Perfect light absorption in monolayer MoS2 empowered by optical Tamm states

Yangwu Li; Hua Lu; Jiadeng Zheng; Shichang Li; Xiao Xuan; Jianlin Zhao

doi:10.3788/COL202119.103801

Abstract

We present the perfect light absorption of monolayer molybdenum disulfide (

{MoS}_{2}

) in a dielectric multilayer system with two different Bragg mirrors. The results show that the strong absorption of visible light in monolayer

{MoS}_{2}

is attributed to the formation of optical Tamm states (OTSs) between two Bragg mirrors. The

{MoS}_{2}

absorption spectrum is dependent on the layer thickness of Bragg mirrors, incident angle of light, and the period numbers of Bragg mirrors. Especially, the nearly perfect light absorption (99.4%) of monolayer

{MoS}_{2}

can be achieved by choosing proper period numbers, which is well analyzed by the temporal coupled-mode theory.

Keywords

light absorption molybdenum disulfide optical Tamm states

1. Introduction

Atomically thinned materials including graphene, transition metal dichalcogenides (TMDCs), hexagonal boron nitride (hBN), MXenes, etc., exhibit fascinating electric, optical, mechanical, and thermal properties and attract broad interests in the fields of electronics, optoelectronics, and photonics^[1–5]. In contrast to graphene, TMDCs are semiconductors with a direct bandgap, as they are exfoliated into monolayers, which advance the development of two-dimensional (2D) material photonics and optoelectronics^[6]. As a typical TMDC, molybdenum disulfide ( ${MoS}_{2}$ ) possesses a direct bandgap of 1.8 eV when reducing from bulk to monolayer^[7]. In the ${MoS}_{2}$ monolayer, a hexagonal layer of Mo atoms is sandwiched between two hexagonal layers of S atoms in a trigonal prismatic arrangement. The atomic thickness and electronic band structure make ${MoS}_{2}$ an excellent candidate for achieving field effect transistors with an ultra-high on/off ratio^[8]. Recently, ${MoS}_{2}$ has achieved numerous applications in photodetection^[9,10], photoluminescence^[11], mode-locking lasers^[12,13], solar cells^[14], and photovoltaic devices^[15]. Nevertheless, the average single-pass light absorption of monolayer ${MoS}_{2}$ is approximately 10% in the visible wavelength range^[16]. The intrinsically weak interaction between light and ${MoS}_{2}$ seriously hinders light harvesting and photoelectric conversion. Enhancing the light absorption of ${MoS}_{2}$ is of vital importance for realizing high-performance photoelectronic devices. Fortunately, some photonic structures have been reported to improve light absorption of monolayer ${MoS}_{2}$ . A photonic crystal on a metallic mirror enabled an average light absorption of about 51% in monolayer ${MoS}_{2}$ ^[16]. Wang et al. employed a Fano-resonant photonic crystal structure to increase the ${MoS}_{2}$ absorption up to 90%^[17]. Bahauddin et al. designed a plasmonic architecture with silver nanoparticles to realize broadband ${MoS}_{2}$ absorption of 37%^[18]. Luo et al. demonstrated an enhanced dual-band ${MoS}_{2}$ absorption of 57% and 87.5% by employing a metallic metamaterial with periodic nanoribbons and a flat substrate^[19]. Based on an ${Al}_{2} O_{3} / Al$ nanocavity, Janisch et al. boosted the light absorption of monolayer ${MoS}_{2}$ up to 70% utilizing the strong interference effect^[20]. Enhancing light absorption has also been investigated in other atomic-layer materials, such as graphene, ${WSe}_{2}$ , black phosphorus, etc.^[21–24]. However, the light absorption efficiencies of the atomic-layer materials are limited in these photonic structures with the excitation of plasmonic or cavity-supported resonances. Optical Tamm states (OTSs) are a kind of highly confined interface modes formed at the interface between two different dielectric Bragg mirrors. The operating wavelength of OTSs exists in the overlapped photonic band gaps of two Bragg mirrors^[25]. The lossless OTS mode can be directly excited in arbitrary polarizations and is independent of the incident angle, providing great potential for reinforcing the ${MoS}_{2}$ interaction with light and device capabilities^[25,26].

