Abstract

In this paper, we propose and demonstrate a switchable terahertz metamaterial absorber with broadband and multi-band absorption based on a simple configuration of graphene and vanadium dioxide (VO2). The switchable functional characteristics of the absorber can be achieved by changing the phase transition property of VO2. When VO2 is insulating, the device acts as a broadband absorber with absorbance greater than 90% under normal incidence from 1.06 THz to 2.58 THz. The broadband absorber exhibits excellent absorption performance under a wide range of incident and polarization angles for TE and TM polarizations. Moreover, the absorption bandwidth and intensity of the absorber can be dynamically adjusted by changing the Fermi energy level of graphene. When VO2 is in the conducting state, the designed metamaterial device acts as a multi-band absorber with absorption frequencies at 1 THz, 2.45 THz, and 2.82 THz. The multi-band absorption is achieved owing to the fundamental resonant modes of the graphene ring sheet, VO2 hollow ring patch, and coupling interaction between them. Moreover, the multi-band absorber is insensitive to polarization and incident angles for TE and TM polarizations, and the three resonance frequencies can be reconfigured by changing the Fermi energy level of graphene. Our designed device exhibits the merits of bi-functionality and a simple configuration, which is very attractive for potential terahertz applications such as intelligent attenuators, reflectors, and spatial modulators.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Terahertz metamaterial technology, has an operational frequency in the range of 0.1–10 THz, and has broad application prospects in many fields such as super-lens [13], perfect absorbers [47], and invisible cloaks [89]. Terahertz absorber, a device with high absorption rate for incident electromagnetic waves, are in an urgent need in wireless security, radar communication, and selective transceiver applications [1014]. Due to the difficulty in achieving perfect absorbing in the terahertz band for natural materials, metamaterial absorbers have become a research focus in this field. The metamaterial absorber based on an electromagnetic resonator design was first proposed in 2008 [15], which achieved a peak absorption greater than 88% at 11.5 GHz. Subsequently, various metamaterial absorbers have been studied and designed [1619], However, the spectral position and corresponding functions of most absorbers are usually fixed, which significantly limits the scope of applications. Therefore, tunable multi-function metamaterials are desirable for terahertz intelligent systems.

Graphene is composed of carbon atoms in the planar hexagonal lattice. It is a two-dimensional monolayer material and a viable candidate for tunable devices owing to its unique features [2021]. The graphene surface can excite surface plasmons in the mid-infrared and terahertz bands, and its surface conductivity can be manipulated by adjusting the Fermi energy level under a fixed structure [2224]. Hence, graphene-based tunable terahertz absorbers have been widely reported in recent years. Various graphene-based broadband absorbers with periodically patterned squares, disks, and ribbons have also been investigated [5,16,25]. Furthermore, dual-band or multi-band absorbers [2628], ultra-bandwidth absorbers with multilayer graphene [2932], and simple structured absorbers [3336] have received considerable attention. However, most of these terahertz absorbers are designed for a single functionality, this inherent drawback has been unfavorable to the development of many intelligent systems. In radar communication systems, for example, it is important to selectively transmit and receive terahertz signals. In electromagnetic radiation interference system, flexibly switching the interference frequency band and adjusting the interference amplitude is also necessary. Thus, developing a switchable multi-functional terahertz absorber is urgently needed.

Besides graphene, vanadium dioxide (VO2) films exhibit excellent and unique reversible insulating-conducting phase transition properties. At room temperature, VO2 has a monoclinic crystal structure, which transforms into a tetragonal rutile structure when the temperature is increased. When the temperature is decreased to the phase transition temperature, VO2 transforms into the monoclinic crystal structure again. Notably, when the insulating-conducting phase transition occurs, the electrical [37] and optical properties [3839] of VO2 change drastically. In 2006, Jepsen et al. reported that at the ambient temperature of approximately 340 K, the conductivity of VO2 can undergo a significant and sudden change of five orders of magnitude (40–4×105 S/m); further, a reversible phase transition simultaneously occurs from the insulating state to the conducting state [40]. In addition to the thermal approach [41], electric field [42], light [43], terahertz field [44], and other external excitations can also cause the insulating-conducting phase transition properties. Recently, based on hybrid VO2 metamaterials, transparent absorbers [4549], dual broadband absorbers [50], absorption and polarization conversion devices [51,52], and broadband and narrowband absorption conversion absorbers [10,53] have been presented. Furthermore, other devices based on active metamaterials have been developed in the last years [5458]. Owing to its excellent photoelectric functional performance, VO2 has tremendous potential applications in THz absorbers.

In this study, we utilized the manipulated property by Fermi energy level of graphene and the phase transition property of VO2 to demonstrate a switchable and tunable bi-functional terahertz absorber with broadband and multi-band absorption. When VO2 is in the insulating state, broadband absorption properties are achieved from 1.06 THz to 2.58 THz with optimized geometry. Alternatively, when VO2 is in the conducting state with the same geometry, the designed device acts as an adjustable multi-band absorber at the absorption frequency of 1 THz, 2.45 THz, and 2.82 THz. The proposed switchable metamaterial absorber exhibits the advantages of bi-functionality, incident angle and polarization independence, and a simple configuration.

2. Design and method

Figure 1 shows a three-dimensional (3D) schematic of the proposed switchable metamaterial absorber, which is periodic in x and y directions. The basic unit cell of the proposed absorber consists of five layers: VO2 hollow ring patch, graphene ring sheet, polysilicon sheet, polyethylene cyclic olefin copolymer (Topas) spacer, and continuous metallic reflector. Topas is a transparent and stiff amorphous thermoplastic copolymer with superior optical properties for advanced terahertz applications, which also has excellent heat resistance, near zero hygroscopicity and high stability. In simulation, the Topas spacer has a relative dielectric permittivity of 2.35 and is assumed to neglect material loss and dispersion. Its thickness (ttopas) and period (p) are set to 27 µm and 44 µm, respectively. Polysilicon, a form of elemental silicon, is an extremely important and excellent semiconductor material, which can act as an electrostatic gating sheet of bias voltage Vg to regulate the Fermi energy level of the graphene sheet in the proposed absorber. In addition, in order not to affect the performance of the absorber, the thickness of polysilicon should be as thin as possible. In simulation, the polysilicon sheet has a relative dielectric permittivity of 3 and is embedded in the Topas spacer 20 nm below the graphene ring sheet, and the thickness of polysilicon is set to tp=20 nm. Furthermore, based on the Drude model [59], a continuous metallic reflector with a thickness of tAu=0.5 µm is set at the bottom layer, and its thickness is considerably greater than the skin depth of electromagnetic waves to ensure complete suppression of the terahertz wave transmission.

 figure: Fig. 1.

Fig. 1. Three-dimensional schematic of the proposed switchable and tunable terahertz metamaterial absorber with broadband and multi-band absorption.

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The finite integration technique (FIT) is used to numerically calculate and analyze the properties of the proposed switchable metamaterial absorber. In the simulation, the Kubo formula of ${\sigma _g} = {\sigma _{intra }} + {\sigma _{inter }}$ (unit:S) with intraband and interband [57] contributions is used to determine the surface conductivity of graphene.

$${\sigma _{\textrm{intra }}}(\omega ,{\mu _c},\Gamma ,T) = \frac{{j{e^2}}}{{\pi {\hbar ^2}(\omega - j2\Gamma )}}\int\limits_0^\infty {\frac{{\partial {f_d}(\xi ,{\mu _c},T)}}{{\partial \xi }}} - \frac{{\partial {f_d}( - \xi ,{\mu _c},T)}}{{\partial \xi }})\xi \partial \xi, $$
$${\sigma _{\textrm{inter }}}(\omega ,{\mu _c},\Gamma ,T) = \frac{{j{e^2}(\omega - j2\Gamma )}}{{\pi {\hbar ^2}}}\int\limits_0^\infty {\frac{{{f_d}(\xi ,{\mu _c},T) - {f_d}( - \xi ,{\mu _c},T)}}{{{{(\omega - j2\Gamma )}^2} - {{4\xi } / {{\hbar ^2}}}}}} \partial \xi, $$
where ${f_d}(\xi ,{\mu _c},T) = {({e^{{{^{(\xi - {\mu _c})}} / {{k_B}T}}}} + 1)^{ - 1}}$ is the Fermi energy-Dirac distribution, $\Gamma = 2{\tau ^{ - 1}}$ is the phenomenological scattering rate, $\tau$ is the relaxation time, $\hbar$ is the reduced Plank’s constant, and ${k_B}$ is the Boltzmann’s constant. Additionally, $\omega$, $\xi$, ${\mu _c}$, and $T$ denote the radian frequency, energy, Fermi energy level, and absolute temperature, respectively. In this study, the graphene is modeled using the commercial CST Microwave Studio simulation software which possess the created graphene material for optical applications. The Fermi energy level of graphene (${\mu _c}$) is assumed to 0.7 eV, and the relaxation time ($\tau$) of graphene is assumed to be 1.0 ps and 0.1 ps for multi-band and broadband absorption, respectively. Further, the thickness of graphene is set to 0.34 nm, and the graphene ring sheet radius (Rg) is 18 µm, which is connected using g=1-µm-width graphene strips for electric-continuity and gating convenience. It should be noted that to clearly depict the structure diagram of the proposed absorber, some scaling is done in Fig. 1, the actual absorber’s geometry is consistent with that given in this paper.

