## Abstract

In-plane anisotropic two dimensional (2D) layered materials, such as ReS_{2}, offer one more degree of freedom than isotropic 2D materials to deliver various physical properties. Optical properties of ReS_{2} flakes are dependent on the layer number (N) from monolayer to multilayer. Here, taking ReS_{2} flakes with an anisotropic-like (AI) stacking order as an example, we probed their optical properties by optical contrast (OC) measurements. The N-dependent and angular-dependent characteristics of ReS_{2} flakes were demonstrated respectively by unpolarized and polarized OC curves. The results facilitate further study on out-of-plane and in-plane complex refraction index properties of ReS_{2} flakes.

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

## 1. Introduction

Two dimensional (2D) materials have attracted great interest, due to their excellent physical properties and remarkable device prospects [1–4]. There is a new class of 2D materials, the transition metal dichalcogenides (TMDs) with the chemical formula MX_{2}. A single monolayer of MX_{2} consists of an atomic plane of a transition metal (M: Ti, Mo, Re, etc.) sandwiched between two chalcogen atomic planes (X: S, Se, Te, etc). This crystalline structure is strongly governed by the group of the transition metal [5] and gives rise to diverse electronic properties [6]. One kind of TMDs has disordered 1T'-structure with representative materials as ReS_{2} [3,4] and ReSe_{2} [7], unlike the commonly seen highly symmetric 1T- or 2H-structure TMDs, e.g., MoS_{2} [8], WS_{2} [8], and WSe_{2} [8]. These disordered 1T'-structure TMDs usually have extra in-plane metal-metal bonds or charge density wave states and exhibit strong in-plane anisotropy in the electronic, vibrational and mechanical properties [3,4,9], regarded as more promising materials for future applications in electronic, optoelectronic and thermoelectric devices [3,4].

For 2D materials, two or more layers can be stacked to form multilayer. These layers are coupled with each other by Van der Waals (vdW) interactions. Interlayer interaction plays an important role in varying physical properties of multilayer 2D materials [10–13]. For example, ReS_{2} remains direct band gap from monolayer to the bulk [13], in contrast to the direct-to-indirect band gap in other TMD materials, e.g. MoS_{2} [10], WS_{2} [12], and WSe_{2} [12]. Moreover, the presence of interlayer coupling in ReS_{2} layers makes their properties dependent on layer number [14–16]. In this paper, we probed optical properties of ReS_{2} flakes by unpolarized and polarized OC measurements. Their N-dependent and angular-dependent characteristics were demonstrated respectively and their out-of-plane and in-plane complex refraction index were studied. 2D layered materials with high anisotropy are currently gaining an increasing interest for polarization-integrated nanodevice applications [17]. The tunable in-plane and vertical optical behaviors in ReS_{2} flakes with different N will expedite further investigations of novel optoelectronic devices.

## 2. Materials and methods

#### 2.1 Preparation and layer number identification of ReS_{2} flakes

Due to the visualization by optical microscopy and facilitating microfabrication of devices in the future research, ReS_{2} flakes were obtained by micromechanical cleavage of a bulk ReS_{2} crystal (2D semiconductors Inc.) on Si(110) substrate covered with a 90-nm SiO_{2}. The layer number and stacking order of ReS_{2} flakes were estimated by ultralow-frequency Raman measurements [14–16]. The shear (S) and breathing (B) modes correspond to the relative parallel and perpendicular motions of the planes themselves, whose frequencies are related to layer number and stacking order of ReS_{2} flakes [14–16]. We selected some flakes with the most stable stacking order, i.e., anisotropic-like (AI) structure [14]. They contained 1L-7L ReS_{2} flakes, whose optical image were shown in Fig. 1(a). The flakes with different N display different colors. The size of samples was above 5 μm to ensure the accuracy of subsequent tests. Details of the S and B modes of these ReS_{2} flakes are presented in Appendix.

#### 2.2 Measuring unpolarized and polarized reflection spectra of ReS_{2} flakes

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 100X (NA = 0.9). Tungsten halogen lamp was used as a light source, with the spot size below 2 μm. The reflection spectra were measured from the samples and bare substrate 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 signals were achieved by focusing the microscope. Schematic diagram of the microscope system to measure reflection spectra was shown in Fig. 1(b).

