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Abstract
In this paper, we theoretically design a dual-band graphene-based terahertz (THz) absorber combining the magnetic resonance with a THz cold mirror without any metallic loss. The absorption spectrum of the all-dielectric THz absorber can be actively manipulated after fabrication due to the tunable conductivity of graphene. After delicate optimization, two ultra-narrow absorption peaks are achieved with respective full width at half maximum (FWHM) of 0.0272 THz and 0.0424 THz. Also, we investigate the effect of geometric parameters on the absorption performance. Coupled mode theory (CMT) is conducted on the dual-band spectrum as an analytic method to confirm the validity of numerical results. Furthermore, physical mechanism is deeply revealed with magnetic and electric field distributions, which demonstrate a totally different principle with traditional plasmonic absorber. Our research provides a significant design guide for developing tunable multi-resonant THz devices based on all-dielectric configuration.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Terahertz (THz) technology has received burgeoning amount of interest, which shows unprecedented potentials in sensing, imaging, detection and communications [1,2]. Since the first experimental work was reported in 2008 [3], THz metamaterial absorbers have attracted tremendous attention attributed to their advantages of controlling electric fields and confining incident waves to the sub-wavelength scale structures. In the following years, various THz metamaterial absorbers with different configurations, such as square rings, cross patterns, circular split rings, have been proposed to achieve single-band, multi-band or broadband absorption [4–9]. Besides, novel materials such as graphene [10], black phosphorus [11,12], WS2 [13], molybdenum disulfide [14,15] and Dirac semimetal [16,17] are also utilized in the THz metamaterial absorbers for various applications. The complex conductivity of graphene depends on its chemical potential which can be altered by electrostatic doping or chemical doping [18]. Thus, metamaterial absorbers based on graphene with tunable characteristics have been extensively investigated and studied [19–23]. For instance, Amin et al. designed an ultra-broadband THz absorber based on multilayered graphene, whose bandwidth of 90% absorption could be extended up to 7 THz with only three layers [19]. In the same year, Andryieuski et al. demonstrated a graphene metamaterial THz absorber with the effective surface conductivity approach [20]. Both narrowband and broadband tunable absorbers were investigated after formulating the requirements to the design. Hajian et al. proposed an electrical tunable metamaterial absorber with omnidirectional and polarization insensitive characteristics [21]. Graphene-hBN-based hole arrays were utilized to achieve nearly perfect resonant absorption. Xia et al. achieved multi-band perfect absorptions with peak absorbance higher than 99% using single-layer graphene-based rectangular gratings [22]. For gratings with bottom-open configuration, both even- and odd-order modes could be realized in the four-band perfect absorptions. In 2020, our group utilized nested graphene loops to achieve multi-band absorption as an ultracompact identification tag in the THz band [23]. Benefiting from the exceptional property of graphene, the proposed THz tag has a uniform geometric size while encoding different bit sequences.
However, all the plasmonic graphene-based structures above suffer from high losses of metallic substrate or gratings, heating, and incompatibility with complementary metal oxide semiconductor (CMOS) fabrication processes [24,25]. Therefore, recent developments have led to a new branch of metamaterial absorber aiming at the manipulation of optically induced magnetic resonances in all-dielectric configuration [26–30]. Nevertheless, the previous works seldom focused on the tunable multi-resonant property in all-dielectric absorber operating in the THz range.
In this paper, we numerically propose a tunable dual-band THz absorber based on all-dielectric configuration, which consists of unpatterned monolayer graphene. The analysis of coupled mode theory (CMT) is conducted on the structure to verify the numerical results. Magnetic and electric field distributions are plotted to reveal the mechanism of magnetic resonance for absorption enhancement. The effects of geometric parameters on the absorption performance are investigated. Also, the angular dependence on the incident THz wave is evaluated. Furthermore, the tunable ability of the proposed absorber is illustrated with various chemical potentials of graphene.
