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# Enhanced superconductivity in spin–orbit proximitized bilayer graphene

## Abstract

In the presence of a large perpendicular electric field, Bernal-stacked bilayer graphene (BLG) features several broken-symmetry metallic phases1,2,3 as well as magnetic-field-induced superconductivity1. The superconducting state is quite fragile, however, appearing only in a narrow window of density and with a maximum critical temperature Tc ≈ 30 mK. Here we show that placing monolayer tungsten diselenide (WSe2) on BLG promotes Cooper pairing to an extraordinary degree: superconductivity appears at zero magnetic field, exhibits an order of magnitude enhancement in Tc and occurs over a density range that is wider by a factor of eight. By mapping quantum oscillations in BLG–WSe2 as a function of electric field and doping, we establish that superconductivity emerges throughout a region for which the normal state is polarized, with two out of four spin-valley flavours predominantly populated. In-plane magnetic field measurements further reveal that superconductivity in BLG–WSe2 can exhibit striking dependence of the critical field on doping, with the Chandrasekhar–Clogston (Pauli) limit roughly obeyed on one end of the superconducting dome, yet sharply violated on the other. Moreover, the superconductivity arises only for perpendicular electric fields that push BLG hole wavefunctions towards WSe2, indicating that proximity-induced (Ising) spin–orbit coupling plays a key role in stabilizing the pairing. Our results pave the way for engineering robust, highly tunable and ultra-clean graphene-based superconductors.

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## Data availability

The data shown in the main figures are available from the CaltechDATA at https://doi.org/10.22002/wecmz-csm13.

## Code availability

The code used to reproduce data plots is available from the CaltechDATA at https://doi.org/10.22002/wecmz-csm13. The code used for the modelling is available upon reasonable request.

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## Acknowledgements

We thank A. Young and A. Macdonald for fruitful discussions. This work has been primarily supported by NSF-CAREER award (no. DMR-1753306), and the Office of Naval Research (grant no. N142112635) and the Army Research Office (grant award no. W911NF17-1-0323). Nanofabrication efforts have been in part supported by the Department of Energy DOE-QIS program (DE-SC0019166). S.N.-P. acknowledges support from the Sloan Foundation (grant no. FG-2020-13716). J.A. and S.N.-P. also acknowledge the support of the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation through grant no. GBMF1250. C.L. and E.L.-H. acknowledge support from the Gordon and Betty Moore Foundation’s EPiQS Initiative, grant no. GBMF8682.

## Author information

Authors

### Contributions

Y.Z. and S.N.-P. designed the experiment. Y.Z., R.P. and H.Z. performed the measurements, fabricated the devices and analysed the data. A.T., E.L.-H. and C.L. developed the theoretical models and performed the calculations supervised by J.A. K.W. and T.T. provided the hBN crystals. S.N.-P. supervised the project. Y.Z., A.T., E.L.-H., C.L., H.Z., R.P., J.A. and S.N.-P. wrote the manuscript with the input of other authors.

### Corresponding author

Correspondence to Stevan Nadj-Perge.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature thanks R. Thomas Weitz and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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## Extended data figures and tables

### Extended Data Fig. 1 Quantum oscillations at selected gate points.

a,b, nD phase diagram with the coloured dots indicating the positions at which the quantum oscillations (left panels in cm) are taken. cm, Left panels show the quantum oscillations at the coloured dots. Right panels show the normalized Fourier transform of the corresponding Rxx(1/B) data.

### Extended Data Fig. 2 Reproducibility of zero-magnetic-field superconductivity in BLG-WSe2.

a, Schematic of a dual-gated device. Doping density n and D field are controlled by tuning top and bottom gate voltage vt and vb. b,c, Optical images of the investigated devices. The scale bar in each panel corresponds to 10 μm. d,e, Rxx versus doping density n and D field measured from the first device D1 between contacts 1—3 (d) and 2—4 (e). Contacts 1—3 were used for the measurements in the main text. f,g, Rxx versus doping density n measured from the second device D2 at D/ϵ0 = 0.9 V/nm (f) and 1 V/nm (g).

