Identification of the monolayer thickness difference in a mechanically exfoliated thick flake of hexagonal boron nitride and graphite for van der Waals heterostructures

Exfoliated flakes of layered materials, such as hexagonal boron nitride (hBN) and graphite with a thickness of several tens of nanometers, are used to construct van der Waals heterostructures. A flake with a desirable thickness, size, and shape is often selected from many exfoliated flakes placed randomly on a substrate using an optical microscope. This study examined the visualization of thick hBN and graphite flakes on SiO2/Si substrates through calculations and experiments. In particular, the study analyzed areas with different atomic layer thicknesses in a flake. For visualization, the SiO2 thickness was optimized based on the calculation. As an experimental result, the area with different thicknesses in a hBN flake showed different brightness in the image obtained using an optical microscope with a narrow band-pass filter. The maximum contrast was 12% with respect to the difference of monolayer thickness. In addition, hBN and graphite flakes were observed by differential interference contrast (DIC) microscopy. In the observation, the area with different thicknesses exhibited different brightnesses and colors. Adjusting the DIC bias had a similar effect to selecting a wavelength using a narrow band-pass filter.


Introduction
Van der Waals heterostructure devices constructed from layered materials of various substances have been studied intensively because of their novel electrical and optical properties. Layered materials can obtain an atomically flat, ideal interface and a precisely controlled number of layers. Nanotechnology Nanotechnology 34 (2023) 295701 (10pp) https://doi.org/10.1088/1361-6528/accf23 * Authors to whom any correspondence should be addressed.
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The exfoliated flakes are prepared from bulk crystals with adhesive tape [8]. After mechanical exfoliation, the flakes are transferred to a substrate. Since a large number of flakes with different size, shape, and thickness are placed randomly on a substrate, appropriate flakes for constructing a heterostructure are found in them. Generally, a large and flat isolated flake with the same number of layers is ideal for use in heterostructures. However, an exfoliated flake often has areas with different atomic layer thicknesses. In addition, residues of tapes for exfoliation are present on the flake surface. These disturb the construction of ideal interfaces in a heterostructure. Thus, an optical technique that can provide high contrast for finding ideal flakes is required for constructing a desirable heterostructure.
The presence of a flake of a layered material placed on a substrate can be visualized by optical microscopy from the difference between the reflection intensity from the flake surface and the substrate surface. In the first paper on mechanically exfoliated graphene, a thermally oxidized Si substrate with 300 nm-thick SiO 2 layer was employed for the visualization [8]. The visibility of an atomically thin flake depends on the thickness of SiO 2 layer by optical interference, where a 90-or 300 nm-thick SiO 2 provides a large difference in reflection intensity [9,10]. Therefore, a thermally oxidized Si substrate with a 90 or 300 nm-thick SiO 2 layer has been commonly used in layer materials research.
The reflection intensity is discussed based on the reflectance (R) governed by interference effects. To visualize a flake of a layered material on a substrate, the optical contrast is defined as C n = (R n − R sub )/R sub = ΔR/R sub = R n /R sub − 1 where n is the number of layers for the layered material, R n is the reflectance for the surface of the flake, R sub is the reflectance for the surface of the substrate, and ΔR = R n − R sub . Since R n and R sub depend on wavelength (λ), the observation at a wavelength that R sub is small leads to the visualization of a flake with high contrast [9][10][11][12][13][14][15][16][17][18][19][20]. For a Si substrate with a 90 or 300 nm-thick SiO 2 layer, R sub is small in the visible wavelength range and ∼10% at λ = ∼550 nm [9,10]. When observing a flake of n = 1-5, C n increases linearly with n [13,14]. Therefore, n can be estimated from the C n value obtained by the observation.
Another issue is to observe the difference in n for a thick flake of n > 5. Although ΔR for a thick flake is large, it is difficult to recognize the difference in n for a flake since (R n − R n−1 )/R n is small. Hence, (R n − R n−1 )/R n rather than C n is a suitable definition of contrast for the difference in n for a thick flake. Therefore, to visualize the difference in n, the thickness of a SiO 2 layer should be adjusted so that R n is small.