In the terahertz range, the dielectric permittivity of VO2 can be represented using the Drude model [60,61]:

$$\varepsilon (\omega ) = {\varepsilon _\infty } - \frac{{{\omega _p}^2(\sigma )}}{{{\omega ^2} + i\gamma \omega }},$$
where ${\varepsilon _\infty }$ = 12 is the dielectric permittivity at infinite frequency, $\gamma =$ 5.75 × 1013 rad/s is the collision frequency, and ${\omega _p}(\sigma )$ is the plasma frequency that depends on conductivity $\sigma$. The plasma frequency can be expressed as ${\omega _p}^2(\sigma ) = {\sigma / {{\sigma _0}}}{\omega _p}^2({\sigma _0})$, where ${\sigma _0}$ = 3 × 105 S/m and ${\omega _p}({\sigma _0})$ = 1.4 × 1015 rad/s. To mimic the insulating-conducting phase transition properties of VO2, we used different conductivities for different photoelectric properties of VO2. In this study, $\sigma$ = 2 × 105 S/m and $\sigma$ = 40 S/m indicate the conducting and insulating states, respectively. The structure of the VO2 layer is obtained by cutting a central VO2 ring patch in a continuous VO2 patch, whose radius is Rv= 22 µm and thickness is tVO2= 0.5 µm. Notably, the center of the cut VO2 ring patch coincides with the center of the graphene ring sheet.

3. Results and discussions

For verification, we examined the unit cell of the proposed switchable metamaterial absorber and obtained its electromagnetic response. In FIT numerical simulations, two Floquet ports are assigned to the unit cell of the absorber in the z-direction, one as the source port and the other as the receiving port. Moreover, the periodic boundary conditions in the x and y directions are used to mimic an infinite array. To calculate the absorbance A of the proposed absorber, the transmission coefficient S21 and reflection coefficient S11 are obtained via FIT numerical simulations. The absorbance is calculated as A=1-R-T, where R = |S11|2 and T = |S21|2. Because the transmittance T can be approximated to 0 using the continuous metallic reflector, maximal absorption efficiency can be achieved by minimizing the reflection coefficient S11.

The absorbance A for TE and TM polarizations under normal incident terahertz waves and Fermi energy level ${\mu _c}$ = 0.7 eV is shown in Fig. 2. The spectra of absorbance A for both TE and TM polarizations completely coincide, indicating an excellent polarization-insensitive property, which is attributed to the axisymmetric structure. When VO2 is in the insulating state, the absorber exhibits broadband absorption with a 90% absorbance bandwidth of 1.52 THz, from 1.06 to 2.58 THz, for both polarizations. The perfect absorption is achieved at frequencies of 1.28 THz and 2.3 THz. Alternatively, when VO2 is in the conducting state under the same geometry, the designed device acts as a multi-band absorber at the absorption frequencies of 1 THz, 2.45 THz, and 2.82 THz. As can be seen, the absorbance of the proposed multi-band absorber is relatively high from 2.45 THz to 2.82 THz. This is interesting, in the sensing system, this type of absorption spectrum has potential flexible applications. For example, the perfect absorption points of 1 THz, 2.45 THz and 2.82 THz can be used to accurately distinguish different objects, detect explosives and absorb surveillance radar, while the frequency band between 2.45 THz to 2.82 THz can be used for sensitivity analysis, terahertz filters and attenuators [16,62,63].

 figure: Fig. 2.

Fig. 2. Simulated absorbance of the proposed switchable metamaterial absorber under different VO2 phase states for both TE and TM polarizations at ${\mu _c} = 0.7\textrm{ }eV$ under normal terahertz incidence.

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The electric field distributions are analyzed at different resonance frequencies to reveal the absorption mechanism of the proposed absorber. As shown in Fig. 3, the electric field distributions for TE polarization on the x-y and y-z planes are simulated under the conditions of Fermi energy level ${\mu _c}$ = 0.7 eV and normal incidence. We used a color map to denote the strength of the field. Figure 3(a) shows the top view and side view of the simulated electric field distributions of the proposed broadband absorber when VO2 is in the insulating state at resonance frequencies of 1.28 THz and 2.3 THz. At these frequencies, the absorber achieves strong field confinement around the graphene ring sheet, which causes high-efficiency terahertz trapping and absorption. The electric fields are forcefully confined in the gap between the two graphene ring sheets at the resonance frequency of 1.28 THz and confined around the edges of the individual graphene ring sheet at 2.3 THz. In other words, the absorption at the 1.28 THz resonance frequency is mainly attributed to the coupling interaction between the graphene ring sheets, whereas that at 2.3 THz resonance frequency is mainly attributed to the fundamental resonance mode (the electric dipole resonance) of the individual graphene ring sheet [10]. These two resonance regions are superimposed to achieve broadband absorption. The electric field distributions of the side view prove the above statement. At 1.28 THz, due to the absorption caused by the coupling interaction between graphene ring sheets, there will be a lot of energy confinement at the edge of the periodic substrate. At 2.3 THz, there is only a small amount of energy bound at the edge of the substrate due to the absorption caused by the resonance of the individual graphene ring sheet. Additionally, this finding indicates that when VO2 is in an insulating state, it is useless for absorption, and its existence does not affect the broadband absorption characteristics of graphene. Figure 3(b) shows the simulated electric field distributions of the proposed multi-band absorber when VO2 is in the conducting state at the three resonance frequencies of 1 THz, 2.45 THz, and 2.82 THz. Similarly, the absorption at 1 THz resonance frequency is mainly attributed to the fundamental resonant modes of the graphene ring sheet and VO2 hollow ring patch, indicating that the VO2 patch also provides absorption in addition to that provided by the graphene sheet. The field distributions at 1.28 THz and 1.0 THz are similar, but the absorption principle is different. Because the electric field distribution of 1.28 THz is only provided by graphene, while that of 1 THz is provided by both graphene and VO2. The electric field distributions at 2.45 THz and 2.82 THz are similar, which are confined in the gap between the graphene sheet and VO2 patch. The two absorption points are mainly attributed to coupling interaction between the graphene ring sheet and VO2 hollow ring patch. In the same way, the side view of the electric field distributions proves the above statement again. At 1 THz, energy confinement can be observed at the edge of the periodic structure due to the fundamental resonance mode of VO2. While at 2.45 THz and 2.82 THz, the energy confined by the coupling interaction between VO2 and graphene is inside the periodic structure and cannot be observed. Notably, because of the axisymmetric structure of the proposed absorber, the TM polarization electric field distributions of the broadband and multi-band absorber is the same as the TE polarization electric field distribution. However, the azimuth angle is rotated by 90°. Furthermore, the type of resonance is the plasmon resonance induced by the designed metamaterial structures, which can effectively enhance the absorption resulted from the Fabry-Perot cavity between the metamaterial structure and the reflector. The effective impedance can be calculated by the transmission coefficient S21 and reflection coefficient S11. In these absorption peaks, the effective impedance perfectly matches the free space impedance with the real part is equal to 1 and the imaginary part is equal to 0.

 figure: Fig. 3.

Fig. 3. Simulated front-view and side-view of the electric field distributions for TE polarization under the conditions of ${\mu _c} = 0.7\textrm{ }eV$ and normal incidence. (a) Electric field distributions of the proposed broadband absorber at 1.28 THz and 2.3 THz. (b) Electric field distributions of the proposed multi-band absorber at 1 THz, 2.45 THz, and 2.82 THz.

