1. Introduction

Infrared (IR) sensors play a vital role across various fields, serving critical functions in applications such as sensing, night vision capabilities, and environmental monitoring for security, safety, and pollution detection. They are also employed in automotive safety enhancement, medical diagnostics, reliable communication, and remote sensing, among others [1–6]. Dual-band IR sensors operate simultaneously in two distinct infrared bands, providing an enhanced perspective of the monitored environment. This capability is crucial in several applications. For instance, automotive night vision systems utilize this technology to improve differentiation between living beings and inanimate objects, thereby enhancing safety [7]. Moreover, optimized dual-band gas sensors can be tailored to multiple spectral fingerprints of their target to boost resolution and accuracy [8]. Additionally, in missile warning systems, dual-band IR sensors enable the detection of distinct incoming targets, aiming to reduce the likelihood of false alarms [9].

The fields of plasmonics and metasurfaces have significantly contributed to the development of highly responsive and selective IR sensors [10–13]. Geometrically tailored and subwavelength structures with desired electromagnetic responses can efficiently absorb infrared light at a desired wavelength with reduced full-width half maximum (FWHM). These attributes help overcome the limitations associated with sensors reliant on bulky multilayer metal-insulator-metal (MIM) absorbers [14–20]. In addition to high responsivity, tunability is another important factor in IR sensing that allows for the manipulation of the targeted wavelengths through electrically tuning material properties or geometry modifications [21–26]. The emergence of graphene, simultaneously exhibiting plasmonic responses and electrical tunability, has enabled the development of reconfigurable and single-band absorbers at infrared frequencies based on merging this material with ultrathin metasurfaces [27–33]. Even though achieving spectrally selective absorption over two simultaneous bands is a challenging task, several works have explored this type of device [34–40]. For instance, dual-band responses can be achieved through electromagnetically induced transparency (EIT) [34]. An alternative option could be to use graphene ribbons, as described in [35], to precisely control the spectral position of the resonances. However, this method is difficult to realize in practice as requires patterning two different graphene layers located nearby. A continuous graphene layer combined with dielectric rods has also been proposed for dual-band IR absorption [36], a design that leads to polarization-sensitive absorption. Other groups have explored different strategies, like depositing gold resonators on top of graphene layers [37] and gathering resonators with fixed geometrical dimensions [38], among others [39,40]. Despite these efforts, there is still a need to develop versatile platforms capable of providing spectrally selective dual-band absorption across the IR spectrum. Such a platform should address the aforementioned challenges, including easy fabrication, polarization insensitivity, spectral selectivity, and significant reconfiguration capabilities.

In this work, we tackle this problem by exploiting hybridized modes arising in gold/graphene metasurfaces. The design relies on an array of cross-shaped gold resonators printed over an ultrathin metasurface. Such structures have been well studied in the literature and provide spectrally selective IR absorption through a fundamental mode together with large polarization insensitivity due to their symmetric geometry [41–43]. We modify the host metasurface by overlaying the gold resonators with graphene cross-ribbons. In the resulting structure, the metasurface fundamental mode hybridizes with the graphene plasmonic ribbons, leading to a pair of spectrally selective IR absorptive modes whose properties can easily be identified and controlled. More specifically, the ribbons’ width determines the spectral distance between the two modes while graphene’s chemical potential controls the absorption level of each mode. It should be stressed that the operation wavelength of the device can be scaled across the IR spectrum by modifying the gold resonators of the host metasurface while maintaining its dual-band behavior. We have studied the performance of this platform using full-wave numerical simulations based on CST Studio [44]. Additionally, we have developed a coupled-mode theory (CMT) [45] approach to capture the fundamental operation principle of this type of device and explain the hybridization process that enables their dual-band IR response.

