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Abstract
Monolayer two-dimensional materials (2DMs) have excellent optical and electrical properties and show great application potential in photodetectors. However, the thickness at the atomic scale leads to weak light absorption, which greatly limits the responsivity of corresponding photodetectors. Here we propose an all-dielectric sub-wavelength zero-contrast grating structure that enables a monolayer of MoS2 with ultra-narrow bandwidth perfect light absorption. The absorption enhancement can be attributed to the critical coupling of guided mode resonances from two specific order diffractions in the structure, as confirmed by the planar waveguide theory and coupled mode theory. Such absorption enhancement can be generalized to any other absorptive atomically thin films, and the wavelength of perfect absorption can be tuned by scaling the dimension of the photonic structure. Our results offer a promising photonic approach to realize ultra-highly sensitive narrow-band photodetectors by using atomically thin materials.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Two-dimensional materials [1–4], such as graphene (Gr) [5,6], transition metal dichalcogenides (TMDCs) [7,8], black phosphorus (BP) [9], hexagonal boron nitride (h-BN) [10], have attracted extensive attention for micro-nano photonic and optoelectronic devices due to their exceptional electrical and optical properties. They have shown great application prospects in the fields of photodetectors [11], field-effect transistors [12], photovoltaic devices [13], etc. However, the light absorption of these two-dimensional materials (2DMs) with atomic scale thickness is very weak for optoelectronic devices [14], even though their absorption coefficient is not low. For example, the thickness of monolayer MoS2 is 6∼7 Å, and the average single-pass absorption is less than 10% in the visible spectral range [15]. The intrinsic drawback of poor light absorption in monolayer 2DMs hinders the performance of 2DMs-based photodetectors. Therefore, boosting the light absorption of 2DMs is essential in extending the applications of 2DMs in photodetectors.
Recently, a large number of methods have been proposed to improve the light absorption of 2DMs [16]. The typical method to enhance 2DMs absorption is to introduce a larger field localization near the absorption layer or increase the interaction distance between the absorption layer and light [17]. Therefore, various forms of artificial optical microstructures are introduced to enhance light absorption. For example, with a multilayer structure supporting highly localized Tamm plasmons, the MoS2-coupled system yields an absorption peak with a bandwidth of 11.5 nm and a peak of 96% [18]. When a planar nanocavity is coupled to a monolayer MoS2 absorber, high absorption of 70% was achieved at a wavelength of 450 nm [14]. However, even higher absorption, which is expected for highly sensitive photodetectors, was hardly accessible by using these kinds of structures. Fortunately, monolayer TMDCs shows perfect absorption with the assistance of a photonic structure that allows meeting the critical coupling condition [19]. In most structures for absorption enhancement, metals are widely used in the form of microstructures or mirrors to greatly increase the absorption of 2DMs [20,21]. However, the inherent ohmic loss of metals limits the upper limit of the monolayer 2DMs absorption and produces thermal effects, which are undesired to the devices. Recently, it has been demonstrated that a grating structure including metal back reflector and dielectric spacer can achieve near-total absorption in a monolayer of 2DMs [22]. However, the metallic Al back reflector still contributes to the total absorption slightly (∼5%) in such a structure. Meanwhile, the complicated structural design also brings about much uncertainty and difficulty in structure fabrication. Therefore, several simple all-dielectric structures were introduced [23]. For example, Qing et al. proposed a grating guided mode resonance structure to enhance the ultra-narrowband absorption of monolayer MoS2, in which both the absorption of TE (transverse electric) and TM (transverse magnetic) mode reached 65% and 53%, respectively [24]. This kind of ultra-narrowband absorption favors monochromatic photodetection with excellent sensitivity [25,26]. However, the electric field generated by the resonance of a single guided mode is mainly localized in the structure and is difficult to be absorbed efficiently, thereby making this method unable to achieve perfect light absorption.