In this Letter, we firstly propose to realize perfect light absorption of ${MoS}_{2}$ in a multilayer structure, consisting of a monolayer ${MoS}_{2}$ embedded between two different Bragg mirrors. Due to the excitation of OTS, the interaction between light and monolayer ${MoS}_{2}$ can be greatly enhanced, and thus the light absorption of ${MoS}_{2}$ can approach 99.4% in the visible range. Both the theoretical and simulation calculations illustrate that the light absorption of ${MoS}_{2}$ can be tailored by altering the layer thickness of Bragg mirrors, angle of incident light, and period number of Bragg mirrors. The temporal coupled-mode theory (TCMT) is employed to effectively analyze the light absorption evolution of ${MoS}_{2}$ with the period numbers of Bragg mirrors. Our results will offer a new way to enhance the interaction between light and atomic-layer materials and realize high-performance 2D material-based optoelectrical devices.

2. Structure and Model

Figure 1 shows the proposed multilayer photonic structure with a monolayer ${MoS}_{2}$ sandwiched in two all-dielectric Bragg mirrors. The Bragg mirrors are composed of the alternatively stacked silicon nitride ( ${Si}_{3} N_{4}$ ) and silica ( ${SiO}_{2}$ ) layers, whose refractive indices are set as $n_{a} = 2.05$ and $n_{b} = 1.46$ , respectively^[27,28]. The ${Si}_{3} N_{4}$ and ${SiO}_{2}$ layers can be deposited in turn by plasma-enhanced chemical vapor deposition^[29]. The deposition rate and time can be controlled for the different layer thickness. Monolayer ${MoS}_{2}$ can be synthesized on a sapphire substrate with a large area by chemical vapor deposition and transferred by the wetting transfer method^[14]. The thicknesses of the ${Si}_{3} N_{4}$ and ${SiO}_{2}$ layers are denoted by $a_{l}$ and $b_{l}$ ( $a_{r}$ and $b_{r}$ ) for the left (right) Bragg mirror, respectively. The period numbers of Bragg mirrors are $P_{l}$ and $P_{r}$ . The relative permittivity of monolayer ${MoS}_{2}$ can be determined by the Lorentz model^[30,31]: $ε_{r} = ε_{\infty} + \sum_{k = 1}^{K} \frac{a_{k}}{ω_{k}^{2} - ω^{2} - j ω b_{k}},$ (1)where $ε_{\infty}$ , $a_{k}$ , $b_{k}$ , $ω$ , and $ω_{k}$ stand for the DC permittivity, oscillation power, damping factor of the $k$ th oscillator, angular frequency of incident light, and resonance frequency of the $k$ th oscillator, respectively. These parameters in the Lorentz model can be found in Ref. [31]. The thickness of the ${MoS}_{2}$ layer can be set as $h = 0.615 nm$ ^[30,31]. With this value, the ${MoS}_{2}$ light absorption matched well with the experimental data^[30,31].

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Figure 1.Schematic diagram of the multilayer structure consisting of two different Bragg mirrors and a monolayer MoS₂ sandwiched in the Bragg mirrors. The thicknesses of Si₃N₄ and SiO₂ layers are denoted by a_l and b_l (a_r and b_r) for the left (right) Bragg mirror, respectively. The period numbers of the left and right Bragg mirrors are P_l and P_r, respectively. The light is incident from the left side of the structure.

The spectral response of multilayer structures can be theoretically calculated by the transfer matrix method (TMM)^[32–34]. Our multilayer structure can be simplified as a combination of $2 (P_{l} + P_{r}) + 1$ dielectric layers with $2 (P_{l} + P_{r}) + 2$ interfaces. When a TM-polarized incident light impinges on the left side of the structure with an incident angle $θ$ , the reflection and transmission coefficients of the $i$ th interface are expressed as $r_{i} = (n_{i} \cos θ_{i - 1} - n_{i - 1} \cos θ_{i}) / (n_{i} \cos θ_{i - 1} + n_{i - 1} \cos θ_{i})$ and $t_{i} = 2 n_{i - 1} \cos θ_{i - 1} / (n_{i} \cos θ_{i - 1} + n_{i - 1} \cos θ_{i})$ , respectively. $n_{i}$ and $θ_{i}$ stand for the refractive index and light propagation angle in the $i$ th layer [ $i = 1, 2, \dots, 2 (P_{l} + P_{r}) + 1$ ], respectively. The relation between the propagation angles can be governed by Snell’s law: $n_{i - 1} \sin θ_{i - 1} = n_{i} \sin θ_{i}$ ( $θ_{0} = θ$ ).

