Identifying Majorana vortex modes via nonlocal transport

Björn Sbierski, Max Geier, An-Ping Li, Matthew Brahlek, Robert G. Moore, and Joel E. Moore

Phys. Rev. B 106, 035413 – Published 14 July 2022

Abstract

The combination of two-dimensional Dirac surface states with $s$ -wave superconductivity is expected to generate localized topological Majorana zero modes in vortex cores. Putative experimental signatures of these modes have been reported for heterostructures of proximitized topological insulators, iron-based superconductors or certain transition metal dichalcogenides. Despite these efforts, the Majorana nature of the observed excitation is still under debate. We propose to identify the presence of Majorana vortex modes using a nonlocal transport measurement protocol originally employed for one-dimensional settings. In the case of an isolated subgap state, the protocol provides a spatial map of the ratio of local charge- and probability-density which offers a clear distinction between Majorana and ordinary fermionic modes. We show that these distinctive features survive in the experimentally relevant case of hybridizing vortex core modes.

Received 2 August 2021
Revised 7 June 2022
Accepted 27 June 2022

DOI:https://doi.org/10.1103/PhysRevB.106.035413

Physics Subject Headings (PhySH)

Majorana bound states Quantum transport

Iron-based superconductors

Scanning tunneling spectroscopy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Björn Sbierski ^1,2, Max Geier ³, An-Ping Li ⁴, Matthew Brahlek⁵, Robert G. Moore ⁵, and Joel E. Moore^1,6

¹Department of Physics, University of California, Berkeley, California 94720, USA
²Department of Physics and Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstr. 37, D-80333 München, Germany
³Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark
⁴Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
⁵Materials Sciences and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
⁶Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

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Vol. 106, Iss. 3 — 15 July 2022

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Images

Figure 1
Schematic of the proposed nonlocal transport setup for a Fu-Kane material with superconducting surface Dirac state in the vortex phase. In addition to the standard scanning-tunneling spectroscopy setup using grounded bulk and tip contact (“T”), an additional contact (“C”) is required. This contact does not need to be realized as a second tip but can be spatially extended.
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Figure 2
Tunneling LDOS $ρ (ω) = \int d r ρ (ω, r) / \int d r$ for the two-dimensional Fu-Kane model in the absence of magnetic field as found from exact diagonalization of the lattice model (6) with parameters $μ = 0.6$ and $Δ_{0} = 0.2$ and system size $L_{x} \times L_{y} = 84 \times 86$ . The oscillations are due to finite-size effects which are incompletely smoothed by the intrinsic level broadening $Γ_{0} = 0.02 Δ_{0} = 0.004$ . The superconducting gap $ω = \pm Δ_{0}$ is indicated by grey vertical lines and the density of states (DOS) of the normal-state Dirac Hamiltonian $D (ω) = \frac{ω}{2 π {(ℏ v)}^{2}}$ is depicted by the red dashed line.
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Figure 3
Numerical lattice-model results from exact diagonalization for a pair of vortices located at a distance $R = 13$ as indicated by the black crosses. The parameters are $μ = 0.6$ , $Δ_{0} = 0.2$ , $ξ = 2$ and $λ = 30$ and the overall system size is $L_{x} \times L_{y} = 60 \times 58$ . The top row shows the LDOS $ρ$ [Eq. (9) with $Γ_{0} = 0.004$ , $ω = E_{0}$ ] in panel (a), intensity $n$ in panel (b), charge $q$ in panel (c) and the ratio $q / n$ in panel (d) for the hybridized MVM state at energy $E_{0} = 0.0045$ , the bottom row with panels (e)–(h) reports the same quantities for the lower one of the two hybridized CdGM-state energies, $E_{1} = 0.09$ .
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Figure 4
One-dimensional cut through the data depicted in Figs. 3 and 3 along the brown line connecting the two vortices. The vertical lines denote the vortex positions. The low energy MVM data from Fig. 3 is shown in the left panel, the right panel displays the hybridized finite energy CdGM-state from Fig. 3.
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Figure 5
Nonlocal transport simulation for a lattice model including a vortex pair. We model the tip “T” as a single-atomic lead with $t = 1$ and $γ_{T} = 0.4$ and the extended contact “C” as a $A_{C} = 4 \times 4$ patch of the same single-atomic leads with $γ_{C} = 0.1$ (green). For a local intensity $n_{C}$ on the order of $0.01$ (c.f. Fig. 3) this results in a broadening $Γ_{C} = n_{C} A_{C} γ_{C}^{2} / t ≃ 0.002$ which is smaller than the MVM hybridization energy (we neglect the intrinsic broadening $Γ_{0}$ in order to obtain a unitary scattering matrix). For the temperature of the leads, we take $T = 0.002$ . The model parameters (see Table 1) are the same as in Fig. 3. The left column with panels (a), (d) shows the bias-dependent nonlocal conductance $g_{C T} (ω, R_{1})$ with the tip positioned at the right vortex. A low-energy peak structure highlighted by the vertical dashed lines appears at $| ω | = E_{0} = 0.0045$ , see panel (a), and $| ω | = E_{1} = 0.09$ , see panel (d). In panels (b) and (e), we depict $g_{C T} (ω = E_{0, 1}, r)$ . The right panels (c) and (f) depict $g_{C T}^{sym} / g_{C T}^{asym} (ω = E_{0, 1})$ , which quantitatively agree to the $q / n$ maps of Figs. 3, 3.
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Figure 6
Nonlocal transport simulation for a lattice model including a set of vortices arranged in a distorted lattice. The parameters are the same as in Fig. 5. The left column with panels (a), (d) shows the bias-dependent nonlocal conductance $g_{C T} (ω, R_{i})$ for 13 vortices $i = 1, 2, ..., 13$ . The colors of the lines correspond to the color of the crosses in the other panels, the latter mark the vortex positions. Panel (a) zooms to small bias voltages around the peak at $E_{0} = 0.0055$ while panel (d) shows a larger bias range including the peak at $E_{1} = 0.105$ . In panels (b) and (e), we depict $g_{C T} (ω, r)$ for $ω = E_{0}$ and $ω = E_{1}$ , respectively. The right panels (c) and (f) depict $g_{C T}^{sym} / g_{C T}^{asym}$ for $ω = E_{0}$ and $ω = E_{1}$ , respectively. As the inner three vortices (red, grey, yellow) do not show a pronounced peak structure at low energies $\sim E_{0}$ in panels (a), (b), we discard their vicinity for the plot in panel (c).
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