Some groups have been studying the visualization of an ultra-thin film using a substrate with low reflectance, such as a Au/SiO 2 /Si [11,12,[16][17][18] or SiN x /Si substrate [15,19,20]. A monolayer (1 L) hBN flake can be visualized with high contrast using such a substrate. A transparent hBN flake with a 1 L thickness of 0.333 nm is generally difficult to visualize. One of the groups realized the C 1 = 12% for 1 L hBN using an optimized SiN x /Si substrate with R sub = ∼0% at a certain λ. The C 1 = 12% value is larger than C 1 = 2.5% for an optimized SiO 2 /Si substrate with R sub = ∼10% [15]. For a hBN flake with n atomic layer thickness, an area with a 1 L difference in the flake with high contrast might be observed by adjusting R n to zero. Therefore, a study that intentionally reduces R n to zero is needed to visualize an area with a thickness difference of a few atomic layers in a thick hBN flake.
This study investigated the visualization of thick hBN and graphite flakes placed on SiO 2 /Si substrates by optical microscopy to visualize areas with different numbers of layers in a thick flake. A flake with a thickness of 4-100 nm is called a thick flake in this study. This is because hBN flakes, approximately 30 nm in thickness, and graphite flakes, a few nm in thickness, are used in van der Waals heterostructure devices [21][22][23][24][25][26]. Thus, this study focused on visualizing thick flakes. The thickness of the SiO 2 layer plays an important role in the visualization of a flake. Thus, for optimizing the SiO 2 thickness, the optical reflectance for a substrate with a flake was examined through calculations. In the experiment, hBN and graphite flakes were observed using a narrow band-pass filter and by differential interference contrast (DIC) microscopy. The images captured with a digital camera are discussed in terms of the color predicted from the calculation. Figure 1(a) shows a schematic diagram of a flake consisting of a layered material placed on a SiO 2 /Si substrate examined in this study. Thermally oxidized n-type Si (100) substrates, the bulk hBN crystals grown at high pressure and temperature [27], and Kish graphite (Graphene Supermarket) were used for the samples. The Si substrate with a 95 nm-thick SiO 2 layer was cleaned by sonication in acetone for 5 min and 2-propanol for 5 min. The SiO 2 thickness was adjusted to 20-77 nm by chemically etching the SiO 2 surface in a 48 wt% HF solution diluted to 5 vol% with deionized water. After exposing the substrate to UV/ozone for 15 min for clearing, single crystal thin flakes of hBN or graphite were prepared by mechanical exfoliation using scotch tape (BK-12N, Scotch) and transferred immediately to the substrate. Although the ideal isolated thick and flat flakes with the same number of layers can be obtained by the preparation, flakes that have atomic steps were focused to study the identification of the monolayer thickness difference in a flake. The image was taken using a monochrome 12-bit camera (CS-63M, Bitran) cooled to 10°C or a color 8-bit camera (EOS Kiss X4, Canon). Superior images were obtained by adjusting the aperture stop so that the light disk at the objective back focal plane was ∼80% of that for fully open. This corresponds to a substantial reduction in NA. A narrow band-pass filter was inserted in the optical path to investigate the wavelength dependence. The full width at half maximum (FWHM) of the narrow band-pass filter was 10 nm. The experimental contrast was calculated from the digital values in the image photographed by the monochrome camera [12,15]. A flake was observed by DIC microscopy using two linear polarizers and a Nomarski prism without a narrow band-pass filter. The linear polarizers were arranged on cross Nicol. The DIC bias (Δ 0 ) was adjusted by changing the position of the Nomarski prism. The reflectance spectra were measured using a spectroscope (BTC-110S, B&W Tek) equipped at the trinocular head of the microscope. Since the spatial resolution of the microscopic measurement is practically less than 3 μm, the reflectance of the area only inside the flake was measured. The morphology of the substrate surface was observed by atomic force microscopy (AFM, NanoNavi, SII) with a Si cantilever (SI-DF3-R, SII Nano-Technology, Japan) in tapping mode at room temperature in ambient air (50% relative humidity, ∼20°C). The root-mean-square roughness of the SiO 2 surface was typically 0.2 nm. The thickness of the SiO 2 film was estimated from the spectra obtained by ellipsometry (Auto SE, Horiba) and reflectance spectroscopy.

