## Abstract

In this paper, a broadband absorber utilizing monolayer molybdenum disulfide (MoS_{2}) is proposed, and a generalized interference theory (GIT) is derived to investigate this absorber. Using the hybrid Lorentz–Drude and Gaussian model of monolayer MoS_{2} and the dyadic Green’s functions, the propagation properties of monolayer MoS_{2} are first investigated. Then, a sandwich-like MoS_{2}-based absorber design is proposed in the visible regime. The sandwich-like structure is mounted on a fully reflective gold mirror, which forms a Fabry-Perot resonator to strengthen light–matter interactions and enhance the absorption. To numerically calculate the absorption performance of this absorber, the GIT is next derived from interference theory. The numerical results indicate that an absorption ≥ 90% is obtained for a range of wavelengths (λ) from 389 to 517 nm, and this absorber can operate well, even with an angle of incidence up to 60°, which also verifies the prediction of the MoS_{2}-based absorber mainly operating at λ < 700 nm. Afterward, the operating mechanism of the proposed design is determined using the theory of destructive interference. Finally, the proposed design and derived GIT are validated by a simulation using commercial electromagnetic software. The derived GIT drives the numerical investigation of the multilayer structure with various polarization types and angles of incidence of the waves, and the MoS_{2}-based absorber can be used in several applications such as photoelectric storage and photoelectric detection.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Absorbers have attracted significant research interest as they have several applications in sensing [1], imaging [2], and energy storage [3] from the microwave to UV regime [4–6]. In recent years, engineers and scientists have developed more efficient absorbers using two-dimensional (2D) materials, e.g., graphene [7–9], black phosphorus (BP) [10–13], and molybdenum disulfide (MoS_{2}) [14,15].

Graphene has been frequently investigated because of its exotic thermal, mechanically flexible, optical, and electrical properties [16–22]. However, the direct application of graphene in some devices is very difficult, because it is characterized by a zero or near-zero band gap and low absorption coefficient, which limits its applications under conditions where high on–off ratios and strong light–matter interactions are required [23]. BP, conversely, possesses a direct band gap; however, its environmentally induced degradation still blocks the development of BP-based devices [24,25]. MoS_{2} also has a direct band gap [26], high current cut-off ratios [27], and tunable optical and electronic properties. Because of these attractive features, MoS_{2} can function as or play a significant role in field-effect tunneling transistors [28], photovoltaic cells [29–31], photodetectors [32,33], and thermal management [34,35]. However, the monolayer MoS_{2} is approximately 0.65-nm thick and can only absorb about 25% of the incident energy in the visible regime [36]. Many researchers have made significant efforts to enhance the absorption of MoS_{2}. Long et al. theoretically increased the optical absorption in monolayer MoS_{2} using multi-order magnetic polaritons [37]. Huo et al. demonstrated that monolayer MoS_{2} and other nano-metal couplings cannot only enhance the absorption but also expand the bandwidth [38]. However, the absorptions are primarily caused by the patterned nano-metal rather than MoS_{2}. Mukherjee et al. numerically revealed that monolayer MoS_{2} and silica in the form of Bragg stacks can obtain the highest average optical absorption of about 94.7% in the visible regime [14], which would, nevertheless, lead to a bulky configuration. Kang et al. showed that it is possible to fabricate multi-stacked monolayer MoS_{2} films, and the optical absorption of single, double, and triple stacks increases almost linearly with the stack number [39]. This work provides tremendous promise of monolayer MoS_{2} in many applications such as photonics and optoelectronics.

In this study, we investigate a broadband MoS_{2}-based absorber by deriving a generalized interference theory (GIT). We first use the dyadic Green’s function to theoretically calculate the reflection and transmission coefficients of the monolayer MoS_{2} between different dielectrics and angles of incidence. Next, we propose a broadband sandwich-like absorber utilizing the intact monolayer MoS_{2}, and derive the GIT from interference theory to numerically investigate the proposed absorber. Comparing with the available interference theory [24,40,41], this derived GIT can accurately calculate the absorption of the sandwich-like absorber illuminated by waves with various polarization types and angles of incidence. Then, the dielectric spacer thickness and number of layers of the sandwich-like structure (*NLSS*) are optimized by the derived GIT, and the absorption mechanism is revealed by the theory of destructive interference. Finally, a simulation is obtained using commercial electromagnetic software. The simulated results are consistent with the numerical results and then, verify the derived GIT.

