Current Flow in Topological Insulator Josephson Junctions due to Imperfections

This paper attributes the unexpected nonzero Josephson currents observed in topological insulator junctions at integer fluxoid states to junction imperfections rather than Majorana zero modes, proposing that low-energy bound states arising from irregularities drive the current and suggesting microwave spectroscopy as a method to experimentally verify this mechanism through distinctive vortex transition selection rules.

The Gist

The Big Picture: A Mystery in a Ring

Imagine a superconductor (a material that conducts electricity with zero resistance) wrapped around a topological insulator (a special material that conducts electricity only on its surface). When you arrange this in a circle (called a Corbino geometry), it acts like a tiny, super-fast racetrack for electrons.

Usually, if you put a magnetic field through the center of this ring, the electrons arrange themselves into specific patterns called "vortices." Physics tells us that if you have a whole number of these magnetic "whirlpools" (flux quanta), the electrical current flowing around the ring should be exactly zero. It's like a perfectly balanced scale.

However, recent experiments showed something strange: even with these perfect numbers of whirlpools, a tiny but measurable current was flowing, but only when the temperature was extremely low. Scientists wondered: Is this a sign of exotic new particles (Majorana modes) doing something magical?

This paper says: No, it's not magic. It's just a little bit of messiness.

The Analogy: The Perfectly Aligned Chorus vs. The Off-Key Singer

To understand the authors' explanation, imagine a choir of singers (the electrons) standing in a perfect circle.

• The Ideal Scenario: If every singer stands at the exact same distance from the center and sings the exact same note, their voices cancel each other out perfectly in a specific direction. The net sound (current) is zero.
• The Real Scenario: In the real world, the stage isn't perfectly round. Maybe the floor is slightly uneven, or the singers are standing a tiny bit closer together in one spot and a bit further apart in another. These are imperfections.

The authors argue that the tiny current observed in the experiment isn't because of some mysterious new physics, but simply because the "stage" (the junction) isn't perfectly uniform. The width of the ring varies slightly, or the chemical properties change a little bit along the path.

The "Atomic" Limit: Isolated Atoms

The paper focuses on a specific condition called the "atomic limit."

• Imagine: A row of isolated trees in a forest. Each tree has its own small patch of grass around it.
• The Physics: The magnetic field creates "vortices" (the trees). In this specific setup, the trees are far enough apart that their grass patches don't overlap. Each vortex acts like its own independent "atom."

In a perfect world, each of these "vortex atoms" would have a special state at zero energy (a Majorana mode) that doesn't carry any current. But because the "forest floor" is uneven (the width of the junction varies), the energy levels of these atoms shift slightly.

How the Current Appears

The authors explain that at very low temperatures, the electrons settle into the lowest energy states available.

1. The Imperfection: Because the junction is slightly wider in some places and narrower in others, the "energy landscape" for the electrons changes as you go around the ring.
2. The Shift: This unevenness breaks the perfect symmetry. It's like if one singer in the choir suddenly sang a slightly different note because they were standing on a bump.
3. The Result: This tiny shift allows a small current to flow. The paper calculates that even a 10% variation in the width of the junction is enough to explain the size of the current seen in experiments (about 10 nano-amperes).

They also looked at the "current profiles" (how the current moves through each vortex). They found that the special zero-energy state (the Majorana mode) stays at zero current, but the other excited states get "kicked" by the imperfections, creating the flow we see.

The Proposed Test: Microwave Spectroscopy

To prove this theory, the authors suggest a way to "listen" to these electrons.

• The Analogy: Imagine tapping a wine glass to hear its specific ring.
• The Method: They propose shining microwaves at the junction. If their theory is right, the electrons will only absorb energy at very specific frequencies.
• The Prediction: They predict a unique "selection rule" (a specific pattern of allowed notes). The electrons can only jump between energy levels in pairs, following a very specific mathematical rhythm ($\sqrt{n} + \sqrt{n-1}$). If scientists see this specific pattern in the microwave data, it confirms that the current is indeed caused by these specific low-energy states reacting to imperfections.

Summary

• The Problem: Experiments showed a tiny electric current in a superconducting ring with a whole number of magnetic whirlpools, which shouldn't happen.
• The Cause: The paper argues this is caused by imperfections (like the ring being slightly uneven in width), not by exotic new physics.
• The Mechanism: These imperfections break the perfect symmetry, allowing low-energy electrons to flow.
• The Proof: The authors propose a microwave test that would reveal a unique "fingerprint" of these electron jumps, confirming that the current comes from these specific, slightly messy states.

In short: The "ghost" current isn't a ghost; it's just the result of a slightly crooked track.

