Abstract

An absorber based on hybrid metamaterial with vanadium dioxide and graphene has been proposed to achieve dynamically switchable dual-broadband absorption property in the terahertz regime. Due to the phase transition of vanadium dioxide and the electrical tunable property of graphene, the dynamically switchable dual-broadband absorption property is implemented. When the vanadium dioxide is in the metallic phase, the Fermi energy level of graphene is set as zero simultaneously, the high-frequency broadband from 2.05 THz to 4.30 THz can be achieved with the absorptance more than 90%. The tunable absorptance can be realized through thermal control on the conductivity of the vanadium dioxide. The proposed device acts as a low-frequency broadband absorber if the vanadium dioxide is in the insulating phase, for which the Fermi energy level of graphene varies from to 0.1 eV to 0.7 eV. The low-frequency broadband possesses high absorptance which is maintained above 90% from 1.10 THz to 2.30 THz. The absorption intensity can be continuously adjusted from 5.2% to 99.8% by electrically controlling the Fermi energy level of graphene. The absorption window can be further broadened by adjusting the geometrical parameters. Furthermore, the influence of incidence angle on the absorption spectra has been investigated. The proposed absorber has potential applications in the terahertz regime, such as filtering, sensing, cloaking objects, and switches.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Terahertz (THz) radiation ranging from 0.1 THz to 10 THz has attracted remarkable interest for its promising application in future wireless communication [1,2], imaging [3,4], sensing [5,6], etc. Metamaterials are artificially constructed electromagnetic (EM) materials composing of periodically arranged subwavelength microstructures. Metamaterial-based absorbers (MMAs) show promising applications in thermal emitters [7,8], solar cells [9,10], and invisible cloaks [11]. Research on terahertz MMAs leads to an attractive branch of terahertz devices due to their scalable features. Conventional MMAs are usually composed of a layer of periodically arranged metallic patterns, a dielectric interlayer, and a metallic ground plane. When the effective impedance matches with the impedance of the free space, maximum absorptance can be realized.

Narrow working bandwidth and fixed operating frequency range absorbers are limited in practical applications. Therefore, MMAs with broadband absorption and reconfigurable characteristics are more expected in recent years. The usual way to realize broadband absorption is combining more resonators in each meta-molecule with different sizes as well as stacking multiple resonator layers [1214]. Actively controlled elements, such as graphene [1517], vanadium dioxide (VO2) [18,19], doped silicon [20], germanium antimony telluride (GST) [21], liquid crystals [22], and liquid metals [23] are introduced in these metamaterials devices to realize the reconfigurable characteristics. Among these materials, graphene has become a massively attractive material for its unique electrical properties, such as tight field confinement, high electron mobility, and flexible tunability [24]. Graphene-based metamaterials [2527] provide an efficient approach to achieve the dynamically tunable devices, for the conductivity of graphene can be dynamically tuned by adjusting the Fermi energy through chemical doping [28] or electrostatic gating [29]. VO2 as a phase transition material undergoes a reversible transition behavior from the insulator phase to the metallic phase triggered by electrical [30], thermal [31], and optical excitation [32]. Consequently, VO2 is significantly attractive for plentiful intelligent devices to realize switchable and tunable functionalities. In recent years, tunable functional devices have been investigated based on the insulator-metal transition characteristic of VO2, such as transforming broadband absorption into polarization conversion [33,34], tunable broadband asymmetric transmission into perfect absorption [35], electromagnetically induced transparency into absorption [36], broadband absorption into narrowband absorption [37], and absorber into transparent conducting metal [38]. To the best of our knowledge, heretofore there are no devices that can achieve the dynamically switchable dual-broadband absorption in the THz regime.

In this work, the metamaterial absorber consisted of a square-shaped graphene layer loaded with identical VO2 square loops has been proposed to achieve dynamically switchable dual-broadband absorbing properties in the THz regime. Utilizing the insulator-to-metal phase transition of VO2 and the electrically tunable property of graphene, the absorbing property can be switched from low-frequency broadband to high-frequency broadband. When the VO2 is in the metallic state, the proposed absorber shows perfect high-frequency broadband absorbing property, and the intensity can be continuously tuned by VO2 under thermal control. When the VO2 is in the insulating state, a low-frequency broadband absorber is achieved with the bandwidth and the absorptance adjusted by the Fermi energy level of graphene. The field analyses, as well as the relative impedance, are also carried out to reveal the absorbing mechanism. Besides, the absorption performances at variable incidence angles for both TE and TM polarizations are investigated. All these results demonstrated that low- and high- frequency broadband with tunable absorptance can be dynamically switched by the same absorber which promises diverse applications in the THz regime.

2. Model construction and simulation

The proposed absorber consists of six layers: the VO2-graphene metamaterial layer, the polysilicon sheet, the first polyethylene cyclic olefin copolymer (Topas) layer, the VO2 film, the second Topas layer, and a 500-nm-thick gold (Au) film at the bottom serving as the metallic ground plane, as depicted in Fig. 1(a). Topas is a transparent and stiff amorphous thermoplastic copolymer with superior optical properties for advanced terahertz applications, which also has excellent heat resistance, high stability, a negligible absorption coefficient, and constant refractive index throughout the THz range [39,40]. The VO2-graphene metamaterial layer is composed of periodically arranged 200-nm-thick VO2 square loops and a square-shaped graphene layer with an ultra-thin (20 nm) Topas film used as the insulating spacer. Polysilicon with a relative dielectric permittivity of 3 is an ultra-pure form of silicon and composed of many small crystals. Attributed to its extremely excellent semiconductor characteristics, polysilicon can be used as an electrode to control the Fermi energy level of the graphene layer [39]. The influence of polysilicon layer on absorption is almost zero for its thickness being set as only 20 nm. The top and side view of the unit cell is shown in Figs. 1(b) and 1(c), respectively. The periods of the unit cell are Px on x-direction and Py on y-direction. The length and the width of the VO2 square loops are set as lVO2 and wVO2, respectively. The distances from the unit boundary of the connected graphene squares are w1 and w2. The Topas layer is separated by a continuous VO2 film, and the thicknesses from top to bottom are hd1, hVO2, and hd2. Unless otherwise specified, the geometric parameters are set as Px = Py = 38 µm, w1 = 4.5 µm, w2 = 18.5 µm, wVO2 = 4 µm, lVO2 = 23 µm, hd1= 15 µm, hd2= 9 µm and hVO2= 1 µm. The relative permittivity of Topas is 2.35 with negligible loss and dispersion in the THz regime.

 figure: Fig. 1.

Fig. 1. (a) Schematic of the proposed absorber based on VO2-graphene metamaterials and the incident light polarization configuration. (b) Top view of the unit cell. (c) Side view of the unit cell.

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The proposed absorber is numerically calculated by the finite-element solver COMSOL Multiphysics. The unit cell both in x and y directions is treated with the periodic boundary condition. The square-shaped graphene is modeled as a surface current, which is defined as the product of graphene conductivity and the tangential electric field on the graphene plane, through applying boundary conditions [4143]. A plane wave polarized parallel to the x-direction is used as the normal incident light. The complex surface conductivity of graphene can be described by Kubo formula: σg(ω, τ, µc)= σintra(ω, τ, µc)+ σinter(ω, τ, µc), where ω is the incident angular frequency, τ is the relaxation time, and µc is the chemical potential. The interband and intraband transition contributions are expressed as [44]

$${\sigma _{\textrm{inter} }}(\omega ,\tau ,{\mu _c}) \approx \frac{{j{e^2}}}{{4\pi \hbar }}\ln\left[ {\frac{{2|{{\mu_c}} |- ({\omega + {j / \tau }} )\hbar }}{{2|{{\mu_c}} |+ ({\omega + {j / \tau }} )\hbar }}} \right],$$
$${\sigma _{\textrm{intra} }}(\omega ,\tau ,{\mu _c}) \approx j\frac{{{e^2}{k_B}T}}{{\pi {\hbar ^2}({\omega + j{\tau^{ - 1}}} )}}\left[ {\frac{{{\mu_c}}}{{{k_B}T}} + 2\ln \left( {\textrm{exp} \left( { - \frac{{{\mu_c}}}{{{k_B}T}}} \right) + 1} \right)} \right],$$
where kB is the Boltzmann constant, $\hbar$ is the reduced Planck constant, and T is the temperature in kelvin. For kB < µc, µc is equal to the Fermi energy level Ef which can be defined as ${E_f} = \hbar {\upsilon _F}\sqrt {\pi n}$, where n is the carrier concentration. The relaxation time $\tau = \mu {E_f}{e^{ - 1}}{\upsilon _F}^{ - 2}$ characterizes the plasmon decay on account of impurities, where the Fermi velocity is νF = 106 m/s. The carrier mobility µ is 1500 cm2V−1s−1 corresponding to τ = 0.1 ps for Ef = 0.7 eV in this study.

