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1 Supporting Information A rigorous and accurate contrast spectroscopy for ultimate thickness determination of micrometre-sized graphene on gold and molecular sensing Joel M. Katzen, Matěj Velický, Yuefeng Huang, Stacey Drakeley, William Hendren, Robert M. Bowman, Qiran Cai, Ying Chen, Lu Hua Li, and Fumin Huang * School of Mathematics and Physics, Queen s University Belfast, BT7 1NN, United Kingdom Institute of Frontier Materials, Deakin University, Waurn Ponds, Victoria, Australia * f.huang@qub.ac.uk S-1

2 r dr d θ 0 dθ θ r Figure S1 top: diagram showing the incident aperture of the objective lens (top view); bottom: schematic showing the focusing cone of light. The correspondence between the incident annular ring and the focusing cone is indicated by filled grey areas. To calculate the proportion of light incident at an angle, we assume light is uniformly incident through the input aperture (top, Figure S1) of the objective lens. This is a reasonable assumption as the size of the incident light beam is much wider than the input aperture of the objective lens. The objective can be approximated as an effective lens with NA=0.9. Following a simplified geometric optical ray path, light passing through an annular aperture (shaded area, Figure S1) between and + is focused onto the sample within the range of incident angles between and +. The amount of light is proportional to the area of the annular ring, given by where is the focusing distance. Input Eq.2 into Eq.1, we get =2 (1) = tan (2) =2 tan (sec ) (3) The reflectivity of light of one polarization (TE or TM) is calculated by S-2

3 = ( ) ( ) (4) ( ) is the reflectivity at the incident angle θ. is the total area of the incident aperture of the objective lens (the top circle in Figure S1), = (tan ) (5) where the maximum incident angle is decided by the numerical aperture of the objective lens, sin = ( =64 for the 100x objective, NA=0.9). Input Eq.5 into Eq.4, we get = ( ) ( ) (6) ( ) In the experiments, non-polarized white light source was used in the reflectivity measurements. Assuming there is equal distribution of the transverse-electric (TE) and transverse-magnetic (TM) polarization component of light, the calculated averaged reflectivity is given by, = ( + ) (7) and is given by Eq.6 for the corresponding polarization. Figure S2 shows the calculated reflectivity of a monolayer graphene ( =2.6, =1.3, = ) on the Au substrate (100 nm Au, 10 nm Ti and Si substrate), using a NA=0.9 objective lens. The reflectivity spectrum of the TE (blue) polarization is different from that of the TM (green) polarization. The averaged reflectivity spectrum (solid red) based on Eq.7 matches excellently with the measured reflectivity spectrum (black) in the wavelength range between nm. This firmly validates the theoretical model. There is notable discrepancy in the short wavelengths region, which is possibly due to several factors. One possibility is that the TE and TM polarization component of light is not exactly equal in the focusing field, which will have a significant impact on the contrast in the short wavelength region, as the reflectivity of the TE and TM polarizations deviates strongly in the short wavelength region. This is likely in reality. Although the white light source is non-polarized in nature, not all optical elements (e.g., beamsplitters, optical fibers, lenses) in the experimental setup are 100% polarization conserved. If we assume there is slightly more TM component (60%) than the TE component (40%), the averaged reflectivity (dashed red) shows an improved agreement with the experimental result in the short wavelength region and keeps the excellent agreement in the long wavelength region. However, some discrepancy still remains near 400 nm. This is possibly due to the dispersion effect of the optical properties of graphene. In the simulation, the refractive index of graphene is assumed to be constant ( = ), independent of the wavelength. In reality, this is not true. It varies with wavelength, and this dependence becomes more pronounced in the short wavelength region 2,3. In addition, the optical system has a low efficiency in the short wavelength region, which may also contribute to the large deviation in the short wavelength end. S-3

