Measuring the Refractive Index of Highly Crystalline Monolayer MoS2 with High Confidence

Monolayer molybdenum disulphide (MoS2) has attracted much attention, due to its attractive properties, such as two-dimensional properties, direct bandgap, valley-selective circular dichroism, and valley Hall effect. However, some of its fundamental physical parameters, e.g. refractive index, have not been studied in detail because of measurement difficulties. In this work, we have synthesized highly crystalline monolayer MoS2 on SiO2/Si substrates via chemical vapor deposition (CVD) method and devised a method to measure their optical contrast spectra. Using these contrast spectra, we extracted the complex refractive index spectrum of monolayer MoS2 in the wavelength range of 400 nm to 750 nm. We have analyzed the pronounced difference between the obtained complex refractive index spectrum and that of bulk MoS2. The method presented here is effective for two-dimensional materials of small size. Furthermore, we have calculated the color contour plots of the contrast as a function of both SiO2 thickness and incident light wavelength for monolayer MoS2 using the obtained refractive index spectrum. These plots are useful for both fundamental study and device application.

I n recent years, the discovery of various kinds of two-dimensional (2D) materials [1][2][3][4] has promoted lowdimensional physics research. Among these materials, monolayer MoS 2 , a stable atomically thin structure with honeycomb lattice, has attracted much attention, because of its remarkable physical properties and novel applications, such as direct bandgap 2 , strong spin-orbit coupling 5,6 , valley-selective circular dichroism 7-10 , valley Hall effect 11,12 , nonlinear optical effect 13 , and two-dimensional heterostructures [14][15][16][17] . Recent progress in the synthesis of highly crystalline monolayer MoS 2 with chemical vapor deposition (CVD) method [18][19][20] has made it a promising candidate for novel electronic and optoelectronic devices. Its complex refractive index in visible range is important, because many of its novel properties are closely related to this wavelength range. Beal et al. measured the complex refractive index of bulk MoS 2 21 in 1979. However, physical properties of a 2D material are usually very different from those of bulk material, especially for MoS 2 , which has an indirect bandgap in bulk form and a direct bandgap in monolayer form 2 . Recently several groups have measured the optical constants of large-area CVD grown thin films of MoS 2 or other transition metal dichalcogenides using spectroscopic ellipsometry technique [22][23][24] . However, so far, it is still difficult to measure the refractive index of highly crystalline monolayer MoS 2 directly, because the highly crystalline monolayer MoS 2 flakes obtained by present methods (mechanical exfoliation, CVD etc.), are usually too small in size (, tens of microns). In 2007, Blake et al. successfully visualized graphene under an optical microscope by utilizing the contrast of graphene on a SiO 2 /Si substrate. Given the refractive index of graphene, the contrast can further be calculated based on the Fresnel law 25 . In our work, by improving the spatial resolution of a reflectance spectrum system via spatial filtering, we managed to obtain the optical contrast spectra of highly crystalline monolayer MoS 2 . Then by curve fitting the relations between the contrast and SiO 2 thickness based on the Fresnel law, we obtained the complex refractive index spectrum of monolayer MoS 2 in the wavelength range of 400 nm to 750 nm, together with the confidence interval, which reflects the accuracy of the obtained refractive index. We have analyzed the pronounced difference between the obtained complex refractive index spectrum and that of bulk MoS 2 . Furthermore, we have calculated the color contour plots of the contrast as a function of both SiO 2 thickness and incident light wavelength for monolayer MoS 2 using the obtained refractive index spectrum. These plots are useful for both fundamental study and device application.

Results
Synthesis and characterization of highly crystalline monolayer MoS 2 . In this work, we synthesized monolayer MoS 2 on various SiO 2 /Si substrates by CVD method. Figures 1a, b show two typical optical images of the material grown on 280 nm SiO 2 /Si substrates. The monolayer MoS 2 tend to form isolated triangular islands, with grain sizes ranging from a few microns to tens of microns. With increased growth time, some monolayer MoS 2 islands overlap with one another to form bilayer or multi-layer structures as indicated with the arrows in Fig. 1b. The thickness of a MoS 2 triangular island depicted in Fig. 1c, measured by an atomic force microscope (AFM), is about 0.79 nm, consistent with data from literatures 18,26 . The photoluminescence (PL) spectra of both monolayer and bilayer MoS 2 are shown in Figure 1d. Raman signals also appear in the PL spectra at higher energy, details of which are shown in the inset. For monolayer MoS 2 , the PL spectrum is dominated by two peaks around 1.85 eV and 2.00 eV, which come from the A and B exciton transitions, respectively. The A and B excitons form at the Kpoint of the Brillouin zone, where strong spin-orbit coupling induces a splitting in the opposite spin valence bands by about 150 meV 2,10,27 . For bilayer MoS 2 , the corresponding two exciton transition peaks are much weaker and exhibit a small red shift 27,28 . The Raman spectrum of monolayer MoS 2 consists of the E 1 2g (384.3 cm 21 ) and A 1g (404.6 cm 21 ) modes. Peak distance (D) of them is about 20.3 cm 21 . The E 1 2g and A 1g Raman modes of bilayer MoS 2 peak at 382.9 cm 21 and 404.8 cm 21 respectively, with a D of about 21.9 cm 21 . Peak distance between these two modes has been used to identify the number of layers of MoS 2 29 . For monolayer and bilayer MoS 2 , the D values are about 18,20 cm 21 and 21,22 cm 21 , respectively 19,29 . Our results are consistent with these criteria.