The dielectric permittivity of VO2 can be described by the Drude model in the terahertz regime, which is expressed as follows [45]

$$\varepsilon (\omega ) = {\varepsilon _\infty } - \frac{{\omega _p^2(\sigma )}}{{{\omega ^2} + i\gamma \omega }},$$
where ɛ=12 is the dielectric permittivity at the infinite frequency, γ=5.75×1013 rad/s is the collision frequency, and ωρ(σ) is the plasma frequency dependent on conductivity, which can be approximately expressed as $\omega _p^2(\sigma ) = {{{\sigma ^2}{\omega _p}({\sigma _0})} / {\sigma _0^2}}$ with σ0=3×105 S/m and ωρ(σ0) = 1.4×1015 rad/s. In the simulation process, insulating phase and the metallic phase are described with conductivities of 0 S/m and 2×105 S/m, respectively. The conductivity of the gold is correspondingly described by the Drude model with the plasma frequency ωp = 1.36×1016 rad/s and the scattering rate Γ=3.33×1013 rad/s in this study [46].

3. Results and discussion

The transmission coefficient T(ω), reflection coefficient R(ω), and absorption coefficient A(ω) are calculated by S-parameters from the simulated results, and given as T(ω)=|S21|2, R(ω)=|S11|2 and A(ω) = 1-|S11|2-|S21|2, respectively. Figure 2 shows the calculated absorption spectrum of the proposed absorber for a different state, and the black rows show the resonance frequencies. The red solid curve shows the absorptance of the proposed absorber when the VO2 is in the insulating phase. Simultaneously, the Fermi energy level of the square-shaped graphene is set as 0.7 eV. The proposed absorber can realize low-frequency broadband from 1.10 THz to 2.30 THz with the absorptance exceeding 90%. When the VO2 is in conducting phase with the conductivity 2×105 S/m and the Fermi energy level of graphene is set as zero, the high-frequency broadband can be achieved in the frequency range of 2.05–4.30 THz, as shown by the blue dashed curve.

 figure: Fig. 2.

Fig. 2. The simulated absorption spectra of the proposed absorber for different VO2 states. The red solid curve illustrates the absorptance when VO2 is in the insulating phase. The blue dash curve illustrates the absorptance with wVO2 = 3 µm when VO2 is in the metallic phase.

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An energy-level diagram is utilized to further reveal the physical mechanism of the high-frequency broadband absorption, as shown in Fig. 3(a). For the specific case of VO2 square loop, the highly geometry-dependent plasmon response can be seen as an interaction between the essentially fixed-frequency plasmon response of a VO2 square and that of a VO2 cavity [29,47,48]. The induced charges distributed on inner and outer interfaces of the VO2 square loop originate from the VO2 square and cavity plasmons. The VO2 square and cavity plasmons interact with each other for finite width of VO2 square loop, and result in the splitting of the plasmon resonances into two new resonances: the dipolar bonding resonance and the dipolar antibonding resonance.

 figure: Fig. 3.

Fig. 3. (a) Energy-level diagram describing the plasmonic hybridization effect in VO2 square loops due to the interaction between the VO2 square and cavity plasmons. Distributions of electric field amplitude |E| (b) at 2.6 THz, and (c) 3.6 THz on the top surface of the VO2 square loops for the high-frequency broadband absorption.

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The distributions of electric field amplitude |E| for the high-frequency broadband absorber at its absorption peaks, i.e., 2.60 THz (first resonance), and 3.60 THz (second resonance), are shown in Figs. 3(b) and 3(c), respectively. Obviously, the electric field is mainly concentrated at the outside (inside) part of the VO2 loop at the first (second) resonance frequency. The white arrows point the direction of the electric field, and the size of the arrows represent the intensity. The directions of electric field indicate clearly that the induced charges of the plasmonic modes are “++−−” at 2.6 THz and “+−+−” at 3.6 THz, respectively. It implies that the first resonance mode at 2.6 THz is a bonding plasmon mode, as shown in Fig. 3(b). While the second resonance mode at 3.6 THz is an antibonding plasmon mode, as shown in Fig. 3(c). The two resonance modes from the VO2 square loops lead to the high-frequency broadband absorption of the proposed absorber.

The electric field and power flow distributions at different resonance frequencies are analyzed to reveal the mechanism of the low-frequency broadband absorption. At the first resonance frequency 1.35 THz, the electric field concentrates around the edges and in the gap between adjacent square-shaped graphene unit cells, as shown in Fig. 4(a). The distributions of the power flow and the z-component electric field Ez further prove the above statement. As shown in Fig. 4(e), the streams are concentrated and enhanced in the gap between adjacent square-shaped graphene unit cells, then curl in the Topas layer, and is finally absorbed. The distribution of z-component electric field Ez is presented in Fig. 4(c), and the intensity is forcefully enhanced in adjacent square-shaped graphene connection. From Fig. 4(g), one can clearly see that the magnetic fields are mainly distributed in the lower Topas spacer region, which demonstrates the excitation of magnetic dipolar resonance. Accordingly, the first resonance of the low-frequency broadband absorption is mainly caused by the magnetic dipolar resonance and the coupling interaction between the neighboring unit cells.

 figure: Fig. 4.

Fig. 4. Distributions of (a), (b) the electric field amplitude |E| and (c), (d) the z-component electric field Ez on the top surface of the square-shaped graphene for the low-frequency broadband absorption at 1.35 THz and 2.1 THz, respectively; Distributions of the power flow at the central cross section of unit cell (e) at 1.35 THz, and (f) 2.1 THz, respectively; Distributions of the magnetic field |H| and surface current at the central cross section of unit cell (g) at 1.35 THz, and (h) 2.1 THz, respectively.

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At the second resonance frequency 2.1 THz, the electric field concentrates entirely almost on the edges of the square-shaped graphene unit cells, as shown in Fig. 4(b), which indicates large charge accumulates at the edges of the unit cell. Most of the streams flow across the center region of the square-shaped graphene, then curl in the Topas layer, and back into the center region, as shown in Fig. 4(f). As shown in Fig. 4(d), opposite charges accumulate at left and right sides of the unit cell, indicating excitation of the electric dipole resonance in the patterned graphene layer. For clarity, we only marked the directions of the currents at the central cross section of unit cell with red arrows, and the distribution of magnetic field in Fig. 4(h) demonstrates the existence of the induced currents on the metallic ground plane. The antiparallel surface currents induce a strong magnetic resonance. Consequently, the second resonance mode of the low-frequency broadband absorption originates from the magnetic resonance and the electric dipole resonance caused by the individual graphene square. In general, electromagnetic energy can be stored by the strong resonances of the proposed absorber, then dissipated completely by ohmic loss of graphene layer, and the low-frequency broadband absorption is finally formed.

To comprehend the absorption characteristics of the proposed absorber, variable absorption spectra are simulated with different geometrical parameters. The parameters remain unchanged as initial settings except for the variable parameter. As shown in Fig. 5(a), the second resonance redshift and the absorptance of the two resonances are both attenuated as the thickness increases, which can be explained well by the effective capacitance between the VO2 loops and the ground plane, and impedance-matching with the free space. The bandwidth of the high-frequency broadband absorber shows a maximum of 2.75 THz when the thickness of the first Topas layer is set as 14 µm. It is clearly observed in Fig. 5(b) that the first resonance redshifts clearly and the absorptance of both the first and second resonance are enhanced with increasing lVO2. According to the LC circuit model, the resonance frequency of the proposed absorber is proportional to the effective length of the patterned structure [49]. When increasing lVO2 and sustaining the period of the unit cell, the localized field between the neighboring loops will enhanced, and the effective distance between the opposite charges will increase significantly as a result of the electric field of the first resonance frequency concentrates at the outside part of VO2 loops, thus leading to a clearly red shift of the first resonance. The simulated absorption spectra as a function of the width of the VO2 loops wVO2 is shown in Fig. 5(c). The first resonance remains nearly stable, while the second resonance appears blue shift, as the width of the VO2 loops wVO2 increases. Analogously, when wVO2 increases, the effective distance between the opposite charges will decrease significantly due to the electric field of the second resonance frequency concentrates at the inside part of VO2 loops, then the second resonance moves to higher frequency. In the meantime, the intensities of the absorption broadband are reduced, which can be explained by the impedance-matching theory for the increasing VO2 loops’ volume. The influence of the conductivity of VO2 on the absorption spectra is described in Fig. 5(d). The results show that the absorber has remarkable absorption performance when the conductivity of VO2 is set as 2×105 S/m, 1×105 S/m and 2×103 S/m. This phenomenon can be elucidated by the impedance matching theory in the Figs. 5(e) and 5(f).

 figure: Fig. 5.

