1. Introduction

Substantial effort has been devoted to the development of devices that operate in the so-called ‘terahertz gap’, i.e., the frequency range from 0.1 to 10 THz, a spectral window that has been less explored than its microwave and optical counterparts. Nevertheless, terahertz spectroscopic tools have not been widely successful due to the lack of response from most naturally occurring materials in this window [1,2]. On the other hand, THz radiation is of great interest to researchers in a variety of fields, owing to its energetic equivalence to various biological, physical and chemical processes, which provides the basis for THz systems for nonionic bio-imaging, security screening and high-bit-rate communications, etc [1,2]. The manipulation of electromagnetic waves by structural materials has been well known for decades, but has only recently attracted renewed attention due to the emergence of metamaterials, which are artificial materials whose response can be tailored by controlling their subwavelength structure [3,4]. Importantly, terahertz metamaterials provide a pathway to creating artificial materials that are capable of filling the so-called “THz gap” [5].

Taking advantage of the subunit engineering offered by metamaterials, researchers have demonstrated versatile components that can manipulate THz radiation efficiently, including THz modulators [6–32], wavefront controllers [23–26] and polarization convertors [33–36]. In particular, metamaterial absorbers (MMAs) have recently attracted significant attention because of their capability for efficient THz radiation manipulation based on their specifically designed absorption properties. Nevertheless, like most conventional structural materials, THz MMAs in general suffer from two shortcomings that limit their usefulness, i.e., a narrow operating bandwidth and their fixed response. A large number of studies have been devoted to THz MMAs with broadband absorptivity or reconfigurability. In particular, three-dimensional resonators created by stacking alternating layers of dielectric/metal [37–39] or by patterning thick dielectric films [40] have been widely adopted for broadband THz MMAs, primarily due to their tailorable cavity modes. Planar THz MMAs with one intricately structured layer have also been reported [41]. On the other hand, active media, such as liquid crystals [13] and semiconductors [14] have been introduced into metallic structure based metamaterial systems to enable tunability, which have led to the development of MMA based THz spatial modulators [42], sensing [43] and imaging [44] devices. Nevertheless, THz MMAs that simultaneously possess broadband absorptivity and tunability over a large scale remain rare.

Vanadium dioxide (VO₂), a common phase-change material that undergoes an insulator-to-metal transition (IMT) near room temperature (~67 °C) has recently been utilized to enable reconfigurable metamaterials at THz [15,16,18] and optical frequencies [45–47]. IMT of VO₂ can be achieved through direct thermal control or by optical excitation. Electrical signals triggered IMT has also been reported. Compared with other phase change materials [48,49], VO₂ is particularly attractive for hybrid THz metamaterial systems due to the huge index change and the interesting hysteresis behavior during its phase transition. The latter has been used to demonstrate THz memory metamaterials [15]. We recently demonstrated that VO₂ in its metallic phase can support strong THz resonance, which can be exploited to achieve highly tunable THz devices [50,51].

In this work, we numerically demonstrate a VO₂ based THz metamaterial absorber exhibiting broadband absorptivity that arises from the multiple resonances supported by an intricate doubly periodic array of planar VO₂ structures. We also demonstrate that the corresponding absorption is strongly dependent on the electrical properties of VO₂ in the THz spectral regime. Field analyses are carried out to identify the localized absorption on different parts of the VO₂ structure at a series of frequencies within the high absorption band. Furthermore, the dependence of absorption on incidence angle indicates that the proposed MMA can maintain its broadband absorption performance over a ± 50° field-of-view for the frequency band ranging from 0.61 to 1.36 THz. Moreover, due to the C4 rotational symmetry of the structure, this design is intrinsically polarization insensitive in case of small incidence angle [52]. Given the dynamic response of VO₂ under external stimulus [53], the proposed systems may facilitate broadband and highly tunable THz metadevices [54,55].

2. Theory and numerical simulations

A close inspection of the reported planar structure comprising broadband MMAs reveals that there are two crucial factors that lead to the desired absorption behavior. First, the resonating structures, which are composed of subunits (for instance, coplanar sectional structures) with different geometrical parameters should be capable of supporting multiple electromagnetic resonances with a continuous spectral distribution. Second, lossy materials should be introduced to the MMA systems to further broaden the resonances and allow the corresponding spectral superposition that gives rise to a flat high absorption band. Employing a genetic algorithm optimization approach, an earlier demonstration of MMAs confirms that a single-layer lossy metal nanostructure array can be used to realize broadband ideal absorption in the mid-infrared regime [56].

