1. Introduction

Graphene flakes exhibit remarkable electronic and optical properties, thus provide an excellent testing ground for the physics of two-dimensional systems [1, 2] and attract much interest for applications of post-silicon electronics and optoelectronics [3, 4]. Graphene flakes have a layered structure. The weak van der Waals (vdW) interactions keep the layers together, in contrast to atoms within each layer which are held together by strong chemical bonds. However, the presence of interlayer coupling in graphene layers makes their properties dependent on the layer number (N) from single-layer to multi-layer [5, 6]. We generally use the notation NLG to indicate graphene flakes with N layers. In this paper, we demonstrated a simple and fast method to probe reflection spectra of NLG layers on SiO₂/Si substrate in the broad wavelength range of 450-750 nm and studied that the reflectivity of NLG as N increases. From them, we noticed that there exists an optimized optical matching between 1LG to 5LG and SiO₂ layer due to interference effect. The relevant parameter in interference system such as the thickness of the SiO₂ layer was discussed. The results show that the combination of 1LG-5LG and SiO₂ layer can be serviced as anti-reflection coating with the nanoscale thickness and the thickness of graphene layers plays an important role in controlling the efficiency of anti-reflection.

2. Materials and methods

2.1 Preparation and layer number identification of NLG flakes up to 100 layers

Graphene flakes were obtained by micromechanical cleavage of natural graphite on SiO₂/Si substrate with SiO₂ thickness determined as 90nm [7, 8]. They can be easily seen by microscope. The layer number of graphene flakes was pre-estimated by the Raman mode intensity from substrate [9]. We prepared 15 graphene flakes with different N from 1 to 94. Figure 1(a) and (b) show the optical image of some graphene flakes. According to the reference [9], layer number identification of NLG up to N = 100 can be realized by Raman measurements of the Si intensity ratio between the Si peak (I(Si_G)) from SiO₂/Si substrate overlying graphene flakes and the Si peak (I(Si₀)) from bare SiO₂/Si substrate. The standard values of the Si intensity ratio of I(Si_G)/ I(Si₀) for NLG flakes deposited on SiO₂/Si substrate have been given in the supplementary data of the reference [10]. For the sake of N identification, Fig. 1(c) gives the layer number of these 15 graphene flakes by Raman measurements by comparing the measured I(Si_G)/ I(Si₀) values with the standard values. According to the standard values, the layer number deviation is almost zero for N ≤ 15, less than 1 for 16 ≤ N ≤ 30, and less than 3 for 31 ≤N < 100 [9].

 

Fig. 1 (a) Optical image of a flake contained 1LG, 2LG, 4LG, 5LG, 6LG, and 8LG on a 90-nm SiO₂/Si substrate. (b) Optical image of a flake contained 16LG, 22LG, 29LG, 34LG, 48LG, 60LG, and 75LG on a 90-nm SiO₂/Si substrate. (c) The standard values [9, 10] and experimental data of I(Si_G)/I(Si₀) for 532-nm excitation and NA = 0.45, by which the thickness of NLG flakes are identified.

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2.2 Measuring and calculating reflectivity from NLGs on the SiO₂/Si substrate

Reflection spectrum measurements were performed in a backscattering geometry at room temperature using a Jobin-Yvon HR800 micro-Raman system, which is equipped with liquid nitrogen cooled charge coupled device and the objective of 50X (NA = 0.45). Tungsten halogen lamp was used as a light source. A Semrock mirror with a high reflectivity up to 99% in the range from 350 to 1100nm was used as a reference to the light intensity from lamp source. The best reflected light signal was achieved by focusing the microscope to get a maximum peak intensity.

We measured reflection spectra from bare SiO₂/Si substrate and from 15 graphene flakes on the SiO₂/Si substrate in the broad wavelength range of 450-750 nm. All reflection spectra were calibrated by measuring the reflection spectrum of the Semrock mirror under the same condition to obtain the reflectivity curves. The reflectivity curves of NLGs are shown by black and green curves in Fig. 2, in which the reflectivity curve of bare SiO₂/Si substrate is given as a reference by blue curve. Figure 2(a) shows the reflectivity curves of bare substrate and 1LG-5LG. It is noted that these 6 curves show a very similar behavior with the lowest reflectivity locating at ~560 nm. The reflectivity monotonically decreases with the increase of N. The lowest reflectivity is ~10% on the bare substrate and down to ~4% on 5LG. Figure 2(b) shows the reflectivity of 8LG, 12LG, 16LG, 22LG, and 29LG. Their reflectivity experiences a process of decrease to increase with a turning point at N = 16. The curves with N = 22 and N = 29 are depicted by green color to avoid confusion and facilitate seeing the turning point. The wavelength of the lowest reflectivity gradually shifts from ~560 nm in N = 8 to ~650 nm in N = 29. Figure 2(c) shows the reflectivity of 34LG, 48LG, 60LG, 75LG, and 94LG. With N increasing from 34 to 94, the lowest reflectivity monotonically increases from ~4% to ~26%, which locates at ~700 nm or above. The reflectivity of these thicker graphene flakes is larger than that of bare substrate.

