Temperature-Dependent Optical Properties of Graphene on Si and SiO 2 /Si Substrates

: Systematic investigations are performed to understand the temperature-dependent optical properties of graphene on Si and SiO 2 /Si substrates by using a variable angle spectroscopic ellipsometry. The optical constants of graphene have revealed changes with the substrate and temperature. While the optical refractive index (n) of monolayer graphene on Si exhibited clear anomalous dispersions in the visible and near-infrared region (400–1200 nm), the modiﬁcation is moderate for graphene on SiO 2 /Si substrate. Two graphene sheets have shown a pronounced absorption in the ultraviolet region with peak position related to the Van Hove singularity in the density of states. By increasing the temperature from 300 K to 500 K, for monolayer graphene on Si, the n value is gradually increased while k decreased. However, the optical constants [n, k] of monolayer graphene on SiO 2 /Si exhibited unpredictable wave variations. In the wavelength range of 400–1200 nm, an experiential formula of a like-Sellmeier equation is found well suited for describing the dispersions of graphene on Si and SiO 2 /Si substrates.


Introduction
Since the seminal experimental realization of one atom thick graphene sheets [1] along with the measurements of quantum Hall effect [2], a great deal of interest has emerged in both the fundamental research and the development of device engineering concepts. Graphene has an extremely high carrier mobility~15,000 cm 2 V −1 S −1 [3] and thermal conductivity 5000 W m −1 K −1 [4] with a very strong Young's modulus~1 TPa [5]. The Dirac Fermions in graphene has caused both integer and fractional quantum Hall effect [6]. Unconventional superconductivity has also been realized in a 2-dimensional superlattice created by stacking two sheets of twisted graphene relative to each other by a small angle [7,8]. Along with the unique electronic features, graphene has displayed extraordinary optical responses. Graphene, being a one-atom-thick sheet of carbon exhibits significant absorption in the visible to infrared wavelength region (2.3%) with reflectance less than 0.1% [9]. This means that a one-atom-thick graphene layer is extremely transparent having a high degree of flexibility with excellent optical properties.
This work aimed to understand the optical and temperature-dependent features of commercial graphene. A series of graphene samples were prepared by ACS Material, LLC using standard manufacturing procedures (product detail found on the website of https: //www.acsmaterial.com/graphene-on-silicon-substrate.html accessed on 29 March 2021). Graphene was first prepared on copper foil by the chemical vapor deposition (CVD) method and then transferred onto Si and on SiO 2 /Si substrates. The graphene qualities are examined through Raman spectroscopy and atomic force microscopy (AFM). The temperature-dependent optical properties are systemically investigated by exploiting a variable angle spectroscopic ellipsometry (VASE) method. The refractive index (n) and extinction coefficient (k) of the graphene sheets having different layers prepared on Si and SiO 2 /Si substrates are carefully analyzed, establishing temperature-dependence between 300 K and 500 K. The anomalous optical dispersions of graphene prepared on Si and SiO 2 /Si in the visible to near-infrared wavelength region (i.e., λ between 400 to 1200 nm), are well described by a modified Sellmeier equation.

