SUPPLEMENTARY INFORMATION

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1 Collapse of superconductivity in a hybrid tin graphene Josephson junction array by Zheng Han et al. SUPPLEMENTARY INFORMATION 1. Determination of the electronic mobility of graphene. 1.a extraction from the field effect curve The electronic mobility of the tested sample can be extracted from conductivity by the relation: ne. The carrier concentration n of electrons or holes in the graphene channel regions can be approximated 1 by g n n n, where n 0 is the residual charge carrier concentration at charge neutrality point, and n C ( V V )/ e is the charge carrier concentration induced by the electrostatic g g d backgate, while V d is the gate voltage at the charge neutrality point, and C is the capacitance per unit area of the Si/SiO 2 backgate. The field effect curve therefore can be fitted by the following formula: L R = + R We n 2 0 +[C (V g V d ) / e] 2 c, (2) where L and W are graphene length and width and R c is the contact resistance. Here C / e cm V ed is the charge carrier concentration induced per volt, with d = 285 nm the thickness of the oxide we used, and the dielectric constant of SiO 2. The field effect curve is fitted as a black dashed line in Fig.S1b, showing an effective electronic mobility of about cm 2.V -1.s -1. n 0 is a fitting parameter characterizing the amplitude of charge fluctuations at the vicinity of charge neutrality point. The fit leads to n 0 = cm -2, which corresponds to the charge induced by a gate voltage shifted by 5.9V from the charge neutrality point. Notice that the collapse of superconductivity happens at about 10 V away from the charge neutrality point, and NATURE PHYSICS 1

2 SUPPLEMENTARY INFORMATION the divergence from theoretically expected BKT transition happens at 20 V away from the charge neutrality point (Fig. 2b in the main text). That is to say, the physical effect observed in the paper takes place at electrostatic doping levels above the threshold where electron-hole puddles dominate. 1.b extraction from classical Hall effect Another way to obtain independently a quantitative estimate of the mobility is provided by performing classical Hall measurement, since the Hall resistance R H is defined as: Vxy 1 R H, (3) I B ne xx where V xy is the transverse voltage, while I xx is the longitudinal biasing current (1 na in our case). In Fig.S1a, R H is measured as a function of the magnetic field at V g = 30 V and 100 mk. The charge n is then extracted by the slope of a linear fit in Fig.S1a, found to be about cm -2, which corresponds to an electronic mobility of ~ 675 cm 2 V -1 s -1, in agreement with the one extracted with Equation (2) at 4 K. Figure S1: a) all resistance of the tested sample obtained at V g =30 V, temperature 100 mk, and a bias current of 1 na. b) Field effect measured at 4 K at zero magnetic field. Dashed line corresponds to the fit using Eq. (2). 2 NATURE PHYSICS

3 SUPPLEMENTARY INFORMATION 2. Determination of mean free path and electronic diffusion coefficient of graphene The electron diffusion coefficient D for a 2D material is given by D v l /2,where l is the mean free path, and v F the Fermi velocity. We assume v F ~10 6 m/s in graphene. The F mean free path l is related to sample conductivity by the formula 2 /2 F l h e k. Here is the Fermi wavelength in graphene, with ~ 3.9, the dielectric constant, and d=285 nm the thickness of SiO 2 wafer. By using the normal state field effect curve at 4 K, gate-dependence of l and D can be extracted as shown in Fig.S2. Figure S2: a) Gate-dependence of mean free path l in graphene, extracted in the normal state field effect curve at 4 K temperature, b) Diffusion coefficient calculated from curve in a) according to the relation D =v F l / 2. NATURE PHYSICS 3

4 SUPPLEMENTARY INFORMATION 3. Estimation of the Andreev conductance at the tin-disk/graphene interface. In the main text, Equations (1)-(2) are valid under the assumption that the Andreev interface between tin metal disks and graphene is transparent, i.e. tin/graphene interface conductance is much higher than the quantum of conductance. To verify that devices follow this assumption, one can estimated the Andreev conductance from the drop of resistivity of the device at the superconducting transition of tin nanoparticles. The area of a unit cell of the triangular array is Aunit 2 3 b /2. The coverage fraction 2 2 of disks decorating graphene is then given by ( a ) /( 3 b / 2) ~ 14.5%. The tin disks decorating graphene can be seen as a set of metallic resistors placed in parallel with the graphene plane and connected to it with an interface conductance G int. When the disks starts to become superconducting, these resistors will shunt the normal graphene underneath and the resulting device resistance drop will depend on both G int and the coverage ratio 2. As seen in Fig. S3, a sharp drop of resistance is measured precisely at the tin superconducting transition, without any temperature delay and its relative amount exceeds the tin dot coverage ratio. Assuming a good Andreev contact, the tin disks should shunt the graphene at the transition, with a relative resistance drop at the superconducting transition of dots that must be at least be equal to the coverage fraction. Experimental data supports a highly transparent interface. The relative drop at the upper transition R/R N is found to depend only on the charge carrier type (hole or electron) as shown in Fig. S3d. A relative drop of about 20% is seen in the electron side, a value exceeding the tin disk coverage fraction. Therefore, it is reasonable to assume G int 2e 2 / h in our system. Note that Sn being 4 4 NATURE PHYSICS

