Coexisting topological hinges and 1D Rashba states in Bi$_{0.97}$Sb$_{0.03}$ revealed by the Josephson effect

TL;DR

This work demonstrates coexisting 1D hinge states and Rashba edge states in Bi$_{0.97}$Sb$_{0.03}$, revealed through Josephson-junction measurements that unveil a $4\pi$-periodic supercurrent associated with hinge modes. Edge-dominated interference patterns, quasi-1D bulk transport, and robust fractional Shapiro steps establish a direct topological origin for the observed transport phenomena, while tight-binding simulations confirm multiple hinge channels arising from natural step edges. The coexistence of Rashba and SOTI states explains the broadened edge current profiles, with only the topologically protected hinge channels responsible for the $4\pi$ component. These results position Bi$_{1-x}$Sb$_x$ as a programmable SOTI platform, where nano-engineering of edges enables multi-channel topological superconducting devices with potential quantum information applications.

Abstract

Second-order topological insulating (SOTI) states in three-dimensional materials are helical one-dimensional hinge states. Inducing superconductivity in these states leads to gapless bound states, characterized by the 4$π$-periodic current-phase relation. Here, we provide evidence of the topologically protected hinge states in Dirac semimetal Bi$_{0.97}$Sb$_{0.03}$ nanoflakes by an unconventional interference pattern in a magnetic field, and the 4$π$-periodic supercurrent carried by these states via the suppressed first and third Shapiro steps. Tight-binding simulations confirm the presence of multiple hinge modes, supporting our interpretation of Bi$_{0.97}$Sb$_{0.03}$ as a prototypical designable SOTI platform. Quantum confinement effect is identified by a quasi-one-dimensional bulk transport, and the confined Rashba states are responsible for the broadened hinge states.

Figures (5)

