Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Probing the quantum metric of 3D topological insulators

Nature Materials (2026) Cite this article

Subjects

Abstract

The surface states of three-dimensional topological insulators possess geometric structures that imprint distinctive signatures on electronic transport. A prime example is the Berry curvature, which controls, for instance, electric frequency doubling via its higher-order moments. In addition to the Berry curvature, topological surface states are expected to exhibit a quantum metric, which plays a key role in nonlinear magnetotransport. Here we provide evidence for a nonlinear response activated by the quantum metric of the topological surface states of Sb2Te3. We measure a time-reversal-odd, nonlinear magnetoresistance that is independent of temperature and scattering time below 30 K, and is thus of intrinsic geometrical origin. This quantum metric magnetoresistance can be controlled by tuning the contributions of the top and bottom topological surface states through voltage gating. Our measurements demonstrate the existence and tunability of quantum geometry-induced transport in topological phases of matter and enable the design of functional topological devices.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on SpringerLink
  • Instant access to the full article PDF.

USD 39.95

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Quantum geometry of a 3D topological insulator.
The alternative text for this image may have been generated using AI.
Fig. 2: Sb2Te3 topological insulator.
The alternative text for this image may have been generated using AI.
Fig. 3: Nonlinear magnetotransport.
The alternative text for this image may have been generated using AI.
Fig. 4: Quantum-metric-driven nonlinear conductivity.
The alternative text for this image may have been generated using AI.
Fig. 5: Gate voltage control of linear and nonlinear transport in Sb2Te3.
The alternative text for this image may have been generated using AI.

Data availability

The data that support the findings of this study are available via Zenodo at https://doi.org/10.5281/zenodo.16910820 (ref. 66) and from the corresponding authors upon reasonable request.

References

  1. Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    Article  CAS  Google Scholar 

  2. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    Article  CAS  Google Scholar 

  3. Nielsen, H. B. & Ninomiya, M. A no-go theorem for regularizing chiral fermions. Phys. Lett. B 105, 219–223 (1981).

    Article  Google Scholar 

  4. Tian, W., Yu, W., Shi, J. & Wang, Y. The property, preparation and application of topological insulators: a review. Materials 10, 814 (2017).

    Article  PubMed  PubMed Central  Google Scholar 

  5. Breunig, O. & Ando, Y. Opportunities in topological insulator devices. Nat. Rev. Phys. 4, 184–193 (2021).

    Article  Google Scholar 

  6. Sato, M. & Ando, Y. Topological superconductors: a review. Rep. Prog. Phys. 80, 076501 (2017).

    Article  PubMed  Google Scholar 

  7. He, M., Sun, H. & He, Q. L. Topological insulator: spintronics and quantum computations. Front. Phys. 14, 43401 (2019).

    Article  Google Scholar 

  8. Huáng, N. J. et al. Quantum anomalous Hall effect for metrology. Appl. Phys. Lett. 126, 040501 (2025).

    Article  Google Scholar 

  9. Pesin, D. & MacDonald, A. H. Spintronics and pseudospintronics in graphene and topological insulators. Nat. Mater. 11, 409–416 (2012).

    Article  CAS  PubMed  Google Scholar 

  10. He, P. et al. Quantum frequency doubling in the topological insulator Bi2Se3. Nat. Commun. 12, 698 (2021).

    Article  CAS  PubMed  Google Scholar 

  11. Xu, N., Xu, Y. & Zhu, J. Topological insulators for thermoelectrics. npj Quantum Mater. 2, 51 (2017).

    Article  Google Scholar 

  12. Bardarson, J. H. & Moore, J. E. Quantum interference and Aharonov–Bohm oscillations in topological insulators. Rep. Prog. Phys. 76, 056501 (2013).

    Article  PubMed  Google Scholar 

  13. Gao, A., Nagaosa, N., Ni, N. & Xu, S.-Y. Quantum geometry phenomena in condensed matter systems. Preprint at https://arxiv.org/abs/2508.00469 (2025).

  14. Verma, N., Moll, P. J. W., Holder, T. & Queiroz, R. Quantum geometry: revisiting electronic scales in quantum matter. Preprint at https://arxiv.org/abs/2504.07173 (2025).

  15. Ahn, J., Guo, G.-Y. & Nagaosa, N. Low-frequency divergence and quantum geometry of the bulk photovoltaic effect in topological semimetals. Phys. Rev. X 10, 041041 (2020).