 

Fig. 2 The experimental and theoretical reflectivity curves of (a) bare SiO₂/Si substrate and 1LG-5LG on the SiO₂/Si substrate, (b) 8LG, 12LG, 16LG, 22LG, and 29LG on the SiO₂/Si substrate, (c) 34LG, 48LG, 60LG, 75LG, and 94LG on the SiO₂/Si substrate.

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The reflectivity of NLGs on the SiO₂/Si substrate as a function of N can be calculated in an Air/NLG/SiO₂/Si structure by the multiple reflection interference method [8–17]. Details of calculations are described in the supporting data. We present the theoretical reflectivity curves of bare substrate (denoted by blue curves) and NLGs (denoted by pink and gray curves and red dash curve) in Fig. 2 to compare with the experimental curves. The reflectivity reaches a turning point at 16LG, as shown by a red dash curve, which is in agreement with the experiment. In order to avoid confusion, the curves with N = 22 and N = 29 are specially marked by gray color.

3. Results and discussion

Figure 2(a) indicates that reflectivity curve from bare SiO₂/Si substrate show a very similar behavior with those five reflectivity curves from 1LG-5LG on SiO₂/Si substrate as we increase N. This means that there is an optimized optical matching existing between 1LG-5LG and SiO₂ layer. The physical mechanism of optical matching is due to interference effect. Based on the multiple reflection interference method, the incident light and reflected light undergo multiple reflection at the interfaces and optical interference within the medium in the Air/NLG/SiO₂/Si structure, as shown in Fig. 5(a). The transmission and reflection of electric field components in the four-layer structure can be described by transfer matrix equation, as shown in Fig. 5(b). As N increases, the phase factor ${\delta }_1$ in NLG is the only variable parameter that relates to the reflectivity. When N is less than 6, ${\delta }_1$is close to zero and the phase change in the optical path due to NLG layers coating can be neglected. It induces the impedance matching of 1LG-5LG and SiO₂ layer. The similar interference behavior has been reported in the monochromatic Rayleigh scattering of 1LG-6LG on SiO₂/Si substrate in which the contrast increases linearly for thinner graphene flakes [12]. In that paper, the phase change in the interference process of graphene layers and SiO₂ layer was studied and the results coincide with our conclusion [12]. However, Rayleigh scattering by some lasers with certain wavelengths have only got discrete experimental signals at the corresponding wavelengths. Here, we measured the reflection spectra in the broad wavelength range of 450-750 nm by using a lamp and spectrometer to replace a laser beam and a photon-counting avalanche photodiode (APD) in Rayleigh scattering measurement, respectively. Our experimental results exhibit the whole feature of the reflectivity in 450-750 nm, in which the impedance matching between 1LG-5LG and SiO₂ layer is clearly shown.

In the Air/NLG/SiO₂/Si structure, the thickness of SiO₂ layer $d_{Si{O}_2}$ is a critical factor in the interference of transmission and reflection of electric field components. Here, we calculated the reflectivity of bare SiO₂/Si substrate as a function of $d_{Si{O}_2}$. Figure 3(a) shows the reflectivity of bare substrate in wavelength $\lambda$ = 450-750 nm with 0nm <$d_{Si{O}_2}$< 500nm. The bright areas and dark areas appear alternately. The first dark area is around $d_{Si{O}_2}$ = 75~105 nm. $d_{Si{O}_2}$ = 90 nm is contained which is used in our experiment. Due to the SiO₂ layer tuned as an anti-reflection coating, graphene flakes become visible in this area. In Casiraghi’s work [12], 1LG-6LG were prepared on SiO₂/Si substrate with $d_{Si{O}_2}$ = 300 nm, which is one data point in the second dark area in Fig. 3(a). Thus, similar interference behavior occurs in these dark areas. Here, we specially show the reflectivity of bare substrate with 75nm <$d_{Si{O}_2}$< 105nm in Fig. 3(b). As $d_{Si{O}_2}$ increases, the lowest reflectivity locates at a straight line as a function of $\lambda$, which is $\lambda =a+b{d}_{Si{O}_2}$ with a = 113 nm and b = 5. It means that the wavelength of the minimum reflectivity is from $\lambda$~488 nm when $d_{Si{O}_2}$ = 75 nm to $\lambda$~638 nm when $d_{Si{O}_2}$ = 105 nm. Thus, the optimal anti-reflection wavelength is selectable in a broad wavelength range by adjusting the thickness of SiO₂ layer.

