Possible Pairing Symmetry of BaPtAs$_{1-x}$Sb$_{x}$ with an Ordered Honeycomb Network

This study investigates the pairing symmetry of superconducting BaPtAs$_{1-x}$Sb$_{x}$, revealing that while the Sb-rich end ($x=1$) exhibits a time-reversal symmetry-breaking chiral $d$-wave state, the As-rich end ($x=0$) favors competitive nodal $f$-wave or conventional $s$-wave states without such symmetry breaking.

The Gist

Imagine a microscopic dance floor where electrons are the dancers. In most materials, these dancers move in a chaotic, individualistic way. But in a superconductor, they pair up and dance in perfect, synchronized harmony, allowing electricity to flow without any resistance (like friction).

The big question scientists have been asking about a specific family of materials called BaPtAs$_{1-x}$Sb$_x$ is: What kind of dance are they doing?

This paper investigates how changing the "ingredients" of this material changes the dance style. Here is the story in simple terms:

1. The Stage: A Honeycomb Dance Floor

The material is built like a honeycomb (think of a beehive or a soccer ball pattern). It's made of Platinum (Pt) and either Arsenic (As) or Antimony (Sb).

• BaPtAs is the version with 100% Arsenic ($x=0$).
• BaPtSb is the version with 100% Antimony ($x=1$).
• BaPtAs$_{1-x}$Sb$_x$ is the mix in between.

Scientists noticed something weird: When they looked at the pure Antimony version (BaPtSb), they detected a tiny, spontaneous magnetic field appearing inside the material just as it started superconducting. This is like seeing the dancers spontaneously start spinning in a circle, creating a whirlwind. This suggests a very exotic, "chiral" (handed) dance that breaks the rules of time symmetry.

However, when they looked at the pure Arsenic version (BaPtAs), that magnetic signal disappeared. It seemed like the dancers were doing a much more boring, standard dance.

2. The Investigation: Building a Virtual Model

The authors (Tsuyoshi Imazu and colleagues) decided to build a computer model to figure out why the dance changes.

• The Map: They used supercomputers to map out the "energy landscape" of the electrons in both materials. Think of this as drawing a topographic map of hills and valleys where the electrons live.
• The Key Difference: They found that in the Antimony version (BaPtSb), the electrons live very close to a "saddle point" (a specific spot on the map that looks like a horse saddle). This spot is a hotspot for electron activity. In the Arsenic version (BaPtAs), the electrons are further away from this hotspot.

3. The Dance Styles (Pairing Symmetries)

In the world of superconductors, the "dance style" is called pairing symmetry. The paper tests a few possibilities:

• The "Standard" Dance ($s$-wave): Like a simple, round ball. Everyone holds hands in a circle. It's stable but boring. It doesn't create magnetic fields.
• The "Figure-8" Dance ($f$-wave): A more complex shape with some gaps (nodes) where dancers aren't holding hands.
• The "Chiral" Dance ($d$-wave): This is the fancy one. Imagine the dancers spinning in a specific direction (clockwise or counter-clockwise) while holding hands. This creates a "handedness" (chirality) and generates that spontaneous magnetic field the scientists saw.

4. The Discovery: The Recipe Matters

By running their computer simulations, the authors found a clear rule:

• At the Antimony end ($x=1$): Because the electrons are right next to that "saddle point" hotspot, the Chiral $d$-wave dance is the most stable. The electrons want to spin and create that magnetic field. This matches the experimental data perfectly.
• At the Arsenic end ($x=0$): Because the electrons are further from the hotspot, the fancy Chiral dance becomes unstable. The system switches to a standard $s$-wave or a nodal $f$-wave. These dances don't create magnetic fields, which explains why the signal disappeared in the experiment.

The Analogy: The Ice Skater

Imagine an ice skater (the electron) trying to spin.

• In BaPtSb, the ice is perfectly smooth and shaped just right (near the saddle point). The skater can easily spin in a perfect, fast circle (Chiral $d$-wave), creating a whirlwind (magnetic field).
• In BaPtAs, the ice is a bit rougher or the shape is different. The skater tries to spin, but it's too hard. Instead, they settle for a simple glide or a wobbly spin (Standard $s$-wave or $f$-wave). No whirlwind is created.

Why Does This Matter?

This paper solves a mystery. It explains why changing a small amount of the chemical ingredient (swapping Arsenic for Antimony) completely changes the physics of the material.

• For Science: It confirms that these materials are "topological superconductors" (a fancy term for materials with special, robust quantum properties) at one end of the spectrum.
• For the Future: If we can control this "dance switch," we might be able to build quantum computers that are much more stable and less prone to errors. The paper suggests that by squeezing the material (applying pressure) or doping it, we could force the electrons to switch back to the "Chiral" dance even in the Arsenic version.

In short: The authors showed that by tweaking the recipe of this honeycomb material, they can turn a "boring" superconductor into a "magical" one that spins on its own, and they proved exactly why that happens using computer models.

Technical Summary

1. Problem Statement

The paper addresses the unresolved nature of the superconducting pairing symmetry in the family of pnictide superconductors with an ordered honeycomb network, specifically the solid solution BaPtAs$_{1-x}$Sb$_x$.