Here, we propose an all-dielectric sub-wavelength zero-contrast grating structure to generate guided mode coupling for enhancing the absorption of monolayer MoS2. The zero-contrast grating can effectively eliminate the influence of local reflection and phase changes on enhanced absorption, because there is no difference in refractive index between the grating layer and the underlying waveguide layer [27]. The rigorous coupled wave analysis (RCWA) [28] calculation shows that the absorption of monolayer MoS2 can reach nearly 100% for both TE and TM modes. It is worth noting that both TE mode and TM mode can exhibit ultra-narrow absorption, with the full widths at half maximum (FWHM) of 0.28 nm and 0.76 nm, respectively, and the quality factor (Q-factor) of 2095 and 710. The wavelength of the perfect absorption peak can be adjusted by tuning the overall scaling factor of the structure, s, defined as the size ratio between each corresponding dimension in the structure. The mechanism of enhanced absorption is attributed to the resonant coupling of the guided modes of two specific orders of diffraction generated by the grating, resulting in a significant field enhancement near MoS2 and leakage of local energy in the waveguide layer downward. When the structural leakage rate is equal to the internal loss rate of MoS2, perfect absorption is achieved. Interestingly, the enhanced absorption effect of this structure is not limited to monolayer MoS2, and can be extended to any other absorptive atomically thin films. This simple but efficient structure for the absorption enhancement of monolayer 2DMs promises ultra-sensitive narrow-band photodetection.
2. Structure and model
As shown in Fig. 1, the proposed photonic structure for the absorption enhancement is composed of two parts, a grating layer connected with a waveguide layer, under which was put the light absorptive thin layer of interest. In a typical case, a MoS2 monolayer was introduced to form a perfect absorber. Both the grating layer and the waveguide layer are made of TiO2 material ($ {\textrm{n}_{\textrm{Ti}{\textrm{O}_2}}}\textrm{ = 2}\textrm{.4}$). In this structure, the grating height is dg, the width is w, the grating period is P, and the waveguide layer height is dw. The incident light with an incident angle θ enters the structure in the TE (the electric field is parallel to grating grooves) or TM (the magnetic field is parallel to grating grooves) polarization state. A numerical code based on the rigorous coupled-wave analysis (RCWA) [28] was used to calculate the absorption properties of monolayer MoS2 and the coupled system with photonic structure. The monolayer MoS2 was treated as an isotropic homogeneous thin film with a thickness of 0.615 nm and the wavelength-dependent complex permittivity of MoS2 was from previous experimental results [29]. Compared to the substrate supported structure, our free-standing all-dielectric grating structure is certainly challenging in fabrication [30–32]. It is still feasible to prepare such a structure experimentally through ion beam etching (IBE) and deep reactive ion etching (DRIE) [33], especially when a hole substrate is used to support the photonic structure.
Fig. 1. Schematic diagram of an all-dielectric sub-wavelength grating for absorption enhancement of monolayer MoS2. The right side is the cross-sectional view of a unit cell.
3. Results and discussion
As shown in Fig. 2, the absorption spectra of the monolayer MoS2 in this structure was calculated by using the RCWA method in TE mode. Here we firstly calculate the reflectance R and transmittance T of the structure. The absorption of monolayer MoS2 can be expressed by the formula, A=1-R-T, since the structure is an all-dielectric system, where only MoS2 causes absorption. An ultra-narrow band perfect absorption peak appeared at a wavelength of 586.6 nm. The absorption at this wavelength was significantly enhanced in comparison with the suspended monolayer MoS2. This perfect absorption effect can be attributed to the coupling of guided modes, resulting in the match of structural leakage rate γ and MoS2 internal loss rate δ, that is, to meet the critical coupling conditions. In the whole process, the incident light is diffracted by the grating. The structure supports various guided modes, whose transverse component of nonzero order diffracted wave vector is equal to the waveguide propagation constant. This external propagating wave will strongly couple with the guided mode, causing resonance phenomenon, so that at the resonance frequency, a significant field localization effect occurs in the structure. Different guided modes are coupled at a specific frequency, which can cause energy leakage in the structure, which is beneficial to enhance the absorption of 2D materials near the structure. At the resonance frequency, due to the guided mode resonance effect, the transmission port is blocked (i. e. T = 0). The structure at this frequency can be approximated as a single-port system, and its absorption can be expressed as: A=1-R. According to coupled mode theory (CMT) [34–36], when the lossy material MoS2 is introduced to the grating structure, the absorption of the system can be expressed as: [19,37,38]
