$p$-wave superconductivity and Josephson current in $p$-wave unconventional magnet/$s$-wave superconductor hybrid systems

This paper demonstrates that hybrid systems combining $p$-wave unconventional magnets and $s$-wave superconductors induce an $s+p$-wave-like superconducting state characterized by zero-energy flat bands, odd-frequency pairing, and tunable Josephson currents driven by the interplay between noncollinear magnetic order and singlet pairing.

The Gist

The Big Picture: Mixing Magic Ingredients

Imagine you are a chef trying to bake a very special cake. You have two main ingredients:

1. A Superconductor (The "Zero-Resistance" Cake): A material where electricity flows perfectly without any friction, like a car driving on a perfectly smooth, endless highway. Usually, these are made of "s-wave" pairs (think of them as dance partners holding hands tightly in a circle).
2. A Weird Magnet (The "P-Wave" Magnet): A new type of magnet recently discovered. Unlike normal magnets where all the tiny internal arrows (spins) point in the same direction, this magnet has a "non-collinear" structure. Imagine a crowd of people where everyone is spinning in a complex, swirling pattern rather than standing still or marching in a straight line.

The scientists in this paper asked: "What happens if we bake these two ingredients together?"

They found that when you mix a standard superconductor with this weird "swirling" magnet, something magical happens: the standard superconductor starts acting like a p-wave superconductor.

The Magic Trick: The "Dance Floor" Transformation

In the world of physics, electrons usually pair up in two main ways:

• Spin-Singlet (The "Circle Dance"): Two electrons hold hands and spin in opposite directions. This is the standard "s-wave" dance.
• Spin-Triplet (The "Line Dance"): Two electrons hold hands and spin in the same direction. This is the rare "p-wave" dance, which is very hard to find in nature.

The Analogy:
Imagine the electrons in the superconductor are dancers on a floor. Normally, they do the "Circle Dance." But when they step onto the floor of the P-wave Magnet, the magnet's swirling magnetic field acts like a DJ who forces the dancers to change their moves.

Suddenly, the "Circle Dancers" are forced to do the "Line Dance" (Spin-Triplet). The paper shows that this magnet doesn't just sit there; it actively transforms the superconductor's behavior, making it behave as if it were a rare, exotic material.

The "Flat Band" Highway: The Zero-Energy Express

One of the most exciting discoveries in the paper is the creation of Zero-Energy Flat Bands.

The Analogy:
Imagine a highway where cars usually have to speed up and slow down (changing energy levels). But at the very edge of this new hybrid material, a special "Flat Band" appears. This is like a magic express lane where cars (electrons) can travel at a constant, zero-energy speed without any bumps or hills.

Why is this cool?

• These "Flat Bands" are topologically protected, meaning they are very stable and hard to destroy.
• They are the playground for Majorana Zero Modes. Think of these as "ghost particles" that are their own antiparticles. They are the holy grail for building quantum computers because they are incredibly stable and can store information without errors.

The paper proves that by tuning the "chemical potential" (basically, how many electrons are in the system, like adjusting the number of dancers on the floor), you can turn these Flat Bands on or off. It's like a light switch for quantum magic.

The Josephson Current: The "Tug-of-War"

The paper also looks at what happens when you put two of these hybrid materials next to each other with a tiny gap in between (a Josephson Junction). This is like a Tug-of-War between two teams of electrons.

• The Old Rule: In normal magnets, if you try to connect a "Circle Dance" superconductor to a "Line Dance" one, they can't hold hands, and the Tug-of-War stops (no current).
• The New Discovery: Because the weird magnet transformed the superconductor into a "Line Dance" style, the two sides can now hold hands!
  • However, the paper found that the "Circle Dance" (the original s-wave pairing) didn't disappear completely. It's still there, lurking in the background.
  • This creates a unique mix: The current flows, but it has a special rhythm. It's not just a simple back-and-forth; it has a complex beat (harmonics) that depends on the temperature and the number of electrons.

Why Does This Matter?

