Spin-resolved Andreev levels and parity crossings in hybrid superconductor–semiconductor nanostructures

Journal name:
Nature Nanotechnology
Volume:
9,
Pages:
79–84
Year published:
DOI:
doi:10.1038/nnano.2013.267
Received
Accepted
Published online

Abstract

The physics and operating principles of hybrid superconductor–semiconductor devices rest ultimately on the magnetic properties of their elementary subgap excitations, usually called Andreev levels. Here we report a direct measurement of the Zeeman effect on the Andreev levels of a semiconductor quantum dot with large electron g-factor, strongly coupled to a conventional superconductor with a large critical magnetic field. This material combination allows spin degeneracy to be lifted without destroying superconductivity. We show that a spin-split Andreev level crossing the Fermi energy results in a quantum phase transition to a spin-polarized state, which implies a change in the fermionic parity of the system. This crossing manifests itself as a zero-bias conductance anomaly at finite magnetic field with properties that resemble those expected for Majorana modes in a topological superconductor. Although this resemblance is understood without evoking topological superconductivity, the observed parity transitions could be regarded as precursors of Majorana modes in the long-wire limit.

At a glance

Figures

  1. Andreev levels in a hybrid N-QD-S system and device description.
    Figure 1: Andreev levels in a hybrid N–QD–S system and device description.

    a, The upper panel shows schematics of a N–QD–S device with asymmetric tunnel couplings to the normal metal (Au) and superconductor (V) leads, ΓN and ΓS, respectively. Δ is the superconducting gap, U is the charging energy, μi is the chemical potential of the i lead and ε0 is the QD energy level relative to μS (in the ΓS right arrow 0 limit, the QD has zero, one or two electrons for ε0 > 0, −U < ε0 <0 or ε0 < −U, respectively). The subgap peaks located at ±ζ represent the Andreev levels. In tunnel spectroscopy measurements the alignment of μN to an Andreev level yields a peak in the differential conductance. This process involves an Andreev reflection at the QD–S interface, which transports a Cooper pair to the S lead and reflects a hole to the N contact. The qualitative phase diagram16, 17, 18, 19, 21 (lower panel) depicts the stability of the magnetic doublet (|Dright fence = | ↑ right fence, | ↓ right fence) versus that of the spin singlet (|Sright fence). b, Low-energy excitations of the QD–S system and their expected evolution in a magnetic field, B. Doublet GS case (left): | ↑ right fence is stabilized by B and Andreev levels related to the transition | ↑ right fence right arrow |Sright fence are observed. Singlet GS case (right): at finite B, the excited spin-split states | ↑ right fence and | ↓ right fence give rise to distinct Andreev levels with energy ζ and ζ, respectively. EZ = |g|μBB is the Zeeman energy, where |g| is the g-factor and μB is the Bohr magneton. c, Device schematic: the N and S leads were made of Ti (2.5 nm)/Au (50 nm) and Ti (2.5 nm)/V (45 nm)/Al (5 nm), respectively. The QD system is tuned by means of three gates: a plunger gate, a barrier gate close to the S contact and a back gate. B is applied in the (x, y) device plane (x being parallel to the NW). d, Scanning electron micrograph of a N–QD–S device.

  2. Andreev levels in different coupling regimes and their magnetic-field dependence.
    Figure 2: Andreev levels in different coupling regimes and their magnetic-field dependence.

    Experimental plots of dI/dV versus (Vpg,V) for different QD–S couplings, ΓS (increasing from top to bottom) and magnetic fields parallel to the NW axis (Bx increasing from left to right). Along the Vpg range of the top-left panel, the system GS changes from singlet (|Sright fence) to doublet (|Dright fence) and back to singlet, following the red trajectory in the qualitative diagram on the right side of the same row. We found that increasing Vbg resulted in larger ΓS, which thereby leads to an upwards shift in the phase diagram. Eventually, the red trajectory is pushed into the singlet region (mid and bottom diagrams). Experimentally, this results in the disappearance of the doublet GS loop structure, as shown in the middle and bottom panels of the first column. The second and third columns show the B evolution of the Andreev levels in the three coupling regimes. For relatively weak coupling (top row), the Andreev levels for a singlet GS split because of the Zeeman effect, whereas those for a doublet GS simply move apart. At intermediate coupling (middle row), B induces a QPT from a singlet to a spin-polarized GS, as denoted by the appearance of a loop structure (middle row, third panel). At the largest coupling (bottom row), the Zeeman splitting of the Andreev levels is clearly visible over all the Vpg range. The splitting is gate dependent with a maximum in the central region.