When measuring the polarized reflection spectra, a polarizer was placed on the light path in front of the sample, which can be seen in schematic diagram of Fig. 1(b). By continuously rotating the polarizer, polarization directions of incident and reflected light were simultaneously varied with polarization angles from 0° to 360°. When measuring the unpolarized reflection spectra, the polarizer on the light path was taken off.

#### 2.3 OC method of NL-ReS_{2} flakes

There is a commonly used reflection spectrometric method, i.e., OC, to probe optical properties of 2D flakes related to layer number [18–21]. The OC of NL-ReS_{2} flakes is defined as $1-{R}_{NL-\mathrm{Re}{S}_{2}}(\lambda )/{R}_{sub}(\lambda )$, that is, the difference between reflected light intensities from NL ReS_{2} flakes deposited on substrate and from the bare substrate divided by reflected light intensity from the bare substrate. OC is dependent on the wavelength (λ) of the light source and the substrate.

## 3. Results and discussion

#### 3.1 Studying N-dependent characteristics by the unpolarized OC method

In order to demonstrate N-dependent properties of ReS_{2} flakes, we first measured unpolarized reflection spectra of 1L-7L ReS_{2} flakes supported on SiO_{2}/Si substrate under the incidence of natural light emitted by tungsten halogen lamp. The integral optical properties of ReS_{2} layers can be obtained with no concern for its in-plane anisotropy. The OC curves of NL-ReS_{2} were shown in Fig. 2(a). All these curves show a peak at long wavelength and a valley at short wavelength, whose wavelength are dependent on N. First, the peaks show redshift with the increase of N from 3 to 7. When N is from 1 to 2, the peaks show almost no shifting. The peak positions of NL ReS_{2} are summarized by black triangles in Fig. 2(b). The peaks locate at the same wavelength of ~610 nm in 1L-ReS_{2} and 2L-ReS_{2} flakes, and they shift from ~621 nm in N = 3 to ~780 nm in N = 7. Second, the valleys show redshift with the increase of N from 3 to 7. When N is from 1 to 2, the valleys are weaker. The valley positions of NL ReS_{2} are summarized by blue circles in Fig. 2(b). The valleys locate at the same wavelength of ~438 nm in 1L-3L ReS_{2} flakes. As N increases from 3 to 7, the valley gradually shifts from ~438 nm to ~497 nm. We specially presented the OC curves of 1L-3L ReS_{2} flakes in the inset of Fig. 2(a) for comparison. More details can be seen in the inset.

In addition, there exists a significant discrepancy of positive (peak) and negative (valley) amplitude in the OC curves of NL ReS_{2} with different N. The peak becomes stronger and the valley becomes deeper with the increase of N. At short wavelength, the valley value increases from 7% in 1L to 16% in 3L, and decreases from 9% in 4L to −67% in 7L. However, at long wavelength, the peak value increases from 1L to 7L, which is ~43% in 1L-ReS_{2}, and increases with N, for example, ~56%, ~71%, ~76%, ~80%, ~83%, ~85% in 2L-7L. The approach also provides a reference for layer-number identification of ReS_{2} flakes on SiO_{2}/Si substrate. It is easy to be used to avoid additional non-standard and expensive instruments in determination of layer number of ReS_{2} flakes by Raman spectroscopy or photoluminescence spectroscopy methods [14–16].

#### 3.2 Studying angular-dependent characteristics by the polarized OC method

Due to strong in-plane anisotropy of ReS_{2} flakes, we further studied their angular-dependent properties by polarized OC method. Taking a 2L-ReS_{2} flake supported on SiO_{2}/Si substrate for example, the sample was mounted on a test platform with the substrate placed in a fixed direction of the upward orientation of (110). We defined this direction as polarization angle 0°, and used it as a starting point to turn the polarizer counterclockwise every 10° for polarized reflection spectra measurements. The Si substrate and the ReS_{2} flake have similar polarization of the form of sin or cos function with the period of π. However, there is a certain angle between the polarization direction of the substrate and that of the ReS_{2} flake. We measured the reflection spectra from both the 2L-ReS_{2} flake and bare substrate for every revolution of the polarizer, and calculated the polarized OC curves by the formula. The polarized OC curves reflected the individual polarization properties of the ReS_{2} layers. Figure 3(a) shows a series of polarized OC curves of 2L-ReS_{2} flake with varying polarization angles from 0° to 180° and the curves are offset for clarity. There are a peak at ~610 nm and a valley at ~438 nm, which show semblable features with unpolarized OC curve of 2L-ReS_{2} flake. The peak and the valley show no significant position change at different polarization angles. The intensities at 610 nm show no significant fluctuation. However, the feature at ~438 nm shows alternate intensity of ups and downs, which are marked by stars. The polar plots of the intensities of the polarization dependent OC values at 610 nm and 438 nm are displayed in Fig. 3(b) by the black and red dots, respectively, with varying polarization angles from 0° to 360°. The intensities at both 610 nm and 438 nm change in the form of sin or cos function with the period of π. The minimum and maximum values at both 610 nm and 438 nm occur at the polarization angle of 40° and 130°, respectively. However, the intensities at 438 nm exhibit stronger fluctuation than those at 610 nm as polarization angle varies. The intensities at 610 nm show amplitude variations from 44% to 62%, while at 438 nm show amplitude variations from 4% to 34%. Schematic diagram of polarization directions with the smallest and largest OC is shown in Fig. 3(c), in which the sample placement is marked.