Our proposed hybrid dual-band absorber can be applied for several applications. For instance, in photodetection, graphene ribbons will harvest hot carriers while the gold nanostructures contribute to enhance the overall IR absorption intensity [46–48]. Another potential application is in the design of dual-band thermal sensors – which can be realized by incorporating the proposed metasurface on top of radiofrequency (RF) microelectromechanical systems (MEMS) [49,50]. Moreover, this design can lead to dual-band emitters in which thermal radiation is modified by the metasurface [51], yielding dual-band IR radiation.

The rest of this paper is organized as follows. Section 2 presents the design of the dual-band hybrid IR absorber. Section 3 investigates the device performance by varying the geometrical dimensions of the gold resonators and graphene ribbons, as well as graphene’s electrical properties. Section 4 describes the coupled-mode theory developed to characterize the hybridization process that enables dual-band absorption. Finally, Section 5 concludes the paper.

2. Design of dual-band and spectrally selective IR absorbers

Figure 1(a) illustrates the schematic of the proposed device, comprising a metasurface resonator overlaid with a patterned graphene sheet. The host metasurface consists of periodic cross-shaped gold resonators with symmetric arms, separated from a continuous gold layer by an aluminum nitride (AlN) substrate. The geometrical dimensions are depicted in a top view of a unit cell in Fig. 1(b), where ‘$\textrm{a}$’ and ‘$\textrm{b}$’ represent the width and length of the cross arms, respectively, ‘$\textrm{P}$’ denotes the period of the structure, and ‘${\textrm{a}_\textrm{g}}$’ denotes the width of the cross-ribbons of graphene. Figure 1(c) provides a side view of this multilayer structure, where ${\textrm{t}_\textrm{c}}$, ${\textrm{t}_\textrm{d}}$_, and ${\textrm{t}_\textrm{g}}$ indicate the thickness of each layer.

 

Fig. 1. Proposed dual-band and spectrally selective IR absorber. (a) Geometrical schematic. The device is composed of periodic cross-shaped gold resonators, graphene patterned in cross-ribbons, an AIN substrate, and a gold ground plate. (b) Top-view and (c) cross-view of a unit cell of this structure. (d) Absorption profile. Results show the response of the host metasurface (dot-dashed red line) and the hybrid graphene/gold metasurface (solid blue line). Inset illustrates the mode profile when the device is illuminated by an incident plane wave with momentum along -z direction (normal to the interference) and linear polarized electric field along x direction (${\vec{E}_i} = {E_0}\hat{x}$) and magnetic field along the -y direction $({\vec{H}_i} ={-} {H_0}\hat{y}$). ${\mathrm{\lambda }_\textrm{g}}$ and ${\mathrm{\lambda }_\textrm{c}}$ represent the resonant wavelength of the dark and bright modes, respectively. Other parameters are $\textrm{a} = 250\textrm{nm}$, $\textrm{b} = 2.5{\;\ \mathrm{\mu} \mathrm{m}}$, $\textrm{P} = 3.5{\;\ \mathrm{\mu} \mathrm{m}}$, ${\textrm{t}_\textrm{g}} = 100\textrm{nm}$, ${\textrm{t}_\textrm{d}} = 250\textrm{nm}$, ${\textrm{t}_\textrm{c}} = 150\textrm{nm}$, ${\mathrm{\mu }_\textrm{c}} = 0.55\textrm{eV}$, and ${\textrm{a}_\textrm{g}} = 330\textrm{nm}$.

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In our numerical and theoretical study, gold and AlN are modeled using the Lorentz-Drude [52] and Lorentz permittivity model [49], respectively, and graphene is characterized by surface conductivity using the well-known Kubo formula [53]. Graphene conductivity (${\mathrm{\sigma }_\textrm{g}}$) is a function of angular frequency $(\mathrm{\omega } )$, temperature (T), relaxation time ($\mathrm{\tau })$, and chemical potential $({{\mathrm{\mu }_\textrm{c}}} )$, where ${\mathrm{\mu }_\textrm{c}}$ can be adjusted by applying a gate voltage. In our study, we set common values of $\mathrm{\tau } = 0.1\textrm{ps},\textrm{T} = 300\textrm{K},$ and chemical potential in the range of 0.1-0.9 $\textrm{eV}$, representing a graphene layer with high mobility ranging from 1000 to 10000 c${\textrm{m}^2}/\textrm{VS}$, as extensively reported in experimental works [54,55,56].

