Ballistic transport in 1D Rashba systems in the context of Majorana nanowires

TL;DR

This work tackles how disorder diminishes or hides the SOC-induced helical gap in 1D Rashba nanowires relevant to Majorana platforms. It develops a minimal normal-wire model with Zeeman splitting and Rashba spin-orbit coupling, plus Gaussian disorder, and computes ballistic conductance via Landauer transport (KWANT); it shows that a robust helical gap yields a $G=\frac{e^2}{h}$ plateau and possible re-entry to $G=\frac{2e^2}{h}$ when $\alpha$ exceeds a critical value $\alpha_c \approx \sqrt{V_Z/m^*}$, while disorder introduces Fabry-Pérot–like resonances that can obscure these signatures. It then extends to superconducting wires with a proximitized self-energy in a BdG framework to study non-local conductance and compares with InAs/Al experiments, finding best agreement for a substantial disorder strength $|V_{dis}|\sim 4$ meV and correlation length $l_{dis}\sim 10$ nm. The results underscore that disorder can strongly suppress helical-gap signatures and emphasize the value of normal-state ballistic conductance measurements to bound SOC, Zeeman parameters, and disorder, guiding experimental design and interpretation for Majorana devices.

Abstract

Recent work on Majorana-bound states in semiconductor-superconductor hybrid structures has elucidated the key role of unintentional (and unknown) disorder (producing low-energy Andreev-bound states) in the system, which is detrimental to the emergence of Majorana-carrying topological superconductivity artificially engineered through the combination of superconductivity, Zeeman spin splitting, and Rashba spin-orbit coupling. In particular, the disorder must be smaller than the superconducting gap for the appearance of Majorana modes, but the disorder-induced appearance of subgap Andreev-bound states suppresses the Majorana modes. We theoretically investigate, as a function of disorder, the normal state ballistic transport properties of nanowires with and without superconductors in order to provide guidance on how to experimentally estimate the level of disorder. Experimentally, the superconductivity is suppressed simply by rotating the magnetic field appropriately, so both physics can be studied in the same set-up. In particular, the presence of spin-orbit coupling and Zeeman splitting produces a helical gap in the 1D electronic band structure, which should have clear signatures in ballistic transport unless these signatures are suppressed by disorder and/or Fabry-Pérot resonances associated with the finite wire sizes. Our work provides a benchmarking of when and what signatures of the putative helical gap (which is essential for the emergence of Majorana modes by leading to a single Fermi surface) could manifest in realistic nanowires.

Figures (6)

• Figure 1: Top row: Energy spectrum for semiconductor nanowire (green) with perpendicular Rashba-type spin-orbit coupling $\alpha$ and Zeeman field $V_Z$ for (a) zero $\alpha$ and $V_Z$; (b) finite $\alpha$ and zero $V_Z$; (c) finite (and sufficiently large) $\alpha$ and $V_Z$. The purple implies the spin degenerate states, while the blue and red correspond to the $s_x$ spin up and down states, while the gradient color indicates the spin hybridization state with an approximate spin direction. Bottom row: Energy spectrum for semiconductor (green)--superconductor (cyan) hybrid nanowire with parallel Rashba type spin-orbit coupling $\alpha$ and Zeeman field $V_Z$ for (d) zero $\alpha$ and $V_Z$; (e) finite $\alpha$ and zero $V_Z$; (f) finite $\alpha$ and $V_Z$. Both (c) and (f) show helical gaps.
• Figure 2: Tunneling conductance with various lead chemical potentials (black curves) and the band structure for approximate $s_x$ up and down (blue and red curves) in a InAs nanowire with a large spin-orbit coupling $\alpha=0.5$ eVÅ, wire chemical potential $\mu=0$ (horizontal dashed line), and a Zeeman field $V_Z=0.2$ meV corresponding to Fig. \ref{['fig:1']}(c), showing a helical gap and the re-entrant of the $2e^2/h$ conductance from $e^2/h$. The disorder strength increases from left to right: pristine wire in (a); $\sigma_\mu=0.05$ meV in (b); $\sigma_\mu=0.2$ meV in (c); $\sigma_\mu=0.3$ meV in (d). The conductance is for a single disorder realization without any ensemble averaging.
• Figure 3: Tunneling conductance with various lead chemical potentials (black curves) and the band structure for approximate $s_y$ up and down (blue and red curves) in a InAs nanowire with a small spin-orbit coupling $\alpha=0.1$ eVÅ, wire chemical potential $\mu=0$ (horizontal dashed line), and a Zeeman field $V_Z=0.2$ meV, showing no helical gap nor the re-entrant of the quantized conductance. The disorder strength increases from left to right: pristine wire in (a); $\sigma_\mu=0.05$ meV in (b); $\sigma_\mu=0.2$ meV in (c); $\sigma_\mu=0.3$ meV in (d). The conductance is for a single disorder realization without any ensemble averaging.
• Figure 4: (a) Numerical non-local conductance $G_{LR}$ as a function of chemical potential $\mu$ (in meV) for different lengths $L$ and disorder strength $|V_{dis}|$. The disorder strength of the different plots in the figure increases from blue to red while the length of the system increases from left to right. Each plot shows that the conductance vanishes below a critical chemical potential. At the same time, the critical chemical potential increases as the disorder strength and length increase. Note that to improve visibility, a chemical potential broadening of $0.14\,\textrm{meV}$ has been introduced to suppress mesoscopic fluctuations. (b) Experimental results for non-local conductance as a function of plunger voltages that are extracted from microsoftquantum2023inasal for increasing lengths from left to right. As described in Sec. \ref{['sec:SC']}, the range of plunger voltage corresponds to the chemical potential range plotted in the x-axis in panel (a). Comparing the profile of the conductance from panel (b) with the various disorders at panel (a), $|V_{dis}|=5\,\textrm{meV}$ appears to match the experiment the closest.
• Figure 5: (a) Numerical normalized non-local conductance $G_{LR}/\sqrt{G_{RR}G_{LL}}$ as a function of chemical potential $\mu$ (in meV) for different lengths $L$ and disorder strength $|V_{dis}|$ (meV). The disorder strength of the different plots in the figure increases from blue to red while the length of the system increases from left to right. Each plot shows that the conductance vanishes below a critical chemical potential. At the same time, the critical chemical potential increases as disorder strength and length increases. Note that to improve visibility, a chemical potential broadening of $0.14\,\textrm{meV}$ has been introduced to suppress mesoscopic fluctuations. (b) Experimental results for normalized conductance as a function of plunger voltages that are extracted from microsoftquantum2023inasal for increasing lengths from left to right. As described in Sec. \ref{['sec:SC']}, the range of plunger voltage corresponds to the chemical potential range plotted in the x-axis in panel (a). Comparing the profile of the conductance from panel (b) with the various disorders at panel (a), $|V_{dis}|=5\,\textrm{meV}$ appears to match the experiment the closest.