Based on Maxwell’s equations and the continuity of tangential components for electric and magnetic field vectors at the boundaries, the transfer matrices $M_{i}$ and $P_{i}$ can be derived to characterize the evolution of electric field amplitudes when light passes through the $i$ th interface and layer. They satisfy the relation as follows: $[\begin{matrix} E_{i - 1}^{+} \\ E_{i - 1}^{-} \end{matrix}] = [\begin{matrix} 1 / t_{i} & r_{i} / t_{i} \\ r_{i} / t_{i} & 1 / t_{i} \end{matrix}] [\begin{matrix} e^{- j ϕ_{i}} & 0 \\ 0 & e^{j ϕ_{i}} \end{matrix}] [\begin{matrix} E_{i}^{+} \\ E_{i}^{-} \end{matrix}] = M_{i} P_{i} [\begin{matrix} E_{i}^{+} \\ E_{i}^{-} \end{matrix}],$ (2)where $E_{i - 1}^{+}$ and $E_{i - 1}^{-}$ represent the electric field amplitudes of incident and reflected light on the left side of the $i$ th interface, respectively. $ϕ_{i}$ is the phase shift for the light propagating over distance $d_{i}$ in the $i$ th layer, which can be expressed as $ϕ_{i} = 2 π d_{i} n_{i} \cos θ_{i} / λ$ . The overall transfer matrix $Q$ of the multilayer structure can be obtained by multiplying all matrices in sequence, $Q = [\prod_{i = 1}^{2 (P_{r} + P_{l}) + 1} M_{i} P_{i}] M_{2 (P_{r} + P_{l}) + 2} = [\begin{matrix} Q_{11} Q_{12} \\ Q_{21} Q_{22} \end{matrix}] .$ (3)

The reflection, transmission, and absorption spectra of the multilayer structure can be calculated by $R = {| Q_{21} / Q_{11} |}^{2}$ , $T = {| 1 / Q_{11} |}^{2}$ , and $A = 1 - R - T$ , respectively. By using the above TMM, we investigate the light propagation characteristics of the multilayer structure with only two dielectric Bragg mirrors (i.e., $h = 0 nm$ ). The dielectric layer thicknesses of Bragg mirrors are set as $a_{l} = 89 nm$ , $b_{l} = 44 nm$ , $a_{r} = 148 nm$ , and $b_{r} = 130 nm$ . Thus, the bandgaps of two Bragg mirrors can be overlapped, providing an essential condition for the excitation of OTSs^[25]. The period numbers of the left and right Bragg mirrors are chosen as $P_{l} = 8$ and $P_{r} = 20$ , respectively. As shown in Fig. 2(a), there is an obvious and narrow dip at the wavelength of 509.5 nm in the reflection spectrum. To verify the theoretical results, we use the finite-difference time-domain (FDTD) method to numerically simulate the light propagation in multilayer structures (Lumerical FDTD Solutions)^[35]. For the convergence of simulations, the custom non-uniform mesh is adopted, and the maximum mesh steps along the $x$ and $y$ directions are set as $Δ x = 1 nm$ and $Δ y = 5 nm$ . The simulation time is set as 8000 fs with an auto shutoff level of $10^{- 10}$ . The mesh size of ${MoS}_{2}$ layer is 0.1 nm. The electric field amplitude of incident light is set as 1 V/m. The simulation results agree well with the TMM results, as shown in Fig. 2(a). According to the TMM calculated reflection spectra of two separate Bragg mirrors, the dip wavelength (509.5 nm) is exactly located in the overlapped region of the two photonic bandgaps (not shown here). To prove the formation of OTS, the field intensity distribution of ${| E |}^{2}$ at the dip wavelength is simulated via the FDTD method, as depicted in the upper inset of Fig. 2(b). The white dotted boxes show the areas of two Bragg mirrors. We can see that the electric field is mostly confined around the interface, and the intensity is enhanced by over 50 folds. Both the reflection dip in the overlapped bandgap and strongly confined electric field indicate the existence of OTS at the interface between two Bragg mirrors. It can contribute to the strong boosting of light–matter interaction^[25]. To explore OTS-assisted light–matter interaction, we introduce a monolayer ${MoS}_{2}$ sandwiched between the two Bragg mirrors, as shown in Fig. 1. Figure 2(b) shows the absorption spectra of the multilayer structure with the ${MoS}_{2}$ monolayer. We can see that the absorption spectrum of ${MoS}_{2}$ exhibits a distinct peak with a height of 43.0% at the wavelength of 510.5 nm. The theoretical calculations agree well with the numerical simulations. Because of the OTS excitation, the electric field intensity in the ${MoS}_{2}$ monolayer at the absorption peak wavelength is enhanced to $5.2 V^{2} / m^{2}$ . Thus, the reinforced light absorption can be attributed to the OTS field confined at the interface of Bragg mirrors.