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Further, we studied the relationship between terahertz absorption behavior and polarization and incident angles. To simplify the analysis, we fixed Fermi energy level ${\mu _c}$ = 0.7 eV and other parameters as the initial values. Figure 4(a) and Fig. 4(d) show the absorption spectra based on the polarization angle under incident angle θ = 0° of the proposed broadband and multi-band absorber. The proposed absorber exhibits absolute polarization insensitivity with polarization angle φ ranging from 0° to 360°. This polarization-insensitive property is ascribed to the symmetrical structure of the designed metamaterial, which is an important property for terahertz absorbers. Figure 4(b) and Fig. 4(c) show the absorption spectra based on the incident angle θ of the proposed broadband absorber for TE and TM polarizations, respectively. When VO2 is in the insulating state, the designed broadband absorber exhibits prominent properties with stable absorbance and operating bandwidths in a wide incident angle range of 0° to 60° for both TE and TM polarizations. Moreover, when the incident angle is greater than 60°, the absorption spectrum becomes sensitive to the incident angle. Figure 4(e) and Fig. 4(f) demonstrate the absorption spectrum based on the incident angle of the proposed multi-band absorber for TE and TM polarizations, respectively. The designed multi-band absorber exhibits excellent absorption performance in the first absorption band for both polarizations when VO2 in the conducting state. For instance, the absorption spectrum with 90% absorbance peak is insensitive to incident angle up to 60°. To this end, the rest of the absorption bands also demonstrate stable absorbance performance in an incident angle range of 0° to 40° for both TE and TM polarizations. When the incident angle is greater than 40°, the absorbance decreases and generates undesirable resonance frequencies. Because as the incident angle θ increases, the tangential electric filed decreases accordingly, and the parasitic resonances of certain parts of the structure increases sharply. This wide-angle behavior is attributed to the absorption properties that are mainly related to the resonances producing the subwavelength structure of the proposed switchable metamaterial absorber, while being less dependent on the incident angle [10]. The subwavelength structure of the proposed absorber can compensate for the momentum mismatch between the metamaterial surface of the graphene-VO2 and the terahertz wave from free space, thereby exhibiting excellent wide-angle behavior.

 figure: Fig. 4.

Fig. 4. Simulated absorption spectrum of the broadband (a-c) and multi-band (d-f) absorber as functions of frequency and the polarization angle φ and incident angle θ under the condition of ${\mu _c} = 0.7\textrm{ }eV$.

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Moreover, the impact of the geometry on the absorption performance of the broadband and multi-band absorber is discussed. To simplify the analysis, we retain Fermi energy level ${\mu _c}$ = 0.7 eV and the thickness of Topas ttopas= 27 µm, where initial ttopas is selected as a quarter wavelength and then slightly optimized to achieve the final value. The analysis is performed under the normal incidence of TE polarization. The absorption performance of the broadband absorber as functions of frequency and graphene radius (Rg), VO2 radius (Rv), and VO2 thickness (tVO2) are shown in Fig. 5(a)-Fig. 5(c), respectively. When VO2 is in the insulating state, the absorption performance of the broadband absorber is unaffected by the geometrical dimensions of the VO2 hollow ring patch. In other words, VO2 does not contribute to the broadband absorption performance when it is in the insulating state. The absorption performance of the absorber when Rg is increased is shown in Fig. 5(a). As Rg increases, the graphene area and absorption bandwidth also increase. Figure 5(d) –Fig. 5(f) show that when VO2 is in the conducting state, the geometrical dimensions of the graphene ring sheet and VO2 hollow ring patch have a significant influence on the multi-band absorbance performance. As Rg increases, the graphene area increases, the first resonance frequency moves to a lower frequency, and the second resonance frequency is relatively fixed. When Rg = 18 µm, the resonance frequency caused by graphene is adjacent with the fundamental resonance frequency of VO2, the interaction between them forms the third perfect absorption frequency point. As shown in Fig. 5(e), when Rv is less than the initial Rg= 18 µm, the graphene is covered by VO2 and only one resonance frequency is generated. It is obvious from Fig. 3(b) that, the electric field energy is mainly confined to the sub-wavelength structure between the edge of graphene and VO2. When VO2 covers the edge of graphene, it will hinder the absorption of the sub-wavelength structure, and most of the terahertz wave will be reflected. When Rv = 22 µm, optimal performance with three-band absorption is achieved. As the Rv increases, the VO2 area decreases and cannot provide sufficient absorption intensity, so that the absorption value gradually decreases. Figure 5(f) shows that the absorption performance of multi-band absorber is generally unaffected by tVO2 when it is greater than 0.5 µm, because this thickness is much greater than the skin depth of electromagnetic waves.

 figure: Fig. 5.

Fig. 5. Simulated absorption performance of the broadband (a-c) and multi-band (d-f) absorber as functions of frequency and graphene radius Rg, VO2 radius Rv, and VO2 thickness tVO2.

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Next, to verify the adjustable absorption performance of the absorber using the bias voltage, we simulated the absorption spectrum as a function of frequency and Fermi energy level ${\mu _c}$ under normal incidence for TE polarization. As shown in Fig. 6(a), when VO2 is in the insulating state, the broadband absorbance peak can be adjusted while maintaining the same center frequency from 20% to nearly 100% by changing ${\mu _c}$ from 0 eV to 0.7 eV, thus demonstrating excellent broadband absorption modulation above 80%. While when VO2 is in conducting state, as shown in Fig. 6(b), due to the interaction between the fundamental resonance frequency of VO2 and the gradually increasing absorption value of graphene, resonance frequencies for multi-band absorber are reconfigurable by changing the Fermi energy level. As ${\mu _c}$ increases from 0.5 eV to 1 eV, the first resonance frequency is reconfigurable from 1 THz to 1.1 THz with over 90% absorbance peak. For the second and third resonance frequencies, the reconfigurable resonance frequency with over 90% absorbance peak switchs from 2.2 THz to 2.65 THz and 2.55 THz to 3.16 THz, respectively. In summary, the broadband and multi-band absorber can be tuned by changing the Fermi energy levels. This absorber can be widely used in terahertz intelligent systems.

 figure: Fig. 6.

Fig. 6. Simulated absorption performances of the broadband (a) and multi-band (b) absorber as functions of frequency and the Fermi energy levels.

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Finally, the potential application of the metamaterial absorber in sensing is studied. According to the above analysis, the metamaterial absorber has a strong local field at the resonance frequency and is very sensitive to the changes of the surrounding medium. Therefore, in the implementation, placing objects above the absorber. Terahertz wave will pass through the objects and then incident inside the absorber. When the parameters of objects such as the permittivity change, the resonance frequency of the absorber will be shifted, and the offset of the resonance frequency can be obtained by theoretical calculation according to the electromagnetic perturbation theory. Therefore, based on the resonance frequency offset, the permittivity of the objects can be measured. However, the bottleneck restricting the industrialization of terahertz metamaterial absorbers is mainly related to the processing of devices, which is also a key point in the future research of terahertz metamaterials devices.

4. Conclusion

In conclusion, we proposed and demonstrated a switchable metamaterial absorber with broadband and multi-band absorption operating in the terahertz regime. Our absorber design is well structured and consists of the VO2 hollow ring patch, graphene ring sheet, polysilicon sheet, Topas spacer, and continuous metallic reflector. By changing the phase transition property of VO2, the switchable functional properties of the absorber can be achieved. When VO2 is in the insulating state, a broadband absorber with absorbance greater than 90% under normal incidence from 1.06 THz to 2.58 THz is achieved. The perfect absorption is achieved at frequencies of 1.28 THz and 2.3 THz. The broadband absorber demonstrates excellent absorption performance with 90% terahertz absorbance in a wide incident angle range of 0°−60° for TE and TM polarizations. The absorption bandwidth and intensity of the broadband absorber can be adjusted by changing the area or Fermi energy level of graphene. When VO2 is in the conducting state, the designed metamaterial device acts as a multi-band absorber with absorption frequencies of 1 THz, 2.45 THz, and 2.82 THz. The simulated electric field distributions show that the multi-band absorption is attributed to the fundamental resonance modes of the graphene ring sheet and VO2 hollow ring patch, as well as the coupling interaction between them. Moreover, the multi-band absorber exhibits more than 90% peak absorbance in an incident angle range of 0°−40° for both TE and TM polarizations. By carefully optimizing the geometric parameters, optimal multi-band absorption characteristics of the absorber can be obtained. Additionally, the resonance frequencies of the multi-band absorber can be reconfigured by changing the Fermi energy level of graphene.

Feasibility of the proposed design and a possible fabrication procedure are discussed here. (1) Use chemical vapor deposition (CVD) to generate graphene on the copper film and use magnetron sputtering technique to fabricate VO2, meanwhile, the metallic reflector can be deposited on the Si wafer by gilding. (2) Topas gum should be spin-coated over the metallic reflector to form an Au-Topas wafer, and use the same process to spin-coat Topas on the polysilicon. (3) Transfer the polysilicon-Topas wafer onto the Au-Topas wafer to obtain an Au-Topas-polysilicon-Topas substrate. (4) Coat polymethyl methacrylate (PMMA) gum on the graphene-copper sample, and use FeCl3 solution to dissolve the copper from the copper-graphene-PMMA sample. (5) Transfer the graphene-PMMA sample onto the Au-Topas-polysilicon-Topas substrate, then use acetone solution to remove the PMMA gum to obtain an Au-Topas-polysilicon-Topas-graphene wafer. (6) Etch patterned graphene using reactive ion etching system, and finally, etch VO2 film onto the patterned graphene to form the proposed absorber.

