Abstract
Two magneticfieldinduced quantum critical behaviors were recently discovered in two dimensional electron gas (2DEG) at LaTiO_{3}/SrTiO_{3} interface and interpreted by disordered superconducting puddles coupled through 2DEG. In this scenario, the 2DEG is proposed to undergo a spontaneous phase separation and breaks up into locally superconducting puddles in a metallic matrix. However, as the inhomogeneous superconducting 2DEG is only illative, this proposal still lacks the direct experimental demonstration. Here, we artificially construct superconducting puddles2DEG hybrid system by depositing tin nanoislands array on single crystalline monolayer graphene, where the two quantum critical behaviors are reproduced. Through the finitesize scaling analysis on magnetoresistivity, we show that the two quantum critical behaviors result from the intraisland and interisland phase coherence, respectively, which are further illustrated by the phase diagram. This work provides a platform to study superconducting quantum phase transitions in a 2D system and helps to integrate superconducting devices into semiconductor technology.
Introduction
Novel quantum phenomena, like high temperature superconductivity^{1,2,3}, Bose metallic state^{4,5}, and quantum griffiths^{6} phase were always observed in 2D superconducting material and their properties were believed to be controlled by a continuous quantum phase transition (QPT) at zero temperature. The QPT occurs at zero temperature as a result of the change of the Hamiltonian under the nonthermal parameters, like disorder, the magnetic field or the doping^{7}. The quantum criticality arising from a continuous QPT can govern the phase diagram up to very high temperature and be described by the theory of finitesize scaling (FSS). The conventional perpendicular magnetic field tuned QPT is always single in the amorphous or granular 2D superconductors^{8,9,10,11,12,13}. While recent experiments instead showed superconductor to insulator phase transitions (SIT) tuned by the magnetic field corresponding to two QPTs at LaTiO_{3}/SrTiO_{3} interface^{14}. Previous theoretical studies suggest that the QPT, even for multiple QPTs, may be induced by inhomogeneities of superconductivity spontaneously developed near the transition in two dimensions^{15,16}. Thus, the two QPTs emerged at LaTiO_{3}/SrTiO_{3} interface were also assumed to be related to the inhomogeneous superconductivity, in which one QPT corresponded to the local ordering of superconducting puddles formed by highmobility carriers (HMCs) and the other QPT was due to the coherence between superconducting puddles coupled by 2DEG^{14}. However, since the existence of superconducting puddles and 2DEG at LaTiO_{3}/SrTiO_{3} interface could not be visually observed, the physical mechanism underlying multiple QPTs in 2D superconductors is still under debate. To elucidate the origin of the observed multiple QPTs, a visualized platform to reproduce the system consisting of the illative superconducting puddles2DEG system at LaTiO_{3}/SrTiO_{3} interface is essentially demanded.
Here, we artificially fabricated the inhomogeneous 2D superconductor consisting of superconducting tin nanoislands and single crystalline monolayer graphene, in which an array of disordered superconducting tin nanoislands can visually mimic the suspected superconducting puddles and single crystalline monolayer graphene provides two dimensional electron gas (2DEG), as schematically illustrated in Fig. 1a. The obtained graphenetin nanoislands array hybrid exhibits the ideal 2D superconductivity at low temperature. As increasing the perpendicular magnetic fields, the graphenetin nanoislands array hybrid is driven from the 2D superconductor to the weakly localizing metal. Along with the transition process, we observe two quantum criticalities, which correspond to the intratin nanoisland superconductivity and the intertin nanoisland superconductivity, respectively. The constructed H–T phase diagram further verifies the presence of two QPTs in graphenetin nanoislands array hybrid.
Results
Fabrication of inhomogeneous 2D superconductor
Four inch intrinsic Ge (110) wafer was chosen as the starting platform for the synthesis of single crystalline monolayer graphene^{17,18} (Supplementary Figure 1). By using the stencil mask, the standard hall bar device matrixes were subsequently fabricated on graphene wafer. Then, 10 nmthick elemental superconductor tin was deposited on 4 inch singlecrystalline graphene wafer by electron beam evaporation. Due to the poor wettability of graphene and the low melting point of tin^{19,20}, an array of selfassembled irregular tin nanoislands with lateral size of ~150 nm and interval of ~40 nm was formed to build graphenetin nanoislands hybrid system (Fig. 1b, Supplementary Figures 2, 3). The single crystalline graphene provides a strict 2DEG platform, which is rather inert in the environment^{21} and has no grain boundary intervalley carrierscattering effect^{22,23}, while an array of disordered superconducting tin islands visually reproduces the hypothesized superconducting puddles^{14}. Both cross sectional scanning transmission electron microscopy (STEM) and STEMenergy dispersive xray spectroscopy (STEMEDS) 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 fourterminal 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 semiconductorlike resistivity behavior (dR/dT<0) where the sheet resistance increases as the temperature decreases (inset in Fig. 2a), then undergoes twostep 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 semiconductorlike resistivity behavior can be understood in terms of a weak localization behavior proposed in 2D metals^{19} which is consistent with the previous observation^{24}. 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 previously^{19}. 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 arrays^{19,25}, 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 superconductivity^{19,26,27,28}. 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.
