SUPPLEMENTARY INFORMATION

SUPPLEMENTARY INFORMATION SUPPLEMENTARY INFORMATION Trilayer graphene is a semimetal with a gate-tuneable band overlap M. F. Craciun, S. Russo, M. Yamamoto, J. B. Oostinga, A. F. Morpurgo and S. Tarucha Thickness determination of single-layer, bilayer and trilayer graphene Recent development in the fabrication of few layer graphene (FLG) devices has established that the optical contrast between FLG and the SiO /Si substrate is an efficient and unambiguous method to identify the number of graphene layers [, ]. The optical contrast of FLG can be maximized by using the appropriate thickness of the SiO layer and the light wavelength. The best contrast is achieved on a 8nm thick SiO using green light []. We have prepared FLG on top of an oxidized Si substrate (8nm SiO ) and determined their thickness by analyzing the intensity of the flakes optical micrographs in the green RGB channel[]. Fig. S shows a selection of FLG flakes formed by regions of different thickness used in our investigations. To identify the number of graphene layers we extract the greenchannel component of the RGB value on the flake (G f ) as well as on the substrate (G s ), and we evaluate the relative (green-channel) shift RGS =(G s G f )/G s. This quantity eliminates the contribution from the substrate and from the light intensity of the source, thus it allows a direct comparison of FLG prepared on different substrates. The plot of the RGS extracted from the images of 9 flakes clearly shows steplike changes with increasing their optical contrast (see Fig. S). Thus we find that all the regions marked with SG in the images of Fig. S have RGS values around RGS SG.6, whereas the BG and TG regions have RGS values respectively RGS BG RGS SG =. and RGS TG RGS SG =.8. Quantum Hall measurements and transport experiments on double-gated FLG with RGS SG.6 conclusively demonstrate that these flakes are single layer graphene (see Fig. S8). Transport measurements through double-gated FLG with RGS BG. reveal an increase in resistance with decreasing temperature (as should be expected from the opening of a gap in the low-energy density of states), stemming from their bilayer nature. FLG with RGS TG.8 exhibit the systematic decrease of resistivity with increasing the electric field that is the subject of this work. This analysis shows that both nature nanotechnology www.nature.com/naturenanotechnology

RGS and transport experiments in double-gated FLG provide two independent methods which can be successfully used as a tool for determining the precise number of graphene layers. Relative Green Shift..8..6 Number of layers. 6 7 8 9 Flake counter TG SG SG TG BG TG BG µm µm SG BG SG µm TG SG BG µm SG BG TG µm BG 8µm TG Figure S : The relative green shift plotted for 9 FLG flakes deposited on a Si substrate with a 8nm SiO layer. The blue, red and yellow dots correspond to the RGS of the single layer, bilayer and trilayer graphene whose transport data are shown and discussed in detail in the main text. The optical images show a selection of FLG flakes (viewed under green light) formed by areas of different thickness. nature nanotechnology www.nature.com/naturenanotechnology

Contact resistance in trilayer graphene devices Our experiments on double gated devices rely mostly on two-probe measurements (R P ), whereas measurements on the Hall bar devices were performed in a four-probe configuration (R P ). The difference between R P and R P is the contact resistance at the graphene/metal interface (R C ). So far little is known on R C, which a priori can be a function of applied V bg. Here we show that, consistently with previous findings by Russo et al. [], R C is a weakly gate dependent quantity in triple layer graphene thus it cannot account for the observed electric field dependence of transport properties in double gated trilayer graphene devices. We successfully determined the value of the contact resistance on the same devices used to extract δε by comparing two-probe measurements in samples fabricated on the same flake, with different aspect ratio (W/L, W sample width and L contacts separation). R P is given by R P (V bg )=R gr (V bg )+R C (V bg ) with R gr = ρ gr (V bg )L/W (ρ gr (V bg ) gate dependent resistivity). We can then determine quantitatively the contact resistance as a function of back-gate voltage from the following expressions: R C =(R Dev P ρ gr (V bg )L Dev /W Dev )/ where ρ gr (V bg )=(R Dev P RP Dev )(L Dev /W Dev L Dev /W Dev ). This analysis conducted on several triple layer graphene devices systematically shows that R C is weakly dependent on the back-gate voltage[] (see green curves in the plots of Fig. Sa and b). We have later also measured the contact resistance on a number of other devices by comparing two terminal and four terminal measurements (see Fig. Sc)and by scaling experiments as a function of contact separation and found identical, gate-voltage independent values with all these different methods. The plot in Fig. d shows the measured R C for each trilayer device. It is apparent that the contact resistance at trilayer graphene-ti/au interface is R C 8Ωµm independent of the temperature at least up to 77K (see Fig. Sd). The fact that R C is weakly dependent on the back gate voltage and typically its value is less than % of the total resistance at the charge neutrality point in our devices, demonstrates that in our experiments R C cannot account for the observed decrease in the maximum resistance as a function of perpendicular electric field applied in double gated trilayer devices. nature nanotechnology www.nature.com/naturenanotechnology