Calculation of optical contrast
The reflectance spectra and photographs were acquired through an objective lens. Therefore, the oblique incidence should be considered in the calculation. Reflectance R(λ) at a wavelength λ is expressed as using θ 0 is defined by sin θ 0 = NA. Here r p (λ, θ) and r s (λ, θ) are the complex reflectivities for the p and s components of the incident light tilted from the surface normal with an angle θ, respectively [14,18,[28][29][30]. NA was set to 0.7 for all calculations. r p (λ, θ) and r s (λ, θ) were calculated based on a multilayer model using the transfer matrix method [13]. The multilayer structure was composed of an hBN or graphite film, a SiO 2 film, and a Si substrate, as shown in figure 1(a). The refractive indices of hBN and graphite were set to 2.2 and 2.6−1.3i [9,31] in the calculation, respectively, where i is an imaginary unit. When the sample was photographed under a light passing through a narrow band-pass filter, the reflectance detected by the camera corresponds to the reflectance averaged for λ in the range of λ′ − λ FWHM /2 to λ′ + λ FWHM /2. λ′ is the central wavelength of the filter, and λ FWHM is the FWHM. Thus, the detected reflectance R ( ) l can be approximated as The contrast of a flake with an n-layer thickness with regards to a substrate is defined as C n = R n /R sub − 1, as described in the introduction. Here, R n is the reflectance for the flake surface, and R sub is the reflectance for the surface of the substrate. Thus, the spectrum of calculated contrast is expressed as - [12,15]. However, the contrast of the area with k-layer thickness in a thick flake with n-layer thickness is defined by The color simulation for micrographs was performed from R(λ). R(λ) was converted to XYZ color space using the following equations: x , ( ) l y , ( ) l and z¯( ) l are the CIE color matching functions; Λ denotes the integration on [380 nm, 780 nm]; I(λ) is the illuminant; K is a constant. The light source of a halogen lamp was assumed to be black-body radiation of 3500 K. K is adjusted and is proportional to the exposure time to reproduce the photograph. The X, Y, and Z values are converted to the standard RGB values. Figure 1(c) presents a schematic diagram of the R sub , R 1 , R n−1 , and R n spectra for a flake with an n-layer thickness shown in figure 1(a). The wavelength at which the spectrum has the minimum shifts to a long wavelength as the atomic layer thickness is increased [17,18]. This study focused on visualizing a hBN flake rather than a graphite flake. This is because the visualization of a hBN flake, which is optically transparent, is more complex than that of a graphite flake, which is not transparent. First, the contrast for a 1 L hBN flake on a SiO 2 /Si substrate was examined to visualize areas with a 1 L difference in a thick hBN flake discussed below. Figure 1(d) shows the contrast calculated for a 1 L hBN flake on SiO 2 /Si substrate (C 1 ) as a function of λ and SiO 2 thickness (d SiO2 ) [10]. As defined in section 2.3, the contrast is given by

Results and discussion
On a SiO 2 film with a thickness of 60-140 nm or 260-330 nm, the contrast is approximately ±3% at a certain λ [9].
The contrast for a thick hBN flake was next examined. In particular, this study focused on the contrast of a hBN flake with d BN = 40 nm (n = 120) as an example. Figure 2(a) shows the C n calculated as a function of d BN and λ for d SiO2 = 90 nm. The C n increased as d BN was increased and reached the maximum at d BN = ∼60 nm since the R sub is constant, which was attributed to the increase in R n . For d BN = 40 nm, R sub , R 120 , and C 120 are calculated to be 11.13%, 58.59%, and 426.6% at λ = 500 nm, respectively. A hBN flake of d BN = 40 nm in an image taken at λ = 500 nm should be brighter than the substrate on which the flake is placed since C 120 is a positive value. Ifthe flake has an area of 1 L thinner, R 119 and C 119 are 58.38% and 424.8%, respectively. The difference of R 120 and R 119 was 0.21%, which was 0.0037 of R 120 = 58.59%. Thus, it is difficult to visualize a 1 L difference if a general 8-bit camera with linear sensitivity to light intensity is used. The difficulty is explained as follows. An 8-bit data image gives 256 shades expressed by digital integers (D) of 0 to 255. The D is a value of ctI(λ)R(λ) as an integer where c is a camera constant, and t is the exposure time. The value of (R 120 − R 119 )/ R 120 , 0.0037, is close to 0.0039 = 1/256. Therefore, the D value for R 119 is equal to or one less than that for R 120 . This causes difficulty in visualization. As described in section 2.3, −(R k /R n − 1) is suitable for quantitatively evaluating the difference in atomic layer thickness in a flake. Thus, the contrast for the difference in atomic layer thickness in a thick flake is defined as C k ′ = −(R k /R n − 1). According to the definition, the contrast of a 1 L thickness difference is expressed as C' n−1 = − (R n−1 /R n − 1). Figure 2(b) shows the C' n−1 calculated as a function of d BN and λ for d SiO2 = 90 nm. For d BN = 40 nm, the C' 120−1 value was 0.37% at λ = 500 nm. Figures 2(a) and (b) show that a SiO 2 /Si substrate of d SiO2 = 90 nm is suitable for visualizing a thick flake. In contrast, the SiO 2 /Si substrate is unsuitable for confirming the area with the difference in atomic layer thickness in the flake. Figures 2(c) and (d) show the C n and C' n−1 for 20 nm-thick SiO 2 as a function of d BN and λ. The C n value for 20 nm-thick SiO 2 is lower than that for 90 nmthick SiO 2 over a wide range, including d BN = 40 nm. In contrast, the absolute value of C' n−1 for a 20 nm-thick SiO 2 is larger than that for a 90 nm-thick SiO 2 over a wide range. In particular, the C' 120−1 value was 12.2% at λ = 500 nm. The small R 120 value of 0.78% contributes to the large C' 120−1 . In addition, the C 120 value was 98% at λ = 500 nm, and a hBN flake of d BN = ∼40 nm should be observed. Thus, a SiO 2 /Si substrate of d SiO2 = 20 nm is suitable for visualizing the area with the difference in atomic layer thickness in the flake.
