#### Abstract

A possibility to realize thermotunable isotropic negative-index metamaterials up to terahertz regime is theoretically investigated. The proposed composite metamaterials consist of dielectric spheres embedded randomly in a semiconductor host medium. As the variation in the intrinsic carrier density in InSb due to a variation in temperature thus changes the plasma frequency, we show theoretically the provided composite metamaterials whose index of refraction is thermally tunable and potentially less propagation loss. Furthermore, the effects of the radius of the dielectric sphere on the modulation frequency and bandwidth are discussed. Finally, we find that the design parameters for the composites can be scaled for application in the higher frequency regions.

#### 1. Introduction

Negative-index metamaterials (NIMs), with simultaneously negative permittivity and permeability, were introduced by Veselago nearly forty years ago [1] and have recently received much attention in the literature owing to great potential for applications, such as perfect lens [2], optical cloaking [3], and perfect absorber [4]. Recently, metamaterials have also been used to regulate the light field [5–7]. There has been much progress in the development of the metamaterials (MMs) from the microwave frequencies to infrared and optical frequencies. In the microwave regime, the MMs can be constructed as a combined arrays consisting of metal wires and split-ring resonators [8]. MMs at the infrared and optical frequencies were accomplished with pairs of metal rods [9] and for the inverted system of pairs of dielectric voids in metal [10]. Most of the MMs designs are constructed by the metallic elements, which incur many drawbacks such as high loss, very narrow bandwidth, and highly anisotropic. An alternative scheme was proposed based on the resonance in individual nonmagnetic dielectric scatterer with very high permittivity [11–14]. However, dielectrics with extremely high permittivity also suffer from large damping. Recently, Seo et al. have shown that isotropic NIMs up to optical frequencies can be realized with dielectric spheres embedded randomly in a negative permittivity host (NPH) medium [15]. As a result, the DS/NPH shows more robust characteristics over the DS/DH in terms of fabrication tolerance, bandwidth, and propagation loss.

Most provided NIMs usually display a narrow operational bandwidth, and the operational frequency is not tunable. The tunable MMs are obviously very highly desired and potential expectations for future optical devices are, for example, spatial light modulators and tunable optical filters. There are various strategies reported for tuning the magnetic resonance of the permeability include electrical control [16, 17], magnetically control [18], and temperature changing [19, 20]. Generally, tunability is introduced by ferromagnetic materials [21, 22], ferroelectric [13, 23], liquid crystal [20, 24, 25], etc. However, the abovementioned tunable NIMs mostly operated at the low frequencies or high losses. In this paper, we will propose the designs of the thermotunable and low loss isotropic negative-index metamaterials up to terahertz region, which consists of dielectric spheres embedded randomly in a semiconductor host medium.

#### 2. The Permittivity of the Semiconductor and the Effective Medium Theory

In the terahertz regime, the complex-valued relative permittivity of InSb is given by the simple Drude model [26] where is the angular frequency, represents the high-frequency permittivity,and is the damping constant; is the plasma frequency and depends on the intrinsic carrier density , the effective mass of free carriers, the electronic charge , and the free-space permittivity . Compared to metals, the plasma frequency of InSb depends strongly on the temperature . The intrinsic carrier density (in m^{−3}) in InSb obeys the relationship [27]where is the Boltzmann constant and the temperature is in Kelvin. A variation in due to a variation in thus changes . Consequently, in the far-infrared portion of the terahertz regime, of InSb is very sensitive to . Hence, for NIMs comprising InSb host and dielectric sphere, we can expect that temperature variations can cause substantial variations in the optical response characteristics.

The composite material consisting of spherical scatters can be described as a homogeneous medium in the subwavelength limit and the effective electric permittivity and effective magnetic permeability are and , respectively. As shown in Figure 1, we assume that the dielectric spheres possessing a relative dielectric permittivity and relative magnetic permeability are embedded in a semiconductor host medium whose relative permittivity and relative permeability are and , respectively. Assuming that the dielectric spheres are located in the host medium randomly and homogeneously, in the quasistatic limit, we can derive the relative effective permittivity and the relative effective permeability from the extended Maxwell-Garnett theory (EMG), as shown in [14]where is the volume fraction representing the ratio of the volume of the dielectric spheres to the total volume and , where denotes the sphere radius and is the unit cell size. and are the electric-dipole and magnetic-dipole components of the scattering T matrix of a single sphere, respectively. They can be written as where is the spherical Bessel function and is the spherical Hankel function for , respectively. . and stand for the sphere size parameter, and ,and and are the wavelengths in the sphere medium and host medium, respectively.

