1. Introduction

Graphene, a planar layer of carbon atoms in a hexagonal lattice, is an attractive two-dimensional material for a wide variety of applications due to its unique electric and optical properties [1–4]. A monolayer of graphene, being only atomically thick (~0.34nm thickness), shows very weak optical response with absorptance of only ~2.3% in the visible to infrared range [5]. The weak optical absorption is beneficial for applications such as light-emitting diodes [6] and LCD screens [7, 8] where graphene is used as transparent conductor. However, it limits the applications in graphene-based active optoelectronic devices, such as photodetectors [9–12], broadband optical modulators [13], antennas [14] and solar cells [15], where a strong light-matter interaction and enhanced absorption are highly desired.

Different approaches have been proposed to enhance graphene absorption in wavelength ranging from visible/near-infrared to mid- and far-infrared. In the mid- to far-infrared range(5~100um in free space), graphene behaves like Drude-type materials and exhibits strong plasmonic response [3, 16]. Therefore, great absorption enhancement can be obtained by shaping the graphene into ribbons [17] or disks [18] to excite plasmonic resonances of graphene. In the visible and near-infrared, however, the plasmonic response of unpatterned graphene is no longer available. To enhance the absorption in graphene, one can combine graphene with plasmonic nanostructures that generate strong near-field localization of surface plasmons [9, 19, 20]. Another method is to integrate graphene with resonant microcavity, in which the increase of both interaction length of light with graphene and the resonant optical field inside the cavity lead to efficient absorption enhancement [21–24]. Recently, increased absorption in graphene has been obtained by employing deep metallic nano-gratings that enable a strong localized electric field at the magnetic resonances or magnetic polaritons (MPs) [25–27]. The nano-grating supports a closed current loop induced by the magnetic field of the incident wave inside the deep trench, the monolayer of graphene covered on top of the trench serves as a pure resistor, therefore the absorption is greatly enhanced. Although the presence of the nano-grating can improve the optical response of graphene with enhanced absorption, fabrication of deep metallic nano-grating with high aspect ratio is still challenging, which is certainly detrimental to practical applications of graphene-based optoelectronic devices.

In this paper, we present an alternate approach to enhancing absorption in graphene in near-infrared region by combining graphene with a multilayer Metal-Dielectric-Metal (MDM) nanostructure. The structure consists of a shallow metal grating with a dielectric spacer deposited on a metallic film. The following three points are the key to obtaining enhanced absorption in graphene. First, in contrast to the above-mentioned deep grating in which the enhancement of absorption in graphene relies on MP excitations inside the deep grating grooves, this structure allows to excite strong MP resonances between the grating strips and the metallic film. The graphene monolayer covered on top of the MDM nanostructure, forming a Graphene-MDM (GMDM) structure, again acts as a resistor, resulting in an enhanced energy absorption at the slit opening. Second, the slits of the shallow gating provides a channel for the incoming electromagnetic field to pass through and store energy inside the spacer. Importantly, the depth of the grating does not matter so much for the absorption performance. Third, since the MP resonances are excited within the dielectric spacer, the resulting absorption enhancement can be easily tuned by simply adjusting the structural parameters such as spacer thickness and the grating period. The simple structural configuration and the flexible tunability in absorption enhancement make this structure beneficial for practical fabrication and future applications in graphene-based active optoelectronic devices.

2. Simulation methods

To demonstrate the validity of this approach and explore the mechanism for the absorption enhancement, we numerically simulated the absorption property of the proposed nanostructure using the commercial finite element (FEM) solver COMSOL Multiphysics. The GMDM structure under study is schematically shown in Fig. 1.

 

Fig. 1 Schematic of the proposed GMDM nanostrcture. A monolayer of graphene is covered on top of a multilayer MDM structure. Here, $p$is the period of the grating, $b$is the slit width and $d$ is slit depth. The thickness of the dielectric spacer is $t$.

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In our simulations, we choose silver as the material for the shallow grating and the film due to its low loss compared to gold in infrared region. Silicon dioxide (SiO₂) is selected as the dielectric spacer. The silver film below the SiO₂ layer is assumed to be sufficiently thick to be opaque. For the modeling of graphene, we consider it as a monolayer with a sheet conductance containing the contribution from intraband and interband transitions, i.e., $\sigma ={\sigma }_{\begin{array}{l}\text{inter}\\ \end{array}}+{\sigma }_{\text{intra}}$, determined by the Kubo formalisms [28]:

3. Results and discussions

We first calculated the absorption of the MDM structure with and without graphene for TM waves at normal incidence. Figure 2 shows the results for the structure with grating period $p=200$nm, slit width $b=60$nm, slit depth $d=30$nm and spacer thickness $t=20$nm.

