## Abstract

Guided wave modes in the uniaxial anisotropic hyperbolic metamaterials (HMMs) based on highly doped semiconductor instead of metal in the mid-infrared region are investigated theoretically. The heavily doped semiconductor is used to overcome the restrictions of the conventional metal-based structures caused by the lake of tunability and high metal loss at mid-infrared wavelengths. The unit cells of our proposed metamaterial are composed of alternating layers of undoped InAs as a dielectric layer and highly doped InAs as a metal layer. We numerically study the linear and nonlinear behavior of such multilayer metamaterials, for different arrangements of layers in the parallel (vertical HMM) and perpendicular (horizontal HMM) to the input wave vector. The effect of doping concentration, metal to dielectric thickness ratio in the unit cell (fill-fraction), and the total thickness of structure on the guided modes and transmission/reflection spectra of the metamaterials are studied. Moreover, the charge redistribution due to band-bending in the alternating doped and undoped layers of InAs is considered in our simulations. We demonstrate that the guided modes of the proposed hyperbolic metamaterial can change by increasing the intensity of the incident lightwave and entering the nonlinear regime. Therefore, the transition from linear to the nonlinear region leads to high-performance optical bistability. Furthermore, the switching performance in the vertical and horizontal HMMs are inspected and an ultrafast, low power, and high extinction ratio all-optical switch is presented based on a vertical structure of nonlinear highly doped semiconductor hyperbolic metamaterials.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Confinement and control of the electromagnetic energy in the nanoscale are required to achieve ultrafast and energy-efficient all-optical nanophotonic devices [1–3]. The use of plasmonic nanostructures and metamaterials are two main approaches, to addressing these requirements [4,5]. Surface plasmon polaritons are the most promising solutions for tight light confinement in deep sub-wavelength scales [6,7]. Plasmonic nanostructures can overcome the diffraction limit of light and increase localized electromagnetic fields [8]. Therefore, the emergence of plasmonics has revolutionized the light-based technologies and enabled low power, compact, and ultra-fast applications of linear and nonlinear optical phenomena [9–11]. Plasmonic nanostructures have been extensively studied and developed in the last decade [12]. Nowadays, the optimization of plasmonic waveguides and the use of new materials have attracted significant interest for improving tunability, reducing losses, and increasing the efficiency of optical devices [13,14]. In addition to developing plasmonics knowledge, metamaterials have also been investigated to overcome the diffraction limit of light and implement high-performance devices [15–18]. Due to the unique and unusual behavior of metamaterials such as extraordinary electromagnetic properties that are not available or not easily obtained in nature, it is expected that the combination of metamaterials and plasmonic nanostructures can lead to the production of novel devices with high efficiency and unprecedented functionalities [19]. One type of plasmonic metamaterials is a uniaxial anisotropic multilayer structure with an isofrequency contour (IFC) in the form of an open hyperboloid. Such metamaterials can be implemented using alternating subwavelength layers of isotropic dielectric and metal [20]. The open hyperboloidal isofrequency contour of these metamaterials arises from the opposite signs of the principal components of their magnetic or electric tensor [21–23]. This unusual IFC profile of HMMs and their ability to manipulate and control the light at sub-wavelength scales can be utilized in high-performance nanophotonics devices [24–27]. For example, the hyperbolic metamaterials have applications in negative refraction [28], hyperlens [29], enhancement of the spontaneous emission [30], and sensing [31]. Furthermore, the plasmonic hyperbolic metamaterials can strongly enhance the nonlinear optical effects due to the highly localized fields. As a result, the plasmonic hyperbolic metamaterials with strong nonlinear Kerr effect and flexible control of electromagnetic waves by tuning the hyperbolic shape of isofrequency surface can be utilized in optical bistability and design the high-performance optical nanoswitches. However, the plasmonic hyperbolic waveguides composed of metal and dielectric layers deal with some limitations such as high metal loss and lack of tunability [32,33]. One approach to overcome these problems is to utilize graphene instead of metal in these structures. The use of graphene in plasmonic hyperbolic metamaterials leads to high tunability and reduction of propagation losses [34–36]. However, the surface plasmons frequencies of graphene typically fall in the terahertz range, and to change the plasma frequency, an external voltage must be applied [37]. In the mid-infrared range, which is widely used in optical communication systems, due to the large imaginary part of the noble metal permittivity, ohmic losses are more critical. Highly doped semiconductors are an alternative to metals that offer lower loss and better tunability of the plasma frequency in the Mid-infrared region [38–40].