### Extended Data Fig. 3 The third superconducting BLG-WSe2 device D3.

a, Rxx versus doping density n and displacement field D measured at zero magnetic field for device D3. b, Rxx versus B and doping density n for D/ϵ0 = 1.1 V/nm. c, Fourier transform of Rxx(1/B) versus n and fν for D/ϵ0 = 1.1 V/nm. Schematics on top depict the possible Fermi surface structures for the different phases given that spin-valley flavours are not mixed. d, Rxx versus n at zero magnetic field for the same D field. The superconducting phase possesses a flavour-polarized normal state with two majority and multiple (m > 2) minority Fermi pockets (denoted as FP(2, m>2)+). e, Rxx versus density n and temperature showing a superconducting dome at D/ϵ0 = 1.15 V/nm. f, Critical current disappearing with B field measured at D/ϵ0 = 1.1 V/nm, n = − 6.55 × 10 11cm−2. g, Rxx versus n and B field around the superconducting region for D/ϵ0 = 1.1 V/nm.

### Extended Data Fig. 4 The fourth superconducting BLG-WSe2 device D4.

a, Rxx versus doping density n and displacement field D measured at zero magnetic field for device D4. b, Rxx versus B and doping density n for D/ϵ0 = 1.1 V/nm. c, Fourier transform of Rxx(1/B) versus n and fν for D/ϵ0 = 1.1 V/nm. Schematics on top depict the possible Fermi surface structures for the different phases given that spin-valley flavours are not mixed. d, Rxx versus n at zero magnetic field for the same D field. The superconducting phase possesses a flavour-polarized normal state with two majority and multiple (m > 2) minority Fermi pockets (denoted as FP(2, m > 2)+). e, Rxx versus density n and temperature showing a superconducting dome at D/ϵ0 = 1.1 V/nm. f, Critical current disappearing with B field measured at D/ϵ0 = 1.1 V/nm, n = − 6.7 × 10 11cm−2. g, Rxx versus n and B field around the superconducting region for D/ϵ0 = 1.1 V/nm.

### Extended Data Fig. 5 Transverse magnetic focusing with out-of-plane and in-plane magnetic field.

a, Non-local resistance Rnl measured as a function of n and B at D/ϵ0 = 0.6 V/nm with the configuration shown in c. b, Non-local resistance Rnl measured as a function n and B at D/ϵ0 = − 0.2 V/nm. Transverse magnetic focusing with an in-plane field is due to imperfect sample alignment. Therefore, we can estimate the field misalignment angle by comparing a and b (see Methods for further discussion).

### Extended Data Fig. 6 Polarized phases.

a,cf, Expected normalized quantum oscillation frequencies as a function of density without interactions (a), allowing for U(1)v unbroken states with C3 preserved (c), allowing for U(1)v unbroken states with C3 broken (d), allowing for IVC ordered states with C3 preserved (e), and allowing IVC ordered states with C3 broken (f). Regions with red (blue) backgrounds correspond to singly polarized states that preserve (spontaneously break) the valley symmetry (even when the initial conditions were chosen to allow IVC order to develop, as in e and f). Regions coloured yellow are multiply polarized (no distinction is made between those with and without IVC order). Insets show the Fermi surfaces corresponding to a select set of fillings. The D field is set to D/ϵ0 = 1 V/nm in all plots. All simulations include SOC (λI = 0.7 meV and λR = 3 meV) except for the red curve in a. b, Plot of the difference between the change in ground state energy induced by Ising SOC for an IVC state and the change in ground state energy induced by Ising SOC for a VP state. The energy is normalized by the number of carriers Nf. The negative values obtained imply that the addition of Ising SOC to a VP state increases its energy more than the addition of Ising SOC increases the ground state energy of an IVC ground state.

### Extended Data Fig. 7 Fan diagrams at D < 0.

a,b, Rxx versus B and doping density n for D/ϵ0 = − 1 V/nm (a) and − 0.9 V/nm (b). c,e, Fourier transform of Rxx(1/B) versus n and fν for D/ϵ0 = − 1 V/nm and − 0.9 V/nm, respectively. Rxx data within 0.05 T < B < 0.6 T are used for converting. The corresponding Rxx data at zero magnetic field are shown in d and f. We note that none of the measured devices showed zero-magnetic-field superconductivity for D < 0 as typically verified down to D/ε0 = − 1 V/nm (and in some instances to D/ε0 = − 1.1 V/nm).

### Extended Data Fig. 8 Zero-magnetic-field superconductivity and FP(2, 2)+ phase at D > 0.

ac, Rxx versus B and doping density n for D/ϵ0 = 1 V/nm (a), 0.9 V/nm (b), and 0.8 V/nm (c). d,f,h, Fourier transform of Rxx(1/B) versus n and fν for D/ϵ0 = 1 V/nm, 0.9 V/nm, and 0.8 V/nm, respectively. Rxx data within 0.05 T < B < 0.6 T are used for converting. The corresponding Rxx data at zero magnetic field are shown below (e,g,i). We see a good match between the doping range exhibiting superconductivity and the FP(2, 2)+ phase region, regardless of D fields.