 

Fig. 2 Calculated absorption of the MDM structure with (black) and without (red) graphene. The structure parameters used here is $p=200$nm, $b=60$nm, $d=30$nm and $t=20$nm.

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We can clearly see a peak with absorption about 14% at 1060nm, which is increased up to around 50% in the presence of graphene overlay without changing the peak wavelength, as denoted by the black dashed line. The absorption peak is associated with the excitation of the fundamental MP mode. In contrast to the MP modes excited inside the trenches of deep gratings in Ref. 26, the main feature of the MDM structure is that the dielectric spacer is introduced and it plays an important role in supporting MP modes between the grating strips and the silver film. The underlying mechanism can be clearly seen from the distribution of magnetic field and current density at MP resonance, as shown in Fig. 3(a). The presence of dielectric spacer enables the incident TM wave with magnetic field oscillating along z direction strongly localize underneath the grating strip. The oscillating magnetic field induces one pair of anti-parallel currents (red arrows around the spacer): one current in the grating strip along the x direction and the other near the sliver film surface in the opposite direction. These two currents cause a diamagnetic response [29], which is then coupled to the metallic film to generate MP modes with a fundamental resonance at 1060 nm.

 

Fig. 3 Distribution of magnetic field and current density in the GMDM structure at MP resonance 1060nm (a) and the schematic of equivalent LC circuit (b). The color contour shows the magnitude of magnetic field along z direction. The red arrows denote the direction and magnitude of the current density.

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To explain the absorption enhancement in the GMDM structure at the MP resonance, an equivalent inductor-capacitor (LC) circuit model is introduced, in which the resonance condition for the MP mode of the magnetic response can be described as follows. First, consider the MDM structure in the absence of graphene overlay, the metal strip and the bottom silver film can be approximated as metal strip pair. This provides three main parts that contribute to the LC circuit: the capacitance $C$, the mutual inductance $L_m$ introduced by the strip pair separated by the spacer thickness $d$, and the corresponding self-inductance $L_e$considering the contribution of the drifting electrons inside the metal strips [30]. Besides, the grating slit (the air gap between silver strips) also introduces another capacitance $C_{air}$ [31], which is neglected in our case since we use shallow grating with the slit depth of several tens nanometers. Therefore, the total impedance can be written as

The wavelength of the fundamental MP resonance ${\lambda }_{MP}=2\pi c_0\sqrt{(L_m+L_e)C}$can be obtained from Eq. (3) by taking$Z_{\text{tot}}=0$. Here, $c_0$ is the speed of light in free space. Using the same structural parameters used for calculation of Fig. 1 with grating slit depth$d=30$nm, we successfully reproduced the fundamental MP resonance for this MDM structure by following the calculation approach used in Ref. 31.

In the GMDM structure, the presence of graphene across the slit opening gives rise to an additional impendence $Z_G$ to the LC circuit [26], which is related to the conductance of graphene: $Z_G=R_G+i\omega L_G=b/\sigma$. Figure 3(b) shows the equivalent circuit for the GMDM structure. At the fundamental MP resonance (1060 nm), our calculation shows that $Z_G$ is dominated by a pure resistance$R_G=9.9\times {10}^{-4}\Omega$. This strongly suggests that the graphene located at the slit opening, acting as a pure resistor, plays an important role in absorbing the incoming energy and thus causing the enhanced absorption at the MP resonance wavelength.

The above statement can be further verified by calculating the power dissipation density at resonance wavelength (1060 nm) for the MDM and GMDM structures with the same geometries. Here, the magnitude of the incident power density is set to unity.

The calculated results are shown in Fig. 4. Two main features can be clearly seen: first, at MP resonance in MDM structure, most incident electromagnetic power flows through the silt opening and is dissipated inside the grating strips and the silver substrate. The amount of power dissipation from this part is not increased even in presence of graphene in GMDM structure, as shown by the red-color region in Figs. 4(a) and 4(b). Second, in the GMDM structure in Fig. 4(b), the local power density in the graphene layer at the silt opening is extremely strong with maximum power dissipation on the order of 1016 W/m³, which is two orders of magnitude higher than the highest power dissipation in the silver region (note that the values of the white region representing the power dissipation on graphene exceeds the scale of the color bar to obtain better contrast in Fig. 4). This reveals the fact that most of the power is absorbed by the graphene, which accounts for the absorption enhancement demonstrated in Fig. 2.

 

Fig. 4 Comparison of the power dissipation density (W/m³) of the MDM with (a) and without (b) graphene at 1060nm for the geometry of $p=200$nm, $b=60$nm and $t=20$nm. The red arrows represent the vector of the power flow that is absorbed by the MDM/GMDM structure.