The switchable metamaterial absorber presented in this study shows the merits of bi-functionality, incident angle and polarization independence, a simple configuration, and is adjustable by changing the Fermi energy level of graphene. For the terahertz intelligent system, this bi-functional device is important, which can be used for objects imaging under the broadband absorption and objects distinguishing under the multi-band absorption. Using one device to achieve two different functions can greatly reduce the size of terahertz system. And this absorber is also very attractive for potential terahertz applications such as intelligent attenuators, reflectors, and spatial modulators.

Funding

National Natural Science Foundation of China (61871072).

Disclosures

The authors declare no conflicts of interest.

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21. L. Ju, B. S. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. G. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011). [CrossRef]  

22. T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8(2), 1086–1101 (2014). [CrossRef]  

23. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene Plasmonics,” Nat. Photonics 6(11), 749–758 (2012). [CrossRef]  

24. F. H. Koppens, D. E. Chang, F. Javier García de Abajo, and G. de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011). [CrossRef]  

25. P. Chen and A. Alu, “Terahertz metamaterial devices based on graphene nanostructures,” IEEE Trans. THz Sci. Technol. 3(6), 748–756 (2013). [CrossRef]  

26. M. L. Huang, Y. Z. Cheng, Z. Z. Cheng, H. R. Chen, X. S. Mao, and R. Z. Gong, “Based on graphene tunable dual-band terahertz metamaterial absorber with wide-angle,” Opt. Commun. 415, 194–201 (2018). [CrossRef]  

27. G. Dayal and S. Anantha Ramakrishna, “Design of multi-band metamaterial perfect absorbers with stacked metal-dielectric disks,” J. Opt. 15(5), 055106 (2013). [CrossRef]  

28. B. X. Wang, G. Z. Wang, and L. L. Wang, “Design of a novel dual-band terahertz metamaterial absorber,” Plasmonics 11(2), 523–530 (2016). [CrossRef]  

29. A. Fardoost, F. G. Vanani, A. Amirhosseini, and R. Safian, “Design of a multilayer graphene-based ultrawideband terahertz absorber,” IEEE Trans. Nanotechnol. 16(1), 1 (2016). [CrossRef]  

30. P. Fu, F. Liu, G. J. Ren, F. Su, D. Li, and J. Q. Yao, “A broadband metamaterial absorber based on multi-layer graphene in the terahertz region,” Opt. Commun. 417, 62–66 (2018). [CrossRef]  

31. W. R. Zhu, F. J. Xiao, M. Kang, D. Sikdar, and M. Premaratne, “Tunable terahertz left-handed metamaterial based on multi-layer graphene-dielectric composite,” Appl. Phys. Lett. 104(5), 051902 (2014). [CrossRef]  

32. L. Liu, W. W. Liu, and Z. Song, “Ultra-broadband terahertz absorber based on a multilayer graphene metamaterial,” J. Appl. Phys. 128(9), 093104 (2020). [CrossRef]  

33. M. Jiang, Z. Song, and Q. Liu, “Ultra-broadband wide-angle terahertz absorber realized by a doped silicon metamaterial,” Opt. Commun. 471, 125835 (2020). [CrossRef]  

34. M. Wei, Z. Song, Y. Deng, Y. Liu, and Q. Liu, “Large-angle mid-infrared absorption switch enabled by polarization-independent GST metasurfaces,” Mater. Lett. 236, 350–353 (2019). [CrossRef]  

35. S. Wu, D. C. Zha, L. Miao, Y. He, and J. J. Jiang, “Graphene-based single-layer elliptical pattern metamaterial absorber for adjustable broadband absorption in terahertz range,” Phys. Scr. 94(10), 105507 (2019). [CrossRef]  

36. G. S. Deng, P. Chen, J. Yang, Z. P. Yin, and L. Z. Qiu, “Graphene-based tunable polarization sensitive terahertz metamaterial absorber,” Opt. Commun. 380, 101–107 (2016). [CrossRef]  

37. J. A. Koza, Z. He, A. S. Miller, and J. A. Switzer, “Resistance switching in electrodeposited VO2 thin films,” Chem. Mater. 23(18), 4105–4108 (2011). [CrossRef]  

38. W. X. Huang, X. Yin, C. Huang, Q. Wang, T. Miao, and Y. Zhu, “Optical switching of a metamaterial by temperature controlling,” Appl. Phys. Lett. 96(26), 261908 (2010). [CrossRef]  

39. J. Du, Y. Gao, H. Luo, L. Kang, and C. Cao, “Significant changes in phase-transition hysteresis for Ti-doped VO2 films prepared by polymer-assisted deposition,” Sol. Energy Mater. Sol. Cells 95(2), 469–475 (2011). [CrossRef]  

40. P. Jepsen, B. Fischer, A. Thoman, H. Helm, J. Suh, R. Lopez, and R. Haglund, “Metal-insulator phase transition in a VO2 thin film observed with terahertz spectroscopy,” Phys. Rev. B 74(20), 205103 (2006). [CrossRef]  

41. Q. Y. Wen, H. Zhang, Q. Yang, Y. Xie, K. Chen, and Y. Liu, “Terahertz metamaterials with VO2 cut-wires for thermal tenability,” Appl. Phys. Lett. 97(2), 021111 (2010). [CrossRef]  

42. C. Ko and S. Ramanathan, “Observation of electric field-assisted phase transition in thin film vanadium oxide in a metal-oxide-semiconductor device geometry,” Appl. Phys. Lett. 93(25), 252101 (2008). [CrossRef]  

43. F. Fan, W. H. Gu, S. Chen, X. H. Wang, and S. J. Chang, “State conversion based on terahertz plasmonics with vanadium dioxide coating controlled by optical pumping,” Opt. Lett. 38(9), 1582–1584 (2013). [CrossRef]  

44. S. B. Choi, J. S. Kyoung, H. S. Kim, H. R. Park, D. J. Park, B. J. Kim, Y. H. Ahn, F. Rotermund, H. T. Kim, K. J. Ahn, and D. S. Kim, “Nanopattern enabled terahertz all-optical switching on vanadium dioxide thin film,” Appl. Phys. Lett. 98(7), 071105 (2011). [CrossRef]  

45. Z. Song, A. Chen, J. Zhang, and J. Wang, “Integrated metamaterial with functionalities of absorption and electromagnetically induced transparency,” Opt. Express 27(18), 25196–25204 (2019). [CrossRef]  

46. L. Chen and Z. Song, “Simultaneous realizations of absorber and transparent conducting metal in a single metamaterial,” Opt. Express 28(5), 6565–6571 (2020). [CrossRef]  

47. Z. Song, M. Jiang, Y. Deng, and A. Chen, “Wide-angle Absorber with Tunable Intensity and Bandwidth Realized by a Terahertz Phase Change Material,” Opt. Commun. 464, 125494 (2020). [CrossRef]  

48. Z. Song, Y. Deng, Y. Zhou, and Z. Liu, “Terahertz Toroidal Metamaterial with Tunable Properties,” Opt. Express 27(4), 5792–5797 (2019). [CrossRef]  

49. Q. Chu, Z. Song, and Q. Liu, “Omnidirectional Tunable Terahertz Analog of Electromagnetically Induced Transparency Realized by Isotropic Vanadium Dioxide Metasurfaces,” Appl. Phys. Express 11(8), 082203 (2018). [CrossRef]  

50. J. Huang, J. Li, Y. Yang, J. Li, J. H. Li, Y. Zhang, and J. Yao, “Active controllable dual broadband terahertz absorber based on hybrid metamaterials with vanadium dioxide,” Opt. Express 28(5), 7018–7027 (2020). [CrossRef]  

51. Z. Song and J. Zhang, “Achieving broadband absorption and polarization conversion with a vanadium dioxide metasurface in the same terahertz frequencies,” Opt. Express 28(8), 12487–12497 (2020). [CrossRef]  

52. M. Zhang, J. Zhang, A. Chen, and Z. Song, “Vanadium dioxide-based bifunctional metamaterial for terahertz waves,” IEEE Photonics J. 12(1), 1–9 (2020). [CrossRef]  

53. Z. Song, A. Chen, and J. Zhang, “Terahertz switching between broadband absorption and narrowband absorption,” Opt. Express 28(2), 2037–2044 (2020). [CrossRef]  

54. A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, “Spatiotemporal light control with active metasurfaces,” Science 364(6441), eaat3100 (2019). [CrossRef]  

55. C. Hail, A. Michel, D. Poulikakos, and H. Eghlidi, “Optical Metasurfaces: Evolving from Passive to Adaptive,” Adv. Opt. Mater. 7(14), 1801786 (2019). [CrossRef]  