Full-wave numerical simulations of the device unit cell are carried out using CST Studio [44] by applying periodic boundary conditions. The unit cell is illuminated by a normally incident plane wave with linear polarization. Initially, we optimize the dimensions of the metasurface resonator without graphene to achieve narrowband absorption with a resonance peak in the mid-IR range as shown in Fig. 1(d) by a dot-dashed red line. Selecting dimensions of $\textrm{a} = 250\textrm{nm},\textrm{b} = 2.5\mathrm{\ \mu m},\textrm{}$ and periodicity $\textrm{P} = 3.5{\;\ \mathrm{\mu} \mathrm{m}},$ results in a spectrally selective absorption peak at $\mathrm{\lambda } = 7.89{\;\ \mathrm{\mu} \mathrm{m}}$ with a FWHM of ∼$440\textrm{nm}$. The material thicknesses are chosen as follows: ${\textrm{t}_\textrm{g}}\textrm{} = 100\textrm{nm}$ (ground layer thickness), ${\textrm{t}_{\textrm{d}}}$= 250 nm (AIN substrate thickness), and ${\textrm{t}_\textrm{c}} = 150\textrm{nm}$ (thickness of the gold cross). These thickness values are consistently used throughout the paper.

It is worth noting that changing the arms’ width slightly modifies the resonance FWHM, whereas adjusting the arm’s length significantly shifts the resonance position in the IR band [57,58,59]. Additionally, as two arms of the gold cross are equal, this resonator is quasi-independent of the polarization of the incident light up to an elevation of approximately ∼ $45^\circ $ measured from the metasurface’s normal direction [41,60,61,62]. Throughout the paper, we refer to this resonance mode arising from the host metasurface as the bright mode – as it refers to localized surface plasmon polaritons (SPPs) generated when the gold surface interacts with incident light. These modes are “bright” because they are readily excited by incoming light.

Next, we incorporate the graphene cross-ribbons on top of the metasurface to have a hybrid design, as illustrated in Fig. 1(a). By optimizing the width of the graphene ribbons, ${\textrm{a}_\textrm{g}}$, it is possible to achieve a dual-band response composed of two spectrally selective absorptive peaks with equal absorption levels (∼82% in Fig. 1) and somewhat similar FWHM (${\sim} \textrm{}353\textrm{nm}$ and ${\sim} \textrm{}455\textrm{nm}$), as shown in Fig. 1(d) by a solid blue line. This panel shows how the device response transitions from a single band to a dual band by adding graphene ribbons. In the hybrid structure, the second resonance peak appears at lower wavelengths $({{\mathrm{\lambda }_\textrm{g}}} )$ than the host metasurface resonance due to the plasmonic coupling between the graphene ribbons and the gold resonators. We denote this resonance peak as the dark mode. Dark modes are edge or surface plasmon polaritons supported by the graphene ribbons. These modes are termed “dark” because they are not directly excited by normally incident light in the absence of the gold resonator. When the graphene ribbon is narrow, a hybrid mode appears between the ribbons and the gold nanostructures. These hybrid modes represent a blending of characteristics from both materials and can exhibit unique properties due to their combined nature. As the width of the ribbon increases, the coupling rate decays, and the absorption peak related to the graphene modes becomes dominant. The coupling process also affects the fundamental mode supported by the host metasurface, leading to a frequency shift to ${\mathrm{\lambda }_\textrm{c}}\textrm{}$ and a change in its absorption level. The inset plots in Fig. 1(d) illustrate the electric field intensity at these two resonances. The electric field distribution at $\textrm{}{\mathrm{\lambda }_\textrm{c}}$ follows a common resonance among gold resonators, whereas the field profile at ${\mathrm{\lambda }_\textrm{g}}$ has additional energy trapped in the graphene ribbon region close to the gold cross-shaped resonators.

