Unveiling the Superconducting Ground State of Heusler alloy Pd2ZrIn via muon spin relaxation and rotation measurement

This study establishes Pd2ZrIn as a weakly coupled, dirty-limit, type-II superconductor with a fully gapped, nodeless s-wave order parameter and preserved time-reversal symmetry, based on comprehensive muon spin relaxation and rotation measurements alongside electrical and magnetic characterization.

The Gist

Imagine a world made of tiny, dancing particles called electrons. In most metals, these electrons zip around chaotically, bumping into each other and the atoms of the metal, creating resistance (like friction). But in a special state called superconductivity, these electrons pair up and dance in perfect unison, gliding without any friction at all. This allows electricity to flow forever without losing energy.

This paper is a detective story about a specific metal alloy called Pd2ZrIn (a mix of Palladium, Zirconium, and Indium). The scientists wanted to know: Is this metal a "clean" dancer, or is it a "messy" dancer that still manages to perform perfectly?

Here is the story of their investigation, broken down into simple concepts:

1. The Messy Ballroom (The Disorder)

Usually, for electrons to dance perfectly, the "ballroom" (the crystal structure of the metal) needs to be perfectly organized. But Pd2ZrIn is a bit of a mess. It belongs to a family of metals called Heusler alloys, which are known for having a "B2-type antisite disorder."

• The Analogy: Imagine a ballroom where the seats are arranged in a perfect grid. In a perfect metal, every person sits in their assigned seat. In Pd2ZrIn, about half the time, the Zirconium guests and Indium guests accidentally swap seats. They are sitting in the wrong chairs!
• The Result: This creates a lot of "noise" and obstacles. In physics terms, this is called the "dirty limit." Usually, scientists worry that if a ballroom is too messy, the dancers (electrons) will trip and the superconductivity will break.

2. The Big Reveal: It's Still Dancing!

Despite the messy seating arrangement, the scientists found that Pd2ZrIn still becomes a superconductor when cooled down to about 2.2 Kelvin (colder than outer space!).

• The Discovery: They measured how electricity flowed and how the metal reacted to magnets. They saw a sharp drop in resistance and a strong repulsion of magnetic fields (the Meissner effect). This confirmed that the whole chunk of metal was superconducting, not just a tiny part of it.
• The Type: It's a Type-II superconductor. Think of this like a sponge. A "Type-I" superconductor is like a solid wall that pushes all water (magnetism) away. A "Type-II" superconductor is like a sponge that lets some water in through tiny channels (called vortices) while still keeping the rest dry. Pd2ZrIn is this sponge-like material.

3. The Microscope: Muon Spin Relaxation (μSR)

To see how the electrons were dancing, the scientists used a super-powerful microscope called muon spin relaxation (μSR).

• The Analogy: Imagine sending tiny, invisible spies (muons) into the metal. These spies have a magnetic needle (spin) that points in a specific direction. If there are any hidden magnetic fields inside the metal (which would mean the electrons are dancing in a weird, broken way), the spies' needles would wiggle or spin out of control.
• The Finding: The spies reported back: "Everything is calm." There were no hidden magnetic fields. The electrons were not breaking any fundamental rules (specifically, Time-Reversal Symmetry was preserved). This means the dance is "conventional" and orderly, even though the ballroom is messy.

4. The Shape of the Dance (The Gap)

In superconductors, the energy required to break the electron pairs is called the "superconducting gap."

• The Question: Is this gap a perfect sphere (like a smooth ball), or does it have holes in it (like a donut)?
• The Finding: The data showed the gap is a perfect, smooth sphere. There are no holes (nodes). This is called an s-wave state.
• The "Dirty" Twist: Because the ballroom was so messy (the Zr and In swapped seats), the electrons were bumping into things constantly. This forced the superconductivity to behave in a specific way called the "dirty-limit s-wave." It's like a dancer who has to perform a perfect routine while running through a crowd of people, yet they still manage to keep the rhythm perfectly.

5. The Final Verdict: A Conventional Hero

The scientists compared Pd2ZrIn to a famous map called the Uemura Plot, which separates "ordinary" superconductors from "exotic" ones.

• The Result: Pd2ZrIn landed firmly in the "Ordinary" camp. It follows the classic rules of physics (BCS theory) perfectly.
• The Takeaway: This is a big deal because it proves that disorder doesn't always ruin superconductivity. Even with a messy atomic structure, if the underlying rules are right, the electrons can still pair up and dance perfectly.

Summary in One Sentence

Pd2ZrIn is a superconductor that manages to perform a flawless, friction-free dance despite living in a chaotic, messy ballroom, proving that even in disorder, perfect order can emerge.

Technical Summary

1. Problem Statement

The interplay between crystallographic disorder and superconducting pairing is a fundamental challenge in condensed matter physics. While Anderson's theorem suggests that weak non-magnetic disorder does not affect isotropic $s$-wave superconductivity, strong disorder can alter the electronic density of states (DOS), enhance quasiparticle scattering, and potentially drive systems into unconventional pairing states or the "dirty-limit" regime.
Full Heusler alloys ($X_2YZ$) are ideal candidates for studying this interplay because they often exhibit intrinsic structural disorder (specifically B2-type antisite disorder) alongside superconductivity. While alloys like $Ni_2NbAl$ and $YPd_2Sn$ show conventional behavior, others like $LuPd_2Sn$ exhibit signs of unconventional or multi-band superconductivity.
The specific gap: The superconducting ground state of the Heusler alloy $Pd_2ZrIn$ remains unresolved. Previous reports suggested superconductivity at low temperatures, but the nature of the superconducting gap (nodeless vs. nodal), pairing symmetry, and the impact of its significant intrinsic antisite disorder on time-reversal symmetry (TRS) had not been examined at the microscopic level.

