The impact of the spacer and metal layer on the absorption of the heterostructures composed of TMDCs to design narrowband and broadband absorbers

Abstract

Owing to their exceptional optical and electronic properties, the two-dimensional transition metal dichalcogenide (TMDC) monolayers have received a great deal of attention. In fact, it has been suggested that heterostructures including spacers, metals, and two of the TMDC monolayers increase absorption of the narrowband and broadband in the visible range. This study investigates the effects of the number, place, and thickness of the spacers; metal thickness; and angle and polarization of the incident light on the absorption. Inserting the spacer into the structure increases the absorption via inducing light localization and enhancing the intensity of the light in the TMDC monolayers. Further, the effect of employing one spacer on improving the absorption is almost equal to making use of double spacers. The proposed structures by enhanced light-material interaction can raise the amount of the absorption over 90% throughout the broadband wavelength range of 300–480 nm and above 65% at the narrowband wavelength of 617 nm. The findings of the study suggest promising prospects of these structures for a variety of applications particularly in narrowband and broadband optical devices.

1 Introduction

Because of their direct bandgap and high optical absorption in their nanometer thickness, two-dimensional transition metal dichalcogenide (TMDC) monolayers have become an appealing candidate in the visible range, increasingly drawing the attention of the researchers over the recent decade (Soleimani-Amiri and Gholami Rudi 2020; Li and Zhu 2015). TMDCs have a great potential to be adopted in various photonic and optoelectronic devices like photonic absorbers, photodetectors, photovoltaic devices, solar cells, bio-sensors, and modulators (Khan et al. 2020; Cheng et al. 2018; Sharma et al. 2020). These materials with the chemical formula of MX₂, where M is a transition metal element from group VI (Mo, W) and X denotes a chalcogen (S, Se) such as MoS₂, MoSe₂, WS₂, WSe₂, and the like. These materials form X–M–X with the chalcogen atoms in two hexagonal planes separated by a plane of transition metal atoms. Bulk TMDCs demonstrate a covalently bonded layered structure in which the adjacent layers are held together by weak van der Waals bonding. The bulk TMDCs are semiconductors with an indirect band gap. When the van der Waals bonding is broken by reactions, the material gradually turns into a mono layer with a direct band gap in the visible range (Manzeli et al. 2017; Choi et al. 2017).

Increasing absorption in narrowband, broadband, narrow-angle or broad-angle is the requirement of achieving appropriate optical response in optoelectronical appliances (Ansari et al. 2020a, 2020, 2021). Different methods have been proposed to improve the absorption of TMDCs, e.g., employing multilayer structures containing TMDC monolayers (Strange et al. 2022; Yang et al. 2022). Owing to a variety of absorption peaks of these monolayers, heterostructures consisting of two TMDC monolayers can be used to increase the absorption over a wide range of wavelengths (Ansari et al. 2021). Placing dielectric with a specific thickness as a spacer between TMDCs monolayers is likely to increase the absorption due to its constructive interference (Ansari et al. 2021, 2020c). Placing a metal layer next to TMDC allows reflecting back the incoming light as well as being an architecture for supporting surface plasmons (SPs) or localized surface plasmons (LSPs), to enhance light absorption (Rahman et al. 2020; Luo et al. 2021). Even in the case of using uniform metallic layers in the absorber structure in which no SPs or LSPs find the chance of excitation, metallic layers can act like reflecting mirrors which increase the light path length inside the TMDC layer and increase the light absorption (Li et al. 2017; Cao et al. 2017). Metal type, refractive index and it’s work function affect the absorption (Rakić et al. 1998). Au layer has high imaginary refractive index that results in increasing absorption in the wavelength region of 300–500 nm. In terms of experimental work, it is easier to thin Au than other metals. Such absorber can be implemented in light emitting diodes (LEDs) and broadband ultraviolet into near infrared convertors for solar panels. Au is used on connectors and contacts because it has excellent corrosion resistance, high electrical conductivity (only copper and silver are better) and, alloyed with small amounts of nickel or cobalt, has good wear resistance. For low voltage, low current and low contact force applications, Au is the best material (Goodman 2002). For this reason, Au is examined in this article.