#### 3.3 Analysis of the complex refraction index of ReS_{2} flakes

The origin of OC is the multiple reflection interference in Air/NL-ReS_{2}/SiO_{2}/Si structure [18–21]. In this model, complex refraction index ($\tilde{n}=n-ik$) of ReS_{2} is an important and key parameter [18–20]. If $\tilde{n}$ is provided, the OC curves of NL-ReS_{2} can be calculated by the multiple reflection interference method [18–20]. However, there are hardly relevant reports about them. In turn, we can give analysis of refractive properties of ReS_{2} by their OC curves. From the unpolarized OC curves of 1L-7L ReS_{2} flakes on SiO_{2}/Si substrate, the integral complex refraction index properties of ReS_{2} along the layer thickness direction can be studied with no concern for its in-plane anisotropy. Moreover, from the polarized OC curves of 2L-ReS_{2} flake, the complex refraction index properties of ReS_{2} with in-plane anisotropy were studied.

The Air/NL-ReS_{2}/SiO_{2}/Si structure contains air(${\tilde{n}}_{0}$), NL-ReS_{2}(${\tilde{n}}_{1}$, ${d}_{1}$), SiO_{2}(${\tilde{n}}_{2}$ [22], ${d}_{2}$), Si(${\tilde{n}}_{3}$ [22], ${d}_{3}$), where${\tilde{n}}_{i}$ and ${d}_{i}$ (i = 0,1,2,3) are the complex refractive index and the thickness of each medium. The incident light passes through interfaces of Air/NL-ReS_{2}, NL-ReS_{2}/SiO_{2} and SiO_{2}/Si, and finally is absorbed by the Si layer. Meanwhile, the reflected light is collected from each interface and finally transmits into the air. Assuming that both incident light and reflected light are perpendicular to the in-plane ReS_{2} atoms, schematic diagram of transmission process of electric field components in the four-layer structure is shown in Fig. 4(a). The OC is expressed as $1-{R}_{NL-\mathrm{Re}{S}_{2}}(\lambda )/{R}_{sub}(\lambda )$, where $R=r{r}^{\ast}$, $r={E}^{-}/{E}^{+}$. ${E}_{NL-Re{S}_{2}}^{+}$ and ${E}_{NL-Re{S}_{2}}^{-}$ are incident and reflected electric field component into or out of the NL-ReS_{2} flake, while ${E}_{Si{O}_{2}}^{+}$ and ${E}_{Si{O}_{2}}^{-}$are incident and reflected electric field component into or out of the bare substrate. Details of calculations are described in Appendix. Once the other parameters in the system are determined, $\tilde{n}$ of ReS_{2} can be calculated by inverse solutions of OC equations [18–20].

