Unveiling Topological Hinge States in the Higher-Order Topological Insulator WTe$_2$ Based on the Fractional Josephson Effect

This study provides experimental evidence for topological hinge states in the higher-order topological insulator WTe$_2$ by observing the absence of the first Shapiro step in Al-WTe$_2$-Al Josephson junctions, a signature attributed to a 4$\pi$-periodic current-phase relationship that supports the pursuit of Majorana zero modes and topological quantum applications.

The Gist

Imagine a giant, three-dimensional chocolate bar. Usually, if you eat a chocolate bar, you can eat the inside (the bulk), the top and bottom (the surfaces), or the edges. But in the world of quantum physics, there's a special kind of "chocolate bar" called a Higher-Order Topological Insulator (HOTI).

Here's the weird rule for this special bar:

• The inside is an insulator (electricity can't flow).
• The surfaces are also insulators.
• But the sharp corners and edges (the "hinges")? They are super highways where electricity flows perfectly.

This paper is about finding proof that these "hinge highways" exist in a material called WTe2 (Tungsten Ditelluride) and that they behave like a secret, topological superhighway.

The Detective Work: The "Shapiro Step" Test

To prove these highways are real and special, the scientists built a tiny bridge (a Josephson Junction) across the WTe2 crystal. They connected two superconductors (materials with zero electrical resistance) to the crystal and sent electricity through it while shining microwave light on it.

Think of the electricity flowing across the bridge like a swing.

• Normal Swings (Trivial Physics): If you push a normal swing with a steady rhythm, it moves in a predictable pattern. In physics terms, the electricity creates "steps" (called Shapiro steps) at every beat of the microwave rhythm. You see Step 1, Step 2, Step 3, and so on.
• The Secret Swing (Topological Physics): The scientists suspected the hinge highways were special. They thought the "swing" here had a secret rule: it only wants to move every second beat. It skips the first beat. So, you would see Step 1 is missing, but Step 2, Step 4, and Step 6 appear. This is called the Fractional Josephson Effect.

The Experiment: Two Different Bridges

The team built two types of bridges to compare:

1. The "Bulk" Bridge (bJJ): This bridge was wide. It allowed electricity to flow through the "insider" parts of the chocolate bar (the bulk) and the edges.

   • Result: The swing behaved normally. They saw Step 1, Step 2, Step 3. The "noise" from the bulk hid the secret hinge behavior.

2. The "Hinge" Bridge (hJJ): This bridge was very narrow, designed so that electricity could only flow through the sharp edges (the hinges).

   • Result: Step 1 vanished! The swing skipped the first beat. They saw Step 2, Step 4, etc., but the first step was completely gone.

Why This Matters: The "Ghost" in the Machine

Why is a missing step so exciting?

In the quantum world, there are particles called Majorana zero modes. You can think of them as "ghosts" that live on the edges of these materials. These ghosts have a special rule: they are "paired up" in a way that makes the electricity flow in a 4-step cycle instead of the usual 2-step cycle.

• Normal world: 1, 2, 1, 2... (2π periodicity)
• Topological world with ghosts: 1, 2, 3, 4, 1, 2, 3, 4... (4π periodicity)

Because the cycle is double, the "first step" of the normal rhythm gets swallowed up. The fact that the scientists saw the first step disappear in the narrow bridge is like finding a fingerprint that proves the "ghosts" (Majorana modes) are actually there, riding the hinge highways.

The Verdict

The scientists didn't just guess; they checked the math. They changed the speed of the microwaves (the rhythm of the push) and the power. They found that the "missing step" only happened when the rhythm was just right, exactly matching the predictions for these topological ghosts.

In simple terms:
They built a tiny, narrow road on a special crystal. When they tested the traffic, they found that the cars (electrons) were skipping the first stop sign. This "skipping" proved that the road is a magical, topological highway protected by the laws of quantum mechanics, not just a regular road.

Why Should You Care?

This isn't just about chocolate bars or swings.

1. Quantum Computers: These "ghost" particles (Majorana modes) are the holy grail for building quantum computers. They are incredibly stable and hard to mess up, which could solve the biggest problem in quantum computing: errors.
2. New Electronics: Understanding these hinge states could lead to new types of super-fast, energy-efficient electronics (spintronics) that use the "spin" of electrons rather than just their charge.

Summary: The team successfully caught the "ghosts" of quantum physics in the act by watching them skip a beat in a tiny electrical bridge, proving that the edges of WTe2 are indeed a special, topological superhighway.

Technical Summary

1. Problem Statement

Higher-order topological insulators (HOTIs) are a novel class of materials characterized by topologically protected conducting states at dimensions two or more lower than the bulk (e.g., 1D hinge states in a 3D material). While theoretical models predict that these hinge states should exhibit unique topological properties, such as spin-momentum locking and the potential to host Majorana zero modes when coupled with superconductors, experimental verification has been challenging.

• The Core Challenge: In materials like tungsten ditelluride (WTe2), the topological hinge states are often obscured by dominant, topologically trivial bulk and surface conduction channels.
• The Gap: Previous studies have identified conducting channels at WTe2 hinges, but conclusive evidence regarding their topological nature (specifically, whether they support a fractional Josephson effect) has remained elusive. Standard transport measurements often fail to distinguish between trivial and topological contributions.