We characterized crystal structure of the monolayer MoS 2 with a high-resolution transmission electron microscope (HRTEM), as shown in Fig. 1e. We can see clearly the {100} planes of MoS 2 , with a lattice spacing of about 0.270 nm 18 . The HRTEM image, together with the corresponding selected area electron diffraction (SAED) shown in the inset, demonstrates that the monolayer MoS 2 is a single crystal with a hexagonal lattice structure.
Measuring the contrast spectra of monolayer MoS 2 on SiO 2 /Si substrates. The schematic diagram of the experimental set-up for measuring the contrast spectrum of monolayer MoS 2 on a SiO 2 /Si substrate is shown in Fig. 2a. An optical microscope equipped with a white light source was used to collect the reflected light signal from the sample. An optical fiber was used to selectively couple part of the reflected light signal into a spectrometer. In our case, the spatial resolution of the reflectance spectrum, which depends on both the objective lens magnification and the fiber diameter, is about 1 mm under 503 objective lens, smaller than the size of the monolayer MoS 2 islands. Figure 2b shows the reflectance spectra measured on monolayer MoS 2 grown on a 280 nm SiO 2 /Si substrate and a bare 280 nm SiO 2 /Si substrate respectively, for wavelengths ranging from 400 nm to 750 nm. Corresponding measured sites are labeled as A and B in the inset. The contrast of monolayer MoS 2 on a SiO 2 /Si substrate can be defined as: where I MoS2 and I substrate are the reflected light intensities from MoS 2 and the substrate, respectively. From the two reflectance spectra, we can calculate the contrast spectrum, which is also shown in Fig. 2b.  We assume normal incidence in the analysis of this work. We need to select a proper objective lens in order to minimize the error caused by the spatial Fourier components of the beam away from normal incidence 30 . On one hand, this requires the numerical aperture (N. A.) to be as small as possible. On the other hand, the lens with a smaller N. A. usually has smaller magnification, which in return reduces spatial resolution. Figures 2c shows three contrast spectra of monolayer MoS 2 on a 280 nm SiO 2 /Si substrate, which were obtained by using objective lenses with N. A. of 0.2 (103), 0.55 (503), and 0.75 (1003), respectively. We can see that, apart from the spectrum collected by the lens with N. A. of 0.75, the contrast spectra collected by the other two lenses are similar, which means that the error caused by the beam away from normal incidence can be ignored when the N. A. is less than 0.55. Therefore, we used the 503 objective lens in our experiment to achieve optimal results.