a) R(KΩ) R(KΩ) 8 TG - - - - TG R p R p Rc - p p Rc R c W(Ω µ m) b) R(KΩ) V bg c) d) 6..8. R p R p Rc T=mK T=.K T=77K V bg 6 TG Number of samples Figure S : a) and b) show in blue and red plots of -probe resistance vs V bg for different devices fabricated on two distinct trilayer graphene flakes and measured at T=mK (a) W =.6µm, W =.9µm and fixed L =.µm; b) W =.66µm, W =.µm and fixed L =.µm; Curves in green are plots of the contact resistance extracted from R P as described in the text. c) Gate-voltage dependence of R C obtained from the comparison of two and four probe measurements at T=.K (W =.µm, L =.9µm). d) Summary plot of the contact resistance R c W determined on 6 different trilayer devices and at three different temperatures T=mK,.K and 77K. Comparison of two- and four-terminal measurements In order to minimize the possible effect of disorder on the experiments, we strived to fabricate as many samples as possible on a same flake, so that the spatial proximity of the devices would help to exclude that the nature and amount of disorder would vary very strongly from device to device. For this reason we conducted most of the transport measurements in a two-terminal configuration, since by including a smaller number of contacts we could fabricate more devices on a given flake. Nevertheless, in several cases we have nature nanotechnology www.nature.com/naturenanotechnology

also extracted δε from four-terminal measurements (R P ). The clear conceptual advantage offered by this device configuration is that R P does not contain the resistance at the metalgraphene interface (R C ), thus it offers a direct way to probe intrinsic transport properties to trilayer graphene. In Fig. Sa and b we present plots of the -probe square resistance as a function of and V bg for the device with top gate labeled by A (L =.µm and W =8.µm) see micrograph picture in the inset of Fig. Sc). It is apparent that the maximum resistance R max decreases with increasing the electric field applied perpendicular to the trilayer graphene, just as observed in the two-terminal measurements (see Fig. in the article). Assuming the two band model with m =.m -see the main text of the article- we can estimate the band overlap δε for each different external electric field E ext (see Fig. Sc). The electric field tunability of δε achieved in a four-terminal device is fully consistent with the one estimated in a two-terminal geometry (see Fig. in the main article). Furthermore, the fact that two and four terminal geometries are equally suitable to investigate the band overlap in double gated trilayer graphene, is conclusively proved by comparing estimates of δε from R P and R P measured on the same device. In Fig. Sa and Fig. Sb we show respectively R P and R P versus for different fixed V bg measured in the multiprobe device in the inset of Fig. Sc. The overall electric field behavior of the measured resistance is similar in both probes configurations, whereas R P differs from R P by a gate independent quantity corresponding to the contact resistance (R C = 8 ± 6Ωµm) -in good agreement with scaling experiments by Russo et al. []. Estimates of δε from both measurements show a remarkable agreement given that R C is much smaller than the total measured resistance in a -probe configuration (see Fig. S). A plot of δε estimated for 6 different devices and for different multi-probe geometries shows that a similar electric field modulation of δε is achieved in all cases Fig. S6. The precise values of external electric field necessary to induce a certain band overlap differ, however, from sample to sample (about %). These variations in the electric field are likely the consequence of different electric field screening at the SiO/graphene interfaces due to unwanted contamination during the fabrication, also reflected in a different doping level of the graphene flakes. nature nanotechnology www.nature.com/naturenanotechnology

a) R (KΩ) b) Vtg 8 R (KΩ) V bg 9 8 7 - - c) - 8 6 V bg - 8 δ ε (mev) 7 6 A µm gate gate B gate C -. -..... E ext (V/nm) Figure S : a) shows a color coded plot of the -terminal square resistance versus V bg and measured in the device with top gate labeled by A (L =.µm and W =8.µm) at T=mK. In the inset in the panel c) it is shown a micrograph picture of the measured device. Panel b) shows sweeps for fixed V bg of the -probe square resistance, corresponding to vertical cuts of the color coded plot of panel a). c) shows a plot of the estimated δε from the measurements of panel a) assuming the two band model with m =.m as described in the main text of the article. 6 nature nanotechnology www.nature.com/naturenanotechnology

a) b) R p (KΩ) c) 8 6 - Vtg V bg 9 8 7 6 R p (KΩ) 8 6 - Vtg V bg 9 8 7 6 δε (mev) µm gate 6 p p p-r c -.... E ext (V/nm) Figure S : a) and b) show respectively -probe and -probe resistance measurements versus for fixed V bg at T=.K. In c) we plot the corresponding estimates for the δε from R P and R P (blue and red dots) together with the estimate of δε from the two probe resistance minus the contact resistance (green dots). Two probe transport measurements in other trilayer and single layer graphene devices In addition to the measurements on the trilayer graphene double-gated device shown in the main text, consistent results have been obtained on other samples. Data from a selected number of other trilayer devices (named TG, TG and TG) are shown in Fig. S7. nature nanotechnology www.nature.com/naturenanotechnology 7

