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# Zero-field superconducting diode effect in small-twist-angle trilayer graphene

## Abstract

The critical current of a superconductor can be different for opposite directions of current flow when both time-reversal and inversion symmetry are broken. Such non-reciprocal behaviour creates a superconducting diode and has recently been experimentally demonstrated by breaking these symmetries with an applied magnetic field or by the construction of a magnetic tunnel junction. Here we report an intrinsic superconducting diode effect that is present at zero external magnetic field in mirror-symmetric twisted trilayer graphene. Such non-reciprocal behaviour, with sign that can be reversed through training with an out-of-plane magnetic field, provides direct evidence of the microscopic coexistence between superconductivity and time-reversal symmetry breaking. In addition to the magnetic-field trainability, we show that the zero-field diode effect can be controlled by varying the carrier density or twist angle. A natural interpretation for the origin of the intrinsic diode effect is an imbalance in the valley occupation of the underlying Fermi surface, which probably leads to finite-momentum Cooper pairing and nematicity in the superconducting phase.

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## References

1. Ginzburg, V. Ferromagnetic superconductors. Sov. Phys. JETP-USSR 4, 153–160 (1957).

2. Matthias, B., Suhl, H. & Corenzwit, E. Spin exchange in superconductors. Phys. Rev. Lett. 1, 92–94 (1958).

3. Anderson, P. Theory of dirty superconductors. J. Phys. Chem. Solids 11, 26–30 (1959).

4. Fulde, P. & Ferrell, R. A. Superconductivity in a strong spin-exchange field. Phys. Rev. 135, A550–A563 (1964).

5. Larkin, A. & Ovchinnikov, Y. N. Nonuniform state of superconductors. Sov. Phys.-JETP 20, 762 (1965).

6. Buzdin, A. I. Proximity effects in superconductor-ferromagnet heterostructures. Rev. Mod. Phys. 77, 935–976 (2005).

7. Bianchi, A., Movshovich, R., Capan, C., Pagliuso, P. G. & Sarrao, J. L. Possible Fulde-Ferrell-Larkin-Ovchinnikov superconducting state in CeCoIn5. Phys. Rev. Lett. 91, 187004 (2003).

8. Radovan, H. A. et al. Magnetic enhancement of superconductivity from electron spin domains. Nature 425, 51–55 (2003).

9. Felner, I., Asaf, U., Levi, Y. & Millo, O. Coexistence of magnetism and superconductivity in R1.4Ce0.6RuSr2Cu2O10−δ(R = Eu and Gd). Phys. Rev. B 55, R3374–R3377 (1997).

10. Aoki, D. et al. Coexistence of superconductivity and ferromagnetism in URhGe. Nature 413, 613–616 (2001).

11. Pfleiderer, C. et al. Coexistence of superconductivity and ferromagnetism in the d-band metal ZrZn2. Nature 412, 58–61 (2001).

12. Ren, Z. et al. Superconductivity induced by phosphorus doping and its coexistence with ferromagnetism in EuFe2(As0.7P0.3)2. Phys. Rev. Lett. 102, 137002 (2009).

13. Dikin, D. A. et al. Coexistence of superconductivity and ferromagnetism in two dimensions. Phys. Rev. Lett. 107, 056802 (2011).

14. Li, L., Richter, C., Mannhart, J. & Ashoori, R. Coexistence of magnetic order and two-dimensional superconductivity at LaAlO3/SrTiO3 interfaces. Nat. Phys. 7, 762–766 (2011).

15. Bert, J. A. et al. Direct imaging of the coexistence of ferromagnetism and superconductivity at the LaAlO3/SrTiO3 interface. Nat. Phys. 7, 767–771 (2011).

16. Ando, F. et al. Observation of superconducting diode effect. Nature 584, 373–376 (2020).

17. Yuan, N. F. & Fu, L. Supercurrent diode effect and finite-momentum superconductors. Proc. Natl Acad. Sci. USA 119, e2119548119 (2022).

18. Daido, A., Ikeda, Y. & Yanase, Y. Intrinsic superconducting diode effect. Phys. Rev. Lett. 128, 037001 (2022).

19. He, J. J., Tanaka, Y. & Nagaosa, N. A phenomenological theory of superconductor diodes. New J. Phys. 24, 053014 (2022).

20. Ichikawa, F. et al. Asymmetric critical currents of Nb and YBCO thin films for the polarity of the transport current in parallel magnetic fields. Physica C Supercond. 235, 3095–3096 (1994).

21. Jiang, X., Connolly, P., Hagen, S. & Lobb, C. Asymmetric current-voltage characteristics in type-II superconductors. Phys. Rev. B 49, 9244–9247 (1994).

22. Broussard, P. & Geballe, T. Critical currents in sputtered Nb-Ta multilayers. Phys. Rev. B 37, 68–74 (1988).

23. Scammell, H. D., Li, J. & Scheurer, M. S. Theory of zero-field superconducting diode effect in twisted trilayer graphene. 2D Mater. 9, 025027 (2022).

