1. Introduction

With the exfoliation technique for preparing atomically thin sheets [1], layered two dimensional (2D) materials have attracted great interest due to their excellent physical properties and remarkable device prospects [1,2]. There is a kind of representative 2D materials, transition metal dichalcogenides (TMDs), with representative materials as molybdenum disulphide (MoS₂) [3,4]. A single monolayer of MoS₂ consists of an atomic plane of a Mo sandwiched between two S atomic planes. Multilayer MoS₂ have layers weakly stacked by van der Waals force with a S-Mo-S covalently bonded sandwich in each layer. Such stacked layer structure makes properties of MoS₂ flakes dependent on its thickness, or layer number (denoted as N) [5–7].

Several optical techniques have been used to probe N-dependent optical properties of MoS₂ flakes, such as Raman spectroscopy [6–8] and photoluminescence (PL) [5]. Raman spectrum consists of E¹_2g and A_1g modes with a peak distance between these two modes used to identify the layer number of 1L-4L MoS₂ layers [6,7], meanwhile contains ultralow-frequency shear (S) and breathing (B) modes with their frequencies used to identify the layer number of 1L-10L MoS₂ layers [7,8]. PL spectrum shows the strong contrast of PL efficiency between 1L and multilayer MoS₂ sheets due to the feature of transferring from indirect-bandgap in bulk material to direct-bandgap in 1L sheet [5]. Although A and B exciton emission peaks can be also displayed in PL spectrum, they coincide with the direct-bandgap PL peak in 1L sheet and separate from the indirect-bandgap PL peak with increasing layer number but with reduction in strength of 10²-10³ orders of magnitude [5], which are all not conducive to studying exciton properties. Recently, optical reflection and transmission spectra have proven to be another powerful technique to study 2D materials [9–14]. Frisenda R et al [9] showed a versatile optical microscope setup to carry out differential reflectance and transmittance spectroscopy in TMDs to determine their number of layers and to characterize their fundamental optical properties such as excitonic resonances. In this paper, we measured reflection spectra of NL MoS₂ flakes on SiO₂/Si substrate in the broad wavelength range of 400-800 nm and studied their exciton properties as a function of N by utilizing the experimental setup with higher spectral resolution for samples with sizes in the micron range. Moreover, we noticed that the substrate-related interference effect influenced excitonic features collected from MoS₂ flakes. Here, we can optimize reflectivity curves of MoS₂ flakes to investigate their exciton properties by selecting proper SiO₂ thickness.

2. Materials and methods

MoS₂ flakes were prepared from monolayer to multilayer on SiO₂/Si substrate with different SiO₂ thickness by micromechanical cleavage of a bulk MoS₂ crystal (2D semiconductors Inc.). We selected 3 types of most commonly used SiO₂/Si substrates with SiO₂ thickness of 89, 100 and 302nm. The thicknesses of the SiO₂ capping layer were measured by a spectroscopic ellipsometer (J A Woollam, M-2000DI) with repeatability of 0.5 nm. MoS₂ flakes were easily seen by naked eyes via microscope. Their layer number were pre-estimated by the ultralow-frequency Raman measurements of S and B modes, as previously done for MoS₂ flakes [8]. Due to the micron-sized samples, reflection spectrum measurements were performed in a backscattering geometry using a Jobin-Yvon HR800 micro-Raman system, which was equipped with liquid nitrogen cooled charge coupled device and the objective of 50X (NA = 0.45). The size of samples was above 5 μm to ensure the accuracy of subsequent tests. Tungsten halogen lamp was used as a light source, with the spot size below 2 μm. A Semrock mirror with a high reflectivity up to 99% in the range from 350 to 1100 nm was used as a reference to the light intensity from lamp source. The reflection spectra were measured from the samples, the bare substrates and the Semrock mirror in the broad wavelength range of 400-800 nm. The 600 lines per mm grating was used, which enables one to have each CCD pixel to cover 1nm. The best reflected light signal was achieved by focusing the microscope to get maximum peak intensity. The reflectivity of MoS₂ flakes on SiO₂/Si substrate were obtained by dividing the reflection spectra from the samples by that from the Semrock mirror.