The obtained findings of this study suggest that using heterostructures including spacers, metal, and two TMDC monolayers increases the absorption in broadband and narrowband wavelength. In addition, the effects of the number, place, and thickness of the spacers; the thickness of the metal; and angle and polarization of the incident light on absorption have been examined in this study. The results confirm that absorption increases by inserting spacer into the structure inducing localization of light, and resulting in the enhancement of the light intensity in the TMDC monolayers. In sum, the research findings may open the door to potential applications of absorbers and solar cells for enhancing light-material interaction in the visible range.

2 Methods

Heterostructures including metal (Au), spacer (S1, S2), and different combinations of TMDCs in the form of TMDC1/TMDC2, TMDC1/Au/TMDC2, TMDC1/S1/Au/TMDC2, and TMDC1/S1/Au/S2/TMDC2 are investigated to increase light-material interaction in TMDCs. SiO₂ was chosen as the substrate and SiO₂ or TiO₂ was selected as S1 and S2. The schematic of TMDC1/Au/TMDC2 (structure IV), TMDC1/S1/Au/TMDC2 (structure V), and TMDC1/S1/Au/S2/TMDC2 (structure VI) are shown in Fig. 1,TMDC1 and TMDC2 monolayers are selected from MoS₂, WSe₂, and WS₂ with the thickness of 0.615, 0.649, and 0.618 nm, respectively (Ansari and Ghorbani 2018). The thicknesses of the S1, S2, and Au layers are represented by d_S1, d_S2, and d_Au, respectively. According to Ansari et al., (Ansari et al. 2021), the two combinations of MoS₂/MoS₂ and WS₂/WSe₂ provide the best condition with the optimal absorption. Therefore, we made use of the proposed combinations in the present study. Light passing obliquely through the air was radiated with the incident angle of ${\uptheta }_{0}$ to the structure, and the propagation angle in each layer was obtained according to Snell’s law.

Fig. 1

The heterostructures include TMDC1, TMDC2, metal (Au), and spacer (S1, S2) a TMDC1/Au/TMDC2, b TMDC1/S1/Au/TMDC2, and c TMDC1/S1/Au/S2/TMDC2

Absorption spectra were calculated on the basis of the transfer matrix method (Liu et al. 2012). The transfer matrix of constituted layer for either of the transverse electric (TE) or transverse magnetic (TM) polarizations is obtained as

$${\text{M}}_{{\text{j}}} = \left( {\begin{array}{ll} {{\text{cos}}\left( {2\pi {\text{n}}_{{\text{j}}} {\text{d}}_{{\text{j}}} \frac{{{\text{cos}}\theta_{{\text{j}}} }}{{\lambda }}} \right)} & { - {\text{i }}\Gamma {\text{ sin}}\left( {2\pi {\text{n}}_{{\text{j}}} {\text{d}}_{{\text{j}}} \frac{{{\text{cos}}\theta_{{\text{j}}} }}{{\uplambda }}} \right)} \\ { - {\text{i }}\Gamma {\text{ sin}}\left( {2\pi {\text{n}}_{{\text{j}}} {\text{d}}_{{\text{j}}} \frac{{{\text{cos}}\theta_{{\text{j}}} }}{{\lambda }}} \right)} & {{\text{cos}}\left( {2{\pi n}_{{\text{j}}} {\text{d}}_{{\text{j}}} \frac{{{\text{cos}}\theta _{{\text{j}}} }}{{\lambda }}} \right)} \\ \end{array} } \right),$$