Figure 2.(a) Reflection spectra in the multilayer structure with P_l = 8, P_r = 20, a_l = 89 nm, b_l = 44 nm, a_r = 148 nm, b_r = 130 nm, and θ = 0°. (b) Light absorption spectra of the multilayer structure with a MoS₂ monolayer. The lines and circles represent theoretical and simulation results, respectively. The upper inset shows the field distribution of |E|² at the wavelength of 509.5 nm in the multilayer structure (h = 0 nm). The lower inset shows the spectrum of MoS₂ light absorption in the full visible range.

3. Results and Analysis

Subsequently, we investigate the relation between the Bragg layer thickness and ${MoS}_{2}$ light absorption. Figure 3(a) depicts the evolution of the absorption spectrum with the thickness of the ${Si}_{3} N_{4}$ layer $a_{l}$ . Obviously, the absorption peak appears as a red shift with increasing $a_{l}$ and reaches the maximum when $a_{l} = 82 nm$ . The simulation results coincide well with the TMM calculations. As shown in Fig. 3(b), the absorption of monolayer ${MoS}_{2}$ can reach 95.3% with a full width at half-maximum (FWHM) of $\sim 1.9 nm$ when $a_{l} = 82 nm$ . To clarify the physical mechanism of the light absorption improvement, we study the electric field intensities in ${MoS}_{2}$ between the Bragg mirrors with different $a_{l}$ , as plotted in the inset of Fig. 3(b). It is found that the electric field intensity exhibits a maximum when $a_{l} = 82 nm$ , which induces the strongest light- ${MoS}_{2}$ interaction and light absorption in the visible range. Furthermore, we investigate the dependence of ${MoS}_{2}$ light absorption on the thickness $a_{r}$ and incident angle $θ$ . Figure 3(c) depicts the evolution of the absorption spectrum with $a_{r}$ when $a_{l} = 82 nm$ . It shows that the wavelength of the absorption peak possesses a red shift with increasing $a_{r}$ , and the absorption efficiency can reach the highest value of 95.7% when $a_{r} = 149 nm$ . With increasing the incident angle, there is a blue shift for the absorption peak, as shown in Fig. 3(d). The energy of OTSs rises when increasing $θ$ ^[25]. Thus, we can observe a blue shift of the absorption peak. Meanwhile, the height of the absorption peak almost remains unchanged. The adjustment of incident angle $θ$ contributes to the flexible selection of the operating wavelength and ${MoS}_{2}$ light absorption.