Materials and Methods
Four graphene samples were meticulously examined by using VASE and variable temperature methods. The samples considered here were: (a) monolayer and bilayer graphene on Si substrate, and (b) monolayer, bilayer graphene on SiO 2 /Si substrate (a Si wafer covered with a 300 nm SiO 2 layer). The dimensions of the graphene sheets were 2.54 cm × 2.54 cm. To examine the graphene qualities, room temperature Raman scattering spectra were measured using a micro-region Raman spectrometer with an excitation laser source of a wavelength of 532 nm and a spot size of 2 µm (iHR550, HORIBA, Kyoto, Japan), and the surface morphology of the samples were examined by the atomic force microscope (Dimension Icon, BRUKER NANO Inc., Billerica, MA, USA). The SE measurements were performed by using a Müeller Matrix Ellipsometer (Wuhan Eoptics Technology Co. Ltd., Wuhan, China) equipped with a heating and cooling system (THMS600, Linkam, Surrey, UK). The optical constants were extracted by modeling and data fitting analysis. The SE has the advantages of being a non-destructive and highly accurate technique. In the VASE studies, we used deuterium and halogen sources and varied the incident angles from 50 • , 55 • , 60 • , 65 • , and 70 • , respectively. The beam size was about 4 mm. To study the temperature-dependent characteristics of graphene, we recorded the angle of polarization psi (ψ) and phase difference delta (∆) of the reflective polarization lights between 300-500 K in the steps of 20 K.
To extract the optical constants, we fitted and modeled the recorded data of psi (ψ) and delta (∆) by using Eometrics (Wuhan Eoptics Technology Co. Ltd., Wuhan, China). In our data analysis, a three-layer model was adopted, which properly included contributions from the substrate, oxide layer, and graphene, and the goodness of fitting was evaluated by the Mean Squared Error (MSE) method. Since the manufacturing procedures would induce a modification of bandgap properties of graphene [28][29][30][31], the dispersion properties of the graphene layers were described by 5 Tauc-Lorentz oscillators and 1 Drude oscillator. As for the oxide layer (SiO 2 ) and the substrate (Si), their dispersions were described by the Sellmeier and parametric models which were provided by the materials database of Eometrics.

Raman Scattering and AFM Measurements
Raman spectroscopy was used to characterize the quality of the transferred graphene. Figure 1 shows the room temperature Raman scattering spectra recorded for four graphene samples, in which all samples displayed typical Raman peaks of graphene. All graphene samples included two intrinsic Raman peaks (G, 2D) and two disorder-induced peaks (D, D+D" peaks for graphene on Si, and D, D'+D" peaks for graphene on SiO 2 ) [32][33][34]. Among the defect-activated peaks, D'+D" and D+D" corresponded to the combination mode of the D' and D" modes as well as the D and D" modes [34]. Si had strong absorption in the visible wavelength, the intensity of Raman scattering light for the graphene on Si substrate was weaker than that of graphene on SiO 2 /Si substrate. The Raman scattering results indicated there existed a degree of defects in the four transferred graphene sheets. The intensity ratio between the 2D band and the G band can determine the number of graphene layers [35][36][37]. The I 2D /I G values were higher than 2, in the range between 1 and 2, and lower than 1, corresponding to the presence of monolayer graphene, bilayer graphene, and three or more layers, respectively [35]. As for our four graphene samples, the I 2D /I G values of two monolayer graphene were respectively 2.1 (on Si substrate) and 3.4 (on SiO 2 /Si substrate) while those of two bilayer graphene were 1.5 (on Si substrate) and 1.7 (SiO 2 /Si substrate).
Raman spectroscopy was used to characterize the quality of the transferred graphene. Figure 1 shows the room temperature Raman scattering spectra recorded for four graphene samples, in which all samples displayed typical Raman peaks of graphene. All graphene samples included two intrinsic Raman peaks (G, 2D) and two disorder-induced peaks (D, D+D" peaks for graphene on Si, and D, D'+D" peaks for graphene on SiO2) [32][33][34]. Among the defect-activated peaks, D'+D" and D+D" corresponded to the combination mode of the D' and D" modes as well as the D and D" modes [34]. Si had strong absorption in the visible wavelength, the intensity of Raman scattering light for the graphene on Si substrate was weaker than that of graphene on SiO2/Si substrate. The Raman scattering results indicated there existed a degree of defects in the four transferred graphene sheets. The intensity ratio between the 2D band and the G band can determine the number of graphene layers [35][36][37]. The I2D/IG values were higher than 2, in the range between 1 and 2, and lower than 1, corresponding to the presence of monolayer graphene, bilayer graphene, and three or more layers, respectively [35]. As for our four graphene samples, the I2D/IG values of two monolayer graphene were respectively 2.1 (on Si substrate) and 3.4 (on SiO2/Si substrate) while those of two bilayer graphene were 1.5 (on Si substrate) and 1.7 (SiO2 /Si substrate). Figure 2 illustrates the AFM images of graphene samples displaying smooth surfaces with a few surface contaminants. In 2 μm × 2 μm AFM images, the Root Mean Square (RMS) roughness was around 1.5-4.5 nm and the average roughness was 0.6-0.9 nm respectively. The samples had enough smooth surfaces for SE measurements and analysis.   Figure 2 illustrates the AFM images of graphene samples displaying smooth surfaces with a few surface contaminants. In 2 µm × 2 µm AFM images, the Root Mean Square (RMS) roughness was around 1.5-4.5 nm and the average roughness was 0.6-0.9 nm respectively. The samples had enough smooth surfaces for SE measurements and analysis.

