Terahertz Metamaterials with Semiconductor Split-Ring Resonators for Magnetostatic Tunability

We studied a metasurface constituted as a periodic array of semiconductor split-ring resonators. The resonance frequencies of the metasurface excited by normally incident light were found to be continuously tunable in the terahertz regime through an external magnetostatic field of suitable orientation. As such metasurfaces can be assembled into 3D metamaterials, the foregoing conclusion also applies to metamaterials comprising semiconductor split-ring resonators.


Introduction
Negatively refracting metamaterials have been a subject of recent enormous interest, in both the physics and the engineering communities, because they possess electromagnetic response characteristics that are not (at least, widely) displayed by natural materials [1, 2,3]. Generally, these metamaterials are composite materials comprising metallic inclusions of specific shapes and sizes immersed in some homogeneous host medium, and convey the promise of a wideranging set of applications.
Metamaterial-based devices could come to include filters, modulators, amplifiers, transistors, and resonators, among others [4,5]. The usefulness of such a device could be extended tremendously if the metamaterial's response characteristics can be dynamically tuned. Tunability strategies examined thus far are electrical control [6,7,8], magnetostatic control [9,10], and optical pumping [11]. In all of cited publications, the inclusions are splitring resonators (SRRs) made of some metal. Tunability is introduced by ensuring that another component (e.g., the substrate on which the SRRs are printed or another type of inclusions) is made of an electro-optic material, liquid crystal, ferrite, etc.
In this letter, we suggest a new type of tunable metamaterials. A metamaterial of this type comprises SRRs that are not metallic but, instead, are made of a material whose electromagnetic response properties can be tuned by an external agent. For the sake of illustration, we chose to focus on a tunable, terahertz, metamaterial that is controllable by an externally applied magnetostatic (or quasimagnetostatic) field.

Analysis and Numerical Results
The essence of the chosen metamaterial is a metasurface [12]: a planar array of semiconductor split-ring resonators (SRRs) periodically printed on an isotropic dielectric substrate. Figure 1a shows a single SRR with linear dimensions in the plane ranging from 2 µm to 36 µm, and with a thickness of 200 nm. The SRRs are printed, as shown in Fig. 1b, on a square lattice of period 60 µm. The SRRs are assumed to be made of a doped semiconductor, such as InAs, whose relative permittivity obeys the Drude model in the absence of an externally applied magnetostatic field [13,14]. As a typical isotropic dielectric substrate would only slightly shift the resonances but not affect other characteristics significantly [15], we set its electromagnetic properties to be that of free space (i.e., vacuum) in our numerical work.
Computer simulations of the spectral response of the chosen metasurface were performed using the commercial software CST Microwave Studio TM 2006B, which is a 3D full-wave solver that employs the finite integration technique. The planar SRR array was taken to be entirely surrounded by air and open boundary conditions were employed along the propagation direction. Without the applied magnetostatic field, the doped semiconductor's relative permittivity scalar ε is given as a function of the angular frequency ω by where ∞ ε represents the high-frequency value; the plasma frequency Let the chosen metasurface be oriented parallel the xy plane with the two gaps of every SRR aligned parallel to the x axis. The SRR gaps would resonate if the incident electric field were to be oriented along the y axis (i.e., perpendicular to the two gaps in each SRR).
We calculated the transmittance spectrum of the metasurface when illuminated by a plane wave with its wave vector oriented parallel to the z axis and its electric field oriented parallel to the y axis. Fig. 1c shows that a transmittance dip exists at 1.59 THz frequency. This dip can be attributed to a resonance of the SRRs: as shown in Fig. 1d, the electric field is concentrated in both gaps of every SRR at this frequency. Fig. 1e displays the vector plot of the current density at 1.59 THz. If both gaps of every SRR were closed, the transmittance dip at 1.59 THz would disappear, as can be gathered from the transmittance spectrum presented in Fig. 1f.
In order to obtain a tunable response from the chosen metasurface, we considered the application of an external magnetostatic field B 0 to affect the resonance in the transmittance spectrum displayed in Fig. 1c. Generally, without the applied magnetostatic field, the dielectric response of a semiconductor is described by a scalar ε(ω), such as provided in Eq.
(1). However, when B 0 is applied, the scalar ε(ω) has to be replaced by the tensor ) ( ω ε . In the Faraday configuration, B 0 is aligned parallel to the wave vector of the incident plane wave (i.e., along the z axis). With the assumption that B 0 is spatially uniform, the nonvanishing components of ) ( ω ε are [13]: where is the cyclotron frequency. The off-diagonal components ε  Denoting the resonance angular frequency by ω 0 , we fitted the data in Fig. 2b to the formula ω 0 = 0.27 + 2.04 exp(−B 0 /2.11) . This formula indicates that the resonance frequency decays exponentially as the applied magnetostatic field increases in magnitude. One way to understand the foregoing trend is as follows. In the Faraday configuration, plane wave propagation in a semiconductor depends on [16], i.e.,  Figure 4a shows that the transmittance spectra in the first Voigt configuration are qualitatively similar to that for the Faraday configuration in Fig. 2a Figure 5 shows that real part of ⊥ ñ decreases with increasing B 0 , which implies the same correlation with the resonance frequency as deduced for the Faraday configuration. In the second Voigt configuration, B 0 is aligned parallel to the electric field of the incident plane wave (i.e., B 0 is oriented along the y axis), so that the nonvanishing components of ) ( ω ε now are [13]