## Abstract

An ultra-broadband multilayered graphene absorber operating at terahertz (THz) frequencies is proposed. The absorber design makes use of three mechanisms: (i) The graphene layers are asymmetrically patterned to support higher order surface plasmon modes that destructively interfere with the dipolar mode and generate electromagnetically induced absorption. (ii) The patterned graphene layers biased at different gate voltages backed-up with dielectric substrates are stacked on top of each other. The resulting absorber is polarization dependent but has an ultra-broadband of operation. (iii) Graphene’s damping factor is increased by lowering its electron mobility to 1000cm^{2}/Vs. Indeed, numerical experiments demonstrate that with only three layers, bandwidth of 90% absorption can be extended upto 7THz, which is drastically larger than only few THz of bandwidth that can be achieved with existing metallic/graphene absorbers.

© 2013 Optical Society of America

## 1. Introduction

There is an increased interest in designing perfect electromagnetic wave absorbers at terahertz (THz) frequencies since they are oftentimes an indispensible component in devices/systems utilized in biosensing, imaging, and communications [1]. Recently graphene has become one of the most attractive materials for designing THz-wave absorbers. The reasons are three-folds: (i) A graphene layer can support surface plasmon polaritons (SPPs) at THz frequencies. Consequently absorption at the resonance frequency of the SPPs is significantly increased. (ii) These frequencies can be dynamically tuned via biasing the graphene layer. (iii) The absorption in an atomically thin graphene sheet is already very high, i.e., “2.3%” per layer.

However, considering its atomic thickness, a single layer of graphene remains practically transparent to THz waves and as a result the overall absorption is significantly reduced. To overcome this problem, patterned graphene layers (as opposed to uniform ones) have been designed [2–6]. Periodic patterns help to increase the coupling of energy from the incident field to the SPPs [2] and therefore increase the overall absorption. Absorber designs based on periodic arrays of graphene nano-disks have been shown to fully absorb electromagnetic waves within the mid-infrared spectrum [2]. The structured graphene in the form of micro-ribbons together with the control of coherent interference effects from back reflector permits complete absorption within the same frequency range [4].

From a practical point of view, the bandwidth of absorption at THz frequencies is important for many of the applications mentioned above [1]. But unfortunately, achieving broadband absorption at THz frequencies still remains a challenging task [7]. The current bandwidth of operation in metal-based plasmonic absorbers is limited to only a few THz (see [7–12] in Table 1). This is attributed to the narrow bandwidth of SPPs generated on metallic surfaces at nanoscales, which are used as a mechanism to achieve absorption. The damping on graphene layer is even smaller than that on the metal surface, therefore graphene SPPs are also narrowband resulting in a bandwidth of absorption smaller than 1.8 THz (see [2–6] in Table 1).

In this paper, an absorber design operating at THz frequencies is proposed. The design makes use of three mechanisms: (i) The asymmetric pattern on the graphene layer(s) allows for generation of higher order SPP modes with resonance frequencies that can be tuned via electrical biasing. (ii) Several patterned graphene layers biased at different voltages backed with dielectric substrates are stacked on top of each other. (iii) Graphene’s damping factor is increased by lowering its electron mobility to 1000cm^{2}/Vs.

It should be emphasized here that mechanisms (i) and (ii) are being used for the first time in absorber design. The use of an asymmetric pattern results in a quadrupolar SPP, which generates electromagnetically induced absorption by destructively interfering with the fundamental dipolar SPP at THz frequencies [13, 14]. Even though the resulting design is polarization dependent, it has drastically increased the bandwidth of near-unity absorption in comparison with graphene absorbers designed to operate with only dipolar mode [4]. The idea of biasing graphene layers at different voltages allows for generation of SPPs at different frequencies and increases the “chance” of destructive interference between reflected and transmitted fields at different graphene layers. Additionally, even though the mechanism (iii) is not novel, increasing graphene’s damping factor broadens the bandwidth of plasmonic resonances and hence that of the absorption as well [15, 16]. It is demonstrated numerically that the combination of these three mechanisms significantly increases the bandwidth of the proposed absorber in comparison with the designs listed in Table 1.

## 2. Design

#### 2.1. Overall design and mathematical model

The principle behind achieving near-unity absorption is the destructive interference of fields reflected from (multiple) layers of (lossy) impedance surfaces, dielectric substrates, and a fully reflective ground plane. The ground plane does not allow any fields to be transmitted and the only possible reflection path is suppressed through the destructive interference of the fields. Obviously, this leads to a highly enhanced absorption [17, 18].

