Fabrication of inhomogeneous 2D superconductor

Four inch intrinsic Ge (110) wafer was chosen as the starting platform for the synthesis of single crystalline monolayer graphene17,18 (Supplementary Figure 1). By using the stencil mask, the standard hall bar device matrixes were subsequently fabricated on graphene wafer. Then, 10 nm-thick elemental superconductor tin was deposited on 4 inch single-crystalline graphene wafer by electron beam evaporation. Due to the poor wettability of graphene and the low melting point of tin19,20, an array of self-assembled irregular tin nanoislands with lateral size of ~150 nm and interval of ~40 nm was formed to build graphene-tin nanoislands hybrid system (Fig. 1b, Supplementary Figures 2, 3). The single crystalline graphene provides a strict 2DEG platform, which is rather inert in the environment21 and has no grain boundary intervalley carrier-scattering effect22,23, while an array of disordered superconducting tin islands visually reproduces the hypothesized superconducting puddles14. Both cross sectional scanning transmission electron microscopy (STEM) and STEM-energy dispersive x-ray spectroscopy (STEM-EDS) also suggest that tin nanoislands are discontinuously formed on graphene/Ge (110) substrate (Fig. 1c). To avoid the possible degradation of tin nanoislands due to oxidation, the device array (Fig. 1e) was immediately diced from the whole wafer (Fig. 1d) and transferred into the dilution refrigerator for four-terminal transport measurements.

Two dimensional superconductivity

As cooling down from 8 K, the sheet resistance (R s ) of the device array at zero magnetic field firstly exhibits a semiconductor-like resistivity behavior (dR/dT<0) where the sheet resistance increases as the temperature decreases (inset in Fig. 2a), then undergoes two-step superconducting transition process, as demonstrated in Fig. 2a. It is worth noting that the intrinsic Ge (110) substrate becomes totally insulating at 10 K (Supplementary Figure 4), so the shunt effect from the substrate can be securely avoided at the temperature below 10 K. The semiconductor-like resistivity behavior can be understood in terms of a weak localization behavior proposed in 2D metals19 which is consistent with the previous observation24. The first abrupt change in the resistance slop occurs at around T Sn = 3.7 K, which is attributed to the superconducting transition of tin nanoislands as observed previously19. While the superconducting fluctuation effect of tin islands emerges far above T Sn , which drops the weak localization resistance and induces a small peak at 5.6 K, as observed in inset of Fig. 2a. When further cooling down, those superconducting tin nanoislands can couple to each other through the 2DEG in graphene as a Josephson junction arrays19,25, where the competition between the charging energy E c and the Josephson coupling energy E j of the superconducting islands is responsible for the global superconductivity19,26,27,28. Note that the coupling effect from the 2DEG in graphene is verified by the comparison experiment (Supplementary Figure 5) in which the device becomes insulating after the removal of graphene between adjacent tin nanoislands by oxygen plasma etching. Thus, the second abrupt change in the resistance slop at lower temperature can be understood as the onset of the global superconducting phase coherence aided by the 2DEG arising from single crystalline monolayer graphene.

Fig. 2 Two dimensional superconductivity of graphene–tin nanoislands array. a The four-terminal sheet resistance (R s ) versus temperature (T). The black arrow indicates the critical temperature of bulk tin T Sn = 3.7 K. Inset: R s -T curve on the semi-logarithmic scale from 4 K to 8 K. The R s peak hints the superconducting fluctuation. b The upper critical fields H c2 as a function of the angle θ. θ is the angle between the magnetic field and the surface normal direction of the device with current flowing always in plane, as sketched in the right inset. The blue solid line and the orange solid line correspond to the theoretical fitting of the H c2 (θ) using the 2D Tinkham formula \((H_{{\mathrm{c}}2}(\theta )\sin \theta /H_{{\mathrm{c}}2}^\parallel )^2 + |H_{{\mathrm{c}}2}(\theta )\cos \theta /H_{{\mathrm{c}}2}^ \bot | = 1\) and the 3D anisotropic mass model\(H_{{\mathrm{c}}2}(\theta ) = H_{{\mathrm{c}}2}^\parallel /(\sin ^2(\theta ) + \gamma ^2\cos ^2(\theta ))^{1/2}\)with \(\gamma = H_{{\mathrm{c}}2}^\parallel /H_{{\mathrm{c}}2}^ \bot\), respectively. \(H_{{\mathrm{c}}2}^\parallel\) and \(H_{{\mathrm{c}}2}^ \bot\) represent the upper critical field parallel and perpendicular to the surface of the device, respectively. c Voltage–current (V–I) curves on logarithmic scale at various temperatures. The black dashed line represents V = I3. Inset shows the extracted power-law fitting exponent α as a function of the temperature. The Berezinskii-Kosterlitz-Thouless (BKT) temperature T BKT = 2.42 K is defined by α = 3 Full size image