56. J. Li, Y. Zhang, J. N. Li, X. Yan, L. Liang, Z. Zhang, J. Huang, J. Li, Y. Yang, and J. Yao, “Amplitude Modulation of Anomalously Reflected Terahertz Beams Using All-optical Active PancharatnamBerry Coding Metasurfaces,” Nanoscale 11(12), 5746–5753 (2019). [CrossRef]  

57. F. Ding, Y. Yang, and I. B. Sergey, “Dynamic Metasurfaces Using Phase Change Chalcogenides,” Adv. Opt. Mater. 7(14), 1801709 (2019). [CrossRef]  

58. J. Li, J. T. Li, Y. Yang, J. N. Li, Y. Zhang, L. Wu, Z. Zhang, M. S. Yang, C. L. Zheng, J. H. Li, J. Huang, F. Y. Li, T. T. Tang, H. T. Dai, and J. Q. Yao, “Metal-graphene hybrid active chiral metasurfaces for dynamic terahertz wavefront modulation and near field imaging,” Carbon 163, 34–42 (2020). [CrossRef]  

59. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef]  

60. Z. Song, K. Wang, J. Li, and Q. H. Liu, “Broadband tunable terahertz absorber based on vanadium dioxide metamaterials,” Opt. Express 26(6), 7148–7154 (2018). [CrossRef]  

61. L. Ye, X. Chen, C. Zhu, W. Li, and Y. Zhang, “Switchable broadband terahertz spatial modulators based on patterned graphene and vanadium dioxide,” Opt. Express 28(23), 33948–33958 (2020). [CrossRef]  

62. D. Hu, T. H. Meng, H. Y. Wang, Y. K. Ma, and Q. F. Zhu, “Ultra-narrow-band terahertz perfect metamaterial absorber for refractive index sensing application,” Results Phys. 19, 103567 (2020). [CrossRef]  

63. A. S. Dhillon, D. Mittal, and R. Bargota, “Triple band ultrathin polarization insensitive metamaterial absorber for defense, explosive detection and airborne radar applications,” Microw Opt Technol Lett 61(1), 89–95 (2019). [CrossRef]  

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  32. L. Liu, W. W. Liu, and Z. Song, “Ultra-broadband terahertz absorber based on a multilayer graphene metamaterial,” J. Appl. Phys. 128(9), 093104 (2020).
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  33. M. Jiang, Z. Song, and Q. Liu, “Ultra-broadband wide-angle terahertz absorber realized by a doped silicon metamaterial,” Opt. Commun. 471, 125835 (2020).
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  34. M. Wei, Z. Song, Y. Deng, Y. Liu, and Q. Liu, “Large-angle mid-infrared absorption switch enabled by polarization-independent GST metasurfaces,” Mater. Lett. 236, 350–353 (2019).
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    [Crossref]
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    [Crossref]
  46. L. Chen and Z. Song, “Simultaneous realizations of absorber and transparent conducting metal in a single metamaterial,” Opt. Express 28(5), 6565–6571 (2020).
    [Crossref]
  47. Z. Song, M. Jiang, Y. Deng, and A. Chen, “Wide-angle Absorber with Tunable Intensity and Bandwidth Realized by a Terahertz Phase Change Material,” Opt. Commun. 464, 125494 (2020).
    [Crossref]
  48. Z. Song, Y. Deng, Y. Zhou, and Z. Liu, “Terahertz Toroidal Metamaterial with Tunable Properties,” Opt. Express 27(4), 5792–5797 (2019).
    [Crossref]
  49. Q. Chu, Z. Song, and Q. Liu, “Omnidirectional Tunable Terahertz Analog of Electromagnetically Induced Transparency Realized by Isotropic Vanadium Dioxide Metasurfaces,” Appl. Phys. Express 11(8), 082203 (2018).
    [Crossref]
  50. J. Huang, J. Li, Y. Yang, J. Li, J. H. Li, Y. Zhang, and J. Yao, “Active controllable dual broadband terahertz absorber based on hybrid metamaterials with vanadium dioxide,” Opt. Express 28(5), 7018–7027 (2020).
    [Crossref]
  51. Z. Song and J. Zhang, “Achieving broadband absorption and polarization conversion with a vanadium dioxide metasurface in the same terahertz frequencies,” Opt. Express 28(8), 12487–12497 (2020).
    [Crossref]
  52. M. Zhang, J. Zhang, A. Chen, and Z. Song, “Vanadium dioxide-based bifunctional metamaterial for terahertz waves,” IEEE Photonics J. 12(1), 1–9 (2020).
    [Crossref]
  53. Z. Song, A. Chen, and J. Zhang, “Terahertz switching between broadband absorption and narrowband absorption,” Opt. Express 28(2), 2037–2044 (2020).
    [Crossref]
  54. A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, “Spatiotemporal light control with active metasurfaces,” Science 364(6441), eaat3100 (2019).
    [Crossref]
  55. C. Hail, A. Michel, D. Poulikakos, and H. Eghlidi, “Optical Metasurfaces: Evolving from Passive to Adaptive,” Adv. Opt. Mater. 7(14), 1801786 (2019).
    [Crossref]
  56. J. Li, Y. Zhang, J. N. Li, X. Yan, L. Liang, Z. Zhang, J. Huang, J. Li, Y. Yang, and J. Yao, “Amplitude Modulation of Anomalously Reflected Terahertz Beams Using All-optical Active PancharatnamBerry Coding Metasurfaces,” Nanoscale 11(12), 5746–5753 (2019).
    [Crossref]
  57. F. Ding, Y. Yang, and I. B. Sergey, “Dynamic Metasurfaces Using Phase Change Chalcogenides,” Adv. Opt. Mater. 7(14), 1801709 (2019).
    [Crossref]
  58. J. Li, J. T. Li, Y. Yang, J. N. Li, Y. Zhang, L. Wu, Z. Zhang, M. S. Yang, C. L. Zheng, J. H. Li, J. Huang, F. Y. Li, T. T. Tang, H. T. Dai, and J. Q. Yao, “Metal-graphene hybrid active chiral metasurfaces for dynamic terahertz wavefront modulation and near field imaging,” Carbon 163, 34–42 (2020).
    [Crossref]
  59. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
    [Crossref]
  60. Z. Song, K. Wang, J. Li, and Q. H. Liu, “Broadband tunable terahertz absorber based on vanadium dioxide metamaterials,” Opt. Express 26(6), 7148–7154 (2018).
    [Crossref]
  61. L. Ye, X. Chen, C. Zhu, W. Li, and Y. Zhang, “Switchable broadband terahertz spatial modulators based on patterned graphene and vanadium dioxide,” Opt. Express 28(23), 33948–33958 (2020).
    [Crossref]
  62. D. Hu, T. H. Meng, H. Y. Wang, Y. K. Ma, and Q. F. Zhu, “Ultra-narrow-band terahertz perfect metamaterial absorber for refractive index sensing application,” Results Phys. 19, 103567 (2020).
    [Crossref]
  63. A. S. Dhillon, D. Mittal, and R. Bargota, “Triple band ultrathin polarization insensitive metamaterial absorber for defense, explosive detection and airborne radar applications,” Microw Opt Technol Lett 61(1), 89–95 (2019).
    [Crossref]

2020 (12)

M. Zhang and Z. Song, “Terahertz bifunctional absorber based on a graphene-spacer-vanadium dioxide-spacer-metal configuration,” Opt. Express 28(8), 11780–11788 (2020).
[Crossref]

L. Liu, W. W. Liu, and Z. Song, “Ultra-broadband terahertz absorber based on a multilayer graphene metamaterial,” J. Appl. Phys. 128(9), 093104 (2020).
[Crossref]

M. Jiang, Z. Song, and Q. Liu, “Ultra-broadband wide-angle terahertz absorber realized by a doped silicon metamaterial,” Opt. Commun. 471, 125835 (2020).
[Crossref]

L. Chen and Z. Song, “Simultaneous realizations of absorber and transparent conducting metal in a single metamaterial,” Opt. Express 28(5), 6565–6571 (2020).
[Crossref]

Z. Song, M. Jiang, Y. Deng, and A. Chen, “Wide-angle Absorber with Tunable Intensity and Bandwidth Realized by a Terahertz Phase Change Material,” Opt. Commun. 464, 125494 (2020).
[Crossref]

J. Huang, J. Li, Y. Yang, J. Li, J. H. Li, Y. Zhang, and J. Yao, “Active controllable dual broadband terahertz absorber based on hybrid metamaterials with vanadium dioxide,” Opt. Express 28(5), 7018–7027 (2020).
[Crossref]

Z. Song and J. Zhang, “Achieving broadband absorption and polarization conversion with a vanadium dioxide metasurface in the same terahertz frequencies,” Opt. Express 28(8), 12487–12497 (2020).
[Crossref]

M. Zhang, J. Zhang, A. Chen, and Z. Song, “Vanadium dioxide-based bifunctional metamaterial for terahertz waves,” IEEE Photonics J. 12(1), 1–9 (2020).
[Crossref]

