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

This study provides experimental evidence for coexisting topological hinge states and 1D Rashba states in Bi$_{0.97}$Sb$_{0.03}$ nanoflakes through Josephson effect measurements, identifying the material as a prototypical second-order topological insulator platform.

The Gist

Imagine a world where electricity doesn't just flow through a wire like water in a pipe, but instead gets "locked" to the very edges of a material, refusing to scatter or get lost. This is the promise of topological materials, a special class of crystals that could revolutionize future computers.

This paper is about a specific material, a mix of Bismuth and Antimony (specifically Bi0.97Sb0.03), and the researchers' discovery of two very special types of "highways" for electricity hidden inside it.

Here is the breakdown of their findings using simple analogies:

1. The "Hinge" Highways (The Main Discovery)

Usually, we think of electricity flowing through the middle of a material. But in this specific crystal, the researchers found that electricity loves to travel along the edges and the corners (or "hinges") of the crystal, like cars sticking to the guardrails of a mountain road.

• The Analogy: Imagine a 3D block of cheese. In a normal block, if you cut a slice, the cheese is soft everywhere. But in this "topological" block, the inside is hard and solid, while the very edges and corners are coated in a slippery, frictionless ice.
• The Superpower: These edge paths are "protected." If the road has a pothole (a defect in the crystal), the electricity doesn't crash; it just flows around it. This is crucial for building stable quantum computers.

2. The "Magic" Current (The Proof)

How did they prove these highways exist? They used a trick called the Josephson Effect, which is like a bridge between two superconductors (materials with zero electrical resistance).

• The Analogy: Think of the current as a wave. In normal materials, the wave repeats every time it goes around a circle (a 360-degree turn, or 2π). But in these special topological highways, the wave is "lazy" and only repeats after two full circles (a 720-degree turn, or 4π).
• The Evidence: When they tested the material with high-frequency signals (like radio waves), they saw a "missing step." It's like a staircase where the 1st and 3rd steps are missing, leaving only the even ones. This "missing step" is the fingerprint of the protected topological state. The paper shows that the more edge current they had, the more obvious these missing steps became.

3. The "Ghost" Highways (The Rashba States)

Here is the twist: The researchers found that the "edge" wasn't just a single, thin line of traffic. It was actually a wide, broad highway.

• The Analogy: They expected a single-lane road (the topological hinge). Instead, they found a multi-lane highway. Why? Because the crystal isn't perfectly smooth; it has tiny "steps" or terraces on its surface, like a staircase.
• The Rashba Effect: These steps created a second type of highway called Rashba states. These are like "ghost lanes" that run alongside the real topological lanes. They are not as protected as the topological ones (they can scatter if they hit a bump), but they carry a lot of current.
• The Result: The "broad" edge current they saw was actually a mix of the protected topological lanes and these extra Rashba lanes. The paper explains that the "missing steps" in their experiment came from the topological lanes, while the extra width of the current came from the Rashba lanes.

4. The "Squeezed" Effect (Quantum Confinement)

The researchers also noticed that when they made the crystal flakes very narrow (like a thin strip), the behavior changed.

• The Analogy: Imagine a wide river. If you build a dam across it, the water slows down and spreads out. But if you squeeze the river into a tiny, narrow channel, the water behaves differently—it becomes a single, focused stream.
• The Finding: When the crystal was very thin, the "bulk" (middle) of the material started acting like a one-dimensional wire. This confirmed that the material's size changes how electricity moves through it, a phenomenon called quantum confinement.

Summary

The paper claims to have found a "designable" material where:

1. Topological Hinge States exist: Protected, frictionless paths along the edges that show a unique "4π" signature (the missing steps).
2. Rashba States coexist: Extra, wider paths caused by tiny steps on the crystal surface, which explain why the edge current looks "fuzzy" or broad.
3. Structure Matters: The natural "steps" and imperfections in the crystal actually create more of these special highways, rather than destroying them.

In short, they found a material that acts like a perfect, protected highway system for electricity, but with a twist: the highway is wider than expected because of the crystal's natural "stairs," and they proved it works by watching how the electricity waves dance.

Technical Summary

Technical Summary: Coexisting Topological Hinges and 1D Rashba States in Bi$_{0.97}$Sb$_{0.03}$

Problem and Motivation
The realization of topological quantum computing relies on inducing superconductivity in topological materials to form Majorana bound states (MBS). While one-dimensional (1D) helical states are promising candidates, their formation and stability in the presence of material defects and geometric irregularities remain subjects of debate. Specifically, second-order topological insulators (SOTIs) are predicted to host helical 1D hinge states protected by crystalline symmetries. However, experimental evidence for these hinge states, particularly their response to local defects and their ability to host a 4$\pi$-periodic supercurrent (a signature of MBS), has been lacking. Furthermore, distinguishing between topologically protected hinge modes and trivial edge states, such as Rashba modes, in real-world materials with structural imperfections is a significant challenge. This study investigates the Bi$_{1-x}$Sb$_x$ alloy system, specifically at 3% antimony doping (Bi$_{0.97}$Sb$_{0.03}$), to identify robust hinge states and understand the interplay between topological and trivial edge transport.

