Unconventional magnetization in the multiphase superconductor PdBi$_2$

This study reports the observation of highly anomalous, linear, and anisotropic magnetization in the layered superconductor $\beta$-PdBi$_2$ above a critical field, attributing these unconventional signatures to a field-induced transition from an s-wave state to a nodal p-wave phase characterized by a spatial separation into superconducting and normal domains.

The Gist

The Big Picture: A Superconductor That Changes Its Mind

Imagine a material called PdBi2 (a mix of Palladium and Bismuth). At very cold temperatures, this material becomes a superconductor. In the world of physics, a superconductor is like a "magic shield" that completely repels magnetic fields, pushing them away so they can't get inside. This is usually called the Meissner effect.

Normally, if you push a magnet hard enough against a superconductor, the magnetic field eventually breaks through in tiny, organized tubes called vortices (think of them like tiny tornadoes of magnetism). In most materials, these tornadoes get stuck on impurities, creating a "sticky" situation where the material's magnetic response changes depending on whether you are increasing or decreasing the magnetic field. This is called hysteresis.

However, the researchers in this paper found that PdBi2 does something very strange and unexpected when you apply a magnetic field parallel to its flat surface.

The Strange Behavior: The "Perfectly Smooth" Slide

In a normal superconductor, as you increase the magnetic field, the material fights back, then gets "stuck," and the magnetic response is messy and unpredictable (hysteretic).

But in PdBi2, once the magnetic field reaches a certain critical point (about 0.3 times the maximum field the material can handle), the behavior changes completely:

1. It becomes perfectly smooth: The magnetic response becomes a straight, linear line.
2. It becomes perfectly reversible: If you increase the field and then decrease it, the material follows the exact same path back. There is no "stickiness" or memory of where it was before.
3. It loses its shield: The material stops repelling the magnetic field as strongly as it should. It only blocks about 15–25% of the field instead of 100%.

The Analogy:
Imagine a crowd of people (the electrons) holding hands to form a solid wall against a wind (the magnetic field).

• Normal Superconductor: As the wind gets stronger, the people get tired, some let go, and the wall gets wobbly. If you stop the wind and start again, the people are in different positions, so the wall looks different.
• PdBi2 (The Anomaly): Suddenly, at a specific wind speed, the crowd doesn't just get wobbly; they split into two distinct groups. One group keeps holding hands (superconducting), but the other group lets go completely and stands still (normal). Because they are separated into neat, distinct zones, the wind flows through the "let go" zones perfectly smoothly, and the "holding hands" zones react in a predictable, linear way. There is no chaos or sticking.

The Discovery: A "Phase Separation"

The researchers propose that this strange behavior happens because the material undergoes a phase transition.

1. Low Field (s-wave): At low magnetic fields, the material is in a standard superconducting state (called s-wave).
2. High Field (p-wave): When the field gets strong enough (above a point they call H*), the material switches to a different, more exotic state called nodal p-wave.

The key finding is that these two states don't just mix together like milk in coffee. Instead, they separate into distinct domains, like oil and water.

• Some parts of the crystal become normal metal (letting the magnetic field in).
• Other parts remain superconducting (blocking the field).

This creates a patchwork quilt inside the crystal. The magnetic field penetrates the "normal" patches, while the "superconducting" patches try to shield the rest. This separation explains why the magnetic response is so linear and reversible: the field isn't fighting a messy, sticky vortex lattice; it's simply filling up the "normal" patches in a very orderly fashion.

The "One-Way" vs. "Two-Way" Street

The paper highlights a fascinating difference depending on which direction the magnetic field is applied:

• Field applied perpendicular (straight down): The material behaves like a normal superconductor. The magnetic field creates the usual "tornadoes" (vortices) that get stuck, causing the messy, sticky behavior we expect.
• Field applied parallel (flat along the surface): The material acts like the "patchwork quilt" described above. The magnetic field creates large, flat islands of normal metal and superconducting metal.

The Analogy:
Think of the crystal as a multi-story building.

• If you push a magnet down through the floors (perpendicular), the magnetic "wind" gets caught on the stairs and railings (vortices), creating a chaotic, sticky mess.
• If you push the magnet sideways along the floors (parallel), the building suddenly reorganizes. Some rooms become empty (normal), and some stay furnished (superconducting). The wind flows through the empty rooms perfectly smoothly, while the furnished rooms stay put. The result is a very clean, predictable flow.

Why This Matters (According to the Paper)

The researchers are not claiming this will lead to new medical devices or faster computers right now. Instead, their goal is to understand the rules of the game.

• They have identified a new "signature" or fingerprint of unconventional superconductivity.
• They showed that this material can switch between different types of superconductivity (s-wave to p-wave) just by changing the magnetic field.
• They proved that this switch creates a spatial separation of phases (domains), which is a rare and specific phenomenon in physics.

In short, they found a material that, under the right conditions, stops acting like a messy, sticky superconductor and starts acting like a perfectly organized, split-personality system. This helps scientists understand how exotic superconductors behave, which is a crucial step in the broader quest to understand quantum materials.