We first measured reflection spectra of MoS₂ flakes on SiO₂/Si substrate with 100nm SiO₂ thickness. Figure 1(a) shows the reflectivity curves of 1L-5L, 9L and 10L MoS₂ on 100nm SiO₂/Si substrate and of bare substrate in the range of 400-800 nm. (The optical images and Raman spectra of samples are presented in the Appendix.) The outlines of these reflectivity curves of MoS₂ flakes are similar on which there are many dips by contrast of the smooth reflectivity curve of bare substrate. Because both MoS₂ flakes and SiO₂ layer are penetrable by incident and reflected lights in reflection spectrum measurements, the incident lights passed through them and finally were absorbed by the Si layer, but the reflected lights were collected from interfaces of Air/NL-MoS₂, NL-MoS₂/SiO₂ and SiO₂/Si. The dips in the reflectivity curves of MoS₂ flakes provide an effective measure of absorbance of MoS₂ flakes, which are the reciprocal of absorption peaks of gap transitions.

 

Fig. 1 (a) The reflectivity curves of 1L-5L, 9L and 10L MoS₂ on SiO₂/Si substrate with 100nm SiO₂ thickness and of bare substrate in the range of 400-800nm. (b) The reflectivity curves in the range of 550-700nm. The curves are offset for clarity. (c) The position of A and B excitons from 1L-5L, 9L and 10L MoS₂ are summarized by red and blue circles respectively.

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3. Results and discussion

A and B excitons arise from direct gap transitions at the K point due to a spin–orbit split valence band and degenerate conduction band at the K point, with the energy difference as an indication of the strength of spin-orbit interaction [5,15,16]. According to previous reports about band structure and exciton transitions in MoS₂ flakes [5,15,16], the dips of our reflectivity curves in the range of 550-700nm derive from the absorption of A and B excitons. The energy of B exciton is higher than A exciton with the energy difference approximate to 0.15 eV [5] or 50 nm [17], and their energy depend on the layer number of MoS₂ flakes [5], all of which can be easily shown in our reflectivity curves with wavelength as horizontal axis. In order to obtain the more detailed emission wavelength of A and B excitons for different N of NL MoS₂, we showed reflectivity curves of bare substrate and MoS₂ flakes in the range of 550-700nm in Fig. 1(b). The curves were offset for clarity. The A and B excitons in 1L MoS₂ locate at 652 nm and 603 nm respectively. Both of them show blueshift as N increases. Because the dips due to excitons are not so sharp, we didn’t read directly the position of A and B excitons. We inverted reflectivity curves by the calculation of 1/x. In the inverted curve, the data of vertical axis were multiplied a lot, and A and B excitonic peaks became a lot sharper. We obtained the position of A and B excitons by using the “peak searching” function in the Labspec 6 software and the error bar for that is ± 1nm. The emission wavelength of A and B excitons from 1L-5L, 9L and 10L MoS₂ were summarized by red and blue circles respectively in Fig. 1(c). There is a big shift in the wavelength of A exciton from ~652 nm for N = 1 to ~662 nm for N = 2. The wavelength spacing is 10 nm between 1L and 2L. However, the shift of A exciton becomes smaller with N increases, which is gradual from ~662 nm for N = 2 to ~665 nm for N = 10. The B exciton shifts more slowly from 1L to 2L than the A exciton. It shifts from ~603 nm for N = 1 to ~605 nm for N = 2, and then it gradually shifts from ~605 nm for N = 2 to ~606 nm for N = 10. Because the energy difference between A and B direct excitonic transitions is due to the spin-orbital splitting of the valence band at the Brillouin zone K point, the position change of A and B excitons as a function of layer number is derived from the change of direct excitonic states with layer thicknesses. Based on the photoluminescence spectra of 1-6L MoS₂ and WS₂ samples in the reference [18], frequency shifts of A and B excitonic peaks are relatively easy to be seen from 1L to 2L. Especially, A exciton movement is more obvious from 1L to 2L maybe due to the influence of the emerging indirect emission peak on the side of it in 2L.