24. Diez-Merida, J. et al. Magnetic Josephson junctions and superconducting diodes in magic angle twisted bilayer graphene. Preprint at https://arxiv.org/abs/2110.01067 (2021).

25. Hooper, J. et al. Anomalous Josephson network in the Ru-Sr2RuO4 eutectic system. Phys. Rev. B 70, 014510 (2004).

26. Siriviboon, P. et al. Abundance of density wave phases in twisted trilayer graphene on WSe2. Preprint at https://arxiv.org/abs/2112.07127v1 (2021).

27. Zinkl, B., Hamamoto, K. & Sigrist, M. Symmetry conditions for the superconducting diode effect in chiral superconductors. Preprint at https://arxiv.org/abs/2111.05340 (2021).

28. Scheurer, M. S. Mechanism, time-reversal symmetry, and topology of superconductivity in noncentrosymmetric systems. Phys. Rev. B 93, 174509 (2016).

29. Liu, X. et al. Tuning electron correlation in magic-angle twisted bilayer graphene using Coulomb screening. Science 371, 1261–1265 (2021).

30. Liu, X., Zhang, N. J., Watanabe, K., Taniguchi, T. & Li, J. Isospin order in superconducting magic-angle twisted trilayer graphene. Nat. Phys. 18, 522–527 (2022).

31. Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene. Nature 590, 249–255 (2021).

32. Hao, Z. et al. Electric field–tunable superconductivity in alternating-twist magic-angle trilayer graphene. Science 371, 1133–1138 (2021).

33. Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moiré heterostructure. Science 367, 900–903 (2020).

34. Lin, J.-X. et al. Spin-orbit–driven ferromagnetism at half moiré filling in magic-angle twisted bilayer graphene. Science 375, 437–441 (2022).

## Acknowledgements

J.-X.L. and J.I.A.L. acknowledge funding from NSF DMR-2143384. Device fabrication was performed in the Institute for Molecular and Nanoscale Innovation at Brown University. P.S. acknowledges support from the Brown University Undergraduate Teaching and Research Awards. M.S.S. acknowledges funding from the European Union (ERC-2021-STG, Project 101040651 - SuperCorr). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. H.D.S. acknowledges funding from the ARC Centre of Excellence FLEET. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan (grant number JPMXP0112101001), and JSPS KAKENHI (grant numbers 19H05790, 20H00354 and 21H05233).

## Author information

Authors

### Contributions

J.-X.L. and P.S. fabricated the devices and performed the measurements. J.-X.L., P.S. and J.I.A.L. analysed the data. J.-X.L., P.S., H.D.S., M.S.S. and J.I.A.L. wrote the article. K.W. and T.T. provided the hBN crystals. S.L., D.R. and J.H. provided the WSe2 crystals.

### Corresponding author

Correspondence to J.I.A. Li.

## Ethics declarations

### Competing interests

A provisional patent application has been filed by Brown University under Serial No. 63/289,563. The inventors include J.I.A.L., J.L., P.S. and M.S.S. The application, which is pending, contains proposals of two-dimensional material architecture designed for realizing the zero-field superconducting diode effect.

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Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work

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## Extended data

### Extended Data Fig. 1 Weak nonreciprocity in normal state transport in sample B.

a, Rxx versus T measured at B = 0 and νtTLG = 2.5 in sample B. Tc is around 0.95 K, defined by the onset in Rxx with increasing temperature. b, dV/dI measured with Idc = + 200nA (red dots) and − 200 nA (blue dots) as a function of temperature in the temperature range T > Tc. c, Renormalized dV/dI showing the difference between Idc = ± 200 nA as a function of T. The difference in dV/dI between Idc = ± 200 nA is around 50Ω, which is consistent with the very weak nonreciprocity at Idc > Ic at low temperature (Extended Data Fig. 5b). This nonreciprocity decreases with increasing temperature and completely vanishes at T > 6 K. This observation is consistent with an underlying fermi surface with partial valley imbalance, which simultaneously breaks time-reversal and inversion symmetries. The valley imbalance, hence time-reversal symmetry breaking, onsets spontaneously at T ≈ 6 K. Even though time-reversal and inversion symmetries are broken in the normal state, nonreciprocity is greatly enhanced by the onset of the superconducting phase.

