1. Introduction

Over the past decade, two-dimension (2D) materials [1–3], due to their significant electric and optical properties, have attracted extensive attention in nanophotonics and optoelectronics, such as graphene [4,5], transition-metal dichalcogenides (TMDCs) [6,7], hexagonal boron nitride [8,9], and GaSe [10]. Among these 2D materials, graphene has been reported to support much stronger binding of surface plasmons in the mid-infrared regions, and the optical response of graphene, originating from its unique gapless band structure, can be tuned in an ultra-wideband range through electrostatic field, magnetic field, and chemical doping [4,5]. Thus, many graphene plasmonic devices and the other graphene-based structures [11,12] have been motivated extensive studied, such as optical absorbers [13–20], filters [21,22], optical sensors [23–28], modulators [29,30], photodetectors [31,32] and antennas [33–36].

Unlike graphene, TMDCs (e.g. MoS₂, MoSe₂, WS₂, and WSe₂) possess special direct band gaps and internal amplification similar to semiconductors, when they are transformed from the multilayers to monolayer, such as the monolayer molybdenum disulfide (MoS₂) with a direct band gap around 1.8 eV [37] for electronic transition. As a special TMDC, MoS₂, due to its unique electric and optical properties, has attracted much interest for applications in the photoluminescence [38], photovoltaic devices [39], and the field-effect transistors [40] as well as optoelectronic detectors [41]. However, due to the inherent atomic thickness, monolayer MoS₂ encounters a severe challenge for the light-matter interaction, which leads to weak light absorption and emission [42,43]. More specifically, in the visible and near-infrared range, monolayer MoS₂ with the thickness of 6-7 Ǻ only absorbs about 10% of light at normal incidence due to its unique electronic band structure [44]. This property is beneficial to develop flexible transparent electrodes, but the weak light-MoS₂ interaction limits its further applications in optoelectronic devices. Thus, remarkably enhance the light-MoS₂ interaction in monolayer MoS₂ will be very useful in future optoelectronic devices, and several physical methods have been reported. Gan et al. demonstrate that by coupling monolayer MoS₂ to a planar photonic crystal nanocavity, it is possible to dramatically enhance its internal quantum efficiency for transitions on resonant with the cavity. The experimental results and theoretical calculations reveal that the maximum enhancement of the MoS₂ spontaneous emission rate by the cavity modes can be higher than 70, with a suppression factor of about 0.4 due to the planar photonic crystal lattice [45]. Lu et al. designed a novel multilayer architecture and demonstrated that the light absorption of the atomically thin MoS₂ layer can reach 96% with a narrow spectral width of 11.5 nm in the visible range due to the generation of highly-confined Tamm plasmons in the structure [46]. Janisch et al. proposed a MoS₂-based absorber coupled with a planar nanocavity, which could enhance the light absorption of monolayer MoS₂ to nearly 70% at the wavelength of 450 nm [42]. Based on the photonic crystal slab coated on an ultra-thick metal back reflector, Li et al. theoretically and numerically demonstrated that the perfect absorption in monolayer TMDCs can be observed by critical coupling [43]. On the other hand, in order to enhance the light-MoS₂ interaction, conventional plasmonic nanostructures based on noble metals can also be combined with monolayer MoS₂ for developing novel nanophotonic devices with high performance. However, related researches are rarely reported in MoS₂-based absorption devices.

In this paper, in consideration of the atomically thin physical structure of MoS₂, we take advantage of the metal-insulator-metal (MIM) plasmonic perfect absorber, which demonstrates tunable zero reflectance due to influences of a plasmon guided mode within a much small dielectric gap [47,48]. Thus, we propose and investigate a MoS₂-based optical absorber inspired by MIM metamaterial in the visible region. Moreover, our proposed MoS₂-based absorber provides higher absorption efficiency than the other plasmonic nanostructures for MoS₂ absorption enhancement [49,50].