Due to the SiO₂ thickness of SiO₂/Si substrate as one of the important influence factors in reflection spectra, we further studied the reflectivity curves of MoS₂ flakes on SiO₂/Si substrate with 89nm SiO₂ thickness and 302nm SiO₂ thickness. (The optical images of samples are presented in Appendix in Fig. 7.) Fig. 2(a) shows A and B excitons of 1L-3L MoS₂ on 89nm SiO₂/Si substrate in the range of 550-700nm. The curves were offset for clarity. These reflectivity curves are similar with those of 1L-3L MoS₂ on 100nm SiO₂/Si substrate. Both A and B excitons shift toward longer wavelengths with increasing N. The position of A and B excitons from 1L-3L MoS₂ on 89nm SiO₂/Si substrate were summarized in Fig. 2(b) by red and blue dotted circles respectively, compared with those on 100nm SiO₂/Si substrate plotted by red and blue circles respectively. The wavelength difference in two cases is smaller than 2nm, which is close to our test error limit. However, the reflectivity curves of MoS₂ flakes with 302nm SiO₂ thickness changed dramatically as shown in Fig. 3(a), compared with the former reflectivity curves with 89nm and 100nm SiO₂ thickness. There are three dips in 3L MoS₂ on 302nm SiO₂/Si substrate in the range of 550-700nm, which means new interference effects occur related with SiO₂ thickness.

 

Fig. 2 (a) The reflectivity curves of 1L-3L MoS₂ on SiO₂/Si substrate with 89nm SiO₂ thickness in the range of 550-700nm. The curves are offset for clarity. (b) The position of A and B excitons from 1L-3L MoS₂ are summarized in the case of 89nm and 100nm SiO₂ thickness.

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Fig. 3 (a) The reflectivity curves of 1L and 3L MoS₂ on SiO₂/Si substrate with 302nm SiO₂ thickness in the range of 550-700nm. (b) The calculated reflectivity curves of 3L MoS₂ on SiO₂/Si substrate with 89nm, 100nm and 302nm SiO₂ thickness. The curves were offset for clarity.

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In order to manifest the above phenomena, we used the multiple reflection interference method [10,11] to calculate the reflectivity curves of 3L MoS₂ on 89nm, 100nm and 302nm SiO₂/Si substrates in the range of 550-700nm, as shown in Fig. 3(b). In the Air/3L-MoS₂/SiO₂/Si structure, the incident light and the emergent light undergoes multiple reflection at the interfaces of Air/3L-MoS₂, 3L-MoS₂/SiO₂ and SiO₂/Si, and optical interference within each medium. We assume normal incidence perpendicular to the MoS₂ atom plane, the reflectivity curves were calculated by using transfer matrix formalism based on classical electrodynamics and Fresnel’s law. Details of calculations are described in Appendix. In Fig. 3(b), there are two dips in 3L MoS₂ on 89nm and 100nm SiO₂/Si substrate but three dips in 3L MoS₂ on 302nm SiO₂/Si substrate, which is consistent with the experimental results.

We further calculated the reflectivity of MoS₂ flakes for different N. Figures 4(a)–4(d) are the color contour plots of the reflectivity of bare substrate and 1L, 3L, 10L MoS₂, respectively, as function of both SiO₂ thickness of 0-500nm and reflection wavelength of 500-750nm. We can see bright areas and dark areas appear alternately in the reflectivity of bare substrate, as shown in Fig. 4(a). Due to the SiO₂ layer tuned as an antireflection coating [11,12], MoS₂ flakes become visible in these dark areas. The first dark area is around d_SiO2 = 50~120 nm. The contained 89nm and 100nm is used in our experiment. The second dark area is around d_SiO2 = 200~300 nm. The contained 300 nm is used in our experiment. The pink dotted lines were marked to show these three used SiO₂ thickness.

 

Fig. 4 The color contour plots of the reflectivity of (a) bare substrate and (b) 1L, (c) 3L, (d) 10L MoS₂ on SiO₂/Si substrate, respectively, as function of both SiO₂ thickness of 0-500nm and reflection wavelength of 500-750nm.

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In Fig. 4(b)-(d), the shapes of bright areas and dark areas change a lot compared with bare substrate due to the absorption of excitons. The shape changes are more noticeable as N increases, which is consistent with the experimental results shown in Fig. 1(a). The green dotted lines were marked to show the emission wavelength of A and B excitons in Fig. 4(b)-(d). Due to the first dark area nearly parallel to the wavelength axis, we can easily identify A and B excitons for the fixed 89nm and 100nm SiO₂ thickness. However, other dark areas have non-negligible angles with the wavelength axis due to the phase changes in different wavelengths caused by thicker SiO₂ thickness in the interference process [11,12]. The higher the order of dark area is, the larger the angle is. Thus, for the fixed 302nm SiO₂ thickness, we fail to identify A and B excitons. The optimized SiO₂ thickness is 50-100 nm to investigate the exciton properties of MoS₂ flakes by reflectivity curves. Using the similar method, we have showed the optimized optical matching between 1L to 5L graphene flakes and SiO₂ layer in terms of interference effect [12], which opens the possibility to exploit the enhancement of anti-reflection coating with the nanoscale thickness. In addition, we have found similar results in the research of N dependent excitons properties in other TMDs [13] and related research can be extended to N dependent optical properties of anisotropic 2D flakes [14], all of which prove optical reflectance techniques to be very powerful techniques to study 2D materials.

