Continuous and reversible tuning of the disorder-driven superconductor–insulator transition in bilayer graphene

The influence of static disorder on a quantum phase transition (QPT) is a fundamental issue in condensed matter physics. As a prototypical example of a disorder-tuned QPT, the superconductor–insulator transition (SIT) has been investigated intensively over the past three decades, but as yet without a general consensus on its nature. A key element is good control of disorder. Here, we present an experimental study of the SIT based on precise in-situ tuning of disorder in dual-gated bilayer graphene proximity-coupled to two superconducting electrodes through electrical and reversible control of the band gap and the charge carrier density. In the presence of a static disorder potential, Andreev-paired carriers formed close to the Fermi level in bilayer graphene constitute a randomly distributed network of proximity-induced superconducting puddles. The landscape of the network was easily tuned by electrical gating to induce percolative clusters at the onset of superconductivity. This is evidenced by scaling behavior consistent with the classical percolation in transport measurements. At lower temperatures, the solely electrical tuning of the disorder-induced landscape enables us to observe, for the first time, a crossover from classical to quantum percolation in a single device, which elucidates how thermal dephasing engages in separating the two regimes.


Contact resistance
Since the measured device resistance includes the contact resistance (Rc), we estimated Rc using a four-probe measurement scheme shown in Fig. S1(a). The measured four-probe contact resistance [Rc,4p = (V+ -V-)/Ibias] of the left and right contacts were -3  and -4 , respectively.
Here, V+ and V-are the electrical potential of the two electrodes and Ibias is the bias current between I+ and I-contact leads. The negative value of Rc,4p in the cross-junction geometry can be understood as that Rc is much smaller than the electrode resistance Rline (~ 7 ), resulting in nonuniform current flow along the junction #1. When Rc is sufficiently larger than Rline, bias current flows uniformly through the junction along the vertical direction and each top and bottom electrode becomes equipotential. This results in voltage difference V = V+ -V-to be positive [ Fig. S1(b)] and Rc,4p well represents Rc. However, when Rc is sufficiently smaller than Rline, the electrode on top and the graphene layer at bottom behave as a single piece with the current flow becoming nonuniform along the junction. In this case, V-gets higher than V+, which leads to a negative value of Rc,4p. This feature is confirmed in the numerical simulation for different values of Rc in Fig. S1(c) and S1(d). Simulation was done by commercial package COMSOL Multiphysics with the same geometrical and electrical parameters of the device. As Rc gets smaller than Rline, Rc,4p becomes negative and saturated to the value of -Rline [ Fig. S1(e)], which is close to the experimentally measured value of Rc,4p. This ensures that Rc of our device was an order of a few ohms, which was negligible compared to the device resistance (a few hundreds ohms).

Coordinate transformation of the resistance map
Experimentally, we constructed a resistance map as a function of the bottom gate and top gate voltages (Vb and Vt, respectively) as shown in Fig. S3

Josephson coupling in the superconducting phase
When the square resistance becomes smaller than the quantum resistance, the superconducting phase emerges in the region of bilayer graphene layer. As discussed in the main text, the superconducting phase is induced by the proximity effect from the superconducting electrodes. In this section, we present the genuine Josephson coupling via the bilayer graphene layer, confirmed by microwave irradiation and applying perpendicular magnetic fields on the bilayer-graphene Josephson junction. When a microwave was irradiated on the Josephson junction, the beating of ac voltage and the ac Josephson effect generated equidistant voltage steps in the current-voltage characteristics [ Fig. S4(a)], which is known as Shapiro steps 1 . In Fig. S4(b), the voltage step size V shows highly linear relationship with the irradiated microwave frequency fmw as V =hfmw/2e with Planck's constant h and electron charge e.
Another unique feature of Josephson junction is periodic oscillation of critical current (Ic) with applied perpendicular magnetic field (B), which is known as Fraunhofer pattern 1

Heat dissipation by electron-phonon coupling in bilayer graphene in low temperature regime
The saturation behaviour of resistance by the dissipative Joule heating shown in Figs.

S5(a), (b), and (c) gives information about the electron-phonon coupling in the bilayer graphene
Josephson junction device. Crossover temperature (T0) and the saturation resistance (R) correspond to the electron temperature in association with the base sample holder temperature and the dissipative power P = I 2 R, respectively, with bias current I = 1 nA r.m.s. Most of the heat generated by the bias current is dissipated via electron-phonon coupling, since hot electron diffusion into the electrodes can be ignored due to the exponentially suppressed quasiparticle density of states of the lead (Pb) superconducting electrode 3 . Also, we can assume that the phonon of bilayer graphene is fully thermalized to the temperature of the silicon oxide substrate since the interfacial thermal resistance is a few orders of magnitude smaller than the thermal resistance between electron and phonon of the bilayer graphene 3,4 . Here, the interfacial thermal resistance at low temperature is estimated by extrapolating the experimental data in Ref. [4]. Fig. S5(d) displays the relation between crossover temperature and P along with the bestfit curve of P = A(Tel  -Tph  ), giving the best-fit value of electron-phonon coupling exponent  = 2.8 ± 0.1 for the (base) phonon temperature Tph = 50 mK and the coefficient A = 77 ± 14 fW•K -2.8 . Here, we assumed that the electron temperature (Tel) at the base temperature is saturated to T0.
The exponent  = 2.7 ± 0.1 was also determined by the slope in double logarithmic plot in Fig.   S5(e), assuming that Tph  term was negligible compared to Tel  for Tel > 100 mK. The exponent  was smaller than 4 and close to 3, which mimicked the electron-phonon coupling in disordered monolayer graphene systems in millikelvin temperature range 3,5 . This low value of the exponent ( < 4) makes bilayer graphene system a unique platform for the bias-dependent finite-size scaling studies for the independent determination of a dynamical critical exponent. This sharply contrasts with ordinary two-dimensional electron systems 6 (with  = 4 − 7), which are easily driven into 'dangerous' regime where Joule heating significantly enhances the electron temperature and thus obscures the quantum critical scaling behaviour.

Finite-size scaling with bias electric field
As discussed in the main text, the bias current (I) dependence of Rsq is also differentiated into two phases and enables finite-size analysis on electric field (E). Analysis similar to the one in Fig. 4(b) is adopted to determine the exponent v(z+1), but now (dR/dx)x=0 is plotted as a function of I as shown in

Estimation of the number of graphene layers
To identify the number of graphene flakes, we used the intensity contrast in the green light range 8 . Fig. S6(a) shows the optical image of graphene flakes exfoliated on a highly electron-doped Si substrate capped with a 300-nm thick SiO2 layer. Green light contrast (Cgreen) of the graphene flakes shows the linear relationship to the number of graphene layers as shown in Fig. S6(b). Bilayer graphene part (region 2) was selected to fabricate the dual-gated bilayer graphene Josephson junction device in this study.