In the following, we will explore how controlling the graphene cross-ribbon width as well as the chemical potential modifies the energy guided in that region, in turn, allows for the manipulation of the dual-band performance of this type of device.

3. Origin of the dual band response and performance

Figure 2 explores the absorption spectrum of the proposed hybrid absorber for various widths of the graphene cross-ribbons (${\textrm{a}_\textrm{g}}$). Starting with Fig. 2(a), it presents the response of the host metasurface in the absence of graphene (or ${\textrm{a}_\textrm{g}} = 0$). As discussed above, here there is only one resonance peak arising from the interaction of the gold resonators within the metasurface and the incident light. Adding graphene cross-ribbons with a width slightly larger than the width of the gold resonators leads to an emerging absorption peak (dark mode) at the wavelength ${\mathrm{\lambda }_\textrm{g}}$ shorter than the bright mode wavelength (${\mathrm{\lambda }_\textrm{c}}),$ as shown in Fig. 2(b). Dark mode arises due to the coupling of the surface plasmons in the graphene ribbon with the modes of the cross-shaped gold resonator. As the width of the graphene layer increases, the coupling between the graphene ribbon and the gold resonator gets stronger, leading to a simultaneous increase of the dark mode absorption level at ${\mathrm{\lambda }_\textrm{g}}$ and a decrease in the bright mode absorption level at ${\mathrm{\lambda }_\textrm{c}}$. The evolution of this coupling process is illustrated in Fig. 2(c–e). It should be noted that, for specific values of the graphene cross-ribbon width, dual-band responses with identical absorption levels for both bright and dark modes are found. This trend continues until the so-called dark mode becomes dominant while the bright peak vanishes for sufficiently large width values of the graphene ribbon. Consequently, the absorption spectrum shifts from dual-band to single-band, as depicted in Fig. 2(f). During this process, the coupling between the bright and dark modes also modifies their operation wavelength. Inset plots in Fig. 2 show the electric field distribution from a top view of a unit cell excited at resonances associated with the bright and dark modes. Results show that increasing the width of the graphene cross-ribbon leads to an increasingly large electric field intensity of the dark mode. Simultaneously, the intensity of the bright mode associated with the gold resonators decreases and eventually disappears as the plasmonic response of the graphene ribbon becomes dominant.

 

Fig. 2. Absorption profile of the proposed IR device and evolution of the dual-band response by varying the width of graphene cross-ribbons (${\textrm{a}_\textrm{g}}$). Inset plots depict the electric-field profile (from a top view of a unit cell) at resonance frequencies corresponding to the dark (${\mathrm{\lambda }_\textrm{g}}$) and bright (${\mathrm{\lambda }_\textrm{c}}$) modes. Solid lines are obtained with full-wave numerical simulations and dashed lines with the CMT model. Graphene chemical potential is set to ${\mathrm{\mu }_\textrm{c}} = 0.55\textrm{eV}$ and other parameters are as in Fig. 1.

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The influence of the graphene ribbon width on the absorption of the device is further explored in the heat map shown in Fig. 3(a). There, red spots represent absorption peaks related to the coupled bright and dark modes. Results show that when the width of the graphene ribbons is smaller than the width of the cross-shaped gold element (i.e., ${\textrm{a}_\textrm{g}} < \textrm{a}\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }250\textrm{nm}$), a single-band response is observed associated with the bright mode resonating at ${\mathrm{\lambda }_\textrm{c}}$. Therefore, the bright mode from the host metasurface dominates at this range. As the width of graphene gradually increases, the device supports two coupled hybrid modes at ${\mathrm{\lambda }_\textrm{c}}$ and ${\mathrm{\lambda }_\textrm{g}}\textrm{}$- highlighted by a white dashed box in the plot and roughly corresponding to the ribbon width range of $300 < {\textrm{a}_\textrm{g}} < 420\textrm{nm}$. Further increasing the ribbon width, the spectrum evolves into a single band dominated by the dark mode at ${\mathrm{\lambda }_\textrm{g}}$. Therefore, selecting a desired graphene ribbon width can enable a single- or dual-band response.