Fitting the complex refractive index of monolayer MoS 2 . For normal light incidence, the reflected light intensity from a sample with the geometric structure shown in the inset of Fig. 2a can be written as 25 :  (1) and (2), the contrast of monolayer MoS 2 on a SiO 2 / Si substrate as a function of both SiO 2 thickness (d 2 ) and complex refractive index of monolayer MoS 2 (n 1 ,k 1 ): In order to obtain the complex refractive index spectrum of monolayer MoS 2 , we measured contrast spectra of 26 monolayer MoS 2 samples on SiO 2 /Si substrates with SiO 2 thickness ranging from ,130 nm to ,370 nm ( Supplementary Fig. S1). From these contrast spectra, we extracted the relation between the contrast (C) and SiO 2 thickness (d 2 ) under a specific incident light wavelength, e.g. 651 nm, as shown in Fig. 3a (scattered open circles). The standard deviation of each point, indicated by its error bar, was obtained by multiple measurements. By curve fitting these data with equation (3) (the solid line), we obtained the complex refractive index of monolayer MoS 2 (n 1 ,k 1 ) at 651 nm. Similarly, we can obtain the C-d 2 relations under different incident light wavelength (from 400 nm to 750 nm). The resulting complex refractive index spectrum of monolayer MoS 2 is shown in Fig. 3b (scattered red circles), together with the confidence interval (red shadow), which corresponds to a 95% confidence level. The overall confidence interval is narrow, especially in the wavelength range from 550 nm to 700 nm, indicating that the result is with high confidence. We expect uniformity in sample properties, such as MoS 2 thickness, crystalline quality, and surface flatness, to contribute to the confidence interval. Therefore the narrow confidence interval in the wavelength range from 550 nm to 700 nm indicates good sample uniformity, which can be attributed to the stable double-temperature-zone CVD synthesis method employed in this work as well as the fundamental thickness uniformity of the layered 2D material (the thickness can only be one, two, or more mono-layers). The observed gradually broadened confidence interval at wavelengths below 450 nm and above 700 nm may result mainly from the poor signal-noise ratio of the measuring system in those wavelength ranges. The broadened confidence interval at the wavelength range of 450-525 nm may result from the poor signal-noise ratio of the measured optical contrast, which is related to the SiO 2 thickness employed. We can see in Fig. S1 that the absolute optical contrast of most of our samples are weaker in the wavelength range of 450-525 nm.
For comparison, we also plotted the complex refractive index spectrum of bulk MoS 2 21 in this figure with the black dashed lines. We can see that, for most wavelengths, the real and imaginary parts of the refractive index of monolayer MoS 2 are both lower than their bulk counterparts. In addition, for monolayer MoS 2 , two peaks of the real part of the refractive index spectrum, located at 663 nm and 621 nm, exhibit a blue shift compared to their bulk counterparts (at 684 nm and 626 nm, respectively). Similarly, two peaks of the imaginary part (at 651 nm and 603 nm) also show a blue shift compared to their bulk counterparts (at 671 nm and 611 nm, respectively). Given that the imaginary part represents the electromagnetic wave absorption in a material, we attribute the two peaks of the imaginary part of the refractive index spectra to A and B exciton absorptions, respectively. The observed blue shifts of exciton absorption of monolayer MoS 2 compared to their bulk counterparts may be due to the difference of A and B excitons energy in bulk and monolayer MoS 2 2,27 . It is worth noting that, compared to the exciton emission peaks (located at 1.85 eV and 2.00 eV) shown in Fig. 1d, the observed A and B exciton absorption peaks of monolayer MoS 2 (located at 651 nm and 603 nm, i.e. 1.90 eV and 2.06 eV) also show a blue shift. This phenomenon results from the well-known Stokes shift between absorption and emission 32 . There also exists a peak near 400 nm for the imaginary part spectrum of monolayer MoS 2 , which may be related to the convoluted C and D excitons 28,33 . The reason why the peak located around 480 nm of the real part of refractive index spectrum for the bulk MoS 2 splits into two peaks for monolayer case is still unknown, and needs further study.
In order to demonstrate the difference of the refractive indices for monolayer and bulk MoS 2 from another point of view, we calculated three representative contrast spectra of monolayer MoS 2 on SiO 2 /Si substrates, with SiO 2 thicknesses of 262, 281, and 150 nm, respectively, labeled as samples (1)-(3). The results are plotted (solid lines) together with those calculated using the refractive index of bulk MoS 2 (dashed lines) in Fig. 4a. Figure 4b shows the optical images of samples (1)-(3), respectively. The optical images show different colors as a result of the different reflectance spectra of the three samples. We also took the monochromatic images of the samples by inserting narrow-band filters into the illuminating optical path. As an example, three monochromatic images of sample (1) are shown in Fig. 4c. The images show quite different contrast, the blue one having the worst contrast, and the green one having the best. From the converted grayscale images, we obtained the contrast data directly. The obtained contrast data for all the three samples are plotted in Fig. 4a (scattered open circles). We can see that the measured contrast data agree well with those calculated with the refractive index of monolayer MoS 2 (solid lines).
Using refractive indices of monolayer and bulk MoS 2 , we calculated the color contour plots of the contrast as a function of both SiO 2 thickness and incident light wavelength for monolayer MoS 2 on SiO 2 /Si substrate. The results were plotted in Fig. 4d under the same color bar. The clear difference between these two plots confirms again that the difference between the refractive indices of monolayer and bulk MoS 2 cannot be ignored. For both fundamental study and device application, it is useful to make the sample visible under an optical microscope. It is common to use 280 nm-300 nm SiO 2 /Si substrates for monolayer MoS 2 in order to visualize it. In principle, with this color contour plot, one can make the monolayer MoS 2 visible on any SiO 2 /Si substrate by selecting the proper incident light wavelength, and vice versa. For convenient to use, we replotted the color contour plot in Fig. 4e using a more appropriate color bar.