The upper critical field (H c2 ) of the global superconductivity as a function of θ at 2 K is demonstrated in Fig. 2b. Here, H c2 is defined as the magnetic field where the sheet resistance becomes 50% of the normal state resistance as shown in Supplementary Figure 6. It is found that the sharp peak of H c2 observed at θ = 90° can be well fitted by the 2D Tinkham model (see the blue line in the left inset of Fig. 2b), but deviates from the 3D anisotropic mass model (orange line in the left inset of Fig. 2b), indicating the behavior of 2D superconductivity. Such a θ dependence of H c2 behavior was also observed in other 2D superconducting thin films, like ZrNCl-EDLT4, SrTiO 3 -EDLT29, niobium-doped SrTiO 3 thin film30. Figure 2c demonstrates the voltage–current (V–I) curves of graphene-tin nanoislands array from 1.87 to 4 K in the log-log scale. A power-law dependence of \(V \propto I^\alpha\) behavior is observed at each temperature and the extracted exponent α increases monotonically with decreasing temperature (inset in Fig. 2c). Since the additional thermometer suggests the temperature variation is less than 0.07 K during the V–I measurement (Supplementary Figures 7-10), the observed power-law V–I behavior is not due to the effect of Joule heating, and corresponds to the Berezinskii-Kosterlitz-Thouless (BKT) type transition in 2D superconductor6,31. The BKT type transition is usually utilized to interpret the unbound vortex–antivortex pairing into bound vortex–antivortex process at low temperature in 2D superconductor, and V–I curves always meet V = I3 at the BKT transition temperature (T BKT )5,6,31. Here the T BKT equals to 2.42 K at α = 3, as indicated by the blue solid line in the inset of Fig. 2c, which is consistent with the reported BKT temperature of tin islands coupled by the exfoliated graphene flake20.

Two SITs observed in the graphene-tin nanoislands array

The measurements conducted above confirm that graphene-tin nanoislands array system behaves as a true 2D superconductor since the feature characteristics of 2D superconductivity are obviously observed. The superconductivity in 2D system can be easily tuned into a weakly localizing metal state by the perpendicular magnetic field6,9,13,14. Figure 3a (Supplementary Figure 11) demonstrates the R s (T) curves at 0 < H < 3 kOe, where the R(T) curve at each magnetic field is separated by the superconducting (dR/dT > 0) and weakly localizing metallic (dR/dT < 0) regions at certain temperature T peak (marked with arrows). As the magnetic field increases, the T peak shifts to a lower temperature and ends at the T-independent resistance region as previously observed in the LaSr 2−x Cu x O 4 thin film1. The T-independent resistance region is indicated by the critical line (the horizontal lines), which is the signature of the continuous QPT. In Fig. 3a, the two critical lines accompanied by the two T peak behaviors (pink and black arrows) suggest the presence of two distinct continuous QPTs.