Z. Song, A. Chen, and J. Zhang, “Terahertz switching between broadband absorption and narrowband absorption,” Opt. Express 28(2), 2037–2044 (2020).
[Crossref]

J. Li, J. T. Li, Y. Yang, J. N. Li, Y. Zhang, L. Wu, Z. Zhang, M. S. Yang, C. L. Zheng, J. H. Li, J. Huang, F. Y. Li, T. T. Tang, H. T. Dai, and J. Q. Yao, “Metal-graphene hybrid active chiral metasurfaces for dynamic terahertz wavefront modulation and near field imaging,” Carbon 163, 34–42 (2020).
[Crossref]

L. Ye, X. Chen, C. Zhu, W. Li, and Y. Zhang, “Switchable broadband terahertz spatial modulators based on patterned graphene and vanadium dioxide,” Opt. Express 28(23), 33948–33958 (2020).
[Crossref]

D. Hu, T. H. Meng, H. Y. Wang, Y. K. Ma, and Q. F. Zhu, “Ultra-narrow-band terahertz perfect metamaterial absorber for refractive index sensing application,” Results Phys. 19, 103567 (2020).
[Crossref]

2019 (13)

A. S. Dhillon, D. Mittal, and R. Bargota, “Triple band ultrathin polarization insensitive metamaterial absorber for defense, explosive detection and airborne radar applications,” Microw Opt Technol Lett 61(1), 89–95 (2019).
[Crossref]

Z. Song, A. Chen, J. Zhang, and J. Wang, “Integrated metamaterial with functionalities of absorption and electromagnetically induced transparency,” Opt. Express 27(18), 25196–25204 (2019).
[Crossref]

A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, “Spatiotemporal light control with active metasurfaces,” Science 364(6441), eaat3100 (2019).
[Crossref]

C. Hail, A. Michel, D. Poulikakos, and H. Eghlidi, “Optical Metasurfaces: Evolving from Passive to Adaptive,” Adv. Opt. Mater. 7(14), 1801786 (2019).
[Crossref]

J. Li, Y. Zhang, J. N. Li, X. Yan, L. Liang, Z. Zhang, J. Huang, J. Li, Y. Yang, and J. Yao, “Amplitude Modulation of Anomalously Reflected Terahertz Beams Using All-optical Active PancharatnamBerry Coding Metasurfaces,” Nanoscale 11(12), 5746–5753 (2019).
[Crossref]

F. Ding, Y. Yang, and I. B. Sergey, “Dynamic Metasurfaces Using Phase Change Chalcogenides,” Adv. Opt. Mater. 7(14), 1801709 (2019).
[Crossref]

Z. Song, Y. Deng, Y. Zhou, and Z. Liu, “Terahertz Toroidal Metamaterial with Tunable Properties,” Opt. Express 27(4), 5792–5797 (2019).
[Crossref]

M. Wei, Z. Song, Y. Deng, Y. Liu, and Q. Liu, “Large-angle mid-infrared absorption switch enabled by polarization-independent GST metasurfaces,” Mater. Lett. 236, 350–353 (2019).
[Crossref]

S. Wu, D. C. Zha, L. Miao, Y. He, and J. J. Jiang, “Graphene-based single-layer elliptical pattern metamaterial absorber for adjustable broadband absorption in terahertz range,” Phys. Scr. 94(10), 105507 (2019).
[Crossref]

X. Huang, F. Yang, B. Gao, Q. Yang, J. Wu, and W. He, “Metamaterial absorber with independently tunable amplitude and frequency in the terahertz regime,” Opt. Express 27(18), 25902–25911 (2019).
[Crossref]

W. W. Meng, J. Lv, L. Zhang, L. Que, Y. Zhou, and Y. Jiang, “An ultra-broadband and polarization-independent metamaterial absorber with bandwidth of 3.7 THz,” Opt. Commun. 431, 255–260 (2019).
[Crossref]

L. Ye, X. Chen, F. Zeng, J. Zhuo, F. Shen, and Q. H. Liu, “Ultra-wideband terahertz absorption using dielectric circular truncated cones,” IEEE Photonics J. 11(5), 1–7 (2019).
[Crossref]

L. Ye, F. Zeng, Y. Zhang, and Q. H. Liu, “Composite graphene-metal microstructures for enhanced multiband absorption covering the entire terahertz range,” Carbon 148, 317–325 (2019).
[Crossref]

2018 (8)

L. Ye, X. Chen, G. Cai, J. Zhu, N. Liu, and Q. H. Liu, “Electrically tunable broadband terahertz absorption with hybrid-patterned graphene meta-surfaces,” Nanomaterials 8(8), 562 (2018).
[Crossref]

L. Ye, F. Zeng, Y. Zhang, X. Xu, X. Yang, and Q. H. Liu, “Frequency-reconfigurable wide-angle terahertz absorbers using single- and double-layer decussate graphene ribbon arrays,” Nanomaterials 8(10), 834 (2018).
[Crossref]

L. Ye, X. Chen, J. Zhuo, F. Han, and Q. H. Liu, “Actively tunable broadband terahertz absorption using periodically square-patterned graphene,” Appl. Phys. Express 11(10), 102201 (2018).
[Crossref]

Y. Cai, Z. Wang, S. Yan, L. Ye, and J. Zhu, “Ultraviolet absorption band engineering of graphene by integrated plasmonic structures,” Opt. Mater. Express 8(11), 3295–3306 (2018).
[Crossref]

M. L. Huang, Y. Z. Cheng, Z. Z. Cheng, H. R. Chen, X. S. Mao, and R. Z. Gong, “Based on graphene tunable dual-band terahertz metamaterial absorber with wide-angle,” Opt. Commun. 415, 194–201 (2018).
[Crossref]

P. Fu, F. Liu, G. J. Ren, F. Su, D. Li, and J. Q. Yao, “A broadband metamaterial absorber based on multi-layer graphene in the terahertz region,” Opt. Commun. 417, 62–66 (2018).
[Crossref]

Q. Chu, Z. Song, and Q. Liu, “Omnidirectional Tunable Terahertz Analog of Electromagnetically Induced Transparency Realized by Isotropic Vanadium Dioxide Metasurfaces,” Appl. Phys. Express 11(8), 082203 (2018).
[Crossref]

Z. Song, K. Wang, J. Li, and Q. H. Liu, “Broadband tunable terahertz absorber based on vanadium dioxide metamaterials,” Opt. Express 26(6), 7148–7154 (2018).
[Crossref]

2017 (1)

2016 (5)

J. Liu, L. Fan, J. Ku, and L. Mao, “Absorber: A novel terahertz sensor in the application of substance identification,” Opt. Quantum Electron. 48(2), 80 (2016).
[Crossref]

I. Escorcia, J. Grant, J. Gough, and D. R. S. Cumming, “Uncooled CMOS terahertz imager using a metamaterial absorber and pn diode,” Opt. Lett. 41(14), 3261–3264 (2016).
[Crossref]

B. X. Wang, G. Z. Wang, and L. L. Wang, “Design of a novel dual-band terahertz metamaterial absorber,” Plasmonics 11(2), 523–530 (2016).
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A. Fardoost, F. G. Vanani, A. Amirhosseini, and R. Safian, “Design of a multilayer graphene-based ultrawideband terahertz absorber,” IEEE Trans. Nanotechnol. 16(1), 1 (2016).
[Crossref]

G. S. Deng, P. Chen, J. Yang, Z. P. Yin, and L. Z. Qiu, “Graphene-based tunable polarization sensitive terahertz metamaterial absorber,” Opt. Commun. 380, 101–107 (2016).
[Crossref]

2014 (2)

W. R. Zhu, F. J. Xiao, M. Kang, D. Sikdar, and M. Premaratne, “Tunable terahertz left-handed metamaterial based on multi-layer graphene-dielectric composite,” Appl. Phys. Lett. 104(5), 051902 (2014).
[Crossref]

T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8(2), 1086–1101 (2014).
[Crossref]

2013 (3)

G. Dayal and S. Anantha Ramakrishna, “Design of multi-band metamaterial perfect absorbers with stacked metal-dielectric disks,” J. Opt. 15(5), 055106 (2013).
[Crossref]

P. Chen and A. Alu, “Terahertz metamaterial devices based on graphene nanostructures,” IEEE Trans. THz Sci. Technol. 3(6), 748–756 (2013).
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F. Fan, W. H. Gu, S. Chen, X. H. Wang, and S. J. Chang, “State conversion based on terahertz plasmonics with vanadium dioxide coating controlled by optical pumping,” Opt. Lett. 38(9), 1582–1584 (2013).
[Crossref]

2012 (1)

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene Plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

2011 (5)

F. H. Koppens, D. E. Chang, F. Javier García de Abajo, and G. de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref]

