Supporting Information for Graphene conductance uniformity mapping Jonas D. Buron 1, 2, Dirch H. Petersen 2, Peter Bøggild 2, David G. Cooke 3, Michael Hilke 3, Jie Sun 4, Eric Whiteway 3, Peter F. Nielsen 5, Ole Hansen 2,6, August Yurgens 4, Peter U. Jepsen 1 1 DTU Fotonik, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark 2 DTU Nanotech, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark 3 Department of Physics, McGill University, Montréal, Québec, Canada H3A 2T8 4 Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden 5 Capres A/S, Diplomvej, Building 373, DK-2800 Kongens Lyngby, Denmark 6 CINF, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark THz-TDS experimental details THz sheet conductance maps were produced from THz-TDS data recorded using a Picometrix T- ray 4000 fiber-coupled spectrometer, cf. Figs. 1(b) and (c) in the main text. The spectrometer uses femtosecond near-infrared laser pulses and LT-InGaAs photoconductive antenna(pca) chips to generate and coherently detect the electric field of ultra short THz electromagnetic pulses in the time domain 1. The samples were raster scanned in the x-y direction in the focal plane between the fiber coupled emitter and detector units to form spatial maps. Two polyethylene (PE) lenses were used to achieve better focusing and thus improved spatial resolution. Partial internal reflections from the O 2 -air interface in the substrate lead to a reflected signal that consists of multiple, periodic echoes with a temporal spacing given by the time-of-flight through the substrate, as illustrated schematically in Fig. 1(c) in the main text. Time windowing and subsequent Fourier transformation gives access to the amplitude and phase of the frequency content of each of the echoes E % out,1, E % out,2, K for every pixel in the map. The more reflections the THz pulse undergoes at the graphene-covered interface, the higher the accumulated graphene response will be. We apply an analysis which relies on the 2 nd transmitted pulse in the transmitted pulse train, as shown in Fig. 1(c) in the main text, as we found this to be the optimum trade-off between graphene signal magnitude and noise level. The inset of Fig. 1(b) in the main text shows the frequencydependent amplitude of the 2 nd transmitted pulse E% ( ),2 ω recorded at a pixel covered by graphene (black traces) and without graphene (red traces). On the basis of an analysis of the Fresnel coefficients for the given sample geometry, where the graphene film is modeled as an infinitely thin conducting film with a complex sheet conductance % σ, transfer functions relating each of the consecutive echoes to the input field can be derived for areas with and without graphene coverage. out s
The graphene sheet conductance is then related to the ratio between measured transmitted fields from sample areas with and without graphene coverage by ( ) 2 ( ) 2 ( n+ 1) n 1 Z0σ s ( 1) + 1+ % σ E% % out,2, G = E% n n Z out,2, 0 s (S.1) Where n = 3.42 is the refractive index of silicon and Z 0 = 377Ω is the vacuum impedance. The dielectric response of the 90-nm O 2 layer is so small that it can safely be ignored in the analysis. Inverting equation (S.1) results in an analytical expression for the complex frequency-dependent sheet conductance: ( ) 2n Z T% 2 2 2 2 ± na na + 4nAnBT% + 4nB T% na 2nAnB T% % σ ω = s B 0 (S.2), where T% E = % E% out,2, G out,2,, E%,2, is the complex Fourier transform of second transmitted out G pulse through a graphene-covered area, E%,2, is the complex Fourier transform of the second out transmitted pulse through an area without graphene, n = n + 1, n = n 1, n = 3. 42 is the refractive index of silicon and Z 0 =377 Ω is the vacuum impedance. We formed the average of 250 subsequent time-domain traces for the imaging, resulting in data with reliable spectroscopic information in the frequency range 0.1-1.5 THz 2. THz-TDS mapping resolution The resolution of the THz-TDS sheet conductance mapping technique is evaluated from in-situ measurements on the THz-TDS sheet conductance mapping of sample 1. The average rise distance on the left edge of the THz sheet conductance image of sample 1 in Fig. 2(b) is evaluated as a function of frequency to produce plot of the spot size FWHM vs. frequency. The result is shown in supporting figure 1. A B
Supporting Figure 1: THz spot size FWHM vs frequency. The spotsize is from the average rise distance on the left edge of the THz sheet conductance image of sample 1 in Fig. 2(b) This analysis reveals a spot size FWHM that is decreasing with increasing frequency, as is expected. The spot size in the 1.3-1.4 THz frequency range of interest is found to be approximately 0.32 mm FWHM. Raman spectroscopy and imaging Supporting Figure 2 shows Raman spectra for the two CVD graphene samples under investigation, showing distinct 2D, G and D peaks, are indicative of monolayer graphene containing few crystal defects. Supporting Figure 2: Representative Raman spectra for CVD graphene films sample 1 and sample 2. Supporting Figure 3 shows u-raman maps of sample 1, showing the spectral positions of D, G and 2D peaks as well as the ratios of the G/d and 2D/G peaks. A region of spectral redshift is observed in all 3 positional maps, indicating local strain in the graphene film. The peak ratio maps are rather uniform and featureless.
