Approaching ballistic transport in suspended graphene. Supplementary Information

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1 Approachig ballistic trasport i suspeded graphee Xu Du, Iva Skachko, Athoy Barker, Eva Y. Adrei Departmet of Physics & Astroomy, Rutgers the State Uiversity of New Jersey Cotets Supplemetary Iformatio A. Graphee preparatio B. Suspedig the graphee device C. Measuremets D. Impact of lead geometry A. Graphee preparatio Graphee depositio is doe usig a method similar to the oe itroduced by the pioeerig work i ref. S1. Prior to graphee depositio, the Si/SiO substrates were baked i formig gas (Ar/H ) at 00C for 1 hour to remove water ad orgaic residue. A thi foil of highly orieted pyrolytic graphite (eutro detector quality) was peeled from the bulk material usig scotch tape ad trasferred oto the Si/SiO substrate with a fie pair of tweezers. Pressure was the applied oto the graphite foil usig compressed high purity itroge gas through a stailess steel eedle, for ~5 secods. The foil was the removed from the substrate ad the substrate was carefully checked uder a optical microscope for cadidates of sigle layer graphee. This process is repeated util a few graphee flakes ca be idetified. The sample was the immediately coated with PMMA resist for further E-beam lithography. 008 Macmilla Publishers Limited. All rights reserved.

2 B. Suspedig the graphee device The suspeded graphee (SG) devices were fabricated from covetioal o-suspeded graphee (NSG) devices with Au/Ti leads deposited o Si/SiO (300m) substrates. After the NSG devices were made, the samples were coated with PMMA. Ad a additioal e- beam lithography step was carried out to ope two small PMMA widows (typically 0. ~ 0.5μm squares) o the two sides of the graphee chael (illustrated i Figure1). The samples were the immersed i 7:1 (NH 4 F: HF) buffered oxide etch. Etchig was doe at 5C 0 for 6.5 mi. Due to the weak couplig of graphee to the substrate, capillary actio draws the etchat udereath the whole graphee film. Hece, the etchig actually starts i the etire graphee chael shortly after the sample is immersed. The isotropic etchig therefore results i the whole device (the graphee ad the leads attached to it) to become suspeded (Figure ). After the etchig, the etchat was replaced by DI water, the hot acetoe (to remove the PMMA) ad fially hot isopropaol, with the sample remaiig the liquid at all times. Fially the sample was take out of the isopropaol ad left to dry. Normally the suspeded device would be destroyed by wickig of the liquid at this poit. However due to the small surface tesio of hot isopropaol, devices with chael legth smaller tha 1μm were foud to survive the process with high success rate (Figure 3). The SG samples were baked i formig gas (Ar/H ) at 00C 0 for 1 hour to further remove orgaic residue ad water right before the measuremets. 008 Macmilla Publishers Limited. All rights reserved.

3 Supplemetary Figure 1. SEM image of a SG device i the middle of its fabricatio process. The sample, started from a covetioal device o substrate, was coated with PMMA. holes were opeed at the sides of the graphee chael i PMMA mask. The BOE etch removes the SiO udereath the device, ad (before the removal of the PMMA) creates a PMMA bridge which protects the graphee from bee collapsed by the surface tesio. I the fabricatio of the actual devices, the samples were left i liquid without take out util the PMMA mask is removed. Supplemetary Figure. SEM image of the same device show i Figure 1. This figure illustrates the possible problems oe may ecouter durig the fabricatio of a SG device. Sectio I: the graphee bridge was too log (~3m), hece got tor apart by the surface tesio whe the solvet dried out. Sectio II: a short sectio of graphee which was successfully suspeded. The gold leads were sigificat sagged by the surface tesio, due to the reduced mechaical stregth as a result of over etchig ad over broadeig of the ditch (this device was etched for 10 mi). Sectio III: without support from the leads, graphee collapsed oto the back gate. For actual workig devices, the edge of the graphee has to be covered ad supported by the leads. Supplemetary Figure 3. SEM image of a workig SG device. The chael legth betwee the voltage leads is ~0.6 µm. The etchig time is 6.5 mi. 008 Macmilla Publishers Limited. All rights reserved.