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 ZrNClEDLT^{4}, SrTiO_{3}EDLT^{29}, niobiumdoped SrTiO_{3} thin film^{30}. Figure 2c demonstrates the voltage–current (V–I) curves of graphenetin nanoislands array from 1.87 to 4 K in the loglog scale. A powerlaw 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 710), the observed powerlaw V–I behavior is not due to the effect of Joule heating, and corresponds to the BerezinskiiKosterlitzThouless (BKT) type transition in 2D superconductor^{6,31}. 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 = I^{3} 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 flake^{20}.
Two SITs observed in the graphenetin nanoislands array
The measurements conducted above confirm that graphenetin 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 field^{6,9,13,14}. 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 Tindependent resistance region as previously observed in the LaSr_{2−x}Cu_{ x }O_{4} thin film^{1}. The Tindependent 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.
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/4e^{2}=6.45 kΩ^{32}. The early experiments on InO_{ x } film showed that the disorder of the system had significant influence on the critical resistance^{13}. 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 graphenetin nanoislands array system (R_{c1}/R_{Q} = 0.15 & R_{c2}/R_{Q} = 0.16) agrees with the weak disorder picture^{13} 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 analysis^{7}. 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) = 1^{32}. 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 } ~ ξ^{z} in the vicinity of the continuous QPT. We thus performed the FFS analysis and replotted 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.
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 replotted 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 loglog 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} interface^{14}, amorphous Nb_{0.}_{15}Si_{0.}_{85}^{12}, amorphous Bi^{10} and ultrathin high T_{c} superconductor^{1}. The dynamical scaling exponent z usually equals to 1 in the consequence of long range Coulomb interaction between charges^{32,33} as observed in ultrathin Bi film^{11}, amorphous MoGe^{9}, unless the particular case in ^{4}He porous media with short range Coulomb interactions^{34,35}. 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 model^{7}. The v = 2/3 is expected in clean (2 + 1)D XY regime^{36}, which frequently describes the QPT induced by magnetic field in metallic superconducting films, such as in the amorphous Bi film^{10,11} or ultrathin Be film^{37}. 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 coupling^{36}, suggesting that the QPT is driven by phase fluctuations at intratin 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 flux^{14,38}. 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} interface^{14}, amorphous InO_{x}^{13} and amorphous MoGe^{9}, which is the consequence of the QPT in 2D dirty regimes^{14,39}. 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 parameter^{40}. Thus, the exponent v = 3.85 is related to the QPT of 2D dirty regimes and attributed to the QPT of intertin nanoislands superconductivity.
It is worth noting that the electrostatic gating can also induce the QPT in 2D superconductor by changing the carrier density^{2,41}. However, the pervious study on the electrostatically gated graphenetin nanoislands hybrid system only exhibits single QPT^{19}, 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 density^{42}, but can not tune the carrier density of metallic tin nanoislands^{40}, therefore, only single QPT emerges corresponding to the intertin islands physics. As the magnetic field can induce the vortex matter in both the intratin and intertin nanoisland superconductors, two QPTs can be observed in the graphenemetallic tin nanoislands hybrid system accordingly, other than single QPT emerging during the electrostatic gating.
Discussion
Despite the two quantum critical behaviors are well built at H_{1}^{*} and H_{2}^{*}, respectively, the resistance scaling behavior at H_{1}^{*} holds only down to 2 K instead of approaching to the lowest temperature, in contrast to the H_{2}^{*}. To account for the quantum critical behaviors of H_{1}^{*} at lowest temperature, the H–T phase diagram has been constructed in Fig. 5. The T_{peak} data are extracted from the Fig. 3a with the black and pink arrows, which separate the superconductivity region (dρ/dT > 0) and the weakly localizing metal region (dρ/dT < 0). The scaling regions for the critical field H_{1}^{*} and H_{2}^{*} are also shown in Fig. 5. The scaling region corresponding to H_{1}^{*} is about 2–3 K, which is slightly less than the tin bulk superconducting transition temperature T_{c} = 3.7 K and larger than the zero resistance temperature (∼1 K), while the scaling region corresponding to H_{2}^{*} is around 0.05 ~ 0.2 K, which is approaching to the lowest temperature (50 mK) in present experiment. By analyzing the resistance data in the Arrhenius plot^{4,5} (Supplementary Figure 15), the blue stars T_{TAFF} inside the superconducting dome in Fig. 5 are obtained, which yield the boundary between the thermally activated flux flow region and the quantum tunneling region. It is striking to note that the boundary exhibits a Vshaped topology with the bottom located at the H_{1}^{*} exactly. The T_{TAFF} line inside the superconducting phase generally reflects the crossover of vortex from the thermal activation to quantum tunneling mechanism^{43}. In particular to the ultrathin superconducting diskshaped nanoislands, the theory proposed that the magnetic field dependent T_{TAFF} line follows \(T_{{\mathrm{TAFF}}} \propto \Phi  \Phi _0\) in the dissipative regime, where Φ and Φ_{0} are the magnetic flux and the quantum flux through a nanoisland, respectively^{44}. Thus, for an ultrathin superconducting nanoisland, the T_{TAFF} is expected to reach the minimum at Φ = Φ_{0} and increases again once Φ is beyond Φ_{0}, which yields a Vshape T_{TAFF} line as a function of H. Meanwhile, according to the previous report^{14} and our work, the critical field \(H_1^ \ast \sim \Phi _0/L_{\mathrm{d}}^2\) at high temperature regime is the dephasing filed, which is determined by the nanopuddle size L_{d} and the quantum flux Φ_{0}. Therefore, we assume the Vshape T_{TAFF} line should be related to the critical field \(H_1^ \ast\). In addition, it is natural to imagine that tin islands start to become superconducting at high temperature regime, and extend to form a superconducting landscape consisting of superconducting islands and coupling regions as the temperature decreases. The dissipation scheme in our disordered system is believed to attribute to the vortices movement in the formed superconducting landscape, so it is plausible to observe the T_{TAFF} line of Vshape in a region where the interislands physics dominates at low temperature. As discussed above, it is concluded the two quantum critical behaviors correspond to the physics related to the intratin and intertin nanoisland length scales, suggesting the multiscale inhomogeneity can gives rise to multicritical behaviors. It may provide a new insight to the quantum manybody effects in the strongly correlated electronic systems like manganites and cuprates where the ground state has an inhomogeneous mixture of phases in nanoscale.
