Advances in Condensed Matter Physics

Volume 2018, Article ID 2794161, 6 pages

https://doi.org/10.1155/2018/2794161

## Thermotunable Terahertz Negative-Index Metamaterials with Dielectric Spheres Embedded in Semiconductor Host

^{1}International Collaborative Laboratory of 2D Materials for Optoelectronics Science and Technology, Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province, College of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China^{2}School of Electronics and Communication Engineering, Yulin Normal University, Yulin 537000, Guangxi, China

Correspondence should be addressed to Xiaoyu Dai; moc.621@iaduyoaix

Received 4 June 2018; Accepted 13 August 2018; Published 20 September 2018

Academic Editor: Charles Rosenblatt

Copyright © 2018 Yu Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.