Spatially Inhomogeneous Triplet Pairing Order and Josephson Diode Effect Induced by Frustrated Spin Textures

This paper demonstrates that frustrated spin textures induce anisotropic, spatially inhomogeneous triplet pairing and a Josephson diode effect in superconductors by generating $d$-vector-dependent couplings that break inversion and time-reversal symmetries, distinct from mechanisms driven by spin-orbit coupling.

The Gist

The Big Picture: Dancing Electrons and Frustrated Spins

Imagine a superconductor as a giant, perfectly synchronized dance floor. In a normal superconductor, all the electrons (the dancers) move in perfect lockstep, holding hands in a specific way. Usually, they hold hands in a simple, uniform pattern (like holding hands with a partner in a circle).

But in spin-triplet superconductors, the electrons are more complex. Instead of just holding hands, they have an internal "orientation" or "pose" (represented in the paper by a d-vector). Think of this as the dancers not just holding hands, but also pointing their noses in a specific direction. In a standard superconductor, everyone points their nose in the same direction.

This paper asks: What happens if the dance floor itself is built on a foundation of "frustrated" magnets?

The Setup: The Frustrated Spin Textures

The authors imagine a scenario where the superconducting dance floor sits on top of a layer of tiny magnets (spins). These magnets are "frustrated."

• The Analogy: Imagine three friends sitting in a triangle, each trying to face away from the other two. If they are in a line, they can easily face opposite ways. But in a triangle, if Friend A faces away from B, and B faces away from C, Friend C is stuck—they can't face away from both A and B at the same time. They are "frustrated."
• In the paper, these frustrated magnets form a complex, swirling pattern (a "spin texture") rather than a simple grid.

The Discovery: The "Pliable" Dance

The paper shows that when the electrons (the dancers) interact with these frustrated magnets, something strange happens to their "nose-pointing" direction (the d-vector).

1. The New Force: Usually, the electrons want to keep their noses pointing in the exact same direction everywhere to save energy. However, the frustrated magnets introduce a new force that acts like a twist.
2. The Metaphor: Imagine the dance floor is made of a stiff rubber sheet. Usually, if you try to twist one part of the sheet, it snaps back to being flat. But the frustrated magnets make the sheet "pliable" (like soft clay).
3. The Result: Instead of everyone pointing their nose in the same direction, the electrons start pointing in different directions depending on where they are. The "nose" of the electron pair twists and turns as you move across the material. The paper calls this a spatially inhomogeneous pairing order. It's a dance where the choreography changes from one spot to the next, creating a swirling pattern of electron orientations.

How It Works: The Tunneling Bridge

How do the magnets talk to the electrons? The paper uses a concept called tunneling.

• The Analogy: Imagine two islands (superconducting grains) separated by a river. Electrons need to jump (tunnel) across the river to stay connected.
• The Twist: Usually, the river is just water. But here, the river is filled with the "frustrated" magnetic spins. When an electron jumps across, its path is influenced by the specific swirl of the magnets in the river.
• The Outcome: This influence creates a special kind of connection between the two islands. It's not just a simple bridge; it's a bridge that forces the dancers on one island to twist their pose relative to the dancers on the other island. This "twist" is what allows the complex, swirling patterns to form.

The "Diode" Effect: One-Way Traffic

The most exciting practical finding in the paper is the Josephson Diode Effect.

• The Analogy: Think of a standard electrical wire as a two-way street. Cars (current) can drive forward or backward with equal ease.
• The Diode: A diode is a one-way street. Cars can go forward easily, but if they try to go backward, they hit a wall.
• The Paper's Claim: The authors show that if the magnetic "river" between the islands has a specific type of twist (called spin chirality), the supercurrent becomes a one-way street.
  • Current can flow easily in one direction.
  • Current is blocked or much harder to push in the other direction.
• Why? The combination of the twisted electron poses (non-collinear d-vectors) and the swirling magnets breaks the rules of symmetry. It's like a lock that only turns one way.

Summary of Key Claims

1. Frustration creates variety: Frustrated magnetic textures (swirling spins) can force superconducting electrons to change their orientation as they move through the material, creating complex, swirling patterns instead of a uniform state.
2. It's not just spin-orbit coupling: Usually, scientists think these effects come from the interaction between an electron's spin and its motion (spin-orbit coupling). This paper proves that frustrated magnets alone can create these effects, even without that specific interaction.
3. The Diode Effect: If the magnetic texture is "chiral" (swirling in a specific direction), the superconductor acts like a diode, allowing current to flow much better in one direction than the other.

In short: The paper describes how a "frustrated" magnetic background can turn a uniform superconductor into a pliable, twisting material that can act as a one-way valve for electricity.

Technical Summary

Technical Summary: Spatially Inhomogeneous Triplet Pairing Order and Josephson Diode Effect Induced by Frustrated Spin Textures

Problem Statement
The interplay between frustrated magnetism and unconventional superconductivity remains an active area of research, particularly regarding how noncollinear or noncoplanar spin textures influence superconducting pairing orders. While systems like the kagome Weyl semimetal Mn$_3$Ge and the transition-metal dichalcogenide 4Hb-TaS$_2$ exhibit coexisting frustrated spin textures and spin-triplet superconductivity, the microscopic mechanisms by which these textures affect the pairing order parameter are not fully understood. Specifically, there is a need to understand how frustrated spin textures can generate anisotropic Josephson couplings that stabilize spatially inhomogeneous pairing orders and induce non-reciprocal transport phenomena, such as the Josephson diode effect, without relying solely on spin-orbit coupling or noncentrosymmetric crystal structures.