2. Methodology

The authors investigated the topological nature of hinge states in multilayer WTe2 by fabricating and analyzing Josephson Junctions (JJs) using a Fractional Josephson Effect approach.

• Device Fabrication:
  • Material: 13-nm-thick multilayer WTe2 crystals mechanically exfoliated onto Si/SiO2 substrates.
  • Junction Types: Two distinct configurations were created to isolate different conduction pathways:
    1. Bulk-Dominant JJ (bJJ): Wide junction ($W = 5.8 \, \mu\text{m}$) where current flows primarily through the bulk.
    2. Hinge-Dominant JJ (hJJ): Narrow junction ($W = 0.65 \, \mu\text{m}$) designed to maximize the ratio of hinge-to-bulk current contribution.
  • Electrodes: Aluminum (Al) superconducting leads were deposited to form Al–WTe2–Al proximity junctions.
• Experimental Setup:
  • Measurements were performed at 20 mK.
  • Shapiro Step Analysis: The devices were irradiated with microwaves of varying frequencies ($f$) and powers. The authors analyzed the current-voltage (I-V) characteristics to observe the formation of Shapiro steps (voltage plateaus).
• Theoretical Framework:
  • The study utilized the Resistively and Capacitively Shunted Junction (RCSJ) model.
  • They modeled the current-phase relationship (CPR) as a superposition of $2\pi$-periodic (trivial) and $4\pi$-periodic (topological) components: $I_J(\phi) = I_{2\pi}\sin(\phi) + I_{4\pi}\sin(\phi/2)$.
  • Key Indicator: In topological JJs with $4\pi$-periodic Andreev bound states (ABS), the first Shapiro step (at voltage $V_0 = hf/2e$) is expected to be missing or suppressed, while even multiples ($2V_0, 4V_0$) remain.

3. Key Contributions

• Experimental Isolation of Hinge States: By comparing bJJ and hJJ geometries, the authors successfully isolated the contribution of hinge states from the bulk, demonstrating that the topological signature is geometry-dependent.
• Observation of the Missing First Shapiro Step: The study provides the first clear experimental evidence of the missing first Shapiro step specifically in hinge-dominated WTe2 junctions, a hallmark of the fractional Josephson effect.
• Quantitative Validation: The authors quantitatively linked the missing step to the $4\pi$-periodic current component ($I_{4\pi}$) and validated their findings against numerical simulations of the RCSJ model, ruling out trivial mechanisms like Landau-Zener transitions or underdamping.
• Current Density Correlation: The estimated $4\pi$-periodic current magnitude derived from Shapiro steps ($\approx 93$ nA) closely matched the hinge current density previously determined via magnetic interference patterns, confirming the spatial origin of the effect.

4. Key Results

• Bulk-Dominant Junction (bJJ):
  • Exhibited standard Shapiro steps at all integer multiples of $V_0$ ($n=1, 2, 3...$).
  • The first step ($n=1$) was prominent, indicating that the transport was dominated by topologically trivial bulk states ($2\pi$-periodic CPR).
• Hinge-Dominant Junction (hJJ):
  • Missing First Step: At low microwave frequencies, the first Shapiro step ($n=1$) was completely absent ($Q_{12} = w_1/w_2 \approx 0.057$), while the second step ($n=2$) was clearly visible.
  • Frequency Dependence: As microwave frequency increased, the first step gradually reappeared. This behavior is consistent with theoretical predictions where the $4\pi$-periodic component is suppressed at higher frequencies due to quasiparticle poisoning or thermal effects, but dominates at low frequencies.
  • Quantitative Analysis: The ratio of the first to second step amplitudes ($Q_{12}$) decreased as frequency decreased, approaching zero. This confirmed the presence of a significant $4\pi$-periodic Josephson current component ($I_{4\pi} \approx 93$ nA).
• Exclusion of Trivial Mechanisms:
  • Landau-Zener Transitions (LZT): Ruled out because the bJJ and hJJ had similar junction transparencies, yet only the hJJ showed the missing step.
  • Underdamping: The McCumber parameter ($\beta \approx 1.6$) was too low to cause a missing first step via underdamping alone (which typically requires $\beta > 3$).
  • Joule Heating: While heating broadened step boundaries at high frequencies, the low-frequency behavior (where the step is missing) could not be attributed to heating.

5. Significance

• Confirmation of HOTI Physics: This work provides robust experimental evidence that multilayer WTe2 hosts topologically protected hinge states that can support a fractional Josephson effect, confirming its status as a 3D Higher-Order Topological Insulator.
• Pathway to Majorana Zero Modes: The observation of $4\pi$-periodic supercurrents is a prerequisite for the existence of Majorana zero modes (MZMs). This study establishes WTe2-based hybrid devices as a viable platform for realizing MZMs, which are crucial for topological quantum computing.
• Spintronics Applications: The spin-momentum locking inherent in these hinge states, combined with superconductivity, opens new avenues for developing superconducting spintronic devices and Josephson diodes.
• Methodological Advance: The study demonstrates that Shapiro step analysis, particularly the frequency dependence of the missing first step, is a powerful tool for distinguishing topological boundary modes from trivial bulk conduction in complex materials.

In conclusion, the paper successfully bridges the gap between theoretical predictions of HOTIs and experimental reality, utilizing the fractional Josephson effect to unambiguously identify topological hinge states in WTe2.