As the calculation results in the above paragraph, R sub and R n are essential factors for visualizing a flake and an area with the difference in atomic layer thickness in the flake, respectively. Thus, the spectra of R sub and R n were measured experimentally. Figure 3 shows the spectra of R sub and R n obtained from measurement and calculation. The calculation results indicated by the light colors roughly reproduce the experimental spectra for all cases. The solid dark green and blue lines in the figure indicate R sub for d SiO2 = 20 nm and 95 nm, respectively. R sub for the 95 nm SiO 2 film was smaller than that for the 20 nm SiO 2 film, as shown in figure 3. The red and black lines show R n for 41 nm hBN on a 20 nm SiO 2 film and 6.3 nm graphite on a 77 nm SiO 2 film, respectively. The SiO 2 thickness was optimized to realize small R n . R n for a thick hBN and a graphite flake on the optimized SiO 2 /Si substrate is close to zero at a certain λ. Figure 4 presents the results obtained for a 45 nm (136 L) hBN flake on a 20 nm SiO 2 film by optical microscopy and AFM. Figure 4(a) is a color image taken under white light. A flake with a navy-blue color was observed. Figure 4(b) is a monochromatic image taken under light passing through a band-pass filter of λ = 530 nm, corresponding to the dotted area in figure 4(a). Some steps are shown in figure 4(b). This result indicates that using an appropriate band-pass filter contributes to the visualization of steps. In addition, the exposure time for capturing the image was adjusted for R n ,  being different from that for R sub since R sub is higher than R n . As a result, the D of the area of the substrate saturates and is fully white in the image. Figure 4(c) shows an AFM height image corresponding to the area enclosed by the solid black line in figure 4(b). Figure 4(d) shows the two height profiles along the green lines labeled A and B in figure 4(b). The step height is 0.30 nm and 0.57 nm for lines A and B, respectively, corresponding to the thicknesses of 1 L and 2 L, respectively. Thus, the downside area for lines A and B is an area of n = 135 and 134, respectively. Figure 4(b) presents the measured n values. The steps can be recognized in an image taken using a commercial color camera. Figures 4(e) and (f) show color images observed under light through a band-pass filter of λ = 520 and 590 nm, respectively. These images have not been processed by software. For λ = 520 nm, the area of 136 L has a bright color compared with the surrounding area. For λ = 590 nm, the area of 136 L has a dark color compared with the surrounding area. The brightness and darkness are explained by C' n−1 , as shown in figure 2(d). At d BN = 45 nm, the C' n−1 value is positive at λ = 520 nm and negative at λ = 590 nm, corresponding to the brightness and darkness, respectively. Figure 5 shows the C' n−1 spectra for d SiO2 = 20 nm obtained by measurements and calculations. The red and black plots represent the experimental spectra for d BN = 41 nm and 45 nm, respectively. The solid lines with a light color, by calculation, roughly reproduce the experimental spectra. The difference between calculation and experimental spectra may be caused by scattering light from the surrounding thick flake. For d BN = 41 nm, the experimental spectrum has the maximum |C' n−1 | of 12% at 570 nm. The C' n−1 value for a 1 L difference in thickness is comparable to the contrast of a 1 L hBN flake on the antireflection of silicon nitride substrate [14]. When a desirable flake is found from a large amount of exfoliated flakes on a substrate, the flake often has small fragments or tape residues on the surface. The present technique can also visualize them (figure S1).
