1. Introduction

Graphene and other two-dimensional (2D) layered materials, such as transition metal dichalcogenides (TMD) and black phosphorus, possessing diverse and unusual electrical and optical properties, have attracted increasing attention in the field of electronics, photonics and optoelectronics over the past few years [1]. Different from the presented devices made of bulk materials, these layered materials may provide exciting opportunities for novel photonic and optoelectronic applications, such as flat lenses [2–4], ultrathin mirrors [5–7], ultrafast photodetectors [8–10], polarization devices [11–14] and nonlinear optics [15–17]. However, determined by their intrinsic properties, the light absorption efficiency of 2D materials is generally weak, which seriously limits their progress towards many practical applications such as photodetectors, modulators and thermal emitters [18–20].

To greatly enhance the light absorption of 2D materials, many methods have been proposed. One typical method is exciting plasmons of materials. For example, graphene supports plasmonic resonances in the mid-infrared to terahertz ranges, which can enormously enhance the near-field of graphene. Many perfect absorption structures based on graphene plasmons thus have been proposed, including narrowband [21], broadband [22,23] and multiband perfect absorption structures [24,25]. Besides plasmons, the excitation of other polaritons (such as phonon-polariton or exciton-polariton) also strongly enhance light-matter interactions in 2D materials [26,27], but they generally work in the mid-infrared to terahertz ranges or at low temperatures. For materials without plasmonic/polaritonic responses or materials that cannot support plasmonic/polaritonic modes in the desired ranges, the combination of 2D materials with resonant nanostructures is a common way to enhance the absorption. Specifically, the perfect absorption of these materials has been demonstrated by utilizing dielectric resonant structures [28], localized surface plasmon of noble metals [29–31], photonic crystal modes [32], Fabry-Pérot (F-P) cavity resonances [33,34] and so on. Although the existing absorption-enhancement structures are manifold enough and cover working bands from visible to terahertz, they rely too much on the excitation of plasmons or resonances of other structures. Especially for those non-plasmons structures, realization of perfect absorption in practical would be a huge challenge if the designed structure is not simple enough. Because the complicated structures will involve too many processing technologies and the fabrication tolerance will be small.

In this paper, we propose a simple absorption-enhancement resonant structure based on subwavelength gratings of monolayer molybdenum disulphide (MoS$_2$). 2D MoS$_2$ is a promising material for photonic and optoelectronic applications such as photodetectors and light-emitting diodes [35]. Here, the feasibility of achieving perfect absorption in the visible ranges is numerically demonstrated. And based on the guided mode resonance of MoS$_2$, this type of structure does not need the participation of plasmonic resonances or the combination with other nanostructured resonant structures [36–41]. Neither does it need many layers of MoS$_2$ with a thickness of at least tens of nanometers to achieve strong Mie resonances [42]. The linewidth of the absorption spectra can be narrow down to 0.1 nm. It also exhibits a linear characteristic of angle-dependence. The structure is also relatively easy for practical frabrication and can be extend to various substrates as the 2D materials can be transfered easily.

2. Results and discussion

Figure 1(a) is the schematic illustration and geometric parameters of proposed structure. An array of periodic MoS$_2$ nanostripes are integrated as a relief grating layer on top of a titanium dioxide (TiO$_2$) waveguide layer and a silica substrate. A 200 nm gold layer is attached at the bottom to reflect all the transmitted light. The period of MoS$_2$ gratings is designed as $\Lambda = 360$ nm and the duty cycle is $f = 0.5$. The depth of waveguide layer is $d_{wg} =220$ nm and the depth of silica $d_{s} = 458$ nm. The structure is illuminated with a linearly polarized wave at normal incidence from vacuum (n = 1).