In this context, the various guided wave modes in a plasmonic hyperbolic metamaterial based on a highly doped semiconductor in the mid-infrared region are investigated. The linear and nonlinear behavior of our proposed structure considering the effects of important parameters such as doping concentration, fill-fraction of the unit cells, and the total thickness of multilayer structure in two different arrangements of layers (vertical and horizontal HMMs) are studied. We show that by increasing the intensity of the input TM lightwave and the transition from linear to the nonlinear regime, the high-performances switching can be realized in our proposed nanostructure. Furthermore, a low power and high extinction ratio all-optical nanoswitch in the mid-infrared region based on nonlinear highly doped semiconductor hyperbolic metamaterials is designed. The finite difference time domain (FDTD) method has been adopted to simulate our proposed nanostructures.

## 2. Hyperbolic metamaterials based on highly doped semiconductors

The usual HMMs structures, which consists of metal and dielectric layers, must be modified to implement the high efficiency plasmonic hyperbolic metamaterials in the Mid-infrared region. Here, considering the high metal losses at the mid-infrared wavelength range, a doped semiconductor has been used instead of metal to improve tunability and reduce the ohmic losses. However, the use of alternating doped and undoped layers of InAs in the implementation of structures would cause the band-bending to become serious. The band-bending can have a significant effect on the performance of semiconductor-based metamaterials. Therefore, to simulate such structures by the FDTD method, we used the constituent materials and first calculated the charge distribution and the width of depletion regions by electrical analysis of the multilayer structures and solving the drift-diffusion and Poisson equations. Next, we entered the charge redistribution profile due to the band-bending into our FDTD code. Figure 1 shows the charge distribution profile for a multilayer structure consists of altering doped and undoped layers of InAs. The doping concentration of doped layers is $1 \times {10^{19}}({c{m^{ - 3}}} )$ and the thickness of layers is 100 nm. Three regions of A, B, and C in Fig. 1 show the undoped, graded, and constant doped regions, respectively. Region A has a constant permittivity and the Drude model can be used to calculate the permittivity of constant carrier density in region C. For region B, the plasma wavelength is a function of depth and is calculated using the depth-dependent charge distribution and considering the effects of the non-parabolic effective mass.

Figure 2 shows two different arrangements of the proposed multilayer structure consisting of altering layers of doped InAs acting as metal and undoped InAs acting as the dielectric with the thickness of ${d_m}$ and ${d_d}$, respectively. The total thickness of structures in the direction of wave propagation is indicated by L_{V} and L_{H}. Figure 2(a) illustrates the schematic in which the direction of TM wave propagation is parallel to the layers (vertical structure) and Fig. 2(b) describes the structure with layers perpendicular to the direction of wave vector (horizontal structure).

The thickness of layers is much smaller than the wavelength and an effective medium approximation (EMA) can be applied to investigate the dispersion behavior of the proposed structures [41,42] and a better understanding of the multilayer structures guided modes. But because the EMA model does not take into account the effect of band-bending, it may not be accurate enough compared to the method of the constituent materials. Our simulations, confirming the results of previous experimental works [42–44], show that with increasing layer thickness and doping density, the effect of band-bending on the metamaterial behavior is significantly reduced and the results of the EMA model will be more consistent with the results of the constituent structure. For this reason, we have chosen layer thicknesses and doping in such a way that the EMA model helps to explain the behavior of our structures with a good approximation.

