1. Introduction

Two-dimension (2D) materials [1–4] have received burgeoning amount of interest in recent years, due to their remarkable electric and optical properties, such as graphene [5,6], hexagonal boron nitride (hBN) [7,8], and transition-metal dichalcogenides (TMDCs) [9,10]. Graphene with only one atom thickness has been reported to support surface plasmon polaritons in the mid-infrared regions, because of the dominated intraband transition in electrons [11–13]. Graphene plasmons, with the extreme field confinement, low damping losses, and advantageous gate-voltage tunability [14], are emerging as a possible platform for new-generation nano-integrated optoelectronic devices including optical modulators [15], metamaterial perfect absorbers [16], transform optics [17,18], optical filters [19] and polarizers [20]. Accompanied by the excited mid-infrared plasmons, the light-matter interaction [21] in monolayer graphene has been boosted dramatically in this spectral range [22–24]. For example, Jiang et al. achieved perfect terahertz absorption in graphene by the modified Otto configuration [25]. Hu et al. utilized the structure of the sub-wavelength multilayer dielectric grating to realize near-unity absorption of light in monolayer graphene at near-infrared frequencies [26]. Wang et al. proposed the Tamm surface plasmons to yield the graphene-based multichannel perfect absorbers [27]. In addition, the guided resonances in the photonic crystal slab combined with multilayer Bragg reflectors were reported to obtain the total absorption of light in monolayer graphene [28–30] and hBN [31,32], even their heterostructures [8]. Different from graphene, TMDCs (e.g. MoS₂, MoSe₂, WS₂, and WSe₂) become the direct-bandgap semiconductors [33,34], when they are transformed from the bulk to monolayer, such as the monolayer MoS₂ with a direct band gap around 1.8 eV [35] for electronic transition. Owing to special direct band gaps and internal amplification, monolayer TMDCs have been considered as the more preferable atomically thin materials for photodetection [36], photoluminescence [37], and the field-effect transistors [38] as well as photovoltaic devices [39]. Notwithstanding the impressive prospect in photonics and optoelectronics applications, the inherent atomic thickness of monolayer TMDCs enables a significant challenge for the light-matter interaction, thereby resulting in weak light emission and absorption [40]. For instance, the average single-pass absorption of light in monolayer MoS₂ with the thickness of 6-7 Ǻ is about 10% in the visible spectral range [41]. The intrinsic drawback of poor absorption of light in monolayer TMDCs hinders assuredly the effective performance of TMDCs-based devices. Therefore, boosting absorption of light in monolayer TMDCs will play a vital role in extending applications of TMDCs in future optoelectronic devices. Thus far, in order to improve absorption of light in monolayer TMDCs, especially in the MoS₂, several physical methods were proposed to enhance the interaction between the monolayer MoS₂ and the incident light. Lu et al. proposed a multilayer photonic structure to achieve nearly perfect absorption of light in monolayer MoS₂, utilizing the strong field confinement of interfacial Tamm plasmons [42]. Cao et al. designed a novel hybrid structure of combining the silver grating with a distributed Bragg reflector to realize the broad-band absorption of light in monolayer MoS₂. The average absorption as high as 59% within wavelength ranging from 420 to 700 nm was observed based on the plasmonic resonant effect of the silver grating [43]. Besides the silver nanoribbons, plasmonic silver nanodisc arrays [35,44] were experimentally used for obtaining broad-band light absorption in monolayer MoS₂. In particular, compared with the plasmonic resonant effect, the magnetic coupling effect [45] was proved to be a more efficient way to enable broad-band absorption enhancement in monolayer MoS₂. The average absorption up to 72.7% within the visible wavelength range was achieved. However, the perfect absorption peak with the dynamical wavelength-selective property is essential for the high-efficiency photocatalysis, photoluminescence and photodetection, but has been rarely reported in MoS₂-based absorption structures.

In this paper, the novel scheme of critical coupling with guided resonances is proposed to realize the tunable total absorption of light in monolayer MoS₂, based on the structure consisting of the monolayer MoS₂ supported by a photonic crystal slab with silver back reflectors. The finite-difference time-domain (FDTD) simulations demonstrate that the total absorption is realized at the optimal ratio of the hole radius to the period of the slab, where the leakage rate of the guided resonance is equal to the absorption rate of the monolayer MoS₂. Simulation results are in excellent agreement with theoretical calculations. The spectral positions of absorption peaks are tuned effectively by a small change in the period and the thickness of the slab. Intriguingly, such absorber exhibits the unusual polarization-insensitive feature and good absorption stability over a wide angle range of incidence around ± 60°. In addition to MoS₂, the total absorption in monolayer MoSe₂, WS₂, and WSe₂ is realized by the same principle. Hence, our results may open up a new avenue for improving the light-matter interaction in atomically thin 2D materials, and find important applications in TMDC-based photoluminescence and photodetection.