Supporting figure 3: µ-raman maps of D, G and 2D peak spectral positions and G/D, 2D/G peak ratios for sample 1 Supporting figure 4 shows µraman maps of sample 2, showing height and spectral position and peak ratios of all 3 main peaks 2D, G and D. Of all 6 maps, the G and 2D peak heights show the strongest correlation with the observed sheet conductance maps. The G and 2D peak height follow each other quite closely, indicating that graphene coverage is the main influence on these two parameters. This is also indicated in the maps of the peak height ratios G/D and 2D/G.
Supporting figure 4: µ-raman maps for sample 2. From top to bottom the maps show peak height (top), peak spectral position (middle) and (bottom) peak height ratios G/D and 2D/G. M4PP sheet conductance map blurring of sample 2 To more easily compare THz-TDS and M4PP sheet conductance maps, the spot size blurring of the THz-TDS conductance map of sample 2 can be compensated for by introducing a blurring of the M4PP sheet conductance map of sample 2. The M4PP sheet conductance map (supporting figure 5) is blurred with a Gaussian profile of 0.32 mm FWHM, which was found to be the resolution of the THz-TDS sheet conductance map. The resulting map shows a very high degree of resemblance with the THz-TDS (supporting figure 6) sheet conductance map of the same sample, except for the bright spot in the upper left corner of the THz-TDS conductance map.
Supporting figure 5: M4PP sheet conductance map, blurred with a Gaussian profile of 0.32 mm FWHM, corresponding to the estimated THz spotsize at 1.3-1.4 THz Supporting figure 6: THz sheet conductance map at 1.3-1.4 THz Variable electrode pitch M4PP measurements Examples of M4PP dual configuration sheet conductance data from a representative highly conducting region and a representative poorly conducting region is presented here. The data is obtained from 7 different A and B configurations of a 12 point probe, each with different electrode pitch. The 12 point probe was moved in 6 steps of 5 µm, measuring 7 dual configuration sheet conductances with different electrode pitches in each engage. Supporting figure 7 shows the mean sheet conductance as a function of probe pitch. Supporting figure 8 shows the full dataset of sheet conductance and R A /R B for individual engages and probe pitches in the two regions. The data shows that in a region of high conductance, where R A /R B is closer to the ideal 2D case of 1.26, the measured four point probe sheet conductance shows no significant dependence on electrode pitch. In contrast, the region with very poor conductance shows a R A /R B ratio that deviates strongly from 1.26 and is often measured to be close to 1.00, and a sheet conductance that increases with smaller electrode pitches.
Supporting figure 7: average M4PP dual configuration sheet conductance of 6 engages as a function of electrode pitch for a highly conducting region and a poorly conducting region on sample 2 Supporting figure 8: M4PP dual configuration sheet conductance and R A /R B values for different electrode pitches and positions along two line-scans in a highly conducting region and a poorly conducting region on sample 2. The data is recorded with a 12 point probe, facilitating 7 different equidistant M4PP configurations, in 6 engages with 5 µm spacing. (1) Baker, C.; Gregory, I. S.; Tribe, W. R.; Bradley, I. V.; Evans, M. J.; Withers, M.; Taday, P. F.; Wallace, V. P.; Linfield, E. H.; Davies, A. G.; Missous, M. Appl. Phys. Lett. 2003, 83, 4113 4115. (2) Jepsen, P. U.; Fischer, B. Opt. Lett. 2005, 30, 29 31.