In the following, the transverse and longitudinal components of permittivity are shown as pairs of $({{\varepsilon_ \bot },{\varepsilon_\textrm{||}}} )$ and $({\varepsilon_ \bot^{\prime},\; \varepsilon_\textrm{||}^{\prime}} )$ for the vertical and horizontal structures, respectively. The dielectric components in the parallel $({{\varepsilon_\textrm{||}}} )$ and perpendicular $({{\varepsilon_ \bot }} )$ directions to incident wave vector can be written as:

Here, ${\varepsilon _\infty }$ is the high frequency permittivity, $\gamma $ is the damping term, and ${\omega _p}$ is the plasma angular frequency given by

${N_d}$, $q$, and ${m^\ast }$ are the doping concentration, the electron charge, and the electron effective mass, respectively, and ${\varepsilon _0}$ is the permittivity of free space. The effective mass of electrons in the doped InAs depends on the doping concentration. Therefore, the effective mass is calculated from Eq. (5) by empirical model as follows [46,47]:The coefficients ${f_m}$ and ${f_d}$ are the metal and dielectric thickness ratios to the total thickness of a unit cell and are defined as:

In such multilayer structures, the effective permittivity is anisotropic and the signs of its longitudinal and transverse components are the function of wavelength. Therefore, by increasing the wavelength to values greater than the resonance wavelength $({{\lambda_p}} )$, and solving the Maxwell equations for an incident TM wave, four different operating modes can be obtained [48]. For both vertical and horizontal structures, the different modes are Effective dielectric $({\varepsilon _ \bot },\varepsilon _ \bot ^{\prime} > 0,\; {\varepsilon _\parallel },\varepsilon _\textrm{||}^{\prime} > 0)$, effective metal $({\varepsilon _ \bot },\varepsilon _ \bot ^{\prime} < 0,\; {\varepsilon _\parallel },\varepsilon _\textrm{||}^{\prime} < 0)$, Type Ι HMM$({\varepsilon _ \bot },\varepsilon _ \bot ^{\prime} > 0,\; {\varepsilon _\parallel },\varepsilon _\textrm{||}^{\prime} < 0)$, and Type ΙΙ HMM $\left( {{\varepsilon_ \bot },\varepsilon_ \bot^{\prime}\left\langle {0,\; {\varepsilon_\parallel },\varepsilon_\textrm{||}^{\prime}} \right\rangle 0} \right)$. In the effective dielectric mode, the incident light can propagate through the structure, while in the effective metal mode, no real k vector satisfies the dispersion relation and the incident light is totally reflected. On the other hand, the signs of the longitudinal and transverse components of permittivity are opposite in both Type Ι HMM and Type ΙΙ HMM, and these modes have hyperboloidal dispersion curves in two orthogonal directions. Type Ι HMM exhibits a negative refractive index and incident light can propagate through the structure, but in Type ΙΙ HMM, such as effective metal mode the incident lightwave is reflected.

Guided modes of metamaterial are affected by some parameters of structure. The fill-fraction (FF) is a parameter that plays an important role in the optical behavior of hyperbolic metamaterials. The fill-fraction represents the thickness ratio of the metal or highly doped semiconductor in a unit cell. According to Eqs. (1) and (2), the permittivity components are the function of the fill-fraction, and as a result, it is possible to modify the guided modes by changing the fill-fraction. Here, the effect of fill-fraction on the metamaterial guided modes is studied for three different fill-fractions in a horizontal structure consisting of InAs/doped InAs layers with a doping concentration of ${N_d} = 1 \times {10^{19}}({c{m^{ - 3}}} )$ and a unit cell thickness of 200 nm. Figure 3 shows the longitudinal $({\varepsilon_\textrm{||}^{\prime}} )$ and transverse $({\varepsilon_ \bot^{\prime}} )$ components of permittivity at various wavelengths for three different fill-fractions. As displayed in Fig. 3, with a fill-fraction of 0.5, depending on the signs of parallel and perpendicular components of permittivity, two modes can occur. For wavelengths higher than the plasma wavelength$({{\lambda_p}} )$, the structure first enters in Type Ι HMM, and with further increasing of the wavelength, the Type ΙΙ HMM has appeared. However, for a fill-fraction smaller than 0.5, the effective dielectric mode has occurred before the emergence of the Type ΙΙ HMM, and for a fill-fraction larger than 0.5, an effective metal mode is observed in a wavelength window between the Type Ι HMM and Type ΙΙ HMM. Therefore, the fill-fraction modification can be applied to control the metamaterial operating modes. Moreover, Figs. 4(a)–4(b) show the transmission and reflection spectra of the mentioned horizontal metamaterial with a thickness of 1.2 μm for the various fill-fractions. In the multilayer structure with a fill-fraction of 0.6, the reflection rate has increased in the wavelength range that effective metal mode has appeared. However, in the structure with the fill-fraction of 0.4, the transmission rate has increased in the effective dielectric mode region.