Fig. 3 Two QPTs observed in graphene–tin nanoislands array. a The sheet resistance R s as a function of temperature T for different magnetic fields H from 0 to 3 kOe. The black and pink arrows indicate the R s peaks. b, c The detailed data collected from the low temperature region (0.05~0.30 K) and high temperature region (2.0~3.0 K) with the magnetic fields varying from 2.4 to 3 kOe and from 1.4 to 2.4 kOe, respectively. The black and pink dashed lines indicate the critical fields H 1 * = 1.75 kOe and H 2 * = 2.58 kOe, respectively, where the R s values are independent of the temperatures Full size image

To clarify the nature of two QPTs, we performed the detailed measurements nearby the critical lines as shown in Fig. 3b, c, respectively. It reveals there exist two critical regions where the sign of dR/dT changes as the magnetic field varies and dR/dT can approach to zero for certain magnetic field. For high temperature critical region (HTCR), the critical resistance R c and the critical magnetic field H* are R c1 = 988.4 Ω and H 1 * = 1.75 kOe, respectively. While, R c2 = 1044.4 Ω and H 2 * = 2.58 kOe are obtained in low temperature critical region (LTCR). It is rather interesting that both critical resistances R c1 = 988.4 Ω and R c2 = 1044.4 Ω are significantly lower than the quantum resistance for Cooper pairs R Q = h/4e2=6.45 kΩ32. The early experiments on InO x film showed that the disorder of the system had significant influence on the critical resistance13. For the strong disordered materials, the critical resistance approaches R Q as predicted by the dirty boson model, while the weak disorder system undergoes the QPT from the superconducting state to an metallic phase with R c < R Q 13. Thus, our graphene-tin nanoislands array system (R c1 /R Q = 0.15 & R c2 /R Q = 0.16) agrees with the weak disorder picture13 due to the weak electronic disorder of single crystalline monolayer graphene.

Two QPTs identified by the FSS

The continuous QPT in Fig. 3 can be further confirmed by the FSS analysis7. The FSS states that the R s (H) near the continuous QPT obeys the relationship \(R_{\mathrm{s}}(H,T)/R_{\mathrm{c}} = F(|H - H_{\mathrm{c}}|T^{ - 1/zv})\), where H is the magnetic field, H c is the critical field, R c is critical resistance and F is an arbitrary function with F(0) = 132. The parameter v is the correlation length exponent, z is the dynamical scaling exponent and \(\delta = |H - H_{\mathrm{c}}|\) is the absolute value of distance from the transition, which determine the spatial correlation length ξ and the temporal correlation length ξ τ in ξ ~ δ−v and ξ τ ~ ξz in the vicinity of the continuous QPT. We thus performed the FFS analysis and re-plotted the same sheet resistance data displayed in Fig. 3 as a function of H in Fig. 4a, c. It is observed that two sets of R s (H) curves exactly converge at the crossing points (R c1 , H 1 *) and (R c2 , H 2 *), respectively (Supplementary Figure 12). The values of crossing points, i.e., R c1 = 988.4 Ω, H 1 * = 1750 Oe, R c2 = 1044.4 Ω and H 2 * = 2580 Oe are consistent with the critical values obtained in Fig. 3. For both HTCR and LTCR, the R s decreases at H < H c * while increases at H > H c * when the temperature decreases, which reveals the critical crossing points are temperature independent and separate different quantum ground states.

Fig. 4 Scaling behavior of the superconductor-insulator quantum phase transition in graphene–tin nanoislands array. a Sheet resistance R s as a function of magnetic field H for different temperatures from 2 to 2.5 K. The black arrow indicates the critical magnetic field and the critical sheet resistance. b Finite-size scaling analysis of QPT by utilizing the same data extracted from a. Inset: the temperature as a function of the scaling parameter t(T) on a log-log scale (see the main text for definition of t). The power-law fitting determines zv = 0.63 ± 0.01. c Sheet resistance R s as a function of magnetic field H for different temperatures from 0.05 to 0.2 K. d Finite-size scaling analysis of QPT utilizing the same data extracted from c. Inset: the temperature as a function of the scaling parameter t on a log-log scale. The power-law fitting determines zv = 3.85 ± 0.10 Full size image