2. Design and modeling
Figure 1 depicts the schematic of the proposed graphene-based all-dielectric dual-band THz absorber (GADTA). The structure is made up of periodic SiO2 ribbons, a monolayer graphene, a polymide layer, a THz cold mirror consisting of alternate Si and Al2O3 layers, and a transparent substrate. In our simulation, the refractive indices (RI) of SiO2, polymide, Si, Al2O3 and substrate are assumed as 2, 1.8, 3.42, 1.79 and 2, respectively [31]. The absorption performance of GADTA is investigated via the finite element method (FEM). Periodic boundary conditions are applied in the x-axis and y-axis directions. The THz incidence is imposed downward from the top surface of GADTA. Meshes of user-defined size are applied on graphene layer due to its localized enhanced electromagnetic fields. Tetrahedral meshes are applied for the remaining domains of the structure. Due to its ultra-thin thickness compared with the incidence wavelength, graphene can be assumed as a two-dimensional surface with isotropic surface conductivity σ(ω). According to the Kubo equation, σ(ω) can be calculated with the following expression [32],
Fig. 1. Schematic of the proposed graphene-based all-dielectric dual-band THz absorber (GADTA), where w1 and w2 represent the different widths of SiO2. The symbols h and t denote the thicknesses of SiO2 and polymide, respectively. p is the periodicity of the periodic unit cell.
Thus, as can be seen from Eq. (2), σ mainly depends on τ, μc and ω. Based on the previous work [33,34], τ and μc are set as 1 ps and 0.7 eV, respectively, in the following discussions.
3. Results and discussions
To demonstrate the role of the cold mirror in the GADTA configuration, we first investigate the reflectance spectra of the THz cold mirror compared with traditional perfect electric conductor (PEC) substrate such as noble metals. The THz cold mirror consists of a stack of dielectric materials without metallic loss to reflect the THz wave as a role of mirror in the THz range. According to the characteristic matrix method for multi-layer dielectric coatings [35], a stack of quarter-wavelength dielectric layers with alternate high RI (Si) and low RI (Al2O3) is adopted. Under normal incidence, the effective admittance of the mirror and its reflection rate can be calculated as:
In Eqs. (3) and (4), nH, nL and ns represent the RI of Si, Al2O3 and substrate, respectively. m is an odd number, which denotes the total number of dielectric layers. These two equations imply the more the amount of layers the higher the reflection rate. The bandwidth of high reflectance Δf = fmax - fmin can be further calculated with Eq. (5),
Here, f0 is the center frequency for the design of quarter-wavelength dielectric layers in the cold mirror. fmin and fmax are the minimum and the maximum frequencies of the high reflectance band, respectively. Through the above discussion, we choose f0 = 4 THz as the center frequency, while the all-dielectric stack is optimized with 5 layers of Si (5 µm thick) and 4 layers of Al2O3 (12.2 µm thick). As a result shown in Fig. 2(a), the near-unity reflectance band is observed in the designed THz range. In this broad THz band, transmission is almost blocked by the THz mirror. In Fig. 2(a), the analytical result demonstrates good agreement with the numerical result based on FEM, which confirms the validity of FEM simulations for further study on the all-dielectric graphene-based THz device.
Fig. 2. (a) The reflectance spectra of the THz cold mirror and PEC are compared in the THz region. (b) Absorption spectra of the cold mirror and PEC with insulator and graphene layers. (c) Absorption spectra of the proposed GADTA and the traditional PEC structure. (d) Absorption spectra of GADTA under TE and TM polarized incidence. The polarization for the incidence in (a)-(c) is TE polarization. The values of these structural parameters are as follows: w1 = 13 µm, w2 = 5 µm, h = 7.8 µm, t = 35 µm, p = 60 µm, under normal incidence.
Next, we deposit the graphene layer on an insulator of polymide supported by the THz cold mirror and PEC, respectively, to obtain the absorption spectra in Fig. 2(b). As can be observed, the reflected wave is partly dissipated inside graphene with both the THz cold mirror and PEC substrate. The absorption rate with the THz cold mirror is a bit lower than that with PEC, which is mainly attributed to the perfect reflection property of PEC.
