Optical methods for determining thicknesses of few-layer graphene flakes

Figure 1 shows a schematic diagram of an optical path when an incident light passes through an m-layer film structure. We denote as R(λ,θ_i;n,d) the global reflectance of the m-layer structure and as T(λ,θ_i;n,d) its transmittance, both of which are functions of the incident light wavelength λ, the incident angle θ_i, the complex refractive index vector n = (n_i,n₁,n₂,...,n_m,n_s)^T, and the thickness vector d = (d_i,d₁,d₂,...,d_m,d_s)^T of the structure. In a typical experimental setup, the incident medium is usually air with a refractive index of n_i = 1.0003, and the thickness of the incident medium, d_i, and the substrate, d_s, are usually assumed to be infinite.

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By following the matrix method of multilayer theory, we calculated the reflectance and transmittance of the m-layer structure as follows [21, 22]:

$$\begin{eqnarray} R(\lambda ,{\theta }_{\mathrm{i}};\mathbf{n},\mathbf{d})&=&\left \vert \frac{{E}_{\mathrm{i}}^{-}}{{E}_{\mathrm{i}}^{+}}\right \vert ^{2}, \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} T(\lambda ,{\theta }_{\mathrm{i}};\mathbf{n},\mathbf{d})&=&{C}_{0}^{2}\frac{\mathrm{Re}({n}_{\mathrm{s}})}{\mathrm{Re}({n}_{\mathrm{i}})}\left \vert \frac{{E}_{\mathrm{s}}^{+}}{{E}_{\mathrm{i}}^{+}}\right \vert ^{2}, \end{eqnarray} \tag{ 2 }$$

where C₀ = cosθ_i/cosθ_s for p-polarized light and C₀ = 1 for s-polarized light, while ${E}_{k}^{+}$ is the amplitude of the electric field vector of the wave traveling in the direction of incidence in the kth layer and ${E}_{k}^{-}$ is that of the negative-going wavevector. Apparently, both reflectance and transmittance are functions of the wavelength and incident angle of the incident light for a given multilayer structure.

In a real optical system, the incident angle of light is not strictly zero, but has a certain distribution [16, 18, 23]. This distribution of incident range is usually described via the numerical aperture (NA) [24] of the objective lens. The maximum incident angle θ_max can be described by the equation [16, 18] NA = n_isinθ_max, where n_i is the refraction index of the medium between samples and the objective lens; n_i = 1.0003 when the medium is air. Assuming that the intensity of light at different incident angles follows a Gaussian distribution, we can calculate the modified total reflectivity of light using the following integration [16]:

$$\begin{eqnarray} &&\fl {R}_{\mathrm{tot}}(\lambda ,\text{NA};\mathbf{n},\mathbf{d})\nonumber\\ &&\tqs =\int \nolimits _{-{\theta }_{\mathrm{max}}}^{{\theta }_{\mathrm{max}}}R(\lambda ,{\theta }_{\mathrm{i}};\mathbf{n},\mathbf{d})\phi ({\theta }_{\mathrm{i}},0,{\theta }_{\mathrm{max}}/3)\mathrm{d}{\theta }_{\mathrm{i}}, \end{eqnarray} \tag{ 3 }$$

where ϕ(θ_i,0,θ_max/3) is the Gaussian distribution with the mean of 0 and standard deviation of θ_max/3, where θ_max = arcsin(NA/n_i).

Once the reflectance of a multilayer structure is known, its color can be calculated by a standard procedure [25]. According to the international standards, color can be described in the International Commission on Illumination (usually abbreviated as CIE, for its French name) color space by using different sets of parameters (or tristimuli), e.g. XYZ parameters, RGB parameters, or L*a*b* parameters [25] (the corresponding color spaces are called CIE XYZ, CIE RGB and CIELAB color spaces respectively). For a given light source with a normalized spectral power distribution (SPD) P(λ) in the wavelength space, the CIE XYZ tristimulus components can be obtained as