Due to the use of doped semiconductor instead of metal in our structure, the optical properties of the metamaterial can be modified by changing the doping concentration. For vertical and horizontal metamaterials with a fill-fraction of 0.5, the transverse $({\varepsilon _ \bot },\varepsilon _ \bot ^{\prime})$ and longitudinal $({{\varepsilon_\textrm{||}},\varepsilon_\textrm{||}^{\prime}} )$ components of permittivity and related guided modes at various wavelengths for different doping concentrations (different plasma wavelengths) are sketched in Fig. 5. Increasing the resonance wavelength $({{\lambda_p}} )$ at lower doping concentrations allows for negative permittivity and hyperbolic dispersion at higher wavelengths. As shown in Fig. 5, for a horizontal structure with fill-fraction of 0.5 and total thickness of 400 nm, by increasing the wavelength of input TM wave to values slightly greater than the resonance wavelength (${\lambda _p})$, the permittivity components can be calculated as $\varepsilon _ \bot ^{\prime} > 0\; \; $ and $\varepsilon _\textrm{||}^{\prime} < 0$, thus the multilayer structure acts as the Type Ι HMM. In this mode, the metamaterial exhibits a negative refractive index, and as a result, the incident light propagates through the structure with little reflection. If the wavelength increases to higher values, the signs of parallel and perpendicular components of the permittivity change $(\varepsilon _ \bot ^{\prime} < 0\; \; $ and $\varepsilon _\textrm{||}^{\prime} > 0)$, and structure acts as Type ΙΙ HMM. In this mode the propagation of incident light is prohibited and the light is reflected. The transmission and reflection spectra of proposed horizontal metamaterial are shown in Figs. 6(a)–6(b). Also, the effects of doping concentration on the behavior of a vertical structure are considered. The Fill-fraction of vertical metamaterial is set to 0.5 and its thickness to 200 nm. In this structure due to displacement of the transverse and longitudinal components of permittivity in comparison to the horizontal structure, by increasing the wavelength to values higher than the resonance wavelength$({{\lambda_p}} )$, Type ΙΙ HMM (reflection mode) appears first, and then with further increase in the wavelength the structure enters in Type Ι HMM (transmission mode). Figures 6(c)–6(d) illustrate the transmission and reflection spectra of the vertical structure.

The thickness of the structure (L) in the direction of the wave vector is another parameter affecting the transmission and reflection spectra of the multilayer hyperbolic metamaterials. Figures 7(a)–7(b) illustrate the transmission and reflection spectra of a horizontal HMM for various thicknesses of 200, 400, and 600 nm. The thickness of each unit cell is 200nm, the fill-fraction is 0.5, and the doping concentration is$7.5 \times {10^{19}}({c{m^{ - 3}}} )$. As expected, increasing the number of unit cells resulting in a higher thickness of structure reduces the transmission rate and increases the reflection rate in different wavelengths. Figure 7(c) shows the sum of the normalized values of transmission and reflection rates, which can be a criterion for measuring losses in the nanostructure. According to Fig. 7(c), in the wavelength range where metamaterial is in the Type Ι HMM (transmission mode), for a structure with higher thickness, due to the increasing losses in the doped semiconductor, the sum of transmission and reflection rates is reduced. Although, a structures with higher thickness in the guided mode of Type ΙΙ HMM (reflection mode) exhibit lower losses because of the higher reflection. Furthermore, the effect of higher doping concentration on losses is explained in Fig. 7(d). The higher doping concentrations result in the larger imaginary part of the permittivity and the losses during transmission are increased.