## 2. Permittivity model and transmission properties of monolayer MoS_{2}

#### 2.1 Permittivity model of monolayer MoS_{2}

Several models have been proposed to describe the complex permittivity of 2D materials such as MoS_{2} [36,42–44]. In this work, the hybrid Lorentz–Drude and Gaussian (HLDG) model [36] is used to characterize monolayer MoS_{2}. The HLDG model can be expressed as

*ω*= 28.3 meV is the plasma frequency and

_{p}*ε*

_{∞}= 4.44 is the electrostatic permittivity.

*ω*,

_{j}*a*, and

_{j}*b*are the resonant frequency, oscillator strength, and damping coefficients for the

_{j}*j*th oscillator, respectively; moreover, the specific values are listed in [36]. In contrast, the imaginary component of the Gaussian term can be expressed by a typical Gaussian distribution function as

*ζ*= 2.7723 is the mean,

*δ*= 0.3089 the standard deviation, and

*α*= 23.224 the maximum value [29]. Furthermore, the real component of the Gaussian term can be calculated by the Kramers–Kronog relation as [45]

_{${\epsilon}_{c,i}^{G}$}is the imaginary component of the Gaussian term and can be expressed as Eq. (3). Given the fixed temperature of 300 K and Fermi energy of 0 eV, the real and imaginary components of the complex permittivity of monolayer MoS

_{2}are plotted in Fig. 1. It can be seen that their values are comparable when the wavelength (

*λ*) is smaller than 700 nm; however, the imaginary component tends to 0 when λ > 700 nm. Theoretically, the imaginary component contributes to the electromagnetic dissipation and we can, therefore, predict that the high absorption of the MoS

_{2}-based absorber mainly occurs at λ less than 700 nm.

#### 2.2 Propagation properties of monolayer MoS_{2}

Figure 2(a) depicts a laterally infinite monolayer MoS_{2} embedded in the *x*–*y* plane at the interface between two different media characterized by *μ _{l}*,

*ɛ*for

_{l}*z*>

*d*

_{0}and

*μ*

_{l}_{+1},

*ɛ*

_{l}_{+1}for

*z*< 0, i.e.,

*d*

_{0}is the thickness of the monolayer MoS

_{2}. In this case, we set

*d*

_{0}to be 0.65 nm [36]. Its reflection coefficient can be calculated by the following formulas from the dyadic Green’s functions [46]:

*R*

_{TE}and

*R*

_{TM}are the reflection coefficients of the transverse electric (TE) and transverse magnetic (TM) waves, respectively. In Eqs. (5) and (6),

*n*= $\sqrt{{\epsilon}_{l+1}/{\epsilon}_{l}}$ and

*m*= $\sqrt{{\mu}_{l+1}/{\mu}_{l}}$, and

*σ*is the surface conductivity of the monolayer MoS

_{s}_{2}and can be obtained by the following formula:Moreover,

*k*is the radial wavenumber,

_{ρ}*k*the wavenumber in free space,

*θ*the angle of incidence, and

*k*the wavenumber in the region

_{l}= ω $\sqrt{{\mu}_{l}{\epsilon}_{l}}$*l*. When a plane wave is incident on the interface, the far scattered field in the dielectric

*l*, which is the reflected field, can be represented as

*E*=

^{r}*âR*exp(–

*jk*) with the reflection coefficient of

_{l}z*R = R*

_{TE}or

*R*

_{TM}. The scattered field in the dielectric

*l*+ 1, which is the transmitted field, can be represented as

*E*exp(

^{t}= âT*jk*with the transmission coefficient of [46]orGiven the free space of the dielectric

_{l + 1}z)*l*and

*l*+ 1 (i.e.,

*ε*=

_{l}*ε*

_{l}_{+1}=

*ε*

_{0}and

*μ*=

_{l}*μ*

_{l}_{+1}=

*μ*

_{0}), the reflection coefficients for the TE and TM waves with different angles of incidence are presented in Figs. 2(b) and 2(c), respectively; Figs. 2(d) and 2(e) present those for the TE and TM waves if we further modify

*ε*

_{l}_{+1}to 3

*ε*

_{0}, respectively. Clearly, |

*R*

_{TE}| steadily increases with

*θ*; however, |

*R*

_{TM}| is different. In Fig. 2(c), |

*R*

_{TM}| monotonically decreases with

*θ*because of the free space of the dielectric

*l*and

*l*+ 1. Because of the difference in the dielectric

*l*and

*l*+ 1 in Fig. 2(e), |

*R*

_{TM}| first decreases and then increases with

*θ*, which is derived from the existence of the Brewster angle for the TM wave. It should be noted that the propagation coefficients for the TE and TM waves are identical when

*θ*= 0°.

