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Tuning topological superconductivity in phase-controlled Josephson junctions with Rashba and Dresselhaus spin-orbit coupling

Benedikt Scharf, Falko Pientka, Hechen Ren, Amir Yacoby, and Ewelina M. Hankiewicz
Phys. Rev. B 99, 214503 – Published 12 June 2019
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  • INTRODUCTION
  • MODEL
  • TOPOLOGICAL PHASE IN JOSEPHSON JUNCTIONS…
  • CONDITIONS FOR THE TOPOLOGICAL PHASE IN…
  • MAJORANA END STATES IN JOSEPHSON…
  • SIGNATURES OF MAJORANA BOUND STATES IN…
  • TUNING AND TESTING TOPOLOGICAL…
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Recently, topological superconductors based on Josephson junctions in two-dimensional electron gases with strong Rashba spin-orbit coupling have been proposed as attractive alternatives to wire-based setups. Here, we elucidate how phase-controlled Josephson junctions based on quantum wells with [001] growth direction and an arbitrary combination of Rashba and Dresselhaus spin-orbit coupling can also host Majorana bound states for a wide range of parameters as long as the magnetic field is oriented appropriately. Hence, Majorana bound states based on Josephson junctions can appear in a wide class of two-dimensional electron gases. We study the effect of spin-orbit coupling, the Zeeman energies, and the superconducting phase difference to create a full topological phase diagram and find the optimal stability region to observe Majorana bound states in narrow junctions. Surprisingly, for equal Rashba and Dresselhaus spin-orbit coupling, well localized Majorana bound states can appear only for phase differences ϕ≠π as the topological gap protecting the Majorana bound states vanishes at ϕ=π. Our results show that the ratio between Rashba and Dresselhaus spin-orbit coupling or the choice of the in-plane crystallographic axis along which the superconducting phase bias is applied offer additional tunable knobs to test Majorana bound states in these systems. Finally, we discuss signatures of Majorana bound states that could be probed experimentally by tunneling conductance measurements at the edge of the junction.

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    • Received 18 April 2019
    • Revised 30 May 2019

    DOI:https://doi.org/10.1103/PhysRevB.99.214503

    ©2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Proximity effectSpintronicsSuperconductivityTopological phase transitionTopological superconductors
    1. Physical Systems
    Josephson junctionsTwo-dimensional electron system
    1. Techniques
    Bogoliubov-de Gennes equationsTight-binding model
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Benedikt Scharf1, Falko Pientka2, Hechen Ren3, Amir Yacoby4, and Ewelina M. Hankiewicz1

    • 1Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Cluster of Excellence ct.qmat, University of Würzburg, Am Hubland, 97074 Würzburg, Germany
    • 2Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
    • 3Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
    • 4Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

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    Vol. 99, Iss. 21 — 1 June 2019

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    Images

    • Figure 1
      Figure 1

      (a) Schematic setup of the phase-controlled Josephson junction. The position of the Majorana bound states γ that appear at the ends of the normal region in the topological phase is also indicated. The angle of the direction of the in-plane Zeeman term with respect to the x direction is given by θZ. Depending on the in-plane crystallographic axes along which the superconducting phase difference is applied, the SOC affects the formation of a topological phase differently. Fermi contours for 2DEGs formed in quantum wells with [001] growth direction: (b) 2DEG with Rashba SOC and 2DEGs with Rashba as well as Dresselhaus SOC if the x axes are chosen along the (c) [100] and (d) [110] directions, respectively.

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    • Figure 2
      Figure 2

      Andreev bound-state spectrum as a function of the transverse momentum ky at [(a), (d)] ϕ=0, [(b), (e)] ϕ=ϕc, given by Eq. (3), and [(c), (f)] ϕ=π. In panels (a)–(c), either θsoc=0 (only Rashba SOC, β=0), EZ=|EZ|ey, or θsoc=π/2 (only Dresselhaus SOC, α=0), EZ=|EZ|ex, that is, EZ⊥nsoc. In panels (d)–(f), either θsoc=0,EZ=|EZ|ex, or θsoc=π/2,EZ=|EZ|ey, that is, EZ∥nsoc. The topological gap Δtop appearing at ϕ=π close to kF+ksoc is indicated in panel (c). No finite topological gap Δtop arises in panel (f), where for any |E|<Δ a state can be found. In all panels, m=0.038m0,W=100 nm, λsoc=α2+β2=16 meV nm, μS=1 meV, μN=0.7 meV, Δ=250μeV, and |EZ|=0.5 meV. The spectra have been computed employing a finite-difference method along the x direction with a width of the entire S/N/S junction of Wtot=2WS+W=1μm.

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    • Figure 3
      Figure 3

      Topological phase diagrams for different configurations of SOC, described by θsoc, and the Zeeman term EZ: (a) θsoc=0 (α=16 meV nm, β=0), EZ=|EZ|ey, (b) θsoc=0.15π (α≈14.3 meV nm, β≈7.3 meV nm), EZ⊥nsoc, (c) θsoc=0.25π (α=β≈11.3 meV nm), EZ⊥nsoc and (d) θsoc=0.05π (α≈15.8 meV nm, β≈2.5 meV nm), EZ=|EZ|ey. The total strength of SOC is λsoc=α2+β2=16 meV nm in all panels and all other parameters are the same as in Fig. 2. The white lines indicate the phase boundaries ϕc computed from Eq. (3) and Δtop only measures the gap inside the topological phase, whereas gaps outside the topological phase have been set to zero. The insets illustrate the directions of nsoc and EZ.