Spectroscopic Ellipsometric Measurements
The optical characteristics of graphene samples were investigated by variable angle spectroscopic ellipsometry. In Figure 3a-d we have displayed the measured psi(ψ) and delta(Δ) at 300 K for monolayer graphene samples on Si and SiO2/Si substrates, respectively by varying incident angles from 50°, 55°, 60°, 65°, and 70°. The experimental data (black solid lines) and the fitted curves (red dotted lines) compared reasonably well. In Table 1, for four samples the thicknesses of graphene and oxide layers and the MSE values are presented. The low MSE values (<15 in Table 1) indicated good fits. The thicknesses of monolayer graphene on Si and on SiO2/Si were 0.38 nm and 0.34 nm, respectively, which was in accordance with those previously reported [13,16,21,25]. In Table 2, the fitting parameters of Tauc-Lorentz and Drude oscillators for monolayer graphene on Si and SiO2/Si substrates are presented, where Ampn, Brn, Eon, and Egn correspond to amplitude, broadening, center energy, and bandgap energy of oscillator n (n is an integer), and Scat. Time is scattering time, respectively. The bandgap energy of Tauc-Lorentz oscillators as shown in Table 2 well agreed with the reports in References [28][29][30][31], indicating the dispersion models for graphene layers we have chosen were reasonable.

