1 Introduction

Since the discovery of mechanical exfoliated graphene in Manchester in 2004 [1, 2], two-dimensional (2D) materials have attracted much interest [3]. However, graphene does not have a native bandgap, leading to the prosperity of other 2D semiconducting materials, such as the transition-metal dichalogenides [4–7]. Transition-metal dichalogenides have demonstrated many extraordinary optical, thermal, magnetic and electrical properties and are ready for various applications [8–10]. Recently, single-layer MoSe2 has been exfoliated onto SiO2/Si substrate. Determined by photoluminescence measurements, MoSe2 displayed good thermal stability with a direct bandgap of 1.55 eV [11]. In addition, single-layer MoSe2 was proved to have electrocatalytic activity [8]. Like graphene, these properties are highly related to thickness [8, 12, 13]. However, 2D MoSe2 produced by mechanical exfoliation not only is single layer but also has a lot of thick flakes. Therefore, it is crucial to find a method to identify the thickness of 2D MoSe2 accurately and rapidly.

Up to now, many methods have been studied, such as atom force microscope (AFM), Raman spectroscopy and optical microscopy (OM) [12–14]. Accurate enough, AFM and Raman spectroscopy [12] are commonly used to measure the thickness of 2D nanosheet, but they consume too much time, which might lead to the pollution of the samples before measurements [4, 13]. Compared with AFM and Raman spectroscopy, the optical technique performs more rapidly and conveniently [10, 15–17].

In this paper, OM method was used to identify the thickness of 2D MoSe2 nanosheet with a few layers. Furthermore, the calculated and measured contrast was integral, which can be used to determine the MoSe2 layer number accurately and rapidly.

2 Theoretical calculation

The key to identifying the thickness of 2D MoSe2 nanosheet rapidly and accurately by OM is to correlate its thickness with its optical contrast. In order to calculate the contrast, it was considered the case of normal light incidence from air onto a three-layer structure consisting of two films (MoSe2/SiO2) on top of a third semi-infinite film (Si), as shown in Fig. 1. The optical contrast (C(λ)) of MoSe2 on SiO2/Si substrate was calculated by Fresnel equation under normal incident conditions [18]

$$C(\lambda ) = \frac{{I_{0} (\lambda ) - I(\lambda )}}{{I_{0} (\lambda )}}$$
(1)

where I 0(λ) = |r 0(λ)|2 and I(λ) = |r(λ)|2 are the reflected light intensities of the SiO2/Si and MoSe2/SiO2/Si system, in which r is complex amplitude reflectance. Both intensities are light wavelength (λ) dependent.

$$r_{0} (\lambda ) = \frac{{r_{2}^{\prime } + r_{3} {\text{e}}^{{ - 2i\delta_{2} }} }}{{1 + r_{2}^{\prime } r_{3} {\text{e}}^{{ - 2i\delta_{2} }} }}$$
(2)
$$r(\lambda ) = \frac{{r_{1} + r_{2} {\text{e}}^{{ - 2i\delta_{1} }} + r_{3} {\text{e}}^{{ - 2i(\delta_{1} + \delta_{2} )}} + r_{1} r_{2} r_{3} {\text{e}}^{{ - 2i\delta_{2} }} }}{{1 + r_{1} r_{2} {\text{e}}^{{ - 2i\delta_{1} }} + r_{2} r_{3} {\text{e}}^{{ - 2i\delta_{2} }} + r_{1} r_{3} {\text{e}}^{{ - 2i(\delta_{1} + \delta_{2} )}} }}$$
(3)

where \(r_{2}^{{\prime }} = \frac{{n_{0} - n_{2} }}{{n_{0} + n_{2} }}\), \(r_{1} = \frac{{n_{0} - n_{1} }}{{n_{0} + n_{1} }}\), \(r_{2} = \frac{{n_{1} - n_{2} }}{{n_{1} + n_{2} }}\), \(r_{3} = \frac{{n_{2} - n_{3} }}{{n_{2} + n_{3} }}\) are the relative indices of refraction. n 0, n 1, n 2 and n 3 are, respectively, the wavelength-dependent complex refractive indices of air, MoSe2, SiO2 and Si,

$$n_{i} = {\text {Re}}(n_{i} ) + \text{Im} (n_{i} )$$
(4)

where \({\text{Re}}(n_{i} )\) is the optical refractive index and \(- \text{Im} (n_{i} )\) is the absorption coefficient (i = 0, 1, 2, or 3). For n 0 (air) and n 2 (SiO2), only the real part of the refractive index was used in the model. δ 1 = 2πd 1 n 1/λ and δ 2 = 2πd 2 n 2/λ are the phase shifts when light passes through MoSe2 and SiO2, respectively. d = N × d 1 is the thickness of the MoSe2 nanosheet, where N is the number of layers, d 1 is the thickness of single MoSe2 layer, and d 2 is the thickness of the SiO2 layer.