  3. Magnetic-field evolution of the Andreev levels at fixed gate voltage and the level-repulsion effect.
    Figure 3: Magnetic-field evolution of the Andreev levels at fixed gate voltage and the level-repulsion effect.

    a, dI/dV(Vpg,V) measurement at B = 0 corresponding to a singlet–doublet–singlet sweep. b, The left panel shows the qualitative B evolution of the low-energy states of a QD–S system as expected for a doublet GS. The corresponding experimental data measured at position 1 in a are shown on the right. ζ increases linearly with B until it approaches the edge of the superconducting gap. The levels then move towards zero following the B suppression of Δ. c, Same as b, but for a singlet GS. The experimental plot in the right panel was taken at position 2 in a. It shows an asymmetric splitting of the ζ and ζ peaks. The weak B dependence of ζ results from the level repulsion between | ↓ right fence and the continuum of quasiparticle states above Δ.

  4. Magnetic-field induced QPT and angle anisotropy.
    Figure 4: Magnetic-field induced QPT and angle anisotropy.

    a, The left panel shows dI/dV(B,V) taken at the position of the vertical line in the inset (same device as in Fig. 3). The right panel shows line traces at equally spaced B values as extracted from the data in the left panel (each shifted by 0.005 × 2e2/h). The QPT induced by the field is observed as a ZBP that extends over a B range of about 150 mT. This apparently large extension is a consequence of the finite width of the Andreev levels. b, dI/dV(V) traces taken with |B| = 0.6 T, at different angles. This field magnitude corresponds to the QPT field when B is aligned to the NW axis at θ = 0. Owing to the g-factor anisotropy, the ZBP associated with the QPT is split and suppressed when B is rotated away from the NW axis. The peak splitting has a maximum at θ = π/2.