Fig. 5. The simulated absorption spectra of the high-frequency broadband absorber as a function of (a) the thickness of the first Topas layer hd1, the length (b) lVO2 and the width (c) wVO2 of VO2 loops. (d) The absorption spectra vary with the conductivity of VO2 from 2×102 S/m to 2×105 S/m. (e) Real parts and (f) imaginary parts of the relative impedance Zr with different conductivities of VO2.

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The impedance matching theory is used to explain the physical mechanism of the absorption performance for the switchable broadband absorber. Due to the ground plane prevents the downward wave propagation, the transmittance T(w) = |S21|2 is very close to zero. The absorptance and the relative impedance with normal incidence can be described as

$$A(\omega ) = 1 - R(\omega ) = 1 - {\left|{\frac{{Z - {Z_0}}}{{Z + {Z_0}}}} \right|^2} = 1 - {\left|{\frac{{{Z_r} - 1}}{{{Z_r} + 1}}} \right|^2},$$
$${Z_r} = \sqrt {\frac{{{{(1 + {S_{11}}(\omega ))}^2} - {S_{21}}{{(\omega )}^2}}}{{{{(1 - {S_{11}}(\omega ))}^2} - {S_{21}}{{(\omega )}^2}}}},$$
where Z is the effective impedance, Z0 is the free space impedance, and Zr = Z / Z0 is the relative impedance between the proposed absorber and the free space. The real and imaginary parts of the relative impedance with varying conductivity of VO2 is shown in Figs. 5(e) and 5(f). It is obvious that the real parts close to 1 and the imaginary parts approach to 0 in the frequency range of 1.8–2.2 THz and 4.0–4.9 THz when the conductivity of VO2 is set as 2×105 S/m (black solid curve), 1×105 S/m (red das curve) and 2×103 S/m (purple dash curve), which means the impedances of the proposed absorber and the free space are nearly matched, as well as the absorptance reaches the maximum. The results were consistent with the absorption spectra in Fig. 5(d).

The absorption spectra of the low-frequency broadband absorber with various parameters are also investigated. This increased thickness of the second Topas layer hd2 leads to the bandwidth grow narrow and intensity of the spectra become progressively weaker, as shown in Fig. 6(a). That’s because, the coupling strength of the magnetic resonance response is mainly determined by the thickness of the dielectric layer. At the optimal value hd2 = 6 µm, the impedance of the proposed absorber matches that of free space, and the maximum bandwidth of the low-frequency broadband absorber 1.5 THz can be achieved. As shown in Fig. 6(b), when the smaller distance from the unit boundary of the connected graphene squares w1 increases, the first resonance has blue shift slightly and the second resonance remain almost unchanged, which results in narrower bandwidth. In Fig. 6(c), the absorption intensity of the first resonance is weakened, and the second resonance is enhanced as the greater distance from the unit boundary of the connected graphene squares w2 increases. The two resonance peaks of the low-frequency broadband absorption originate from the electric dipole resonance caused by the neighboring and individual graphene square, respectively. Thus, the various w1 and w2 affect the absorption spectra.

 figure: Fig. 6.

Fig. 6. The simulated absorption spectra of the low-frequency broadband absorber as a function of (a) the thickness of the second Topas layer hd2, the distances from the unit boundary of the connected graphene squares are (b) w1 and (c) w2.

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The simulated absorption spectra of the high-frequency broadband absorber as a function of the Fermi energy level are presented in Fig. 7. It is obvious that the bandwidth is broadening as the Fermi energy level increases. The maximum absorptance decreases to 5.2% when the Fermi energy level is set as 0.01 eV, and the maximum absorptance of 99.8% is obtained at the Fermi level of 0.6 eV. It indicates that the proposed low-frequency broadband absorber has a potential application in optical switch. When the Fermi energy level of graphene varied from 0.01 eV to 0.7 eV, the state of the proposed absorber can be switched from reflection (“OFF” > 94.8%) to absorption (“ON” > 90%) in the frequency range from 1.05 to 2.35 THz. The comparison of the absorption bandwidth (BW), fractional bandwidth, and the adjustment range of the absorption (from the maximum absorption intensity of state “OFF” to minimum absorption intensity of state “ON”) with some similar works are shown in Table 1. In addition, the fabrication and performance stability maybe the potential challenges of these graphene-based devices.

 figure: Fig. 7.

Fig. 7. The absorption spectra vary with the Fermi energy level from 0.01 eV to 0.7 eV.

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Tables Icon

Table 1. Comparison between the proposed device and the similar works.

Finally, the influences of oblique incidence on the absorption performance for both transverse electric (TE) and transverse magnetic (TM) polarizations are investigated. For high-frequency broadband absorber for TE polarization, the intensity remains larger than 80% as the incidence angle is below 60°, and the bandwidth broadened a little with the increasing incidence angle, as shown in Fig. 8(a). Nevertheless, the stable absorption of the high-frequency broadband absorber for TM polarization is disturbed when the incidence angle is over 20°, as shown in Fig. 8(b). The reason is that for TE polarization, the electric field is always parallel to x-axis, while the tangential component of the electric field decreases with the increasing incidence angle for TM polarization. Thus, the dipolar bonding resonance and the dipolar antibonding resonance of VO2 can be effectively excited for TE polarization, and the proposed absorber can matin a relatively more stable broadband absorption performance for TE polarization than that for TM polarization. The similar phenomenon of absorption spectrum with various incident angle can also be observed in previously reported work based on VO2 [37]. As shown in Figs. 8(c) and 8(d), the low-frequency broadband absorber exhibits excellent absorption performance with an intensity of more than 80% until the incidence angle varies up to 50° for TE polarization and 60° for TM polarization. The center frequency blue shifts slightly with the increase of incidence angle. The ratio of the center operating frequency to the period is λ/P = 176.5/38 = 4.6 at 1.7 THz, and this subwavelength property contributes to the wide-angle behavior. In general, the switchable broadband absorber indicates good polarization-independent property, i.e., the absorption properties for both TE and TM polarizations are almost the same. These results also show that good performance can be achieved for both TE and TM polarizations over a wide range of oblique incident angles, except that for the high-frequency broadband absorber for TM polarization, it is a little sensitive to the incidence angle.

 figure: Fig. 8.

Fig. 8. The absorption spectra of the high-frequency broadband absorber with different incidence angles for (a) TE polarization, (b) TM polarization, when VO2 in metallic phase. The absorption spectra of the low-frequency broadband absorber with different incidence angles for (c) TE polarization, (d) TM polarization, when VO2 in insulating phase. The 80% absorptance is indicated by the white contour curves.

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4. Conclusions

In conclusion, dynamically switchable dual-broadband absorber based on multilayer VO2-graphene metamaterials is numerically investigated in the THz regime. The proposed switchable absorber is attributed to the merging of the phase transition of VO2 and the electrical tunable property of graphene. When the VO2 loops are in the conducting state, and the Fermi energy level of graphene is set as zero simultaneously, the high-frequency broadband absorber can be achieved from 2.05 THz to 4.30 THz with the maximum absorptance nearly to 100%. When the Fermi energy level varies from 0.1 eV to 0.7 eV, for which the VO2 is in the insulating phase, the low-frequency broadband can be realized from 1.10 THz to 2.30 THz with an excellent absorptance of more than 90%. The absorptance can be continuously adjusted from 5.2% to 99.8% through adjusting the Fermi energy level of the graphene layer. The physical mechanism of the switchable dual-broadband absorption is investigated in detail, and the bandwidths can be further broadened by optimizing the geometrical parameters. In addition, the simulated results indicate that both the low- and high- frequency broadband have perfect absorption in a wide range of incidence angle for both TE and TM polarizations, but the high-frequency broadband is a little sensitive to the incidence angle for TM polarization comparatively. Profiting from these attractive properties, the dynamically switchable absorber may have promising applications in tunable filtering, sensing, and switches.