1. Quantum Computing: The "Flat Bands" and "Majorana particles" found here are the building blocks for future quantum computers. This paper shows a new, potentially easier way to create them using these weird magnets.
2. New Materials: It proves that you don't need to find rare, exotic materials in nature to get "p-wave" superconductivity. You can just mix a common superconductor with a specific type of magnet to get the same effect.
3. Control: The scientists showed that you can tune this effect. By changing the chemical potential (the "dancer count"), you can control whether the special quantum states appear or disappear.

Summary in One Sentence

By mixing a standard superconductor with a swirling, weird magnet, the scientists created a new hybrid material that forces electrons to dance in a rare way, creating a stable "express lane" for quantum particles that could power the computers of the future.

Technical Summary

1. Problem Statement

The search for topological superconductivity and Majorana zero modes is a central goal in condensed matter physics, often requiring spin-triplet $p$-wave pairing. However, intrinsic spin-triplet $p$-wave superconductors are rare and difficult to realize in materials. While hybrid systems (e.g., nanowires with spin-orbit coupling and ferromagnetism) have been proposed to induce effective $p$-wave pairing, the role of p-wave unconventional magnets (PUMs) remains an emerging area of study.

PUMs are characterized by non-collinear spin structures and $p$-wave-like spin splitting on the Fermi surface, distinct from conventional ferromagnets or altermagnets. The specific problem addressed in this work is:

• How does a conventional spin-singlet $s$-wave superconductor (SC) behave when proximitized by a PUM?
• Can the non-collinear spin structure of a PUM induce effective $p$-wave superconductivity and topological edge states (zero-energy flat bands) in an $s$-wave SC?
• How do these induced states affect the Josephson current in hybrid junctions, particularly regarding the coupling of spin-singlet and spin-triplet components?

2. Methodology

The authors employ a theoretical framework based on the tight-binding model and the Bogoliubov-de Gennes (BdG) formalism.

• Model Hamiltonian: They utilize an effective 4$\times$4 Hamiltonian for the PUM (based on Ref. [52]) which includes:
  • Kinetic energy ($\epsilon(k)$).
  • Spin-dependent hopping terms ($t_x, t_y$) that generate spin-helix structures.
  • Local $sd$-interaction ($J$) that creates the non-collinear spin structure characteristic of PUMs.
  • The system is coupled to a conventional spin-singlet $s$-wave SC via the proximity effect.
• Green's Function Approach:
  • Surface Density of States (SDOS): Calculated using the recursive Green's function method for semi-infinite systems to identify edge states at the [100] boundary.
  • Pair Amplitude Analysis: The anomalous Green's functions are decomposed into four symmetry classes: Even-frequency Spin-singlet Even-parity (ESE), Even-frequency Spin-triplet Odd-parity (ETO), Odd-frequency Spin-singlet Odd-parity (OSO), and Odd-frequency Spin-triplet Even-parity (OTE).
  • Josephson Current: Calculated for Josephson junctions (PUM-SC/Normal Metal/SC and PUM-SC/PUM-SC) in both high-transparency and low-transparency limits. The current-phase relation $I(\phi)$ is analyzed via the Fourier series expansion $I(\phi) = \sum I_m \sin(m\phi)$.

3. Key Contributions

• Induction of $s+p$-wave-like State: The paper demonstrates that the non-collinear spin structure of a PUM, when combined with an $s$-wave SC, effectively induces a spin-triplet $p$-wave superconducting state alongside the original $s$-wave component.
• Topological Edge States: The authors confirm the emergence of zero-energy flat bands at the [100] edge, which are topologically protected by winding numbers. These bands are a signature of the induced $p$-wave pairing and are relevant to Majorana zero modes.
• Symmetry Analysis of Pairing: A detailed classification of pair amplitudes at the edge reveals that while the bulk is dominated by ETO (spin-triplet) pairing, the edge hosts a coexistence of Odd-frequency Spin-triplet Even-parity (OTE) pairing (enhanced by zero-energy flat bands) and the original Even-frequency Spin-singlet Even-parity (ESE) pairing.
• Josephson Current Modulation: The study reveals that the presence of the PUM prevents the vanishing of the first harmonic ($I_1$) in Josephson junctions, a feature typically seen in pure $p$-wave/$s$-wave junctions. This is due to the residual ESE component.