(1)

where n_j, ${\uptheta }_{\mathrm{j}}$, and dj are, refractive index, light propagation angle, and thickness of the j^th layer, respectively, and λ is the light wavelength on the air and Γ for TE and TM polarizations is correspondingly defined as $\frac{1}{{\mathrm{n}}_{\mathrm{j}}\mathrm{cos}{\uptheta }_{\mathrm{j}}}$ and $\frac{\mathrm{cos}{\uptheta }_{\mathrm{j}}}{{\mathrm{n}}_{\mathrm{j}}}$. Multiplying tranfer matrix of each constituent layer can make the final transfer matrix M for the whole structure. The tangential electric and magnetic fields can be expressed for the first (${E}_{0t}$, ${H}_{0t})$ and last ($E_{{{\text{s}}t}}$, $H_{{{\text{s}}t}}$) surrounding layers as follows

$$\left( {\begin{array}{*{20}c} {E_{st} \left( z \right)} \\ {H_{st} \left( z \right)} \\ \end{array} } \right) = \left( M \right){ }\left( {\begin{array}{*{20}c} {E_{0t} \left( 0 \right)} \\ {H_{0t} \left( 0 \right)} \\ \end{array} } \right),$$

(2)

and amplitudes of the transmitted (c⁺) and reflected (c⁻) fields in each layer are calculated as

$$\left( {\begin{array}{*{20}c} {E_{0t} \left( z \right)} \\ {H_{0t} \left( z \right)} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\psi e^{{i2r\pi n_{0} \frac{{\cos \theta_{0} }}{\lambda }}} } & {\psi e^{{ - i2r\pi n_{0} \frac{{\cos \theta_{0} }}{\lambda }}} } \\ {n_{0} \varphi e^{{i2r\pi n_{0} \frac{{\cos \theta_{0} }}{\lambda }}} } & { - n_{0} \varphi e^{{ - i2r\pi n_{0} \frac{{\cos \theta_{0} }}{\lambda }}} } \\ \end{array} } \right){ }\left( {\begin{array}{*{20}c} {c_{0}^{ + } } \\ {c_{0}^{ - } } \\ \end{array} } \right),$$

(3)

$$\left( {\begin{array}{*{20}c} {E_{st} \left( z \right)} \\ {H_{st} \left( z \right)} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\psi e^{{i2r\pi n_{s} \frac{{\cos \theta_{s} }}{\lambda }}} } & {\psi e^{{ - i2r\pi n_{s} \frac{{\cos \theta_{s} }}{\lambda }}} } \\ {n_{{\text{s}}} \varphi e^{{i2r\pi n_{s} \frac{{\cos \theta_{s} }}{\lambda }}} } & { - n_{{\text{s}}} \varphi e^{{ - i2r\pi n_{s} \frac{{\cos \theta_{s} }}{\lambda }}} } \\ \end{array} } \right){ }\left( {\begin{array}{*{20}c} {c_{s}^{ + } } \\ 0 \\ \end{array} } \right),$$

(4)

where 0 and s refer to air and substrate, and ψ and φ for the TE (TM) polarization are equal to $cos{\theta }_{j}$ (1) and 1 ($cos{\theta }_{j}$), respectively. Next, transmission, reflection, and absorption are determined as follows T = $\frac{{\mathrm{n}}_{\mathrm{s}}\mathrm{cos}{\uptheta }_{\mathrm{s}}}{{\mathrm{n}}_{0}\mathrm{cos}{\uptheta }_{0}}{\left|\frac{{c}_{s}^{+}}{{c}_{0}^{+}}\right|}^{2}$, R = ${\left|\frac{{c}_{0}^{-}}{{c}_{0}^{+}}\right|}^{2}$, and A = 1–T–R. The refractive index of SiO₂, TiO₂, Au, and TMDCs monolayer are obtained using formulas ${n}_{{SiO}_{2}}=\sqrt{1.28604141+\frac{1.07044083{\lambda }^{2}}{{(\lambda \times {10}^{-3})}^{2}-1.00585997\times {10}^{-2}}+\frac{1.10202242{\lambda }^{2}}{{(\lambda \times {10}^{-3})}^{2}-100}}$, (Ghosh 1999) ${n}_{{TiO}_{2}}=\sqrt{5.913+\frac{0.2441}{{\lambda }^{2}-0.0803}}$ (DeVore 1951) ${N}_{Au}=\sqrt{1-\frac{{\omega }_{p}^{2}}{{\omega }^{2}+i\gamma \omega },}$ (Johnson and Christy 1972) and ${N}_{TMDC}=\mathrm{ n }+\mathrm{ ik }= \sqrt{{\varepsilon }_{\mathrm{TMDC}}}=\sqrt{({\varepsilon }_{real}+i {\varepsilon }_{imag})}=\sqrt{{\varepsilon }_{\infty }+\sum_{j=1}^{N}\frac{{a}_{j}}{{\omega }_{j}^{2}-{\omega }^{2}-i\omega {b}_{j}}}$, where $\lambda$, ${\varepsilon }_{\infty ,}$ ${\omega }_{p,}$ $\gamma$, and ω are the light wavelength in nanometers, DC permittivity, plasma frequency, damping frequency, and incident light frequency, respectively, and ${a}_{j}$, ${\omega }_{j}$, and ${b}_{j}$ are oscillation power, resonant frequency, and damping factor of the ${j}^{th}$ oscillator whose quantities are expressed in Ref. (Ansari and Ghorbani 2018). Parameters ${\varepsilon }_{\infty ,}$ ${a}_{j}$, ${\omega }_{j}$, and ${b}_{j}$ for the TMDC monolayers are listed in Table 1.