• Experimental Context: Previous experiments using muon-spin relaxation ($\mu$SR) have detected a spontaneous internal magnetic field below the critical temperature ($T_c$) in BaPtSb ($x=1$), suggesting a time-reversal symmetry-breaking (TRSB) state. Conversely, this signal is suppressed in BaPtAs ($x=0$) and intermediate concentrations.
• Theoretical Controversy: While some theories propose a chiral $d$-wave state (which breaks TRS) for these materials, others suggest conventional $s$-wave or nodal $f$-wave states. The mechanism driving the transition between these states as the Sb-concentration ($x$) changes remains unclear.
• Objective: The authors aim to theoretically determine the most stable pairing symmetry for $x=0$ (BaPtAs) and $x=1$ (BaPtSb) and explain the concentration-dependent symmetry change observed in experiments.

2. Methodology

The study employs a multi-step theoretical approach combining ab initio calculations with effective model analysis:

• First-Principles Calculations (DFT):
  • Used the Quantum ESPRESSO package with plane-wave pseudopotentials and the PBE exchange-correlation functional.
  • Calculated electronic structures for pristine BaPtAs and BaPtSb, including Spin-Orbit Coupling (SOC).
  • Identified that the lack of an inversion center in the unit cell leads to antisymmetric spin-orbit coupling (ASOC), lifting spin degeneracy.
• Effective Tight-Binding Model Construction:
  • Constructed a 16-orbital tight-binding model (8 orbitals per spin) using maximally localized Wannier functions (via Wannier90).
  • Orbitals included: Pt $5d$ ($d_{3z^2-r^2}, d_{xz}, d_{yz}, d_{x^2-y^2}, d_{xy}$) and As/Sb $p$ ($p_z, p_x, p_y$).
  • Validated that the model reproduces the DFT band structures and Fermi surfaces accurately.
• Superconducting Gap Equation Analysis:
  • Formulated the pairing interaction within the point group $D_{3h}$.
  • Considered both spin-singlet and spin-triplet channels, restricting analysis to $S_z=0$ states due to approximate $S_z$ conservation.
  • Solved the zero-temperature gap equation to calculate the condensation energy ($E_{con}$) for various irreducible representations (symmetries).
  • Constructed phase diagrams in the parameter space of coupling constants (intra-band vs. inter-band) and Fermi level shifts.

3. Key Contributions and Results

A. Electronic Structure Differences

• Fermi Surfaces (FS): Both compounds exhibit three types of Fermi surfaces:
  • FS-1 & FS-2: Cylindrical surfaces around the $\Gamma$-A line (quasi-2D).
  • FS-3: Three-dimensional surfaces around the zone boundary (K, K').
• Van Hove Singularity (VHS): A saddle point at the M point lies close to the Fermi level ($\varepsilon_F$) in both materials, enhancing the Density of States (DOS).
  • Crucial Difference: In BaPtSb ($x=1$), the FS-3 extends much closer to the M-point (VHS) than in BaPtAs. In BaPtAs, the FS-3 is smaller and further from the M-L line.

B. Phase Diagrams and Stability

By solving the gap equation, the authors mapped the most stable pairing symmetry based on coupling constants:

• At $x=1$ (BaPtSb): The chiral $d$-wave state ($d_{x^2-y^2} \pm i d_{xy}$) is the most stable over a broad parameter range.
  • Reasoning: The basis function of the chiral $d$-wave has a large amplitude near the M-point. Since the FS-3 in BaPtSb is close to the M-point (VHS), this state gains significant condensation energy.
  • Symmetry: This state breaks Time-Reversal Symmetry (TRSB), consistent with the observed spontaneous magnetic field.
• At $x=0$ (BaPtAs): The chiral $d$-wave is less stable. Instead, states without TRSB compete effectively, specifically:
  • Nodal $f$-wave: Stable when inter-band coupling is weak.
  • Conventional $s$-wave: Becomes dominant when inter-band coupling is strong.
  • Reasoning: The FS-3 in BaPtAs is further from the M-point, reducing the weight of the chiral $d$-wave contribution in the gap equation.

C. Mechanism of Symmetry Change

The paper demonstrates that the transition from a TRSB state ($x=1$) to a non-TRSB state ($x=0$) is driven by the rigid shift of the Fermi level relative to the Van Hove singularity at the M-point.

• As $x$ decreases (from Sb to As), the Fermi level shifts away from the M-point.
• This shift reduces the overlap between the Fermi surface and the high-amplitude region of the chiral $d$-wave basis function, destabilizing the TRSB state in favor of $s$-wave or nodal $f$-wave states.

4. Significance

• Resolution of Controversy: The study provides a theoretical explanation for the experimental $\mu$SR results, confirming that the spontaneous magnetic field in BaPtSb arises from a chiral $d$-wave state, while its absence in BaPtAs is due to a symmetry change to non-TRSB states.
• Role of VHS: It highlights the critical role of the proximity of the Fermi surface to a Van Hove singularity (saddle point) in stabilizing topological superconducting states in non-centrosymmetric crystals.
• Predictive Power: The authors predict that hydrostatic pressure could alter the Fermi surface to bring the VHS closer to the Fermi level, potentially stabilizing the chiral $d$-wave state even in BaPtAs.
• Future Directions: The paper outlines specific experimental probes (Knight shift, spin-lattice relaxation, specific heat, upper critical field anisotropy, and polar Kerr effect) required to definitively confirm the pairing symmetry across the entire concentration range.

In conclusion, the paper successfully links the microscopic electronic structure (specifically the position of the Fermi surface relative to the M-point saddle point) to the macroscopic pairing symmetry, offering a coherent scenario for the symmetry-breaking transition in the BaPtAs$_{1-x}$Sb$_x$ system.