2. Structure and model

Figure 1(a) shows the schematic of the proposed structure consisting of a monolayer MoS₂ coated on photonic crystal slabs with a silver back reflector. The periods of the photonic crystal slab in the x and y directions are P_x, and P_y, respectively. The circular air hole is introduced into each unit cell. The x-z cross section of the unit cell is depicted in Fig. 1(b). The thickness of monolayer MoS₂ is d. The thickness of the photonic crystal slab is h, and that of the silver reflector is D. The radius of the air hole is R. In all numerical calculations, the monolayer MoS₂ is considered as a thin film with thickness of d = 0.615 nm. The wavelength-dependent complex permittivity of the monolayer MoS₂ measured experimentally by Li et al. is used for our FDTD simulations [46]. The photonic crystal slab is assumed to the silicon with the refractive index of 3.45. The refractive index of the air hole is 1. The complex relative permittivity of silver is characterized by the typical Drude model [47,48]: ${\epsilon }_m(\omega )={\epsilon }_{\infty }-{\omega }_p^2/({\omega }^2+i\omega \gamma ).$ Here, ε_∞ = 3.7 is the dielectric constant at infinite angular frequency, and ω_p = 9.1 eV is the bulk plasma frequency representing the natural frequency of the oscillations of free conduction electrons. The γ = 0.018 eV stands for the damping frequency of the oscillations, and ω is the angular frequency of the incident light. The plane wave with the electrical polarization along the x direction irradiates normally this system.

 

Fig. 1 (a) Schematic illustration of the absorption enhancing system composed of the upper monolayer MoS₂, the sandwiched photonic crystal slab with circular air holes, and the ultra-thick metal back reflector. (b) Cross-sectional view of the unit cell. Relative structural parameters are indicated.

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3. Results and analysis

It is well known that the unique guided resonances are supported by a photonic crystal slab, in addition to the usual guided modes [49]. Unlike guided modes without any coupling to external radiations, a guided resonance not only has its electro-magnetic power confined strongly within the slab but also can couple to external radiations. When the atomically thin monolayer MoS₂ with high light transmission is placed on the photonic crystal slab, the configuration of guided resonances is hardly affected. However, based on the tightly confined electro-magnetic resonances, the extensive field enhancement occurs at the monolayer MoS₂, thereby improving prominently its light-matter interaction. Moreover, if the photonic crystal slab with specific structural dimensions, such as the thickness and the period much smaller than the wavelength of interest, is designed, the higher-order guided resonances will be prohibited, thereby only the existence of the zero-order guided resonance. Considering the proposed structure shown in Fig. 1, the transmission from the system is very close to zero (T = 0), because the thickness (D = 400 nm) of the silver back reflector is much larger than the penetration depth of electromagnetic waves. Such structure hence behaves as the configuration of the single port coupled with a resonator. The versatile coupled-mode theory (CMT) [50–52] can illustrate successfully the inherent property of this construction. As shown in Fig. 1(b), the amplitudes of normalized input and output waves transmitted through the port are given by S₊, and S_-, respectively. The normalized amplitude of the guided resonance with the resonant frequency ω₀ is assumed as a. The time rate of the amplitude change in the guided resonance of the photonic crystal slab without input waves is depicted as the external leakage rate γ. When the lossy monolayer MoS₂ is placed on the photonic crystal slab, the introduced dissipative losses in the guided resonance is characterized by a small intrinsic loss rate δ. If the plane wave with the frequency ω is launched normally into this port, the entire system can be described by the following equations, according to the energy conservation and the time reversal symmetry [50].