Table 1 Coefficients aj, bj, and ωj Correspond to the Lorentz Model for Four Monolayers of WSe₂, MoS₂, and WS₂

3 Result

The effects of the thickness of the Au layer as well as the thickness, substance, and number of the spacer layers in the heterostructures including TMDCs on the absorption are examined here. At first, WS₂ and WSe₂ monolayers were selected as TMDC1 and TMDC2. To achieve the aim, an Au layer was placed between the WS₂ and WSe₂ monolayers (structure IV) and its absorption was plotted as a function of Au thicknesses and the wavelength, as displayed in Fig. 2a. According to Fig. 2a, an absorption of 60.2%, is observed at λ = 444 nm and ${{\text{d}} }_{\text{Au}}$= 65 nm and an absorption of 18.6% is detected at λ = 617 nm and ${{\text{d}} }_{\text{Au}}$= 16 nm. The skin depth of Au layer is between 10 ~ 15 nm for wavelength of 300 to 500 nm. When the thickness of the Au layer is several times the skin depth of Au, the light does not reach the TMDC layer. Further, we selected SiO₂ as spacer1 and placed it between WS₂ monolayer and Au (structure V11),the curve of the absorption versus Au and spacer thicknesses at λ = 444 nm is depicted in Fig. 2b. Optimal absorption in this structure is 92% which is observed when ${{\text{d}} }_{\text{Au}}\text{ = 65}$ nm and ${{\text{d}} }_{\text{S1}}$ = 40 nm. Moreover, the curve of the absorption versus Au and spacer thicknesses at λ = 617 nm is displayed in Fig. 2c (structure ${\text{V}}{21}$). The optimal amount of absorption at this wavelength is 62% which is observed when ${{\text{d}} }_{\text{Au}}\text{ = 43}$ nm and ${{\text{d}} }_{\text{S1}}$ = 67 nm.

Fig. 2

a Absorption of structure IV as a function of Au thicknesses and the wavelength. b Absorption spectra of structure ${\text{V11}}$ versus Au and spacer thicknesses in λ = 444 nm. c Absorption spectra of structure V21 versus Au and spacer thicknesses in λ = 617 nm. d The absorption as a function of the thicknesses of the spacer1 and spacer2 for structure ${\text{VI13}}$ at λ = 444 nm

Figure 2d displays the absorption at λ = 444 nm, as the function of the spacer1 and spacer2 thickness, for the structure ${\text{VI13}}$ (in which spacer TiO₂ was placed between WS₂ monolayer and Au and spacer SiO₂ was put between Au and WSe₂ monolayer). The maximum absorption of 96.5% is observed when ${{\text{d}} }_{\text{Au}}$ = 65 nm $, {{\text{d}} }_{\text{S1}}$ = 21 nm, and ${{\text{d}} }_{\text{S2}}$ = 56 nm. The absorption behavior at the wavelength of 617 nm resembles the one at the wavelength of 444 nm.