## 3. Results and discussion

#### 3.1 Absorber model

Here, we propose a broadband absorber that utilizes a multilayered MoS_{2} material/dielectric sandwich-like structure on a gold mirror. The gold mirror blocks all transmission and forms a Fabry–Perot resonator to enhance the light–matter interaction. A schematic of the design with *NLSS* = 5 is illustrated in Fig. 3(a). Five layers of the monolayer MoS_{2} (located along the interfaces labeled from 1st to 5th) and the gold mirror (located on the right of the 6th interface) are separated by the dielectric substrates with a relative permittivity *ε*_{r} = 3 and thickness *d _{l}* (

*l*= 1–5). Moreover, the total thickness of MoS

_{2}and the dielectric substrate is

*d*, and each layer of MoS

_{2}is an intact 2D structure without any patterns.

We can theoretically calculate the absorption of the sandwich-like absorber using a GIT derived from interference theory [24,40,41] (see the next subsection). The destructive interference between the directly reflected wave and the following multiple emergent waves effectively traps the wave in the sandwiched absorber, resulting in a high absorption.

#### 3.2 GIT and performance of the proposed absorber

Now, the GIT will be presented. The absorption associated with the multilayered design can be expressed as [24]

where*ϕ*(

_{l}*l*= 1–5) represents the

*l*th delayed phase through the dielectric substrates and the monolayer MoS

_{2}. Furthermore,

*ϕ*can be calculated bywherein which Γ

_{l}*(*

_{l}*l*= 1–6) represents the total reflection coefficients at the

*l*th interface.

*t*

_{l,l+}_{1},

*t*

_{l+}_{1}

_{,l}, r_{l,l+}_{1}, and

*r*

_{l+}_{1}

*respectively denote the transmission and reflection coefficients across the*

_{,l}*l*th interface, where

*r*

_{l},_{l}_{+1}and

*r*

_{l}_{+1}

*can be calculated using Eqs. (5) and (6), respectively. Then, we obtain*

_{,l}*t*

_{l,l+}_{1}and

*t*

_{l+}_{1}

*according to Eqs. (9) and (10), respectively. The gold mirror acting as a ground does not allow the transmission of waves, such that*

_{,l}*t*

_{6,7}= 0 and

*r*

_{6,7}= ± 1 (positive sign and negative sign respectively for TM and TE wave); therefore, Γ

_{6}=

*r*

_{6,7}= ± 1 can be derived from Eq. (12). By reverse iteration, Γ

_{1}can be calculated, and the absorption can then be obtained by Eq. (11). This is the GIT derived in this paper.

Using the above GIT, we can calculate the absorption spectra of the sandwich-like absorber with different *NLSS*s and dielectric thicknesses. Assuming *NLSS* = 1 and a normally incident wave, we first change the dielectric spacer thickness *d* from 10 to 100 nm and calculate the absorption at the wavelength range of 300–900 nm, as shown in Fig. 3(b). The result indicates that the highest absorption is about 72.4% and the resonant wavelength is approximately 457 nm when *d* is 58.65 nm. Keeping *d* = 58.65 nm and the normally incident wave, we increase the *NLSS* from 2 to 10 systematically, and the absorption is presented in Fig. 3(c). It can be seen that the absorption bandwidth broadens with the increase in the *NLSS*; however, the absorption peak clearly drops, and dual-band absorption appears when *NLSS* is greater than 5. In order to show more accurate results, we present the wavelength-dependent absorption for *NLSS* = 1, 3, 5 and 6 in Fig. 3(d). We see that *NLSS* = 5 can be selected to optimize the absorption, where the bandwidth of the absorption greater than 90% is located between 389 and 517 nm, which suggests that the proposed absorber achieves broadband performance. Illuminated by the normally incident wave, it should be noted that the result for the TE wave is coincident with that for the TM wave.

In addition, the angle of incidence *θ* has a great influence on the absorption performance. The absorptions for the TE and TM waves are presented in Figs. 3(e) and 3(f), respectively. For the TE wave, the proposed sandwich-like absorber realizes wideband performance while *θ* < 30°, and dual-band absorption can be realized at θ between 30° and 60°. Conversely, there is no dual-band absorption for the TM wave. Moreover, the absorption performance for the TM wave is very close to that for the TE wave while *θ* < 30°.

From Figs. 3(b)–3(f), we notice that the absorption mainly occurs at λ < 700 nm, which verifies the prediction in subsection 2.1.