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    • Figure 4
      Figure 4

      Dependence of the topological gap Δtop at ϕ=π (dashed curves) and ϕ=1.2π (solid curves) on θsoc for different configurations of EZ: (a) EZ=|EZ|ex (blue) and EZ=|EZ|ey (red) as well as (b) EZ rotated with θsoc such that EZ⊥nsoc (blue). The parameters are the same as in Fig. 2 with |EZ|=1.1 meV kept constant for the different configurations of EZ. The lower panels (c)–(e) show schemes of the three different configurations presented in panels (a) and (b).

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    • Figure 5
      Figure 5

      Andreev bound-state spectrum as a function of the transverse momentum ky for (a) α=β (that is, θsoc=0.25π) at ϕ=π,ϕ=1.1π, and ϕ=1.2π and for (b) a fixed ϕ=π with θsoc=0.25π,θsoc=0.225π, and θsoc=0.2π. Here, the position ky=k0 of a generic gap closing and the low-energy two-level system described by Eq. (16) are also indicated. In both panels, EZ⊥nsoc with |EZ|=0.5 meV in panel (a) and |EZ|=1.1 meV in panel (b). All other parameters are the same as in Fig. 2.

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    • Figure 6
      Figure 6

      Dependence of the topological gap Δtop at ϕ=π (solid curve) and ϕ=1.2π (dashed curve) on θsoc for a Josephson junction with a phase bias along the [110] direction. Here, EZ=|EZ|ey is kept constant with |EZ|=1.1 meV. The other parameters are the same as in Fig. 2. The inset shows a scheme of the setup.

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    • Figure 7
      Figure 7

      Probability densities ρ(r)=|ψ(r)|2 (in μm−2) of Majorana bound states in a finite system with W=100 nm, WS=450 nm, and finite length L=2μm. Here, a superconducting phase difference ϕ=π and several different configurations of SOC and EZ are shown: (a) θsoc=0 (α=16 meV nm, β=0), EZ,x=0,EZ,y=0.5 meV, (b) θsoc=0.15π (α≈14.3 meV nm, β≈7.3 meV nm), EZ,x=−0.21 meV, EZ,y=0.41 meV, (c) θsoc=0.25π (α=β≈11.3 meV nm), EZ,x=−EZ,y=−0.35 meV. Schemes of the configurations investigated are shown below the density plots. In all panels, |EZ|=0.5 meV, λsoc=16 meV nm, m=0.038m0,μS=1 meV, μN=0.7 meV, and Δ=250μeV. The x direction is chosen along the crystallographic [100] direction.

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    • Figure 8
      Figure 8

      Local density of states D(E) for a Josephson junction with W=100 nm, WS=450 nm, L=2μm, m=0.038m0,μS=1 meV, μN=0.7 meV, Δ=250μeV, θsoc=0.15π (α=14.3 meV nm, β=7.3 meV nm), and EZ⊥nsoc. [(a)–(d)] D(E) (in a.u.) measured at the top edge of the N region for different Zeeman terms: (a) |EZ|=0, (b) |EZ|=0.23 meV, (c) |EZ|=0.46 meV, (d) |EZ|=0.69 meV. [(e), (f)] ∂2D/∂E2|E=0 (in a.u.) measured (e) at the top and (f) in the center of the N region as a function of |EZ| and ϕ. The dashed black lines denote the phase boundaries obtained from Eq. (3) for L→∞. (g) Scheme of the setup investigated with the position of the edge and bulk probes indicated.

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    • Figure 9
      Figure 9

      Curvature of the edge LDOS, ∂2D/∂E2|E=0 (in a.u.), as a function of the Zeeman field |EZ| and the superconducting phase difference ϕ for different widths W of the N region: (a) W=100 nm, (b) W=300 nm, and (c) W=500 nm. The other parameters are chosen as in Fig. 7: WS=450 nm, L=2μm, m=0.038m0,μS=1 meV, μN=0.7 meV, Δ=250μeV, λsoc=16 meVnm, θsoc=0, and EZ=|EZ|ey. In all panels, the dashed black lines denote the phase boundaries obtained from Eq. (3) for L→∞.

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    • Figure 10
      Figure 10

      Topological gap Δtop for an InAs quantum well with fixed β=4 meV nm and different strengths of Rashba SOC: (a) α=0, (b) α=β=4 meV, and (c) α=16 meV nm. The junction is set up with phase bias along the [110] direction and a magnetic field parallel to the S/N interfaces, B=Bey. Here, W=200 nm, WS=500 nm, m=0.026m0,μS=1 meV, μN=0.8 meV, g=10, and Δ=150μeV, corresponding to a InAs/Al heterostructure [46]. For g=10, a magnetic field B=1 T corresponds to a Zeeman energy EZ=gμBB/2≈0.29 meV.

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    • Figure 11
      Figure 11

      Phase boundaries of a ultranarrow junction of width W=10 nm obtained from the scattering approach for a finite barrier and a δ-barrier model. In panel (a), μS=1 meV and V0=0.3 meV, while μS=10 meV and V0=3 meV in panel (b). Here, the S regions are assumed to be semi-infinite in the x direction. The other parameters are m=0.038m0,Δ=250μeV, and EZ⊥nsoc.

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    • Figure 12
      Figure 12

      Phase boundaries of (a) a junction of width W=100 nm and (b) one of width W=300 nm obtained from the scattering approach for a finite barrier and a δ-barrier model. In panel (a), μS=1 meV and V0=0.3 meV, while μS=10 meV and V0=3 meV in panel (b). All other parameters are similar to Fig. 11.

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