## 3. Nonlinear response and bistability in the semiconductor-based hyperbolic metamaterials

The strong field confinement achieved by plasmonic hyperbolic metamaterials enhances the optical nonlinear effects which are utilized in many different applications. Here, the effect of nonlinear coefficients on the permittivity of highly doped InAs layers of metamaterial are inspected. Moreover, the optical bistability and switching performance in such epsilon-near-zero (ENZ) metamaterials are investigated. The nonlinear susceptibilities that appear with increasing amplitude of the input electric field, lead to changing the permittivity components. Therefore, transition from linear to nonlinear regime can modify the guided modes of the metamaterial. Furthermore, the optical bistability and high-performance switching can be implemented by moving from a reflection mode to a transmission mode or vice versa. Our simulations demonstrate the nonlinear Kerr effect is responsible for the metamaterial behavior in the nonlinear region. The Kerr effect modifies the corresponding permittivity of InAs nanolayers as [49]

where ${\chi ^{(3 )}}$ is the third-order nonlinear optical susceptibility of InAs, which is dependent on doping concentration, and ${E_0}$ is the field intensity. Therefore, according to Eqs. (1) and (2), the transverse and longitudinal components of permittivity of metamaterial are modified by increasing the field intensity in the nonlinear regime.In this section, the nonlinear behavior of both vertical and horizontal HMMs at the same wavelength (12 μm) and doping concentration of $1 \times {10^{19}}c{m^{ - 3}}$ are studied. The thickness of the vertical structure in the direction of the input wave propagation is 150 nm, and the fill-fraction is 0.5. In these conditions, for low input intensities (linear regime) the metamaterial is biased in the Type ΙΙ HMM (reflection mode). By increasing the intensity of the input electric field and entering the nonlinear regime, the permittivity of doped InAs becomes positive and as a result, the operating mode changes to dielectric mode. In dielectric mode the incident TM wave propagates through the multilayer structure. Considering the nonlinear coefficients in the permittivity of layers, Fig. 8(a) shows the transverse and longitudinal components of permittivity at different wavelengths, for the linear regime with low electric field amplitude $({1\; V/m} )$ and nonlinear regime with high electric field amplitude$({{{10}^6}\; V/m} )$. Figure 8(b) illustrates the transmission and reflection spectra of the vertical structure. As seen, in the nonlinear region, in contrast to the linear regime, the predominant phenomenon is the transmission.

The second structure to be studied is an InAs/doped InAs multilayer metamaterial in which the layers are arranged horizontally (horizontal structure) in the direction of the incident TM wave. This structure is composed of 4 unit cells with a fill-fraction of 0.8 and a total thickness of 800 nm. The fill-fraction is selected so that for low input intensity in the linear regime a reflection mode can be realized at the wavelength of 12 μm. At this wavelength, the only reflection mode that can be achieved in this structure is the effective metal mode. An analysis similar to that performed for the vertical structure was repeated for this structure and results are reported in Figs. 8(c)–8(d). According to Fig. 8(c), in the nonlinear regime, with an electric field of order${10^6}({V/m} )$, the operating mode of structure changes from effective metal (reflection mode) to dielectric mode (transmission mode) and as stated in Fig. 8(d), the incident light propagates through the structure. The transmission and reflection spectra in Fig. 8(d) reveal the high losses in the horizontal multilayer structure due to high thickness of structure and high metal loss in the transition from effective metal mode to dielectric mode.

Figure 9 shows the operating modes of vertical and horizontal structures for different fill-fraction at some wavelengths in the mid-infrared region. The doping concentration for two structures is$1 \times {10^{19}}c{m^{ - 3}}$. According to Fig. 8, the effective dielectric mode, the effective metal mode, and the dielectric region (where the permittivity of highly doped semiconductor is positive) are the same for both structures, but Type Ι and Type ΙΙ HMM, which are the transmission and reflection modes respectively, have replaced. Therefore, to design an optical device such as a switch at a given wavelength using these structures, the initial mode can be determined by selecting the structure type and the fill-fraction ratio. In this study, the transition is assumed from a reflection mode in linear regime to a transmission mode in nonlinear regime, so the arrows show the transition path from reflection mode to transmission mode in Fig. 9.

However, it should be noted that the horizontal HMM structure that needs negative longitudinal component of permittivity ($\varepsilon _\textrm{||}^{\prime}$) for transition to Type Ι HMM, utilizes resonance to realize negative $\varepsilon _\textrm{||}^{\prime}$ and suffers high losses from it. On the other hand, high speed mode transition is required to achieve a very fast switch. Our simulations reveal that the transition from reflection mode to transmission mode due to increasing the input intensity at a given wavelength in vertical structure is faster than horizontal structure because the vertical metamaterial is a non-resonant structure that can be achieved with smaller thickness than horizontal structure. A structure with the desired modes can be selected using Fig. 9 to design a switch at a given wavelength in the mid-infrared range.