    CAS  Google Scholar 

  16. Ahn, J., Guo, G.-Y., Nagaosa, N. & Vishwanath, A. Riemannian geometry of resonant optical responses. Nat. Phys. 18, 290–295 (2022).

    Article  CAS  Google Scholar 

  17. Jiang, Y., Holder, T. & Yan, B. Revealing quantum geometry in nonlinear quantum materials. Rep. Prog. Phys. 88, 076502 (2025).

    Article  Google Scholar 

  18. Yu, J. et al. Quantum geometry in quantum materials. npj Quantum Mater. 10, 101 (2025).

    Article  Google Scholar 

  19. Liu, T., Qiang, X.-B., Lu, H.-Z. & Xie, X. C. Quantum geometry in condensed matter. Natl Sci. Rev. 12, nwae334 (2025).

    Article  CAS  PubMed  Google Scholar 

  20. Tian, H. et al. Evidence for Dirac flat band superconductivity enabled by quantum geometry. Nature 614, 440–444 (2023).

    Article  CAS  PubMed  Google Scholar 

  21. Gao, A. et al. Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure. Science 381, 181–186 (2023).

    Article  CAS  PubMed  Google Scholar 

  22. Wang, N. et al. Quantum-metric-induced nonlinear transport in a topological antiferromagnet. Nature 621, 487–492 (2023).

    Article  CAS  PubMed  Google Scholar 

  23. Kim, S. et al. Direct measurement of the quantum metric tensor in solids. Science 388, 1050–1054 (2025).

    Article  CAS  PubMed  Google Scholar 

  24. Kang, M. et al. Measurements of the quantum geometric tensor in solids. Nat. Phys. 21, 110–117 (2025).

    Article  CAS  Google Scholar 

  25. Sala, G. et al. The quantum metric of electrons with spin-momentum locking. Science 389, 822–825 (2025).

    Article  CAS  PubMed  Google Scholar 

  26. Zhao, T.-Y. et al. Magnetic field induced quantum metric dipole in Dirac semimetal Cd3As2. Phys. Rev. Lett. 135, 026601 (2025).

    Article  CAS  PubMed  Google Scholar 

  27. Chen, W. Quantum geometrical properties of topological materials. J. Phys. Condens. Matter 37, 025605 (2025).

    Article  CAS  Google Scholar 

  28. Mera, B., Zhang, A. & Goldman, N. Relating the topology of Dirac Hamiltonians to quantum geometry: when the quantum metric dictates Chern numbers and winding numbers. SciPost Phys. 12, 018 (2022).

    Article  Google Scholar 

  29. Mercaldo, M. T., Cuoco, M. & Ortix, C. Nonlinear planar magnetotransport as a probe of the topology of surface states. Phys. Rev. B 111, 155442 (2025).

    Article  CAS  Google Scholar 

  30. Liu, C.-X. et al. Model Hamiltonian for topological insulators. Phys. Rev. B 82, 045122 (2010).

    Article  Google Scholar 

  31. Fu, L. Hexagonal warping effects in the surface states of the topological insulator Bi2Te3. Phys. Rev. Lett. 103, 266801 (2009).

    Article  PubMed  Google Scholar 

  32. Kaplan, D., Holder, T. & Yan, B. Unification of nonlinear anomalous Hall effect and nonreciprocal magnetoresistance in metals by the quantum geometry. Phys. Rev. Lett. 132, 026301 (2024).

    Article  CAS  PubMed  Google Scholar 

  33. Das, K., Lahiri, S., Atencia, R. B., Culcer, D. & Agarwal, A. Intrinsic nonlinear conductivities induced by the quantum metric. Phys. Rev. B 108, L201405 (2023).

    Article  CAS  Google Scholar 

  34. Zhang, H. et al. Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nat. Phys. 5, 438–442 (2009).

    Article  CAS  Google Scholar 

  35. Rimoldi, M. et al. Epitaxial and large area Sb2Te3 thin films on silicon by MOCVD. RSC Adv. 10, 19936–19942 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  36. Shafiei, A. et al. Wafer-scale synthesis of topological insulator Sb2Te3 thin films. Adv. Mater. Interfaces 12, 2400961 (2025).

    Article  CAS  Google Scholar 

  37. Anderson, T. L. & Krause, H. B. Refinement of the Sb2Te3 and Sb2Te2Se structures and their relationship to nonstoichiometric Sb2Te3−ySey compounds. Acta Crystallogr. B 30, 1307–1310 (1974).