Figure 3.(a) Evolution of MoS₂ light absorption spectrum with the Si₃N₄ layer thickness a_l in the structure with a_r = 148 nm and θ = 0°. (b) Corresponding absorption spectrum of MoS₂ monolayer when a_l = 82 nm and θ = 0°. The inset shows the dependence of electric field intensity in MoS₂ on a_l. (c) Evolution of MoS₂ light absorption spectrum with the thickness a_r when a_l = 82 nm and θ = 0°. (d) Evolution of MoS₂ light absorption spectrum with θ when a_l = 82 nm and a_r = 149 nm. The circles denote the simulation results. Here, P_l = 8, P_r = 20, b_l = 44 nm, and b_r = 130 nm.

Moreover, we can see that the light absorption of ${MoS}_{2}$ relies on the period number of Bragg mirror $P_{l}$ , as depicted in Fig. 4(a). There is a slight blue shift for the ${MoS}_{2}$ absorption peak with increasing $P_{l}$ . The blue shift of the absorption peak arises from the deviation of OTS wavelengths with varying $P_{l}$ . The Bloch-wave-expansion method (BWEM) is introduced to obtain the precise OTS wavelengths and its dependence on the period numbers of two Bragg mirrors^[36]. The occurrence of OTSs can be characterized by matching the surface impedances at the interface of two Bragg mirrors, namely, $ξ_{left}$ and $ξ_{right}$ . The match condition can be specified by the wavelength-dependent function, $| ξ_{left} - ξ_{right} |$ . It is found that the OTS wavelength coincides with the dip wavelength, where $| ξ_{left} - ξ_{right} |$ is minimum. The OTS wavelengths (red dashed line with dots) calculated with BWEM are plotted in Fig. 4(b). Since the period number of the left Bragg mirror $P_{l}$ is relatively small, the increase of $P_{l}$ will cause a non-negligible change for $ξ_{left}$ in the overlapped band gap of two Bragg mirrors, resulting in a blue shift. As shown in Fig. 4(b), the BWEM theoretical results agree well with the TMM calculations. The minor difference ( $\sim 1 nm$ ) between the OTS wavelength and ${MoS}_{2}$ absorption peak could be attributed to the insertion of the ${MoS}_{2}$ monolayer. The absorption peak gradually increases as $P_{l}$ changes from 7 to 9, while it decreases when $P_{l}$ further increases, as shown in Fig. 4(b). To clarify the mechanism, we analyze the ${MoS}_{2}$ light absorption using the TCMT^[37]. According to TCMT, the multilayer system can be treated as a lossy cavity with two ports, as depicted in the inset of Fig. 4(c). The cavity mode with the resonant frequency $ω_{0}$ radiates to the left and right ports with the decay rates $γ_{1}$ and $γ_{2}$ , respectively. $γ_{3}$ is the internal loss rate due to the dissipative loss of ${MoS}_{2}$ . $S_{i \pm}$ ( $i = 1$ and 2) are the amplitudes of incoming and outgoing waves coupled with the cavity mode, respectively. There is no light inputting from the right port, namely, $S_{2 +} = 0$ . The coupled equations can be described as $\frac{d a}{d t} = (- j ω_{0} - γ_{1} - γ_{2} - γ_{3}) a + \sqrt{2 γ_{1}} S_{1 +},$ (4) $S_{1 -} = - S_{1 +} + \sqrt{2 γ_{1}} a,$ (5) $S_{2 -} = \sqrt{2 γ_{2}} a .$ (6)

Figure 4.(a) Light absorption spectra of MoS₂ monolayer in the multilayer structure with different period numbers P_l when P_r = 20. (b) Absorption peak values of MoS₂ as a function of P_l, and the wavelengths of MoS₂ absorption peak and OTS as a function of P_l. (c) Absorption spectra of MoS₂ obtained by the TMM calculation (dots) and fitting (line) when P_l = 9. The inset shows the TCMT model of OTS in the structure. (d) Fitting parameters γ₁, γ₂, and γ₃ versus P_l. (e) Absorption spectra of MoS₂ in the structure with different P_r when P_l = 9. The inset shows the absorption spectra around the peaks. (f) Fitting parameters γ₁, γ₂, and γ₃ versus P_r. Here, a_l = 82 nm, b_l = 44 nm, a_r = 149 nm, b_r = 130 nm, and θ = 0°.