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle X=X({\theta }_{\mathrm{i}};\mathbf{n},\mathbf{d})=\int \nolimits _{0}^{\infty }P(\lambda )R(\lambda ,{\theta }_{\mathrm{i}};\mathbf{n},\mathbf{d})\bar {x}(\lambda )\mathrm{d}\lambda \\ \displaystyle Y=Y({\theta }_{\mathrm{i}};\mathbf{n},\mathbf{d})=\int \nolimits _{0}^{\infty }P(\lambda )R(\lambda ,{\theta }_{\mathrm{i}};\mathbf{n},\mathbf{d})\bar {y}(\lambda )\mathrm{d}\lambda \\ \displaystyle Z=Z({\theta }_{\mathrm{i}};\mathbf{n},\mathbf{d})=\int \nolimits _{0}^{\infty }P(\lambda )R(\lambda ,{\theta }_{\mathrm{i}};\mathbf{n},\mathbf{d})\bar {z}(\lambda )\mathrm{d}\lambda , \end{array} \end{eqnarray} \tag{ 4 }$$

where R(λ,θ_i;n,d) is the reflectance of the multilayer structure, $\bar {x}(\lambda ),\bar {y}(\lambda )$ and $\bar {z}(\lambda )$ are the CIE color-matching functions, to account for the response of the eyes [20]. For a color to be captured and displayed, its XYZ tristimulus needs to be converted to a device-dependent color scheme (such as RGB), which is accomplished by multiplying the XYZ vector by a transformation matrix ${\mathbf{M}}_{\mathrm{d}}^{-1}$ as indicated in equation (5):

$$\begin{eqnarray} &&\left [\begin{array}{@{}c@{}} \displaystyle R\\ \displaystyle G\\ \displaystyle B \end{array}\right ]={\mathbf{M}}_{\mathrm{d}}^{-1}\left [\begin{array}{@{}c@{}} \displaystyle X\\ \displaystyle Y\\ \displaystyle Z \end{array}\right ]. \end{eqnarray} \tag{ 5 }$$

Conversely, the RGB parameters can be converted to the XYZ components by multiplying by M_d. Because the transformation matrix is generated using the phosphor chromaticity coordinates and the reference white point of the recording device (more details can be found in [25–27]), M_d is device dependent; thereby the RGB parameters are also device dependent. Correctly estimating M_d requires obtaining detailed knowledge about the optical recording devices, which is usually very difficult if not infeasible. For a standard RGB (sRGB) system, the illuminant D65 is usually adopted as the virtual reference light source [28] and the M_d can be calculated from a standard procedure [25–27].

Because the CIELAB color space has several distinct advantages [16, 29], e.g. its gamut is relatively uniform and it has been extensively studied with several standardized definitions of color difference, the XYZ parameters are often expressed in the CIELAB color space [29, 30] using L*,a*,b* parameters as follows:

$$\begin{eqnarray} \begin{array}{@{}rcl@{}} \displaystyle {L}^{\ast }&\displaystyle =&\displaystyle 1 1 6 \Phi (Y/{Y}_{0})-1 6\\ \displaystyle {a}^{\ast }&\displaystyle =&\displaystyle 5 0 0[\Phi (X/{X}_{0})-\Phi (Y/{Y}_{0})]\\ \displaystyle {b}^{\ast }&\displaystyle =&\displaystyle 2 0 0[\Phi (Y/{Y}_{0})-\Phi (Z/{Z}_{0})], \end{array} \end{eqnarray} \tag{ 6 }$$

where L* represents the lightness, a* and b* represent the chromaticity, X₀,Y₀ and Z₀ are the CIE XYZ tristimulus values of the standard illuminant and Φ(t) is given by

$$\begin{eqnarray*} \Phi (t)=\left \{ \begin{array}{@{}ll@{}} \displaystyle {t}^{1/3}&\displaystyle \tqs t\gt \left (\frac{6}{2 9}\right )^{3}\\ \displaystyle \frac{1}{3}\left (\frac{2 9}{6}\right )^{2}t+\frac{4}{2 9}&\displaystyle \tqs t\leq \left (\frac{6}{2 9}\right )^{3}. \end{array}\right. \end{eqnarray*}$$

On the basis of the CIE color spaces, we can now quantitatively define color differences/contrasts between two colors. In this work, we will consider two standard color differences in the CIELAB color space and two color contrasts in the CIE RGB color space, whose definitions are given in the following.

