Helical phases and Bogoliubov Fermi surfaces probed by superconducting diode effects

This study employs the quasiclassical Eilenberger formalism to demonstrate that noncentrosymmetric superconductors with Rashba spin-orbit coupling and in-plane magnetic fields exhibit tunable superconducting and Josephson diode effects, wherein the emergence of Bogoliubov Fermi surfaces at the Lifshitz transition not only maximizes diode efficiency but also induces a strong current anisotropy that serves as a novel detection method for these exotic states.

The Gist

The Big Picture: Superconducting Diodes and "Helical" Dance Floors

Imagine a superconductor as a perfectly smooth dance floor where electrons (the dancers) can move without any friction. Normally, these dancers move in pairs (Cooper pairs) and flow equally well in both directions, like a two-lane road without traffic jams.

However, this paper examines a special type of superconductor called a non-centrosymmetric superconductor (NCS). Imagine this dance floor as having a built-in "twist" or "rotation" (called Rashba spin-orbit coupling). If you add a magnetic field (like a strong wind blowing over the floor), the dancers begin to move in spirals. The paper calls this a helical phase.

Due to this spiral motion, the dancers find it easier to move in one direction than the other. This creates a superconducting diode effect: current flows easily in one direction but is blocked in the other, just like a one-way street.

The Two Main Experiments

The researchers investigated this phenomenon in two different scenarios:

1. The Bulk System (The Entire Dance Floor)
They considered the superconductor as a whole block of material. They found that as the magnetic field increased, the dancers passed through two different "modes":

• The weak helical phase: A gentle spiral where the dancers are still mostly paired.
• The strong helical phase: A wilder, tighter spiral where the pairing momentum is very high.

The "Perfect" Diode Moment:
The paper discovered a very specific "sweet spot" exactly at the boundary where the dancers switch from the gentle spiral to the wild one. At this exact moment (a critical end point), the diode effect becomes nearly perfect. It is like finding the exact moment a door swings open so easily that it lets 100% of people through in one direction and 0% in the other.

2. The Josephson Junction (The Bridge)
They also examined a bridge connecting two superconductors with a gap in the middle (a "normal" region). This is like a bridge connecting two dance floors.

• Short Bridges: When the bridge is short, the diode effect is driven by how the dancers are already rotating on both sides.
• Long Bridges: When the bridge is long, the magnetic field in the middle gap becomes the main driver. The researchers found that the "one-way-ness" of the bridge oscillates back and forth like a tuning fork as the magnetic field is adjusted (switches). This means you could turn the diode on or off by changing the field strength.

The Mystery of "Ghost" Surfaces (Bogoliubov-Fermi Surfaces)

The most exciting part of the paper concerns the strong helical phase. In this state, the researchers predict the appearance of something called Bogoliubov-Fermi surfaces (BFS).

The Analogy:
Imagine the dance floor normally has a "gap" in the middle where no one can dance (this is the energy gap in a normal superconductor).

• In the strong helical phase, this gap doesn't just get smaller; it gets pierced.
• These piercings form a ring or a surface within the gap where "ghost" dancers (quasiparticles) can exist, even though the superconductor should theoretically be completely gapped. The paper calls these Bogoliubov-Fermi surfaces.

The "Anisotropic" Discovery:
Here lies the key insight: These ghost surfaces are not round; they are shaped like a specific track on the dance floor.

• If you try to push the electric current along the track where these ghosts live, the current gets crushed. The "one-way" effect (the diode) disappears, and the bridge no longer conducts well.
• If you push the current across the track, the current flows effortlessly.

This creates a strong anisotropy (directional dependence). It is like a road that is open when you drive from north to south, but if you try to drive from east to west, the road suddenly transforms into a wall of traffic.

Why This Matters (According to the Paper)

Proving the existence of these "ghost" surfaces (BFS) has been very difficult. Normally, scientists look for them by measuring heat or how much current leaks through, but these methods are tricky because "dirty" materials (disorder) can falsify these signals.

The authors propose a new, cleaner way to find them: Look at the direction of the current.
If you have a superconductor with these ghost surfaces, the electric current will behave very differently depending on which direction you orient your magnetic field or your current. If you see this specific "directional wall" where the current is blocked, it is a strong sign that these Bogoliubov-Fermi surfaces are present.

Summary of Claims

• Diode Efficiency: The ability to let electric current flow only in one direction is maximized exactly at the moment the superconductor switches from a "weak" spiral state to a "strong" spiral state.
• Tunable Bridges: In long bridges, the diode effect can be switched on and off by changing the magnetic field strength.
• Directional Blockade: In the strong spiral state, the presence of "ghost" surfaces (BFS) causes electric current to be blocked when it tries to move in a specific direction relative to the magnetic field.
• New Detection Method: This directional blockade (anisotropy) offers a new way to prove the existence of these ghost surfaces, distinct from other methods relying on heat or leakage.

The paper does not claim that these findings can already be used for medical devices, quantum computers, or specific commercial products; it focuses exclusively on understanding the fundamental physics of how these electrons behave and how we can detect these exotic states.