In summary, the 2D superconductor consisting of the superconducting tin nanoislands together with single crystalline monolayer graphene is synthesized on intrinsic Ge (110) wafer. This hybrid system can visually reproduce the superconducting puddles2DEG system assumed at oxide interface, and the expected two QPTs induced by the magnetic field reappear as identified through FSS analysis on magnetoresistivity. The constructed H–T phase diagram shows that both QPTs correspond to the intratin nanoisland superconductivity and the intertin nanoisland superconductivity, respectively. This study provides an ideal visualized platform to investigate 2D superconducting QPT. In addition, the superconducting device arrays built on semiconductor germanium wafer sheds light on a great potential to enable the integration of superconducting devices with the existing semiconductor technology.
Methods
Graphene synthesis and characterization
Single crystalline waferscale monolayer graphene was synthesized on the intrinsic Ge (110) wafer by chemical vapor deposition (CVD). The Ge(110) wafer was loaded in a 4inch CVD furnace. Afterwards, the chamber was evacuated to high vacuum. Then, the chamber was filled with the gas mixture of argon (Ar, 99.9999% purity) and hydrogen (H_{2}, 99.9999% purity) to make the chamber to reach the atmospheric pressure. During the growth process, a certain proportion gas mixture of methane (CH_{4}, 99.99% purity), hydrogen (H_{2}, 99.9999% purity), argon (Ar, 99.9999% purity) was flowed in the chamber at 910 ^{o}C for 300 min. Finally, the chamber was rapidly cooled down to room temperature under the flowing gas mixture of H_{2} and Ar.
The surface topography of graphene was measured by the tapping mode AFM (Multimode 8, Bruker) with RTESP AFM tips. Raman spectra measurement (HORIBA Jobin Yvon HR800) was carried out with the laser wavelength 514 nm and a spot size of 1 μm.
Device fabrication and measurement
To avoid the possible contamination of the graphene surface by chemical polymer resist, we fabricated the device using the stencil mask which could maintain the intrinsic property of graphene. To exclude the possible shunt effect from the substrate, 4 inch intrinsic Ge (110) wafer (TaiCrystal, >50 ohm cm, 400 μm thickness) was chosen as the starting platform for the synthesis of graphene and the fabrication of Hall bar devices. The Hall bar device fabrication process started with the 10 nm Ti/100 nm Au electrode pattern, which was deposited utilizing the stencil mask to form ohmic contact with graphene. To define the graphene channel, the graphene except the channel graphene was etched using oxygen plasma aligned with the stencil mask. Finally, the 10 nm thick tin film was deposited on the sample by electron beam evaporation to complete the fabrication of hall bar device. R–T curve of Hall device was measured in physical property measurement system with He^{3}He^{4} dilution refrigerator (PPMS, Quantum design) with a base temperature of 50 mK.
Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.
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Acknowledgements
The authors thank the Program of Shanghai Academic/Technology Research Leader (16XD1404200), Key Research Project of Frontier Science, Chinese Academy of Sciences (QYZDBSSWJSC021), National Science and Technology Major Project (2016ZX02301003), NSFC Grant No. 11574338, NSAF Grant No. U1530402 and the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB04040300).
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Z.D., X.X., T.H. and X.W. conceived the project. Y.S. and H.X. designed and performed the experiments. Y.S., Y.M., Z.X., and M.Z. conducted the material growth. Y.S., Z.D., H.X., and T.H. performed the data analysis and cowrote the paper. All authors discussed the results and commented on the manuscript.
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Correspondence to Tao Hu or Zengfeng Di.
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Further reading

The transport properties in graphene/singleunitcell cuprates van der Waals heterostructure
Superconductor Science and Technology (2019)
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