7 µm δε(mev) 6 Gate -.... E ext (V/nm) p - R c p Figure S : a) show δε estimated from R P and R P R C at T=mK (i.e. -probe resistance minus the contact resistance) for the sample of Fig. a in the main article. Figures S7a and S7b show data from sample TG, plotted in the same way as in Fig. a and b in the main text -i.e. the dependence of the square resistance (R ) on the voltage applied to one of the gates, with the other gate kept at a constant potential. Furthermore, Figures S7c and S7d show R for samples TG and TG as a function of the voltage applied to the back gate (V bg ), with the voltage on the top gate ( ) kept constant. All these measurements consistently show that the maximum resistance decreases as the top and back gates are biased with opposite voltages of increasing magnitude (i.e. the external electric field increases) and it also shifts as a function of applied voltage to both gates. Fig. S8a shows transport measurements through double-gated single layer graphene devices. In contrast to trilayer graphene, the value of the maximum resistance for single layer graphene remains constant when increasing the electric field consistent with previous reported results on similar devices []. The single layer character of the FLG used in this device was confirmed by quantum Hall effect (QHE) measurements. The conductance versus V bg measurement presented in figure S8b shows clearly Hall plateaus at half-integer 8 nature nanotechnology www.nature.com/naturenanotechnology

δε(mev) 8 7 6 dev, p dev, p dev, p dev, p dev, p dev6, p dev6, p -. -..... E ext (V/nm) Figure S 6: Summary plot of the estimated δε for the 6 different devices measured in two and four probe configuration (dev: T=mK, W =.78µmL =.µm; dev: T=.K, W =.6µmL =.µm; dev: T=.K, W =µml =.µm; dev: T=.K, W =.8µmL =.µm; dev: T=mK, W =8.µmL =.µm; dev6, P: T=.K, W =.µml =.9µm; dev6, P: T=.K, W =.µml =.µm). multiples of e /h, characteristic of the QHE in single layer graphene [6, 7]. Note on supplementary figure S9 As discussed in the main text, the temperature dependence of the total carrier density in the mixed state n is used to estimate the band overlap δε and its dependence on the perpendicular electric field. Specifically, the fitting of n(t )/n(.k) at all the different values of the perpendicular electric field relies on δε as the only free parameter. To complement Fig.b of the main text we show in Fig. S9 the temperature dependence of n(t )/n(.k) and its fitting for positive top gate voltage. nature nanotechnology www.nature.com/naturenanotechnology 9

a) V V -V b) V -V V V V V bg -V -V -V R (KΩ) TG - - V bg R (KΩ) V TG -. -. -. -...... R (KΩ) 6 c) V TG 6 d) V -V V V -V V V -V -V V R (KΩ) V V -V -V TG -V - - V bg - - V bg Figure S 7: Additional transport measurements in double-gated trilayer graphene devices. a,c & d, Square resistance of the trilayer versus back-gate voltage (V bg ) measured for different values of the top gate voltage ( ). The value of R max with increasingly large opposite voltages. systematically decreases when both gates are biased b, Trilayer square resistance versus top gate voltage measured for different values of the back gate voltage showing a similar gate-voltage dependence of the height of R max. All the data have been measured at T=.K. nature nanotechnology www.nature.com/naturenanotechnology

a) V V V -V V bg - V R (KΩ) - -. b). G(e /h).... - - V bg Figure S 8: (a) Square resistance of the single-layer graphene versus top-gate voltage ( ) measured for different values of the back-gate voltage (V bg ) at T=mK. The value of the maximum resistance remains constant when increasing the electric field. (b) Two terminal conductance versus back gate voltage (V bg ) at high magnetic field (B=8T) and T=mK showing plateaus at half-integer values of e /h, characteristic of the QHE in single-layer graphene. nature nanotechnology www.nature.com/naturenanotechnology

n(t)/n(.k).... =V =.8V =.6V =.V =.V =V. T(K) Figure S 9: Temperature dependence of carrier concentration in the mixed state for positive top gate voltage. Note on the two quadratic bands approximation In our analysis we have assumed that the low energy TG band structure is described by two quadratic bands which have an overlap δ. This approximation disregards the detailed low energy TG band structure, and it will be important to check if an analysis of the data in terms of the precise band structure of TG (i.e., not relying on the two parabolic bands only) is consistent with our observations. Theoretical results based on tight binding model -containing the detailed low energy electronic properties-[8] appear to confirm that also when the fully detailed low energy TG band structure is taken into account, good agreement with our experimental results is obtained. These calculations also shed light on the origin of the band overlap, which is due to the non-zero value of the transfer integrals γ and γ, rationalizing why a sizable band overlap appears in trilayers, but not in singleand bi-layers. nature nanotechnology www.nature.com/naturenanotechnology

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