Technical Summary

Technical Summary: Helical Phases and Bogoliubov Fermi Surfaces Investigated via Superconducting Diode Effects

Problem Statement
Non-centrosymmetric superconductors (NCS) with Rashba spin-orbit coupling (SOC) and in-plane magnetic fields are theoretically predicted to host helical superconducting states characterized by finite-momentum Cooper pairing. These states are of considerable interest for realizing the bulk superconducting diode effect (SDE) and the Josephson diode effect (JDE), phenomena defined by unequal critical currents ($|I_{c+}| \neq |I_{c-}|$) in opposite directions. While the SDE has been extensively studied in the "weak" helical phase regime, the "strong" helical phase—characterized by the emergence of gapless Bogoliubov Fermi surfaces (BFS)—remains experimentally elusive. Furthermore, the signatures of these helical phases in Josephson contacts, particularly the interplay between finite-momentum pairing in the leads and the Zeeman field in the normal region, have not been systematically investigated. A major challenge in identifying BFS is that most experimental observables (e.g., residual density of states) are indirect and can be mimicked by disorder.

Methodology
The authors apply the quasiclassical Eilenberger formalism to systematically investigate both the bulk SDE and the JDE in a clean NCS with Rashba SOC and in-plane magnetic fields.

• Model: They consider a two-dimensional electron gas with strong Rashba SOC and a Zeeman term due to an in-plane magnetic field. The system is modeled as a superconductor-normal conductor-superconductor (SNS) junction.
• Formalism: The Eilenberger equations are solved self-consistently for the superconducting order parameter, accounting for the finite-momentum pairing ($q$) induced by the magnetic field. The authors treat the two helical bands ($\lambda = \pm$) separately and note that they decouple in the limit of strong SOC.
• Calculations: The study calculates the supercurrent $j(q)$, the critical currents $I_{c\pm}$, the diode efficiency $\eta = (I_{c+} - I_{c-})/(I_{c+} + I_{c-})$, and the current-phase relations (CPR) for both short and long contact limits. The analysis covers the transition between weak and strong helical phases as well as the system's behavior as a function of magnetic field strength, temperature, and contact length.

Main Contributions and Results

1. Bulk SDE and the Critical End Point (CEP):

   • The study confirms that the bulk SDE is always present when the magnetic field is non-zero.
   • The authors identify a first-order Lifshitz-type phase transition between the weak helical phase (gapped spectrum) and the strong helical phase (gapless spectrum with BFS).
   • A central result is that the diode efficiency $\eta$ nominally approaches its maximum value ($\pm 1$) at the critical end point (CEP), where this first-order transition terminates. This enhancement is attributed to the vanishing of the free energy barrier between local minima of the supercurrent, similar to behavior observed near tricritical points in FFLO phases.

2. Mechanisms of the Josephson Diode Effect (JDE):

   • Short Contacts: In the short-contact limit, the JDE is dominated by finite-momentum pairing in the superconducting leads. The diode efficiency reaches an extremum before being suppressed upon entering the strong helical phase.
   • Long Contacts: In the long-contact limit, the JDE is primarily determined by the Zeeman field in the normal region. The diode efficiency oscillates as a function of the magnetic field with a period related to the Thouless energy ($E_T$). This oscillation arises from a relative phase shift between the two helical bands, driven by the magnetic field in the normal conductor and finite-momentum pairing in the superconductor. This offers a pathway to highly tunable Josephson diodes.

3. Suppression and Anisotropy in the Strong Helical Phase:

   • Upon entering the strong helical phase, the formation of BFS leads to a gapless spectrum. This results in a significant suppression of both the Josephson current and the diode efficiency.
   • Crucially, the authors find a strong anisotropy in this suppression. The Josephson current is most strongly suppressed when the current direction in momentum space intersects the BFS. When the current direction does not intersect the BFS, the suppression is weaker.
   • This anisotropy is a direct consequence of the gapless nature of the strong helical phase and the specific location of the BFS in momentum space.

Significance and Claims
The work claims that long Josephson contacts represent a promising platform for investigating helical phases in NCS, particularly the transition into the strong helical phase. The primary significance of the work lies in proposing a new method for detecting Bogoliubov Fermi surfaces (BFS).

• Alternative Detection Method: In contrast to conventional probes based on residual density of states (which can be mimicked by disorder), the authors propose that the strong anisotropic suppression of the Josephson current relative to the current direction offers a unique signature of BFS.
• Generality: The authors assume that this anisotropic suppression is a generic feature of multi-band models with BFS, regardless of whether a JDE is present.
• Experimental Applicability: The work suggests that this effect could be detected via Josephson scanning tunneling microscopy (JSTM), which would enable partial mapping of BFS.

The authors maintain a modest tone regarding the "perfect" diode efficiencies at the CEP and acknowledge that phase coexistence and temperature fluctuations in a real system would likely reduce the efficiency, although the enhancement near the critical point should remain robust. They further note that while disorder is a known mimic for gapless superconductivity, the specific twofold anisotropy predicted here, linked to the magnetic field direction in helical superconductors, offers a potential distinguishing feature.