Methodology
The authors employ a theoretical framework combining the Ambegaokar-Baratoff formalism for Josephson coupling with a microscopic $s$-$d$ exchange model on geometrically frustrated lattices (kagome and triangular).

1. Free Energy Formulation: The study begins by decomposing the total free energy into a homogeneous bulk term and a variation term arising from spatially varying pairing orders. For spin-triplet superconductors, the pairing order is characterized by a $U(1)$ phase and a momentum-dependent $\mathbf{d}$ vector. The authors derive the Josephson free energy between adjacent superconducting grains, identifying three distinct coupling terms analogous to magnetic interactions:

   • Heisenberg-like symmetric coupling ($J_{nm}$).
   • Dzyaloshinskii-Moriya (DM)-like antisymmetric coupling ($D_{nm}$).
   • $\Gamma$-type symmetric traceless coupling ($\Gamma_{nm}$).

2. Microscopic Derivation: Using a $T$-matrix expansion, the authors derive the effective tunneling of itinerant electrons across a barrier containing frustrated local spin moments. They model the barrier as a local exchange field composed of static classical spins on a three-sublattice system. The effective tunneling matrix is expanded perturbatively in the $s$-$d$ coupling strength ($J_{sd}$) up to third order to capture effects of scalar spin chirality.

3. Pairing Correlations: The paper analyzes the generation of spin-triplet pairing correlations in isolated grains arising from either $s$-$d$ exchange or Ising-type spin-orbit coupling. This is done by calculating the anomalous Green's function within the Bogoliubov-de Gennes (BdG) formalism for systems on kagome and triangular lattices.

4. Diode Effect Analysis: The Josephson current is calculated up to second order to determine the critical currents for forward and backward bias. The efficiency of the Josephson diode effect is evaluated based on the symmetry breaking properties (inversion and time-reversal) induced by the spin texture and the relative orientation of $\mathbf{d}$ vectors.

Key Contributions and Results

• Origin of Anisotropic Couplings: The paper demonstrates that anisotropic Josephson couplings ($D_{nm}$ and $\Gamma_{nm}$) can arise from the coupling of itinerant electrons to frustrated local spin moments. Crucially, the authors show that these couplings cannot originate from spin-orbit coupling alone (which typically yields vanishing DM-like terms under time-reversal symmetry for real tunneling amplitudes) but require a local exchange field that breaks time-reversal symmetry.
• Effective Tunneling Mechanism: Through the $T$-matrix expansion, the authors derive that effective tunneling depends on the underlying spin configuration. Specifically:
  • The Heisenberg-like term depends on the dot product of spins ($\mathbf{s}_i \cdot \mathbf{s}_j$).
  • The DM-like term depends on the cross product ($\mathbf{s}_i \times \mathbf{s}_j$).
  • The scalar spin chirality ($\chi_{ijk} = \mathbf{s}_i \cdot (\mathbf{s}_j \times \mathbf{s}_k)$) and higher-order terms contribute to antisymmetric components that break inversion and time-reversal symmetries.
• Spatially Inhomogeneous Pairing Order: The competition between the negative-valued Heisenberg-like coupling (favoring collinear $\mathbf{d}$ vectors) and the finite DM-like coupling (favoring noncollinear textures) leads to an effective "pliability" of the pairing order. This competition can stabilize spatially varying $\mathbf{d}$ vector textures, such as skyrmion-like configurations, particularly in systems with small superconducting grains where Josephson coupling energy competes with bulk condensation energy.
• Josephson Diode Effect: The paper establishes two distinct mechanisms for a Josephson diode effect in spin-triplet superconductors:
  1. Nonvanishing Spin Chirality: In the barrier region, finite scalar spin chirality breaks both inversion and time-reversal symmetries, leading to a phase shift in the first-order Josephson current.
  2. Antisymmetric Coupling between Noncollinear $\mathbf{d}$ Vectors: Even without scalar spin chirality, if the $\mathbf{d}$ vectors of adjacent grains are noncollinear, the antisymmetric Josephson coupling ($D_{nm}$) can break inversion symmetry, resulting in a diode effect.
     The diode efficiency is shown to scale with the magnitude of the spin chirality or the degree of noncollinearity of the $\mathbf{d}$ vectors.

Significance and Claims
The authors claim that their work provides a distinct microscopic route to engineering inhomogeneous superconducting textures and non-reciprocal supercurrents, independent of spin-orbit coupling or noncentrosymmetric crystal lattices. The results suggest that frustrated magnetic textures in materials like Mn$_3$Ge and 4Hb-TaS$_2$ can act as local exchange fields that mediate these anisotropic couplings.

The paper posits that the observed phenomena in these materials, such as long-range coherent Josephson supercurrents and hysteretic behavior, may be underpinned by the frustrated spin textures generating effective tunneling dependent on spin configurations. Furthermore, the study highlights that the interplay between bulk pairing order and Josephson coupling is crucial; while large grains favor homogeneous order, smaller grains allow the anisotropic couplings to dominate, potentially leading to complex spatial textures. The work concludes that these findings offer a theoretical basis for understanding the origin of nontrivial spatial textures in unconventional superconductors and the emergence of the Josephson diode effect in systems with frustrated magnetism.