Funding

Natural Science Foundation of Guangdong Province (2020A1515011154); SZSF (20180123, JCYJ20190808151017218); National Natural Science Foundation of China (60877034, 61275043, 61307048, 61605128).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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24. F. J. G. D. Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics 1(3), 135–152 (2014). [CrossRef]  

25. H. Li, W. H. Xu, Q. Cui, Y. Wang, and J. Yu, “Theoretical design of a reconfigurable broadband integrated metamaterial terahertz device,” Opt. Express 28(26), 40060–40074 (2020). [CrossRef]  

26. H. Li and J. Yu, “Bifunctional terahertz absorber with a tunable and switchable property between broadband and dual-band,” Opt. Express 28(17), 25225–25237 (2020). [CrossRef]  

27. R. Mishra, A. Sahu, and R. Panwar, “Cascaded graphene frequency selective surface integrated tunable broadband terahertz metamaterial absorber,” IEEE Photonics J. 11(2), 1–10 (2019). [CrossRef]  

28. R. Yu, V. Pruneri, and F. J. G. D. Abajo, “Resonant visible light modulation with graphene,” ACS Photonics 2(4), 550–558 (2015). [CrossRef]  

29. Z. Y. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. L. Ma, Y. M. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013). [CrossRef]  

30. D. J. Park, J. H. Shin, K. H. Park, and H. C. Ryu, “Electrically controllable THz asymmetric split loop resonator with an outer square loop based on VO2,” Opt. Express 26(13), 17397–17406 (2018). [CrossRef]  

31. J. Liu and L. Fan, “Development of a tunable terahertz absorber based on temperature control,” Microw Opt Technol Lett 62(4), 1681–1685 (2020). [CrossRef]  

32. X. Tian and Z. Y. Li, “An optically-triggered switchable mid-infrared perfect absorber based on phase-change material of vanadium dioxide,” Plasmonics 13(4), 1393–1402 (2018). [CrossRef]  

33. H. He, X. Shang, L. Xu, J. Zhao, W. Cai, J. Wang, C. Zhao, and L. Wang, “Thermally switchable bifunctional plasmonic metasurface for perfect absorption and polarization conversion based on VO2,” Opt. Express 28(4), 4563–4570 (2020). [CrossRef]  

34. 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]  

35. T. L. Wang, H. Y. Zhang, Y. Zhang, Y. P. Zhang, and M. Y. Cao, “Tunable bifunctional terahertz metamaterial device based on Dirac semimetals and vanadium dioxide,” Opt. Express 28(12), 17434–17448 (2020). [CrossRef]  

36. 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]  

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

38. 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]  

39. H. Zhu, Y. Zhang, L. Ye, Y. Li, and R. Xu, “Switchable and tunable terahertz metamaterial absorber with broadband and multi-band absorption,” Opt. Express 28(26), 38626–38637 (2020). [CrossRef]  

40. Y. Harada, M. S. Ukhtary, M. Wang, S. K. Srinivasan, E. H. Hasdeo, A. R. T. Nugraha, G. T. Noe, Y. Sakai, R. Vajtai, P. M. Ajayan, R. Saito, and J. Kono, “Giant Terahertz-Wave Absorption by Monolayer Graphene in a Total Internal Reflection Geometry,” ACS Photonics 4(1), 121–126 (2017). [CrossRef]  

41. S. H. Mousavi, I. Kholmanov, K. B. Alici, D. Purtseladze, N. Arju, K. Tatar, D. Y. Fozdar, J. W. Suk, Y. F. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, “Inductive tuning of fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared,” Nano Lett. 13(3), 1111–1117 (2013). [CrossRef]  

42. N. Dabidian, S. Dutta-Gupta, I. Kholmanov, K. Lai, F. Lu, J. Lee, M. Jin, S. Trendafilov, A. Khanikaev, B. Fallahazad, E. Tutuc, M. A. Belkin, and G. Shvets, “Experimental demonstration of phase modulation and motion sensing using graphene-integrated metasurfaces,” Nano Lett. 16(6), 3607–3615 (2016). [CrossRef]  

43. E. Galiffi, J. B. Pendry, and P. A. Huidobro, “Broadband tunable THz absorption with singular graphene metasurfaces,” ACS Nano 12(2), 1006–1013 (2018). [CrossRef]  

44. B. S. Rodriguez, R. Yan, M. M. Kelly, T. Fang, K. Tahy, W. S. Hwang, D. Jena, L. Liu, and H. G. Xing, “Broadband graphene terahertz modulators enabled by intraband transitions,” Nat. Commun. 3(1), 780 (2012). [CrossRef]  

45. M. Liu, H. Y. Hwang, H. Tao, A. C. Strikwerda, K. Fan, G. R. Keiser, A. J. Sternbach, K. G. West, S. Kittiwatanakul, J. Lu, S. A. Wolf, F. G. Omenetto, X. Zhang, K. A. Nelson, and R. D. Averitt, “Terahertz-field-induced insulator-to-metal transition in vanadium dioxide metamaterial,” Nature 487(7407), 345–348 (2012). [CrossRef]  

46. G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008). [CrossRef]  

47. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003). [CrossRef]  

48. N. L. Mou, S. L. Sun, H. X. Dong, S. H. Dong, Q. He, L. Zhou, and L. Zhang, “Hybridization-induced broadband terahertz wave absorption with graphene metasurfaces,” Opt. Express 26(9), 11728–11736 (2018). [CrossRef]  

49. Y. Q. Ye, Y. Jin, and S. He, “Omnidirectional, polarization-insensitive and broadband thin absorber in the terahertz regime,” J. Opt. Soc. Am. B 27(3), 498 (2010). [CrossRef]  

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

51. Y. T. Zhao, B. Wu, B. J. Huang, and Q. Cheng, “Switchable broadband terahertz absorber/reflector enabled by hybrid graphene-gold metasurface,” Opt. Express 25(7), 7161–7169 (2017). [CrossRef]  

52. H. Feng, Z. X. Xu, K. Li, M. Wang, W. L. Xie, Q. P. Luo, B. Y. Chen, W. J. Kong, and M. J. Yun, “Tunable polarization-independent and angle-insensitive broadband terahertz absorber with graphene metamaterials,” Opt. Express 29(5), 7158–7167 (2021). [CrossRef]  

References

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  28. R. Yu, V. Pruneri, and F. J. G. D. Abajo, “Resonant visible light modulation with graphene,” ACS Photonics 2(4), 550–558 (2015).
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  30. D. J. Park, J. H. Shin, K. H. Park, and H. C. Ryu, “Electrically controllable THz asymmetric split loop resonator with an outer square loop based on VO2,” Opt. Express 26(13), 17397–17406 (2018).
    [Crossref]
  31. J. Liu and L. Fan, “Development of a tunable terahertz absorber based on temperature control,” Microw Opt Technol Lett 62(4), 1681–1685 (2020).
    [Crossref]
  32. X. Tian and Z. Y. Li, “An optically-triggered switchable mid-infrared perfect absorber based on phase-change material of vanadium dioxide,” Plasmonics 13(4), 1393–1402 (2018).
    [Crossref]
  33. H. He, X. Shang, L. Xu, J. Zhao, W. Cai, J. Wang, C. Zhao, and L. Wang, “Thermally switchable bifunctional plasmonic metasurface for perfect absorption and polarization conversion based on VO2,” Opt. Express 28(4), 4563–4570 (2020).
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  35. T. L. Wang, H. Y. Zhang, Y. Zhang, Y. P. Zhang, and M. Y. Cao, “Tunable bifunctional terahertz metamaterial device based on Dirac semimetals and vanadium dioxide,” Opt. Express 28(12), 17434–17448 (2020).
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  37. Z. Song, A. Chen, and J. Zhang, “Terahertz switching between broadband absorption and narrowband absorption,” Opt. Express 28(2), 2037–2044 (2020).
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  38. L. Chen and Z. Song, “Simultaneous realizations of absorber and transparent conducting metal in a single metamaterial,” Opt. Express 28(5), 6565–6571 (2020).
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    [Crossref]
  43. E. Galiffi, J. B. Pendry, and P. A. Huidobro, “Broadband tunable THz absorption with singular graphene metasurfaces,” ACS Nano 12(2), 1006–1013 (2018).
    [Crossref]
  44. B. S. Rodriguez, R. Yan, M. M. Kelly, T. Fang, K. Tahy, W. S. Hwang, D. Jena, L. Liu, and H. G. Xing, “Broadband graphene terahertz modulators enabled by intraband transitions,” Nat. Commun. 3(1), 780 (2012).
    [Crossref]
  45. M. Liu, H. Y. Hwang, H. Tao, A. C. Strikwerda, K. Fan, G. R. Keiser, A. J. Sternbach, K. G. West, S. Kittiwatanakul, J. Lu, S. A. Wolf, F. G. Omenetto, X. Zhang, K. A. Nelson, and R. D. Averitt, “Terahertz-field-induced insulator-to-metal transition in vanadium dioxide metamaterial,” Nature 487(7407), 345–348 (2012).
    [Crossref]
  46. G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
    [Crossref]
  47. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
    [Crossref]
  48. N. L. Mou, S. L. Sun, H. X. Dong, S. H. Dong, Q. He, L. Zhou, and L. Zhang, “Hybridization-induced broadband terahertz wave absorption with graphene metasurfaces,” Opt. Express 26(9), 11728–11736 (2018).
    [Crossref]
  49. Y. Q. Ye, Y. Jin, and S. He, “Omnidirectional, polarization-insensitive and broadband thin absorber in the terahertz regime,” J. Opt. Soc. Am. B 27(3), 498 (2010).
    [Crossref]
  50. L. Ye, Y. Chen, G. Cai, N. Liu, J. Zhu, Z. Song, and Q. H. Liu, “Broadband absorber with periodically sinusoidally-patterned graphene layer in terahertz range,” Opt. Express 25(10), 11223–11232 (2017).
    [Crossref]
  51. Y. T. Zhao, B. Wu, B. J. Huang, and Q. Cheng, “Switchable broadband terahertz absorber/reflector enabled by hybrid graphene-gold metasurface,” Opt. Express 25(7), 7161–7169 (2017).
    [Crossref]
  52. H. Feng, Z. X. Xu, K. Li, M. Wang, W. L. Xie, Q. P. Luo, B. Y. Chen, W. J. Kong, and M. J. Yun, “Tunable polarization-independent and angle-insensitive broadband terahertz absorber with graphene metamaterials,” Opt. Express 29(5), 7158–7167 (2021).
    [Crossref]