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To quantitively evaluate the amount of energy that is absorbed by the graphene, we performed integration for the power density along the line where graphene is located in xy-plane. Then the absorption in graphene can be calculated by further dividing it by the total incoming power received by the same area. For the geometry used for Fig. 2, absorption in graphene is enhanced up to 37% at MP resonance. Compared to the general absorption of about 2.3% for the graphene monolayer, enhancement of over 16 times was achieved. It is noted in Fig. 2 that the total absorption in the GMDM structure is around 50% (black curve). This indicates that the MDM structure underneath the graphene layer absorbs the rest about 13% of the incoming energy, which is consistent with the simulated result (red curve).

Now we discuss the effects of geometry on absorption in graphene. It is clear that the absorption enhancement in graphene is directly related to the excitation of MP modes, and the graphene overlay has no influence on the MP resonance wavelength. Therefore, one can readily obtain the enhanced absorption at desired wavelength by tuning MP resonances which is dependent on geometry of the proposed structure.

We first discuss the effect of the grating period on the absorption enhancement of graphene. We consider the GMDM structures with grating period ranging from 200 nm to 400 nm by increasing the width of the metal strips while keeping a constant slit width $d=30$ nm.

The calculated results are shown in Fig. 5(a), where the absorption in graphene increases from 37% to 51% with a redshift of MP resonances from 1060 nm to 2180 nm. It is noted that over half of the incoming energy (over 22 times higher than the absorption of free-standing monolayer graphene) is absorbed by the graphene layer at 2180 nm with the grating period of 400 nm. Apparently, graphene exhibits higher absorption efficiency with larger grating period, in other words, with larger width of the metal strip. This can be qualitively explained by considering the current loop generated in the LC circuit. The increasing grating period leads to larger effective width of the metal strip pair, which causes stronger magnetic field localization within the dielectric spacer. The enhanced magnetic response then induces increased current loop. Therefore, as a pure resistor in the LC circuit, the graphene at the slit opening absorbs more incoming energy. In principle, even higher absorption in graphene at the fundamental MP wavelength can be obtained with the increase of the grating period (results are not shown here). In this case, higher order MP modes are also excited in the near-infrared region of interest, which is out of the scope of current study. In addition, the redshift of the MP resonance can be easily understood by the expression of${\lambda }_{MP}=2\pi c_0\sqrt{(L_m+L_e)C}$. The increase of effective width of the metal strip pair results in larger equivalent capacitance and inductance involved in ${\lambda }_{MP}$. This gives rise to longer MP resonance wavelength for larger grating period.

 

Fig. 5 Graphene absorption of GMDM structure for (a): spacer thickness t = 20 nm with varying period from 200 nm-400 nm, and (b): grating period $p=200$nm with spacer thickness t ranging from 5 nm to 20 nm. Here slit width $b=60$nm and slit depth $d=30$nm are used for both cases.

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We then investigate the influence of the dielectric thickness on the absorption in graphene. for the grating parameter $p=200$nm, slit width $b=60$nm and slit depth $d=30$nm, we reduced the dielectric thickness $d$ from 20 nm down to 5 nm. The calculated results are given in Fig. 5(b). We can see that the MP resonances are slowly red shifted from 1060 nm to 1220 nm as $t$ reduces from 20 nm to 10 nm, but when $t$ reduces down to 5 nm, the MP resonance dramatically shifts to 1540 nm. This nonlinear shift with respect to the dielectric thickness can be explained as follows. For the dielectric thickness that is much smaller than the width of the metal strip, i.e., $t<<p-b$, the mutual inductance ($L_m$) introduced by the metal strip pair is far less than $L_e$, therefore, the expression of the MP wavelength can be written as ${\lambda }_{MP}=2\pi c_0\sqrt{L_{e}c\text{'}{\epsilon }_d{\epsilon }_0(p-b)l/t}$ [31], where${\epsilon }_d$ and ${\epsilon }_0$ are the permittivity of the dielectric layer and vacuum permittivity respectively.$c\text{'}$represents a numerical factor that accounts for the effective area of the capacitor on the metal strip [32]. $l$ denotes the metal strip length in the $z$ direction. In this regard, ${\lambda }_{MP}$ is inversely proportional to the square root of $t$(${\lambda }_{MP}\propto \sqrt{1/t}$), which accounts for the calculated nonlinear red shift of the MP resonances.

4. Summary

In summary, a MDM nanostructure consisting of a shallow metal grating with a dielectric spacer deposited on a metallic film is proposed to enhance absorption in graphene in near-infrared region. The absorption enhancement in graphene is directly related to the excitation of the fundamental MP mode between the grating strips and the silver substrate. The graphene located at the grating slit opening behaves like a pure resistor and leads to enhanced energy absorption at the MP resonance wavelength. Moreover, the strong magnetic response induced within the dielectric layer can be controlled by easily adjusting the geometry of the proposed MDM structure, such as grating period and the thickness of the spacer, thus enabling a flexible control over the desired wavelength at which absorption enhancement occurs.