## 4. High-performance switch based on nonlinear semiconductor hyperbolic metamaterials

All-optical switching is an essential function to realize optical signal processing and optical communication systems. Small footprint, high-speed operation, and low power consumption are the features of an ideal optical switch [50]. In this section, a high-performance all-optical nanoswitch based on a multilayer InAs/doped InAs hyperbolic metamaterial is presented. According to the results extracted from the previous sections, an optimal structure is considered to design our proposed ultrafast and low power all-optical switch. As shown in Fig. 10(a) a vertical multilayer hyperbolic structure consists of 3 unit cells with 200** **nm thickness of each cell is utilized to design the proposed switch at 13 μm. The thickness (L) of metamaterial in the direction of incident TM wave propagation is 150** **nm and the doping concentration of the doped InAs layer is $1 \times {10^{19}}({{ c}{{ m}^{ - 3}}} )$. The charge distribution due to band-bending in the multilayer structure is calculated by electrical analysis and then the FDTD method is used to simulate the proposed switch. Our structure switches between the reflection mode (Off) in the linear regime and the transmission mode (On) in the nonlinear regime. In the linear regime, to set the initial mode in Type ΙΙ HMM (reflection mode), the fill-fraction is assigned to (FF=0.5). In the nonlinear regime, when the electric field amplitude increases to about ${10^6}{ \; }({{ V}/{ m}} )$, the permittivity of the doped InAs becomes positive and the operating mode changes to dielectric mode (transmission mode). This transition from Type ΙΙ HMM to dielectric mode provides a high-performance optical switch. To accurately simulate the performance of the proposed switch and prevent the field scattering, we assume that the metal contacts extend as a waveguide to the sides of the metamaterial. The electric field intensity profiles in linear and nonlinear regimes are illustrated in Figs. 10(b) and 10(c), respectively. Considering the electrical analysis and solving the drift-diffusion and Poisson equations, the effect of charge redistribution on the electric field profile has been calculated. The R and T factors indicate the normalized reflected and transmitted power (the ratio of reflection and transmission power to the input power). The results show our proposed structure is a low power and high extinction ratio all-optical switch. In terms of practical implementation, the maximum amplitude of the electric field in the nonlinear regime is notably less than the breakdown threshold of all components. Our proposed semiconductor-based metamaterial is compatible with the fabrication of semiconductor devices. Moreover, because the layers are of the same material in the multilayer structure, no issues are arising from the lattice mismatch in the deposition process. The structure consisted of alternating layers of doped and undoped InAs that can be grown by molecular beam epitaxy (MBE) [43,51]. The sputtering methods and electron-beam deposition have been reported for creating metal contacts on the semiconductor layers [44,52].

## 5. Conclusion

We numerically investigate the linear and nonlinear behavior of multilayer hyperbolic metamaterials based on heavily doped semiconductors instead of metal to reduce the high ohmic losses in the mid-infrared range. The effects of fill-fraction, the doping concentration, and the thickness of multilayer on the guided modes and transmission/reflection spectra of our proposed semiconductor-based metamaterial in the vertical and horizontal structures are studied. We demonstrate that tuning the operating wavelength and guided modes are attained by changing the fill-fraction and the doping concentration. Also, the transmission and reflection rates in various wavelengths are affected by the thickness of the structure, fill-fraction, and the doping concentration. Furthermore, we show by increasing the field intensity and changing the operating regime from linear to nonlinear, the components of permittivity are modified due to the Kerr effect. Consequently, by changing the guided modes of metamaterial, optical bistability and high-performance switching are achievable. We also find that switching from reflection to transmission mode at a given wavelength needs more thickness in the horizontal structure than the vertical to realize acceptable transmission/reflection spectra and high extinction ratio. Besides, the propagation losses in horizontal structures are higher in comparison to vertical structures due to the greater thickness, higher metal loss and the need for resonance mode to obtain negative refraction. Finally, relying on the results of this research, we presented an ultrafast, low power, and high extinction ratio all-optical switch based on a vertical hyperbolic metamaterial in the mid-infrared range at the wavelength of 13 μm.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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