In order to confirm the total absorption, FDTD simulations are performed by using the commercial software of the Lumerical FDTD Solutions. The special structural parameters of P_x = P_y = P = 300 nm, h = 100 nm, and R = 60 nm are chosen. 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 computing time. Simulated absorption spectrum is shown in Fig. 2(a), and indicated by the black line. Compared to the absorption curve of the monolayer MoS₂ suspended in air, we see the near-unity absorption peak at the wavelength of λ₀ = 679.2 nm. The magnetic field |H_z|² distributions at the resonant wavelength are presented in Figs. 2(b) and 2(c). We note that the guided resonance with extensive field confinement occurs at the sandwiched slab. The transmission and reflection channels are suppressed, resulting from the ultra-thick metal substrate, and the critical coupling, respectively. All incident power is confined and absorbed by the entire structure. Meanwhile, the fitted absorption curve by the CMT is consistent with that of FDTD simulations, as shown in Fig. 2(a). The fitted γ = δ = 9.02 × 10¹² Hz is obtained. The quality factors related to the external leakage and intrinsic loss of the guided resonance are$Q_{\gamma }={\omega }_0/2\gamma ,$ and $Q_{\delta }={\omega }_0/2\delta ,$ respectively. The total quality factor of the guided resonance is calculated from the equation$Q=Q_{\gamma }{Q}_{\delta }/(Q_{\gamma }+Q_{\delta }).$ The Q = 76.86 hence is obtained theoretically. The total quality factor can also be estimated from $Q={\lambda }_0/\Delta \lambda ,$ where the Δλ is the FWHM (full width at half maximums) of the simulated absorption peak. The Δλ = 8.72 nm and the Q = 77.89 are obtained based on numerical results. The ting difference of Q between theoretical calculations and numerical estimations is the direct evidence of the existence of the critical coupling, which enables the outstanding total absorption. On the other hand, from the macroscopic electromagnetic point of view, when the total absorption by critical coupling is realized, the impedance of the perfect absorber should match to that of the free space (Z₀ = 1) at the resonant wavelength. The effective impedance of the perfect absorber is calculated using the expression below [53,54].

 

Fig. 2 (a) FDTD-simulated and CMT-fitted absorption spectra with critical coupling and the absorption curve of the monolayer MoS₂ suspended in air. Inset shows the effective impedance of the perfect absorption peak. (b) Magnetic field |H_z|² distributions of the guided resonance at the resonant wavelength of λ₀ = 679.2 nm. (c) Magnetic field |H_z|² distributions of the cross section of proposed structure at the resonant wavelength of λ₀ = 679.2 nm. (d) Simulated absorption spectra for different radii of the air holes with P = 300 nm and h = 100 nm. Inset illustrates the dependence of the maximum absorption on the radii of air holes, when the system is on resonance.

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As mentioned above, to keep the external leakage rate same as the intrinsic loss rate of the guided resonance is the key to realize the critical coupling. Herein, the intrinsic loss results mainly from the lossy monolayer MoS₂. In the interesting wavelength range, the absorption in monolayer MoS₂ is considered approximately to be frequency-independent, as suggested by the blue line in Fig. 2(a). With the intrinsic loss rate almost unchanged, controlling the external leakage rate hence plays a dominant role in achieving the critical coupling. In the photonic crystal slab, the external leakage rate of the guided resonance depends mostly on the ratio of the hole radius to the period, namely, the R / P. When the period is fixed, the external leakage rate γ increases as the hole radius increases. Because the intrinsic loss rate resulting from the monolayer MoS₂ is radius-independent, the FWHM of the guided resonance should increase with the hole radius increase, as shown in Fig. 2(d). At the same time, we find that the total absorption with the critical coupling (γ = δ) occurs at the R = 60 nm, according to the inset in Fig. 2(d). With the radius ranging from 40 to 80 nm, this system evolves from undercoupling, through critical coupling, to overcoupling regime. Therefore, the critical coupling can be realized handily only by engineering suitably the ration of the radius to the period. In addition, we see that the resonant wavelength of the absorption peak tends to exhibit a blue shift, as the hole radius increases, which is because the effective refractive index of the guided resonance decreases when the area of the air hole is extended.

Once the specific ration of the hole radius to the period is chosen and the critical coupling is achieved, we could tune the spectral position of the absorption peak by changing the period and the thickness of the photonic crystal slab. FDTD simulated absorption spectra with different periods of P = 250, 275 and 300 nm are shown in Fig. 3(a), and indicated by the black, red, and blue curves, respectively. As expected, the total absorption is unchanged, but the increase of the period enables the regular red shift of the absorption wavelength. Similarly, the change in the thickness of the slab provides another kind of tunability. As shown in Fig. 3(b), the spectral position of the absorption peak tends to exhibit a red shift with the thickness increase, because the effective refractive index of the guided resonance increases with the thickness increase. Therefore, such perfect absorption system owns the advantageous tunability, thereby finding important applications in nano-scale wavelength-demultiplexing metamaterial absorbers. In particular, when the studied wavelength range is extended, ranging from 500 to 800 nm, the high-order guided resonance yields at the shorter wavelength. For example, in addition to the fundamental resonance mode with the wavelength of 649 nm, another absorption peak appears at the wavelength of 527 nm with h = 106 nm, which is the high-order guided resonance confirmed by the magnetic field distributions shown in the inset of Fig. 3(b). However, the high-order guided resonances are not the emphasis of this paper.

