1. Introduction

Metamaterials (MMs) are a distinct kind of electromagnetic materials and have been intensively studied in the past several years for some fascinating phenomenon, such as negative refraction [1, 2], perfect lens [3, 4], and transparency [5–7]. Its properties mainly result from subwavelength details of the metallic or dielectric element rather than their chemical composition. Recently dynamic control of the MMs’ response attracts lots of interest through the electric or optical methods [8–11]. It is well known that vanadium dioxides ($V{O}_2$) show the transition behavior from insulator to metal at around 340 K. The lattice structure is transformed from monoclinic to tetragonal structures as the temperature rises, and the conductivity of $V{O}_2$ increases by several orders of magnitude during the transition. So $V{O}_2$ has wide potential for temperature sensitive photonic, electronic, and thermic devices. Thus, combining MMs with $V{O}_2$ thin film or small particles is a promising way to dynamically modulate electromagnetic properties of some practical devices [12–24]. In 2010, W. Huang et al. fabricated a composite metamaterial of $goldstrip/V{O}_{2}spacer/goldstrip$ sandwich structure to study the optical response [15]. Their results indicated that the designed nanostructure with $V{O}_2$ spacers can be used as a dynamically temperature-controlling optical switch. In 2012, M. A. Kats et al. showed perfect absorption in a system comprising a single lossy $V{O}_2$ layer with ultra-thin thickness much smaller than the incident wavelength on the sapphire substrate [16]. The design leads to 99.75% absorption at $\lambda =11.6\mu m$ in the vicinity of the $V{O}_2$ insulator-to-metal phase transition. In 2015, H. Kocer et al. demonstrated thermally tunable short-wavelength infrared resonant absorbers by heating up hybrid gold-$V{O}_2$ nanostructure arrays above the phase transition temperature of $V{O}_2$ [17]. In 2017, Z. Zhu et al. realized an efficient absorber meta-device with each unit cell consisting of a gold bow-tie antenna with a small $V{O}_2$ patch placed in its feed gap [18]. The device has an experimentally measured tuning range as large as 360 nm in the near infrared region and a modulation depth of 33% at the resonant wavelength. In the terahertz frequency region, the dielectric constant of $V{O}_2$ varies by several orders of magnitude when the film undergoes phase transition. In this paper, we propose a composite MM with sandwich nanostructures $metalcross/Si{O}_{2}spacer/V{O}_{2}film$ to dynamically modulate the absorption property.

2. Numerical calculations and discussions

In this study, we demonstrate conductivity-tunable terahertz resonant absorbers by utilizing a phase change material ($V{O}_2$) to achieve dynamic control of metadevices. As shown in Fig. 1, the metadevice is based on the perfect absorber architecture with each unit cell consisting of a metal cross and a $V{O}_2$ film [25–27]. These two layers are separated by a $Si{O}_2$ spacer layer. The thicknesses of metal crosses, $Si{O}_2$ layer, and $V{O}_2$ film is 0.02 $\mu m$, 70 $\mu m$, and 0.5 $\mu m$. The chosen width, length, and period of metal crosses in simulation are 2.9 $\mu m$, 155 $\mu m$, and 174 $\mu m$. In order to avoid the Fabry-Perot resonance caused by the finite thickness of the underlying $Si{O}_2$ substrate, the thickness of the underlying $Si{O}_2$ substrate is assumed to be infinite in simulation. A variable conductivity of $V{O}_2$ is assumed to simulate the phase transition effect. The optical permittivity of $V{O}_2$ in terahertz range can be described by Drude model as follows:$\epsilon (\omega )={\epsilon }_{\infty }-\frac{{\omega }_p^2(\sigma )}{{\omega }^2+i\gamma \omega }$, where ${\epsilon }_{\infty }=12$ is the permittivity at high frequency, ${\omega }_p(\sigma )$ is the conductivity dependent plasma frequency and $\gamma$ is the collision frequency [22–24]. In addition, both ${\omega }_p^2(\sigma )$ and $\sigma$ are proportional to free carrier density. The plasma frequency at $\sigma$ can be approximately expressed as ${\omega }_p^2(\sigma )=\frac{\sigma }{{\sigma }_0}{\omega }_p^2({\sigma }_0)$ with${\sigma }_0=3\times {10}^3{\Omega }^{-1}c{m}^{-1}$, ${\omega }_p({\sigma }_0)=1.4\times {10}^{15}rad/s$, and $\gamma =5.75\times {10}^{13}rad/s$ which is assumed to be independent of $\sigma$. The relative permittivity of $Si{O}_2$ with negligible loss is set to 3.8 at terahertz frequencies [28, 29]. The relative permittivity of gold is described by a Drude model ${\epsilon }_{Au}=1-{\omega }_p^2/\omega (\omega +i\Gamma )$ with plasma frequency ${\omega }_p=1.37\times {10}^{16}rad/s$ and collision frequency $\Gamma =1.2\times {10}^{14}rad/s$ [30].