4. Key Results

A. Surface Density of States (SDOS) and Edge States

• Zero-Energy Flat Bands: For specific chemical potentials ($\mu = -4t$) and spin-dependent hopping parameters ($t_x \neq 0, t_y=0$ for $p_x$-wave PUM), the SDOS exhibits a sharp peak at zero energy, corresponding to flat bands.
• Parameter Dependence: The existence of these flat bands depends on the chemical potential and the strength of the magnetic order ($J$). When $J > \Delta$ (pair potential), nodal structures emerge, and zero-energy flat bands appear.
• Anisotropy: The emergence of flat bands is sensitive to the orientation of the edge relative to the spin structure. For example, $p_x$-wave PUMs show flat bands at the [100] edge, while $p_y$-wave PUMs do not under the same conditions due to the alignment of the non-collinear spins.

B. Pair Amplitude at the Edge

• OTE Enhancement: In the presence of zero-energy flat bands, the Odd-frequency Spin-triplet Even-parity (OTE) pairing amplitude is significantly enhanced (orders of magnitude larger than ESE) at the edge.
• Coexistence: Crucially, the ESE (conventional $s$-wave) pairing does not vanish at the edge. This coexistence is the key to the unique Josephson current behavior.
• Bulk vs. Edge: In the bulk, ETO pairing (equal-spin triplet) is dominant. At the edge, the symmetry breaking allows for the emergence of OTE and the persistence of ESE.

C. Josephson Current Characteristics

• PUM-SC / $s$-wave SC Junctions:
  • Unlike pure $p$-wave/$s$-wave junctions where the first harmonic ($I_1$) vanishes, these hybrid junctions exhibit a non-zero $I_1$ component.
  • This is because the ESE component in the PUM-SC couples directly to the $s$-wave SC on the other side.
  • Depending on parameters, the junction can behave as a $\phi$-junction (where the ground state phase is $\pm \phi$) or a standard 0-junction.
• PUM-SC / PUM-SC Junctions:
  • When two PUM-SCs are coupled, the current phase relation is dominated by the coupling of spin-triplet components.
  • Zero-Energy Flat Band Resonance: In the high-transparency limit, the Josephson current is significantly enhanced at low temperatures due to the resonance of zero-energy flat bands on both sides of the junction.
  • Temperature Dependence: The maximum Josephson current ($I_c$) shows distinct temperature dependencies. In the presence of flat bands, $I_c$ is enhanced at low $T$; in their absence, it follows the standard Ambegaokar-Baratoff saturation behavior.
• Tunability: The transition between different current-phase behaviors (e.g., from $I_1$-dominated to $I_2$-dominated or $\phi$-junctions) can be tuned by the chemical potential, which controls the generation of zero-energy flat bands.

5. Significance and Conclusion

• Realization of Topological States: The paper provides a realistic pathway to realize topological superconductivity and Majorana zero modes using PUMs, which have recently been experimentally identified (e.g., in NiI$_2$ and Gd$_3$(Ru,Rh)$_4$Al$_{12}$).
• New Class of Hybrid Systems: It establishes that PUM-SC hybrid systems effectively behave as $s+p$-wave superconductors. The coexistence of $s$-wave and induced $p$-wave components leads to unique transport properties not found in pure $p$-wave systems.
• Experimental Signatures: The study predicts specific signatures for tunneling spectroscopy (zero-bias peaks from flat bands) and Josephson junctions (tunable current-phase relations and $\phi$-junctions). These provide concrete targets for experimental verification.
• Theoretical Robustness: The results are shown to be robust across different PUM models, suggesting that the non-collinear spin structure is the essential ingredient for inducing these topological and transport phenomena.

In summary, this work bridges the gap between unconventional magnetism and topological superconductivity, demonstrating that PUMs can effectively induce $p$-wave pairing in $s$-wave superconductors, leading to rich Josephson physics and potential platforms for topological quantum computing.