2021 (2)

2020 (12)

J. Liu and L. Fan, “Development of a tunable terahertz absorber based on temperature control,” Microw Opt Technol Lett 62(4), 1681–1685 (2020).
[Crossref]

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

H. Li, W. H. Xu, Q. Cui, Y. Wang, and J. Yu, “Theoretical design of a reconfigurable broadband integrated metamaterial terahertz device,” Opt. Express 28(26), 40060–40074 (2020).
[Crossref]

H. Li and J. Yu, “Bifunctional terahertz absorber with a tunable and switchable property between broadband and dual-band,” Opt. Express 28(17), 25225–25237 (2020).
[Crossref]

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

A. Chen and Z. Song, “Tunable isotropic absorber with phase change material VO2,” IEEE T. Nanotechnol. 19, 197–200 (2020).
[Crossref]

H. He, X. Shang, L. Xu, J. Zhao, W. Cai, J. Wang, C. Zhao, and L. Wang, “Thermally switchable bifunctional plasmonic metasurface for perfect absorption and polarization conversion based on VO2,” Opt. Express 28(4), 4563–4570 (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]

T. L. Wang, H. Y. Zhang, Y. Zhang, Y. P. Zhang, and M. Y. Cao, “Tunable bifunctional terahertz metamaterial device based on Dirac semimetals and vanadium dioxide,” Opt. Express 28(12), 17434–17448 (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]

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]

H. Zhu, Y. Zhang, L. Ye, Y. Li, and R. Xu, “Switchable and tunable terahertz metamaterial absorber with broadband and multi-band absorption,” Opt. Express 28(26), 38626–38637 (2020).
[Crossref]

2019 (4)

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]

R. Mishra, A. Sahu, and R. Panwar, “Cascaded graphene frequency selective surface integrated tunable broadband terahertz metamaterial absorber,” IEEE Photonics J. 11(2), 1–10 (2019).
[Crossref]

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

Y. Liu, R. B. Zhong, J. B. Huang, Y. L. Lv, and S. G. Liu, “Independently tunable multi-band and ultra-wide-band absorbers based on multilayer metal-graphene metamaterials,” Opt. Express 27(5), 7393–7404 (2019).
[Crossref]

2018 (5)

Y. Liu, R. B. Zhong, Z. Lian, C. Bu, and S. G. Liu, “Dynamically tunable band stop filter enabled by the metal-graphene metamaterials,” Sci. Rep. 8(1), 2828 (2018).
[Crossref]

D. J. Park, J. H. Shin, K. H. Park, and H. C. Ryu, “Electrically controllable THz asymmetric split loop resonator with an outer square loop based on VO2,” Opt. Express 26(13), 17397–17406 (2018).
[Crossref]

X. Tian and Z. Y. Li, “An optically-triggered switchable mid-infrared perfect absorber based on phase-change material of vanadium dioxide,” Plasmonics 13(4), 1393–1402 (2018).
[Crossref]

N. L. Mou, S. L. Sun, H. X. Dong, S. H. Dong, Q. He, L. Zhou, and L. Zhang, “Hybridization-induced broadband terahertz wave absorption with graphene metasurfaces,” Opt. Express 26(9), 11728–11736 (2018).
[Crossref]

E. Galiffi, J. B. Pendry, and P. A. Huidobro, “Broadband tunable THz absorption with singular graphene metasurfaces,” ACS Nano 12(2), 1006–1013 (2018).
[Crossref]

2017 (5)

2016 (3)

P. Dean, O. Mitrofanov, J. Keeley, I. Kundu, L. Li, E. H. Linfield, and A. G. Davies, “Apertureless near-field terahertz imaging using the self-mixing effect in a quantum cascade laser,” Appl. Phys. Lett. 108(9), 091113 (2016).
[Crossref]

H. K. Kim, D. Lee, and S. Lim, “Wideband-switchable metamaterial absorber using injected liquid metal,” Sci. Rep. 6(1), 31823 (2016).
[Crossref]

N. Dabidian, S. Dutta-Gupta, I. Kholmanov, K. Lai, F. Lu, J. Lee, M. Jin, S. Trendafilov, A. Khanikaev, B. Fallahazad, E. Tutuc, M. A. Belkin, and G. Shvets, “Experimental demonstration of phase modulation and motion sensing using graphene-integrated metasurfaces,” Nano Lett. 16(6), 3607–3615 (2016).
[Crossref]

2015 (2)

X. Ni, Z. J. Wong, M. Mrejen, Y. Wang, and X. Zhang, “An ultrathin invisibility skin cloak for visible light,” Science 349(6254), 1310–1314 (2015).
[Crossref]

R. Yu, V. Pruneri, and F. J. G. D. Abajo, “Resonant visible light modulation with graphene,” ACS Photonics 2(4), 550–558 (2015).
[Crossref]

2014 (2)

F. J. G. D. Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics 1(3), 135–152 (2014).
[Crossref]

I. F. Akyildiz, J. M. Jornet, and C. Han, “Terahertz band: Next frontier for wireless communications,” Phys. Commun. 12, 16–32 (2014).
[Crossref]

2013 (4)

B. Vasic and R. Gajic, “Graphene induced spectral tuning of metamaterial absorbers at mid-infrared frequencies,” Appl. Phys. Lett. 103(26), 261111 (2013).
[Crossref]

Z. Y. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. L. Ma, Y. M. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013).
[Crossref]

A. Andryieuski and A. V. Lavrinenko, “Graphene metamaterials based tunable terahertz absorber: effective surface conductivity approach,” Opt. Express 21(7), 9144–9155 (2013).
[Crossref]

S. H. Mousavi, I. Kholmanov, K. B. Alici, D. Purtseladze, N. Arju, K. Tatar, D. Y. Fozdar, J. W. Suk, Y. F. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, “Inductive tuning of fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared,” Nano Lett. 13(3), 1111–1117 (2013).
[Crossref]

2012 (5)

F. Alves, D. Grbovic, B. Kearney, and G. Karunasiri, “Microelectromechanical systems bimaterial terahertz sensor with integrated metamaterial absorber,” Opt. Lett. 37(11), 1886–1888 (2012).
[Crossref]

F. Alves, B. Kearney, D. Grbovic, and G. Karunasiri, “Narrowband terahertz emitters using metamaterial films,” Opt. Express 20(19), 21025–21032 (2012).
[Crossref]

Y. Wang, T. Sun, T. Paude, Y. Zhang, Z. Ren, and K. Kempa, “Metamaterial-plasmonic absorber structure for high efficiency amorphous silicon solar cells,” Nano Lett. 12(1), 440–445 (2012).
[Crossref]