    Article  CAS  Google Scholar 

  38. Locatelli, L. et al. Magnetotransport and ARPES studies of the topological insulators Sb2Te3 and Bi2Te3 grown by MOCVD on large-area Si substrates. Sci. Rep. 12, 3891 (2022).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  39. Longo, E. et al. Spin–charge conversion in Fe/Au/Sb2Te3 heterostructures as probed by spin pumping ferromagnetic resonance. Adv. Mater. Interfaces 8, 2101244 (2021).

    Article  CAS  Google Scholar 

  40. Longo, E. et al. Large spin-to-charge conversion at room temperature in extended epitaxial Sb2Te3 topological insulator chemically grown on silicon. Adv. Funct. Mater. 32, 2109361 (2022).

    Article  CAS  Google Scholar 

  41. Hikami, S., Larkin, A. I. & Nagaoka, Y. Spin–orbit interaction and magnetoresistance in the two dimensional random system. Prog. Theor. Phys. 63, 707–710 (1980).

    Article  Google Scholar 

  42. Steinberg, H., Laloë, J.-B., Fatemi, V., Moodera, J. S. & Jarillo-Herrero, P. Electrically tunable surface-to-bulk coherent coupling in topological insulator thin films. Phys. Rev. B 84, 233101 (2011).

    Article  Google Scholar 

  43. He, P. et al. Bilinear magnetoelectric resistance as a probe of three-dimensional spin texture in topological surface states. Nat. Phys. 14, 495–499 (2018).

    Article  CAS  Google Scholar 

  44. He, P. et al. Nonlinear planar Hall effect. Phys. Rev. Lett. 123, 016801 (2019).

    Article  CAS  PubMed  Google Scholar 

  45. Fu, Y. et al. Bilinear magnetoresistance in HgTe topological insulator: opposite signs at opposite surfaces demonstrated by gate control. Nano Lett. 22, 7867–7873 (2022).

    Article  CAS  PubMed  Google Scholar 

  46. Tang, J. et al. Electrical detection of spin-polarized surface states conduction in (Bi0.53Sb0.47)2Te3 topological insulator. Nano Lett. 14, 5423–5429 (2014).

    Article  CAS  PubMed  Google Scholar 

  47. Che, X. et al. Strongly surface state carrier-dependent spin–orbit torque in magnetic topological insulators. Adv. Mater. 32, 1907661 (2020).

    Article  CAS  Google Scholar 

  48. Binda, F., Fedel, S., Alvarado, S. F., Noël, P. & Gambardella, P. Spin–orbit torques and spin Hall magnetoresistance generated by twin-free and amorphous Bi0.9Sb0.1 topological insulator films. Adv. Mater. 35, 2304905 (2023).

    Article  CAS  Google Scholar 

  49. Korzhovska, I. et al. Spin memory of the topological material under strong disorder. npj Quantum Mater. 5, 39 (2020).

    Article  CAS  Google Scholar 

  50. Pauly, C. et al. Probing two topological surface bands of Sb2Te3 by spin-polarized photoemission spectroscopy. Phys. Rev. B 86, 235106 (2012).

    Article  Google Scholar 

  51. Hong, S. S., Cha, J. J., Kong, D. & Cui, Y. Ultra-low carrier concentration and surface-dominant transport in antimony-doped Bi2Se3 topological insulator nanoribbons. Nat. Commun. 3, 757 (2012).

    Article  PubMed  Google Scholar 

  52. Steinberg, H., Gardner, D. R., Lee, Y. S. & Jarillo-Herrero, P. Surface state transport and ambipolar electric field effect in Bi2Se3 nanodevices. Nano Lett. 10, 5032–5036 (2010).

    Article  CAS  PubMed  Google Scholar 

  53. Yoshimi, R. et al. Quantum Hall effect on top and bottom surface states of topological insulator (Bi1−xSbx)2Te3 films. Nat. Commun. 6, 6627 (2015).

    Article  CAS  PubMed  Google Scholar 

  54. Chen, J. et al. Gate-voltage control of chemical potential and weak antilocalization in Bi2Se3. Phys. Rev. Lett. 105, 176602 (2010).

    Article  CAS  PubMed  Google Scholar 

  55. Feng, X. et al. Quantum metric nonlinear spin–orbit torque enhanced by topological bands. Mater. Today Quantum 6, 100040 (2025).