2. Structure and model

As shown in Fig. 1, the proposed absorber consists of periodic silver nanoribbons on a silica layer coated with monolayer MoS₂ supported by a flat silver substrate, and monolayer MoS₂ is sandwiched between two lithium oxide (Li₂O, n = 1.65) layers. Compared with only using a silica spacer, the Li₂O layers are used as insulating spacers to prevent carrier transport between metal nanoribbons and MoS₂, and also retain higher carrier mobility of MoS₂ in structure. The thickness (z direction) of each Li₂O layer is 5 nm, and the thickness of silica layer is D_S = 25nm. Meanwhile, the width (x direction) and thickness of the silver nanoribbon are described by W and t, respectively. The filling factor of silver nanoribbons, which is defined as $F=W/P$, where P is the period of silver nanoribbons in the x direction. $\theta$ is the angle of incident light. In our calculation, the monolayer MoS₂ whose wavelength-dependent complex permittivity has been measured experimentally by Li et al. [51], is employed as a thin film with thickness of 0.615 nm in our design. Simultaneously, the permittivity of silver is described by the Drude formula [52]. By using the finite-difference time-domain (FDTD) simulations, dual-band perfect absorption peaks are realized in the visible light regime, and the absorptions of monolayer MoS₂ are enhanced up to 57% and 80.5% at the peak wavelengths, respectively. Moreover, the peak wavelengths of MoS₂ absorption can be separately tuned in a wide wavelength range by manipulating related structural parameters. Furthermore, the proposed absorber tolerates a relatively wide range of incident angles and demonstrates polarization-dependence. In our calculation, the perfectly matched layer absorbing boundary condition is applied along the z direction. The periodic boundary conditions are employed in the x, and y directions, respectively. The non-uniform mesh is adopted, and the minimum mesh size inside the MoS₂ monolayer equals 0.1 nm and gradually increases outside the MoS₂ sheet, for saving storage space and computational time. Although this work focuses on numerical investigation, the proposed MoS₂-based absorber can be relatively easier to realize experimentally compared with the other MoS₂-based absorbers. Subwavelength metamaterials based on silica nanostructures is easy for integration to the current CMOS technology, and chemical vapor deposition (CVD) grown MoS₂ can be transferred over the Li₂O layer using standard transfer techniques [53].

 

Fig. 1 Schematic diagram of the MoS₂-based perfect absorber with dimensions specified.

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3. Results and discussion

To clearly illustrate the physical mechanism, we firstly investigate the absorption of the proposed MoS₂-based absorber and monolayer MoS₂ in the system under illumination of s-polarized and p-polarized normal incident light ($\theta =0$), respectively, as shown in Fig. 2. Since the thickness (D = 300 nm) of the silver substrate is much larger than the penetration depth of electromagnetic waves, the transmission from the proposed structure is very close to zero (T = 0). Then, the absorption of the proposed MoS₂-based absorber can be described by A = 1-R, where R is reflection from the proposed structure. Meanwhile, in our calculation, the absorption of monolayer MoS₂ represents the power ratio of electromagnetic energy absorbed by monolayer MoS₂ in the system and incident electromagnetic energy, which is calculated by the following equation [54]:

 

Fig. 2 Absorption of proposed MoS₂-based absorber and monolayer MoS₂ in the system under illumination of s-polarized and p-polarized normal incident light, respectively, where W = 300nm, P = 800nm, t = 15nm.