Based on the above principle, the absorber shown in Fig. 1 is designed. Three impedance surfaces (at 1^{st}, 2^{nd} and 3^{rd} interface respectively) are patterned layers of graphene (Section 2.2) separated by dielectric substrates with relative permittivity *ε _{r}* = 2.5 and thickness

*d*

_{1},

*d*

_{2}, and

*d*

_{3}, respectively. The reflector at the bottom is gold. The electrostatic biasing of each graphene layer is done via applying a gate voltage by placing conductive contacts between each layer and the gold reflector (electrostatic ground). The bias allows for controlling the chemical potential

*μ*of each graphene layer independently [19]. It should be noted here that the biasing method used here eliminates the need for designing transparent electrodes [2, 20, 21] and reduces the overall complexity of the design significantly.

_{c}The absorber in Fig. 1 is assumed to be excited with an incident electromagnetic plane wave traveling in *y*-direction with the electric field vector polarized along *x*-direction. Using the transmission line model [17] to take into account the field interactions between graphene layers, the absorption associated with the multilayered design is expressed [22]

*,*

_{i}*i*= 1 : 3 represent the total reflection coefficients at the first, second, and third interfaces,

*ϕ*=

_{i}*knd*are the phase lengths across the first, second, and third dielectric substrates, $n=\sqrt{{\epsilon}_{r}}$ is their refractive index, and

_{i}*t*and

_{ij}*r*represent individual transmission and reflection coefficients of the impedance surfaces (See Fig. 2). Several comments about the mathematical model described by Eqs. (1) and (2) are in order: (i) Reflection and transmission coefficients of the interfaces with impedance surfaces,

_{ij}*t*and

_{ij}*r*are computed using a full-wave numerical method as described in Section 2.2 (Fig. 2). (ii) It is assumed here that the gold ground plane is a perfect reflector, i.e.,

_{ij}*r*

_{45}= −1 and therefore the total transmission through the structure is zero. (iii) It has been shown before in [17] that transmission line model used here is valid when the capacitive coupling between two graphene layers is small. This means that, for Eqs. (1) and (2), to produce accurate results, near field coupling between two layers should be small. Indeed, for the frequency band considered here and dimensions of the periodic pattern on the graphene layers (Section 2.2) (Fig. 3), the near fields stay tightly bounded to the graphene surface. Therefore, restricting substrate thicknesses to

*d*

_{1,2,3}≥ 0.5

*μm*reduces the capacitive coupling and makes the mathematical model described by Eqs. (1) and (2) accurate. It should also be noted here that the transmission line model is validated via comparison by a full-wave numerical method.

#### 2.2. Design and simulations of patterned graphene layer

The purpose of using patterned graphene layers in the design of the multilayered absorber, which is shown in Fig. 1 and described in Section 2.1, is to increase the bandwidth of near-unity absorption. The pattern on the layers is generated by repeating a unit cell with a rectangular void periodically in *x* and *y*-directions (Fig. 3). It should be emphasized here that void is located asymmetrically (with respect to the excitation) to generate higher order plasmon modes on the graphene surface [23, 24].

Transmission and reflection coefficients, *t _{ij}* and

*r*, which are required by Eq. (2), are evaluated using the fields scattered from the patterned graphene layer under the same excitation that is shown in Fig. 1 and described in Section 2.1. Scattered fields are computed using the commercially available finite element simulator COMSOL [25]. In these simulations, the complex (bulk) permittivity of graphene is modeled using [26].

_{ij}*σ*represents graphene’s bulk conductivity, ${\omega}_{P}^{2}=({q}^{2}{\mu}_{c})/(\mathrm{\Delta}{\epsilon}_{0}\pi {\hslash}^{2})$ is its equivalent plasma frequency, Δ is the thickness of the layer,

_{G}*q*is the electron charge,

*ħ*is the reduced Plank’s constant,

*μ*is the chemical potential, $\gamma =-(q\hslash {v}_{f}^{2})/(\mu {\mu}_{c})$ is the damping constant,

_{c}*μ*is the electron mobility,

*v*=

_{f}*c*/300m/s is the Fermi velocity, and

*c*is the speed of light in free space. First, reflectance, transmittance, and absorption spectra of the patterned graphene layer are computed for