Here, $a$ is the amplitude of the cavity mode. The reflection and transmission spectra can be achieved by $R (ω) = {| S_{1 -} / S_{1 +} |}^{2}$ and $T (ω) = {| S_{2 -} / S_{1 +} |}^{2}$ , respectively. Then, the absorption spectrum of ${MoS}_{2}$ can be obtained as $A (ω) = \frac{4 γ_{1} γ_{3}}{{(ω - ω_{0})}^{2} + {(γ_{1} + γ_{2} + γ_{3})}^{2}} .$ (7)

The above parameters $γ_{1}$ , $γ_{2}$ , and $γ_{3}$ can be obtained by fitting the ${MoS}_{2}$ absorption spectra. Figure 4(c) shows the fitting absorption spectrum when $P_{l} = 9$ , which is in good agreement with the theoretical result. The fitting results in Fig. 4(d) illustrate that the decay rate $γ_{1}$ and loss rate $γ_{3}$ exhibit a distinct drop with increasing $P_{l}$ . $γ_{2}$ almost remains unchanged and is two orders of magnitude smaller than $γ_{1}$ and $γ_{3}$ (namely, $γ_{2} ≪ γ_{1}$ , $γ_{3}$ ). $γ_{1}$ is closest to $γ_{3}$ when $P_{l} = 9$ , thereby the ratio of $γ_{1}$ and $γ_{3}$ can approach a maximum. Therefore, the ${MoS}_{2}$ light absorption can reach the highest value of 96.2% when $P_{l} = 9$ .

Finally, we discuss the influence of period number $P_{r}$ on the ${MoS}_{2}$ light absorption. As shown in Fig. 4(e), the ${MoS}_{2}$ absorption shows an upward trend, and the increase of peak value slows down as $P_{r}$ increases. The peak value can approach 99.4% with an FWHM of 1.6 nm when $P_{r} = 26$ . The peak wavelength remains unchanged with varying $P_{r}$ from 20 to 26. Compared to $P_{l}$ , $P_{r}$ is sufficiently large. The surface impedance at the interface of the right Bragg mirror $ξ_{right}$ barely changes in the overlapped band gap of Bragg mirrors. Thus, the wavelength with a minimum $| ξ_{left} - ξ_{right} |$ (corresponding to the OTS wavelength) remains at 504.9 nm, and the wavelength of the ${MoS}_{2}$ absorption peak is unchanged (506.0 nm). According to the BEWM calculation, the offset of the $| ξ_{left} - ξ_{right} |$ dip can be neglected when $P_{r}$ increases from 18. In other words, the position of the ${MoS}_{2}$ absorption peak will remain unchanged for $P_{r} \geq 18$ . The fitting parameters in Fig. 4(f) show that $γ_{2}$ still remains two orders of magnitude smaller than $γ_{1}$ and $γ_{3}$ . Meanwhile, $γ_{1}$ and $γ_{3}$ get closer to each other, giving rise to a higher ratio of $γ_{1}$ and $γ_{3}$ . Thereby, the ${MoS}_{2}$ absorption will be infinitely close to one with gradually increasing $P_{r}$ ( $γ_{1} / γ_{3} \approx 1$ ).

4. Conclusions

We have investigated the enhanced visible light absorption of the ${MoS}_{2}$ monolayer sandwiched between two different Bragg mirrors with the excitation of the OTS mode. It is distinct from the cavity-supported light absorption enhancement in previous few-layer systems^[20,38]. The wavelength and efficiency of ${MoS}_{2}$ light absorption are dependent on the layer thickness of Bragg mirrors, light incident angle, and period numbers of Bragg mirrors. The nearly perfect absorption (99.4%) with an ultra-narrow FWHM of 1.6 nm can be obtained in the structure. The theoretical results are in excellent agreement with the numerical simulations. The BWEM has been applied to analyze the precise alternation of OTS wavelengths with periodic numbers of Bragg mirrors. The TCMT has been used to clarify the mechanism of tailoring ${MoS}_{2}$ light absorption with the period numbers. Our results will pave a new way for the ultra-strong light–matter interaction and favorable applications of atomic-layer materials in highly efficient optoelectronic functionalities, such as photodetection and photoluminescence.

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