Inspired by the design reported by Bossard et al [56], here we propose a broadband THz MMA based on a doubly periodic array of VO₂ structures, as illustrated in Fig. 1. The VO₂ layer is embedded in a SiO₂ matrix that is located on top of an optically thick gold film. It has been previously established that the THz dielectric property of VO₂ in its metallic phase can be described by a Drude model, $\epsilon \left(\omega \right)= {\epsilon }_{\infty }-\frac{{({\omega }_{\text{p}}({\sigma }_1))}^2}{{\omega }^2+i\gamma \omega }$, where ${\epsilon }_{\infty }$ is the permittivity at high frequency, ${\omega }_{\text{p}}({\sigma }_1)$ is the conductivity dependent plasmon frequency and $\gamma$ is the collision frequency [57]. Moreover, it can be shown that the plasmon frequency at ${\sigma }_1\text{'}$ can be approximately expressed as ${\omega }_{\text{p}}^2\left({\sigma }_1\text{'}\right)=(\frac{{\sigma }_1\text{'}}{{\sigma }_1}) {\omega }_{\text{p}}^2\left({\sigma }_1\right)$. From measured data [16], we have determined that when σ₁ = 3 × 10³ Ω⁻¹cm⁻¹ (${\epsilon }_{\infty }$ = 12), it follows that the corresponding ${\omega }_{\text{p}}$ = 1.40 × 10¹⁵ rad/s, while $\gamma$ = 5.75 × 10¹³ rad/s is assumed to be independent of ${\sigma }_1$. On the right-hand side of Fig. 1, we show the THz permittivity of VO₂ with a conductivity of σ₁ = 2 × 10³ Ω⁻¹ cm⁻¹, in which the negative value of the real part and the large imaginary part of ε_VO2 indicate that VO₂ in its metallic phase behaves like a ‘poor’ metal in the frequency range of interest. Consequently, as shown in Fig. 1, a combination of the sophisticated VO₂ pattern and the resonant cavity created by the bottom three layers (VO₂/SiO₂/Au) is expected to result in multiple overlapping resonances responsible for the broadband absorptivity. Furthermore, a SiO₂ layer with optimized thickness is applied on the top to further improve the impedance matching condition. Importantly, the pronounced tuning capability of the proposed MMA is afforded by the conductivity dependent dielectric properties of VO₂ during the phase transition [50,51]. It should be noted that IMT of VO₂ is sensitive to its lattice structure. The phase transition of the VO₂ layer (embedded in SiO₂) in the proposed metamaterials is expected to have characteristics slightly different from that of VO₂ films on the sapphire substrate. Nevertheless, previous studies have demonstrated that the basic properties of VO₂ films integrated with complex nanostructures can be well reserved [47,58]. We note that, compared with the absorbers reported by Bossard et al [56], MMAs in this work come with their own set of unique features and advantages. First, exploiting the phase transition of VO₂, response of the proposed MMAs can be actively tuned, while tunability and reconfigurability are of great importance for designing metadevices with practical functionalities. Second, the proposed MMAs are capable of manipulating THz radiation based on VO₂’s interesting material property in the same range, while Ref [56] was focused on mid-IR region. Considering the rare response from naturally occurring materials in this spectral window, THz metamaterials open a pathway to structural materials that fill the so-called “THz gap”. It should be noted that the phonon response may have significant influence on the optical properties of VO₂. For instance, it has been reported that the IMT in VO₂ is dominated by strongly anharmonic phonons rather than electronic contributions [59]. In addition, it has been demonstrated that structural defects can greatly affect the IMT of VO₂ [60,61]. The control of structural defects (such as oxygen vacancy) may provide more flexibility in the design of VO₂ based metadevices.

 

Fig. 1 A vanadium dioxide based broadband THz metamaterial absorber. The schematic depicts the structure of the MMA based on an elaborate VO₂ pattern inspired by an earlier demonstration in the mid-infrared regime [56]. The 1-μm-thick VO₂ layer is embedded in a SiO₂ matrix that is located on top of an optically thick Au film. The schematic on the left shows the unit cell of the metamaterial. The geometric parameters (all unit in μm) are: P = 120, L = R = 48, g = 3, w = 9, L_c = 20, w_c = 5.5, d₁ = 42 and d₂ = 24. The permittivity dispersion of VO₂ with conductivity ${\sigma }_1$ = 2 × 10³ Ω⁻¹cm⁻¹ in the THz frequency range of interest is shown on the right.

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It should be noted that, although MMAs are usually based on delicately designed meta-atoms, finding their response which can be treated as the reflection of multiple interfaces is a classic problem. Given the great importance of MMAs in both fundamental and applied research, four theoretical approaches including effective medium theory [62], transmission line modelling [63], coupled mode theory [64], and interference theory [65] have been developed to describe the underlying physics of MMAs. The distinct advantages and limitations of these approaches have been comprehensively discussed in a recent review paper [66].

