Skip to main content

Thank you for visiting 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.

Perfect Andreev reflection due to the Klein paradox in a topological superconducting state


In 1928, Dirac proposed a wave equation to describe relativistic electrons1. Shortly afterwards, Klein solved a simple potential step problem for the Dirac equation and encountered an apparent paradox: the potential barrier becomes transparent when its height is larger than the electron energy. For massless particles, backscattering is completely forbidden in Klein tunnelling, leading to perfect transmission through any potential barrier2,3. The recent advent of condensed-matter systems with Dirac-like excitations, such as graphene and topological insulators, has opened up the possibility of observing Klein tunnelling experimentally4,5,6. In the surface states of topological insulators, fermions are bound by spin–momentum locking and are thus immune from backscattering, which is prohibited by time-reversal symmetry. Here we report the observation of perfect Andreev reflection in point-contact spectroscopy—a clear signature of Klein tunnelling and a manifestation of the underlying ‘relativistic’ physics of a proximity-induced superconducting state in a topological Kondo insulator. Our findings shed light on a previously overlooked aspect of topological superconductivity and can serve as the basis for a unique family of spintronic and superconducting devices, the interface transport phenomena of which are completely governed by their helical topological states.

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

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type



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

Fig. 1: Perfect Andreev reflection due to the Klein paradox.
Fig. 2: Sensitivity of perfect Andreev reflection to compromised topological superconductivity.
Fig. 3: Andreev reflection process under the Dirac Hamiltonian.

Data availability

The data that support the findings of this study are available within the paper. Additional data are available from the corresponding authors upon reasonable request.


  1. Dirac, P. A. M. The quantum theory of the electron. Proc. R. Soc. Lond. A 117, 610–624 (1928).

    Article  ADS  Google Scholar 

  2. Klein, O. Die reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac. Z. Phys. 53, 157–165 (1929).

    Article  ADS  CAS  Google Scholar 

  3. Calogeracos, A. & Dombey, N. History and physics of the Klein paradox. Contemp. Phys. 40, 313–321 (1999).

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  5. Beenakker, C. W. J. Colloquium: Andreev reflection and Klein tunneling in graphene. Rev. Mod. Phys. 80, 1337–1354 (2008).

    Article  ADS  CAS  Google Scholar 

  6. Fu, L. & Kane, C. L. Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys. Rev. Lett. 100, 096407 (2008).

    Article  ADS  Google Scholar 

  7. Stander, N., Huard, B. & Goldhaber-Gordon, D. Evidence for Klein tunneling in graphene p–n junctions. Phys. Rev. Lett. 102, 026807 (2009).

    Article  ADS  CAS  Google Scholar 

  8. Young, A. F. & Kim, P. Quantum interference and Klein tunneling in graphene heterojunctions. Nat. Phys. 5, 222–226 (2009).

    Article  CAS  Google Scholar 

  9. Blonder, G. E., Tinkham, M. & Klapwijk, T. M. Transition from metallic to tunneling regimes in superconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion. Phys. Rev. B 25, 4515–4532 (1982).

    Article  ADS  CAS  Google Scholar 

  10. Daghero, D. & Gonnelli, R. S. Probing multiband superconductivity by point-contact spectroscopy. Supercond. Sci. Technol. 23, 043001 (2010).

    Article  ADS  Google Scholar 

  11. Lee, W.-C. & Greene, L. H. Recent progress of probing correlated electron states by point contact spectroscopy. Rep. Prog. Phys. 79, 094502 (2016).

    Article  ADS  Google Scholar 

  12. Adroguer, P. et al. Probing the helical edge states of a topological insulator by Cooper-pair injection. Phys. Rev. B 82, 081303 (2010).

    Article  ADS  Google Scholar 

  13. Dzero, M., Sun, K., Coleman, P. & Galitski, V. Theory of topological Kondo insulators. Phys. Rev. B 85, 045130 (2012).

    Article  ADS  Google Scholar 

  14. Syers, P., Kim, D., Fuhrer, M. S. & Paglione, J. Tuning bulk and surface conduction in the proposed topological Kondo insulator SmB6. Phys. Rev. Lett. 114, 096601 (2015).