 

Fig. 1 Schematic of the $V{O}_2$-based tunable terahertz MM absorber studied in this paper. The red (cyan) part is $V{O}_2$ ($Si{O}_2$) with the thickness of $t_2=0.5\mu m$ ($t_1=70\mu m$). The yellow part is gold with the thickness of 0.02$\mu m$, width of 2.9 $\mu m$, length of 155 $\mu m$. The substrate is $Si{O}_2$ which is infinite in simulation.

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Full-wave electromagnetic simulations are performed using finite-element-method (comsol multiphysics). All simulations use extremely adaptive fine mesh settings. A unit cell of the investigated structure is simulated using periodic boundary condition. The total absorptance (A) in the designed structure is calculated by $A=1-R-T=1-{\left|S_{11}\right|}^2-{\left|S_{21}\right|}^2$, where R is the total reflectance and T is the transmittance. As shown in Fig. 2, the simulated results clearly tell that the values of reflectance, transmittance, and absorptance undergoe a noticeable shift as the conductivity increases. Figure 2(c) shows the absorptance spectra as a function of frequency and conductivity under normal incidence. It is obvious that as the conductivity increases, the absorptance increases without changing the operating frequency range. The corresponding absorptance increases from 30% to nearly 100% as conductivity increases from $10{\Omega }^{-1}c{m}^{-1}$ to$2000{\Omega }^{-1}c{m}^{-1}$. In the case of$\sigma =2000{\Omega }^{-1}c{m}^{-1}$, the absorptance can reach 100%. The perfect absorptance is realized by the interference of the fields between metal crosses and $V{O}_2$ film. From the absorptance spectra, it is observed that 90% absorptance bandwidth reaches 0.33 THz with a central frequency of 0.55 THz under normal incidence when $\sigma =2000{\Omega }^{-1}c{m}^{-1}$. The normalized 90% bandwidth with respect to the central frequency is 60%. From the calculated results, we can see that the conductivity of $V{O}_2$ has a remarkable impact on the absorptance properties of the designed system, which enables the creation of terahertz absorber with dynamically flexible behaviors. As shown in Fig. 3, The reason leading to this phenomenon is mainly caused by the variations of permittivity of $V{O}_2$, it experiences an optical transition from dielectric to metal when the conductivity changes from $10{\Omega }^{-1}c{m}^{-1}$ to $2000{\Omega }^{-1}c{m}^{-1}$. The larger the conductivity, the better the metallic behavior, and the absorptance becomes more striking. Therefore, such a MM absorber with the flexible ability can be used as a terahertz broadband attenuator or modulator in absorption or transmission mode.

 

Fig. 2 Simulated reflectance (a), transmittance (b), and absorptance (c) spectra of the designed system at a series of conductivity values of $V{O}_2$ under normal incidence.

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Fig. 3 The calculated real part of permittivity of $V{O}_2$ at a series of conductivities.

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Furthermore, the dependence of absorptance of the proposed absorber on the thickness ($t_2$) of $V{O}_2$ is investigated. To briefly present this property, we only discuss the absorptance analysis under normal incidence. Figure 4 illustrates the relation between absorptance and $t_2$ with the conductivity of $V{O}_2$ fixed as $\sigma =2000{\Omega }^{-1}c{m}^{-1}$. The calculated results tell that absorptance of the designed system under normal incidence is closely depend on $t_2$ when it is smaller than 0.5 $\mu m$. As $t_2$ increases from 0.01 $\mu m$ to 0.5 $\mu m$, absorptance gradually enhances. When the thickness of $V{O}_2$ is larger than 0.5 $\mu m$, absorptance becomes stable because $V{O}_2$ is enough thick to prevent transmission.