To compare the optimal thicknesses for maximum absorption at the wavelength of 444 nm (617 nm), absorption spectra of the structures as the function of the wavelength are plotted in the Fig. 3a, b. A narrowband wavelength at 617 nm and the broadband wavelength range of 300–480 nm can be observed in this figure. Structures consisting of one spacer and two spacers are displayed by a line and a dashed line, respectively. Absorption of the different combinations for optimal thicknesses of Au, spacer1, and spacer2 at the narrowband wavelength of 617 nm and the broadband wavelength range of 300–480 nm are listed in Table 2. In Fig. 1a, a general schematic of the structure IV is shown. The number after the IV (indicated as " IV1" and " IV2") is a representative of Au layer that has different thicknesses (for example IV1 that means Au layer has thickness equal to 65 nm while IV2 indicates that Au layer has thickness equal to16 nm). Also, structure V and VI are shown in Fig. 1b, c. The first number after the V and VI (indicated as "V12" and "VI11") stands for Au layer with different thicknesses (for V12 ${\text{d}}_{\text{Au}}=66 nm$ while for V22 ${\mathrm{d}}_{\mathrm{Au}}=52 nm$). Here, the second number represents spacer layer (1 indicates SiO₂ and 2 indicates TiO₂). Thus, V22 is for Au layer with thickness equal to 52 nm and spacer is TiO2 or VI11 addresses Au layer thickness is 46 nm and spacer is SiO₂. Figure 1 shows a general configuration as schematics of the structures in V11, V21 can be the same and their difference is in the spacer type and thickness. This naming is placed in the second column in Tables 2 and 3. This nomenclature of samples is written for when WS₂ and WSe₂ monolayers were selected as TMDC1 and TMDC2 (addressed in Table 2, and if MoS₂ and MoS₂ monolayers were selected as TMDC1 and TMDC2, only the thickness of Au layer is different that are addressed in Table 3. As shown in Table 2 and Fig. 3a, the absorptions of the structures ${\text{VI14}}$ and ${\text{VI13}}$ (structures ${\text{VI11}}$ and ${\text{VI12}}$) correspond to each other. The structure ${\text{VI14}}$ in the wavelength range of 300 to 480 nm shows the maximum absorption. Nevertheless, due to the minute difference between this absorption and that of the structure ${\text{VI13}}$, among all the structures with double spacers, structure ${\text{VI13}}$ is selected as the optimal structure with maximum amount of absorption in this wavelength range. Compared to all other structures with one spacer, structure ${\text{V12}}$ demonstrates the maximum absorption. The maximum absorption of the one spacer (structure ${\text{V12}}$) is almost equal to that of the double spacers (structure ${\text{VI13}}$). Thus, considering the fact that fewer layers are more suitable for experimental works, structure ${\text{V12}}$ is selected as the best structure in this wavelength range. Examining the absorption at the peak of 617 nm, reveals that the structure with a spacer, structure ${\text{VI11}}$, is the superior structure with the maximum absorption.

Fig. 3

Absorption spectra as function of wavelength for thicknesses where the absorption is increased at wavelength a 444 nm, b 617 nm for WS₂ and WSe₂ monolayers were selected as TMDC1 and TMDC2. (Color figure online)

Table 2 Absorption of different combinations in narrowband wavelength at 617 nm and broadband wavelength in 300–480 nm for optimal thickness of Au, spacer1, and spacer2

Table 3 Absorption of different combinations in narrowband wavelength at 617 nm and broadband wavelength in 300–480 nm for optimal thickness of Au, spacer1, and spacer2

According to Table 2 and Fig. 3b, structure ${\text{V}}{21}$ in the broadband wavelength range of 300–480 nm and structure ${\text{V}}{22}$ at the narrowband wavelength of 617 nm are selected as the best options demonstrating maximum absorption. Comparing Fig. 3a, b reveals that structure ${\text{V12}}$ (the green line in Fig. 3a and structure ${\text{V}}{22}$ (the green line in Fig. 3b are superior options for designing broadband and narrowband wavelength, respectively.

