## 3 Results and Discussion
From the plots in Figures 4 and 5 the following conclusions can be drawn, as also summarized in Table 1: the substrate with a 70 nm SiO\({}_{2}\) thickness is the one that shows the highest contrast and contrast derivative in the range of 0-10 nm, thus representing the most suitable candidate for the exfoliation and identification of thin flakes with a specific layer number. The 280 nm substrate is also a relatively good one for thin flakes. In the range of 10-40 nm, the substrates with SiO\({}_{2}\) thicknesses of 90 nm and 297 nm are good candidates, as they show both a high contrast and a high-contrast derivative. Increasing the hBN thickness into the range of 40-50 nm, a SiO\({}_{2}\) thickness of 271 nm is suggested, followed by thicknesses of 70 nm and 280 nm; while substrates with 280 nm and 297 nm SiO\({}_{2}\) thicknesses are suitable in the 50-65 nm range and in the 65-100 nm range, respectively. Figure 1:0 (**a**) 2D maps displaying the magnitude of the RGB/gray contrast, C\({}_{X}\) (where X = R, G, B, or L), in absolute value as a function of hBN thickness in the 0–200 nm range, for several SiO\({}_{2}\)/Si substrates with SiO\({}_{2}\) thicknesses of 70 nm, 90 nm, 150 nm, 215 nm, 271 nm, 280 nm, and 297 nm (the contrast data for the 90 nm substrate were obtained through an analysis of the RGB values measured by [PERSON] et al. in Supplementary Table SI of Ref. [59], while those for the 215 nm, 271 nm and 297 nm were obtained by analyzing the optical micrographs displayed by [PERSON] et al. in Figure 15d–f of Ref. [55]). (**b**) Analogous 2D maps for the contrast derivative, D\({}_{X}\). The gray areas correspond to missing data in that hBN thickness range. The 150 nm substrate could also represent an option, but only in the 80-95 nm range. If the determination of the exact thickness is not crucial, however, the 90 nm substrate shows the highest contrast across the entire range of 10-100 nm, followed by the 297 nm substrate. Above 100 nm, the 90 nm SiO\({}_{2}\) is generally the most suitable substrate featuring a sizable contrast, followed by the 280 nm one. However, their derivative is not always high. More specific information on the 100-200 nm range can be found in Figure 1:1  
A comparison between the experimental results and the calculated optical contrast suggests that the use of a proper substrate leading to a high-contrast derivative allows for the identification of hBN thicknesses with very high precision of just a couple of layers \begin{table}
\begin{tabular}{c c c} \hline
**hBN Thickness** & **Optimum SiO\({}_{2}\) Thickness** & **Second Choice SiO\({}_{2}\) Thickness** \\ \hline
0–10 nm & 70 nm & 280 nm \\
10–40 nm & 90 nm & 297 nm \\
40–50 nm & 271 nm & 70 nm, 280 nm, (90 nm, 297 nm) \\
50–65 nm & 280 nm & (90 nm, 297 nm) \\
65–100 nm & 297 nm & (90 nm, 271 nm, 70 nm) \\
100–115 nm & 70 nm & (90 nm, 280 nm) \\
115–128 nm & 90 nm & 280 nm \\
128–137 nm & 70 nm & 150 nm, 280 nm, (90 nm) \\
137–141 nm & 150 nm & 280 nm, (90 nm) \\
141–150 nm & 90 nm & (280 nm) \\
150–155 nm & (90 nm) & (280 nm) \\
155–180 nm & 90 nm & (280 nm) \\
180–200 nm & 70 nm & (90 nm, 280 nm) \\ \hline \end{tabular}
\end{table} 
Table 1: Summary of the most suited substrates for the observation of hBN flakes. For every hBN thickness range, the substrate giving the highest contrast and derivative is displayed. The substrates in the last column are also good and represent the second-choice substrates. The substrates within brackets are characterized by a relatively high contrast, but a relatively low derivative. Figure 5: Zoom-in of the 2D plots from Figure 4 in the hBN thickness ranges between 0.3 nm (1 layer) and 30 nm in log-scale. (**a**) 2D maps displaying the magnitude of the contrast, C\({}_{X}\). (**b**) Analogous 2D maps for the contrast derivative D\({}_{X}\). (while a much larger uncertainty up to tens of layers is obtained when the derivative is small).