 

Fig. 3. IR absorption response of the proposed device calculated using full-wave numerical simulations. (a)-(b) Density plot absorption spectrum versus (a) ribbon width ${\textrm{a}_\textrm{g}}$, setting the chemical potential value to ${\mathrm{\mu }_\textrm{c}} = 0.55\textrm{eV}$; and (b) graphene’s chemical potential ${\mathrm{\mu }_\textrm{c}}$, setting the ribbon width to ${\textrm{a}_\textrm{g}} = 330\textrm{nm}$. Black-dashed lines correspond to the resonance wavelengths of bright (${\mathrm{\lambda }_\textrm{c}}$) and dark $({{\mathrm{\lambda }_\textrm{c}}} )$ modes obtained using CMT. (c-e) IR absorption spectrum for the three specific chemical potentials labeled as M, K, L in panel (b), and dashed lines are obtained with CMT.

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Next, we evaluate the effect of graphene’s chemical potential on the response of the proposed device. It is well-known that an external gate bias allows for the manipulation of graphene’s chemical potential and in turn, the electrical and plasmonic properties of the material [57]. In this context, Fig. 3(b) studies the absorption spectra of the device versus graphene’s chemical potential while the graphene ribbon width is fixed to ${\textrm{a}_\textrm{g}} = 330\textrm{nm}$. This ribbon width was selected from the ${\textrm{a}_\textrm{g}}$ range of the white dashed box in Fig. 3(a), where a strong dual-band response is anticipated. Results show that the dark mode associated with the graphene ribbon becomes dominant for low chemical potentials. This is because low Fermi energies enhance the overall graphene’s plasmonic response in the IR band, thus leading to a dominant behavior of the graphene ribbon. As the chemical potential increases, the coupling between the dark and bright modes strengthens. This phenomenon occurs because the rising chemical potential enables more charge carriers within the graphene, leading to an enhanced overlap in electronic states between gold and graphene.

For some specific potential values, the contributions from gold and graphene become comparable, enhancing the overall coupling between the dark and bright modes. This leads again to a region in which dual-band IR absorption responses can be found. Furthermore, graphene’s chemical potential controls the relative absorption levels of the dark and bright modes. The resulting absorption tunability is illustrated in Fig. 3(b-e). As the chemical potential increases further, graphene plasmonic response decreases and thus the influence of gold resonators becomes dominant. Thus, a single-band bright mode response is recovered for sufficiently large chemical potential values. Additionally, adjusting graphene’s chemical potential allows for the electrical control of the spectral position of the resonances (see Fig. 3(b)), with tuning ranges up to 25%. Since the proposed design relies on a series of interconnected graphene ribbons, the gate voltage required to tune them can be applied between any point of the graphene structure and the bottom gold layer. The gate voltage required to obtain a specific chemical potential can be found as described elsewhere [54,56,63]. In brief, it follows the relation ${\mathrm{\mu }_\textrm{c}}^2 = \mathrm{\pi }{\hbar ^2}{\mathrm{\upsilon }_\textrm{F}}{\textrm{C}_\textrm{g}}({{\textrm{V}_\textrm{g}} - {\textrm{V}_{\textrm{Dirac}}}} )/\textrm{e}$, where $\textrm{e}$ is the electron charge, ${\mathrm{\upsilon }_\textrm{F}}$ is graphene’s Fermi velocity, ${\textrm{C}_\textrm{g}}$ is the gate capacitance given by ${\textrm{C}_\textrm{g}} = {\mathrm{\varepsilon }_0}{\mathrm{\varepsilon }_{\textrm{AlN}/}}{\textrm{t}_\textrm{d}},$ and ${\textrm{V}_{\textrm{Dirac}}}$ is the gate voltage when the conductivity of graphene is lowest [54].