For HTCR, to yield the best collapse, the method described in ref.11 was utilized to obtain the appropriate exponent product zv (Supplementary Figures 13, 14). Each magnetoresistivity isotherms curve in Fig. 4a was re-plotted in the form \(R_{\mathrm{s}}(H,T)/R_{\mathrm{c}} = F(|H - H_{\mathrm{c}}|,T)\) and multiplied by the factor t to acquire the best collapse into the lowest temperature curve with the t(T 0 ) = 1 at the lowest temperature T 0 . The temperature dependence of the parameter t is the formula: t = (T/T 0 )−1/zv. The product zv is determined by plotting t dependence of T on a log-log scale with the straight line’s slope equaling to the −zv. The T versus t plot reveals the exponent product value zv = 0.63 (Fig. 4b inset). Using this zv value, the measured R–H curves (Fig. 4a) can be scaled into one single curve as shown in the Fig. 4b with respect to the single scaling variable \(|H - H_{\mathrm{c}}|T^{ - 1/zv}\). zv = 0.63 is consistent with zv = 2/3 observed in QPT induced by perpendicular magnetic field in 2D superconductors, like LaTiO 3 /SrTiO 3 interface14, amorphous Nb 0. 15 Si 0. 85 12, amorphous Bi10 and ultrathin high T c superconductor1. The dynamical scaling exponent z usually equals to 1 in the consequence of long range Coulomb interaction between charges32,33 as observed in ultrathin Bi film11, amorphous MoGe9, unless the particular case in 4He porous media with short range Coulomb interactions34,35. Hence we can obtain correlation length exponent v = 0.63 approximately to 2/3, if takes z = 1. Actually, the continuous QPT in 2D superconducting system can be described by the (2+1)D XY model7. The v = 2/3 is expected in clean (2 + 1)D XY regime36, which frequently describes the QPT induced by magnetic field in metallic superconducting films, such as in the amorphous Bi film10,11 or ultrathin Be film37. Thus the critical behavior in HTCR (zv = 0.63) reflects that the QPT in metallic tin nanoislands belongs to the clean (2 + 1)D XY model. In the clean (2 + 1)D XY model, the external magnetic field causes a frustration in the phase coupling36, suggesting that the QPT is driven by phase fluctuations at intra-tin nanoislands. In the superconducting LaTiO 3 /SrTiO 3 interface, the average size of assumed superconducting puddle L d is determined by the critical magnetic field as described in \(L_{\mathrm {d}}{\mathrm{\sim }}(\Phi _0/H_1^ \ast )^{1/2}\), where Φ 0 is the quantum flux14,38. We thus obtain L d = 118 nm according to\(H_1^ \ast = 1750\,{\mathrm{Oe}}\), which is the same order of the typical size of tin nanoislands deposited in our case as shown in Fig. 1 (Supplementary Figure 3).

The magnetoresistivity curves in LTCR (Fig. 4c) can be scaled into one single curve as in Fig. 4d and we can determine zv = 3.85, which suggests that v = 3.85 and z = 1 as discussed above. The exponent product \(v \ge 1\) is also observed in the disordered system such as LaTiO 3 /SrTiO 3 interface14, amorphous InO x 13 and amorphous MoGe9, which is the consequence of the QPT in 2D dirty regimes14,39. As shown in Fig. 1b, the tin islands with irregular shapes are randomly distributed on single crystalline monolayer graphene. The random electric potentials of the tin nanoislands can contribute to the disorder degree in our hybrid system, which induces the macroscopic inhomogeneous superconducting order parameter40. Thus, the exponent v = 3.85 is related to the QPT of 2D dirty regimes and attributed to the QPT of inter-tin nanoislands superconductivity.

It is worth noting that the electrostatic gating can also induce the QPT in 2D superconductor by changing the carrier density2,41. However, the pervious study on the electrostatically gated graphene-tin nanoislands hybrid system only exhibits single QPT19, which is different from two QPTs induced by the magnetic field in our work. The possible reason is that the electrostatic gating can alter the Fermi level of graphene and the consequent carrier density42, but can not tune the carrier density of metallic tin nanoislands40, therefore, only single QPT emerges corresponding to the inter-tin islands physics. As the magnetic field can induce the vortex matter in both the intra-tin and inter-tin nanoisland superconductors, two QPTs can be observed in the graphene-metallic tin nanoislands hybrid system accordingly, other than single QPT emerging during the electrostatic gating.