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In the following, we will illustrate the physical mechanisms of the proposed absorber. As mentioned above, the suppressed reflectance around the resonant wavelength is due to the typical MIM waveguides. In consideration of the metamaterial theory in an MIM plasmonic perfect absorber [55], the absorption peak at the resonance wavelength corresponds to the reflection dip in the MIM waveguide. The reflection dip is due to the matching of the wave impedance between air and the metamaterial, which is further attributed to the superposition of inverse optical fields that are induced by electric and magnetic surface current excited by the incident wave. Thus, we can explain the absorption increment of MoS₂ in the proposed absorber by investigating electric and magnetic field distributions, as shown in Figs. 3(a)-3(f). At the absorption peak of λ₁ = 560 nm, the complex refractive index of sliver is ${\text{n}}_{Ag}=0.01722+3.9866i$, then the skin depth of sliver is $D_1\approx \lambda /2\pi \mathrm{Im}(Ag)\approx 22.37nm$. Meanwhile, at the absorption peak of λ₂ = 672 nm, the complex refractive index of sliver is ${\text{n}}_{Ag}=0.02485+4.8596i$, then the skin depth of sliver is $D_2\approx \lambda /2\pi \mathrm{Im}(Ag)\approx 22.15nm$. Thus, both $D_1$ and $D_2$ are much larger than the thickness of sliver nanoribbons, which is result in electromagnetic coupling of both optical fields at the air/nanoribbon interface and Li₂O/nanoribbon interface, respectively. Consequently, as shown in Fig. 3(a) and 3(c), the electric field around the sliver nanoribbons is gathered and enhanced. However, for resonance mode λ₁, we can find that parallel electromagnetic field bands stretching along the z-axis are formed, although they are also concentrated around the sliver nanoribbons, as shown in Fig. 3(a) and 3(b). In fact, such electromagnetic field characteristics correspond to the excitation of surface plasmon mode. Simultaneously, for resonance mode λ₂, magnetic field is guided between each sliver nanoribbon and the sliver substrate as shown in Fig. 3(d). In other words, resonance mode λ₂ enhances the near field and concentrates the light energy, which is trapped surrounding the MoS₂ layer as the localized gap-plasmon mode. Therefore, both resonance modes λ₁ and λ₂ can enhance optical fields surrounding the MoS₂ layer. As enhanced optical fields scatter and disappear in the lossy substance, the absorption inside MoS₂ layer is increased accordingly. On the contrary, at the wavelength of 500 nm, since it is far away from resonance wavelength, there is no enhanced optical field for increasing the absorption of MoS₂ layer, as illustrated in Fig. 3(e) and 3(f).

 

Fig. 3 Contour profiles of normalized field of the proposed MoS₂-based absorber for p-polarized light. (a) Electric field and (b) magnetic field at λ₁ = 560 nm. (c) Electric field and (d) magnetic field at λ₂ = 672 nm. (e) Electric field and (f) magnetic field at λ₃ = 500 nm.

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Next, we investigate the influences of changing the width of sliver nanoribbons on the optical absorption of MoS₂, as shown in Fig. 4(a). Since each sliver nanoribbon performances as a Fabry-Perot resonator for the localized gap-plasmon mode, and the resonant wavelength is remarkably sensitive to the width of nanoribbons. Thus, when W is increased, the resonance mode λ₂ will be obvious red-shifted due to the increment of effective resonance wavelength of localized gap-plasmon mode. Moreover, the filling factor F will increase with W, which further reinforces the intensity of field enhancement and concentration between neighboring nanoribbons and inside MoS₂. Thus, the absorption efficiency will firstly increase with W. However, with the continuous increment of filling factor F, too many areas of MoS₂ will be covered by silver nanoribbons. As a result, the absorption efficiency will subsequently decrease with the increment of W. On the other hand, it is worth noting that the resonance mode λ₁ almost has no shift with changing the width of sliver nanoribbons. This is because the resonant wavelength of surface plasmon mode is not sensitive to the width of nanoribbons, but mainly depends on the period of silver nanoribbons. As shown in Fig. 4(b), the resonance mode λ₁ will also be noticeably red-shifted with the increment of P, because the resonant wavelength of surface plasmon mode becomes larger. However, the resonance mode λ₂ almost has no shift with altering the period of sliver nanoribbons, since the resonant wavelength of localized gap-plasmon mode is insensitive to the period of nanoribbons. For the same reason, as mentioned before, the absorption efficiency will increase firstly and then decrease with the increment of P.