*μ*= 1000meV and

_{c}*μ*= 10, 000cm

^{2}/Vs. Two dominant resonant modes are clearly identified in the spectra provided in Fig. 4(a). As expected, the geometrical asymmetry results in generation of a higher order mode. Surface charge distribution computed at 10THz on the graphene unit cell [Fig. 4(b)] clearly demonstrates that this mode is quadrupolar. It is originally dark, but, because of the unit cell’s geometrical asymmetry, it becomes dipole-active permitting efficient energy coupling from the incident field. The characteristic asymmetric Fano lineshape and a narrow electromagnetically induced transparency (EIT), which result from destructive interference of this mode with the dipolar mode, are clearly observed in the reflectance spectra around 9THz. It is also observed in Fig. 4(a) that the reflectance of the quadrupolar resonance is stronger compared to that of the dipolar resonance. Furthermore, it is noticed that two additional relatively weak higher order modes are observed beyond 15THz.

Equation (3) shows that graphene’s dielectric permittivity *ε _{G}* depends on its chemical potential

*μ*. As a natural consequence, graphene layer’s electromagnetic response can be controlled by varying

_{c}*μ*. It should be noted here that varying

_{c}*μ*is equivalent to varying the bias voltage of the layer [26]. Graphene layer’s reflectance and transmittance spectra are computed for

_{c}*μ*= 10, 000cm

^{2}/Vs and varying values of

*μ*between 100meV and 1000meV. Results are presented in Figs. 4(c) and 4(d), respectively. It can be deduced from Eq. (3) that higher values of

_{c}*μ*increase

_{c}*ε*and make the graphene layer a better scatterer, i.e., more reflective. This is demonstrated by Fig. 4(d), where the maximum reflectance is clearly higher for higher values of

_{G}*μ*. Also, Figs. 4(c) and 4(d) show that increasing

_{c}*μ*blue-shifts the quadrupolar and dipolar resonances and the EIT band as a consequence. Controllability of the reflectance and transmittance through voltage biasing opens up possibility of increasing the absorption bandwidth by stacking graphene layers biased at different voltages as described in Sections 2.4 and 2.5.

_{c}#### 2.3. One-layer graphene absorber

For this design, it is assumed that the graphene layer and the gold reflector are located at the first and second interfaces, respectively (Fig. 2). The absorption of the design is computed using Eq. (1), where now Γ_{1} is obtained by setting Γ_{2} = *r*_{23} = −1 in Eq. (2). The design parameters to be selected are the thickness of the dielectric substrate *d*_{1}, chemical potential and electron mobility of the graphene layer, *μ _{c}* and

*μ*, respectively. The dielectric substrate could be thought of as a Fabry-Perot interferometer terminated at its ends with a gold reflector and a partially reflective graphene layer [22]. Interferometer parameters free spectral range FSR =

*c*/2

*nd*

_{1}and finesse = FSR/FWHM represent the frequency difference between successive resonant transmission peaks and their sharpness, respectively [27]. Similar to this concept, here the FSR and finesse of the dispersive and lossy cavity, which is formed by the dielectric substrate terminated by a graphene layer on one end, can be used to characterize its absorption properties. Clearly,

*d*

_{1}should be selected carefully since it determines the resonance locations, where maximum absorption occurs. It also has an effect on the bandwidth of absorption since the patterned graphene layer supports two peaks in the reflectance spectrum with different bandwidths.

It was shown in Section 2.2 that the graphene layer becomes more reflective, i.e., dipolar and quadrapolar resonances in reflectance spectrum become stronger, when *μ _{c}* is increased. As a consequence, the dielectric substrate maintains a greater finesse; alike the finesse of a laser cavity terminated with highly reflecting mirrors. As

*μ*is lowered, finesse of the dielectric layer decreases and dipolar and quadrupolar resonances start overlapping to merge together. The above statements are clearly shown in Figs. 5(a) and 5(b), where the absorption spectra of the one-layer design is provided for various values of

_{c}*d*

_{1}and

*μ*.For

_{c}*d*

_{1}= 100

*μ*m, FSR = 0.95THz, a relatively smaller value, which makes the multiple resonances in the absorption spectrum more visible [see Fig. 5(c)]. Note that these absorption resonances are dispersive unlike the modes of a lasing cavity. Also, note that the multiple (narrowband) resonances are not desirable to achieve broadband (flat top) absorption spectrum. Smaller values of

*d*

_{1}are more suitable for this purpose.