The numerical simulations were conducted using a fully three-dimensional finite element technique (in COMSOL Multiphysics). The optical constants of MoS$_2$ used in simulations originate from the data measured in experiment [43] (Fig. 1(b)) and the thickness of monolayer MoS$_2$ was set as $d_g$ = 0.65 nm [44]. The refractive index of TiO$_2$ and SiO$_2$ were set as 2.36 [45] and 1.47, respectively. The absorption coefficient of TiO$_2$ and SiO$_2$ are both neglectable in the desired band. The maximum computational mesh size of MoS$_2$ was set as 2 nm for in-plane and swept in 10 slices for out-of-plane to make the simulation results accurate enough. The permittivity of gold was described by the Drude model with plasma frequency $\omega _p = 1.37 \times 10^{16}$ s$^{-1}$ and the damping constant $\omega _\tau = 1.23 \times 10^{14}$ s$^{-1}$, which was three times larger than the bulk value considering the increased scattering by surface and grain boundary effects in the thin film.

Figure 2(a) depicts the simulated results of the monolayer MoS$_2$ based perfect absorption structure. For transverse magnetic (TM) polarization, the absorption spectrum exhibits extremely narrow resonance at around 702.73 nm and reaches a nearly perfect absorption fo 99.71%. A more detailed analysis by calculating the Joule losses in the gold reflection mirror and MoS$_2$ (simply by an integration of resistive losses in these two layers in the numerical simulations) shows that only 2.95% of light is absorbed in gold layer while 96.76% of light is absorbed in MoS$_2$ (see the dashed blue line in the inset of Fig. 2(a)). Although the MoS$_2$ gratings is only monolayer thick, the guided mode resonance structure makes it possible to manipulate the incident light strongly [46], which has been proved in the previous work [47]. And with the existence of Salisbury screen structure, the optical response of the structure then relies on interplay of GMR resonance of the monolayer MoS$_2$ and the F-P effect of the cavity (the thickness of the dielectric layer is generally designed to be around $\lambda$/4). In this context, if the parameters are designed properly, the energy that coupled into the structure through the MoS$_2$ gratings could be equal to the absorption loss of the structure. In other words, the critical coupling condition of the structure is achieved. As the MoS$_2$ grating is atomically thick, the coupling rate between free space light and the guided mode in the proosed resonant structure is very small, thus a small absorption in MoS$_2$ can lead to perfect absorption with critical coupling and the linewidth of the absorption spectrum can be narrow down to 0.1 nm. Nevertheless, for transverse electric (TE) polarization, the structure could not show strong resonance but just a wide and weak peak in the range of 600-900 nm, which is caused by the F-P effect. This is because our parameters are designed for TM polarization and it causes the mismatch of critical coupling condition for the TE polarization. Figure 2(b) maps the distribution of the electric field at resonant wavelength for TM polarization, showing that most of the energy is trapped and propagated along the waveguided layer.

One of the characteristic of this GMR based perfect absorption structure is that the quality (Q) value of the structure can be tuned easily. To show this characteristic, the spectra of absorption with varied layer numbers of MoS$_2$ gratings are calculated and shown in Fig. 3. In the simulations, the grating depths of n-layer MoS$_2$ are set to be $n\times d_g$, neglecting the gaps between layers. And the other parameters keeps unchanged. When the layer number of MoS$_2$ improves from 1 to 8, the resonant wavelength redshifts from 702.73 nm to 707.71 nm and the linewidth of full width at half maximum (FWHM) broadens from 0.098 to 3.63 nm, which can be translated to the change of Q value from 7170.71 to 194.96 (Q = $\lambda _0$/FWHM, $\lambda _0$ denotes the resonant wavelength). In the GMR structure, a deeper grating depth determines the higher efficiency of the diffracted waves that coupled into the structure. Simultaneously, it is accompanied by the higher out-coupling efficiency, which relates to larger scattering loss and therefore broader FWHM.

 

Fig. 1. (a) Schematic of the MoS$_2$ based perfect absorption structure. The structure has a multilayer configuration, from top to bottom, consisting of MoS$_2$ gratings, a TiO$_2$ waveguide layer, a silica substrate layer and a Au mirror. (b) Experimental optical constants of MoS$_2$ in the visible ranges obtained from Ref. [43] .