4. Conclusion

We demonstrated a simple and fast technique to probe reflectivity of MoS₂ flakes as a function of layer number and studied the emission wavelength of A and B excitons dependent on layer number. However, the substrate-related interference effect was very likely to hamper the detection of excitonic features collected from MoS₂ flakes. Based on our results, the optimized SiO₂ thickness is 50-100 nm to investigate the exciton properties of MoS₂ flakes by reflectivity curves.

5 Appendix

5.1 The optical images and Raman spectra of MoS₂ flakes on a SiO₂/Si substrate with 100nm SiO₂ thickness

We prepared two sets of 1L-5L, 9L, 10L MoS₂ flakes on 100nm SiO₂/Si substrate by micromechanical cleavage of a bulk MoS₂ crystal. Samples of different layers are stacked in the same position to avoid measurement errors as much as possible in the reflection spectra measurements, as presented in Fig. 5. Figure 6 shows Raman spectra of MoS₂ flakes. Fig. 6(a) shows S and LB modes located below 60cm⁻¹ in MoS₂. The S and LB modes do not exist in 1L MoS₂. With increasing N, the frequency of the LB mode decreases and that of the S mode increases, as shown in Fig. 6(c). The S and LB vibrations are an intrinsic property of NL MoS₂. The frequencies of these modes are linked with N by a linear chain model (LCM) in which each atom layer is considered as a single ball and there is mutual coupling between them. Therefore, the S and LB modes can be utilized to identify N of NL MoS₂ flakes. The layer number of samples in this paper were pre-estimated by the Raman measurements of S and B modes. Fig. 6(b) shows that E¹_2g and A_1g modes are respectively located at ∼ 380cm⁻¹ and ∼ 402cm⁻¹ in MoS₂. With increasing N, the frequency of the A_1g mode decreases and that of the E¹_2g mode increases, as shown in Fig. 6(d).

 

Fig. 5 The optical images of 1L-5L, 9L, 10L MoS₂ flakes on SiO₂/Si substrate with 100nm SiO₂ thickness.

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Fig. 6 Raman spectra of 1L-5L, 9L, 10L MoS₂ flakes with the frequency (a) below 60 cm⁻¹ and (b) between 370 cm⁻¹ and 420 cm⁻¹. The position of (c) S and LB modes and (d) E¹_2g and A_1g modes were summarized with the increasing N.

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5.2 The optical images of MoS₂ flakes on SiO₂/Si substrate with 89nm and 302nm SiO₂ thickness

 

Fig. 7 The optical images of 1L-3L MoS₂ flakes on SiO₂/Si substrate with 89nm SiO₂ thickness, and 1L and 3L MoS₂ flakes on SiO₂/Si substrate with 302nm SiO₂ thickness.

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5.3 The reflectivity calculation in an Air/NL-MoS₂/SiO₂/Si structure

In the paper, we used the multiple reflection interference method to calculate the reflectivity curves of MoS₂ on SiO₂/Si substrates in the range of 500-750nm and studied substrate-related interference effect. The Air/NL-MoS₂/SiO₂/Si structure contains air (${\tilde{n}}_0$), NL-MoS₂ (${\tilde{n}}_1$ [19], $d_1$), SiO₂(${\tilde{n}}_2$ [20], $d_2$), Si(${\tilde{n}}_3$ [20], $d_3$), where ${\tilde{n}}_i$ and $d_i$ (i=0,1,2,3) are the complex refractive index and the thickness of each medium. The light in this four-layer structure undergoes multiple reflection at the interfaces $i$ and $j$, and optical interference within the medium $j$, as shown in Fig. 8(a). The calculation is based on classical electrodynamics and on the transfer matrix formalism. We assume normal incidence in the z direction. The transmission and reflection of total electric and magnetic fields in the four-layer structure can be described by characteristic matrices $A_{ij}$ and $B(z_j)$, where $A_{ij}$ describes the propagation across the interface from $i$ to $j$ layer applying the boundary conditions, and $B(z_j)$ denotes the propagation through the $j$ layer at depth $z_j$. The transverse electric field component and transverse magnetic field component are all perpendicular to the graphene c-axis, and they are associated by $\overrightarrow{H}=\tilde{n}\overrightarrow{E}$. $A_{ij}$ and $B(z_j)$ can be expressed as follows:

$$A_{ij}=\frac{1}{t_{ij}}\left(\begin{array}{cc}1& r_{ij}\\ r_{ij}& 1\end{array}\right),B(z_j)=\left(\begin{array}{cc}{e}^{i{\delta }_j}& 0\\ 0& e^{-i{\delta }_j}\end{array}\right).$$