 

Fig. 4 Thickness dependence of absorptance under normal incidence when $\sigma =2000{\Omega }^{-1}c{m}^{-1}$.

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To easily and clearly understand the effect of perfect absorption when$\sigma =2000{\Omega }^{-1}c{m}^{-1}$, the effective electromagnetic parameters of MM absorber are retrieved [31]. The magnetic component of incident wave couples to this system and thus generates antiparallel currents between metal crosses and $V{O}_2$ film, resulting in resonant $\mu$ response. By changing the conductivity of $V{O}_2$, we can simultaneously tune $\epsilon$ and $\mu$ such that it is possible to approximately match the impedance ($z=\sqrt{\mu /\epsilon }$) to free space, and then minimize the reflectance at a specific frequency. From Fig. 5, it is found that at the frequency of the resonant absorption peak, the permittivity and permeability is approximately equal which means that the impedance of the designed system is almost matched to that of free space, so the reflectance is negligible. Meanwhile, the large imaginary part of the refractive index would generate high absorption.

 

Fig. 5 Retrieved effective physical parameters (a) permittivity, (b) permeability, (c) refractive index, and (d) impedance in the case of perfect absorption when$\sigma =2000{\Omega }^{-1}c{m}^{-1}$.

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Figure 6 gives the absorptance evolution of the structure by tuning of polarization angle from $0^{°}$ to ${90}^{°}$ in steps of $5^{°}$. With increasing polarization angle, the absorptance is completely unchanged. The symmetry of the designed system leads to the polarization-insensitive behavior, and this polarization-insensitive absorber would be helpful in numerous applications. Angular independence is very important for an absorber since incident wave in general case is randomly oblique. Figure 7 plots the spectral absorptance of the MM absorber as a function of incident angle and frequency. As shown in Fig. 7, it is found that the designed absorber shows excellent performances with the stable absorptance and working bandwidth over a wide range of oblique incident angle for both transverse electric (TE) and transverse magnetic (TM) polarizations. The absorptance is almost insensitive to incident angle variations up to ${60}^{°}$ for both TE and TM polarizations, which is beneficial to practical application over a wide range of incident angle. With increasing angles of TE polarization, beyond ${60}^{°}$ there is a noticeable decrease in the absorptance as the incident magnetic field can no longer efficiently drive circulating currents between metal crosses and $V{O}_2$ film. For TM polarization, the maximum absorptance remains great even for the angle of ${75}^{°}$. This is because the direction of magnetic field of the incident wave is unchanged with various incident angles and it can efficiently drive the circulating currents at all angles of incidence, which is important to maintain impedance matching. The incident angle- and polarization-roust characteristics could have great potential applications in terahertz sensing, detecting, and optoelectronic devices.

 

Fig. 6 Simulated absorptance spectrum of perfect absorption phenomenon with tuning of polarization angles when $\sigma =2000{\Omega }^{-1}c{m}^{-1}$ under normal incidence.

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Fig. 7 Incident angular dependence of perfect absorptance phenomenon for TE polarization (a) and TM polarization (b) when $\sigma =2000{\Omega }^{-1}c{m}^{-1}$.

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3. Conclusions

To summarize, we propose a broadband conductivity-tunable absorber by utilizing $V{O}_2$ in the terahertz range. A terahertz-tunable absorptance involving the large shift in resonant peak absorptance has been observed from 30% to 100% at a wide band, and 60% normalized bandwidth of 90% absorptance can be obtained. The absorption of the proposed MM absorber is insensitive to the incident angle of both TE and TM polarizations. And the absorber design in our work can be easily scalable to other terahertz and infrared regions. This work may provide the potential of highly active and tunable MM devices in the terahertz regime. A wide range of applications such as modulator, sensor, and active filter can be expected.