L. Ju, B. S. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. G. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
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J. A. Koza, Z. He, A. S. Miller, and J. A. Switzer, “Resistance switching in electrodeposited VO2 thin films,” Chem. Mater. 23(18), 4105–4108 (2011).
[Crossref]

S. B. Choi, J. S. Kyoung, H. S. Kim, H. R. Park, D. J. Park, B. J. Kim, Y. H. Ahn, F. Rotermund, H. T. Kim, K. J. Ahn, and D. S. Kim, “Nanopattern enabled terahertz all-optical switching on vanadium dioxide thin film,” Appl. Phys. Lett. 98(7), 071105 (2011).
[Crossref]

J. Du, Y. Gao, H. Luo, L. Kang, and C. Cao, “Significant changes in phase-transition hysteresis for Ti-doped VO2 films prepared by polymer-assisted deposition,” Sol. Energy Mater. Sol. Cells 95(2), 469–475 (2011).
[Crossref]

2010 (2)

Q. Y. Wen, H. Zhang, Q. Yang, Y. Xie, K. Chen, and Y. Liu, “Terahertz metamaterials with VO2 cut-wires for thermal tenability,” Appl. Phys. Lett. 97(2), 021111 (2010).
[Crossref]

W. X. Huang, X. Yin, C. Huang, Q. Wang, T. Miao, and Y. Zhu, “Optical switching of a metamaterial by temperature controlling,” Appl. Phys. Lett. 96(26), 261908 (2010).
[Crossref]

2009 (2)

N. I. Landy, C. M. Bingham, T. Tyler, N. Jokerst, D. R. Smith, and W. J. Padilla, “Design, theory, and measurement of a polarization-insensitive absorber for terahertz imaging,” Phys. Rev. B 79(12), 125104 (2009).
[Crossref]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref]

2008 (2)

C. Ko and S. Ramanathan, “Observation of electric field-assisted phase transition in thin film vanadium oxide in a metal-oxide-semiconductor device geometry,” Appl. Phys. Lett. 93(25), 252101 (2008).
[Crossref]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref]

2006 (4)

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref]

P. Jepsen, B. Fischer, A. Thoman, H. Helm, J. Suh, R. Lopez, and R. Haglund, “Metal-insulator phase transition in a VO2 thin film observed with terahertz spectroscopy,” Phys. Rev. B 74(20), 205103 (2006).
[Crossref]

2005 (2)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref]

Y. B. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438(7065), 201–204 (2005).
[Crossref]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref]

Ahn, K. J.

S. B. Choi, J. S. Kyoung, H. S. Kim, H. R. Park, D. J. Park, B. J. Kim, Y. H. Ahn, F. Rotermund, H. T. Kim, K. J. Ahn, and D. S. Kim, “Nanopattern enabled terahertz all-optical switching on vanadium dioxide thin film,” Appl. Phys. Lett. 98(7), 071105 (2011).
[Crossref]

Ahn, Y. H.

S. B. Choi, J. S. Kyoung, H. S. Kim, H. R. Park, D. J. Park, B. J. Kim, Y. H. Ahn, F. Rotermund, H. T. Kim, K. J. Ahn, and D. S. Kim, “Nanopattern enabled terahertz all-optical switching on vanadium dioxide thin film,” Appl. Phys. Lett. 98(7), 071105 (2011).
[Crossref]

Alu, A.

P. Chen and A. Alu, “Terahertz metamaterial devices based on graphene nanostructures,” IEEE Trans. THz Sci. Technol. 3(6), 748–756 (2013).
[Crossref]

Amirhosseini, A.

A. Fardoost, F. G. Vanani, A. Amirhosseini, and R. Safian, “Design of a multilayer graphene-based ultrawideband terahertz absorber,” IEEE Trans. Nanotechnol. 16(1), 1 (2016).
[Crossref]

Anantha Ramakrishna, S.

G. Dayal and S. Anantha Ramakrishna, “Design of multi-band metamaterial perfect absorbers with stacked metal-dielectric disks,” J. Opt. 15(5), 055106 (2013).
[Crossref]

Avouris, P.

T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8(2), 1086–1101 (2014).
[Crossref]

Bargota, R.

A. S. Dhillon, D. Mittal, and R. Bargota, “Triple band ultrathin polarization insensitive metamaterial absorber for defense, explosive detection and airborne radar applications,” Microw Opt Technol Lett 61(1), 89–95 (2019).
[Crossref]

Bechtel, H. A.

L. Ju, B. S. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. G. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref]

Bingham, C. M.

N. I. Landy, C. M. Bingham, T. Tyler, N. Jokerst, D. R. Smith, and W. J. Padilla, “Design, theory, and measurement of a polarization-insensitive absorber for terahertz imaging,” Phys. Rev. B 79(12), 125104 (2009).
[Crossref]

Brongersma, M. L.

A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, “Spatiotemporal light control with active metasurfaces,” Science 364(6441), eaat3100 (2019).
[Crossref]

Cai, G.

L. Ye, X. Chen, G. Cai, J. Zhu, N. Liu, and Q. H. Liu, “Electrically tunable broadband terahertz absorption with hybrid-patterned graphene meta-surfaces,” Nanomaterials 8(8), 562 (2018).
[Crossref]

L. Ye, Y. Chen, G. Cai, N. Liu, J. Zhu, Z. Song, and Q. Liu, “Broadband absorber with periodically sinusoidally patterned graphene layer in terahertz range,” Opt. Express 25(10), 11223–11232 (2017).
[Crossref]

Cai, Y.

Cao, C.

J. Du, Y. Gao, H. Luo, L. Kang, and C. Cao, “Significant changes in phase-transition hysteresis for Ti-doped VO2 films prepared by polymer-assisted deposition,” Sol. Energy Mater. Sol. Cells 95(2), 469–475 (2011).
[Crossref]

Chang, D. E.

F. H. Koppens, D. E. Chang, F. Javier García de Abajo, and G. de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref]

Chang, S. J.

Chen, A.

M. Zhang, J. Zhang, A. Chen, and Z. Song, “Vanadium dioxide-based bifunctional metamaterial for terahertz waves,” IEEE Photonics J. 12(1), 1–9 (2020).
[Crossref]

Z. Song, A. Chen, and J. Zhang, “Terahertz switching between broadband absorption and narrowband absorption,” Opt. Express 28(2), 2037–2044 (2020).
[Crossref]

Z. Song, M. Jiang, Y. Deng, and A. Chen, “Wide-angle Absorber with Tunable Intensity and Bandwidth Realized by a Terahertz Phase Change Material,” Opt. Commun. 464, 125494 (2020).
[Crossref]

Z. Song, A. Chen, J. Zhang, and J. Wang, “Integrated metamaterial with functionalities of absorption and electromagnetically induced transparency,” Opt. Express 27(18), 25196–25204 (2019).
[Crossref]

Chen, H. R.

M. L. Huang, Y. Z. Cheng, Z. Z. Cheng, H. R. Chen, X. S. Mao, and R. Z. Gong, “Based on graphene tunable dual-band terahertz metamaterial absorber with wide-angle,” Opt. Commun. 415, 194–201 (2018).
[Crossref]

Chen, K.

Q. Y. Wen, H. Zhang, Q. Yang, Y. Xie, K. Chen, and Y. Liu, “Terahertz metamaterials with VO2 cut-wires for thermal tenability,” Appl. Phys. Lett. 97(2), 021111 (2010).
[Crossref]

Chen, L.

Chen, P.

G. S. Deng, P. Chen, J. Yang, Z. P. Yin, and L. Z. Qiu, “Graphene-based tunable polarization sensitive terahertz metamaterial absorber,” Opt. Commun. 380, 101–107 (2016).
[Crossref]

P. Chen and A. Alu, “Terahertz metamaterial devices based on graphene nanostructures,” IEEE Trans. THz Sci. Technol. 3(6), 748–756 (2013).
[Crossref]

Chen, S.

Chen, X.

L. Ye, X. Chen, C. Zhu, W. Li, and Y. Zhang, “Switchable broadband terahertz spatial modulators based on patterned graphene and vanadium dioxide,” Opt. Express 28(23), 33948–33958 (2020).
[Crossref]

L. Ye, X. Chen, F. Zeng, J. Zhuo, F. Shen, and Q. H. Liu, “Ultra-wideband terahertz absorption using dielectric circular truncated cones,” IEEE Photonics J. 11(5), 1–7 (2019).
[Crossref]

L. Ye, X. Chen, J. Zhuo, F. Han, and Q. H. Liu, “Actively tunable broadband terahertz absorption using periodically square-patterned graphene,” Appl. Phys. Express 11(10), 102201 (2018).
[Crossref]

L. Ye, X. Chen, G. Cai, J. Zhu, N. Liu, and Q. H. Liu, “Electrically tunable broadband terahertz absorption with hybrid-patterned graphene meta-surfaces,” Nanomaterials 8(8), 562 (2018).
[Crossref]

Chen, Y.