### Extended Data Fig. 2 The robustness of superconductivity in trained and untrained configurations.

a, dV/dI as a function of Idc and νtTLG measured at B = 0 and D = − 100 mV/nm for the electron doped superconducting phase. The measurement is performed with the superconducting diode effect after field training (left panel), and without the superconducting diode effect after ‘un-training’ with a large DC current (right panel). b, The reciprocal (top panel) and non-reciprocal (bottom panel) component of the critical current, $$({I}_{c}^{+}+{I}_{c}^{-})/2$$ and ΔIc, as a function of νtTLG extracted from a. It is worth noting that several experimental works have reported interplay between DC current flow and the sign of magnetic order: in orbital ferromagnetic states, a large DC current is shown to induce sign-reversal in the magnetic order 33,34. It is hypothesized that the mechanism underlying current-induced switching stems from the interaction between different magnetic domains and current flow around the edge of the domain. Our observation that a large DC current couples to the underlying time-reversal symmetry is consistent with previous experimental results. However, the sample interior of an orbital ferromagnet is insulating and current flows along the edge of the magnetic domain. Whereas the sample interior in the nonreciprocal superconducting phase is highly conductive. As such, we anticipate the interplay between DC current and the underlying time-reversal symmetry breaking to be different. Notably, how DC current interacts with the magnetic order remains an open question for graphene moiré systems in general 33,34.

### Extended Data Fig. 3 Optimal doping and critical temperature dependence on displacement field.

a,b, Longitudinal resistance Rxx as a function of filling fraction νtTLG and temperature. The optimal doping is indicated via dash lines in a, and the temperature dependence at optimal doping is plotted in b. c,d, Critical current Ic as a function of filling fraction νtTLG and displacement field D. The optimal doping for each displacement field is plotted in d overlaying with the Rxx map. Both the temperature and the dc current dependence suggest the increase in optimal doping as the displacement field increases.

### Extended Data Fig. 4 The absence of zero-field superconducting diode effect near the magic angle.

a, Differential resistance dV/dI as a function of Idc measured at B = 0, T = 20 mK and νtTLG = − 2.64 for Sample C. The I-V curve is symmetric with DC bias current. b, Current switching experiment taken at the same condition as in a. DC current bias is switched between positive and negative 30 nA (blue), 35 nA (orange) and 40 nA (green). No difference in resistance is observed for different current directions, in either superconducting or normal states. c, dV/dI as a function of Idc measured at B = − 10 mT (blue trace) and + 10 mT (red trace). d, Current switching experiment taken at the same condition as in c. DC current bias is switched between + − 10 nA at B = − 10 mT (blue), and + − 14 nA at B = + 10 mT (red). The non-reciprocity in the presence of a symmetry breaking field shows a superconducting diode effect. The fact that superconducting diode effect is absent at B = 0 shows that time reversal symmetry is preserved in both the normal and superconducting phase in sample C. This is distinctly different compared to the observation in sample A.

### Extended Data Fig. 5 Zero-field superconducting diode effect in sample B.

a, Differential resistance dV/dI as a function of DC current bias Idc measured at B = 0, T = 20 mK and νtTLG = 2.5. The blue vertical stripe marks the peak position in the differential resistance, Ic, where the superconducting phase transitions into normal state behaviour. Blue and red traces denote measurement with positive and negative DC current bias, respectively. The fact that dV/dI measured with different signs of DC current deviate from each other points towards zero-field nonreciprocity in the superconducting transport behaviour. Notably, the nonreciprocity diminishes as DC current exceeds the critical current Ic. b, dV/dI measured with alternating DC current bias at ± 100nA, ± 42nA, ± 30 nA and ± 25 nA. Nonreciprocity is apparent in the DC current range of Idc < Ic. Above the critical current Ic, nonreciprocity is substantially suppressed. This observation suggests that the fermi surface underlying the superconducting phase is partially valley imbalanced. In this scenario, time-reversal and inversion symmetries are simultaneously broken in the normal state, and SOC is not required to enable zero-field superconducting diode effect. However, the presence of SOC could still enhance the nonreciprocity through the following mechanisms: (i) SOC enhances the valley polarization in the partially valley imbalanced fermi surface 34, which gives rise to a more pronounced nonreciprocity in the zero-field superconducting transport behaviour; (ii) according to a recent theoretical work23, the presence of the SOC is essential for the trainability of the zero-field superconducting diode effect. Without the SOC-induced trainability, the sample is expected to have multiple domains of opposite valley polarization, diminishing the observed nonreciprocity. These expectations are consistent with our measurement result, where the zero-field diode effect is much weaker in the sample without WSe2. Since two samples do not offer statistical significance to confirm the influence of SOC on the robustness of the zero-field diode effect, we will leave a more systematic discussion on the influence of WSe2 to a separate work. It is worth noting that the collection of samples studied in our work is sufficient to confirm the main phenomenology, which is the interplay between orbital ferromagnetism, superconductivity, and Coulomb correlation.

## Supplementary information

### Supplementary Information

Supplementary Sections 1–6, Tables 1–4 and Figs. 1–6

## Source data

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### Source Data Extended Data Fig. 1

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Lin, JX., Siriviboon, P., Scammell, H.D. et al. Zero-field superconducting diode effect in small-twist-angle trilayer graphene. Nat. Phys. (2022). https://doi.org/10.1038/s41567-022-01700-1

• Accepted:

• Published:

• DOI: https://doi.org/10.1038/s41567-022-01700-1

• ### Superconducting diode effects

• Kun Jiang
• Jiangping Hu

Nature Physics (2022)