Here, using interference theory, we present the determination of the thickness of the two dielectric layers and the high-absorption band of the proposed MMA. Note that the layered structure illustrated in Fig. 1 is slightly different from most reported MMAs which have three (air-structure, spacer-structure, and spacer-substrate) interfaces. In our design the resonating VO₂ structure is embedded in a dielectric matrix, which results in the fact that the response of the MMA is largely determined by the thickness (d₁ and d₂) of the two dielectric layers. It has been shown that, based upon the interference between the multiple reflections at the interfaces, the interference theory can accurately predict the reflection of the three-interface MMAs via the equation r = r₁₂ + r₂₃t₁₂t₂₁/(r₂₁ + e^-i2β), where r₁₂, t₁₂, r₂₁, and t₂₁ are the complex reflection and transmission coefficients at the air-structure and spacer-structure interfaces, and, due to the high conductivity of metals at THz frequencies, the reflection coefficient at the spacer-substrate interface r₂₃ is assumed to be −1 [65,67]. Moreover, a multi-step approach can be utilized to calculate the overall reflection of a multilayer structure with more interfaces. Consequently, to simplify the calculation without loss accuracy, here we obtain the reflection of the proposed MMA from a two-step approach. In particular, as the schematic illustrated in Fig. 2(a) shows, the reflection (r^S1) of ‘System 1’ (S1) which can be treated as a MMA surrounded by SiO₂ is firstly calculated. And then, the reflection (r^S2) of ‘System 2’ (S2), which represents the overall reflection of the proposed MMA, can be further obtained by considering r₂₃^S2 = r^S1.

 

Fig. 2 Interference theory based reflection analysis. (a) Schematic of the interference analysis. (b) Simulated complex reflection and transmission coefficients (r₁₂^S1 and t₁₂^S1) for the structure including the VO₂-pattern embedded in a SiO₂ matrix at normal incidence when σ₁ = 2 × 10³ Ω⁻¹ cm⁻¹. (c) Absorption spectra of the MMA at normal incidence at a series of VO₂ conductivity (σ₁ in unit of Ω⁻¹ cm⁻¹). (d) The corresponding spectra of amplitude modulation.

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In Fig. 2(b) we show the simulated complex reflection and transmission coefficients (r₁₂^S1 and t₁₂^S1) for the structure of patterned VO₂ (σ₁ = 2 × 10³ Ω⁻¹ cm⁻¹) embedded in SiO₂ for y-polarization of THz wave at normal incidence. Due to the reciprocal property of the structure, we have r₂₁^S1 = r₁₂^S1 and t₂₁^S1 = t₁₂^S1, respectively. By applying the two-step approach mentioned above, we obtain the corresponding theoretical overall absorption (A = 1 - |r|² and T = 0) spectrum of the proposed MMA (red curve in Fig. 2(c)), in which a highly absorptive band in the frequency range from ~0.6 THz to ~1.3 THz is evident. Furthermore, by using the simulated r₁₂^S1 and t₁₂^S1 at series of VO₂ conductivity, we calculate the absorption spectra of the MMA as a function of σ₁ (Fig. 2(c)). It can be seen that the change in VO₂’s property may lead to dynamically tunable absorption property of the MMAs, implying the potential applicability of the proposed metamaterial for phase transition enabled active THz meta-devices. Another interesting observation is the near-unity absorption around 1.7 THz when VO₂ shows intermediate conductivity (e.g. σ₁ = 100 Ω⁻¹ cm⁻¹). This absorption peak is attributed to the interference induced strong absorption in the VO₂ structure. Moreover, to manifest the corresponding tuning of the reflection signal that can be directly detected in practical applications, we define amplitude modulation (AM) as the variation in the MMA’s overall reflection magnitude relative to the case of insulator VO₂, i.e., AM = |r(insu)| - |r(σ₁)| and show the corresponding results in Fig. 2(d). It should be noted that the relatively large periodicity of the VO₂ pattern (P in Fig. 1) leads to diffraction effect at frequencies higher than ~1.3 THz. In our interference theory based calculation for normal incidence case only the zero-order wave is considered for two reasons: (i) the waves of higher-order mode carry rather limited power, and (ii) the high-absorption band of interest is located in the lower frequency range. Nevertheless, the theoretical approach used can still provide important guidance to the design of the proposed MMAs. It should be also noted that, to perform an accurate reflection analysis of metamaterials involving diffraction effect, interference theory taking into account higher-order diffracted waves especially for the case of oblique incidence is desired, but is beyond the scope of this work.