Cheng, Y. Z.

M. L. Huang, Y. Z. Cheng, Z. Z. Cheng, H. R. Chen, X. S. Mao, and R. Z. Gong, “Based on graphene tunable dual-band terahertz metamaterial absorber with wide-angle,” Opt. Commun. 415, 194–201 (2018).
[Crossref]

Cheng, Z. Z.

M. L. Huang, Y. Z. Cheng, Z. Z. Cheng, H. R. Chen, X. S. Mao, and R. Z. Gong, “Based on graphene tunable dual-band terahertz metamaterial absorber with wide-angle,” Opt. Commun. 415, 194–201 (2018).
[Crossref]

Choi, S. B.

S. B. Choi, J. S. Kyoung, H. S. Kim, H. R. Park, D. J. Park, B. J. Kim, Y. H. Ahn, F. Rotermund, H. T. Kim, K. J. Ahn, and D. S. Kim, “Nanopattern enabled terahertz all-optical switching on vanadium dioxide thin film,” Appl. Phys. Lett. 98(7), 071105 (2011).
[Crossref]

Chu, Q.

Q. Chu, Z. Song, and Q. Liu, “Omnidirectional Tunable Terahertz Analog of Electromagnetically Induced Transparency Realized by Isotropic Vanadium Dioxide Metasurfaces,” Appl. Phys. Express 11(8), 082203 (2018).
[Crossref]

Cumming, D. R. S.

Dai, H. T.

J. Li, J. T. Li, Y. Yang, J. N. Li, Y. Zhang, L. Wu, Z. Zhang, M. S. Yang, C. L. Zheng, J. H. Li, J. Huang, F. Y. Li, T. T. Tang, H. T. Dai, and J. Q. Yao, “Metal-graphene hybrid active chiral metasurfaces for dynamic terahertz wavefront modulation and near field imaging,” Carbon 163, 34–42 (2020).
[Crossref]

Dayal, G.

G. Dayal and S. Anantha Ramakrishna, “Design of multi-band metamaterial perfect absorbers with stacked metal-dielectric disks,” J. Opt. 15(5), 055106 (2013).
[Crossref]

de Abajo, G.

F. H. Koppens, D. E. Chang, F. Javier García de Abajo, and G. de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref]

Deng, G. S.

G. S. Deng, P. Chen, J. Yang, Z. P. Yin, and L. Z. Qiu, “Graphene-based tunable polarization sensitive terahertz metamaterial absorber,” Opt. Commun. 380, 101–107 (2016).
[Crossref]

Deng, Y.

Z. Song, M. Jiang, Y. Deng, and A. Chen, “Wide-angle Absorber with Tunable Intensity and Bandwidth Realized by a Terahertz Phase Change Material,” Opt. Commun. 464, 125494 (2020).
[Crossref]

Z. Song, Y. Deng, Y. Zhou, and Z. Liu, “Terahertz Toroidal Metamaterial with Tunable Properties,” Opt. Express 27(4), 5792–5797 (2019).
[Crossref]

M. Wei, Z. Song, Y. Deng, Y. Liu, and Q. Liu, “Large-angle mid-infrared absorption switch enabled by polarization-independent GST metasurfaces,” Mater. Lett. 236, 350–353 (2019).
[Crossref]

Dhillon, A. S.

A. S. Dhillon, D. Mittal, and R. Bargota, “Triple band ultrathin polarization insensitive metamaterial absorber for defense, explosive detection and airborne radar applications,” Microw Opt Technol Lett 61(1), 89–95 (2019).
[Crossref]

Ding, F.

F. Ding, Y. Yang, and I. B. Sergey, “Dynamic Metasurfaces Using Phase Change Chalcogenides,” Adv. Opt. Mater. 7(14), 1801709 (2019).
[Crossref]

Du, J.

J. Du, Y. Gao, H. Luo, L. Kang, and C. Cao, “Significant changes in phase-transition hysteresis for Ti-doped VO2 films prepared by polymer-assisted deposition,” Sol. Energy Mater. Sol. Cells 95(2), 469–475 (2011).
[Crossref]

Eghlidi, H.

C. Hail, A. Michel, D. Poulikakos, and H. Eghlidi, “Optical Metasurfaces: Evolving from Passive to Adaptive,” Adv. Opt. Mater. 7(14), 1801786 (2019).
[Crossref]

Escorcia, I.

Fan, F.

Fan, L.

J. Liu, L. Fan, J. Ku, and L. Mao, “Absorber: A novel terahertz sensor in the application of substance identification,” Opt. Quantum Electron. 48(2), 80 (2016).
[Crossref]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref]

Fardoost, A.

A. Fardoost, F. G. Vanani, A. Amirhosseini, and R. Safian, “Design of a multilayer graphene-based ultrawideband terahertz absorber,” IEEE Trans. Nanotechnol. 16(1), 1 (2016).
[Crossref]

Fischer, B.

P. Jepsen, B. Fischer, A. Thoman, H. Helm, J. Suh, R. Lopez, and R. Haglund, “Metal-insulator phase transition in a VO2 thin film observed with terahertz spectroscopy,” Phys. Rev. B 74(20), 205103 (2006).
[Crossref]

Fleischhauer, M.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref]

Fu, P.

P. Fu, F. Liu, G. J. Ren, F. Su, D. Li, and J. Q. Yao, “A broadband metamaterial absorber based on multi-layer graphene in the terahertz region,” Opt. Commun. 417, 62–66 (2018).
[Crossref]

Gao, B.

Gao, Y.

J. Du, Y. Gao, H. Luo, L. Kang, and C. Cao, “Significant changes in phase-transition hysteresis for Ti-doped VO2 films prepared by polymer-assisted deposition,” Sol. Energy Mater. Sol. Cells 95(2), 469–475 (2011).
[Crossref]

Geng, B. S.

L. Ju, B. S. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. G. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref]

Giessen, H.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref]

Girit, C.

L. Ju, B. S. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. G. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref]

Gong, R. Z.

M. L. Huang, Y. Z. Cheng, Z. Z. Cheng, H. R. Chen, X. S. Mao, and R. Z. Gong, “Based on graphene tunable dual-band terahertz metamaterial absorber with wide-angle,” Opt. Commun. 415, 194–201 (2018).
[Crossref]

Gough, J.

Grant, J.

Grigorenko, A. N.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene Plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

Gu, W. H.

Haglund, R.

P. Jepsen, B. Fischer, A. Thoman, H. Helm, J. Suh, R. Lopez, and R. Haglund, “Metal-insulator phase transition in a VO2 thin film observed with terahertz spectroscopy,” Phys. Rev. B 74(20), 205103 (2006).
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Figures (6)

Fig. 1.
Fig. 1. Three-dimensional schematic of the proposed switchable and tunable terahertz metamaterial absorber with broadband and multi-band absorption.
Fig. 2.
Fig. 2. Simulated absorbance of the proposed switchable metamaterial absorber under different VO2 phase states for both TE and TM polarizations at ${\mu _c} = 0.7\textrm{ }eV$ under normal terahertz incidence.
Fig. 3.
Fig. 3. Simulated front-view and side-view of the electric field distributions for TE polarization under the conditions of ${\mu _c} = 0.7\textrm{ }eV$ and normal incidence. (a) Electric field distributions of the proposed broadband absorber at 1.28 THz and 2.3 THz. (b) Electric field distributions of the proposed multi-band absorber at 1 THz, 2.45 THz, and 2.82 THz.
Fig. 4.
Fig. 4. Simulated absorption spectrum of the broadband (a-c) and multi-band (d-f) absorber as functions of frequency and the polarization angle φ and incident angle θ under the condition of ${\mu _c} = 0.7\textrm{ }eV$.
Fig. 5.
Fig. 5. Simulated absorption performance of the broadband (a-c) and multi-band (d-f) absorber as functions of frequency and graphene radius Rg, VO2 radius Rv, and VO2 thickness tVO2.
Fig. 6.
Fig. 6. Simulated absorption performances of the broadband (a) and multi-band (b) absorber as functions of frequency and the Fermi energy levels.

Equations (3)

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σ intra  ( ω , μ c , Γ , T ) = j e 2 π 2 ( ω j 2 Γ ) 0 f d ( ξ , μ c , T ) ξ f d ( ξ , μ c , T ) ξ ) ξ ξ ,
σ inter  ( ω , μ c , Γ , T ) = j e 2 ( ω j 2 Γ ) π 2 0 f d ( ξ , μ c , T ) f d ( ξ , μ c , T ) ( ω j 2 Γ ) 2 4 ξ / 2 ξ ,
ε ( ω ) = ε ω p 2 ( σ ) ω 2 + i γ ω ,

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