#### 3. Numerical Results and Discussion

Figure 2 shows the simulation results of for K, , and um using [3–5]; here we have assumed that , kg, , and THz [27], and the gallium phosphide (GaP) is assumed for the nonmagnetic dielectric sphere, and the permittivity .

**(a)**

**(b)**

**(c)**

We can see from the figures that the real part of the effective permittivity is negative from 0.58 THz to 0.9 THz, and the real part of the effective permeability is negative from 0.65 THz to 1.0 THz; hence the double region is between about 0.65THz and 0.9 THz and the bandwidth is about 0.25 THz, as shown in the Figure 2(c). Also we can find that the value of is far larger than ; however the imaginary parts of are all much smaller than the composite MMs constructed by the metallic sphere when the incident wave frequency is far away from the resonance regions.

In order to discuss the effects of the temperature on the properties of , in Figure 3, we have plotted the frequency-dependent refractive-index of composite MMs at various temperatures. We can find from Figure 3(a) that the resonance dips are almost kept fixed as the temperature increases; however, the value of the real part of will become bigger and bigger as the temperature increases. Clearly, the optical properties of the MMs, such as , all change significantly with temperature. For proving these properties clearly, in Figure 3(b), we have shown the real part of as a function of temperature from 210 K to 350 K. As expected, can be significantly enlarged by raising the temperature. The temperature-dependent optical properties of composite MMs can be attributed directly to the thermally sensitiveness of the plasma frequency for the permittivity of the semiconductor InSb.

**(a)**

**(b)**

Further, we find that the properties of in our composite structure depend on the radius of the dielectric sphere strongly; hence the spectral position of the resonance point can be tuned by varying the radius of the dielectric sphere. For proving this, we have shown the dependence of the real part of on the radius of the dielectric sphere in Figure 4. It is shown that when the radius of the sphere changes from 50 um to 80 um, the resonance frequency points of the negative refractive-index will move from 0.8 THz to 0.5 THz and the value of the resonance negative refractive-index point will become larger and larger as the radius of the dielectric sphere increases.

We have shown that the low loss and thermotunable NIMs can be realized at the terahertz frequencies; however the design of the material is scalable within a large dynamic range of operating frequencies as long as we change the radius of the dielectric sphere and the temperature simultaneity. First, the intrinsic carrier density in InSb can be changed from m^{−3} at 220 K to at 350 K, which will lead to the plasma frequency of the semiconductor host altering from at 220 K to at 350 K; therefore if only we adjust the scale of the radius of the dielectric sphere, we can obtain the double negative metamaterials at higher frequency. Figure 5, for example, shows the corresponding real part of the refractive-index from 8 THz to 56 THz, where um and K, the other parameters are the same as in Figure 2. Again, it clearly demonstrates that one can obtain a bulk material of double negative metamaterials by embedding the dielectric spheres in a semiconductor host medium randomly at the terahertz even higher frequency. Moreover, the frequency bandwidth also becomes very large to reach 32 THz.

**(a)**

**(b)**

**(c)**

#### 4. Conclusions

In conclusion, we have provided a scheme to realize a thermotunable and less loss isotropic negative-index metamaterials up to terahertz regime by embedding the dielectric spheres in a semiconductor host medium randomly. The composite metamaterial is sensitive to the outer temperature, which makes the double negative frequency regions tunable. Furthermore, the radius of the dielectric sphere will also make the frequency ranges of the double regions tunable. Moreover, by scaling the size of the spheres and outer temperature, we can vary the operating frequency to higher frequency. Such reconfigurable and thermotunable metamaterials have very promising future for designing tunable devices, such as tunable optical cloaking, tunable spatial light modulators, and tunable novel microwave antenna.

#### Data Availability

The [DATA TYPE] data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants nos. 61505122, 61505124), the Natural Science Foundation of SZU (Grant nos. 2016031, 2017018), the program of Fundamental Research of Shenzhen Science and Technology Plan (Grant no. JCYJ20160422152152634), the project supported by Guangdong Natural Science Foundation (Grant no. 2016A030310065), the Natural Science Foundation Guangdong Education Department (2015KQNCX146), and Yulin Normal University Research Grant (Grant no. 2016YJKY06).