It can be seen from Eq. (1) that perfect absorption is achieved when the leakage loss rate and the internal loss rate are equal at the resonance frequency ($\omega = {\omega _0}$). At this time, the transmission and reflection of the structure are blocked by guided mode resonance and critical coupling, respectively. As shown in Fig. 2, the absorption spectrum fitted by using the CMT (dashed line) and RCWA (red line) methods is consistent at the resonance position. At the non-resonance, the deviation occurs, as the CMT method assumes a lossless direct (non-resonant) pathway and gives zero loss away from the resonance [34]. The fitted $\delta = \gamma = 4.017 \times {10^{11}}$ Hz is relatively low compared with other similar structures [19]. According to CMT, the quality factors related to the external leakage of the guided mode resonance and intrinsic loss of the monolayer MoS2 are ${Q_\gamma } = {\omega _0}/2\gamma $ and ${Q_\delta } = {\omega _0}/2\delta $, respectively. The total quality factor can be obtained by ${Q_{\textrm{CMT}}} = {Q_\delta }{Q_\gamma }/({{Q_\delta } + {Q_\gamma }} )$, and the value of ${Q_{\textrm{CMT}}} $ is 2090. The total quality factor can also be estimated from $Q = {f_0}/\triangle f$, where ${f_0}$ and $\triangle f$ are center frequency and FWHM of the absorption peak, respectively. The $Q$ = 2095 is obtained based on the calculation results of RCWA. The Q and ${Q_{\textrm{CMT}}}$ are almost equal, suggesting that the perfect absorption can be attributed to critical coupling. Such a high-quality factor can be attributed to the proper structure that supports the guided mode resonance with a low leakage rate (i. e. $\gamma = 4.017 \times {10^{11}}\; \textrm{Hz}$). According to the CMT, the FWHM of the absorption peak is determined by the sum of the leakage rate and loss rate, i. e $\textrm{FWHM} = 2({\gamma + \delta } )$ [39]. Therefore, such low structural leakage rate leads to ultra-narrowband perfect absorption under the critical coupling.Fig. 2. Absorption spectra of monolayer MoS2 in the proposed structure and in the air calculated by RCWA in the TE mode (red lined, black lined). Critical coupling method (dotted lined) is used to fit the curve. The structural parameters: dg = 0.328 µm, w = 0.25 µm, dw = 0.472 µm, P = 0.5 µm, θ = 0°. The inset is a partially enlarged view around the resonance wavelength.
We begin to unveil the origin of energy leakage. It is well known that sub-wavelength grating supports guided mode resonance at a specific frequency, which yields field localization significantly in the structure. However, it is difficult to use this effect to facilitate absorption directly. Therefore, the single resonance mode is not ideal for enhancing the absorption of monolayer MoS2. We optimize the structure so that the second-order diffractive guided mode is coupled with the first-order diffractive guided mode simultaneously, resulting in an increase of the leakage rate of the structure and effectively enhancing the absorptivity of MoS2. We can find that at the coupling wavelength of the two modes, that is, at a wavelength of 586.6 nm, perfect absorption of a narrow band can be achieved.
Next, we explore the generation mechanism of the perfect absorption peak. To understand the mechanism of perfect absorption, the absorption spectra of monolayer MoS2 of various structural periods are calculated, as shown in Fig. 3(a). The absorption peak nearly linear red shifts with the increase of period, but it is worth noting that the intensity of the absorption peak rises firstly and then decreases. In the study of guided mode resonance [40], it has been proved that the reflection peak of guided mode resonance has a nearly linear red shift behavior with the increase of period, and the reflection of the peak does not decrease significantly, so the perfect absorption here is not only caused by a single resonance guided mode. Due to the high transmittance and the atomic scale thickness of monolayer MoS2, its effect on structural resonance is negligible. To further verify the coupling mode effect, we calculate the reflectivity of the structure (without the monolayer MoS2) of different periods, as shown in Fig. 3(b). After changing the period, the peak at the original mode coupling wavelength of 586.6 nm splits into two reflection peaks due to the different offset of the two reflection peaks with the period. With the increase of the period, the two reflection peaks show a nearly linear red shift, and the reflectivity remains, which is compatible with the theory of guided mode resonance.
Fig. 3. (a) Absorption spectra of monolayer MoS2 under different structures periods (other parameters remain unchanged) when the incident light is normally incident in the TE polarization state. (b) Reflectance spectra of the structure without MoS2 under different periods.