Spectroscopic Ellipsometric Measurements
The optical characteristics of graphene samples were investigated by variable angle spectroscopic ellipsometry. In Figure 3a-d we have displayed the measured psi(ψ) and delta(∆) at 300 K for monolayer graphene samples on Si and SiO 2 /Si substrates, respectively by varying incident angles from 50 • , 55 • , 60 • , 65 • , and 70 • . The experimental data (black solid lines) and the fitted curves (red dotted lines) compared reasonably well. In Table 1, for four samples the thicknesses of graphene and oxide layers and the MSE values are presented. The low MSE values (<15 in Table 1) indicated good fits. The thicknesses of monolayer graphene on Si and on SiO 2 /Si were 0.38 nm and 0.34 nm, respectively, which was in accordance with those previously reported [13,16,21,25]. In Table 2, the fitting parameters of Tauc-Lorentz and Drude oscillators for monolayer graphene on Si and SiO 2 /Si substrates are presented, where Amp n , Br n , Eo n, and Eg n correspond to amplitude, broadening, center energy, and bandgap energy of oscillator n (n is an integer), and Scat. Time is scattering time, respectively. The bandgap energy of Tauc-Lorentz oscillators as shown in Table 2 well agreed with the reports in References [28][29][30][31], indicating the dispersion models for graphene layers we have chosen were reasonable.
The extracted optical constants (n and k) in the wavelength range of 218-1200 nm are shown in Figure 3e,f. It is to be noted that graphene sheets on Si and on SiO 2 /Si exhibited markedly different optical properties. In the range of λ between 400-1200 nm, the monolayer graphene on Si exhibited anomalous dispersions with a larger extinction coefficient than those on SiO 2 /Si. In the ultraviolet wavelength range λ between 220-400 nm, the n of monolayer graphene on Si displayed a sharp peak while on SiO 2 /Si it revealed a moderate feature. Likewise, the k of monolayer graphene on Si exhibited a pronounced peak at~4.64 eV while a weak peak was noticed at~4.78 eV for graphene on SiO 2 /Si. The peak position of k, which is considered as a van Hove singularity of graphene density of state [11], showed an energy difference of 0.14 eV for two samples. Obviously, the interaction of graphene with substrate had significantly affected the optical properties of graphene.    Moreover, we compared the optical constants of monolayer and bilayer graphene. In Figure 4 the optical constants extracted by SE measurements are displayed for four graphene samples along with the fitted results. We noticed that for graphene samples on Si substrate, the n decreased while k increased with the increase in the number of layers. Moreover, the peak positions of k were found redshifted. This was possibly due to the increase of the number of layers-the layer-to-layer interaction decreased the energy of π-to-π* exciton transition near the M point of the Brillouin zone [38,39]. Unlike graphene on Si, the bilayer graphene on SiO 2 /Si displayed a greater n than that of monolayer graphene as wavelengths were longer than 600 nm. Moreover, it exhibited significant dispersion features compared to the monolayer graphene on SiO 2 /Si. It has been suggested earlier that Crystals 2021, 11, 358 6 of 11 the substrates modulate the dispersion of graphene sheets and the enhanced layer-to-layer interaction of graphene degrades the excitonic effects [38][39][40].  In Figure 5 we have displayed the results of temperature-dependent optical constants for two monolayer graphene on Si and SiO2/Si substrates from 300 K to 500 K with a step of 20 K. The optical constants of monolayer graphene on Si varied consistently with temperature. In the region of λ between 400 to 1200 nm, we noticed the n increased while k decreased with the increase of temperature because the coupling between light and electrons was intensified in graphene. However, the optical constants [n, k] of monolayer graphene on SiO2/Si exhibited unpredictable variations, i.e., the refractive index n and the extinction coefficient k displayed fluctuations with the increase of temperature.
Figure 5e,f illustrates respectively the difference of the extracted optical constants by VASE measurements for four samples of graphene sheets prepared on Si substrate at 300 K and at 500 K. It was revealed that the difference of optical constants between 300 K and 500 K for the monolayer graphene on Si were bigger than those of bilayer graphene, suggesting more temperature sensitivity of the monolayer graphene. It could be caused by the fact that monolayer graphene has more defects, stronger graphene-substrate interactions, and lower thermal stability. The temperature effect was relatively smaller and the In Figure 5 we have displayed the results of temperature-dependent optical constants for two monolayer graphene on Si and SiO 2 /Si substrates from 300 K to 500 K with a step of 20 K. The optical constants of monolayer graphene on Si varied consistently with temperature. In the region of λ between 400 to 1200 nm, we noticed the n increased while k decreased with the increase of temperature because the coupling between light and electrons was intensified in graphene. However, the optical constants [n, k] of monolayer graphene on SiO 2 /Si exhibited unpredictable variations, i.e., the refractive index n and the extinction coefficient k displayed fluctuations with the increase of temperature. Graphene is usually applied in photoelectric devices. The optical constant n(λ) reflects the dispersion properties of graphene. It is essential to understand the dependence of dispersion properties of graphene on wavelength and temperature in the transparent region. To further explore the specific dispersion relationship of graphene on Si and on SiO2/Si substrates, we fitted the n by using a like-Sellmeier equation based on the extracted data in the wavelength range of λ between 400-1200 nm at room temperature. The traditional Sellmeier equation describing the dispersion is given by [41]: Figure 5e,f illustrates respectively the difference of the extracted optical constants by VASE measurements for four samples of graphene sheets prepared on Si substrate at 300 K and at 500 K. It was revealed that the difference of optical constants between 300 K and 500 K for the monolayer graphene on Si were bigger than those of bilayer graphene, suggesting more temperature sensitivity of the monolayer graphene. It could be caused by the fact that monolayer graphene has more defects, stronger graphene-substrate interactions, and lower thermal stability. The temperature effect was relatively smaller and the thermal stability was enhanced with the increasing number of layers.
Graphene is usually applied in photoelectric devices. The optical constant n(λ) reflects the dispersion properties of graphene. It is essential to understand the dependence of dispersion properties of graphene on wavelength and temperature in the transparent region. To further explore the specific dispersion relationship of graphene on Si and on SiO 2 /Si substrates, we fitted the n by using a like-Sellmeier equation based on the extracted data in the wavelength range of λ between 400-1200 nm at room temperature. The traditional Sellmeier equation describing the dispersion is given by [41]: where λ is the wavelength, A j and B j are the fitting parameters, and j is an integer. The fitting results for monolayer and bilayer graphene are displayed in Figure 6. The dispersion curves of graphene on Si were fitted well by a like-Sellmeier equation with j = 1 and for graphene on SiO 2 /Si with j = 2. The fitting parameters are recorded in Table 3 and the fitting parameter B 1 was negative which mainly reflected an anomalous dispersion of graphene. It is worth noting that graphene on SiO 2 /Si possessed more complex dispersion features. It was possibly caused by an enhancing layer-to-layer interaction which complicated the dispersion relationship. where λ is the wavelength, Aj and Bj are the fitting parameters, and j is an integer. The fitting results for monolayer and bilayer graphene are displayed in Figure 6. The dispersion curves of graphene on Si were fitted well by a like-Sellmeier equation with j = 1 and for graphene on SiO2/Si with j = 2. The fitting parameters are recorded in Table 3 and the fitting parameter B1 was negative which mainly reflected an anomalous dispersion of graphene. It is worth noting that graphene on SiO2/Si possessed more complex dispersion features. It was possibly caused by an enhancing layer-to-layer interaction which complicated the dispersion relationship.  Furthermore, for graphene on Si substrate, the parameters A1 and B1 as a function of temperature (T) were well represented by a polynomial which is given by.
where a0, a1, a2, b0, b1, and b2 are constants. By substituting the testing temperature and values of A1 and B1 into Equation (2) the fitted results are shown in Figure 7. The red solid lines are the fitted curves using Equation (2), with parameter values of a0, a1, a2, b0, b1, and b2 tabulated in Table 4. Combining the fitting results of expressions (1) and (2), the complete dispersion and temperature-dependent properties of graphene were revealed.  Furthermore, for graphene on Si substrate, the parameters A 1 and B 1 as a function of temperature (T) were well represented by a polynomial which is given by.
where a 0 , a 1 , a 2 , b 0 , b 1 , and b 2 are constants. By substituting the testing temperature and values of A 1 and B 1 into Equation (2) the fitted results are shown in Figure 7. The red solid lines are the fitted curves using Equation (2), with parameter values of a 0 , a 1 , a 2 , b 0 , b 1 , and b 2 tabulated in Table 4. Combining the fitting results of expressions (1) and (2), the complete dispersion and temperature-dependent properties of graphene were revealed.