The CIEDE76 color difference [16, 29] is defined as

$$\begin{eqnarray} &&\Delta {E}_{7 6}=\sqrt{(\Delta {L}^{\ast })^{2}+(\Delta {a}^{\ast })^{2}+(\Delta {b}^{\ast })^{2}}, \end{eqnarray} \tag{ 7 }$$

where ΔL*,Δa* and Δb* represent the differences between the L*a*b* color parameters of the sample and those of the reference substrate. It is noted that the CIEDE76 color difference is essentially the same as the total color difference adopted by Gao et al in their OM method [16].

The CIEDE2000 color difference [31] is defined as

$$\begin{eqnarray} &&\fl \Delta {E}_{0 0}\nonumber\\ &&\mbox { $ \fl =\sqrt{\left (\frac{\Delta {L}^{\prime}}{{k}_{\mathrm{L}}\cdot {S}_{\mathrm{L}}}\right )^{2}+\left (\frac{\Delta {C}^{\prime}}{{k}_{\mathrm{C}}\cdot {S}_{\mathrm{C}}}\right )^{2}+\left (\frac{\Delta {H}^{\prime}}{{k}_{\mathrm{C}}\cdot {S}_{\mathrm{H}}}\right )^{2}+{R}_{\mathrm{T}}\frac{\Delta {C}^{\prime}}{{k}_{\mathrm{C}}\cdot {S}_{\mathrm{C}}}\frac{\Delta {H}^{\prime}}{{k}_{\mathrm{H}}\cdot {S}_{\mathrm{H}}}},$ } \nonumber\\ \end{eqnarray} \tag{ 8 }$$

where ΔL',ΔC' and ΔH' represent the modified differences between the L,C (chroma) and H (hue) parameters of the sample and those of the reference substrate. S_L,S_C and S_H are compensation weight functions for lightness, chroma and hue respectively; R_T is a rotation term for the hue and k_L,k_C and k_H are parametric weighting factors (more details can be found in [31, 32]).

In addition to color differences defined in the CIELAB space, color contrasts defined in the CIE RGB space were also proposed in prior OM methods [17, 18]. For example, Jung et al [18] used the color contrast defined as

$$\begin{eqnarray} \Delta {E}_{\mathrm{RGB}}&=&(\vert R-{R}_{0}\vert +\vert G-{G}_{0}\vert \nonumber\\ &&+\vert B-{B}_{0}\vert )/({R}_{0}+{G}_{0}+{B}_{0}), \end{eqnarray} \tag{ 9 }$$

where (R₀,G₀,B₀) and (R,G,B) are the RGB components of the reference substrate and those of the sample respectively; while the method by Wang et al [17] was based on a color contrast involving the green (G) channel alone:

$$\begin{eqnarray} &&\Delta {E}_{\mathrm{G}}=({G}_{0}-G)/{G}_{0}. \end{eqnarray} \tag{ 10 }$$

From equations (3)–(10), it is evident that the color differences/contrasts generally depend not only on R(λ,θ_i;n,d), i.e. the multilayer structure, but also on P(λ) and M_d, i.e. the characteristics of the light source and the optical recording system. In a typical experimental setup, the precise SPD of the incident light source and the characteristics of the optical recording system are usually not known. How color differences/contrasts change with the light source and optical system for an OM method will determine its robustness. In section 3, we will explore the influences of these uncertain factors on color-based OM methods by performing virtual and real experiments.

As indicated in figure 3, the most critical step in determining the sample thickness is theoretically computing the dependence of the color differences/contrasts on the sample thickness. Since SiO₂/Si is the most common substrate for graphene applications [13, 14, 16–18, 24], we will focus our discussions on graphene/SiO₂/Si systems in this work; however, it should be noted that the findings of this work are generally applicable for other multilayer structures. For a graphene/SiO₂/Si multilayer structure, the system consists of four components: air (incident medium), the few-layer graphene flake (sample), the SiO₂ layer (middle layer), and Si (substrate). The complex refractive indices of Si and SiO₂ are characterized by λ-dependent functions as suggested in [33]. The complex refractive index of FLG is more subtle and it is still an ongoing research subject [9, 13, 34]. Different refractive index values have been deduced indirectly for few-layer graphene samples [9, 13, 35]. In this work, we chose a constant value of the refractive index, as suggested by Blake et al [13]. As will be shown in a later attempt to calibrate the OM method, when we let the refractive index of FLG be a free variable to fit our experimental data, the optimal value of the refractive index is closer to that of bulk graphite, which is consistent with the results of Blake et al [13] for a similar system.