The modes of the two split reflection peaks are confirmed by analyzing the field distribution. When the grating period is 510 nm, the field distribution at the wavelength of 589.0 nm and the field distribution contributed by each diffraction order are calculated, as shown in Fig. 4(a). It can be seen from the electric field distribution of the first-order diffraction that the reflection peak is generated by the first-order diffraction of the incident light to form a fourth-order guided mode resonance in the structure. At a period of 510 nm, the field distribution at the 597.6 nm and the field distribution contributed by each diffraction order are calculated, as shown in Fig. 4(b). It can be seen from the electric field distribution of the second-order diffraction that the reflection peak is generated by the second-order diffraction of the incident light to form a zero-order guided mode resonance in the structure. It can also be understood from the field distribution of the above two reflection peaks when a single guided mode resonance is generated, the electric field is strongly localized in the structure. When lossy material MoS2 is introduced, the enhancement factor will decrease.
Fig. 4. Electric field distribution at two reflection peaks. (a) When the monolayer MoS2 was removed and the structure period was 510nm, the field distribution diagram at a wavelength of 589.0nm was obtained. (b) The field distribution diagram at the wavelength of 597.6nm. From left to right are the total field distribution, 0th order field distribution, 1st order field distribution, 2nd order field distribution.
Starting from the planar waveguide theory [41,42], the origin of the two peaks is further determined. In the TE mode, the transverse component of the wave vector diffracted by the grating, that is x component, can be determined by the Floquet condition [40].
where k0 is the wave vector of the incident light, n1 is the refractive index of the incident medium (here is air), θ is the incident angle, λ0 is the incident wavelength, and P is the structural period, i is the diffraction order. According to effective media theory (EMT), the grating layer can be equivalent to an anisotropic homogeneous medium, and the equivalent refractive index of the grating layer in the TE mode is [43] where nm and nd are the refractive index of the grating ridge and grating groove, respectively, and f is the fill factor of the grating layer. Combined with the waveguide layer with refractive index n3 and the substrate with refractive index n4 (here is air), the intrinsic equation of the four-layer waveguide is solved [44].Fig. 5. Absorption property and guided mode analysis in TE mode. (a) RCWA calculated absorption of dw-wavelength map. (b) Estimated resonance peaks using the dispersion relation of the slab waveguide. Structural parameters are the same as in Fig. 2.
In the TM mode, the perfect absorption effect can also be achieved by optimizing the size parameter of the structure. The above structure is optimized in the TE mode, so neither the mode coupling effect nor the perfect absorption effect can be achieved in the TM mode. According to the above mechanism for enhanced absorption, the structure is optimized to meet the above mode coupling conditions under the TM mode incidence. By optimizing the thickness of both the grating layer and the waveguide layer to adjust the guided-mode resonance wavelengths of different diffraction orders, the coupling effect is achieved and a nearly perfect absorption peak is generated, as shown in Fig. 6(a). The optimized structural parameters in TM mode are: dg = 0.308 µm, w = 0.25 µm, P = 0.5 µm and dw = 0.480 µm.
Fig. 6. The absorption spectrum in TM mode and the tunability of absorption peaks in two modes. (a) Absorptivity of monolayer MoS2 under normal incidence in TM mode. (b) In TE mode, adjust the position of the absorption peak by the overall scaling factor s. (c) In TM mode, adjust the position of the absorption peak by the overall scaling factor s.
The perfect absorption wavelength is tunable in both modes, which is essential to various narrowband photodetectors. However, changing only one parameter of the structure will make the two resonant peak offsets different, resulting in the decoupling of the resonances. For example, as the structural period deviates from the well-coupled condition two guided modes decouple, the absorption peak decreases (Fig. 3(a)). Fortunately, in TE mode and TM mode, by adjusting the overall scaling factor s, two resonance modes can be adjusted simultaneously to achieve resonance coupling in the new structure. In both modes, a large range of perfect absorption peaks can be adjusted by scaling the dimension, and the results are shown in Fig. 6(b) and 6(c). The other absorption peaks in the absorption spectra are due to the structure supporting multiple resonance modes. For example, in Fig. 6(b), when s = 1.0, the two stronger absorption peaks near the wavelengths of 550 nm and 650 nm can be attributed to the excitation of TE13 and TE21 modes. However, the wavelength range of the perfect absorption peak can only be achieved at the regime with a modest extinction coefficient, which is limited by the 2D materials intrinsically. Fortunately, the large available 2D material library offers such an opportunity to extend the range of perfect absorption to the near-infrared and middle-infrared region.