In the unpolarized OC curves of 1L-7L ReS_{2} flakes on SiO_{2}/Si substrate, the valley at short wavelength and the peak at long wavelength exhibit redshift from 1L to 7L. This is because the phase changes in ReS_{2} layers due to the effect of the increasing optical path as N increases in the multiple reflection interference of Air/NL-ReS_{2}/SiO_{2}/Si structure. The negative OC occurs in 5L-7L. This can be understood that 5L-7L samples are so thick that the reflections from their surface become more intense than that from the SiO_{2}/Si substrate, resulting in negative value. Considering negative OC occurs only at short wavelength range, $\tilde{n}$ of ReS_{2} at short wavelength region is different from that at long wavelength region. Thus, taking ~550nm as a division point, $\tilde{n}$ of ReS_{2} just slowly changes the reflectivity of the air-ReS_{2} interface at long wavelength, while at short wavelength it induces that the reflectivity of the air-ReS_{2} interface first increases and saturates, then decreases, and finally varies from positive to negative. According to the valley and peak values of 1L-7L ReS_{2}, $\tilde{n}$ of ReS_{2} was calculated by inverse solutions of OC equations from adjacent two ReS_{2} layers and finally the effective values ${\tilde{n}}_{eff}$ at the valley and peak were determined based on the average of 6 data points. Here, the integral effect of both the s-polarization and p-polarization (perpendicular to the NL-ReS_{2} axis) electric field components were calculated. The complex refraction index of ReS_{2} was taken as (4.0 ± 0.3)-i(1.5 ± 0.3) at short wavelength and (2.0 ± 0.3)-i(2.5 ± 0.3) at long wavelength, which embodied the total refraction properties along the layer thickness direction regardless of in-plane anisotropy.

The in-plane anisotropy of complex refraction index of ReS_{2} was studied by the polarized OC curves of 2L-ReS_{2} flake. The polarization directions with the smallest and largest OC are at the polarization angle of 40° and 130° respectively. In order to facilitate the calculation, we defined these two directions as the s-polarization and p-polarization of electric field component respectively, which were marked in Fig. 4(b). Because the complex refraction index $\tilde{n}$ of ReS_{2} calculated in the previous section was the result of average value from both the s-polarization and p-polarization directions, we defined the complex refraction index of ReS_{2} in the s-polarization and p-polarization direction as $\tilde{n}\pm \Delta \tilde{n}$ respectively. According to the OC curves of 2L-ReS_{2} flake at the polarization angle of 40° and 130° as shown in Fig. 4(b), $\Delta \tilde{n}$ of ReS_{2} was calculated by inverse solutions of OC equations from these two polarization directions. The effective values $\Delta {\tilde{n}}_{eff}$ were determined based on the average of data from short wavelength and long wavelength regions. The difference of complex refraction index between the s-polarization and p-polarization direction was taken as $2\Delta {\tilde{n}}_{eff}$~(0.6 ± 0.05)-i(0.4 ± 0.05).

## 4. Conclusion

Unlike the most TMDs, ReS_{2} flakes have in-plane anisotropic features, which offer them one more dimension than isotropic 2D materials to tune their physical properties. ReS_{2} layers can be stacked to form multilayer ReS_{2} flakes most easily in an AI-stacked structure. These ReS_{2} layers in ReS_{2} flakes are coupled with each other by vdW interaction, which induces ReS_{2} flakes to exhibit different optical properties dependent on the layer number. The N-dependent characteristics of ReS_{2} were investigated by taking 1L-7L ReS_{2} flakes for unpolarized OC measurements. The angular-dependent characteristics of ReS_{2} were studied by taking a 2L-ReS_{2} flake for polarized OC measurements. Finally, we give analysis of refractive properties of ReS_{2} by these OC curves. The results show that complex refraction index of ReS_{2} shows different properties at short wavelength and long wavelength regions in out-of-plane direction, and shows different properties in the s-polarization and p-polarization directions in plane.

## 5 Appendix

#### 7.1 The S and B modes of AI-stacked 1L-7L ReS_{2} flakes

Because the interlayer coupling in multilayer 2D materials is mainly the weak vdW interactions, the frequencies of Raman modes relevant with the interlayer coupling are very low and usually are below 100 cm^{−1}. There are two types of interlayer vibration modes, the S modes and B modes, which correspond to the relative motions of the atom planes parallel and perpendicular to the plane, respectively. The ultralow-frequency Raman spectra were measured at room temperature using a Jobin-Yvon HR800 micro-Raman system equipped with a liquid nitrogen cooled charge coupled device, an objective of 100X (NA=0.9) and a 1800 lines per mm grating. The excitation wavelength was 633 nm from a He–Ne laser with the spot size below 1 μm. The spectral resolution was ∼0.60 cm^{−1}. A typical laser power of 0.3 mW was used to avoid sample heating. The laser plasma lines were removed using a BragGrate bandpass filter. Measurements down to 5 cm^{−1} were enabled by three BragGrate notch filters with optical density 4 and with a FWHM of 5 cm^{−1}.