Fig. 1
figure 1

Schematic depiction of light reflection in three-layered Fresnel’s law model

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3 Results and discussion

Figure 2 shows the color plots of the calculated single-layer MoSe2 optical contrast on SiO2/Si as a function of incident light wavelength with SiO2 thickness ranging from 0 to 350 nm on Si substrate. From the calculations, it is clear that there are two bands with high, positive contrast for visible light with SiO2 thickness of less than 350 nm. And the two bands with high contrast correspond to the thickness of SiO2 in the range of 50–130 and 170–350 nm roughly. Considering that the red light and the green light are the major part when calculating the contrast, the silicon wafer around 90 and 275 nm should give the largest change of contrast. So, 275-nm SiO2/Si substrates were selected in this experiment.

Fig. 2
figure 2

Color plot of calculated optical contrast as a function of incident light wavelength from 400 to 750 nm and thickness of SiO2 from 0 to 350 nm for single MoSe2 nanosheet. Color bar representing optical contrast

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Figure 3 gives the theoretically calculated results of optical contrast spectra of MoSe2 nanosheet with various atomic planes on SiO2 (275 nm)/Si substrate. It is clear that the contrast value changes a lot along the incident light wavelength, depending on the number of MoSe2 layers, and the most obvious contrast is in the range of green light wavelength. The integration of the contrast values in visible light regions is performed. The average contrast values of 1- to 4-layer MoSe2 are 0.22, 0.33, 0.36 and 0.34, respectively, given by the integrated value divided by the visible light range. It makes the identification of the MoSe2 thickness possible that the contrast value depends on the number of MoSe2 layers.

Fig. 3
figure 3

Theoretically calculated optical contrast of 1- to 4-layer MoSe2 nanosheets as a function of incident light wavelength on SiO2 (275 nm)/Si substrate

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To determine the relation between the spectra contrast and the thickness of the MoSe2 nanosheet, natural MoSe2 crystals with different layers were exfoliated mechanically and transferred onto the freshly cleaned 275-nm-thick SiO2-coated Si substrates. The optical images of the MoSe2 nanosheet were captured at the exposure time of 80 ms by a Nikon MM-400 optical microscopy (OM). Figure 4a shows the surface image of a MoSe2 nanosheet using an OM equipped with a color camera. Then, the sample was imaged using a BRUKER Dimension Icon atomic force microscopy (AFM) to measure the accurate thickness of the MoSe2 nanosheet. Figure 4b, c is the corresponding 2D AFM images of the areas marked by white rectangles in Fig. 4a. The height values measured from the white solid lines (labeled as 1, 2 3, 4) in Fig. 4b, c are 0.72, 1.46, 2.23 and 2.88 nm (Fig. 5a1–a4), respectively, and the thickness of single MoSe2 layer is about 0.72 nm. The thickness value is consistent with that reported by others [19]. These height values obtained in the samples correspond to 1- to 4-layer MoSe2 nanosheets, respectively. Figure 5b1–b4 is the contrast profile of the yellow dashed rectangles highlighted in Fig. 4b, c generated by Image software. The contrast values of the 1- to 4-layer MoSe2 nanosheets are 0.24, 0.33, 0.38 and 0.35, respectively. In addition to AFM, the spectrum of MoSe2 nanosheets (1–4 layers) was also measured by spectroscopic ellipsometer (Horiba Uvisel FUV) under normal white light illumination. The optical spectra of 1- to 4-layer MoSe2 of different illuminated wavelength fit well with the theoretically calculated results shown in Fig. 3.

Fig. 4
figure 4

OM image of 1- to 4-layer MoSe2 nanosheets on SiO2 (275 nm)/Si substrate a and corresponding 2D AFM images of selected areas b, c

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Fig. 5
figure 5

Height profiles across white a 1 solid Line 1, a 2 solid Line 2, a 3 solid Line 3 and a 4 solid Line 4 (Δh being height difference of different layers) marked in Fig. 4b, c and contrast values of yellow b 1 dashed Rectangle 1, b 2 dashed Rectangle 2, b 3 dashed Rectangle 3 and b 4 dashed Rectangle 4 marked in Fig. 4b, c

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Knowing the accurate thickness of the MoSe2 nanosheet, it is able to make a correspondence with the optical contrast values. Figure 6 shows that the contrast of the theoretical calculation and experimental data changes with the number of MoSe2 nanosheet layers. The calculated optical contrast fits the contrast analyzed from OM pictures very well. In addition, for the layer number of less than 3, contrast values can be fitted as C = 0.07N + 0.177. One can see that the contrast linearly increases with MoSe2 layer number increasing, making the identification of MoSe2 thickness by OM possible.

Fig. 6
figure 6

Theoretical and experimental contrast (C) values as a function of MoSe2 layer number (N)

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4 Conclusion

The contrast of different layers MoSe2 on 275 nm SiO2/Si was calculated, and the theoretical calculation was verified by AFM and spectroscopic ellipsometer. The results show that the contrast values acquired by OM are consistent with the theoretically calculated contrast values, leading to identifing the thickness of MoSe2 nanosheet accurately and rapidly possible. This work establishes a quantitative framework for detecting single and multiple layers of MoSe2 and other 2D atomic crystals on top of various substrates. It is hoped to provide a lot of convenience on identifying the thickness of 2D materials in fundamental research and applications.