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For a graphene/SiO₂(285 nm)/Si structure illuminated with five standard illuminants, the theoretical variations of the color differences/contrasts with the number of graphene layers are shown in figures 3(a)–(d). In order for a sample thickness to be determined unambiguously, the color differences/contrasts have to vary monotonically with the thickness. For example, if the thickness of an unknown graphene sample is less than ten layers, then its value can be determined unambiguously from all four color differences/contrasts, as suggested by figure 3. However, if the thickness of an unknown sample can be more than ten layers, one has to carefully specify the possible range of graphene thickness to ensure that only one thickness value can be determined ambiguously. Variations of color differences/contrasts with a larger range of graphene thickness can be found in the supplementary information (SI figure S1, available at stacks.iop.org/Nano/24/505701/mmedia).

As shown in figure 3, the variation of the color differences/contrasts changes when different incident light sources are considered. This dependence of the color differences/contrasts on the light source is relatively weak when the sample thickness is small (less than five layers) and it becomes more significant for thicker samples. For example, in figure 3(d), the color difference of 10 corresponds to a sample thickness of five layers if we choose light source F4; but the thickness will become ∼9 layers if we choose light source D65. Ideally, an exact light source should be chosen to ensure a reliable calculation when using color-based optical methods.

We also explored the dependence of color differences/contrasts on the NAs of the optical system (see SI figures 2 and 3, available at stacks.iop.org/Nano/24/505701/mmedia). Generally, the influence of the NAs is much smaller than that of the light sources. For simplicity, we will focus the following discussions on the cases where the NA is assumed to be zero.

In addition to the light source, color differences/contrasts should depend on the reference substrate (i.e. the thickness of the SiO₂ layer). In order to investigate the influence of the reference substrate, we calculated the color differences/contrasts between one-, two-, three- and four-layer graphene and the reference substrate as a function of the SiO₂ layer thickness. The results, assuming an incident light source of D65, are as shown in figure 4.

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As clearly shown in figure 4, the color differences/contrasts are very sensitive to the thickness of the SiO₂ layer. For example, the color contrast ΔE_G between one-, two-, three- and four-layer graphene and the reference substrate only peaks when the SiO₂ layer has certain thicknesses, and decays quickly becoming almost indistinguishable elsewhere. In order to achieve better optical visibility, one should choose an SiO₂ substrate with a proper thickness so that the values of the color differences/contrasts are maximized. Previously, people have used the SiO₂(285 nm)/Si substrate to enhance the reflectivity contrast of graphene for the given monochromatic light source [13, 14], which was similar to the present finding. The structure of the SiO₂ layer affects not only the values of the color differences/contrasts but also their variations with the graphene thickness. For example, the variation trends shown in figure 3 would be drastically different if the SiO₂ layer thickness (which was chosen to be 285 nm) was changed (see SI figure S4, available at stacks.iop.org/Nano/24/505701/mmedia). Because of the strong dependence of the color differences/contrasts on the thickness of the SiO₂ layer, one should accurately measure the thickness of the SiO₂ layer when implementing color-based optical methods.

The above theoretical discussions provide a few strategies for optimizing the performance of color-based optical methods: (a) choosing a color difference/contrast to ensure that its value varies monotonically with the sample thickness; (b) choosing a proper thickness of the SiO₂ layer to enhance the values of the chosen color difference/contrast; (c) measuring the SPD of the light source (if possible) and the thickness of the SiO₂ layer accurately.

