Exotic vortex states at high magnetic fields in a quasi-two-dimensional FeSe-based superconductor

Through comprehensive high-field transport measurements up to 33 T, this study reveals that the quasi-two-dimensional FeSe-based superconductor (TBA+)xFeSe exhibits exotic vortex states, including fragile superconductivity and a unique intermediate regime with finite longitudinal but vanishing Hall resistance, driven by the interplay of strong electronic correlations, thermal fluctuations, and high magnetic fields.

The Gist

The Big Picture: A Superconductor That "Fragilely" Holds Together

Imagine a superconductor as a super-highway where electricity travels without any friction or traffic jams. Usually, we think of this highway as being very strong and stable. However, this paper studies a specific material called (TBA+)xFeSe (a type of iron-based superconductor) that behaves like a very fragile highway.

When you put this material in a strong magnetic field (like a giant magnet), the "traffic" (electricity) starts to get messy. The researchers found that this material doesn't just stop working; it enters strange, exotic states that look like a mix between a solid road, a flowing river, and a chaotic crowd.

The Cast of Characters

1. The Superconductor (The Highway): This is the material that lets electricity flow perfectly.
2. Vortices (The Traffic Jams): When you apply a magnetic field to a superconductor, tiny whirlpools of magnetic force called "vortices" poke through the material. Think of these as traffic jams or whirlpools in the river.
   • In a normal superconductor, these whirlpools line up neatly in a grid (like cars in a parking lot).
   • In this material, because the layers are so thin (quasi-2D), these whirlpools are more like pancakes stacked loosely on top of each other.
3. The "Fragile" State: This is the main discovery. The superconducting highway is so weak in this material that even a tiny push (a small electric current) can knock the traffic jams out of place, causing the electricity to lose its perfect flow.

What They Found: Three Strange States

The researchers used very strong magnets (up to 33 Tesla, which is incredibly powerful) and cooled the material down to near absolute zero. They discovered three distinct "moods" or states the material goes through as the magnetic field gets stronger:

1. The "Fragile Superconductor" (The Glassy Ice)

At low temperatures and high magnetic fields, the material acts like a superconductor that is barely holding on.

• The Analogy: Imagine a sheet of ice that is so thin it cracks if you step on it too hard.
• What happened: When they used a tiny electric current, the material acted like a perfect superconductor (zero resistance). But when they increased the current just a little bit, the "ice" cracked, and resistance appeared.
• Why it matters: This is similar to what happens in cuprate superconductors (another family of high-temperature superconductors) where competing electronic orders (like charge density waves) break the superconductor into tiny, isolated islands. The current has to jump between these islands, and if the jump is too hard, the connection breaks.

2. The "Phase-Fluctuating Vortex State" (The Silent River)

As they warmed the material up slightly, the perfect superconductivity melted, but something weird happened.

• The Analogy: Imagine a river that is flowing fast (resistance is present), but if you drop a leaf in it, the leaf doesn't spin or drift sideways (no Hall effect).
• What happened: The material had electrical resistance (it wasn't a perfect superconductor anymore), but it showed zero Hall resistance. In physics, the Hall effect is like a sideways push on moving charges. Usually, if there is resistance, there is a sideways push. Here, the sideways push vanished.
• The Theory: The researchers suggest that the "whirlpools" (vortices) are still pinned down tightly, but the phase of the superconducting wave is fluctuating wildly. It's like a crowd of people trying to march in step; they are all moving forward, but their steps are so out of sync that they cancel out any sideways movement.

3. The "Anomalous Vortex Liquid" (The Chaotic Slush)

At even higher temperatures or fields, the material turned into a standard "vortex liquid."

• The Analogy: The ice has completely melted into a slushy soup. The whirlpools are now floating freely and chaotically.
• What happened: Now, the material showed normal resistance and a normal sideways Hall effect. The "magic" of the zero-Hall state was gone.

The "Why": A Battle for Control

The paper suggests that this strange behavior happens because of a tug-of-war between two things:

1. Superconductivity: The desire for electrons to pair up and flow perfectly.
2. Competing Orders: Other electronic patterns (like Charge Density Waves) that want to organize the electrons differently.

In this material, the magnetic field forces these two enemies to coexist. The researchers propose that the superconductivity gets chopped up into tiny "puddles" surrounded by these competing patterns. The current has to hop from puddle to puddle. Because the connections are weak, the whole system is incredibly sensitive to how hard you push (the current) and how much the atoms are jiggling (temperature).

The "Pancake" Effect

A key feature of this material is that it is extremely "flat" (quasi-2D). The layers of iron and selenium are separated by large organic molecules, making the distance between them huge compared to other superconductors.

• The Analogy: Think of a stack of pancakes with a lot of syrup between them. The magnetic whirlpools (vortices) don't form long, continuous sticks through the stack; they form individual "pancake" vortices on each layer. This makes the material extremely sensitive to heat and magnetic fields, leading to the "fragile" behavior.

Summary

This paper maps out a new, strange map of how electricity behaves in a very thin, iron-based superconductor under strong magnets. They found that instead of just being "on" or "off," the material goes through a fragile state where it barely conducts, and a silent state where it conducts but with no sideways push. These findings suggest that high-temperature superconductors might share a universal "fragile" nature when pushed to their limits, likely due to a battle between different electronic orders.

Note: The paper does not discuss any medical applications, future commercial uses, or clinical uses. It is purely a study of fundamental physics and how these materials behave under extreme conditions.