The S and B modes have been observed in bilayer to multilayer ReS_{2} [14–16]. In a NL-ReS_{2}, there are N-1 S and N-1 B modes, which are denoted as S_{N,N-i} and B_{N,N-i} (i = 1, 2, ..., N-1), where the S_{N,1}(B_{N,1}) (i.e., i = N-1) is the one with highest frequency and S_{N,N-1} (B_{N,N-1}) (i.e., i = 1) is the one with lowest frequency. According to Ref [14], in the anisotropic AI-stacked layered ReS_{2} flakes, the S modes are nondegenerate and each pair of the S modes should be denoted as S^{X} and S^{Y} modes. However, the B modes are degenerate. The frequencies of these modes as a function of N can be calculated by a linear chain model (LCM).

The AI-stacked 1L-7L ReS_{2} flakes supported on SiO_{2}/Si substrates are determined by Raman measurements of the ultralow-frequency S and B modes. Raman spectra of the S and B modes for AI-stacked 1L-7L ReS_{2} are displayed in Fig. 5(a). Here, the B_{N,N-1} phonon branch and the S^{X}_{N,1} and S^{Y}_{N,1} phonon branches are shown. The theoretical and experimental frequencies of S and B modes as a function of N for AI-stacked 1L-7L ReS_{2} are summarized in Fig. 5(b). Here, the theoretical data calculated by LCM are shown by lines [14], and the experiment data are shown by blue open squares, green open circles and red open circles.

#### 7.2 The OC calculation in Air/NL-ReS_{2}/SiO_{2}/Si structure

The OC curves in an Air/NL-ReS_{2}/SiO_{2}/Si structure can be calculated by the multiple reflection interference method, which has been widely used to quantify OC [18–21] of 2D layered materials. The Air/NL-ReS_{2}/SiO_{2}/Si structure contains air(${\tilde{n}}_{0}$), NL-ReS_{2}(${\tilde{n}}_{1}$, ${d}_{1}$), SiO_{2}(${\tilde{n}}_{2}$ [22], ${d}_{2}$), Si(${\tilde{n}}_{3}$ [22], ${d}_{3}$), 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 $j$. The OC calculation is based on classical electrodynamics and on the transfer matrix formalism [19, 20]. 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 c-axis of NL-ReS_{2}, 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.

The OC is expressed as $1-{R}_{NL-\mathrm{Re}{S}_{2}}(\lambda )/{R}_{sub}(\lambda )$, where $R=r{r}^{\ast}$, $r={E}^{-}/{E}^{+}$. ${E}_{NL-Re{S}_{2}}^{+}$ and ${E}_{NL-Re{S}_{2}}^{-}$ are incident and reflected electric field component into or out of the NL-ReS_{2} flake, while ${E}_{Si{O}_{2}}^{+}$ and ${E}_{Si{O}_{2}}^{-}$ are incident and reflected electric field component into or out of the bare substrate. The incident electric field components pass through interfaces, and finally are absorbed by the Si layer. Meanwhile, the reflected electric field components are collected from each interface and finally transmit into the Air. The transfer matrix equation to calculate ${R}_{NL-\mathrm{Re}{S}_{2}}(\lambda )$ is$\left(\begin{array}{c}{E}_{NL-Re{S}_{2}}^{+}\\ {E}_{NL-Re{S}_{2}}^{-}\end{array}\right)={A}_{01}B({d}_{1}){A}_{12}B({d}_{2}){A}_{23}\left(\begin{array}{c}{E}_{Si}^{+}\\ 0\end{array}\right)$, and that to calculate ${R}_{sub}(\lambda )$ is $\left(\begin{array}{c}{E}_{Si{O}_{2}}^{+}\\ {E}_{Si{O}_{2}}^{-}\end{array}\right)={A}_{12}B({d}_{2}){A}_{23}\left(\begin{array}{c}{E}^{\text{'}}{}_{Si}^{+}\\ 0\end{array}\right)$. Here, ${E}_{Si}^{+}$ and ${E}^{\text{'}}{}_{Si}^{+}$ are electric field components absorbed into the Si layer. Once the other parameters in the system are determined, $\tilde{n}$ of ReS_{2} can be calculated by inverse solutions of OC equations.

## Funding

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

## Acknowledgements

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

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