Technical Summary

Technical Summary: Unconventional Magnetization in the Multiphase Superconductor PdBi₂

Problem Statement
Unconventional superconductors are typically characterized by specific magnetic signatures, such as intrinsic magnetization at interfaces, fractional vortices, or field-induced phase transitions. However, identifying these signatures through magnetization measurements is often obscured by dominant vortex pinning and flux trapping effects. The material $\beta$-PdBi$_2$ is a candidate for topological superconductivity due to strong spin-orbit coupling and "hidden" inversion symmetry breaking. Previous studies suggested a magnetic field-induced transition from conventional $s$-wave to nodal $p$-wave superconductivity in high in-plane fields, yet the macroscopic magnetization response of bulk $\beta$-PdBi$_2$ to magnetic fields remained unreported. The central problem addressed is the characterization of the magnetization and surface susceptibility of $\beta$-PdBi$_2$ to determine if and how this field-induced phase transition manifests in bulk magnetic properties.

Methodology
The authors investigated bulk single crystals of $\beta$-PdBi$_2$ grown via a melt-growth method and exfoliated into platelets with thicknesses ranging from 15 to 140 $\mu$m. To ensure surface integrity, all handling was performed in an argon-filled glovebox. Measurements were conducted using a commercial SQUID magnetometer (MPMS XL7) at temperatures between 1.8 K and 4.5 K.

The study employed two primary measurement techniques:

1. DC Magnetization ($M(H)$): Measured for magnetic fields applied both parallel ($H \parallel ab$) and perpendicular ($H \parallel c$) to the crystal's $ab$-plane. Both zero-field-cooled (ZFC) and field-cooled (FC) protocols were utilized.
2. AC Susceptibility ($\chi$): Measured to probe the surface screening currents. Both in-phase ($\chi'$) and out-of-phase ($\chi''$) components were recorded as a function of DC field and temperature, with varying AC excitation amplitudes ($h_{ac}$) and frequencies.
3. Transport Measurements: Complementary resistance and critical current ($I_c$) measurements were performed on thinner ($\sim$10 $\mu$m) crystals patterned into Hall-bar-like devices to map the phase diagram and verify the persistence of superconductivity.

Key Results
The study reveals a highly anomalous magnetic response in $\beta$-PdBi$_2$, particularly for fields applied parallel to the $ab$-plane ($H \parallel ab$):

• Linear and Reversible DC Magnetization: While low-field behavior is consistent with a Type-II superconductor, above a critical field $H^* \approx 0.3 H_{c2}$, the DC magnetization becomes strictly linear and fully reversible (non-hysteretic) up to the upper critical field $H_{c2}$. This contrasts with the typical logarithmic dependence and large hysteresis expected from vortex pinning in Type-II superconductors.
• Sharp Transition at $H_{c2}$: The transition to the normal state at $H_{c2}$ is marked by a sharp kink where the linear magnetization crosses zero, rather than the smooth approach to zero typical of Type-II materials.
• Anisotropic AC Susceptibility:
  • For $H \parallel c$, the AC susceptibility behaves conventionally, with full diamagnetic screening ($\chi' \approx -1/4\pi$) maintained up to $H_{c2}$.
  • For $H \parallel ab$, full screening is lost abruptly at $H^*$. Above this field, $\chi'$ drops to a plateau representing only 15–25% of the ideal diamagnetic response, accompanied by a peak in the dissipative component $\chi''$. This suppression is frequency-independent but amplitude-dependent.
• Finite Critical Currents: Despite the reversible DC magnetization (which usually implies zero critical current in a vortex picture), transport measurements show that critical currents remain finite throughout the entire field range up to $H_{c2}$ for both field orientations.
• Phase Diagram: A new critical field $H^*(T)$ is identified, separating a low-field region from a high-field region. A weak but distinct kink in the $H_{c2}(T)$ curve is observed near $H^*$, consistent with a phase transition.

Proposed Mechanism
The authors propose that the observed features are not due to vortex melting (which is ruled out by the low Ginzburg number and lack of temperature/frequency dependence) but rather a spatial phase separation into superconducting and normal domains.

Specifically, the transition at $H^*$ is interpreted as a shift from a conventional $s$-wave state to a nodal $p$-wave state. In the bulk crystals, this transition drives the formation of a domain structure where $p$-wave superconducting regions coexist with normal regions. This "intermediate-like" state explains:

• The linear, reversible DC magnetization (reminiscent of the intermediate state in Type-I superconductors).
• The drastic reduction in AC screening for in-plane fields (as normal domains disrupt the surface screening currents required for in-plane fields).
• The persistence of finite critical currents (due to the remaining superconducting volume).

The authors note that for $H \parallel c$, the observation of similar linear magnetization is likely an artifact of the crystal's large aspect ratio, where demagnetization effects create a significant in-plane field component that exceeds $H^*$, inducing the same domain structure.

Significance
The paper claims to identify new experimental signatures of unconventional multiphase superconductivity. By correlating the anomalous magnetization and susceptibility with the previously observed $s$-wave to nodal $p$-wave transition, the work offers a macroscopic view of how field-induced phase transitions manifest in bulk materials. The findings suggest that $\beta$-PdBi$_2$ exhibits a unique state where spatial phase separation occurs between superconducting and normal domains, distinct from the coexistence of different pairing channels within a uniform volume (as seen in MgB$_2$). This provides insight into the magnetic field response of nodal states and the complex interplay between topology and superconductivity in centrosymmetric materials with hidden symmetry breaking. The authors emphasize that while this domain model explains the data, a full theoretical understanding of the vortex evolution and domain boundaries in this system remains an open challenge.