4 Reflectivity Measured Simulated, TE Simulated, TM Simulated,0.5(TE+TM) Simulated, 0.4TE+0.6TM Wavelength (nm) 1000 Figure S2 Comparison between the measured (black) and calculated optical reflectivity (color) of a monolayer graphene on a 100nm Au film, blue: TE polarization; green: TM polarization; solid red: averaged equally between TE and TM; dashed red: averaged with 40% TE and 60% TM. The numerical aperture of objective is 0.9. Refractive index of graphene is = , thickness nm. Refractive index of Au is adopted from ref.4. Figure S3 shows the calculated contrast spectra of a monolayer graphene on an oxidized Si substrate (90 nm SiO 2 ), with the adsorption of amorphous carbon (AC) films of various thicknesses. The maximum contrast increases and the peak wavelength redshifts with the thickness of AC film (Figure S3, b). The contrast curve can be fitted with a quadratic function, = To the first order approximation, the contrast increment gradient is 17.7%/nm. Assuming the contrast detection limit of the system is 0.5%, this indicates the graphene system can detect the adsorption of a sub-monolayer amorphous carbon with an average thickness of nm (0.5/17.7=0.028). Such a high sensitivity can be exploited to develop ultrasensitive molecular sensors. S-4

5 (a) Maximum contrast (%) (b) Contrast (%) Contrast Wavelength Peak wavelength (nm) Wavelength (nm) Carbon Thickness (nm) 2.0 Figure S3 (a) Calculated optical contrast spectra of a monolayer graphene on an oxidized Si (90 nm SiO 2 ) substrate, with the adsorption of a thin layer amorphous carbon (from bottom to top, 0-2 nm, 0.2 nm incremental step). (b) The maximum optical contrast and the peak wavelength as a function of the thickness of amorphous carbon. Refractive index of amorphous carbon is adopted from ref.5. The graphene device can also be used to detect small variations of the refractive index of the surrounding medium. For a monolayer graphene deposited on a 90 nm SiO 2 /Si substrate, simulations indicate that the contrast changes at a rate of 309% per refractive index unit (RIU), when light is incident normal to the graphene surface, as shown in Figure S4. The estimated contrast detection limit in the experiments was about 0.5%, which corresponds to a sensitivity of RIU. This is about one order of magnitude better than most localized surface plasmon resonance (LSPR) sensors, which usually have sensitivity at the order of 10-2 RIU 6. The contrast detection limit was constrained by the noise levels of microscope light source and the sensitivity of spectrometer. With a more stable light source and a more sensitive photon detector, the detection limit can be much better, hence the sensitivity of the contrast spectroscopy of 2D materials can still be significantly improved. Figure S4 (a) calculated contrast change of a monolayer graphene deposited on 90nm SiO 2 /Si substrate, as a function of the refractive index of the surrounding medium, under normal incidence condition. (b) a linear dependency of the contrast change on the variation of refractive index, indicating a contrast sensitivity of 309% per refractive index unit. S-5

6 REFERENCES 1. Born, M.; Wolf, E. Principles of Optics. 7th Ed., University Press: Cambridge, Wang, X.; Chen, Y. P.; Nolte, D. D. Strong Anomalous Optical Dispersion of Graphene: Complex Refractive Index Measured by Picometrology. Opt. Express. 2008, 16, Falkovsky, L. A. Optical Properties of Graphene. J. Phys.: Conf. Ser. 2008, 129, Rakić, A. D.; Djurišic, A. B.; Elazar, J. M.; Majewski, M. L. Optical Properties of Metallic Films for Vertical-Cavity Optoelectronic Devices. Appl. Opt. 1998, 37, Duley, W. W. Refractive Indices for Amorphous Carbon. Astrophys. J. 1984, 287, Anker, J. N.; Hall, W. P.; Lyandres, O.; Shah, N. C.; Zhao, J.; Van Duyne, R. P. Biosensing with Plasmonic Nanosensors. Nat. Mater. 2008, 7, S-6

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