• Figure 1: Evidences of edge supercurrent in Bi$_{0.97}$Sb$_{0.03}$ Josephson junctions.a, left: Schematic representation of multiple helical 1D hinge modes with spin up (blue) and spin down (red) localized along the step-edges and terraces in a 3D second-order topological insulator (SOTI). The hinge modes can host exotic $4\pi$-periodic Andreev bound states (Majorana bound states). In contrast, the bulk states host $2\pi$-periodic Andreev bound states, with a gap opening due to the lack of topological protection. right: Classification of various material phases of Bi$_{1-x}$Sb$_{x}$ alloy based on Sb doping percentage. Experimental observations have confirmed the existence of higher-order hinge states in all three phases- Bi: semimetal Schindler2018Exp, Bi$_{0.97}$Sb$_{0.03}$: Dirac semimetal (this work), and Bi$_{0.92}$Sb$_{0.08}$: topological insulator Aggarwal2021. b, Schematic top-view drawing of sample F1: the bulk JJ (on the top surface, avoiding edges) and the edge JJ (spanning the full width of flake, including edges). The active junction area in bulk JJ is indicated by diagonal hatching. $S$: superconductor (Nb), $N$: normal region (Bi$_{0.97}$Sb$_{0.03}$). Both junctions have length $L=800$ nm. c, Simulated Fraunhofer pattern (FP) for a JJ with homogeneous current distribution, following $I_c = I_{c,\mathrm{max}} \left| \mathrm{sinc}\left(\phi/\phi_0\right) \right|$, where $\phi = BL_{\mathrm{eff}}W$ is the magnetic flux through the junction and $\phi_0$ is the magnetic flux quantum. Inset shows the extracted $J_c(x)$ profile corresponding to the conventional FP, calculated from $I_cB$ using the Dynes–Fulton method Dynes1971, indicating the uniform current distribution with no edge enhancement. d, e, Comparison of $I_c(B)$ interference patterns for bulk and edge JJs on flake F1. The edge JJ (d) exhibits SQUID-like oscillations, while the bulk JJ (e) shows a monotonically decaying non-oscillating $I_c(B)$ pattern. Inset in d depicts the extracted $J_c(x)$ for the edge JJ, with strongly enhanced edge currents and negligible bulk contribution. Inset in e depicts a "gaussian-like" $J_c(x)$ peaked at the center, suggesting (quasi-) 1D ballistic transport in confined geometries.
• Figure 2: Ballistic transport in Bi$_{0.97}$Sb$_{0.03}$ junctions.a, Quantum interference pattern recorded at 70 mK for the F2$\_600$ junction, displaying SQUID-like oscillations in $I_c(B)$. The colormap represents the differential resistance $dV/dI$. Inset: SEM image of the flake F2 with four junctions of lengths 600, 1000, 900, and 800 nm (from top to bottom). Scale bar: 2 $\mu$m. Color scheme matches Fig. \ref{['fig:fig1']}b. b, Extracted $J_c$ distribution across the junction width, calculated from a , showing enhanced edge supercurrent density—characteristic of a SQUID-like interference. Contribution from hinge modes is shown by the gray shaded area. Residual bulk supercurrent is indicated by the black dashed line. c, Temperature dependence of $I_c$ for device F2$\_600$. Measured data points (dark blue) can be fitted with Eilenberger model (solid dark red) for ballistic junctions with near-unity interface transparency. The estimated clean limit superconducting coherence length $\xi_s$ is 240 nm, which provides $L/\xi_s=2.5$. d, Temperature dependence of the extracted $I_c$ for the edge modes (green diamonds) from $J_c$ distribution using the method described in Supplementary Section S2. The inset shows the same for extracted bulk $I_c$ (orange diamonds). Both edge and bulk $I_c(T)$ can be fitted with Eilenberger model (solid dark red). The error bars are estimated from the standard deviation between experimentally and numerically calculated $J_c$ distribution as described in Supplementary Section S1. e, $I_cR_N$ scaling with inverse of junction length for different devices on flake F2 at 70 mK. The red dashed line highlights the linear trend. f, $I_c$ scaling with junction length for same devices as in e, where experimental values (green dots) lie within the Eilenberger framework with upper (solid dark red) and lower (solid dark blue) bounds set by the interface transparency D = 0.9 and 1.0, respectively.
• Figure 3: Fractional a.c. Josephson effect and topological protection of the 1D hinge modes.a, Shapiro steps in device F2$\_600$ measured under radio-frequency (RF) excitation of 0.9 GHz at 70 mK. The colormap shows the Shapiro step size calculated from the d.c. voltage bins as a function of d.c. voltage normalized to $\frac{hf}{2e}$ and RF power. White arrows mark the missing odd ($n = 1, 3$) Shapiro steps. b, Frequency dependence of the ratio between $n=1$ and $n=2$ step size ($Q_{12}$). c, Suppression factor ($1-Q_{12}/Q_{12}^{max}$) as a function of the edge-mediated supercurrent for different devices, highlighting an important observation that the $n=1$ Shapiro step is more diminished in junctions with higher edge state contribution. This suggests overall topological protection of the 1D hinge modes. d, $I_{c,\ edge}/I_{c,\ bulk}$ ratio for device F2$\_600$ showing an increasing trend with temperature up to 2K, suggesting that the bulk $I_c$ decays more rapidly than that of the ballistic edge states. The dashed curve is guide-to-the-eye.
• Figure 4: Experimental and theoretical signatures of the effect of multiple structural irregularities on hinges.a, Cross-sectional SEM image of the F2$\_1000$ device showing the additional step (black arrow) present near one of the junction edge (also visible in AFM image, Supplementary Section S11). b, Overlayed AFM height profile (solid turquoise) showing the $\sim15$ nm high step near the left edge of the junction, designed $J_c^{\text{set}}(x)$ model (shaded region) including this step (black arrow) and experimental $J_c^{\text{exp}}(x)$ distribution (dashed darkred) showing the enhanced width of left edge due to this step. $J_c^{\text{set}}(x)$ is scaled as per the $J_c^{\text{exp}}(x)$ data. c, Tight-binding band structure of a 20 bilayer Bi$_{0.97}$Sb$_{0.03}$ slab (8 nm thick by 16 nm wide) with translation symmetry in the $[1\overline10]$ direction, confinement along the top $(111)$ and side $(11\!-\!2)$ surfaces, and with four identical artificial step edges on one side. The color intensity represents the localization defined as the sum of the wavefunction density over the 10 sites around the site where the wavefunction density is maximal. The most strongly localized bands are four times degenerate (Rashba). d, The summed wavefunction density of the highlighted degenerate Rashba (blue) and SOTI (red) modes at the Fermi level $(E_\text{F}=0)$.
• Figure 5: Finite-size effect on the SQI pattern.a-c, Comparing the $I_cB$ for a, $W = 1800$ nm (300 mK), b, $W = 700$ nm (100 mK) and c, $W = 300$ nm (100 mK) wide junctions reveal the quantum confinement effect due to width of an edge JJ. A gradual shift from the SQUID-like response in widest flake to a monotonically decaying $I_cB$ in the narrowest flake was observed.