The proposed structure allows for obtaining dual-band responses with equal absorption levels by adjusting graphene’s chemical potential and cross-ribbon width. Figure 4(a) shows such a response for three different pairs of parameters. Remarkably, the spectral distance between the two modes is adjustable by finely tuning ${\textrm{a}_\textrm{g}}$ and ${\mathrm{\mu }_\textrm{c}}$ while keeping an identical absorption response. Interestingly, we observe a linear correlation between graphene’s chemical potential and ribbon width to maintain a dual-band response with equal absorption levels (see inset of Fig. 4(a)). Such response appears due to the interplay between graphene plasmonic response, controlled through the chemical potential, and the modes supported by the ribbon width. The responses outlined above can be scaled across the IR spectrum by tailoring the dimensions of the gold resonators without modifying the graphene’s pattern or chemical potential. Figure 4(b) confirms that large tunability can be obtained, easily covering from ${\sim} \textrm{}4\mathrm{\ \mu m}$ up to 9 $\mathrm{\mu m}$. It should be emphasized that to maintain a dual-band IR response with identical absorption levels at both resonances, the length and width of the gold-cross must be adjusted simultaneously. The length of the cross arm primarily dictates the frequency shift, while the width regulates the level of absorption adjustment. Furthermore, in the hybrid design, unlike the bare metasurface (without graphene coating), variations in the width of the cross contribute more significantly to resonance shift, owing to the coupling with graphene. Notably, we observe an almost linear correlation between the length and width of the cross-shaped gold resonator to achieve equal dual-band responses (see inset of Fig. 4(b)). It should also be mentioned that an additional absorption peak appears around $11\mathrm{\;\ \mu m\;\ }$ in all our results, associated with the excitation of phonons in the AlN dielectric.

 

Fig. 4. Dual-band IR absorption response. (a) Tuning the spectral separation distance between dark and bright modes involves adjusting both graphene’s chemical potential and ribbon width, simultaneously. The dimensions of the gold resonators are fixed to $\textrm{a} = 250\textrm{nm}$ and $\textrm{b} = 2.5{\;\ \mathrm{\mu} \mathrm{m}}$ with a periodicity of $\textrm{P} = 3.5{\;\ \mathrm{\mu} \mathrm{m}}.$ Inset plot shows the relationship between graphene’s chemical potential and ribbon width to maintain equal absorption levels in both bands. (b) Scaling the response across the IR spectrum by adjusting the dimensions of the gold resonators while keeping a periodicity of $\textrm{P} = 3.5{\;\ \mathrm{\mu} \mathrm{m}}$. Graphene’s chemical potential and ribbon width are set to $0.5$eV and $325\textrm{nm},$ respectively. Inset plot shows the relationship between the arm width and length of the gold resonator required to maintain equal absorption levels in both bands.

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4. Coupled-mode theory

In this section, we apply coupled-mode theory to analytically explore the fundamental response of the proposed absorbers. In our formulation, the bright mode is provided by the host metasurface as it appears even in the absence of graphene ribbons. We consider that the dark mode is provided by the graphene ribbons, as the resonance that they provide is drastically different in the absence of gold resonators. When these two resonators are coupled together, they form a hybrid structure whose overall response can be modeled by considering that the bright mode is excited by the incident light and then exchanges energy with the dark mode. This behavior can be described by the following coupled equations associated with a two-oscillator system [64–66].

In coupled mode theory, there is a direct proportionality between the square of resonant amplitudes and the energy stored at each resonant mode (represented as ${\textrm{W}_\textrm{c}} \equiv {|{{\textrm{x}_\textrm{c}}} |^2},\textrm{}{\textrm{W}_\textrm{g}} \equiv {|{{\textrm{x}_\textrm{g}}} |^2}$) [45]. Furthermore, resonant systems exhibit a characteristic where the absorption of each mode is in direct proportion to its specific energy reflecting a fundamental resonance principle where energy absorption is heightened when the incoming energy matches the resonant mode's energy. Therefore, by comparing the theoretical values of the square of resonant amplitudes for each mode with their corresponding absorption values obtained from simulations, we can effectively compare theoretical predictions with simulation results.