In Fig. 2(c), the absorption spectrum of the proposed GADTA is plotted compared with a PEC-based structure. The PEC-based structure has the same configuration as GADTA except the substrate. GADTA exhibits two obvious absorption peaks at 3.56 THz and 4.59 THz with high quality factors. The full width at half maximum (FWHM) is Δf1 = 0.0272 THz and Δf2 = 0.0424 THz for corresponding absorption peak, indicating the extremely narrow bandwidth for spectral selective absorption. The quality factor Q which can be defined as Q = f1,2/Δf1,2 reaches the value of 130.88 and 108.25, respectively. Similarly, the PEC-based structure also possesses dual absorption peaks at 3.42 THz and 4.60 THz. The frequency deviations between these two structures originate from the fact that the phase of reflected wave from the THz cold mirror has slight difference with that from PEC substrate.
All the discussions above are based on the transverse electric (TE) polarized incidence, whose magnetic field is along the x-axis direction. On the other hand, we also depict the absorption spectrum of GADTA under transverse magnetic (TM) polarization in Fig. 2(d) for comparison. For TM polarization, resonance cannot be excited due to the fact that the magnetic field is parallel to the SiO2 ribbon. Hence, polarization sensitivity of GADTA can be utilized to design THz reflective polarizers or polarized wave filters.
In order to further guarantee the reliability of the numerical results, the absorption spectrum of GADTA is also investigated by CMT as shown in Fig. 3 [36–39]. Based on the CMT analysis, the model can be described by the following formulas:
where a denotes the amplitude of resonance, ω0 represents the resonant frequency, S+ and S− are the input and outgoing wave amplitudes, γ and δ represent the external leakage rate and intrinsic loss, respectively. The reflection coefficient of the system is:Fig. 3. Absorption spectra of FEM-based numerical simulation (black curve) and CMT theoretical analysis (red and blue curves). All studies are based on the normal TE incidence.
As can be observed from Eq. (9), at resonance frequency (ω = ω0), if γ is equal to δ, perfect absorption occurs due to satisfaction of critical coupling. Additionally, under the critical coupling condition, the impedance matching between the absorber and the free space should be satisfied (Z = Z0 = 1). Thus, the effective absorber impedance can be expressed according to Ref. [40] as:
In our analytic model, the loss and external leakage of the GADTA configuration are δ = 3.23 × 1010 Hz and γe = 5.31× 1010 Hz at f1 = 3.56 THz, respectively. According to the following formula QCMT1 = QδQγ / (Qδ + Qγ), the theoretical quality factor QCMT1 is calculated as 130.96, where Qδ = ω0 / (2δ) is defined as the inherent loss and Qγ = ω0 / (2γe) as the external leakage. Correspondingly, the value of theoretical QCMT2 is calculated as 108.01 through the loss (δ = 4.25 × 1010 Hz) and external leakage (γe = 9.10 × 1010 Hz) of the structure at f2 = 4.59 THz. By comparison, the values of the theoretical quality factor QCMT1 and QCMT2 are almost consistent with the simulated Q values for the two resonant peaks. Furthermore, the effective impedance at the resonance frequencies fails to match the free space impedance, which means GADTA cannot completely absorb the incident wave at the resonances as shown in the absorption spectra. Notably, the analytical result demonstrates good agreement with the numerical result, which confirms the reliability of simulation for further study on GADTA.