For figure 5, the SiO 2 thickness, d SiO2 , is fixed at 20 nm. Next, the dependence of d SiO2 on C' n−1 based on calculation results for the realization of a large contrast was investigated. The maximum |C' n−1 | in a wavelength range is used to evaluate the contrast since the C' n−1 value depends on λ. The observation of a flake is often conducted in a visible wavelength range using a halogen lamp. Thus, the maximum |C' n−1 | was defined in the range of 450-750 nm as the maximum contrast, C' max . The wavelength range of λ < 450 nm was excluded because of the low luminance of a halogen lamp and the low sensitivity of a common camera in the range.   C' max is limited in the range of 450-750 nm. Figure S3 shows the C' n−1 spectra for a certain d BN .
A consumer color camera is often used when a hBN flake with a desirable thickness is found. The color in an image of a hBN flake captured by a color camera can be predicted by calculations. Such a predicted color is useful for finding a hBN flake. Figure 6(b) represents a color map calculated for a hBN flake on a SiO 2 /Si substrate as a function of d BN and d SiO2 . The color gradually changes with d BN and d SiO2 . The color at d BN = 0 corresponds to that of the SiO 2 /Si substrate. For example, the color for d SiO2 = 20 nm is a wheat color at d BN = 0 and a navy color at d BN = 45 nm. The thickness corresponds to the sample shown in figure 4(a). The calculated color roughly reproduces the color in the image captured with the actual digital camera. The data of C' max and color simulation of hBN for thicker SiO 2 are shown in figure S4.
A narrow band-pass filter enables the visualization of a 1 L thickness difference in a thick hBN flake, as shown in figure 4(b). In this technique, a band-pass filter with the appropriate λ for d BN of a hBN flake was selected. Another technique that does not require a narrow band-pass filter was proposed. A DIC microscope also enables the visualization of 1 L thickness difference. Figure 1(b) presents the setup for DIC microscopy. The optical components installed on a commercial reflected microscope were used for DIC microscopy. Figures 7(a)-(c) shows an image of a 45 nm hBN flake on 20 nm SiO 2 film observed under white light using a DIC microscope with different DIC bias, Δ 0 . The flake is the same as that for figure 4. The observed color of the flake and substrate depends on Δ 0 , which an operator can adjust. The color of the flake is black for Δ 0 = 0, blue for Δ 0 = ∼750 nm, and wine-red for Δ 0 = ∼300 nm. Figures 7(b) and (c) show that the flake has some areas colored with different colors shown in figures 4(e) and (f). This result indicates the presence of atomic steps.
Generally, DIC microscopy converts the gradient (Δ 1 ) of the refractive index and/or the optical path length in an observation specimen into intensity in an image. Therefore, when a thick hBN flake on a substrate is observed by DIC microscopy, edges with thickness differences are highlighted in an image. The flake and substrate colors are black in figure 7(a) for Δ 0 = 0. By contrast, the edges of thickness difference for the flake and wrinkles are highlighted by the white and yellow lines because of the large Δ 1 , respectively. Therefore, the shape of the thick hBN flake is recognized by the white shape outlines. However, the presence of atomic steps cannot be recognized in the image since the highlighted  lines of the atomic step edge were not observed. This result suggests that Δ 1 of 1 L hBN is insufficient to change the intensity in the DIC image.