### Extended Data Fig. 9 In-plane magnetic field dependence of superconductivity.

a,b, Rxx versus doping density n and temperature for the superconducting domes at D/ϵ0 = 1 V/nm (a) and 1.1 V/nm (b). c,d, Rxx versus n and B for the superconducting domes at D/ϵ0 = 1 V/nm (c) and 1.1 V/nm (d). Red dots indicate the critical temperature at zero magnetic field. e,f, Pauli violation ratio (PVR) calculated from $${B}_{c\parallel }^{T\approx 30\,{\rm{mK}}}/{B}_{p}$$ as a function of doping density n. Both curves feature strong Pauli-limit violation at low n. Inset of e shows coherence length $${\xi }_{{\rm{GL}}}=\sqrt{{\Phi }_{0}/(2\pi {B}_{c\perp })}$$ and ħvF/πΔ versus n at D/ϵ0 = 1 V/nm. ħvF/πΔ is estimated with a weak-coupling assumption: Δ ≈ 1.76kBTc and vF = ħkf/m* ($${k}_{f}=\sqrt{2\pi {f}_{\nu }| n| }$$ with fν being the normalized frequency of minority Fermi pockets, m* ~ 0.15me). gk, Rxx versus temperature and B at different densities for D/ϵ0 = 1.1 V/nm. In-plane critical fields Bc (the field at which Rxx is 50% of the normal state resistance) are marked by dots, and all the data are well-fit by the phenomenological relation. We see a clear evolution of the PVR as a function of doping.

### Extended Data Fig. 10 Estimating different SOC strengths through quantum Hall effect and quantum oscillations.

ae, Rxx versus $$\nu =2\pi {{\ell }}_{B}^{2}n$$ (B is the Landau magnetic length) and D field at B = 1 T, 2 T, 3 T, 5 T, and 7 T, respectively. Arrows mark the transition of ν = 3 quantum Hall states with D field. Because the Ising SOC is oriented out of plane, an out-of-plane Zeeman splitting will cancel it when $$2{E}_{Z}=2g{\mu }_{B}{B}_{\perp }^{* }={\lambda }_{I}$$ ($${B}_{\perp }^{* }$$ is the magnetic field at which yellow and green arrows are at the same D field; $${B}_{\perp }^{* }\approx 3$$ T here). f,g, ΔRxx versus 1/B (measured up to B = 5 T) at D/ϵ0 = 0.2 V/nm, n = − 3.5 × 1011cm−2 (f) and D/ϵ0 = − 0.1 V/nm, n = − 20 × 1011 cm−2 (g). The corresponding FFT data are shown in h and i. Inset of h shows the FFT splitting Bsplit (marked by black arrows in the main panel) versus doping density n measured at D/ϵ0 = 0.2 V/nm. Coloured lines show the FFT splitting predicted from band structure calculations for the same D field, using Ising SOC λI = 0.7 meV with Rashba SOC λR = 0 meV (purple line) and λR = 4 meV (yellow line).

### Extended Data Fig. 11 Depairing model for doping-dependent Pauli violation ratio.

a,b, Pauli violation ratio expected in a system with: Ising gI and Rashba gRkF coupling (a), Ising gI and orbital $${\widetilde{g}}_{{\rm{orb}}}$$ coupling (b). Note that $${\widetilde{g}}_{{\rm{orb}}}$$ is a dimensionless quantity: the corresponding orbital energy scale is $${\widetilde{g}}_{{\rm{orb}}}{\mu }_{B}B$$. c,d, Fitting the model described in SI, section 9 to the experimental data in Fig. 4c (c) and Extended Data Fig. 9f (d). e,f, Evolution of the extracted parameters gRkF and $${\widetilde{g}}_{{\rm{orb}}}$$ as a function of hole density n, for the four models that are used in the fitting procedure (see SI, section 10); gI = 0.7 meV in all the plots.

## Supplementary information

### Supplementary Information

Two Supplementary figures, text that describes theoretical analysis, Table 1 listing parameters of different models used for the fitting procedure and references.

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Zhang, Y., Polski, R., Thomson, A. et al. Enhanced superconductivity in spin–orbit proximitized bilayer graphene. Nature 613, 268–273 (2023). https://doi.org/10.1038/s41586-022-05446-x

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