    Article  ADS  Google Scholar 

  15. Eo, Y. S., Sun, K., Kurdak, Ç., Kim, D.-J. & Fisk, Z. Inverted resistance measurements as a method for characterizing the bulk and surface conductivities of three-dimensional topological insulators. Phys. Rev. Appl. 9, 044006 (2018).

    Article  ADS  CAS  Google Scholar 

  16. Zhang, X. et al. Hybridization, inter-ion correlation, and surface states in the Kondo insulator SmB6. Phys. Rev. X 3, 011011 (2013).

    Google Scholar 

  17. Jiang, J. et al. Observation of possible topological in-gap surface states in the Kondo insulator SmB6 by photoemission. Nat. Commun. 4, 3010 (2013).

    Article  ADS  CAS  Google Scholar 

  18. Neupane, M. et al. Surface electronic structure of the topological Kondo-insulator candidate correlated electron system SmB6. Nat. Commun. 4, 2991 (2013).

    Article  ADS  CAS  Google Scholar 

  19. Dai, W. et al. Proximity-effect-induced superconducting gap in topological surface states – a point contact spectroscopy study of NbSe2/Bi2Se3 superconductor-topological insulator heterostructures. Sci. Rep. 7, 7631 (2017).

    Article  ADS  Google Scholar 

  20. Xu, S.-Y. et al. Momentum-space imaging of Cooper pairing in a half-Dirac-gas topological superconductor. Nat. Phys. 10, 943–950 (2014).

    Article  CAS  Google Scholar 

  21. Lee, S. et al. Observation of the superconducting proximity effect in the surface state of SmB6 thin films. Phys. Rev. X 6, 031031 (2016).

    Google Scholar 

  22. Borisov, K., Chang, C.-Z., Moodera, J. S. & Stamenov, P. High Fermi-level spin polarization in the (Bi1−xSbx)2Te3 family of topological insulators: a point contact Andreev reflection study. Phys. Rev. B 94, 094415 (2016).

    Article  ADS  Google Scholar 

  23. Shoman, T. et al. Topological proximity effect in a topological insulator hybrid. Nat. Commun. 6, 6547 (2015).

    Article  ADS  CAS  Google Scholar 

  24. Hutasoit, J. A. & Stanescu, T. D. Induced spin texture in semiconductor/topological insulator heterostructures. Phys. Rev. B 84, 085103 (2011).

    Article  ADS  Google Scholar 

  25. Szabó, P. et al. Superconducting energy gap of YB6 studied by point-contact spectroscopy. Physica C 460–462, 626–627 (2007).

    Article  ADS  Google Scholar 

  26. Alexandrov, V., Coleman, P. & Erten, O. Kondo breakdown in topological Kondo insulators. Phys. Rev. Lett. 114, 177202 (2015).

    Article  ADS  Google Scholar 

  27. Wang, M.-X. et al. The coexistence of superconductivity and topological order in the Bi2Se3 thin films. Science 336, 52–55 (2012).

    Article  ADS  CAS  Google Scholar 

  28. Soulen Jr, R. J. et al. Measuring the spin polarization of a metal with a superconducting point contact. Science 282, 85–88 (1998).

    Article  ADS  CAS  Google Scholar 

  29. Strijkers, G. J., Ji, Y., Yang, F. Y., Chien, C. L. & Byers, J. M. Andreev reflections at metal/superconductor point contacts: Measurement and analysis. Phys. Rev. B 63, 104510 (2001).

    Article  ADS  Google Scholar 

  30. Tkachov, G. & Hankiewicz, E. M. Helical Andreev bound states and superconducting Klein tunneling in topological insulator Josephson junctions. Phys. Rev. B 88, 075401 (2013).

    Article  ADS  Google Scholar 

  31. Janvier, C. et al. Coherent manipulation of Andreev states in superconducting atomic contacts. Science 349, 1199–1202 (2015).