References

  1. De Franceschi, S., Kouwenhoven, L. P., Schönenberger, C. & Wernsdorfer, W. Hybrid superconductor–quantum dot devices. Nature Nanotechnol. 5, 703711 (2010).
  2. Hofstetter, L., Csonka, S., Nygård, J. & Schönenberger, C. Cooper pair splitter realized in a two-quantum-dot Y-junction. Nature 461, 960963 (2009).
  3. Herrmann, L. G. et al. Carbon nanotubes as Cooper pair splitters. Phys. Rev. Lett. 104, 026801 (2010).
  4. Das, A. et al. High-efficiency Cooper pair splitting demonstrated by two-particle conductance resonance and positive noise cross-correlation. Nature Commun. 3, 1165 (2012).
  5. Cleuziou, J., Wernsdorfer, W., Bouchiat, V., Ondarcuhu, T. & Monthioux, M. Carbon nanotube superconducting quantum interference device. Nature Nanotechnol. 1, 5359 (2006).
  6. Sau, J. D., Lutchyn, R. M., Tewari, S. & Sarma, S. D. Generic new platform for topological quantum computation using semiconductor heterostructures. Phys. Rev. Lett. 104, 040502 (2010).
  7. Lutchyn, R. M., Sau, J. D. & Sarma, S. D. Majorana fermions and a topological phase transition in a semiconductor–superconductor. Phys. Rev. Lett. 105, 077001 (2010).
  8. Oreg, Y., Refael, G. & von Oppen, F. Helical liquids and Majorana bound states in quantum wires. Phys. Rev. Lett. 105, 177002 (2010).
  9. van Dam, J. A., Nazarov, Y. V., Bakkers, E. P. A. M., De Franceschi, S. & Kouwenhoven, L. P. Supercurrent reversal in quantum dots. Nature 442, 667670 (2006).
  10. Buitelaar, M. R., Nussbaumer, T. & Schönenberger, C. Quantum dot in the Kondo regime coupled to superconductors. Phys. Rev. Lett. 89, 256801 (2002).
  11. Jorgensen, H. I., Novotny, T., Grove-Rasmussen, K., Flensberg, K. & Lindelof, P. E. Critical current 0–π transition in designed Josephson quantum dot junctions. Nano Lett. 7, 24412445 (2007).
  12. Maurand, R. et al. First-order 0–π quantum phase transition in the Kondo regime of a superconducting carbon nanotube quantum dot. Phys. Rev. X 2, 011009 (2012).
  13. Doh, Y. J., De Franceschi, S., Bakkers, E. P. A. M. & Kouwenhoven, L. P. Andreev reflection versus Coulomb blockade in hybrid semiconductor nanowire devices. Nano Lett. 8, 40984102 (2008).
  14. Yamada, Y., Tanaka, Y. & Kawakami, N. Interplay of Kondo and superconducting correlations in the nonequilibrium Andreev transport through a quantum dot. Phys. Rev. B 84, 075484 (2011).
  15. Glazman, L. I. & Matveev, K. Resonant Josephson current through Kondo impurities in a tunnel barrier. JETP Lett. 49, 659 (1989).
  16. Rozhkov, A. V. & Arovas, D. P. Josephson coupling through a magnetic impurity. Phys. Rev. Lett. 82, 27882791 (1999).
  17. Vecino, E., Martín-Rodero, A. & Yeyati, A. L. Josephson current through a correlated quantum level: Andreev states and π junction behavior. Phys. Rev. B 68, 035105 (2003).
  18. Oguri, A., Tanaka, Y. & Hewson, A. C. Quantum phase transition in a minimal model for the Kondo effect in a Josephson junction. J. Phys. Soc. Jpn 73, 24942504 (2004).
  19. Bauer, J., Oguri, A. & Hewson, A. C. Spectral properties of locally correlated electrons in a Bardeen–Cooper–Schrieffer superconductor. J. Phys. Condens. Matter 19, 486211 (2007).
  20. Choi, M. S., Lee, M., Kang, K. & Belzig, W. Kondo effect and Josephson current through a quantum dot between two superconductors. Phys. Rev. B 70, 020502(R) (2004).
  21. Meng, T., Florens, S. & Simon, P. Self-consistent description of Andreev bound states in Josephson quantum dot devices. Phys. Rev. B 79, 224521 (2009).
  22. Domański, T., Donabidowicz, A. & Wysokiński, K. I. Meservey–Tedrow–Fulde effect in a quantum dot embedded between metallic and superconducting electrodes. Phys. Rev. B 78, 144515 (2008).
  23. Futterer, D., Swieboddzinski, J., Governale, M. & König, J. Renormalization effects in interacting quantum dots coupled to superconducting leads. Phys. Rev. B 87, 014509 (2013).
  24. Kanai, Y. et al. Electrical control of Kondo effect and superconducting transport in a side-gated InAs quantum dot Josephson junction. Phys. Rev. B 82, 054512 (2010).
  25. Sand-Jespersen, T. et al. Kondo-enhanced Andreev tunneling in InAs nanowire quantum dots. Phys. Rev. Lett. 99, 126603 (2007).
  26. Eichler, A. et al. Even–odd effect in Andreev transport through a carbon nanotube quantum dot. Phys. Rev. Lett. 99, 126602 (2007).
  27. Pillet, J. D. et al. Andreev bound states in supercurrent-carrying carbon nanotubes revealed. Nature Phys. 6, 965969 (2010).
  28. Deacon, R. S. et al. Tunneling spectroscopy of Andreev energy levels in a quantum dot coupled to a superconductor. Phys. Rev. Lett. 104, 076805 (2010).
  29. Dirks, T. et al. Transport through Andreev bound states in a graphene quantum dot. Nature Phys. 7, 386390 (2011).
  30. Lee, E. J. H. et al. Zero-bias anomaly in a nanowire quantum dot coupled to superconductors. Phys. Rev. Lett. 109, 186802 (2012).
  31. Chang, W., Manucharyan, V. E., Jespersen, T. S., Nygåard, J. & Marcus, C. M. Tunneling spectroscopy of quasiparticle bound states in a spinful Josephson junction. Phys. Rev. Lett. 110, 217005 (2013).
  32. Pillet, J. D., Joyez, P., Zitko, R. & Goffman, M. F. Tunneling spectroscopy of a single quantum dot coupled to a superconductor: from Kondo ridge to Andreev bound states. Phys. Rev. B 88, 045101 (2013).
  33. Kumar, A. et al. Temperature dependence of Andreev spectra in a superconducting carbon nanotube quantum dot. Preprint at http://lanl.arXiv.org/1308.1020v1 (2013).
  34. Giazotto, F. et al. A Josephson quantum electron pump. Nature Phys. 7, 857861 (2011).
  35. Prada, E., San-Jose, P. & Aguado, R. Transport spectroscopy of N–S nanowire junctions with Majorana fermions. Phys. Rev. B 86, 180503(R) (2012).
  36. Rainis, D., Klinovaja, J. & Loss, D. Towards a realistic transport modeling in a superconducting nanowire with Majorana fermions. Phys. Rev. B 87, 024515 (2013).
  37. Sarma, S. D., Sau, J. D. & Stanescu, T. D. Splitting of the zero-bias conductance peak as smoking gun evidence for the existence of the Majorana mode in a superconductor–semiconductor nanowire. Phys. Rev. B 86, 220506(R) (2012).
  38. Liu, J., Potter, A. C., Law, K. T. & Lee, P. A. Zero-bias peaks in the tunneling conductance of spin-orbit-coupled superconducting wires with and without Majorana end-states. Phys. Rev. Lett. 109, 267002 (2012).
  39. Mourik, V. et al. Signatures of Majorana fermions in hybrid superconductor–semiconductor nanowire devices. Science 336, 10031007 (2012).
  40. Das, A. et al. Zero-bias peaks and splitting in an Al–InAs nanowire topological superconductor as a signature of Majorana fermions. Nature Phys. 8, 887895 (2012).
  41. Deng, M. T. et al. Anomalous zero-bias conductance peak in a Nb–InSb nanowire–Nb hybrid device. Nano Lett. 12, 64146419 (2012).
  42. Finck, A. D. K., van Harlingen, D. J., Mohseni, P. K., Jung, K. & Li, X. Anomalous modulation of a zero-bias peak in a hybrid nanowire-superconductor device. Phys. Rev. Lett. 110, 126406 (2012).
  43. Churchill, H. O. H. et al. Superconductor-nanowire devices from tunnelling to multi-channel regime: zero-bias oscillations and magnetoconductance crossover. Phys. Rev. B 87, 242401(R) (2013).
  44. Stanescu, T. D., Lutchyn, R. M. & Sarma, S. D. Dimensional crossover in spin-orbit-coupled semiconductor nanowires with induced superconducting pairing. Phys. Rev. B 87, 094518 (2013).
  45. Nadj-Perge, S., Drozdov, I. K., Bernevig, B. A. & Yazdani, A. Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor. Phys. Rev. B 88, 020407(R) (2013).
  46. Klinovaja, J., Stano, P., Yazdani, A. & Loss, D. Topological superconductivity and Majorana fermions in RKKY systems. Phys. Rev. Lett. 111, 186805 (2013).
  47. Vazifeh, M. M. & Franz, M. Self-organized topological state with Majorana fermions. Phys. Rev. Lett. 111, 206802 (2013).
  48. Braunecker, B. & Simon, P. Interplay between classical magnetic moments and superconductivity in quantum one-dimensional conductors: towards a self-sustained topological Majorana phase. Phys. Rev. Lett. 111, 147202 (2013).
  49. Pientka, F., Glazman, L. & von Oppen, F. Topological superconducting phase in helical Shiba chains. Phys. Rev. B 88, 155420 (2013).
  50. Jiang, X., Xiong, Q., Qian, F., Li, Y. & Lieber, C. M. InAs/InP radial nanowire heterostructures as high electron mobility devices. Nano Lett. 7, 32143218 (2007).

Download references

Author information

Affiliations

  1. SPSMS, CEA-INAC/UJF-Grenoble 1, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France

    • Eduardo J. H. Lee,
    • Manuel Houzet &
    • Silvano De Franceschi
  2. Harvard University, Department of Chemistry and Chemical Biology, Cambridge, Massachusetts 02138, USA

    • Xiaocheng Jiang &
    • Charles M. Lieber
  3. Instituto de Ciencia de Materiales de Madrid, Consejo Superior de Investigaciones Científicas (CSIC), Sor Juana Inés de la Cruz 3, 28049 Madrid, Spain

    • Ramón Aguado

Contributions

E.J.H.L. and S.D.F. conceived the experiment. X.J. grew the semiconductor NWs under C.M.L.'s supervision. E.J.H.L. fabricated the devices and performed all the measurements under S.D.F.'s supervision. R.A. performed the Hartree–Fock calculations, and M.H. carried out the analytical study of the level-repulsion effect. E.J.H.L., S.D.F., R.A. and M.H. analysed and interpreted the results. All authors co-wrote the manuscript.

Competing financial interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to:

Author details

Supplementary information

PDF files

  1. Supplementary information (1,240 KB)

    Supplementary information

Additional data