## 3 Results and Discussion
The fluoresce from the QD was attenuated in the GO region (depicted in blue in Figure 3c) compared to the SiO\({}_{2}\) region (depicted in red in Figure 3c). We evaluated the fluorescence quenching efficiency (\(E\)) from the fluorescence intensities of the QD at the GO and SiO\({}_{2}\) regions, where the QD was under and free from the quenching effect of GO, respectively, by Equation (2):  
\[E=1-\frac{I_{\text{GO}}}{I_{\text{SiO2}}} \tag{2}\]  
where \(I_{\text{GO}}\) and \(I_{\text{SiO2}}\) are the fluorescence intensity at the GO and SiO\({}_{2}\) regions, respectively. Using the average fluorescence intensities in Figure 3c for \(I_{\text{GO}}\) and \(I_{\text{SiO2}}\), we obtained \(E=0.29\) for the QD in Figure 3. Note that \(I_{\text{GO}}\) and \(I_{\text{SiO2}}\) of a single QD are needed to calculate \(E\) because each QD has different brightness as mentioned above. We calculated \(E\) of five other QDs at different positions from their fluorescence intensities at the GO and SiO\({}_{2}\) regions (Figure S1 in the Supplementary Material). The values of \(I_{\text{GO}}\), \(I_{\text{SiO2}}\), and \(E\) of six QDs in total are summarized in Table 1. QDs #1 (Figure 3c) and #2-#4 (Figure S1a-c) showed \(E\) in the range between 0.1-0.3, whereas \(E\) of QDs #5 and #6 (Figure S1d,e) were nearly zero. A negative \(E\) value was numerically obtained for QD #6 because the variation in the fluorescence intensity was larger than the difference between \(I_{\text{GO}}\) and \(I_{\text{SiO2}}\). The difference in \(E\) among the QDs is attributed to the difference in the distance between the GO flake and the QD. The relationship between \(E\) and the distance from GO (\(d\)) based on Equation (1) is plotted in Figure 4a, and the values of \(E\) of QDs #1-#5 (Table 1) are indicated. The estimated \(d\) of QDs #1-4 were 9.4, 10.9, 10.1, and 11.9 nm, respectively, and \(d=17.4\) nm was obtained for QD #5. These values are included in Table 1. We did not calculate \(d\) for QD #6 that had a negative value of \(E\). Figure 3: (**a**) A single particle image of QDs conjugated on DOPC-SLB including DPPTE (\(5\times 10^{-7}\) mol%) and PEG-DSPE (\(C_{\text{PEG}}\) = 5.0 mol%) on the GO/SiO\({}_{2}\)/Si substrate. Representative mobile and immobile QDs are indicated with the white arrow and black arrows, respectively. Scale bar = 5 \(\upmu\)m. (**b**) The diffusion trajectory of the QD indicated by the white arrow in (**a**). Blue and red parts depict the trajectory in the GO region, and the SiO\({}_{2}\) region without GO, respectively. (**c**) Time trace of the fluorescence intensity of the QD during the diffusion in (**b**). The intensities obtained in the GO region and the SiO\({}_{2}\) region without GO are depicted in blue and red, respectively. The histogram of the fluorescence intensity in each region is illustrated on the right. The average intensity of each region is indicated with the dotted line. The QDs used in this study have a cylindrical shape, with a longer axis of 10 nm and a diameter of 5 nm, approximately [46]. Considering the size of the QD and the thickness of a single lipid bilayer (-5 nm), \(d\) = -10 nm obtained for QDs #1-#4 is reasonable as a distance from the GO to the center of a QD existing on a single lipid bilayer (Figure 4b, the left image). As shown in the AFM topographies in Figure 2a, double lipid bilayers also stacked on GO. The center of the QD on the double SLB positions at \(d\) = -15 nm (Figure 4b, the right image), where the QD is rarely affected by quenching of GO as shown in Figure 4a. QDs #5 and #6, whose \(E\) was nearly zero, existed on the double lipid bilayers on GO. The difference in the quenching efficiency was attributed to the layer numbers of the lipid bilayers. QDs are used as donor fluorophores for various FRET-based biosensing [47, 48, 49, 50]. In the FRET theory, the energy of a molecule in the excited state is transferred by the dipole-dipole interaction from a donor to an acceptor.