It is worth noting that the absorption enhanced structure can also be applied to other monolayer 2DMs, such as Gr, WS2, MoSe2, WSe2, and not only limited to MoS2. The universal applicability of this structure to 2D materials also provides a convenient way to expand the adjustable range of the perfect absorption peak. The absorption spectra of WSe2 and Gr were calculated by the RCWA method, as shown in Fig. 7. In the waveband of interest, they are also regarded as isotropic homogeneous thin films. The complex refractive index of WSe2 and Gr comes from the experimental measurement results of Li et al. [29] and Weber et al. [45]. For the monolayer WSe2, the ultra-narrow perfect absorption peak still appears at 586.6 nm in the original structure parameters, as shown in Fig. 7(a). For monolayer Gr, due to low extinction coefficient both at visible to near-infrared region, re-optimization of the structural parameters is necessary to obtain ultra-narrow band perfect absorption peaks, as shown in Fig. 7(b). Compared with monolayer 2DMs in air, the absorption of 2DMs in the structure is significantly enhanced at coupling wavelengths. This result shows that the proposed structure is universal for the absorption enhancement of two-dimensional materials.
Fig. 7. Absorption spectra of other 2DMs. (a) Absorption spectra of monolayer WSe2 in structure and in air. Structural parameters are the same as in Fig. 2. (b) Absorption spectra of monolayer Gr in structure and in air. The structural parameters: dg = 0.2 µm, w = 0.75 µm, dw = 0.41 µm, P = 1.5 µm, θ = 0°. Both the grating layer and the waveguide layer are made of Si material (${\textrm{n}_{\textrm{Si}}}\textrm{ = 3}\textrm{.4}$). The thicknesses of monolayer Gr and WSe2 are 0.34, 0.649 nm, respectively. The incident light is used as the TE mode.
Finally, we evaluate the role of the underlying planar waveguide. We found the absorption become weak once the waveguide layer under the grating is removed. We further calculated the absorption spectra of MoS2 coupled with planar waveguide-free structure as a function of grating thickness by using RCWA (Fig. 8(a)), and analyzed the supporting modes within the grating waveguide. The results are shown in Fig. 8(a) and 8(b). It can be seen from the dispersion relationship of the planar waveguide that in this structure, only the first-order guided mode resonance exists in the wavelength range of interest because of the relatively low effective refractive index of the grating (Fig. 8(b)). Therefore, there is no mode coupling in the structure in the wavelength range of interest. At this time, at the resonance wavelength, the electric field is strongly confined in the structure, which makes it difficult to couple and utilize, thus limiting the enhancement of absorption. The presence of a waveguide layer below the grating layer increases the overall effective refractive index and so supports the high-order guided modes in the structure, where mode coupling occurs by appropriate design to enhance the absorption of 2D material.
Fig. 8. When the waveguide layer in the structure of Fig. 1 is removed and only the grating layer and 2DM are retained, the absorption performance and resonance position at normal incidence in TE mode. The refractive index of the grating is the same as in Fig. 2. (a) Absorption dg-wavelength map using the RCWA. (b) Estimated resonance peaks using the dispersion relation of slab waveguide.
4. Conclusion
In summary, we theoretically proved that an all-dielectric sub-wavelength grating can achieve the ultra-narrow perfect absorption of a monolayer MoS2. According to RCWA and planar waveguide theory, the mechanism of perfect absorption of monolayer MoS2 is revealed, and the perfect absorption is attributed to the coupling of two specific diffraction order guided modes. The absorption spectrum was fitted by CMT to verify that under the critical coupling condition, the monolayer MoS2 achieved perfect absorption. Since the enhanced absorption is facilitated by the grating structure, such photonic structure can be widely used in any other absorptive thin film and is not limited to MoS2. The enhanced absorption effect of this structure will produce an ultrahigh quality factor, which has huge application prospects in various photoelectric sensors. Due to its ultra-narrow band perfect absorption, adjustability of the absorption peak, and the absorption enhanced structure is simple and easy to process, it has great application prospects in micro-nano optoelectronic devices, and is expected to significantly increase two-dimensional materials in optoelectronic devices applications.
Funding
National Natural Science Foundation of China (12064025); Fundamental Research Funds for the Central Universities (19lgpy21).
Disclosures
The authors declare no conflicts of interest.
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