5 SUPPLEMENTARY INFORMATION an electron donor, the Schottky barrier at the interface is higher on the hole side 2 thus explaining why a smaller relative drop of resistance is measured under the regime of graphene hole conduction. Figure S3: a) Temperature dependence of resistance measure at gate voltage of 10 V from 300 K to 60 mk. b) Temperature dependence of resistance measured at several gate voltages, plotted as line-cuts along the temperature axis in Figure 2b of main text. c) Field effect curves before (4K) and after (3.3 K) the superconducting transition of Sn disks. d) R/R N obtained by subtracting curves at 4 K and 3.3 K in b). charge neutrality point of the sample is V d ~ -13 V. R/R N on the hole side is about half of the one on electron side. NATURE PHYSICS 5

6 SUPPLEMENTARY INFORMATION 4. Non saturation of the electronic temperature at 60 mk A levelling-off of the device resistance while cooling is the hallmark of a zero-temperature metallic state. A direct way of excluding some spurious heating effects is to apply a weak magnetic field, thus suppressing the saturation effect. An example is given in Fig. S4, for V g ~ -8 V, where this levelling-off behaviour disappears, and the resistance keeps on increasing with lowering temperature under a perpendicular magnetic field of 30 mt. Figure S4: emperature dependence of resistance at 0 mt and 30 mt for gate voltage V g = -8 V. No levelling-off is observed when a 30 mt magnetic field is applied. 6 NATURE PHYSICS

7 SUPPLEMENTARY INFORMATION 5. Enhancement of critical current in the glassy superconductor region 6 In the theory describing a low-temperature superconducting glassy state, 4-6 critical current is strongly reduced in the reentrant region, as a consequence of weakened Josephson couplings. We have shown in the manuscript a reentrant behaviour of H c2 at 60 mk and V g = 30 V. here, we show the measurement of dv/di-i bias at the same temperature and gate voltage. As shown in Fig. S5a, the supercurrent is almost totally suppressed at ~ 2.7 mt, as indicated by the green dashed line. However, a re-entrant behaviour, though very weak, is also found under an upper magnetic field, at ~ 3 mt, indicated by the red dashed line. Three bias current linecut of dv/di-i bias map of Fig. S5a are shown in Fig. S5b, highlighting a resurgence of the critical current. Figure S5: a) Magnetic field dependence of current-biased differential resistance at V g = 30 V, 60 mk. b) Individual dv/di curves at 2, 2.7, and 3.3 mt. Critical current is first suppressed at around 2.7 mt, then get enhanced again at around 3.3 mt.and in the reentrant region (H ~ 3mT), critical current is much weaker in comparison with the main superconductive region (0 mt< H < 2.2 mt). NATURE PHYSICS 7

8 SUPPLEMENTARY INFORMATION References: 1 Kim, S. et al, Realization of a high mobility dual-gated graphene field-effect transistor with Al 2 O 3 dielectric, Appl. Phys. Lett.94, (2009). 2. Courtois, H., Gandit, P., Mailly, D. & Pannetier, B. Long-range coherence in a mesoscopic metal near a superconducting interface. Phys. Rev. Lett. 76, (1996). 3. Huard, B., Stander, N., Sulpizio, J. A. & Goldhaber-Gordon, D. Phys. Rev. B 78, (R) (2008). 4. Feigel man, M. V. & Ioffe, L. B. Theory of Diamagnetism in Granular Superconductors. Phys. Rev. Lett. 74, (1995). 5. Kagan, D. M., Ioffe, L. B. & Feigel man, M. V. Quantum glass transition in a periodic long-range Josephson array. Zh. Éksp. Teor. Fiz. 116, (1999) [JETP 89, 781 (1999)] Galitski, V. M. & Larkin, A. I. Disorder and Quantum Fluctuations in Superconducting Films in Strong Magnetic Fields. Phys. Rev. Lett. 87, (2001). 8 NATURE PHYSICS