In order to elucidate the physical origin of absorption enhancement inside GADTA, the distributions of magnetic and electric field amplitudes are depicted in Fig. 4. As can be observed in Fig. 4(a), there are obvious magnetic dipole resonances at f = 3.56 THz and f = 4.59 THz, trapping the incident magnetic field inside the gap between SiO2 ribbons and the THz cold mirror. Magnetic resonances are induced along the direction of incident magnetic field, leading to the enhancement and confinement of magnetic field. The concentration of the magnetic field is mainly located close to the corners of SiO2 inside the polymide layer. On the contrary, at f = 4.43 THz, no resonance is excited since this frequency is far away from either of the two absorption peaks. Different from electric dipole resonance, the resonant frequency of magnetic resonance is not totally inversely proportional to the width of ribbon as demonstrated before in Ref. [28]. In GADTA, SiO2 ribbons act as dielectric antennas to excite the magnetic resonance mode, which confines the magnetic field in the polymide layer rather than SiO2 ribbons. The mode area of the magnetic resonance mode is relatively large in the polymide layer, leading to the weak dependence on w for the resonant frequency. Therefore, for SiO2 ribbons with different widths, each of the magnetic dipole is formed away from each other within a certain distance at a certain resonance frequency. However, the strength of the induced magnetic dipole resonance is related to the width of SiO2 ribbons. Thus, combining SiO2 ribbons with two different widths within a given period can achieve dual resonance modes in the THz frequency range.
Fig. 4. (a) Magnetic field intensity and (b) electric field intensity distributions for the proposed GADTA at f = 3.56 THz, f = 4.59 THz and f = 4.43 THz, respectively. The values of these parameters are as follows: w1 = 13 µm, w2 = 5 µm, h = 7.8 µm, t = 35 µm, p = 60 µm. All studies are based on the normal TE incidence.
Energy consumption inside graphene layer can be obtained by the following surface integral:
Here, El is the tangential component of the electric field inside graphene, S is the surface area of graphene, and ε’’ is the imaginary part of graphene permittivity. Therefore, as depicted in Fig. 4(b), the electric field is mainly concentrated around graphene and dissipates in it, which contributes to the enhancement of THz absorption inside GADTA. The incident and reflected electric fields with almost same amplitudes and opposite phases cancel each other, leading to the field reduction around graphene at f = 4.43 THz.As demonstrated in Fig. 2 and Fig. 4, in the designed frequency range, the THz cold mirror acts like a traditional metal substrate which reflects the THz wave entirely. It enhances the strength of resonance and THz absorption inside graphene. The physical insight of GADTA is mainly based on the magnetic resonance mode in all-dielectric metamaterials [24,25]. This mechanism is an analog of the electric resonance mode found in the plasmonic absorber consisting of metal/dielectric/metal (MDM) configuration [41]. The roles of magnetic field and electric field are exchanged in these two mechanisms. However, all-dielectric absorber exhibits overwhelming advantages versus the plasmonic perfect absorber and solves the issues that plasmonic perfect absorber is facing, such as metallic heating and incompatibility with CMOS fabrication processes. Furthermore, in the all-dielectric configuration, the incident electric field is always parallel to the graphene surface, which is of significant importance for the enhanced absorption in two-dimensional materials.
Furthermore, the effects of geometric parameters on the THz absorption of GADTA are also investigated. Figure 5(a) indicates that as the thickness of SiO2 ribbon increases, the left absorption peak exhibits a slight redshift, while the right absorption resonance shows a dramatic redshift. As h increases from 5 µm to 19 µm, the right absorption peak is red-shifted from 4.59 THz to 4.12 THz. When h is larger than 19 µm, the right resonance disappears. Therefore, for the dual-band THz absorber, h should be chosen at a relatively low value.
Fig. 5. Absorption spectra of GADTA as a function of (a) f and h, (b) f and p, (c) f and t, (d) f and θ, respectively. All studies are based on the TE incidence.
The effect of p on the THz absorption inside GADTA is depicted in Fig. 5(b). It is observed that with the increase of p, the central frequencies of resonances exhibit red shifts to some extent. Within the range from 52 µm to 67 µm, two obvious absorption peaks can be observed. When p = 60 µm, both of the absorption peaks can reach up to 87% at the resonance. As p increases from 76 µm to 100 µm, the left absorption peak gradually disappears, while the right resonance tends to be stronger and stronger. Hence, the filling factor calculated as (w1+w2)/p should be chosen at an appropriate value between 0.23 and 0.35.