Next, the effect of *μ* on the absorption spectrum of the one-layer design is characterized. Figure 6(a) presents absorption spectra computed for various values of *μ* and *d*_{1}. The figure clearly shows that for a given value of *d*_{1}, decreasing *μ* (i.e. high losses) significantly increases the bandwidth of absorption. But at the same time, the maximum value of absorption also decreases [Fig. 6(b)]. It should also be noted here that the presence of multiple resonances due to the decrease in FSR for large values of *d*_{1} is clearly identified in Fig. 5(c).

Based on the results presented above, we choose: (i) *d*_{1} = 5*μ*m to avoid separated multiple bands of absorption due to Fabry-Perot cavity resonances, (ii) *μ _{c}* = 1000meV to make dipolar and quadrapolar modes overlap and increase the bandwidth of absorption, and (iii)

*μ*= 1000cm

^{2}/Vs to ensure absorption is near-unity without sacrificing too much from bandwidth. The absorption spectrum of the one-layer design with these parameters is computed and shown in Fig. 6(c). The bandwidth of 90% absorption reaches approximately 3.7 THz making the absorber suitable for practical applications. Introducing additional layers as described next in Sections 2.4 and 2.5 can further increase the bandwidth.

#### 2.4. Two-layer graphene absorber

For the two-layer design, the two graphene layers and the gold reflector are located at the first, second, and third interfaces, respectively (Fig. 2). The absorption of the design is computed using Eq. (1), where now Γ_{1} and Γ_{2} are obtained by setting Γ_{3} = *r*_{34} = −1 in Eq. (2) (Fig. 2). To simplify the problem of selecting design parameters, the chemical potentials of the first and second graphene layers are set to *μ _{c}* = 1000meV and

*μ*= 700meV, respectively and electron mobility of both layers is set to

_{c}*μ*= 1000cm

^{2}/Vs. Then, the design parameters to be selected are the thicknesses of the dielectric substrates,

*d*

_{1}and

*d*

_{2}. The absorption spectrum is computed for various values of

*d*

_{1}and

*d*

_{2}; the results are presented in Figs. 7(a) and 7(b). These figures clearly show that largest bandwidth of absorption can be obtained between 4THz and 12THz and for smaller values of

*d*

_{1}and

*d*

_{2}. Keeping those in mind, to be able to maximize the bandwidth,

*d*

_{1}and

*d*

_{2}that maximize it are sought for. FOM is computed for various values of

*d*

_{1}and

*d*

_{2}and plotted in Fig. 8(a). The figure shows that FOM has a maximum for

*d*

_{1}= 0.5

*μ*m and

*d*

_{2}= 5.45

*μ*m. The absorption spectrum of the two-layer design with these values is plotted in Fig. 8(b). The bandwidth of the 90% absorption reaches 5.8 THz.

#### 2.5. Three-layer graphene absorber

For the three-layer design, three graphene layers and the gold reflector are located at the first, second, third, and fourth interfaces, respectively (Fig 2). The absorption of the design is computed using Eq. (1) (Fig. 2). The chemical potential of the graphene layers are set to *μ _{c}* = 1000meV,

*μ*= 700meV, and

_{c}*μ*= 500meV, respectively and the electron mobility of all three layers is set to

_{c}*μ*= 1000cm

^{2}/Vs. The design parameters to be selected are the thicknesses of the dielectric substrates,

*d*

_{1},

*d*

_{2}, and

*d*

_{3}. It is found that

*d*

_{1}= 0.67

*μ*m,

*d*

_{2}= 0.5

*μ*m and

*d*

_{3}= 4.78

*μ*m maximize

## 3. Conclusion

An ultra-broadband multilayered graphene absorber that operates at THz frequencies is described. The graphene layers are asymmetrically patterned to generate quadrapolar SPPs that destructively interfere with the dipolar ones. The patterned graphene layers biased at different voltages backed-up with dielectric substrates are stacked on top of each other. Additionally, graphene’s damping factor is increased by lowering its electron mobility to 1000cm^{2}/Vs. Numerical experiments demonstrate that combination of these three mechanisms has significantly increased the design’s bandwidth of operation: 6.9 THz bandwidth is obtained for 90% absorption.

Obviously, the output of the proposed absorber design is polarization dependent due to asymmetric pattern of the graphene layers. This limitation can be overcome by hybridizing symmetric graphene and metal layers as described in [20] and the absorber design could be made polarization independent.

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