    Article  Google Scholar 

  56. Wang, H. et al. Intrinsic nonlinear spin Hall effect and manipulation of perpendicular magnetization. Phys. Rev. Lett. 134, 056301 (2025).

    Article  CAS  PubMed  Google Scholar 

  57. Ma, Q., Krishna Kumar, R., Xu, S.-Y., Koppens, F. H. L. & Song, J. C. W. Photocurrent as a multiphysics diagnostic of quantum materials. Nat. Rev. Phys. 5, 170–184 (2023).

    Article  Google Scholar 

  58. Bhalla, P., Das, K., Culcer, D. & Agarwal, A. Resonant second-harmonic generation as a probe of quantum geometry. Phys. Rev. Lett. 129, 227401 (2022).

    Article  CAS  PubMed  Google Scholar 

  59. Rimoldi, M. et al. Effect of substrates and thermal treatments on metalorganic chemical vapor deposition-grown Sb2Te3 thin films. Cryst. Growth Des. 21, 5135–5144 (2021).

    Article  CAS  Google Scholar 

  60. Longo, E. et al. Exploiting the close-to-Dirac point shift of the Fermi level in the Sb2Te3/Bi2Te3 topological insulator heterostructure for spin–charge conversion. ACS Appl. Mater. Interfaces 15, 50237–50245 (2023).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  61. Ren, Z., Taskin, A. A., Sasaki, S., Segawa, K. & Ando, Y. Fermi level tuning and a large activation gap achieved in the topological insulator Bi2Te2Se by Sn doping. Phys. Rev. B 85, 155301 (2012).

    Article  Google Scholar 

  62. Kölling, S., Westerhof, F. R. & Brinkman, A. Gate-electrode-induced nonreciprocal resistance in topological insulators. Phys. Rev. Appl. 24, 014024 (2025).

    Article  Google Scholar 

  63. Lin, J. J. & Bird, J. P. Recent experimental studies of electron dephasing in metal and semiconductor mesoscopic structures. J. Phys. Condens. Matter 14, R501 (2002).

    Article  CAS  Google Scholar 

  64. Xu, G. et al. Weak antilocalization effect and noncentrosymmetric superconductivity in a topologically nontrivial semimetal LuPdBi. Sci. Rep. 4, 5709 (2014).

    Article  PubMed  PubMed Central  Google Scholar 

  65. Shrestha, K. et al. Extremely large nonsaturating magnetoresistance and ultrahigh mobility due to topological surface states in the metallic Bi2Te3 topological insulator. Phys. Rev. B 95, 195113 (2017).

    Article  Google Scholar 

  66. Sala, G. Probing the quantum metric of 3D topological insulators. Zenodo https://doi.org/10.5281/zenodo.16910820 (2026).

Download references

Acknowledgements

We thank J.-M. Triscone for fruitful discussions and M. Lopes for technical support. We thank A. Morpurgo and I. Gutiérrez for support with the device fabrication. G.S. acknowledges support from the Swiss National Science Foundation (grant no. PZ00P2_223542). E.L. acknowledges the financial support from project nos. PID2023-152225NB-I00 and Severo Ochoa MATRANS42 (no. CEX2023-001263-S) of the Spanish Ministry of Science and Innovation (grant no. MICIU/AEI/10.13039/501100011033 and FEDER, EU), project nos. TED2021-129857B-I00 and PDC2023-145824-I00 funded by MCIN/AEI/10.13039/501100011033 and European Union NextGeneration EU/PRTR and by project no. 2021 SGR 00445 by the Generalitat de Catalunya. M.C. acknowledges support from the PNRR MUR project PE0000023-NQSTI and by the Italian Ministry of University and Research (MUR) PRIN 2022 under the grant no. 2022LP5K7 (BEAT). R.M. acknowledges the financial support from the PNRR MUR project PE0000023-NQSTI and the SPIGA project funded by the European Union – NextGeneration EU – PNRR – M4C2, Investment Line 1.1 – PRIN 2022 – ID P2022LXNYN. M.T.M. and C.O. acknowledge partial support by the Italian Ministry of Foreign Affairs and International Cooperation PGR12351 (ULTRAQMAT) and from PNRR MUR project no. PE0000023-NQSTI (TOPQIN). This work was supported by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract no. MB22.00071, by the Gordon and Betty Moore Foundation (grant no. 332 GBMF10451 to A.D.C.), by the European Research Council (ERC).