#### 3.3 Absorption mechanism

At the 1st interface (i.e., *l* = 1), *r*_{12} in Eq. (12) represents the direct reflection from this interface, and *Λ*_{21} represents the multiple emergent waves resulting from the superposition of the multiple reflections among the sandwich-like structure and gold mirror. Under the ideal conditions of the amplitude difference (i.e., Δ*A* = abs(|*r*_{12}| − |*Λ*_{21}|) = 0) and phase difference (i.e., Δ*φ* = arg(*r*_{12}) − arg(*Λ*_{21}) = ± π), an ideal absorption would be obtained, which satisfies the strongest destructive interference of the electromagnetic field [24,40]. Illuminated by a normally incident wave, Δ*A* and Δ*φ* for the proposed absorber structure with different *NLSS*s are presented in Fig. 3(g). For λ between 389 and 517 nm, the Δ*A* is far from 0; when *NLSS* = 3, Δ*φ* is closest to π; however, Δ*A* is approximately zero at only one point. Despite Δ*φ* for *NLSS* = 5 being slightly farther from π than that for *NLSS* = 3, Δ*A* is closer to zero. With increasing *NLSS*, Δ*φ* deviates from π significantly. This indicates that the absorption is determined by both Δ*φ* and Δ*A*. Figures 3(h) and 3(i) present the Δ*A* and Δ*φ* of the direct reflection and multiple reflections for the TE and TM polarizations with different angles of incidence when *NLSS* = 5, respectively. While *θ* ≥ 60°, Δ*A* or Δ*φ* deviate from the ideal value, which weakens the destructive interference. An abnormally large change emerges in Fig. 3(i), i.e., when *θ* = 80°, Δ*A* is closest to 0; however, Δ*φ* significantly deviates from the ideal condition of destructive interference. From Fig. 2 (d), we know that the trend of |*R*_{TM}| is broken when *θ* is greater than the Brewster angle, i.e., arg(*r*_{12}) will change from 0 to π when *θ* crosses the Brewster angle, which causes the large change in Δ*φ*.

#### 3.4 Simulation verification

The absorber was further simulated using CST Microwave Studio to verify the feasibility of the derived GIT [25]. Here, the thickness of the monolayer MoS_{2} was still set as 0.65 nm and meshed using fine girds. Periodic boundary conditions were used in the *x* and *y* directions. The incident plane wave points downward normal to the top surface of the absorber. The wavelength (*λ*)-dependent absorption *A*(*λ*) is expressed as 1 – *R*(*λ*) – *T*(*λ*), where *R*(*λ*) = |*S*_{11}(*λ*)|^{2} and T(*λ*) = |S_{21}(*λ*)|^{2} are respectively the spectral reflection and transmission with the S-parameter *S*_{11} and *S*_{21}. Owing to the gold mirror used in this platform, the light transmission is completely inhibited. Therefore, the absorption can be simplified as *A*(*λ*) *=* 1 – *R*(*λ*). Figures 4(a)–4(c) respectively show the absorption for the proposed absorber model with *NLSS* = 1, 3, and 5 illuminated by the TE wave with different angles of incidence, and Figs. 4(d), 4(e)–4(f) respectively show those for the TM wave. We can see that the theoretical numerical results are completely consistent with the simulated results, which validates the GIT derived in this paper.

## 4. Conclusion

In conclusion, a broadband sandwich-like absorber utilizing the intact monolayer MoS_{2} has been proposed and the GIT that can be used to calculate the sandwich-like structure has been derived. Based on the investigation of the electrical model of monolayer MoS_{2} via the HLDG model, we can predict that the proposed absorber primarily operates at λ less than 700 nm, which is verified by the following numerical results. Through the dyadic Green’s functions, the propagation properties of monolayer MoS_{2} are then obtained. Based on interference theory, the GIT with oblique angles of incidence is derived for a TM wave and TE wave. Moreover, because any wave can be superimposed by the TE wave and TM wave, the derived GIT is validated for the excited wave with any polarization. By optimizing the dielectric spacer thickness and the *NLSS* of monolayer MoS_{2} in the proposed absorber, we obtain the optimization bandwidth range of the absorption greater than 90% from 389 to 517 nm. The absorption mechanism contributes to the strong destructive interference. Finally, the verification of the derived GIT is presented using a CST simulation. This work not only demonstrates the potential of the monolayer MoS_{2} in the application of the functional device but also provides a generalized approach to numerically calculating multilayer structures.

## Funding

National Natural Science Foundation of China (NSFC) (61661012); Natural Science Foundation of Guangxi (GXNSF) (2017GXNSFBA198121); Dean Project of Guangxi Key Laboratory of Wireless Wideband Communication and Signal Processing (GXKL06170104, GXKL06160108); Dean Laboratory of Cognitive Radio and Information Processing.

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