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

Authors: Yu Chen; Zhihao Chen; Binyi Qin; Xiaoyu Dai
Journal: Advances in Condensed Matter Physics
 (2018)

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## 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.


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## Body

## 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] (1)εω=ε∞-ωp2ω2+iγω,where ω is the angular frequency, ε∞ represents the high-frequency permittivity,and γ is the damping constant; ωp is the plasma frequency and ωp=Ne2/ε0m∗ depends on the intrinsic carrier density N, the effective mass m∗ of free carriers, the electronic charge e, and the free-space permittivity ε0. Compared to metals, the plasma frequency ωp of InSb depends strongly on the temperature T. The intrinsic carrier density N (in m−3) in InSb obeys the relationship [27](2)N=5.76×1020T3/2exp⁡-0.262kBT,where kB is the Boltzmann constant and the temperature is in Kelvin. A variation in N due to a variation in T thus changes ωp. Consequently, in the far-infrared portion of the terahertz regime, εω of InSb is very sensitive to T. 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 areεeffω and μeffω, respectively. As shown in Figure 1, we assume that the dielectric spheres possessing a relative dielectric permittivity ε1 and relative magnetic permeability μ1 are embedded in a semiconductor host medium whose relative permittivity and relative permeability are ε2ω and μ2=1, 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 εeffω and the relative effective permeability μeffω from the extended Maxwell-Garnett theory (EMG), as shown in [14](3)εeffω=ε2x3-3ifT1Ex3+3/2ifT1Eμeffω=μ2x3-3ifT1Hx3+3/2ifT1Hwhere f is the volume fraction representing the ratio of the volume of the dielectric spheres to the total volume and f=4/3πr3/a3, where r denotes the sphere radius and a is the unit cell size. T1E and T1H are the electric-dipole and magnetic-dipole components of the scattering T matrix of a single sphere, respectively. They can be written as (4)T1Eω=j1x1x2j1x2′ε1-j1x2x1j1x1′ε2h1+x2x1j1x1′ε2-j1x1x2h1+x2′ε1,(5)T1Hω=j1x1x2j1x2′μ1-j1x2x1j1x1′μ2h1+x2x1j1x1′μ2-j1x1x2h1+x2′μ1,where j1 is the spherical Bessel function and h1+ is the spherical Hankel function for l=1, respectively. Xx′=dX/dx. x1 and x2 stand for the sphere size parameter, x1=ω/crε1μ1=2πr/λ1 and x2=ω/crε2μ2=2πr/λ2,and λ1 and λ2 are the wavelengths in the sphere medium and host medium, respectively.Figure 1
A schematic of a dielectric sphere is embedded into the semiconductor host.r is the radius of the dielectric sphere and a is the unit cell size.


## 3. Numerical Results and Discussion

Figure2 shows the simulation results of εeff,μeff,neff for T=220K, f=0.2, and r=60um using [3–5]; here we have assumed that ε∞=15.68, m∗=0.015me kg, me=9.1×10-31, and γ=2π×0.05 THz [27], and the gallium phosphide (GaP) is assumed for the nonmagnetic dielectric sphere, and the permittivity ε1=12.25×1+i×10-3.Figure 2
The numerical results of the real part (Re) and imaginary (Im) part ofεeff,μeff,neff of the composite DS/NPH effective metamaterials.
(a) (b) (c)We can see from the figures that the real part of the effective permittivityεeff is negative from 0.58 THz to 0.9 THz, and the real part of the effective permeability μeff 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 εeff is far larger than μeff; however the imaginary parts of εeff,μeff,neff 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 temperatureT on the properties of εeff,μeff,neff, 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 neff will become bigger and bigger as the temperature increases. Clearly, the optical properties of the MMs, such as εeff,μeff,neff, all change significantly with temperature. For proving these properties clearly, in Figure 3(b), we have shown the real part of neff as a function of temperature from 210 K to 350 K. As expected, neff 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.Figure 3
The thermotunable negative-index metamaterials. (a) The real part of the refractive-indexneff at different temperatures ranging from 220 to 350 K. (b) The real part of the refractive-index neff at ω/2π=0.7THz for different temperatures ranging from 210 to 350 K.
(a) (b)Further, we find that the properties ofεeff,μeff,neff 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 neff 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.Figure 4
The real part ofneff as a function of the radius of the dielectric sphere, where f=0.2 and T=220K.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 fromN=1.977×1021m−3 at 220 K to N=5.065×1022 at 350 K, which will lead to the plasma frequency of the semiconductor host altering from ωp=2.046×1013 at 220 K to ωp=1.036×1014 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 r=1 um and T=350K, 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.Figure 5
Calculated real and imaginary part of the complex permittivity, permeability, and refractive-index of the composite MMs in the terahertz region between 8 and 56 THz. Wherer=1um and T=350K, the other parameters are same as in Figure 2.
(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.



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