\min },{\lambda }_{\max }$) of the HMM area $\mathrm{Re}{\epsilon }_{\perp }<0$ versus the exponent $m$value for nanowires with the Ag and Au cores.${\epsilon }_{r0}=5,{\epsilon }_{\phi 0}=5,{\epsilon }_z=5$,$b/a=0.5,f=0.4$.

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Dependence of the width of the area $\mathrm{Re}{\epsilon }_{\perp }<0$ (${\lambda }_{\min },{\lambda }_{\max }$) and the short-wave border of the area $\mathrm{Re}{\epsilon }_{\parallel }<0$on the nanowire fill fraction$f$for nanowires with the Ag core in the LC 5CB matrix is presented in Fig. 5. The width of both areas increases with an increase of$f$. A decrease of the exponent $m$ leads to the shift of the area $\mathrm{Re}{\epsilon }_{\perp }<0$ to the long-wave side and an increase of its width. The similar dependence takes also place for nanowires with the Au core and $m<0$.

 

Fig. 5 The width (${\lambda }_{\min },{\lambda }_{\max }$) of the HMM areas $\mathrm{Re}{\epsilon }_{\perp }<0$ (a) and $\mathrm{Re}{\epsilon }_{\parallel }<0$(b) versus the volume fill fraction $f$of the nanowires with Ag core. ${\epsilon }_{r0}=5,{\epsilon }_{\phi 0}=5,{\epsilon }_z=5$,$b/a=0.5$. For $\mathrm{Re}{\epsilon }_{\perp }<0$: $m=0,2,-2$- solid, dashed, dot-dashed lines, respectively.

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The width of the area $\mathrm{Re}{\epsilon }_{\perp }<0$ and the short-wave border of the area $\mathrm{Re}{\epsilon }_{\parallel }<0$increases with an increase of the nanowire shell external to internal radii ratio. As an example, we show this in Fig. 6 for the case of the nanowire with the Ag core. However, as it is seen from Fig. 6(a) the dependence of the position and width of the area $\mathrm{Re}{\epsilon }_{\perp }<0$on the exponent$m$becomes small for the nanowire shells with small thickness ($b/a\to 1$).

 

Fig. 6 The width (${\lambda }_{\min },{\lambda }_{\max }$) of the HMM areas $\mathrm{Re}{\epsilon }_{\perp }<0$(a) and $\mathrm{Re}{\epsilon }_{\parallel }<0$(b) versus the external to internal radii ratio of the nanowire shell with the Ag core. ${\epsilon }_{r0}=5,{\epsilon }_{\phi 0}=5,{\epsilon }_z=5$;$f=0.4$. For$\mathrm{Re}{\epsilon }_{\perp }<0$: $m=0,2,-2$- solid, dashed, dot-dashed lines, respectively.

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Dependence of the width of the area $\mathrm{Re}{\epsilon }_{\perp }<0$ on the shell permittivity parameters ${\epsilon }_{r0}$, ${\epsilon }_{\phi 0}$ for the nanowires with an Ag core in the LC 5CB matrix is shown in Fig. 7 at several values of the exponent $m$. One can see that the width of the HMM area$\mathrm{Re}{\epsilon }_{\perp }<0$increases with an increase of the parameter${\epsilon }_{r0}$ (Fig. 7(a)) and with a decrease of the parameter ${\epsilon }_{\phi 0}$ (Fig. 7(b)). The same dependence is obtained for nanowires with the Au core and $m<0$.

 

Fig. 7 The width (${\lambda }_{\min },{\lambda }_{\max }$) of the HMM area $\mathrm{Re}{\epsilon }_{\perp }<0$ versus the parameter ${\epsilon }_{r0}$ (a) at fixed ${\epsilon }_{\phi 0}=5$and the parameter ${\epsilon }_{\phi 0}$(b) at fixed${\epsilon }_{r0}=5$ in the case of the nanowire shell with Ag core. $m=0,2,-2$- solid, dashed, dot-dashed lines, respectively. $b/a=0.5$,$f=0.4$,${\epsilon }_z=5$.

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In Fig. 8 we present $\mathrm{Re}{\epsilon }_{\perp }$and$\mathrm{Re}{\epsilon }_{\parallel }$versus the wavelength for the LC 5CB containing the core-shell nanowires with either Ag core or Au core for different values of the shell permittivity components ${\epsilon }_r,{\epsilon }_{\phi }$, but with the same value of the shell permittivity tensor trace. Such a situation can take place when the nanowire shell is created by molecules undergoing the trans-cis transitions. In this case, using such transitions we can switch a position of the HMM frequency interval $\mathrm{Re}{\epsilon }_{\perp }<0$(the case of the Ag core in Figs. 8(a), 8(b)) or create this interval (the case of the Au core in Figs. 8(c), 8(d)). However, it is necessary to note that for the nanowires with the Au core the interval $\mathrm{Re}{\epsilon }_{\perp }<0$can appear only if the exponent $m$is negative (see Fig. 4(b)) with sufficiently large $\left|m\right|$.