DIC bias colorizes the flake and substrate in addition to the highlighted lines of the thickness difference. The color of the flake and substrate in figure 7(c) for Δ 0 = ∼300 nm is wine-red and yellow, respectively. Furthermore, the edges of large thickness differences are also highlighted in the image. The white edge lines in figure 7(a) were changed to green by the DIC bias in figure 7(c). These colors that depend on Δ 0 are interference colors. In addition, DIC bias visualizes the atomic steps. Unlike the steps of large thickness difference, the presence of atomic steps can be recognized in figures 7(b) and (c) by the color difference, not outlines ( figure S5). In this case, the color difference in atomic steps might be attributed to the sum of differences in R(λ) at various λ, as shown in figures 1 and 2. Thus, the color of the substrate and flake were simulated. The intensity of each λ is dominated by Δ 0 and R (λ), where the effect of Δ 1 is included in R(λ). The light intensity detected by a camera, I DIC (λ, Δ 0 ), can be expressed by [32] The light intensity with λ is zero at Δ 0 = λl (l = 0, 1, 2, K). Therefore, a Nomarski prism is an optical component that continuously tunes the light source spectrum with a trigonometric function by the DIC bias. Consequently, images of a flake with colors depending on atomic layer thickness were obtained by adjusting the DIC bias instead of using a bandpass filter. A similar investigation with a hBN flake on a SiO 2 /Si substrate of d SiO2 = 20 nm was also conducted for a graphite flake on a SiO 2 /Si substrate of d SiO2 = 77 nm. Figures 8(a)  and (b) show the photographs of a graphite flake with areas of various thicknesses taken under white light and λ = 490 nm illumination, respectively. For the flake, the minimum difference in the atomic layer thickness was 2 L. The step on the surface of the graphite was observed compared with that of the hBN flake with a 2 L difference in figure 4(b). The clear observation was attributed to the extinction coefficient in the refractive index of graphite [9]. The |C' n−1 | value for graphite is larger than that for hBN. The C' n−1 for d gra > 5.5 nm and d gra < 5.5 nm is positive and negative at λ = 490 nm, respectively. This explains the contrast pattern in figure 8(b), where the brightness at the position of 14 L (4.66 nm) and 20 L (6.66 nm) is higher than that of 16 L (5.33 nm) and 18 L (6.00 nm). Note that the positive and negative contrast region in the contour map for C' n−2 is similar to that for C' n−1 (figure S6). Figure 8(d) shows the experimental C' n−1 spectrum for the different graphite flake of d gra = 4.66 nm having a 1 L thickness difference with calculations. The black dots for the experiment indicate an enhancement of the contrast of 33% at λ = 510 nm, which is reproduced with the calculation represented with the solid line. Figure 8(e) shows the calculated C' max as a function of d gra and d SiO2 . The SiO 2 /Si substrate of d SiO2 = ∼75 nm is suitable for finding a flat graphite flake with d gra = ∼5 nm. Figure 8(f) shows the simulated color for graphite on SiO 2 /Si substrate as a function of d gra and d SiO2 . The color of the graphite and substrate is approximated with the simulation.

Conclusions
We attempted the visualization of thick hBN and graphite flakes on SiO 2 /Si substrates, particularly the area with different thicknesses. The reflectance of a thick flake on SiO 2 /Si substrate depends on the SiO 2 thickness, and approaches zero at a certain λ by adjusting the SiO 2 thickness with respect to the flake thickness. The reflectances of 41 nm hBN on 20 nm SiO 2 and 6.3 nm graphite on 77 nm SiO 2 were close to zero at λ = ∼530 nm. At the wavelength that the reflectance from an area of a flake is close to zero, the contrast is sensitive to the thickness difference. Therefore, the difference in the atomic layer thickness in the flake was visualized. The contrast spectra of a 1 L thickness difference were measured using narrow band-pass filters of various λ. The maximum absolute contrast measured for a 1 L difference in thickness was 12% and 33% for 41 nm hBN on 20 nm SiO 2 at λ = 570 nm and 4.3 nm graphite on 77 nm SiO 2 at λ = 510 nm, respectively. The high contrast is sufficient to evaluate the atomic thickness difference with a consumer eight-bit color camera.
A rapid inspection technique that does not use a bandpass filter was proposed. Using the conventional setup of DIC microscopy, the same h-BN flake on 20 nm SiO 2 film was observed under white light illumination. The colors of the substrate and flakes in the DIC image are determined mainly by the DIC bias and R(λ) because of the small difference in the optical path length corresponding to the 1 L difference in thickness in the flake. In this case, DIC bias serves as an adjustable band-pass filter and restricts a light source spectrum to the wavelength determined by a trigonometric function. Therefore, the differences in atomic thickness were visualized similarly because of the sum of the difference in R(λ) for various λ. The colors of the substrate and flake in the photographed image were approximated by the calculation.
Generally, thick exfoliated flakes used for van der Waals heterostructures are obtained on a Si substrate with a ∼90 or ∼300 nm SiO 2 layer. This study shows that optimizing the SiO 2 thickness instead of using a SiO 2 layer with such a thickness contributes to selecting a desirable flake from a large number of exfoliated flakes on the substrate. The SiO 2 thicknesses optimized for ∼40 nm hBN and ∼5 nm graphite was ∼20 and ∼75 nm, respectively. The hBN and graphite flakes had purple and dark red-purplish colors in the digital camera image. The proposed observation technique is useful for constructing ideal van der Waals heterostructures.

Acknowledgments
This work was partly supported by JSPS KAKENHI Grant Numbers 21K04195, 21H04655, Kansai Research Foundation, Chubei Itoh Foundation, Iketani Science and Technology Foundation, and Hyogo Science and Technology Association.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).