Download Full Size | PDF

 

Fig. 2. Simulated results of the monolayer MoS$_2$ based perfect absorption resonant structure. (a) Absorption spectra for both TE (y-polarized) and TM (x-polarized) polarization. Besides total absorption, optical spectra of absorption in MoS$_2$ and Au are also plotted in the inset, respectively. (b) Distributions of electric field in the x-direction at the resonance wavelength $\sim$702.73 nm.

Download Full Size | PDF

 

Fig. 3. Simulated absorption spectra of the MoS$_2$ gratings based resonant structure with different layers for TM polarization, ranging from one layer to eight layers.

Download Full Size | PDF

The figure also depicts the decline of absorption peak, from 99.71% to 61.44% (with an absorption of 60.14% in MoS$_2$), when the layer number of MoS$_2$ improves from 1 to 8. This is the consequence of over-coupling. According to the coupled wave theory [48], FWHM has a square relation with the depth of GMR gratings, so the increment of grating layers means the multiple augment of the coupling efficiency. In the absence of material loss, the reflection peak at the resonant wavelength could actually stay at 100% with the variety of grating depths [47] because the rate of in-coupling is always equal to that of out-coupling. But with the exsitence of material loss, perfect absorption requires that in-coupling energy is equal to material loss. Since the rate of material loss could not follow up the coupling efficiency when the grating depth increases, the critical coupling conditions is broken and the absorption peak fails to keep at 100%. However, perfect absorption for multilayer MoS$_2$ grating can be realized by changing the geometric parameters. For example, if we change the duty cycle to 0.2, the depth of silica $d_{s}$ to 448 nm and keeps other parameters unchanged, perfect absorption can be realized in the 4 layer MoS$_2$ grating based resonant structure (at the wavelength of about 703.6 nm).

Figure 4 illustrates the angular dependence of the monolayer MoS$_2$ gratings based resonant absorber (the geometric parameters are the same as those in Fig. 2). Distinct from plasmonic absorbers or nanostructures based on localized resonances whose relation with the incident angle is usually weak, the GMR structures is sensitive to the angle. As shown in the figure, the oblique incidence of the structure divides the absorption spectrum into two resonances and the distance between them increases when the angle tilts more. This angle-dependence comes from the phase matching. At normal incidence, the two resonances are indistinguishable because they have the same phase while the non-zero angle makes them different so the degeneracy vanishes [49]. The wavelengths of these two resonant absorption peaks shift almost linearly with the angle as it is small. The angle-dependence property could be applied for optical filtering [49], directional thermal emitting [50] and so on.

 

Fig. 4. Angular dispersions of the monolayer MoS$_2$ gratings based resonant structure for TM polarization. Considering the angular sensitivity of GMR structures, the calculated range of angle is fixed in 0 to 1 degree.

Download Full Size | PDF

3. Conclusions

In summary, we have proposed a resonant nanostructure employing subwavelength gratings of monolayer molybdenum disulphide (MoS$_2$) to excite guided modes in the visible range. Perfect absorption is achieved at the resonant peak. The resonances exhibit very high Q-factors and the linewidth of the absorption spectrum can be narrow down to 0.1 nm. The resonant wavelength exhibits an almost linear dependence on the incidence angle. The proposed structure can be fabricated with standard nanofabrication technology. The Au, SiO₂ and TiO₂ films can be deposited by e-beam deposition. Single- and few-layer MoS₂ can be fabricated by mechanical exfoliation or Chemical Vapor Deposition (CVD) and transferred to the top of the structure. The gratings can be fabricated by e-beam lithography or focused ion beam (FIB) milling. The proposed structure provides an alternative way of achieving perfect absorption in 2D materials without the plasmonic resonances or the participation of metamaterials in the visible ranges. It also provides a method to design ultra-narrow band absorbers and similar designs can be applied to other 2D materials. It may find applications for optical filters, directional thermal emitters, 2D materials based lasers and others.