To qualitatively capture the operation principle of the proposed absorber with this approach, the following steps are applied. First, we consider a single-oscillator system to model the bright mode of the host metasurface, i.e., the gold resonators in the absence of graphene. The amplitude of this single oscillator model yields ${|{{\textrm{x}_\textrm{c}}} |^2} = {({\textrm{g}{\textrm{E}_0}} )^2}/({{{({\mathrm{\omega }_{0\textrm{c}}^2 - {\mathrm{\omega }^2}} )}^2} + {{({{\mathrm{\gamma }_\textrm{c}}\mathrm{\omega }} )}^2}} )$. This equation can easily be employed to fit the absorption spectra of the host metasurface (i.e., Fig. 2(a) - dash line) and then determine the parameters $\textrm{g}$, ${\mathrm{\omega }_{0\textrm{c}}}$, and ${\mathrm{\gamma }_\textrm{c}}$, related to the bright mode. Next, we calculate the parameters that define the dark mode in terms of damping, resonant wavelength, and the coupling between the bright and dark modes by fitting Eqs. (3) and (4) to the absorption spectra related to each mode. It should be stressed that these parameters will change depending on the graphene’s pattern dimension $({{\textrm{a}_\textrm{g}}} )$ and the chemical potential ${\mathrm{\mu }_\textrm{c}}$. Arguably, the most important parameters in this process are the coefficient $\mathrm{\kappa }$ that describes the energy interchange process between the modes and the plasmon frequency of the dark mode (${\mathrm{\omega }_{0g}}$) that is correlated to the location of the resonance of the main mode. To account for the influence of key parameters on the structure's optical response, we determined the values of both $\mathrm{\kappa }$ and ${\mathrm{\omega }_{0g}}$ considering first variations in graphene's width, and then the chemical potential. Then, we generate the absorption spectrum by considering the calculated coupling parameters. The corresponding spectrum is shown together with the dual-band numerical spectrum as dashed-line in Fig. 2, and Fig. 3(e-f) shows the fitted CMT dual-band spectrum.

Figure 5 shows the variations of a coupling coefficient $\mathrm{\kappa }$ and plasmon frequency of the dark mode ${\mathrm{\omega }_{0g}}$ obtained through the fitting by considering different values for the width of the graphene cross-ribbons and the chemical potential of graphene. As depicted in Fig. 5(a), by increasing the graphene width, the coupling coefficient (blue line) also increases and gradually saturates to a constant value, which is consistent with the numerical results shown in Fig. 2. Particularly, when the graphene width is large enough (with a width approximately two times larger than the width of the metasurface elements), increasing the graphene width will not change the absorption spectrum and the coupling effect remains stable. On the other hand, at smaller graphene widths, the coupling between the bright and dark modes is not significant, resulting in almost zero. Figure 5(b) confirms that an increase in chemical potential enhances the coupling coefficient (blue line). This response correlates well with our previous reasoning: lower chemical potential increases graphene’s plasmonic response and makes it dominant in the device. As the chemical potential increases, the coupling between the graphene ribbon and the gold resonators increases. Finally, Fig. 5(a) and (b) also show that ${\mathrm{\omega }_{0g}}$, i.e., the resonance frequency related to the dark mode, decreases as the graphene width (chemical potential) increases (decreases). This response is fully consistent with the simulations shown in Fig. 3(a) and (b), in which the absorption of the dark mode is red-shifted with increased/decreased graphene ribbons’ width/chemical potential. We remark that Figs. 2 and 3 demonstrate that couple mode theory captures the response of the system, providing good agreement with full-wave numerical simulations. Moving beyond, we can further utilize the CMT model with the obtained parameters to analytically estimate the response or generate rapid initial designs. This model is highly beneficial for parametric sweeps and to generate quasi-optimal designs that will be tuned using full-wave numerical simulations.