    Article  ADS  CAS  Google Scholar 

  32. Kornev, V. K., Kolotinskiy, N. V., Levochkina, A. Y. & Mukhanov, O. A. Critical current spread and thermal noise in Bi-SQUID cells and arrays. IEEE Trans. Appl. Supercond. 27, 1601005 (2017).

    Google Scholar 

  33. Zhang, C., Lu, H.-Z., Shen, S.-Q., Chen, Y. P. & Xiu, F. Towards the manipulation of topological states of matter: a perspective from electron transport. Sci. Bull. (Beijing) 63, 580–594 (2018).

    Article  CAS  Google Scholar 

  34. Yong, J. et al. Robust topological surface state in Kondo insulator SmB6 thin films. Appl. Phys. Lett. 105, 222403 (2014).

    Article  ADS  Google Scholar 

  35. Li, Y., Ma, Q., Huang, S. X. & Chien, C. L. Thin films of topological Kondo insulator candidate SmB6: strong spin–orbit torque without exclusive surface conduction. Sci. Adv. 4, eaap8294 (2018).

    Article  ADS  Google Scholar 

  36. Ohring, M. Materials science of thin films 2nd edn (Academic, 2001).

  37. Schneider, R., Geerk, J. & Rietschel, H. Electron tunnelling into a superconducting cluster compound: YB6. Europhys. Lett. 4, 845–849 (1987).

    Article  ADS  CAS  Google Scholar 

  38. Sluchanko, N. et al. Lattice instability and enhancement of superconductivity in YB6. Phys. Rev. B 96, 144501 (2017).

    Article  ADS  Google Scholar 

  39. Dynes, R. C., Narayanamurti, V. & Garno, J. P. Direct measurement of quasiparticle-lifetime broadening in a strong-coupled superconductor. Phys. Rev. Lett. 41, 1509–1512 (1978).

    Article  ADS  CAS  Google Scholar 

  40. Mazin, I. I., Golubov, A. A. & Nadgorny, B. Probing spin polarization with Andreev reflection: a theoretical basis. J. Appl. Phys. 89, 7576–7578 (2001).

    Article  ADS  CAS  Google Scholar 

  41. Wolgast, S. et al. Low-temperature surface conduction in the Kondo insulator SmB6. Phys. Rev. B 88, 180405 (2013).

    Article  ADS  Google Scholar 

  42. Taskin, A. A. et al. Planar Hall effect from the surface of topological insulators. Nat. Commun. 8, 1340 (2017).

    Article  ADS  CAS  Google Scholar 

  43. Wang, L.-X. et al. Zeeman effect on surface electron transport in topological insulator Bi2Se3 nanoribbons. Nanoscale 7, 16687–16694 (2015).

    Article  ADS  CAS  Google Scholar 

  44. Chang, C.-Z., Wei, P. & Moodera, J. S. Breaking time reversal symmetry in topological insulators. MRS Bull. 39, 867–872 (2014).

    Article  CAS  Google Scholar 

  45. Fu, Y.-S. et al. Observation of Zeeman effect in topological surface state with distinct material dependence. Nat. Commun. 7, 10829 (2016).

    Article  ADS  CAS  Google Scholar 

  46. Erten, O., Ghaemi, P. & Coleman, P. Kondo breakdown and quantum oscillations in SmB6. Phys. Rev. Lett. 116, 046403 (2016).

    Article  ADS  Google Scholar 

  47. Wolgast, S. et al. Reduction of the low-temperature bulk gap in samarium hexaboride under high magnetic fields. Phys. Rev. B 95, 245112 (2017).

    Article  ADS  Google Scholar 

  48. Analytis, J. G. et al. Transport in the quantum limit by two-dimensional Dirac fermions in a topological insulator. Nat. Phys. 6, 960–964 (2010).

    Article  CAS  Google Scholar 

  49. Thomas, S. et al. Weak antilocalization and linear magnetoresistance in the surface state of SmB6. Phys. Rev. B 94, 205114 (2016).

    Article  ADS  Google Scholar 

  50. Biswas, S. et al. Robust local and nonlocal transport in the topological Kondo insulator SmB6 in the presence of a high magnetic field. Phys. Rev. B 92, 085103 (2015).