B. S. Rodriguez, R. Yan, M. M. Kelly, T. Fang, K. Tahy, W. S. Hwang, D. Jena, L. Liu, and H. G. Xing, “Broadband graphene terahertz modulators enabled by intraband transitions,” Nat. Commun. 3(1), 780 (2012).
[Crossref]

M. Liu, H. Y. Hwang, H. Tao, A. C. Strikwerda, K. Fan, G. R. Keiser, A. J. Sternbach, K. G. West, S. Kittiwatanakul, J. Lu, S. A. Wolf, F. G. Omenetto, X. Zhang, K. A. Nelson, and R. D. Averitt, “Terahertz-field-induced insulator-to-metal transition in vanadium dioxide metamaterial,” Nature 487(7407), 345–348 (2012).
[Crossref]

2011 (1)

2010 (2)

J. Federici and L. Moeller, “Review of terahertz and subterahertz wireless communications,” J. Appl. Phys. 107(11), 111101 (2010).
[Crossref]

Y. Q. Ye, Y. Jin, and S. He, “Omnidirectional, polarization-insensitive and broadband thin absorber in the terahertz regime,” J. Opt. Soc. Am. B 27(3), 498 (2010).
[Crossref]

2009 (1)

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[Crossref]

2008 (2)

E. Rephaeli and S. Fan, “Tungsten black absorber for solar light with wide angular operation range,” Appl. Phys. Lett. 92(21), 211107 (2008).
[Crossref]

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
[Crossref]

2006 (1)

D. Seliuta, I. Kasalynas, V. Tamosiunas, S. Balakauskas, Z. Martunas, and S. Asmontas, “Silicon lens-coupled bow-tie ingaas-based broadband terahertz sensor operating at room temperature,” Electron. Lett. 42(14), 825–827 (2006).
[Crossref]

2003 (1)

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref]

Abajo, F. J. G.

Z. Y. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. L. Ma, Y. M. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013).
[Crossref]

Abajo, F. J. G. D.

R. Yu, V. Pruneri, and F. J. G. D. Abajo, “Resonant visible light modulation with graphene,” ACS Photonics 2(4), 550–558 (2015).
[Crossref]

F. J. G. D. Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics 1(3), 135–152 (2014).
[Crossref]

Ajayan, P. M.

Y. Harada, M. S. Ukhtary, M. Wang, S. K. Srinivasan, E. H. Hasdeo, A. R. T. Nugraha, G. T. Noe, Y. Sakai, R. Vajtai, P. M. Ajayan, R. Saito, and J. Kono, “Giant Terahertz-Wave Absorption by Monolayer Graphene in a Total Internal Reflection Geometry,” ACS Photonics 4(1), 121–126 (2017).
[Crossref]

Z. Y. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. L. Ma, Y. M. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013).
[Crossref]

Akyildiz, I. F.

I. F. Akyildiz, J. M. Jornet, and C. Han, “Terahertz band: Next frontier for wireless communications,” Phys. Commun. 12, 16–32 (2014).
[Crossref]

Alici, K. B.

S. H. Mousavi, I. Kholmanov, K. B. Alici, D. Purtseladze, N. Arju, K. Tatar, D. Y. Fozdar, J. W. Suk, Y. F. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, “Inductive tuning of fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared,” Nano Lett. 13(3), 1111–1117 (2013).
[Crossref]

Alves, F.

Andryieuski, A.

Arju, N.

S. H. Mousavi, I. Kholmanov, K. B. Alici, D. Purtseladze, N. Arju, K. Tatar, D. Y. Fozdar, J. W. Suk, Y. F. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, “Inductive tuning of fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared,” Nano Lett. 13(3), 1111–1117 (2013).
[Crossref]

Asmontas, S.

D. Seliuta, I. Kasalynas, V. Tamosiunas, S. Balakauskas, Z. Martunas, and S. Asmontas, “Silicon lens-coupled bow-tie ingaas-based broadband terahertz sensor operating at room temperature,” Electron. Lett. 42(14), 825–827 (2006).
[Crossref]

Averitt, R. D.

M. Liu, H. Y. Hwang, H. Tao, A. C. Strikwerda, K. Fan, G. R. Keiser, A. J. Sternbach, K. G. West, S. Kittiwatanakul, J. Lu, S. A. Wolf, F. G. Omenetto, X. Zhang, K. A. Nelson, and R. D. Averitt, “Terahertz-field-induced insulator-to-metal transition in vanadium dioxide metamaterial,” Nature 487(7407), 345–348 (2012).
[Crossref]

Balakauskas, S.

D. Seliuta, I. Kasalynas, V. Tamosiunas, S. Balakauskas, Z. Martunas, and S. Asmontas, “Silicon lens-coupled bow-tie ingaas-based broadband terahertz sensor operating at room temperature,” Electron. Lett. 42(14), 825–827 (2006).
[Crossref]

Belkin, M. A.

N. Dabidian, S. Dutta-Gupta, I. Kholmanov, K. Lai, F. Lu, J. Lee, M. Jin, S. Trendafilov, A. Khanikaev, B. Fallahazad, E. Tutuc, M. A. Belkin, and G. Shvets, “Experimental demonstration of phase modulation and motion sensing using graphene-integrated metasurfaces,” Nano Lett. 16(6), 3607–3615 (2016).
[Crossref]

Bu, C.

Y. Liu, R. B. Zhong, Z. Lian, C. Bu, and S. G. Liu, “Dynamically tunable band stop filter enabled by the metal-graphene metamaterials,” Sci. Rep. 8(1), 2828 (2018).
[Crossref]

Cai, G.

Cai, W.

Cao, M. Y.

Chen, A.

Chen, B. Y.

Chen, L.

Chen, Q.

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

Chen, Y.

Cheng, Q.

Cui, Q.

Cumming, D. R.

Dabidian, N.

N. Dabidian, S. Dutta-Gupta, I. Kholmanov, K. Lai, F. Lu, J. Lee, M. Jin, S. Trendafilov, A. Khanikaev, B. Fallahazad, E. Tutuc, M. A. Belkin, and G. Shvets, “Experimental demonstration of phase modulation and motion sensing using graphene-integrated metasurfaces,” Nano Lett. 16(6), 3607–3615 (2016).
[Crossref]

Davies, A. G.

P. Dean, O. Mitrofanov, J. Keeley, I. Kundu, L. Li, E. H. Linfield, and A. G. Davies, “Apertureless near-field terahertz imaging using the self-mixing effect in a quantum cascade laser,” Appl. Phys. Lett. 108(9), 091113 (2016).
[Crossref]

Dean, P.

P. Dean, O. Mitrofanov, J. Keeley, I. Kundu, L. Li, E. H. Linfield, and A. G. Davies, “Apertureless near-field terahertz imaging using the self-mixing effect in a quantum cascade laser,” Appl. Phys. Lett. 108(9), 091113 (2016).
[Crossref]

Deng, Y.

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

Diem, M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[Crossref]

Dong, H. X.

Dong, S. H.

Dutta-Gupta, S.

N. Dabidian, S. Dutta-Gupta, I. Kholmanov, K. Lai, F. Lu, J. Lee, M. Jin, S. Trendafilov, A. Khanikaev, B. Fallahazad, E. Tutuc, M. A. Belkin, and G. Shvets, “Experimental demonstration of phase modulation and motion sensing using graphene-integrated metasurfaces,” Nano Lett. 16(6), 3607–3615 (2016).
[Crossref]

Fallahazad, B.

N. Dabidian, S. Dutta-Gupta, I. Kholmanov, K. Lai, F. Lu, J. Lee, M. Jin, S. Trendafilov, A. Khanikaev, B. Fallahazad, E. Tutuc, M. A. Belkin, and G. Shvets, “Experimental demonstration of phase modulation and motion sensing using graphene-integrated metasurfaces,” Nano Lett. 16(6), 3607–3615 (2016).
[Crossref]

Fan, K.

J. Y. Suen, K. Fan, W. J. Padilla, and X. Liu, “All-dielectric metasurface absorbers for uncooled terahertz imaging,” Optica 4(6), 601–604 (2017).
[Crossref]

M. Liu, H. Y. Hwang, H. Tao, A. C. Strikwerda, K. Fan, G. R. Keiser, A. J. Sternbach, K. G. West, S. Kittiwatanakul, J. Lu, S. A. Wolf, F. G. Omenetto, X. Zhang, K. A. Nelson, and R. D. Averitt, “Terahertz-field-induced insulator-to-metal transition in vanadium dioxide metamaterial,” Nature 487(7407), 345–348 (2012).
[Crossref]

Fan, L.