Another combination that increases the absorption is the placement of MoS₂ monolayer, instead of TMDC1 and TMDC2, in the structures. An Au layer was placed between the two MoS₂ monolayers (structure IV) and its absorption was plotted as a function of Au thicknesses and the wavelength, as displayed in Fig. 4a. According to the Fig. 4a, an absorption is 68.3% is observed at λ = 349 nm and ${{\text{d}} }_{\text{Au}}$= 65 nm and an absorption of 16.1% is detected at λ = 617 nm and ${{\text{d}} }_{\text{Au}}$= 15 nm. By selecting SiO₂ as spacer1 and placing its between MoS₂ and the Au monolayer (structure V11), the curve of the absorption versus Au and spacer thicknesses at λ = 349 nm is depicted in Fig. 4b. Optimal absorption in this structure is 96% which is observed when ${{\text{d}} }_{\text{Au}}\text{ = 46}$ nm and ${{\text{d}} }_{\text{S1}}$ = 29 nm. Moreover, the curve of the absorption versus Au and spacer thicknesses at λ = 617 nm is displayed in Fig. 4c (structure ${\text{V}}{21}$). The optimal amount of absorption at this wavelength is 39% which is observed when ${{\text{d}} }_{\text{Au}}\text{ = 29}$ nm and ${{\text{d}} }_{\text{S1}}$ = 63 nm. Figure 4d displays the absorption at λ = 349 nm, as the function of the spacer1 and spacer2 thickness, for the structure ${\text{VI13}}$ (in which spacer TiO₂ was placed between MoS₂ monolayer and Au and spacer SiO₂ was put between Au and MoS₂ monolayer). The maximum absorption of 94.5% is observed when ${{\text{d}} }_{\text{Au}}\text{ = 46}$ nm,${{\text{d}} }_{\text{S1}}$ = 16 nm, and ${{\text{d}} }_{\text{S2}}$ = 44 nm. The absorption behavior at the wavelength of 617 nm resembles the one at the wavelength of 349 nm.

Fig. 4

a Absorption structure IV as a function of Au thicknesses and wavelength. b Absorption spectra structure ${\text{V11}}$ versus Au and spacer thicknesses in λ = 349 nm. c Absorption spectra structure V21 versus Au and spacer thicknesses in λ = 617 nm. d The absorption as a function of the thicknesses of the spacer1 and spacer2 for structure ${\text{VI13}}$ at λ = 349 nm

To compare the optimal thicknesses for maximum absorption at the wavelength of 349 nm (617 nm), absorption spectra of the structures as the function of the wavelength are plotted in the Fig. 5a, b. A narrowband wavelength at 617 nm and the broadband wavelength range of 300–480 nm can be observed in this figure. Structures consisting of one spacer and two spacers are displayed by a line and a dashed line, respectively. Absorption of the different combinations for optimal thicknesses of Au, spacer1, and spacer2 at the narrowband wavelength of 617 nm and the broadband wavelength range of 300–480 nm are listed in Table 3. As shown in Table 3 and Fig. 5a, nevertheless, due to the minute difference between this absorption and that of the structure ${\text{VI13}}$, among all the structures with double spacers, structure ${\text{VI13}}$ is selected as the optimal structure with maximum amount of absorption in this wavelength range. Compared to all other structures with one spacer, structure ${\text{V12}}$ demonstrates the maximum absorption. The maximum absorption of the one spacer (structure ${\text{V12}}$) is almost equal to that of the double spacers (structure ${\text{VI13}}$). Thus, considering the fact that fewer layers are more suitable for experimental works, structure ${\text{V12}}$ is selected as the best structure in this wavelength range. Examining the absorption at the peak of 617 nm, reveals that the structure with a spacer, structure ${\text{V12}}$, is the superior structure with the maximum absorption.

Fig. 5

Absorption spectra for chosen structures as function of wavelength for thicknesses where the absorption is increased at wavelength a 349 nm, b 617 nm for MoS₂ and MoS₂ monolayers were selected as TMDC1 and TMDC2

According to Table 3 and Fig. 5b, similar to Fig. 5a, the structure ${\text{V}}{21}$ in the broadband wavelength of 300–480 nm and the structure ${\text{V}}{22}$ at the narrowband wavelength of 617 nm are selected as the best structures with the maximum absorption. Comparing Fig. 5a, b, reveals that the structure ${\text{V12}}$(the green line in Fig. 5a is the superior option for designing broadband wavelength and structure ${\text{V}}{22}$ (the green line in Fig. 5b is the better choice for designing narrowband wavelength.