 

Fig. 4 Light absorption of monolayer MoS₂ under normal incident p-polarized light with different (a) widths W and (b) periods P of silver nanoribbons, respectively. The other parameters are the same as Fig. 2.

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Then, we consider effects of changing the thickness of sliver nanoribbons on the absorption of MoS₂, as shown in Fig. 5(a). When the thickness of sliver nanoribbons is 15 nm, which is much less than the skin depth, the optical fields at the air/nanoribbon interface and Li₂O/nanoribbon interface are effectively coupled to each other and enhance the optical absorption of MoS₂, as mentioned before. When the thickness of sliver nanoribbons increases from 15 nm to 30 nm, the absorption of MoS₂ at λ₁ is decreased from 57% to 39.2%, and the absorption of MoS₂ at λ₂ is decreased from 80.5% to 61%. Moreover, both the absorption peaks of λ₁ and λ₂ are blue-shifted slightly. This is because the coupling strength between the top and bottom surfaces of sliver nanoribbons is reduced, and the optical loss ratio inside the sliver nanoribbons becomes higher, which reduces the absorption of MoS₂. However, the thickness of sliver nanoribbons is too small to effectively enhance the absorption of MoS₂. This is because the light diffraction surrounding the sliver nanoribbons is strengthened, and the optical fields surrounding the MoS₂ layer is weakened. Thus, when the thickness of sliver nanoribbons decreases to 10 nm, the absorption of MoS₂ is drastically reduced. On the other hand, similar to the MIM nanostructure, the absorption characteristics of proposed absorber are significantly sensitive to the thickness change of the silica spacer layer. As shown in Fig. 5(b), when the thickness of silica layer increases from 15 nm to 40 nm, the absorption peak of MoS₂ is blue-shifted. Moreover, there is an optimal thickness of silica layer, which makes the absorption peak of MoS₂ up to the maximum. As mentioned above, the maximum absorption of MoS₂ is consistent with the minimal reflection of the proposed absorber, which depends on the plasmon coupling of near fields between each sliver nanoribbon and the sliver substrate. Furthermore, when sliver nanoribbons are gradually far away from the sliver substrate, the proposed structure cannot be considered as a homogeneous effective medium. Thus, as D_S increases, the maximum absorption corresponding to the strongest strength of plasmon coupling is reached at an optimum D_S and weakens subsequently.

 

Fig. 5 Light absorption of monolayer MoS₂ under normal incident p-polarized light with different thicknesses (a) t and (b) D_S, respectively. The other parameters are the same as Fig. 2.

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In addition, since all the above results are based on normal incident light, however, the proposed absorber should insure high absorption efficiency working on the relatively wide range of incident angles in the application, we illustrate optical absorption of MoS₂ as a function of incident light wavelength and angle of incidence, as shown in Fig. 6. When the incident angle increases to 30°, the absorption peaks of MoS₂ at λ₁ and λ₂ are still higher than 25% and 36%, respectively. Thus, the proposed absorber can tolerate a relatively wide range of incident angles in the application.

 

Fig. 6 Light absorption of monolayer MoS₂ as a function of the wavelength and angle of incidence under p-polarized light. The other parameters are the same as Fig. 2.

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4. Conclusion

To conclude, a dual-band MoS₂-based perfect absorber is proposed in the visible light band, and the corresponding absorption characteristics are investigated by using the FDTD method. Simulated results exhibit that the absorptions of monolayer MoS₂ are enhanced up to 57% and 80.5% at the related peak wavelengths, respectively. By changing related structural parameters, the peak wavelengths of MoS₂ absorption can be separately tuned in a wide wavelength range. Meanwhile, the proposed absorber can tolerate a relatively wide range of incident angles and demonstrates polarization-dependence. Moreover, it is worth noting that the physical mechanism of our proposed absorber could be applied to enhance absorption of the other transition-metal dichalcogenides, not just MoS₂. Thus, we believe our designed absorber can find some potential applications in the dual-band and frequency-selective photodetectors working in the visible region.