    Article  ADS  Google Scholar 

  51. Gonnelli, R. S. et al. Temperature and junction-type dependency of Andreev reflection in MgB2. J. Phys. Chem. Solids 63, 2319–2323 (2002).

    Article  ADS  CAS  Google Scholar 

  52. Li, Z.-Z. et al. Andreev reflection spectroscopy evidence for multiple gaps in MgB2. Phys. Rev. B 66, 064513 (2002).

    Article  ADS  Google Scholar 

  53. Park, W. K., Greene, L. H., Sarrao, J. L. & Thompson, J. D. Andreev reflection at the normal-metal/heavy-fermion superconductor CeCoIn5 interface. Phys. Rev. B 72, 052509 (2005).

    Article  ADS  Google Scholar 

  54. Zhang, X. et al. Evidence of a universal and isotropic 2Δ/k B T C ratio in 122-type iron pnictide superconductors over a wide doping range. Phys. Rev. B 82, 020515 (2010).

    Article  ADS  Google Scholar 

  55. Sheet, G., Mukhopadhyay, S. & Raychaudhuri, P. Role of critical current on the point-contact Andreev reflection spectra between a normal metal and a superconductor. Phys. Rev. B 69, 134507 (2004).

    Article  ADS  Google Scholar 

Download references


We thank Y. S. Eo for discussions on the properties of SmB6, F. C. Wellstood for discussions on the possible applications of superconducting Klein tunnelling devices, and H. M. Iftekhar Jaim for assistance with X-ray measurements. This project was funded by ONR N00014-13-1-0635; ONR N00014-15-1-2222; AFOSR number FA9550-14-10332; NSF (DMR-1410665); and C-SPIN, one of six centers of STARnet, a Semiconductor Research Corporation (SRC) programme sponsored by MARCO and DARPA. We acknowledge support from the Maryland NanoCenter. J.P. acknowledges support from the Gordon and Betty Moore Foundation's EPiQS Initiative through grant number GBMF4419. V.G. was supported by DOE-BES (DESC0001911) and the Simons Foundation. This work was also supported in part by the Center for Spintronic Materials in Advanced infoRmation Technologies (SMART), one of the centers in nCORE, an SRC programme sponsored by NSF and NIST. The work at University of California, Irvine, was carried out using the electron microscopy facilities of the Irvine Materials Research Institute (IMRI) and was supported by the National Science Foundation through grant DMR-1506535 and by DOE-BES under grant DE-SC0014430. We acknowledge support from the National Institute of Standards and Technology Cooperative Agreement 70NANB17H301.

Reviewer information

Nature thanks Ewelina Hankiewicz, David Goldhaber-Gordon and Jinfeng Jia for their contribution to the peer review of this work.

Author information

Authors and Affiliations



S.L., X.Z. and I.T. conceived the experiment. S.L. fabricated thin films and devices, and performed their characterization—including point-contact spectroscopy measurements—with assistance from X.Z. and J.S.H. V.S., V.M.Y. and V.G. performed the theoretical calculations. D.S. analysed the compositions of the films. J.F. performed the literature survey on previous Andreev reflection experiments. S.D., T.B. and X.P. performed TEM measurements. V.M.Y., J.P., R.L.G. and V.G. helped with data interpretation and analysis and manuscript preparation. S.L., V.S., X.Z. and I.T. wrote the paper. I.T. supervised and coordinated the project. All authors discussed the results and commented on the manuscript

Corresponding author

Correspondence to Ichiro Takeuchi.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

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

Extended data figures and tables

Extended Data Fig. 1 Structural characterization of SmB6 thin films.

a, High-resolution cross-sectional transmission electron microscopy image of a SmB6 thin film. The yellow squares correspond to the regions of the SAED measurements shown in bd. bd, SAED measurements of SmB6 (b), Si substrate (c) and SmB6/Si interface regions (d). ZA, zone axis. e, Epitaxial relationship between the SmB6 and the Si substrate. f, Aberration-corrected scanning transmission electron microscopy cross-sectional image of a SmB6 thin film. g, θ−2θ X-ray diffraction pattern of a SmB6 thin film on a Si (001) substrate.