2. Methodology

The study employed a comprehensive multi-technique approach combining bulk thermodynamic measurements with microscopic probes:

• Synthesis & Structural Analysis: Polycrystalline $Pd_2ZrIn$ was synthesized via arc melting. Structural characterization was performed using X-ray Diffraction (XRD) with Rietveld refinement to quantify the degree of B2-type antisite disorder between Zr and In sites.
• Bulk Transport & Thermodynamics:
  • Electrical Resistivity ($\rho$): Measured as a function of temperature and magnetic field to determine the critical temperature ($T_C$), residual resistivity ratio (RRR), and upper critical field ($H_{C2}$).
  • Magnetization (DC Susceptibility): Zero-field cooled (ZFC) and field-cooled (FC) measurements to confirm the Meissner effect and type-II behavior.
  • Specific Heat ($C_p$): Measured to determine the Sommerfeld coefficient ($\gamma$), Debye temperature ($\theta_D$), and the magnitude of the specific heat jump at $T_C$.
• Muon Spin Relaxation/Rotation ($\mu$SR): This was the core microscopic technique used:
  • Zero-Field (ZF) $\mu$SR: Performed to detect spontaneous internal magnetic fields, which would indicate Time-Reversal Symmetry Breaking (TRSB).
  • Transverse-Field (TF) $\mu$SR: Performed to probe the magnetic field distribution within the vortex lattice, allowing for the extraction of the magnetic penetration depth ($\lambda$) and superfluid density ($\rho_s$) to determine the superconducting gap structure.

3. Key Results

Structural and Bulk Properties

• Disorder: XRD confirmed a cubic $L2_1$ structure with significant B2-type antisite disorder (approx. 50% disorder between Zr and In sites).
• Superconductivity: The alloy exhibits bulk type-II superconductivity with a transition temperature $T_C \approx 2.2$ K.
• Disorder Regime: The Residual Resistivity Ratio (RRR) is low ($\sim 1.28$), confirming the material is in the dirty-limit regime where the electron mean free path is shorter than the coherence length.
• Carrier Type: Hall effect measurements indicate holes are the dominant charge carriers with a high concentration ($n \approx 6.8 \times 10^{27} m^{-3}$).
• Thermodynamics: Specific heat analysis yielded a Sommerfeld coefficient $\gamma \approx 9.83$ mJ/mol-K$^2$ and a Debye temperature $\theta_D \approx 221$ K. The electron-phonon coupling constant was estimated as $\lambda_{e-ph} \approx 0.56$, indicating weak coupling. The specific heat jump ratio $\Delta C_{el}/\gamma T_C \approx 0.19$ is suppressed compared to the BCS limit (1.43), attributed to disorder-induced broadening.

Microscopic $\mu$SR Findings

• Time-Reversal Symmetry (ZF-$\mu$SR): No spontaneous internal magnetic fields were detected below $T_C$. The relaxation rate remained temperature-independent across the transition. This confirms the preservation of Time-Reversal Symmetry, ruling out TRSB mechanisms often associated with unconventional superconductivity.
• Vortex State & Gap Structure (TF-$\mu$SR):
  • A well-defined flux-line lattice (vortex state) was observed, confirming type-II superconductivity.
  • The temperature dependence of the superfluid density ($\sigma_{sc}$) was analyzed. The data fits the dirty-limit $s$-wave model significantly better than anisotropic or $d$-wave models (lowest $\chi^2$).
  • The superconducting gap was determined to be fully gapped and nodeless with a magnitude $\Delta(0) \approx 0.33 \pm 0.01$ meV.
  • The normalized gap ratio $\Delta(0)/k_B T_C \approx 1.73$ is consistent with weak-coupling BCS theory.

Electronic Classification

• Uemura Plot: The ratio of critical temperature to Fermi temperature ($T_C/T_F$) was calculated to be $\sim 4.2 \times 10^{-4}$. This value falls well below the range typical for unconventional superconductors ($0.01 - 0.1$), placing $Pd_2ZrIn$ firmly in the conventional BCS superconducting regime.

4. Key Contributions

1. Microscopic Confirmation of Conventional State: The study provides the first microscopic evidence (via $\mu$SR) that $Pd_2ZrIn$ is a conventional, nodeless $s$-wave superconductor, despite its high level of structural disorder.
2. Role of Disorder: It demonstrates that significant B2-type antisite disorder in Heusler alloys does not necessarily drive the system toward unconventional pairing or TRSB. Instead, the system adheres to Anderson's theorem, where disorder renormalizes parameters (like the gap and coherence length) but preserves the underlying $s$-wave symmetry.
3. Resolution of Previous Ambiguities: The work clarifies the nature of the superconducting state, distinguishing it from the double-transition behavior seen in annealed samples and confirming bulk superconductivity in the as-synthesized disordered state.

5. Significance

This research is significant for the broader field of superconductivity in complex intermetallics:

• It establishes $Pd_2ZrIn$ as a prototype for disorder-influenced but electronically conventional superconductors within the Heusler family.
• It highlights that disorder alone is insufficient to induce unconventional superconductivity; the underlying electronic structure and symmetry of the parent compound play a more decisive role.
• The findings reinforce the validity of the BCS framework even in the "dirty limit" for Heusler alloys, providing a baseline for understanding more complex, potentially unconventional Heusler systems where disorder and electronic correlations compete.