 

Fig. 3 (a) Absorption spectra for different periods with R / P = 0.2 and h = 100 nm. (b) Absorption spectra with different slab thicknesses, where R = 55 nm and P = 275 nm. Inset shows the magnetic field |H_z|² distributions of the high-order guided resonance at the resonant wavelength of λ₀ = 527 nm.

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We have further investigated the absorption stability of the perfect absorber under the changes in the polarization angle and the incident angle. At the normal incidence, the electrical polarization along the x direction is 0°, and that along the y direction is 90°. When it is varied smoothly from the 0 to 90°, the absorption property of the coupling system is unchanged, as shown in Fig. 4, thanks to the strict central symmetry of the photonic crystal slab with circular air holes. At the same time, absorption as a function of the wavelength and the angle of incidence is shown in the inset of Fig. 4. The incidence along the z direction is 0°. Absorption spectra with the incident angle ranging from 0 to 60° are offered. We note that the resonant wavelength is no related to the incident angle, and that the maximum absorption decreases slightly with the angle increase. With the incident angle increase, the external leakage rate of the guided resonance increases tardily, but the intrinsic loss rate arising from the lossy monolayer MoS₂ is unchanged, thereby giving rise to the breaking of the critical coupling. Thus, the FWHM of the resonance increases and the maximum absorption decreases, as the incident angle increases. However, the maximum absorption is still great than 80%, even the incident angle is up to 60°. The polarization-insensitive property, combined with the good absorption stability under oblique incidence, undoubtedly enables this absorber to be more feasible in practical applications.

 

Fig. 4 Absorption as a function of the wavelength and the polarization angle. Inset illustrates the simulated absorption spectra with different angles of incidence. The slab with P = 275 nm, R = 55 nm, and h = 106 nm is unchanged.

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Finally, it should be pointed out that the proposed scheme of critical coupling can also be applied to obtain total absorption in other monolayer TMDCs, such as MoSe₂, WS₂ and WSe₂, not just the MoS₂. FDTD simulated absorption spectra for coupling systems with respectively monolayer MoSe₂, WS₂, and WSe₂, are demonstrated in Fig. 5. Compared with the single-pass absorption spectra of monolayer TMDCs suspended in air, we witness the appearance of multiple absorption peaks within wavelengths ranging from 400 to 800 nm, by means of guided resonances in photonic crystal slabs. The total absorption by critical coupling occurs at the wavelengths of 632, 505, and 505 nm, corresponding to the monolayer MoSe₂, WS₂, and WSe₂, respectively. Certainly, we can observe the critical coupling at other guided resonances by adjusting the ration of the radius to the period, and can tune the spectral position of the resonant wavelength with critical coupling by varying the thickness and the period of the slab, as same as that used for monolayer MoS₂. Hence, the method of the critical coupling is a general way to realize the total absorption in any lossy atomically thin 2D materials.

 

Fig. 5 Absorption spectra of monolayer TMDCs (e.g. MoSe₂, WS₂, and WSe₂) suspended in air, and that coated on photonic crystal slabs with ultra-thick metal back reflectors. The thicknesses of the monolayer MoSe₂, WS₂, and WSe₂ are 0.646, 0.618, and 0.649 nm respectively. The complex permittivity of monolayer TMDCs measured experimentally by Li et al. is used for our numerical simulations. The photonic crystal slab with P = 275 nm, R = 65 nm, and h = 106 nm is chosen.

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4. Conclusions

In conclusion, we have both theoretically and numerically demonstrated that the total absorption in monolayer TMDCs is observed by critical coupling, with the help of the photonic crystal slab coated on an ultra-thick metal back reflector. The critical coupling of guided resonances depends mainly on the ratio of the hole radius to the period of the slab. Once the specific ration is given, the spectral position of the absorption peak can be tuned effectively by a small change in the period and thickness of the photonic crystal slab. In particular, such device exhibits the original polarization-insensitive property and the good absorption stability under oblique incidence. The structure is simple and the scheme is universal, thereby providing a forceful way to improve the light-matter interaction for future atomically thin 2D materials. Our results may play a significant role in the TMDC-based photoluminescence, photodetection, even the wavelength-selective metamaterial absorbers.