Technical Summary

Technical Summary: Exotic Vortex States in Quasi-2D FeSe-based Superconductor

Problem Statement
High-temperature superconductors, particularly those with strong electronic correlations, exhibit complex intertwinements between superconductivity and competing electronic orders (e.g., spin/charge density waves). In cuprate superconductors, this competition leads to "fragile superconductivity" at high magnetic fields and low temperatures, characterized by a drastic suppression of the critical current ($J_c$) and critical temperature ($T_c$), alongside the emergence of exotic vortex states beyond classical paradigms. While these phenomena are well-documented in cuprates, their existence and mechanisms in iron-based superconductors remain underexplored. Specifically, there is a need to determine if quasi-two-dimensional (quasi-2D) FeSe-based systems, which possess large thermal fluctuations, host similar fragile superconducting states and non-classical vortex phases under high magnetic fields.

Methodology
The study focuses on the quasi-2D superconductor $(TBA^+)_xFeSe$, where tetrabutylammonium ($TBA^+$) cations are intercalated into pristine FeSe single crystals. This intercalation expands the interlayer spacing to ~1.6 nm, creating a highly anisotropic electronic structure with a resistivity anisotropy ($\rho_c/\rho_{ab}$) of $\sim 10^5$ and raising the zero-resistivity transition temperature ($T_c$) from 9 K to above 40 K.

The researchers performed comprehensive high-field transport measurements up to $\mu_0H \approx 33$ T. Key experimental techniques included:

• Isothermal Magnetoresistance (MR): Measured across a wide temperature range (1.55 K to 60 K) using excitation currents spanning three orders of magnitude ($2 \mu A$ to $1000 \mu A$) to probe current-dependent transport behaviors.
• Simultaneous Hall and Longitudinal Resistance: Measurements of Hall resistance ($R_{xy}$) and longitudinal resistance ($R_{xx}$) to distinguish between different vortex states and phase coherence.
• Phase Diagram Reconstruction: Determination of irreversibility lines ($H_{irr}$), upper critical fields ($H_{c2}$), and characteristic transition fields based on resistance criteria (e.g., 0.01% $R_{60K}$).
• Theoretical Analysis: Fitting of data to Berezinskii–Kosterlitz–Thouless (BKT) transitions, 2D Ginzburg–Landau equations, and power-law relations to characterize vortex glass states and thermal fluctuations (quantified by the Ginzburg number, $Gi$).

Key Contributions and Results

1. Extreme Type II Superconductivity with Massive Thermal Fluctuations:
   The study establishes $(TBA^+)_xFeSe$ as an extreme type II superconductor. The estimated Ginzburg number ($Gi > 320$) is exceptionally large, comparable to high-$T_c$ cuprates like Bi-2212, indicating that thermal fluctuations dominate the superconducting properties. The material exhibits a 3D-to-2D crossover of vortices at low fields ($\sim 0.2$ T), transitioning from a Bragg glass to a regime dominated by 2D pancake vortices. The upper critical field is estimated at $\mu_0H_{c2}^c(0) \approx 75$ T.

2. Observation of Fragile Superconductivity:
   At low temperatures and high magnetic fields, the material exhibits "fragile superconductivity." The zero-resistance state is highly sensitive to excitation currents; modest currents significantly suppress the irreversibility field ($H_{irr}$) and induce resistance. This current-dependent behavior, where the critical current is drastically reduced in high fields, mirrors the fragile superconductivity observed in charge-ordered cuprates. The authors attribute this to the spatial fragmentation of superconductivity by competing electronic orders.

3. Emergence of Exotic Vortex States:
   The study maps a complex $H-T$ phase diagram revealing states beyond the classical vortex matter paradigm:

   • 2D Vortex Glass (VG): At the zero-temperature limit and high fields ($>31$ T), the system enters a 2D VG state characterized by power-law resistance scaling ($R \propto T^\alpha$).
   • Phase-Fluctuating Vortex State (PFVS): An intermediate state emerges between the vortex solid and the vortex liquid. This state is characterized by finite longitudinal resistance ($R_{xx} > 0$) but vanishingly small Hall resistance ($R_{xy} \approx 0$). It persists over a broad range of magnetic fields and is robust against changes in excitation current, suggesting strong global vortex pinning despite the quasi-2D nature.
   • Anomalous Vortex Liquid: At higher temperatures or fields, the system transitions to a state with finite Hall resistance, where thermal fluctuations dominate.

4. Mechanism: Competing Orders and Vortex Halos:
   The authors propose that the exotic states arise from an emergent electronic order (potentially charge density waves or pair density waves) that nucleates within vortex halos. As magnetic field increases, the density of vortices rises, causing these competing orders to overlap and fragment the global superconducting phase. This creates a granular superconducting state where global coherence is mediated by weak Josephson couplings between "superconducting puddles," leading to the observed fragility and the unique zero-Hall resistive state.

Significance
The paper claims that $(TBA^+)_xFeSe$ provides a new platform for studying universal high-field vortex physics in high-$T_c$ superconductors. The observation of fragile superconductivity and phase-fluctuating vortex states in an iron-based superconductor, which closely parallels phenomena in cuprates, suggests that the interplay between superconductivity and competing electronic orders is a universal mechanism governing vortex matter in strongly correlated systems. These findings challenge the classical understanding of vortex melting and suggest that emergent granularity driven by competing orders is a key factor in the loss of phase coherence. The authors conclude that while the transport data strongly supports the existence of these exotic states, further microscopic investigations (e.g., NMR, STS, RIXS) are required to unambiguously identify the nature of the emergent electronic order within the vortex halos.