To elucidate the absorption response of the proposed MMAs, we performed full-wave simulations using the commercial finite integration package CST Microwave Studio. A unit cell of the structure was simulated by employing periodic boundary conditions, in which the SiO₂ was modeled as a lossless dielectric with an index of 1.9. Figure 3(a) shows the absorption spectra of the MMA for a series of VO₂ conductivity values. An absorptivity larger than 90% is observed in the frequency range from 0.61 to 1.36 THz, when VO₂ possesses a conductivity higher than 400 Ω⁻¹cm⁻¹. In sharp contrast, the system absorbs less than 30% of the power of incident waves in the same band when the VO₂ is in its insulator state (σ₁ = 10 Ω⁻¹ cm⁻¹). Another interesting observation is that perfect absorption is achieved around 1.73 THz when σ₁ = 100 Ω⁻¹cm⁻¹, which corresponds to the intermediate transition status of VO₂. Different from the metallic-phase VO₂ (e.g. σ₁ = 2000 Ω⁻¹cm⁻¹) systems in which the planar VO₂ structures’ resonance dominate the absorption behavior, this absorption peak is primarily determined by the Fabry-Perot resonance in the air/SiO₂/VO₂/SiO₂/Au cavity system in which the VO₂ layer behaves like a thin film. This is more evident in the field analysis as seen in a later discussion. The corresponding reflection amplitude modulation is shown in Fig. 3(b), which indicates the potential of the MMA for use in broadband THz reflection modulators. This is in good agreement with the theoretical approach based results shown in Fig. 2. To further identify the absorption contribution, the total absorption (open squares) of the MMA at four frequencies of interest is plotted in Fig. 3(c), along with the corresponding absorption in the gold ground plane (open circles), as a function of σ₁. This reveals that almost all energy is dissipated as dielectric loss within the VO₂ structures. Furthermore, Fig. 3(d) shows the absorption spectrum of the same system but with the constituent material of the patterned layer replaced by gold, which indicates the presence of multiple resonances within the spectral range of interest. However, due to the rather high conductivity of gold in the THz regime, which is roughly one order of magnitude higher than that of metallic-phase VO₂, the gold structure does not show the broadband absorption characteristics. The main results of this work are summarized in Fig. 3.

 

Fig. 3 The THz response of the metamaterial absorber. (a) Simulated absorption and (b) reflection modulation amplitude spectra for a series of VO₂ conductivity (σ₁) values. (c) The corresponding absorption spectra at several frequencies of interest as a function of σ₁. The absorption spectrum of a similar metamaterial with a gold patterned layer is shown in (d).

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The finite conductivity of metallic-phase VO₂, on the other hand, causes the corresponding electromagnetic resonance to be sensitive to the VO₂ structures’ thickness. Given the fact that the skin depth of metallic-phase VO₂ around 1.0 THz is roughly 1 μm, we calculated the absorption behavior of the proposed MMA as a function of the VO₂ layer’s thickness (t_VO2) by varying it from 0.2 to 2 μm, as illustrated in Fig. 4(a). It can be seen that the absorptivity in the frequency range from 0.6 to 1.4 THz decreases when t_VO2 is decreased from 1 μm to 0.2 μm, while it becomes saturated if t_VO2 is larger than 1 μm. Furthermore, the absorption spectra of the identical system with an unpatterned VO₂ film calculated over the same series of thickness values are shown in Fig. 4(b). It can be seen that an absorption peak can be identified around 1.0 THz when the unpatterned film thickness is small (e.g. 0.1 μm), while the absorption of the system vanishes with increasing film thickness. This observation can be explained by the interference effect: a thin VO₂ film is partially transparent, which allows the destructive interference condition to be complied with at certain frequencies, while, with larger thickness (e.g. when t_VO2 is thicker than the skin depth), the unpatterned VO₂ film behaves like a reflector allowing almost no transmitted wave. A comparison unambiguously reveals the greatly enhanced absorption corresponding to the MMA, originating from the resonance properties of the complex VO₂ structure. We note that it remains challenging to grow thick VO₂ films on arbitrary substrates such as gold and SiO₂ due to the many stable members of the vanadium oxide family, including VO₂, V₂O₃, V₂O₅, etc. It should also be noted that patterning of VO₂ films based on standard lithographic processes may general involve acid mixture solution enabled wet-etching, which can possibly degrade the properties of VO₂. Nevertheless, a number of studies have verified that with carefully selected fabrication conditions the electronic and optical properties associated with the phase transition can be well reserved in the patterned VO₂ structures [16,58,68,69], which shows the practicality of the proposed VO₂ based MMAs.