Extended Data Fig. 2 Superconducting transition temperature (Tc) of YBx thin films.

a, Temperature-dependent resistance curves of YBx thin films with different stoichiometric B/Y ratios. b, Change in Tc as a function of stoichiometric B/Y ratio (x).

Extended Data Fig. 3 Au-SmB6/YB6 thin film junctions.

a, Cross-sectional schematic of a Au-SmB6 (20 nm)/YB6 (100 nm) structure. b, Optical microscopy image of the device. c, Normalized dI/dV spectra of the Au-SmB6/YB6 structure at different temperatures. The red lines are fits using the Dirac–BTK model. The normalized dI/dV curves at 1.8 K are plotted using the obtained values, whereas the other curves are vertically shifted for clarity.

Extended Data Fig. 4 Yttrium-substituted SmB6 thin films.

a, Comparison of logR against 1/T plots of SmB6, and 20% and 50% Y-substituted SmB6 (that is, Sm0.8Y0.2B6 and Sm0.5Y0.5B6, respectively). The resistance values are normalized by their values at 300 K. The positive linear slopes at in the relatively high-temperature regions are roughly proportional to the activation energy. b, GGsurface (logarithmic scale, normalized by the conductance at 300 K) plotted against 1/T for pure SmB6 (black squares) and Sm0.8Y0.2B6 (red circles). The slopes of the linear fits (black and red lines) correspond to the activation energies (Ea) of pure SmB6 and Sm0.8Y0.2B6, and are 3.0 meV and 2.2 meV, respectively.

Extended Data Fig. 5 Robustness of perfect Andreev reflection.

Point-contact spectra obtained at different positions (1, 2 and 3, which are roughly 1 mm apart from each other) on SmB6/YB6 heterostructures with 20-nm-thick SmB6 (left) and 30-nm-thick SmB6 (right). Conductance doubling is consistently observed at all positions in the dI/dV spectra of the SmB6/YB6 heterostructures.

Extended Data Fig. 6 Standard BTK compared with Dirac–BTK models.

a, Comparison of calculated dI/dV spectra with the standard BTK and the Dirac–BTK models for Z = 0.2, 0.4 and 0.8 ( = 1 meV). b, Comparison of the Dirac–BTK and the standard BTK fits to the experimental dI/dV spectrum of a PtIr-SmB6 (20 nm)/YB6 contact (Fig. 1c). The red curve is the theoretical conductance curve in the Dirac–BTK model and the standard BTK model with Z = 0. Both appear identical, as expected, for the same (here 0.77). The blue curve is the theoretical standard BTK curve with ∆ = 0.77 and Z = 0.39, this Z value is assessed from contacts to other heterostructures in this study that do not exhibit perfect Andreev reflection (that is, those with thin SmB6 (10 nm) and Y-substituted SmB6). The effect of nullifying Z by incorporation of a Dirac material in the Andreev reflection process is clearly seen.

Extended Data Fig. 7 Magnetic-field-dependent dI/dV spectra.

a, dI/dV spectra of Au-SmB6/YB6 device under a magnetic field applied along the in-plane and out-of-plane directions. b, Normalized dI/dV at zero bias as a function of magnetic field. The inset shows superconducting order parameter () as a function of magnetic field normalized by at 0 T ((0)). was estimated as the bias voltage point at which the maximum first derivative of each dI/dV spectrum occurs under different magnetic fields.

Extended Data Fig. 8 Experimentally observed conductance doubling.

Comparison of the normalized dI/dV spectrum obtained from the PtIr-SmB6 (20 nm)/YB6 junction in this work (red line, experimental data) with the reported point-contact spectra obtained from Nb-Cu junctions28,29. The arrows indicate conductance dips near the . Such dips are not present in our spectrum.

Supplementary information

Supplementary Information

This file contains a Supplementary Discussion and Supplementary Figure S1.

Supplementary Tables

This file contains Supplementary Table S1.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lee, S., Stanev, V., Zhang, X. et al. Perfect Andreev reflection due to the Klein paradox in a topological superconducting state. Nature 570, 344–348 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


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