Author information

Author notes

  1. These authors contributed equally: Giacomo Sala, Emanuele Longo.

Authors and Affiliations

  1. Department of Quantum Matter Physics, University of Geneva, Geneva, Switzerland

    Giacomo Sala, Stefano Gariglio & Andrea D. Caviglia

  2. Institute of Materials Science of Barcelona (ICMAB-CSIC), Bellaterra, Spain

    Emanuele Longo

  3. Dipartimento di Fisica ‘E. R. Caianiello’, Universitá di Salerno, Fisciano, Italy

    Maria Teresa Mercaldo & Carmine Ortix

  4. CNR-SPIN c/o Universitá di Salerno, Fisciano, Italy

    Mario Cuoco

  5. CNR-IMM, Agrate Brianza, Italy

    Roberto Mantovan

Authors
  1. Giacomo Sala
    View author publications

    Search author on:PubMed Google Scholar

  2. Emanuele Longo
    View author publications

    Search author on:PubMed Google Scholar

  3. Maria Teresa Mercaldo
    View author publications

    Search author on:PubMed Google Scholar

  4. Stefano Gariglio
    View author publications

    Search author on:PubMed Google Scholar

  5. Mario Cuoco
    View author publications

    Search author on:PubMed Google Scholar

  6. Roberto Mantovan
    View author publications

    Search author on:PubMed Google Scholar

  7. Carmine Ortix
    View author publications

    Search author on:PubMed Google Scholar

  8. Andrea D. Caviglia
    View author publications

    Search author on:PubMed Google Scholar

Contributions

E.L. proposed the use of the Sb2Te3 topological insulator to probe the quantum metric. R.M. coordinated the production of Sb2Te3. G.S. and E.L. fabricated the devices, performed the transport measurements and data analysis with a contribution by S.G. The theory calculations were performed by M.T.M. with help from M.C. and C.O. The experimental work was supervised by A.D.C., and the theoretical work was supervised by C.O. All authors contributed to writing the paper with a first presentation provided by G.S.

Corresponding authors

Correspondence to Giacomo Sala, Emanuele Longo, Carmine Ortix or Andrea D. Caviglia.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Materials thanks Su-Yang Xu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Linear transport.

a, Temperature dependence of the resistivity of Sb2Te3(30 nm) showing a metallic behavior. b, Temperature dependence of the total hole carrier density p and hole mobility μ estimated from measurements of the ordinary Hall effect.

Extended Data Fig. 2 Current dependence of the linear and nonlinear magnetoresistances.

a, Linear magnetoresistance measured at 10 K at increasing current with the field applied in the sample plane and perpendicular to the current direction. b, Nonlinear magnetoresistance measured simultaneously to the linear magnetoresistance in a. c Two-probe linear I-V curve demonstrating ohmic contacts.

Extended Data Fig. 3 Frequency dependence of the linear and nonlinear magnetoresistances.

Frequency dependence of the linear a and nonlinear b magnetoresistances measured at 10 K with a 500 μA current. The magnetic field was applied in the sample plane and perpendicular to the current direction.

Extended Data Fig. 4 Temperature dependence of the nonlinear magnetoresistance.

Temperature dependence of the nonlinear magnetoresistance at several magnetic fields and at gate voltages Vg = 0 V and Vg = −40 V in a and b, respectively.

Extended Data Fig. 5 Quantum-metric nonlinear conductivity as a function of the electronic mobility.

Quantum-metric nonlinear conductivity as a function of the electronic mobility at selected magnetic fields and at gate voltages Vg = 0 V and Vg = −40 V in a and b, respectively. The graphs on the right are a zoom-in on the high-mobility range, which corresponds to temperatures between 2 and 20 K.

Extended Data Fig. 6 Dependence of the linear and nonlinear magnetoresistances on the top-gate voltage.

a,b, Linear and nonlinear magnetoresistances, respectively, measured with an in-plane magnetic field oriented orthogonal to the current direction as a function of the top-electrode voltage, at a temperature T = 2.5 K and with an electric current Iω = 400 μA.

Supplementary information

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sala, G., Longo, E., Mercaldo, M.T. et al. Probing the quantum metric of 3D topological insulators. Nat. Mater. (2026). https://doi.org/10.1038/s41563-026-02617-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Version of record:

  • DOI: https://doi.org/10.1038/s41563-026-02617-3

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing