Normal, superconducting and topological regimes of hybrid double quantum dots

Journal name:
Nature Nanotechnology
Volume:
12,
Pages:
212–217
Year published:
DOI:
doi:10.1038/nnano.2016.227
Received
Accepted
Published online

Epitaxial semiconductor–superconductor hybrid materials are an excellent basis for studying mesoscopic and topological superconductivity, as the semiconductor inherits a hard superconducting gap while retaining tunable carrier density1. Here, we investigate double-quantum-dot structures made from InAs nanowires with a patterned epitaxial Al two-facet shell2 that proximitizes two gate-defined segments along the nanowire. We follow the evolution of mesoscopic superconductivity and charging energy in this system as a function of magnetic field and voltage-tuned barriers. Interdot coupling is varied from strong to weak using side gates, and the ground state is varied between normal, superconducting and topological regimes by applying a magnetic field. We identify the topological transition by tracking the spacing between successive co-tunnelling peaks as a function of axial magnetic field3 and show that the individual dots host weakly hybridized Majorana modes.

At a glance

Figures

  1. Double quantum dot with controllable coupling and superconductivity.
    Figure 1: Double quantum dot with controllable coupling and superconductivity.

    a, Device schematic showing quantum dots defined by etching Al (blue) from the InAs (green) nanowires, with normal metal (Ti/Au) contacts and electrostatic side gates. Conductance g versus gate voltages VL and VR, at zero source–drain voltage (VSD = 0) forms a two-dimensional charge stability diagram (CSD). b, Applying B = 0.7 T quenches superconductivity. At a middle gate voltage of VM = –5 V, a single quantum dot is formed, with diagonal stripes corresponding to 1e-periodic Coulomb blockade peaks. c, Applying VM = –5.5 V separates the dots, resulting in a honeycomb-pattern CSD. d, Lowering B drives the system into the superconducting state, yielding 2e-periodic honeycomb vertices. e, Differential conductance g as a function of VSD and VL (VR) at VM = –5 V shows 2e-periodic Coulomb diamonds for eVSD < Δ and 1e-periodic Coulomb diamonds for eVSD > Δ. Note that VL and VR tune the joint superconducting dot with similar efficiency.

  2. Tuning interdot coupling.
    Figure 2: Tuning interdot coupling.

    a, Electron micrograph with false colour of device 2 (similar to device 3) with epitaxial Al regions indicated schematically (blue). The patterned dots have dimensions of 0.5 μm (left) and 1.5 μm (right). b, Peak conductance gpeak as a function of fractional peak splitting, f = 2δSP, obtained from device 2 in the normal state, with B = 0.7 T. Note that gpeak decreases as f increases. ce, CSDs with various f values, controlled by VM. δS and δP are shown in c. f, gpeak (blue, right axis) and f (black, left axis) versus middle side gate VM, obtained from device 3 in the superconducting regime. Note the non-monotonic behaviour of f versus VM. g,h, CSDs for different values of f. i, Conductance g versus VRC, shows an alternating on/off pattern, corresponding to the width and spacing of the right junction dot energy levels. The high-frequency oscillations are Coulomb peaks associated with the right patterned dot. The results in fi were obtained at finite d.c. bias, VSD = 10 μV.

  3. Addition energy as a function of interdot coupling.
    Figure 3: Addition energy as a function of interdot coupling.

    a, The two set-ups used to measure the addition energy Eadd of the left dot as a function of its coupling to the right superconductor. Set-up 1 has three closed valves. Here f is measured in the normal state as a function of VM. In set-up, 2 the right valve is open, quenching the charging energy of the right dot to well below Δ. Here, Eadd is measured as a function of VM. b, Eadd as a function of f, as both quantities were mapped to VM. c, A typical normal-state double-dot CSD from which f is extracted d, CSD from set-up 2. With the right right valve open, sweeping VRT has no effect while VL changes the number of electrons on the left dot. e,f, VSD versus VL for a relatively closed middle valve, Eadd > Δ (e) and a relatively open middle valve, Eadd < Δ (f). Eadd is defined as the value of eVSD at the apex of the Coulomb diamonds.

  4. Signatures of Majorana modes via even–odd peak spacing along co-tunnelling lines.
    Figure 4: Signatures of Majorana modes via even–odd peak spacing along co-tunnelling lines.

    ah, Charge stability diagrams at increasing magnetic field. Conductance g is shown as a function of gate voltages VR and VL for device 1 (Fig. 1a) (ad) and as a function of VL and VRT for device 2 (eh) (Fig. 2a). Hexagons of charge stability are outlined (dashed green) in e. Note that the interdot coupling strength is roughly unaffected by field. For example, f(B = 0) = 0.61 and f(B = 850 mT) = 0.67 in device 1. Co-tunnelling lines for the left (right) dot are shown in a and e as dashed (solid) white lines. i, Even–odd co-tunnelling peak spacing, S, of device 1 for the left (circles, LD) and right (squares, RD) quantum dots as a function of magnetic field. Data points were calculated by averaging along several co-tunnelling lines of the right and left dots. Peaks from the right dot become evenly spaced around B|| = 750 mT, above which they oscillate once about Seven = Sodd. Inset: zoom-in of spacing oscillation with the extracted energy amplitude. j, As in i, but for device 2. In device 2, the oscillation is observed in the left dot. k, Differential conductance as a function of B|| for device 3 along a co-tunnelling axis, Vcot, corresponding to the left dot (0.5 μm). Increasing B|| halves the peak spacing from 2e to 1e. The overshoot in peak spacing around B|| = 350 mT is consistent with i and j.

References

  1. Chang, W. et al. Hard gap in epitaxial semiconductor–superconductor nanowires. Nat. Nanotech. 10, 232236 (2015).
  2. Krogstrup, P. et al. Epitaxy of semiconductor–superconductor nanowires. Nat. Mater. 14, 400406 (2015).
  3. Albrecht, S. M. et al. Exponential protection of zero modes in Majorana islands. Nature 531, 206209 (2016).
  4. Oreg, Y., Refael, G. & von Oppen, F. Helical liquids and Majorana bound states in quantum wires. Phys. Rev. Lett. 105, 177002 (2010).
  5. Lutchyn, R. M., Sau, J. D. & Das Sarma, S. Majorana fermions and a topological phase transition in semiconductor–superconductor heterostructures. Phys. Rev. Lett. 105, 077001 (2010).
  6. Aasen, D. et al. Milestones toward Majorana-based quantum computing. Phys. Rev. X 6, 031016 (2015).
  7. Freedman, M. H., Kitaev, A., Larsen, M. J. & Wang, Z. Topological quantum computation. Bull. Am. Math. Soc. 40, 3138 (2003).
  8. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 10831159 (2008).
  9. Mourik, V. et al. Signatures of Majorana fermions in hybrid superconductor–semiconductor nanowire devices. Science 336, 10031007 (2012).
  10. Deng, M. T. et al. Anomalous zero-bias conductance peak in a Nb–InSb nanowire–Nb hybrid device. Nano Lett. 12, 64146419 (2012).
  11. Das, A. et al. Zero-bias peaks and splitting in an Al–InAs nanowire topological superconductor as a signature of Majorana fermions. Nat. Phys. 8, 887895 (2012).
  12. Churchill, H. O. H. et al. Superconductor–nanowire devices from tunneling to the multichannel regime: zero-bias oscillations and magnetoconductance crossover. Phys. Rev. B 87, 241401 (2013).
  13. Nadj-Perge, S. et al. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346, 602607 (2014).
  14. Fasth, C., Fuhrer, A., Björk, M. T. & Samuelson, L. Tunable double quantum dots in InAs nanowires defined by local gate electrodes. Nano Lett. 5, 14871490 (2005).
  15. van der Wiel, W. G. et al. Electron transport through double quantum dots. Rev. Mod. Phys. 75, 122 (2002).
  16. Tuominen, M. T., Hergenrother, J. M., Tighe, T. S. & Tinkham M. Experimental evidence for parity-based 2e periodicity in a superconducting single-electron tunneling transistor. Phys. Rev. Lett. 69, 19972000 (1992).
  17. Bibow, E., Lafarge, P. & Lévy, L. P. Resonant Cooper pair tunneling through a double-island qubit. Phys. Rev. Lett. 88, 017003 (2002).
  18. Higginbotham, A. P. et al. Parity lifetime of bound states in a proximitized semiconductor nanowire. Nat. Phys. 11, 10171021 (2015).
  19. Waugh, F. R. et al. Single-electron charging in double and triple quantum dots with tunable coupling. Phys. Rev. Lett 75, 705708 (1995).
  20. Kouwenhoven, L. P. et al. Electron Transport in Quantum Dots (Kluwer Academic, 1997).
  21. Livermore, C., Crouch, C. H., Westervelt, R. M., Campman, K. L. & Gossard, A. C. The Coulomb blockade in coupled quantum dots. Science 274, 13321335 (1996).
  22. Hu, Y. et al. A Ge/Si heterostructure nanowire-based double quantum dot with integrated charge sensor. Nat. Nanotech. 2, 622625 (2007).
  23. Waugh, F. R. et al. Measuring interactions between tunnel-coupled quantum dots. Phys. Rev. B 53, 14231420 (1996).
  24. Larsen, T. W. et al. A semiconductor nanowire-based superconducting qubit. Phys. Rev. Lett. 115, 127001 (2015).
  25. De Franceschi, S., Kouwenhoven, L., Schönenberger, C. & Wernsdorfer, W. Hybrid superconductor–quantum dot devices. Nat. Nanotech. 5, 703711 (2010).
  26. Lafarge, P., Joyez, P., Esteve, D., Urbina, C. & Devoret, M. H. Two-electron quantization of the charge on a superconductor. Nature 365, 422424 (1993).
  27. Feigelman, M. V., Kamenev, A., Larkin, A. I. & Skvortsov, M. A. Weak charge quantization on a superconducting island. Phys. Rev. B 66, 054502 (2002).
  28. Lafarge, P., Joyez, P., Esteve, D., Urbina, C. & Devoret, M. H. Measurement of the even–odd free-energy difference of an isolated superconductor. Phys. Rev. Lett. 70, 994997 (1993).
  29. Averin, D. V. & Nazarov, Y. V. Single-electron charging of a superconducting island. Phys. Rev. Lett. 69, 19931996 (1992).
  30. van Heck, B., Lutchyn, R. M. & Glazman, L. I. Conductance of a proximitized nanowire in the Coulomb blockade regime. Phys. Rev. B 93, 235431 (2016).
  31. Das Sarma, S., 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 (2012).
  32. Kitaev, A. Y. Unpaired Majorana fermions in quantum wires. Phys. Usp. 44, 131136 (2001).
  33. Rainis, D., Trifunovic, L., Klinovaja, J. & Loss, D. et al. Towards a realistic transport modeling in a superconducting nanowire with Majorana fermions. Phys. Rev. B 87, 024515 (2013).
  34. Stanescu, T. D., Lutchyn, R. M. & Das Sarma, S. Dimensional crossover in spin–orbit-coupled semiconductor nanowires with induced superconducting pairing. Phys. Rev. B 87, 094518 (2013).

Download references

Author information

Affiliations

  1. Center for Quantum Devices and Station Q Copenhagen, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

    • D. Sherman,
    • J. S. Yodh,
    • S. M. Albrecht,
    • J. Nygård,
    • P. Krogstrup &
    • C. M. Marcus

Contributions

P.K. and J.N. developed the nanowire materials. D.S. fabricated the devices. D.S., J.S.Y. and S.M.A. carried out the measurements with input from C.M.M. D.S. analysed the data. D.S., J.S.Y. and C.M.M. wrote the paper. All authors discussed the results and commented on the manuscript.

Competing financial interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to:

Author details

Additional data