Similarly with the traditional plasmonic PEC-based absorber [10,14], the thickness of the insulator polymide plays a critical role in the absorption performance. As can be seen from Fig. 5(c), both of the dual absorption peaks show obvious red shifts with the increase of t. As t increases from 16 µm to 44 µm, the left absorption peak is red-shifted from 4.30 THz to 3.40 THz while the central frequency of the right absorption peak gradually decreases from 5.00 THz to 4.06 THz as t increases from 25 µm to 50 µm. This is due to the fact that the magnetic resonance at the absorption peak has a dominant mode area in the insulator layer. Thus, as the thickness of insulator rises up, a longer resonant wavelength is required for the formation of magnetic dipole. Particularly, the absorption rates at both resonances can reach above 80% in the thickness range from 32 µm to 40 µm.
Next, in order to investigate the angular dependence for the performance of GADTA, we plot the absorption spectra under oblique incidences in Fig. 5(d). The angle between the incidence and the normal direction in the incident plane is denoted as θ. As θ increases from 0° to 7°, the left absorption peak at 3.56 THz is slightly divided into two separated peaks, then the spectrum gradually exhibits four obvious absorption bands with narrow bandwidths. If θ is extended to 21°, the left two absorption bands disappear in the spectrum. As the incident angle continues to increase, the resonance strength around 4 THz gets weakened first and then contributes to a broad absorption band. When θ>60°, the resonance becomes exceptionally strengthened leading to absorption rates larger than 80%. The absorption performance of GADTA is strongly sensitive to the incident angle, which is totally different from the traditional plasmonic absorber based on electric dipole resonance [41]. The above analysis implies that the manipulation of incident angle provides a flexible method to control the absorption spectrum for GADTA, showing the potential for developing reconfigurable all-dielectric THz absorbers.
Variation of the physical dimensions after final design and implementation of the THz absorber is inconvenient and not feasible. Therefore, an active tuning method to control the absorber characteristics after fabrication is indispensable. By varying the external voltage applied on graphene, the proposed GADTA is expected to achieve active tuning ability of magnetic resonance. This is attributed to the fact that the conductivity of graphene is determined by its chemical potential μc according to Eq. (2). Therefore, as demonstrated in Fig. 6, both of the absorption peaks can be tuned if μc is altered. When μc = 0.1 eV, the left absorption peak is located at 3.43 THz and the right absorption peak appears at 4.39 THz. As μc increases to 0.9 eV, which means graphene is highly doped, the left peak moves to 3.59 THz and the right peak is blue-shifted to 4.65 THz. The frequency shift of the electrical tuning reaches as large as 0.201 THz/eV for the left peak and 0.325 THz/eV for the right peak, which demonstrates the remarkable tunability of GADTA. Additionally, there is a relatively-low absorption peak around f = 3.25 THz when μc is smaller than 0.3 eV. This is due to the effect of diffraction order which can be immensely avoided by optimizing the thickness of polymide and SiO2 ribbon [28].
Fig. 6. Absorption spectra of the proposed GADTA with various chemical potentials of graphene. The values of these parameters are as follows: w1 = 13 µm, w2 = 5 µm, h = 7.8 µm, t = 35 µm, p = 60 µm. All studies are based on the TE normal incidence.
4. Conclusions
In summary, we have demonstrated a tunable dual-band graphene-based THz absorber with all-dielectric configuration. By combining magnetic resonance and a THz cold mirror, the absorber can achieve two absorption peaks with quality factors as high as 130.88 and 108.25, respectively, under TE incidence. The analytical result of the CMT method has demonstrated good agreement with the simulation result. The absorption spectrum can be adjusted with the manipulation of geometric parameters and incident angles. Moreover, via varying the chemical potential of graphene, both of the resonance can be actively tuned after fabrication attributed to the tunability of graphene conductivity. This easy-to-fabricate structure paves the way for developing various promising THz devices, such as modulators, sensors and detectors.
Funding
Key Laboratory of Microelectronic Devices Integrated Technology, Chinese Academy of Sciences; Natural Science Basic Research Plan in Shaanxi Province of China (2020JM-515); National Natural Science Foundation of China (62005232); Xiamen Science and Technology Project (3502Z20173042); NSAF Joint Fund (U18301160).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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