Technical Summary

Technical Summary: Current Flow in Topological Insulator Josephson Junctions due to Imperfections

Problem Statement
Recent experiments on planar superconductor-topological insulator-superconductor (S-TI-S) junctions, specifically in Corbino geometry, have reported non-zero Josephson currents at low temperatures in states with integer fluxoid numbers. In an ideal system with perfect translational symmetry, the Josephson current should vanish at these integer flux quanta (the zeros of the Fraunhofer pattern). While these observations have been linked to Majorana zero modes localized in Josephson vortices, the mechanism generating a finite critical current in these specific states remained to be elucidated. The authors aim to explain this phenomenon by attributing it to structural imperfections, specifically focusing on the "atomic limit" where low-energy bound states of different vortices do not overlap.

Methodology
The authors model a planar Josephson junction using a Bogoliubov-de Gennes (BdG) Hamiltonian for a topological insulator with proximity-induced superconductivity. They consider the system in the presence of a perpendicular magnetic field, which creates a lattice of Josephson vortices.

• Atomic Limit: The study operates in a regime where the localization length of low-energy orbitals ($\lambda_B$) is much smaller than the distance between vortices ($\ell_B$). This allows the vortex lattice to be treated as an array of nearly independent "vortex atoms."
• Effective Hamiltonian: The authors derive an effective 2$\times$2 Dirac Hamiltonian for the $k$-th vortex, considering only the two lowest energy bound states of the transverse microscopic Hamiltonian. This effective Hamiltonian describes a supersymmetric oscillator.
• Imperfection Modeling: To explain the non-vanishing current, the authors introduce $y$-dependent perturbations, specifically variations in the junction width $W(y)$ or the chemical potential $\mu_N(y)$. They analyze how these irregularities break the translational symmetry that would otherwise cause the current to cancel out.
• Spectroscopy Analysis: The paper employs microwave spectroscopy techniques to analyze transitions between Andreev bound states, calculating matrix elements of the current density operator to determine selection rules.

Key Contributions and Results

1. Mechanism for Non-Zero Current:
   The authors demonstrate that in the absence of irregularities, the total Josephson current for an integer number of flux quanta vanishes exactly due to the ability to remove the external phase offset via coordinate transformation. However, introducing slow variations in the junction width $W(y)$ or chemical potential $\mu_N(y)$ prevents this cancellation. The non-vanishing critical current arises from the contribution of low-energy Caroli-de Gennes-Matricon (CdGM) states (Andreev states) whose energies become dependent on the external phase $\phi_0$ due to the spatial variation of the vortex parameters.

2. Quantitative Agreement with Experiment:
   Using the derived expressions for the current contribution of individual CdGM states, the authors provide estimates that reproduce the magnitude of currents observed in recent experiments (e.g., Ref. [11]). They show that a width variation of approximately 10% ($\delta W/W \sim 0.1$) is sufficient to generate critical currents in the range of nano-amperes ($\sim 10$ nA) at low temperatures ($T \ll E_{1,k}$), consistent with experimental data where currents increase significantly as temperature drops from 1.6 K to 0.25 K.

3. Current Profiles of CdGM States:
   The study investigates the current density profiles of individual CdGM levels under the influence of width gradients.

   • The zero-energy Majorana state remains uncorrected; its current profile remains zero.
   • All excited states ($n \geq 1$) acquire non-zero current profiles due to the gradient. The total current contribution from the $n$-th level scales as $I_n \propto \sqrt{n} I_1$.

4. Microwave Spectroscopy and Selection Rules:
   The authors propose using microwave spectroscopy to probe these low-energy excitations. By analyzing the matrix elements of the current operator, they derive distinctive selection rules for transitions between CdGM levels:

   • Transitions are allowed only between neighboring levels ($n \leftrightarrow n \pm 1$).
   • At absolute zero, absorption transitions occur at frequencies $\Omega_{n,k} = \omega_{Bk}(\sqrt{n} + \sqrt{n-1})$.
   • These transitions involve the pairwise population of neighboring excited states, effectively breaking Cooper pairs in the ground state.

Significance and Claims
The paper claims to provide a robust explanation for the lifting of the zero-nodes in the Fraunhofer pattern of Corbino junctions, attributing it to structural imperfections rather than exotic topological effects alone. The authors emphasize that their simple estimates correctly reproduce the magnitude of experimentally observed currents, validating the "atomic limit" approach with imperfections.

Furthermore, the paper proposes a clear experimental test via microwave spectroscopy, predicting a specific pattern of resonances characteristic of Josephson vortices in long topological junctions. The authors note that while a related study (Ref. [31]) discusses similar effects in open-ended geometries linked to boundary conditions, their work specifically addresses the Corbino geometry, suggesting that irregularities play a crucial role in lifting the zeros even in closed-loop systems. The study concludes by identifying future directions, such as exploring the overlap of CdGM states beyond the atomic limit and the role of phase fluctuations, but does not claim to have resolved these complex interactions.