BCS-BEC crossover driven by small Fermi pockets of a high-Tc cuprate superconductor

The Gist

The Big Mystery: The "Ghost" vs. The "Puddle"

Imagine you are looking at a crowded dance floor (the material) where electrons are the dancers. In high-temperature superconductors (materials that conduct electricity with zero resistance at high heat), scientists have been arguing for decades about what the dance floor looks like.

• The Old Theory (The Large Floor): They thought the dancers were spread out in one giant, continuous circle.
• The New Theory (The Small Puddle): Others thought the dancers were stuck in tiny, isolated puddles.

The problem is that the "dance floor" in these materials is weird. It looks like a broken circle (called a "Fermi arc"). It's hard to tell if that broken piece is just a fragment of a giant circle or a tiny, complete puddle on its own. This confusion made it impossible to understand how the electrons pair up to become superconductors.

The Solution: A Clean Room in a Messy House

Most of these materials are like messy houses. The "dopants" (chemicals added to make them work) are scattered randomly, creating disorder. This mess makes it hard to see the true nature of the electrons.

The researchers in this paper found a special type of material: a four-layer cuprate (specifically Ba2Ca3Cu4O8(F,O)2).

Think of this material as a four-story apartment building.

• The outer floors are messy, right next to the noisy construction zone (the dopants).
• The inner floors are tucked away in the middle. They are shielded from the noise and mess.

By focusing their microscope (a technique called ARPES) only on the inner floors, the researchers found a "clean room." Here, the electrons behave exactly as theory predicts, without the noise of disorder.

The Discovery: Tiny Puddles with Giant Energy

In this clean inner room, the researchers found two surprising things happening at the same time:

1. Small Fermi Pockets: The electrons are indeed stuck in tiny, isolated puddles (small Fermi pockets), not a giant circle.
2. Huge Superconducting Gap: Usually, when electrons are in a tiny puddle with very few of them, they pair up weakly. But here, the pairing is massive.

The Analogy: Imagine a tiny campfire (the small puddle). Usually, a small fire has weak heat. But in this experiment, the tiny campfire is burning as hot as a massive bonfire. The energy holding the electron pairs together is incredibly strong, reaching the theoretical maximum limit for this type of material.

The Twist: More Dancers, Stronger Fire

There is a second surprise. In most physics theories, if you want to move from "weak pairing" to "strong pairing" (a transition called the BCS-BEC crossover), you usually have to remove dancers (reduce the number of electrons).

However, in this experiment, the researchers found the opposite. As they added just a tiny bit more doping (increasing the number of electrons by less than 1%), the system suddenly jumped from a standard state to this extreme, strong-pairing state.

The Analogy: It's like a crowded elevator. Usually, adding more people makes it chaotic. But here, adding just one extra person caused the elevator to instantly transform into a perfectly synchronized dance troupe. This switch happened so fast it was like flipping a light switch.

The Coexistence: Enemies Becoming Partners

Another major finding involves Antiferromagnetism (AF). This is a magnetic state where electrons want to stand still and face opposite directions (like soldiers in a rigid formation). Usually, this "rigid formation" kills superconductivity (the dancing).

In this clean inner layer, the rigid soldiers (AF order) and the dancing pairs (superconductivity) are living in the same room. Instead of fighting, they seem to be helping each other. The rigid formation actually helps the tiny puddles form, and the superconductivity is stronger than in the messy outer layers.

Why This Matters

This paper solves a long-standing puzzle:

1. It proves that small pockets of electrons can exist in these materials.
2. It proves that these small pockets can host extremely strong superconductivity.
3. It shows that this happens in a clean environment (the inner layers), suggesting that the "messiness" of other materials was hiding the true potential of high-temperature superconductors.

In short, the researchers found a hidden, clean layer in a complex material where electrons form tiny, super-strong pairs, offering a new blueprint for understanding how high-temperature superconductivity works.