The intensity versus position curves of the structures IV1, V11, V12, and ${\text{VI13}}$, in which WS₂ and WSe₂ monolayers were respectively selected as TMDC1 and TMDC2 at the wavelength of 444 nm are displayed in Fig. 6a–d. The spacer layer localized the light in the WS₂ monolayer. The intensity of the WS₂ monolayer in the structure V11 is greater than that in other structures. Structures V11, V12, and ${\text{VI13}}$ show the highest amounts of absorption which are almost the same.

Fig. 6

The intensity versus position at the wavelength of 444 nm for structures a IV1, b V11, c V12, and d ${\text{VI13}}$ that TMDC1 and TMDC2 are selected WS₂ and WSe₂ monolayers

Similarly, the intensity versus position curves of the structures IV1, V11, V12, and ${\text{VI13}}$, in which MoS₂ monolayer is selected as both TMDC1 and TMDC2 at the wavelength of 349 nm are presented in Fig. 7a–d. The intensity of the MoS₂ monolayer in the structure V11 is greater than that in other structures. Therefore, structure V11 demonstrates the maximum absorption. Improvement of the absorption should be attributed to the localization of the light as well as light intensity enhancement of the MoS₂ monolayer.

Fig. 7

The intensity versus position at the wavelength of 349 nm for structures a IV1, b V11, c V12, and d $\text{VI13 that TMDC1 and TMDC2 are chosen MoS2 monolayer}$

To study the effect of polarization and incident angle, the absorption versus wavelength and light incident angle was depicted for both of the transverse electric (TE) and transverse magnetic (TM) polarization. This depiction is based on the data provided in Tables 2 and 3 for the structure V22, when WS₂ and WSe₂ were respectively chosen as TMDC1 and TMDC2 Fig. 8a, b and MoS₂ was selected as both TMDC1 and TMDC2 Fig. 8c, d. According to Fig. 8a, b, absorption peak at 617 nm is not sensitive to the angle and polarization of the incident light and narrowband absorption is fixed, which can be useful for the devices which need to be insensitive to the incident angle (tolerance). However, over the wavelength range of 400–455 nm, absorption depends on the angle and polarizations of the incident light and the absorption widths above 70% are observed in the angle range of 0°–53°. Notwithstanding, by increasing the incident angle in Fig. 8c, d, the peak at 617 nm disappears and the absorption in the broadband wavelength range becomes dependent on the angle and polarization. The absorption above 70% for TE (TM) polarization is observed in the wavelength range of 400–460 nm and angle range of 0 to 46° (0°–70°).

Fig. 8

Absorption spectra versus light incident angle for the data in Tables 2 and 3 for structure V22 when WS₂ and WSe₂ chosen as TMDC1 and TMDC2 for both a TE and b TM polarization and when MoS₂ was selected as both TMDC1 and TMDC2 for both c TE and d TM polarization

Finally, the studied structure is compared with other articles and the results are given in the Table 4.

Table 4 Absorption maximum and wavelength range in which absorption maximum structures studied in similar articles

4 Conclusions

Findings of this study suggest that heterostructures including spacers, metal, and two TMDC monolayers in the visible range achieve high absorption in broadband and narrowband wavelength range. In addition, the effects of the type of TMDC; the number, place, and thickness of the spacers; metal thickness; and the angle and polarization of the incident light on the amount of absorption and peak wavelength were explored in this study. It was found that maximum amounts of absorption in the structures with one spacer are almost equal to those in the structures with double spacers. Further, the results of the current study affirms that absorption increases due to the localization and enhancement of the light intensity in the TMDC monolayers induced by inserting a spacer into the structure. In fact, the proposed structure increases absorption over 90% in broadband and above 65% in the narrowband wavelength range. Moreover, the absorption peak of the structure containing WS₂ at 617 nm is not sensitive to the angle and polarization of the incident light. The proposed structures have implications for designing methods to improve light-matter interaction in broadband and narrowband wavelength ranges.