J. Liu and L. Fan, “Development of a tunable terahertz absorber based on temperature control,” Microw Opt Technol Lett 62(4), 1681–1685 (2020).
[Crossref]

Fan, S.

E. Rephaeli and S. Fan, “Tungsten black absorber for solar light with wide angular operation range,” Appl. Phys. Lett. 92(21), 211107 (2008).
[Crossref]

Fang, T.

B. S. Rodriguez, R. Yan, M. M. Kelly, T. Fang, K. Tahy, W. S. Hwang, D. Jena, L. Liu, and H. G. Xing, “Broadband graphene terahertz modulators enabled by intraband transitions,” Nat. Commun. 3(1), 780 (2012).
[Crossref]

Fang, Z. Y.

Z. Y. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. L. Ma, Y. M. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013).
[Crossref]

Federici, J.

J. Federici and L. Moeller, “Review of terahertz and subterahertz wireless communications,” J. Appl. Phys. 107(11), 111101 (2010).
[Crossref]

Feng, H.

Fozdar, D. Y.

S. H. Mousavi, I. Kholmanov, K. B. Alici, D. Purtseladze, N. Arju, K. Tatar, D. Y. Fozdar, J. W. Suk, Y. F. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, “Inductive tuning of fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared,” Nano Lett. 13(3), 1111–1117 (2013).
[Crossref]

Gajic, R.

B. Vasic and R. Gajic, “Graphene induced spectral tuning of metamaterial absorbers at mid-infrared frequencies,” Appl. Phys. Lett. 103(26), 261111 (2013).
[Crossref]

Galiffi, E.

E. Galiffi, J. B. Pendry, and P. A. Huidobro, “Broadband tunable THz absorption with singular graphene metasurfaces,” ACS Nano 12(2), 1006–1013 (2018).
[Crossref]

Ge, S.

Grant, J.

Grbovic, D.

Halas, N. J.

Z. Y. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. L. Ma, Y. M. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013).
[Crossref]

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref]

Han, C.

I. F. Akyildiz, J. M. Jornet, and C. Han, “Terahertz band: Next frontier for wireless communications,” Phys. Commun. 12, 16–32 (2014).
[Crossref]

Hanson, G. W.

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
[Crossref]

Hao, Y. F.

S. H. Mousavi, I. Kholmanov, K. B. Alici, D. Purtseladze, N. Arju, K. Tatar, D. Y. Fozdar, J. W. Suk, Y. F. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, “Inductive tuning of fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared,” Nano Lett. 13(3), 1111–1117 (2013).
[Crossref]

Harada, Y.

Y. Harada, M. S. Ukhtary, M. Wang, S. K. Srinivasan, E. H. Hasdeo, A. R. T. Nugraha, G. T. Noe, Y. Sakai, R. Vajtai, P. M. Ajayan, R. Saito, and J. Kono, “Giant Terahertz-Wave Absorption by Monolayer Graphene in a Total Internal Reflection Geometry,” ACS Photonics 4(1), 121–126 (2017).
[Crossref]

Hasdeo, E. H.

Y. Harada, M. S. Ukhtary, M. Wang, S. K. Srinivasan, E. H. Hasdeo, A. R. T. Nugraha, G. T. Noe, Y. Sakai, R. Vajtai, P. M. Ajayan, R. Saito, and J. Kono, “Giant Terahertz-Wave Absorption by Monolayer Graphene in a Total Internal Reflection Geometry,” ACS Photonics 4(1), 121–126 (2017).
[Crossref]

He, H.

He, Q.

He, S.

Hu, W.

Huang, B. J.

Huang, J. B.

Huidobro, P. A.

E. Galiffi, J. B. Pendry, and P. A. Huidobro, “Broadband tunable THz absorption with singular graphene metasurfaces,” ACS Nano 12(2), 1006–1013 (2018).
[Crossref]

Hwang, H. Y.

M. Liu, H. Y. Hwang, H. Tao, A. C. Strikwerda, K. Fan, G. R. Keiser, A. J. Sternbach, K. G. West, S. Kittiwatanakul, J. Lu, S. A. Wolf, F. G. Omenetto, X. Zhang, K. A. Nelson, and R. D. Averitt, “Terahertz-field-induced insulator-to-metal transition in vanadium dioxide metamaterial,” Nature 487(7407), 345–348 (2012).
[Crossref]

Hwang, W. S.

B. S. Rodriguez, R. Yan, M. M. Kelly, T. Fang, K. Tahy, W. S. Hwang, D. Jena, L. Liu, and H. G. Xing, “Broadband graphene terahertz modulators enabled by intraband transitions,” Nat. Commun. 3(1), 780 (2012).
[Crossref]

Jena, D.

B. S. Rodriguez, R. Yan, M. M. Kelly, T. Fang, K. Tahy, W. S. Hwang, D. Jena, L. Liu, and H. G. Xing, “Broadband graphene terahertz modulators enabled by intraband transitions,” Nat. Commun. 3(1), 780 (2012).
[Crossref]

Jiang, M.

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

Jin, M.

N. Dabidian, S. Dutta-Gupta, I. Kholmanov, K. Lai, F. Lu, J. Lee, M. Jin, S. Trendafilov, A. Khanikaev, B. Fallahazad, E. Tutuc, M. A. Belkin, and G. Shvets, “Experimental demonstration of phase modulation and motion sensing using graphene-integrated metasurfaces,” Nano Lett. 16(6), 3607–3615 (2016).
[Crossref]

Jin, Y.

Jornet, J. M.

I. F. Akyildiz, J. M. Jornet, and C. Han, “Terahertz band: Next frontier for wireless communications,” Phys. Commun. 12, 16–32 (2014).
[Crossref]

Karunasiri, G.

Kasalynas, I.

D. Seliuta, I. Kasalynas, V. Tamosiunas, S. Balakauskas, Z. Martunas, and S. Asmontas, “Silicon lens-coupled bow-tie ingaas-based broadband terahertz sensor operating at room temperature,” Electron. Lett. 42(14), 825–827 (2006).
[Crossref]

Kearney, B.

Keeley, J.

P. Dean, O. Mitrofanov, J. Keeley, I. Kundu, L. Li, E. H. Linfield, and A. G. Davies, “Apertureless near-field terahertz imaging using the self-mixing effect in a quantum cascade laser,” Appl. Phys. Lett. 108(9), 091113 (2016).
[Crossref]

Keiser, G. R.

M. Liu, H. Y. Hwang, H. Tao, A. C. Strikwerda, K. Fan, G. R. Keiser, A. J. Sternbach, K. G. West, S. Kittiwatanakul, J. Lu, S. A. Wolf, F. G. Omenetto, X. Zhang, K. A. Nelson, and R. D. Averitt, “Terahertz-field-induced insulator-to-metal transition in vanadium dioxide metamaterial,” Nature 487(7407), 345–348 (2012).
[Crossref]

Kelly, M. M.

B. S. Rodriguez, R. Yan, M. M. Kelly, T. Fang, K. Tahy, W. S. Hwang, D. Jena, L. Liu, and H. G. Xing, “Broadband graphene terahertz modulators enabled by intraband transitions,” Nat. Commun. 3(1), 780 (2012).
[Crossref]

Kempa, K.

Y. Wang, T. Sun, T. Paude, Y. Zhang, Z. Ren, and K. Kempa, “Metamaterial-plasmonic absorber structure for high efficiency amorphous silicon solar cells,” Nano Lett. 12(1), 440–445 (2012).
[Crossref]

Khalid, A.

Khanikaev, A.

N. Dabidian, S. Dutta-Gupta, I. Kholmanov, K. Lai, F. Lu, J. Lee, M. Jin, S. Trendafilov, A. Khanikaev, B. Fallahazad, E. Tutuc, M. A. Belkin, and G. Shvets, “Experimental demonstration of phase modulation and motion sensing using graphene-integrated metasurfaces,” Nano Lett. 16(6), 3607–3615 (2016).
[Crossref]

Khanikaev, A. B.

S. H. Mousavi, I. Kholmanov, K. B. Alici, D. Purtseladze, N. Arju, K. Tatar, D. Y. Fozdar, J. W. Suk, Y. F. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, “Inductive tuning of fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared,” Nano Lett. 13(3), 1111–1117 (2013).
[Crossref]

Kholmanov, I.