Here, $t_{ij}$ and $r_{ij}$ are transmission and reflection coefficients from the medium $i$ to the medium $j$ respectively. $t_{ij}=\frac{2{\tilde{n}}_i}{{\tilde{n}}_i+{\tilde{n}}_j}$, $r_{ij}=\frac{{\tilde{n}}_i-{\tilde{n}}_j}{{\tilde{n}}_i+{\tilde{n}}_j}$. ${\delta }_j=2\pi {\tilde{n}}_j{d}_j/\lambda$ are phase factors.

 

Fig. 8 Schematic diagrams of (a) multiple reflection and optical interference and (b) the electric field component transfer process in the Air/NL-MoS₂/SiO₂/Si structure.

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The incident electric field component passes through interfaces of Air/NL-MoS₂, NL-MoS₂/SiO₂ and SiO₂/Si, and finally is absorbed by the Si layer. Meanwhile, the reflective electric field component is collected from each interface and finally transmits into the Air. Schematic diagram of the electric field component transfer process is shown in Fig. 8(b). The total transfer matrix equation in this four-layer structure is:

$$\left(\begin{array}{c}{E}_{Air}^{+}\\ E_{Air}^{-}\end{array}\right)=A_{01}B(d_1)A_{12}B(d_2)A_{23}\left(\begin{array}{c}{E}_{Si}^{+}\\ 0\end{array}\right).$$

The reflectivity can be expressed as: $R=r{r}^{\ast }$, where $r=E_{Air}^{-}/E_{Air}^{+}$. $E_{Air}^{+}$ and $E_{Air}^{-}$ are incident and reflected electric field component into or out of the NL-MoS₂ flakes. The calculated $R$ values of the bare 100nm SiO₂/Si substrate (with N=0, denoted as black curve) and 1L-5L, 9L, 10L MoS₂ flakes on 100nm SiO₂/Si substrate (denoted as colorized curves) with d_SiO2=100 nm and λ=500-750 nm are showed in Fig. 9 to compare with the measured values. The calculation results are basically consistent with the experiment. In fact, it was limited when we take ${\tilde{n}}_1$ as the complex refractive index of bulk MoS₂. Some articles have reported that the complex refractive index of bulk MoS₂ is different from that of 1L MoS₂ [21]. However, quantitative analysis of the calculated emission wavelength of A and B excitons of ultrathin NL-MoS₂ flakes on SiO₂/Si substrate was difficult because there exist abundant features in the λ dependent ${\tilde{n}}_1$ for MoS₂ flakes and ${\tilde{n}}_1$ of MoS₂ flakes itself significantly depends on N due to the indirect-to-direct transition from multilayer to 1L. We can only give a qualitative explanation about the energy difference between A and B direct excitonic transitions. Due to the spin-orbital splitting of the valence band at the Brillouin zone K point, the emission wavelength of A and B excitons as a function of layer number was derived from the change of direct excitonic states with layer thicknesses.

 

Fig. 9 (a) The experimental and (b) the calculated reflectivity values of the bare 100nm SiO₂/Si substrate (with N = 0, denoted as black curve) and 1L-5L, 9L, 10L MoS₂ flakes on 100nm SiO₂/Si substrate (denoted as colorized curves).

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On the other hand, the calculations revealed that there is an optimized SiO₂ thickness to avoid substrate-related interference effect influencing the investigation of exciton properties. The optimized SiO₂ thickness is 50-100 nm.

Funding

Youth Project of the National Natural Science Foundation of China (11504077); National Natural Science Foundation of China (61774053); Youth Project of Hebei Province Natural Science Foundation (A2017201012); Key Project of Hebei Province Department of Education Fund (ZD2017007).

Acknowledgements

The authors are grateful to P.H.Tan for the Raman spectra measurements.

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