4 C. Measuremets The trasport properties of the SG samples were measured with a stadard 4-probe locki techique, usig a 17Hz 100A excitatio curret. To prevet bedig of graphee from electrostatic force ad breakdow of the gas dielectric (helium exchage gas), we limit the gate voltage to a rage of ±0V. Figure 4 shows the gate voltage depedece of resistivity at various temperature from 300K to 4.K, for a SG device (L=0.6μm, W=1.6μm). The asymmetry betwee the electro ad hole braches revealed by the ρ V ) curves was observed i all the SG devices we fabricated thus far. We also observed the asymmetry i NSG samples with similar geometry (voltage leads placed across the curret path) for short lead separatio (L< 1 μm). We fid that the asymmetry decreases with icreasig lead separatio ad becomes quite small i samples with the largest separatio or whe the voltage leads are placed i a o-ivasive maer (such as i the Hall bar geometry). This suggests a lik betwee the asymmetry ad the presece of the leads. However, the mechaism by which the leads ca itroduce such asymmetry is still poorly uderstood. Some possible scearios iclude: cotamiatio associated with the depositio of leads; dopig at the cotact betwee the metal ad graphee, which may alter the Desity of states o oe of the carrier braches; a geometrical effect associated with the boudary coditios imposed by the voltage leads. A detailed study of the effect of leads will be udertake i the future. At low temperatures (T<0K), oscillatio i ρ V ) were observed. These oscillatios are due to iterferece of carriers boucig back ad forth betwee the leads with wavelegth tued by the gate voltage. ( g ( g 008 Macmilla Publishers Limited. All rights reserved.

5 Supplemetary Figure 4. Gate voltage depedece of resistivity at various temperature idicated i the leged, for a SG device with L=0.6m, W=1.6m. D. Impact of lead geometry The geometry of the voltage leads imposes the boudary coditios o the electroic wavefuctios ad therefore it is oe of the most importat factors i the measured mobility of ear-ballistic samples. I the case of ideal ballistic juctios measured i a - lead geometry, the coductace is cotrolled by the trasmissio probabilities, T, of the allowed electroic modes which are defied by the boudary coditios (Tworzydlo, J. et. al., PRL. 96, 4680 (006)): L 4e σ = T (1) W h here: T = k μ k cos( kl) i si( kl) () + v F where k = ( q ), = ( + 1/ ) μ v F 1 q π, ad W is the width of the juctio. W 008 Macmilla Publishers Limited. All rights reserved.

6 Whe k is imagiary, the modes are evaescet ad the trasmissio is egligible, so we μ μw oly cosider the terms where k is real, hece > q, or <. vf v F π With k beig real, T = μ μ ( vf q ) ( v q ) cos ( k L) F hece: σ ~ L W 4e h μw it v π F = 0 1 μ μ ( vf q ) ( v q ) cos ( k L) F L 4e μw 1 x ~ dx W h π vf 0 μl 1 x cos 1 x vf (4) π vf here x = ( +1/ ). μw μ L π For large, the itegral i equatio (4) approaches, which yields: v F 4 e Lμ σ ~ h v F e vf N( EF ) τ comparig with the semiclassical Boltzma result: σ =, we obtai the mea free path mfp = vfτ ~ L /. Thus the boudary coditios limit the mea free path of the device to ½ the lead separatio. This result is almost idepedet of the Fermi eergy except at the Dirac poit, as demostrated i the more precise umerical calculatios show i supplemetary Figure 5a. Cosequetly, the mobility of ballistic -lead devices is fiite (3) ad reflects a device mobility which depeds o gate voltage as μ V 1 / g ( show i supplemetary Figure 5b). By cotrast, for juctios measured i a Hall-bar geometry, the mea free path is ot limited by the voltage leads so the mobility usually measures the perfectio of the material, or the material mobility. 008 Macmilla Publishers Limited. All rights reserved.

7 I our devices, the voltage leads ru across the width of the sample (- lead geometry) to provide structural support for the suspeded graphee. Therefore, the measured mobility i our suspeded graphee devices reflects the device mobility. Its value is affected by the disorder i graphee but it is bouded by the value imposed by the voltage lead separatio. This is supported by the observed gate voltage depedece of the mobility i our geometry. Whe the lead separatio i a -lead measuremet is much smaller tha the mea free path of the material, the measured mobility is the device mobility ad does ot give access to the material-limited mobility. I the opposite limit, whe the lead separatio is much larger tha material mfp, the measuremet does probe the material mobility. The experimets described here are at the crossover betwee the two limits. While the two lead geometry does ot give a direct measure of the material-limited mobility, it does provide experimetal access to the physics of the D Dirac Fermio system, icludig geometrically defied modes ad iterferece effects, which have thus far oly bee addressed theoretically a L=1μm L=0.5μm b 10 6 L=1μm L=0.5μm mfp (m) mobility (cm /Vs) Vg (V) Vg (V) Supplemetary Figure 5. Numerically calculated gate voltage depedece of mobility ad mfp for ideal ballistic juctios with idicated geometries, for graphee juctios o 300m SiO. 008 Macmilla Publishers Limited. All rights reserved.

8 Supportig Referece S1. K.S. Novoselov et.al., Sciece 306, 666 (004) 008 Macmilla Publishers Limited. All rights reserved.