 

Fig. 5. Coupling coefficient $\mathrm{\kappa }$ and ${\mathrm{\omega }_{0\textrm{g}}}$ as a function of (a) graphene width ${\textrm{a}_\textrm{g}}$ where ${\mathrm{\mu }_\textrm{c}} = 0.55\textrm{eV}$ ; (b) the chemical potential ${\mathrm{\mu }_\textrm{c}}$ where ${\textrm{a}_\textrm{g}} = 330\textrm{nm}$. Other parameters are $\textrm{a} = 250\textrm{nm}$, $\textrm{b} = 2.5{\;\ \mathrm{\mu} \mathrm{m}}$, $\textrm{P} = 3.5{\;\ \mathrm{\mu} \mathrm{m}}$, ${\textrm{t}_\textrm{g}} = 100\textrm{nm}$, ${\textrm{t}_\textrm{d}} = 250\textrm{nm}$, ${\textrm{t}_\textrm{c}} = 150\textrm{nm}$.

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5. Conclusion

This paper has proposed an IR absorber based on integrating patterned graphene and a metasurface composed of cross-shaped gold resonators. The system exhibits dual-band absorption responses, electronic tunability, and can be scaled across the infrared spectrum. Throughout comprehensive simulations and theoretical analysis, we show that the absorption profile of the device appears due to the hybridization of two modes. On one hand, the bright mode is dominated by the fundamental resonance appearing in the gold metasurface, and on the other, the dark mode arises from the graphene ribbons. Adjusting the geometry of the gold resonator and the ribbon dimensions/chemical potential permits scaling the overall operation wavelength of the absorber as well as fine-tuning the absorption level and spectral separation between the bands, respectively. We also found that this response can be obtained using different values of graphene’s relaxation time, thus relaxing fabrication constraints and the need for high-quality graphene. Coupled-mode theory has been applied to capture the fundamental operation principle of the device, and to provide a deeper understanding of how each parameter modifies its response. Our finding paves the way for the development of flexible, electrically reconfigurable, spectrally selective and dual-band, ultrathin IR absorbers with significant applications in military and civilian applications, including surveillance, spectroscopy, sensing, and imaging, among others.

Appendix

Figure 6(a–b) illustrates the intraband conductivity of graphene for various chemical potentials, calculated using the Kuba formula. The chemical potential can be attained through doping and electrostatic gating. Figure 6(c) displays the desired gate voltage applied to doped graphene layers with different Fermi levels versus chemical potentials, derived from the relation reported in [67,68].

 

Fig. 6. (a) The real and (b) imaginary parts of the conductivity of graphene, normalized to the universal conductivity of σ₀= e²/4ℏ, versus wavelengths for chemical potentials 0.1, 0.3, and 0.5 eV. Other parameters are set to τ=0.1 ps, and T = 300 K. (c) Gate voltage applied to the doped graphene versus the chemical potential of graphene. The Fermi doped levels are set to be E_F1= 0, E_F2= 0.30, E_F3= 0.42, E_F4= 0.52, E_F5= 0.60, E_F6= 0.67, E_F7= 0.73, and E_F8= 0.79 eV, achievable by chemical doping.

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The breakdown voltage of the AlN substrate is reported to be 12 MV/cm [69,70]. In our specific design, the AlN substrate with a thickness of t_d= 250 nm, demonstrates a breakdown voltage of approximately 300 volts, well above the applied gate voltage to graphene. It is worth mentioning that certain techniques, such as optimizing thickness, controlling environmental conditions, and applying surface treatments, have the potential to increase the breakdown voltage threshold.

Funding

This work was supported in part by the U.S. Department of Defense – Army with grant (W911NF2210067).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.

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