Methodology
The authors fabricated Josephson junctions (JJs) on exfoliated Bi$_{0.97}$Sb$_{0.03}$ nanoflakes (thickness 50–250 nm) using electron-beam lithography and in-situ sputter deposition of Niobium (Nb) electrodes. The study employed two primary experimental configurations:

1. Bulk vs. Edge Junctions: Junctions were designed to either span the entire flake width (edge JJ) or be confined to the center, avoiding edges (bulk JJ), to spatially resolve supercurrent distribution.
2. Transport Characterization: Critical current ($I_c$) was measured as a function of magnetic field ($B$) to analyze interference patterns (SQUID-like vs. Fraunhofer-like) and as a function of temperature ($T$) and junction length ($L$) to determine transport regimes (ballistic vs. diffusive).
3. Radio-Frequency (RF) Measurements: Shapiro step measurements were conducted under RF irradiation to detect the periodicity of the current-phase relation (CPR). The suppression of odd Shapiro steps serves as a signature of 4$\pi$-periodic supercurrents.
4. Theoretical Modeling: Tight-binding (TB) simulations using the Kwant package, based on a 16-band model, were performed to simulate band structures with structural irregularities (steps) and to analyze the localization of wavefunctions.

Key Results

• Evidence of Edge Supercurrent and SOTI States:

  • Edge JJs exhibited a pronounced SQUID-like interference pattern in $I_c(B)$, indicating a supercurrent density ($J_c$) strongly enhanced at the edges. In contrast, bulk JJs showed a non-oscillating, monotonically decaying $I_c(B)$, indicative of quasi-1D ballistic transport in the bulk.
  • The extracted $J_c$ profiles revealed multiple edge channels, with critical currents exceeding the theoretical maximum for a single helical mode, suggesting the presence of multiple hinge modes.
  • Temperature dependence analysis ($I_c(T)$) fitted well with the Eilenberger model for ballistic junctions with high interface transparency ($D \approx 0.99$), confirming the ballistic nature of both edge and bulk transport.

• Topological Protection and 4$\pi$ Supercurrent:

  • Shapiro step measurements revealed the suppression of the first and third odd steps (n=1, 3) at 0.9 GHz, a hallmark of 4$\pi$-periodic supercurrents associated with topologically protected gapless states.
  • A direct correlation was established between the degree of odd-step suppression and the percentage of edge-mediated supercurrent. Junctions with higher edge contributions showed stronger suppression, linking the fractional Josephson effect directly to the hinge modes.
  • The ratio of edge to bulk critical current ($\alpha = I_{c,edge}/I_{c,bulk}$) increased with temperature up to 2 K, suggesting that edge modes maintain coherent transport while bulk contributions become diffusive, consistent with topological protection.

• Role of Structural Irregularities and Rashba States:

  • The observed supercurrent was carried by multiple, broadened edge channels rather than a single sharp hinge. Scanning electron microscopy (SEM) and atomic force microscopy (AFM) revealed atomic steps and irregularities at the flake edges.
  • TB simulations demonstrated that these structural steps activate additional hinge modes. Crucially, the simulations identified a coexistence of topological SOTI modes and trivial 1D Rashba states at the edges.
  • The Rashba states, while not topologically protected, are responsible for the broadening of the edge current profile (extending 100–200 nm). The 4$\pi$ component is attributed to the SOTI modes, which constitute a fraction (20%) of the total edge channels, while the majority are Rashba-type.
  • In narrow junctions (W = 300 nm), the edge supercurrent disappeared. The authors attribute this not to the direct coupling of hinge wavefunctions (which are localized within a few nanometers), but to the opening of an effective gap in the surrounding Rashba states, which reduces the transparency for the topological hinge states.

Significance and Claims
The paper claims to provide the first direct experimental correlation between 4$\pi$-periodic Josephson effects and the presence of hinge modes in a Dirac semimetal (Bi$_{0.97}$Sb$_{0.03}$). The study establishes Bi$_{1-x}$Sb$_x$ as a prototypical, designable SOTI platform. Key contributions include:

1. Validation of SOTI Nature: Confirming the topological nature of hinge states through the observation of suppressed odd Shapiro steps and their correlation with edge transport.
2. Mechanism of Edge Broadening: Identifying that the commonly observed broadened edge currents in SOTI systems arise from the coexistence of topological hinge modes and structural-irregularity-induced 1D Rashba states.
3. Design Implications: Demonstrating that structural features (steps) can activate multiple hinge channels, offering a pathway for engineering topological quantum devices via nano-engineering of artificial edges.
4. Robustness: Showing that while Rashba states dominate the spatial extent of the edge current, the topological protection of the SOTI modes ensures the stability of the 4$\pi$ supercurrent component against thermal noise and geometric variations.

The work concludes that the Bi$_{0.97}$Sb$_{0.03}$ system offers a robust platform for advancing topological quantum devices, leveraging the interplay between engineered structural defects and intrinsic topological properties.