 

Fig. 4 The absorption spectra of (a) the MMA with a patterned VO₂ structure and (b) a VO₂ thin film for a series of VO₂ layer thicknesses.

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To further investigate the underlying absorption mechanism, in Fig. 5 we present the simulated power loss density (PLD) distribution in the VO₂ layer of the MMA under THz illumination. With its value expressed in units of W/m³, PLD corresponds to the electric power dissipated inside the materials. As illustrated in Figs. 5(a)-5(e), the evolution of PLD distribution at frequencies within the high absorption band when σ₁ = 2 × 10³ Ω⁻¹ cm⁻¹ explicitly identifies the localized absorption in different parts of the VO₂ pattern, which corresponds to the multiple structure-supported resonances. In sharp contrast, when the VO₂ system is in the insulator-phase, the corresponding PLD is low (Figs. 5(f)-5(j)), which reveals the non-resonant response of the structure. Furthermore, Fig. 5(k) shows the PLD distribution of the system with σ₁ = 100 Ω⁻¹cm⁻¹ at 1.73 THz, i.e., the perfect-absorption frequency illustrated in Fig. 3(a). In contrast to the observation made in Figs. 5(a)-5(e), the energy dissipation is rather non-localized, indicating the thin-film like absorption characteristics of the VO₂ structure in this case. This comparison clearly reveals the distinct absorption mechanisms of the proposed MMA for the different material phases of VO₂.

 

Fig. 5 Simulated power loss density (PLD) distribution in the VO₂ layer at a series of frequencies when (a)-(e) σ₁ = 2 × 10³ Ω⁻¹ cm⁻¹ and when (f)-(j) VO₂ is in its insulator phase. (k) PLD distribution at 1.73 THz when σ₁ = 100 Ω⁻¹ cm⁻¹.

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Besides the geometrically tailorable resonances, MMAs constructed from subwavelength building blocks can potentially provide absorption with large field-of-view. Here, the angle of incidence (θ_in) was varied to study the angular dependence of the absorption behavior of the proposed MMA. In particular, to capture the complete absorption response of the proposed MMA as a function of θ_in, total absorption in the VO₂ structure and the Au substrate was extracted from simulations. Importantly, it should be noted that, the total absorption spectra can also be estimated based on the simulated reflection coefficients, but, in that case, one need to make sure to include the reflection information of all the diffraction modes. It can be seen from Figs. 6(a) and 6(b) that the overall absorption performance of the MMA with metallic-phase VO₂ is insensitive to the incidence angle for both transverse electric (TE) and transverse magnetic (TM) polarized illuminations when the incidence angle varies up to ~50°. In particular, as indicated by the 80% absorptivity contour curves (white), the metamaterial maintains its broadband absorption performance over a ± 50° field-of-view for frequencies ranging from 0.61 to 1.36 THz. On the other hand, for the system with σ₁ = 100 Ω⁻¹ cm⁻¹ (Figs. 6(c) and 6(d)), the observed absorption peak around 1.7 THz blueshifts with increasing incidence angle and, when θ_in approaches ~70°, an obvious additional absorption band appears around 0.6 THz (1.4 THz) for TE (TM) polarized incident waves. A similar dependence of absorption spectra on the incident angle was observed in previous reported optical absorbers [70,71]. It should be noted that a theoretical description of these observations can be achieved using the interference theory that can take into account the higher-order diffraction modes. These results again show the distinct absorption behavior of the MMA with VO₂ in its metallic phase and intermediate transition status.

 

Fig. 6 The incident angular dependence of absorption in the MMA for (a) transverse electric (TE) and (b) transverse magnetic (TM) polarized illumination when σ₁ = 2 × 10³ Ω⁻¹ cm⁻¹. Angular dependence for the (c) TE and (d) TM case when σ₁ = 100 Ω⁻¹ cm⁻¹.

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3. Conclusion

A vanadium dioxide based broadband THz metamaterial absorber was proposed and numerically validated. More importantly, it was demonstrated that the absorptivity can be tuned over a wide range by VO₂’s THz electrical conductivity, which varies dramatically during an insulator-to-metal transition. Field analyses show that the observed broadband absorption behavior arises from the localized resonance supported by the delicate doubly periodic VO₂ structures. By conducting a study of the dependence of absorption on incidence angle, we have shown that the proposed MMAs can maintain their broadband absorption performance over ± 50° field-of-view over the frequency range from 0.61 to 1.36 THz. Given the reported phase transition dynamics of VO₂ under external stimulus, we envision applications of the proposed metamaterial absorber to highly tunable THz metadevices, such as broadband modulators.