Figure 7.
The parameters A1 and B1 as a function of temperature from 300 K to 500 K for the graphene on Si substrate.

Conclusions
In summary, the results of the comprehensive study for the optical properties of graphene revealed interesting characteristics which are not only substrate-, layer-but also temperature-dependent. The graphene samples on Si substrate exhibited significant dispersion features and higher exciton transition energy than those on SiO2/Si substrate. With the increase of temperatures from 300 K to 500 K, the optical constants of monolayer graphene on Si varies regularly while the monolayer graphene on SiO2/Si has fluctuations. An experiential formula of a like-Sellmeier expression is demonstrated well exemplifying the wavelength dependence of refractive indices of monolayer and bilayer graphene samples. In addition, for monolayer graphene on Si, the temperature-dependent dispersion properties are well described by combining the like-Sellmeier equation and a quadratic expression.

Conclusions
In summary, the results of the comprehensive study for the optical properties of graphene revealed interesting characteristics which are not only substrate-, layer-but also temperature-dependent. The graphene samples on Si substrate exhibited significant dispersion features and higher exciton transition energy than those on SiO 2 /Si substrate. With the increase of temperatures from 300 K to 500 K, the optical constants of monolayer graphene on Si varies regularly while the monolayer graphene on SiO 2 /Si has fluctuations. An experiential formula of a like-Sellmeier expression is demonstrated well exemplifying the wavelength dependence of refractive indices of monolayer and bilayer graphene samples. In addition, for monolayer graphene on Si, the temperature-dependent dispersion properties are well described by combining the like-Sellmeier equation and a quadratic expression.