Discussion
In this work, we have synthesized highly crystalline monolayer MoS 2 on 26 different SiO 2 /Si substrates with SiO 2 thickness ranging from ,130 nm to ,370 nm, and have devised a method to measure the contrast spectra of monolayer MoS 2 on these substrates. Using these contrast spectra, we extracted the complex refractive index spectrum of monolayer MoS 2 in the wavelength range of 400 nm to 750 nm. We have analyzed the pronounced difference between the obtained complex refractive index spectrum and that of bulk MoS 2 . Furthermore, we have calculated the color contour plots of the contrast as a function of both SiO 2 thickness and incident light wavelength for monolayer MoS 2 using the obtained refractive index spectrum. These plots are useful for both fundamental study and device application. The measurement method presented here, with the advantage over conventional methods for 2D materials with small size, can be applied to other 2D materials which can be synthesized with good repeatability.

Methods
Preparation of SiO 2 /Si substrates. We chemically etched the thermally oxidized SiO 2 capping layers on the Si substrates to various thicknesses (from 130 nm to 370 nm) using buffered hydrogen fluoride (HF) (HF (40%): NH 4 F (8 M) 5 1:10). The original thickness of the SiO 2 capping layer was 600 nm. The whole etching process was carried out in ice-bath with magnetic stirring. The thickness of the SiO 2 layer was measured by a thin film thickness measurement system (SpectraThick Series, ST2000-DLXn) with repeatability of 0.5 nm. The refractive index of the p-Si substrate (resistivity: 8-12 V?cm) was measured by an ellipsometer (Horiba Jobin Yvon Uvisel) (shown in Fig. S2).
Synthesis of monolayer MoS 2 . We synthesized monolayer MoS 2 on SiO 2 /Si substrates with a double-temperature-zone CVD method at ambient pressure. The MoO 3 (99.99%) and S (99.999%) powders, serving as the source, were loaded onto two quartz boats, respectively, which were later inserted into a 1-inch diameter quartz tube placed in a tubular furnace. The MoO 3 located at the downstream of high-purity argon (Ar) carrier gas. SiO 2 /Si substrates were placed with faces down above MoO 3 . During the synthesis process, the temperatures at MoO 3 and S sources were first raised to 100uC and kept there for 1 h with Ar gas flow rate of 120 sccm to exhaust water and air. Then the temperatures at MoO 3 and S were ramped to 650uC and 220uC, respectively, in 40 min, and kept there for 10 min, with Ar gas flow rate of 10 sccm. After that, the furnace was cooled down to room temperature without feedback with Ar gas flow rate of 10 sccm.
Characterization of monolayer MoS 2 . The thickness and crystal structure of monolayer MoS 2 were characterized by AFM (Bruker Dimension Icon-PT) and TEM (Tecnai F30), respectively. For TEM sample preparation, we first spun a layer of poly (methyl methacrylate) (PMMA) onto a MoS 2 /SiO 2 /Si sample. Then we etched off the SiO 2 layer by KOH aqueous solution (2 M) and cleaned the floating PMMA/MoS 2 membrane several times with deionized water. Finally the membrane was scooped onto a TEM grid and dried. The PMMA was removed by annealing the TEM sample at 400uC for 3 h in an Ar/H 2 ambient. The PL and Raman spectra of monolayer MoS 2 were measured by a confocal Raman microscopic system (Horiba Labram HR800) at room temperature. The excitation laser wavelength was 488 nm. Spatially resolved spectrum system. The spatially resolved reflectance spectrum was measured by using an optical microscopic spectrum system, which included an optical microscope (Zeiss Axio Imager. A2m) equipped with a halogen lamp (Zeiss Hal 100, 12 V, and 100 W), and a spectrometer (Horiba Jobin Yvon Triax 320). One end of an optical fiber was placed at the image plane of the microscope to selectively couple part of the light signal there to the spectrometer. The diameter of the optical fiber is 9 mm.
Obtaining Monochromatic Images. Monochromatic image was obtained by inserting a color filter with bandwidth of ,10 nm into the illuminating optical path of the optical microscope. The central wavelengths of the three filters used in Fig. 4 are 445 nm, 526 nm, and 672 nm, respectively.