To investigate the intrinsic performance of each optical method, we carried out a set of virtual experiments in this section. First, we generated a series of images (containing sRGB information) of virtual graphene samples using a standard illuminant, D65. These graphene samples were assumed to be on an SiO₂(285 nm)/Si substrate with their thicknesses ranging from one layer to ten layers. After obtaining the virtual images, we used four optical methods based on the color differences/contrasts defined in section 2 to estimate the number of graphene layers. The light source and the thickness of the SiO₂ layer were intentionally varied during the calculation; and their effects were explored by comparing the estimated thicknesses of the graphene samples with the initial 'true' values.

Figures 5(a)–(d) show the comparison between the calculated graphene thicknesses obtained from four optical methods and their initial 'true' thickness values. In each of these figures, five sets of data obtained assuming five different standard illuminants (i.e. F4, F6, F10, E and D65) are presented. From figure 5, one can easily find that when the original illuminant (D65) is assumed, all four methods could correctly calculate (recover) the graphene thickness. However, if other illuminants are assumed during the calculation, the estimated graphene thickness can deviate from the true values. This deviation depends on the choice of the color differences/contrasts and generally becomes more significant for thicker samples. According to the results shown in figures 5(a)–(d), the optical method based on color difference ΔE₀₀ seems to be less affected by the choice of illuminants.

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Like figures 5(a)–(d), 6(a)–(d) also illustrate the comparison between the calculated graphene thicknesses obtained from four optical methods and their initial 'true' thickness values. However, in each plot of figures 6(a)–(d), three sets of data obtained assuming three different thicknesses of the SiO₂ layer (i.e. 270, 285 and 300 nm) are presented. When the original thickness of the SiO₂ layer (285 nm) is considered, all four methods could correctly calculate (recover) the graphene thickness. However, if different SiO₂ layer thicknesses (even only 5% off the original value) are assumed, the calculated graphene thickness can deviate substantially from the true values, as shown in figures 6(a)–(d). This high sensitivity of optical methods to the SiO₂ layer thickness is consistent with the theoretical analysis given in figure 4. Therefore, the accurate measurement of the thickness of the SiO₂ layer is a prerequisite for the implementation of color-based optical methods.

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In this section, we will use the optical methods to estimate the thickness of the graphene samples in a real experimental setup. The graphene samples were produced from Kish graphite by mechanical exfoliation [10] and then transferred to SiO₂/Si substrate. The thickness of the SiO₂ was found to be 299.3 ± 1.7 nm, as independently measured by ellipsometry and further confirmed by scanning electron microscopy. The optical images of the graphene samples were obtained with an Olympus BX-51 microscope under two different light sources (xenon light, Olympus AH2-RH, and halogen light, AX LH100).

Figures 7(a) and (b) are the optical images of FLG recorded under xenon light and halogen light respectively. We can see that the color varies slightly for different regions of graphene, indicating changes in sample thickness. Figure 7(c) shows a Raman G band intensity image (the brightness corresponds to the G peak intensity) obtained by Raman microscopy (NTEGRA Spectra, NT-MDT, excitation laser wavelength 532 nm) of the same selected area. From the intensity contrast and the detailed Raman spectra (figure 7(f)), the thicknesses of different regions of graphene were determined and labeled correspondingly in figures 7(a)–(c). Using the optical method based on the CIEDE2000 color difference and assuming the ideal sRGB scheme for the optical recording system, we calculated the thicknesses of different regions and compared them with the true values obtained by Raman microscopy. The results are shown in figures 7(d) and (e), based on the images collected under xenon light and halogen light respectively. Because the SPD of the real incident light was not known, we can use the known real thickness of SiO₂ to find an effective light source (details can be found in SI, available at stacks.iop.org/Nano/24/505701/mmedia). Using the 'calibrated' or 'gauged' effective light source, the thicknesses of both the graphene and the SiO₂ substrate can be correctly recovered, as shown by the filled circles in figures 7(d) and (e). Theoretically, this effective light source once calibrated should also work for other samples if the light source and the recording system are kept the same. To validate this hypothesis, we prepared another FLG sample and used the same effective light source to estimate the sample thickness. The results turned out to be accurate and consistent with independent Raman measurements (see SI for more details, available at stacks.iop.org/Nano/24/505701/mmedia).

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