 

Fig. 8 Real parts of the effective permittivity components, $\mathrm{Re}{\epsilon }_{\perp }$(solid lines) and $\mathrm{Re}{\epsilon }_{\parallel }$(dashed lines), versus the wavelength in the case of the shell permittivity tensor trace to be constant. Nanowires with Ag core - (a), (b); with Au core – (c), (d); ${\epsilon }_{r0}=2,{\epsilon }_{\phi 0}=8$- (a), (c); ${\epsilon }_{r0}=8,{\epsilon }_{\phi 0}=2$- (b), (d). $b/a=0.5,f=0.4,{\epsilon }_z=5$.

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4. Influence of the LC orientational state on the HMM area position

By heating we can transfer the LC into the isotropic state. The cylindrical symmetry of the system is conserved and we can use the formulae presented above putting here ${\epsilon }_{\perp }^{LC}={\epsilon }_{\parallel }^{LC}={\epsilon }_{is}^{LC}$. As ${\epsilon }_{is}^{LC}$one can use ${\epsilon }_{is}^{LC}=({\epsilon }_{\parallel }^{LC}+2{\epsilon }_{\perp }^{LC})/3$. In Fig. 9 we demonstrate an influence of the orientational state of the LC 5CB comparing the positions of the HMM area$\mathrm{Re}{\epsilon }_{\perp }<0$ when the LC director is parallel to the nanowires and when the LC director is disordered (an isotropic state) for nanowires with the Ag core (Fig. 9(a)) or the Au core (Fig. 9(b)). It is seen that in both cases the area $\mathrm{Re}{\epsilon }_{\perp }<0$are sensitive to the LC orientational state: it broadens into the long wavelength side when the LC becomes isotropic. Influence of the LC orientational state on a position of the HMM area $\mathrm{Re}{\epsilon }_{\parallel }<0$ is very weak.

 

Fig. 9 Real part of the effective permittivity components $\mathrm{Re}{\epsilon }_{\perp }$versus the wavelength for the system of the core-shell nanowire array in the LC matrix 5CB: Ag core - (a), Au core - (b); the LC director parallel to the nanowires (solid lines), the LC in an isotropic state (dashed lines). ${\epsilon }_{r0}=8,{\epsilon }_{\phi 0}=2,{\epsilon }_z=5,b/a=0.5$.

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5. Conclusions

We studied a system of the core-shell nanowire arrays in the LC matrix 5CB, supposing that the nanowire shell permittivity is anisotropic with radially power-law dependent permittivity components${\epsilon }_r,{\epsilon }_{\phi }$. The nanowire core material is assumed to be Ag or Au. We show that this system can possess two HMM areas: an area where $\mathrm{Re}{\epsilon }_{\parallel }<0$ and an area where $\mathrm{Re}{\epsilon }_{\perp }<0$. A position of the area $\mathrm{Re}{\epsilon }_{\parallel }<0$depends on the metallic fill fraction $\rho =b^2/a^2$, the volume fill fraction of the nanowires $f$, the ${\epsilon }_z$ component of the shell permittivity, and weakly on the type of the core metal, Ag or Au. It does not depend on the shell permittivity components ${\epsilon }_r,{\epsilon }_{\phi }$ and the exponent $m$value. The HMM area $\mathrm{Re}{\epsilon }_{\perp }<0$ is observed at smaller wavelengths and is more narrow than the area $\mathrm{Re}{\epsilon }_{\parallel }<0$. Its position depends on the metallic fill fraction$\rho$, the volume fill fraction of the nanowires$f$, the shell permittivity components ${\epsilon }_r,{\epsilon }_{\phi }$, and strongly on the exponent $m$value and the type of the core metal, Ag or Au. It does not depend on the ${\epsilon }_z$ component of the shell permittivity. In the case of the Au core the HMM area $\mathrm{Re}{\epsilon }_{\perp }<0$does not appear for the exponent $m\ge 0$.

We show that transition of the LC matrix director from the ordered state to the isotropic state influences a position of the HMM area $\mathrm{Re}{\epsilon }_{\parallel }<0$ weakly, while the area $\mathrm{Re}{\epsilon }_{\perp }<0$are more sensitive to the LC orientational state: it broadens into the long wavelength side when the LC becomes isotropic.

Thus, choosing the parameters of the core-shell nanowires we can obtain a system with HMM properties in a desired visible or infrared spectral region. By changing the orientational state of the LC matrix, one can tune the position of the HMM area.