 

Fig. 3 The reflectivity of bare SiO₂/Si substrate in the broad wavelength range of 450-750 nm with (a) 0nm <$d_{Si{O}_2}$< 500nm and (b) 75nm <$d_{Si{O}_2}$< 105nm.

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Due to the impedance matching of 1LG-5LG and SiO₂ layer, the efficiency of anti-reflection can be adjusted by 1LG-5LG coating on SiO₂ layer. We further calculated the minimum reflectivity of 1LG-5LG at different $d_{Si{O}_2}$ with the corresponding wavelengths, as shown in Fig. 4. All the curves show a monotone decreasing from $d_{Si{O}_2}$ = 75 nm to $d_{Si{O}_2}$ = 105 nm. We defined the reflectivity relative to bare substrate as $D_R=R_{NLG+Si{O}_2/Si}/R_{Si{O}_2/Si}$, in which $R_{Si{O}_2/Si}$ and $R_{NLG+Si{O}_2/Si}$ is the minimum reflectivity from bare substrate and from NLG on bare substrate respectively. $D_R$ is independent with $d_{Si{O}_2}$. From 1LG to 5LG, $D_R$ decreases in proportion, which is ~84% in 1LG, ~71% in 2LG, ~59% in 3LG, ~48% in 4LG, and ~38% in 5LG. The results show that it is feasible for 1LG-5LG to exploit the enhancement of anti-reflection coating with the help of SiO₂ layer.

 

Fig. 4 The lowest reflectivity of 1LG-5LG on SiO₂/Si substrate with 75nm <$d_{Si{O}_2}$< 105nm.

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4. Conclusion

In summary, we found the optimized optical matching between 1LG-5LG and SiO₂ layer by the reflectivity of NLGs on SiO₂/Si substrate. The interference effect is critical in extracting and controlling their optical properties. The combination of 1LG-5LG and SiO₂ layer can be serviced as anti-reflection coating with the nanoscale thickness. The optimal anti-reflection wavelength is selectable by adjusting the thickness of SiO₂ layer, for example, 488-638 nm with $d_{Si{O}_2}$ from 75nm to 105nm. With 1LG-5LG coating on SiO₂ layer, the efficiency of anti-reflection is enhanced as the thickness of graphene layers increases. The minimum reflectivity decreases from 84% of 1LG to 38% of 5LG relative to the SiO₂ layer. Thus, reflection spectra provide a simple and quick way to study optical properties of graphene layers on a SiO₂/Si substrate. These properties are promising in the development of highly functionalized optical film.

Appendix

Reflectivity intensity in an Air/NLG/SiO₂/Si structure can be calculated by the multiple reflection interference method, which has been widely used to quantify optical contrast [8, 11, 12, 16] and Raman intensities [9, 10, 13, 14, 15, 17] of ultrathin flakes of two-dimensional layered materials. The Air/NLG/SiO₂/Si structure contains air(${\tilde{n}}_0$), NLG(${\tilde{n}}_1$ [18], $d_1$), SiO₂(${\tilde{n}}_2$ [19], $d_2$), Si(${\tilde{n}}_3$ [19], $d_3$), 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. 5(a). The calculation is based on classical electrodynamics and on the transfer matrix formalism [8–10, 12, 15, 16]. 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. 5 Schematic diagrams of (a) multiple reflection and optical interference and (b) the electric field component transfer process in the Air/NLG/SiO₂/Si structure.

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The incident electric field component passes through interfaces of Air/NLG, NLG/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. 5(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 NLG flakes. The calculated $R$ values of the bare SiO₂/Si substrate (with N=0, denoted as blue curves) and 1LG-94LG flakes on SiO₂/Si substrate (denoted as pink and gray curves and red dash curve) with $d_{Si{O}_2}$=90 nm and $\lambda$=450-750 nm are showed in Fig.2 to compare with the measured values.

Funding

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

Acknowledgements

The authors are grateful to B. L. Liang for discussions in the writing of the manuscript and P. H. Tan for the reflection spectra measurements.

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