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.

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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.


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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.

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

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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

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Correspondence to Ichiro Takeuchi.

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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.

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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).

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