N. Dabidian, S. Dutta-Gupta, I. Kholmanov, K. Lai, F. Lu, J. Lee, M. Jin, S. Trendafilov, A. Khanikaev, B. Fallahazad, E. Tutuc, M. A. Belkin, and G. Shvets, “Experimental demonstration of phase modulation and motion sensing using graphene-integrated metasurfaces,” Nano Lett. 16(6), 3607–3615 (2016).
[Crossref]

S. H. Mousavi, I. Kholmanov, K. B. Alici, D. Purtseladze, N. Arju, K. Tatar, D. Y. Fozdar, J. W. Suk, Y. F. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, “Inductive tuning of fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared,” Nano Lett. 13(3), 1111–1117 (2013).
[Crossref]

Kim, H. K.

H. K. Kim, D. Lee, and S. Lim, “Wideband-switchable metamaterial absorber using injected liquid metal,” Sci. Rep. 6(1), 31823 (2016).
[Crossref]

Kittiwatanakul, S.

M. Liu, H. Y. Hwang, H. Tao, A. C. Strikwerda, K. Fan, G. R. Keiser, A. J. Sternbach, K. G. West, S. Kittiwatanakul, J. Lu, S. A. Wolf, F. G. Omenetto, X. Zhang, K. A. Nelson, and R. D. Averitt, “Terahertz-field-induced insulator-to-metal transition in vanadium dioxide metamaterial,” Nature 487(7407), 345–348 (2012).
[Crossref]

Kong, W. J.

Kono, J.

Y. Harada, M. S. Ukhtary, M. Wang, S. K. Srinivasan, E. H. Hasdeo, A. R. T. Nugraha, G. T. Noe, Y. Sakai, R. Vajtai, P. M. Ajayan, R. Saito, and J. Kono, “Giant Terahertz-Wave Absorption by Monolayer Graphene in a Total Internal Reflection Geometry,” ACS Photonics 4(1), 121–126 (2017).
[Crossref]

Koschny, T.

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[Crossref]

Kundu, I.

P. Dean, O. Mitrofanov, J. Keeley, I. Kundu, L. Li, E. H. Linfield, and A. G. Davies, “Apertureless near-field terahertz imaging using the self-mixing effect in a quantum cascade laser,” Appl. Phys. Lett. 108(9), 091113 (2016).
[Crossref]

Lai, K.

N. Dabidian, S. Dutta-Gupta, I. Kholmanov, K. Lai, F. Lu, J. Lee, M. Jin, S. Trendafilov, A. Khanikaev, B. Fallahazad, E. Tutuc, M. A. Belkin, and G. Shvets, “Experimental demonstration of phase modulation and motion sensing using graphene-integrated metasurfaces,” Nano Lett. 16(6), 3607–3615 (2016).
[Crossref]

Lavrinenko, A. V.

Lee, D.

H. K. Kim, D. Lee, and S. Lim, “Wideband-switchable metamaterial absorber using injected liquid metal,” Sci. Rep. 6(1), 31823 (2016).
[Crossref]

Lee, J.

N. Dabidian, S. Dutta-Gupta, I. Kholmanov, K. Lai, F. Lu, J. Lee, M. Jin, S. Trendafilov, A. Khanikaev, B. Fallahazad, E. Tutuc, M. A. Belkin, and G. Shvets, “Experimental demonstration of phase modulation and motion sensing using graphene-integrated metasurfaces,” Nano Lett. 16(6), 3607–3615 (2016).
[Crossref]

Li, H.

Li, K.

Li, L.

P. Dean, O. Mitrofanov, J. Keeley, I. Kundu, L. Li, E. H. Linfield, and A. G. Davies, “Apertureless near-field terahertz imaging using the self-mixing effect in a quantum cascade laser,” Appl. Phys. Lett. 108(9), 091113 (2016).
[Crossref]

Li, Y.

Li, Z. Y.

X. Tian and Z. Y. Li, “An optically-triggered switchable mid-infrared perfect absorber based on phase-change material of vanadium dioxide,” Plasmonics 13(4), 1393–1402 (2018).
[Crossref]

Lian, Z.

Y. Liu, R. B. Zhong, Z. Lian, C. Bu, and S. G. Liu, “Dynamically tunable band stop filter enabled by the metal-graphene metamaterials,” Sci. Rep. 8(1), 2828 (2018).
[Crossref]

Lim, S.

H. K. Kim, D. Lee, and S. Lim, “Wideband-switchable metamaterial absorber using injected liquid metal,” Sci. Rep. 6(1), 31823 (2016).
[Crossref]

Linfield, E. H.

P. Dean, O. Mitrofanov, J. Keeley, I. Kundu, L. Li, E. H. Linfield, and A. G. Davies, “Apertureless near-field terahertz imaging using the self-mixing effect in a quantum cascade laser,” Appl. Phys. Lett. 108(9), 091113 (2016).
[Crossref]

Liu, J.

J. Liu and L. Fan, “Development of a tunable terahertz absorber based on temperature control,” Microw Opt Technol Lett 62(4), 1681–1685 (2020).
[Crossref]

Liu, L.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (8)

Fig. 1.
Fig. 1. (a) Schematic of the proposed absorber based on VO2-graphene metamaterials and the incident light polarization configuration. (b) Top view of the unit cell. (c) Side view of the unit cell.
Fig. 2.
Fig. 2. The simulated absorption spectra of the proposed absorber for different VO2 states. The red solid curve illustrates the absorptance when VO2 is in the insulating phase. The blue dash curve illustrates the absorptance with wVO2 = 3 µm when VO2 is in the metallic phase.
Fig. 3.
Fig. 3. (a) Energy-level diagram describing the plasmonic hybridization effect in VO2 square loops due to the interaction between the VO2 square and cavity plasmons. Distributions of electric field amplitude |E| (b) at 2.6 THz, and (c) 3.6 THz on the top surface of the VO2 square loops for the high-frequency broadband absorption.
Fig. 4.
Fig. 4. Distributions of (a), (b) the electric field amplitude |E| and (c), (d) the z-component electric field Ez on the top surface of the square-shaped graphene for the low-frequency broadband absorption at 1.35 THz and 2.1 THz, respectively; Distributions of the power flow at the central cross section of unit cell (e) at 1.35 THz, and (f) 2.1 THz, respectively; Distributions of the magnetic field |H| and surface current at the central cross section of unit cell (g) at 1.35 THz, and (h) 2.1 THz, respectively.
Fig. 5.
Fig. 5. The simulated absorption spectra of the high-frequency broadband absorber as a function of (a) the thickness of the first Topas layer hd1, the length (b) lVO2 and the width (c) wVO2 of VO2 loops. (d) The absorption spectra vary with the conductivity of VO2 from 2×102 S/m to 2×105 S/m. (e) Real parts and (f) imaginary parts of the relative impedance Zr with different conductivities of VO2.
Fig. 6.
Fig. 6. The simulated absorption spectra of the low-frequency broadband absorber as a function of (a) the thickness of the second Topas layer hd2, the distances from the unit boundary of the connected graphene squares are (b) w1 and (c) w2.
Fig. 7.
Fig. 7. The absorption spectra vary with the Fermi energy level from 0.01 eV to 0.7 eV.
Fig. 8.
Fig. 8. The absorption spectra of the high-frequency broadband absorber with different incidence angles for (a) TE polarization, (b) TM polarization, when VO2 in metallic phase. The absorption spectra of the low-frequency broadband absorber with different incidence angles for (c) TE polarization, (d) TM polarization, when VO2 in insulating phase. The 80% absorptance is indicated by the white contour curves.

Tables (1)

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Table 1. Comparison between the proposed device and the similar works.

Equations (5)

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σ inter ( ω , τ , μ c ) j e 2 4 π ln [ 2 | μ c | ( ω + j / τ ) 2 | μ c | + ( ω + j / τ ) ] ,
σ intra ( ω , τ , μ c ) j e 2 k B T π 2 ( ω + j τ 1 ) [ μ c k B T + 2 ln ( exp ( μ c k B T ) + 1 ) ] ,
ε ( ω ) = ε ω p 2 ( σ ) ω 2 + i γ ω ,
A ( ω ) = 1 R ( ω ) = 1 | Z Z 0 Z + Z 0 | 2 = 1 | Z r 1 Z r + 1 | 2 ,
Z r = ( 1 + S 11 ( ω ) ) 2 S 21 ( ω ) 2 ( 1 S 11 ( ω ) ) 2 S 21 ( ω ) 2 ,

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