Technical Summary

Technical Summary: BCS-BEC Crossover Driven by Small Fermi Pockets in High-$T_c$ Cuprates

Problem Statement
The electronic nature of underdoped high-$T_c$ cuprates remains a fundamental unresolved issue, specifically regarding the distinction between a large Fermi surface (carrier density $1+p$) and small Fermi pockets (carrier density $p$). This ambiguity stems from the observation of "Fermi arcs" in angle-resolved photoemission spectroscopy (ARPES), which could represent fragments of either a large surface or small pockets. This uncertainty critically impacts the determination of the Fermi energy ($\varepsilon_F$) and, consequently, the pairing strength ($\Delta/\varepsilon_F$). If the arc is part of a large surface, $\varepsilon_F$ is high ($\sim 1$ eV), suggesting weak-coupling Bardeen-Cooper-Schrieffer (BCS) physics. If it is part of a small pocket, $\varepsilon_F$ is low, potentially placing the system in the strongly coupled Bose-Einstein condensation (BEC) crossover regime.

Furthermore, prior investigations into this regime have been hindered by disorder. In single- and double-layer cuprates, the proximity of dopant layers introduces random potentials and lattice distortions, creating inhomogeneous electronic states that obscure intrinsic physics. While multilayer cuprates ($n \ge 3$) offer "clean" inner CuO$_2$ planes (IPs) shielded from dopant disorder, previous studies on five- and six-layer compounds found small Fermi pockets with very small superconducting gaps ($\sim 5$ meV), potentially due to AF competition or doping levels near the edge of the superconducting dome. The challenge remained to identify a system where small Fermi pockets coexist with a large superconducting gap to definitively probe the BCS-BEC crossover.

Methodology
The authors investigated the four-layer cuprate Ba$_2$Ca$_3$Cu$_4$O$_8$(F,O)$_2$ (F0234), which contains a single set of double inner planes per unit cell. This structure allows for higher doping levels in the inner layers compared to higher-$n$ compounds. Three underdoped single crystals with critical temperatures ($T_c$) of 71 K, 74 K, and 78 K were synthesized under high pressure (4.5 GPa) and characterized via magnetic susceptibility.

The study employed a multi-modal experimental approach:

1. Angle-Resolved Photoemission Spectroscopy (ARPES): Utilizing both laser-based (6.994 eV) and synchrotron-based (55 eV) sources, the team performed band-selective measurements. By exploiting matrix element effects with specific light polarizations, they isolated the spectral signals of the inner planes (IPs) from the outer planes (OPs).
2. Quantum Oscillations: De Haas-van Alphen (dHvA) oscillations were measured via torque magnetometry in pulsed magnetic fields up to 60 T. This provided independent confirmation of the Fermi surface topology and carrier density.
3. Temperature-Dependent Spectroscopy: Energy distribution curves (EDCs) were analyzed across a wide temperature range to distinguish between the superconducting gap, the pairing pseudogap, and the competing pseudogap.

Key Results

1. Coexistence of Small Pockets and Large Gaps: The study confirmed the existence of small Fermi pockets in the inner CuO$_2$ planes of F0234, with a carrier density of $p \approx 5.5\%$. Crucially, these pockets host a remarkably large superconducting gap. At the pocket tip, the gap ($\Delta_{pocket}$) reaches $\sim 30$ meV in the 78 K sample, extrapolating to an antinodal gap ($\Delta_0$) of $\sim 60$ meV. This is approximately twice the gap magnitude of the outer planes and comparable to the highest $T_c$ cuprates (e.g., HgBa$_2$Ca$_2$Cu$_3$O$_{8+\delta}$).
2. Strong Pairing Strength: The small Fermi energy ($\varepsilon_F \approx 50$ meV) combined with the large gap yields a pairing strength ratio $\Delta_{pocket}/\varepsilon_F \approx 0.6$. This value approaches the theoretical upper bound for two-dimensional superconductivity and significantly exceeds that of conventional BCS superconductors and other cuprate compounds.
3. BCS-BEC Crossover Signatures:
   • Pseudogap: A pairing pseudogap (indicative of preformed pairs) persists well above $T_c$, with a closing temperature ($T_{pair}$) significantly higher than $T_c$.
   • Bogoliubov Flat Band: The ARPES data reveals a Bogoliubov flat band spanning from $+k_F$ to $-k_F$. This feature, indicative of spatially localized pair states where the pair size $\xi$ is comparable to the inter-pair distance $d_p$, is a hallmark of the BCS-BEC crossover.
4. Unconventional Doping Dependence: The transition toward the BEC regime occurs abruptly as the carrier density increases (within a narrow window of $\sim 0.6\%$), contrary to the conventional expectation that the crossover occurs upon decreasing carrier density.
5. Role of Antiferromagnetism: Despite the coexistence of static antiferromagnetic (AF) order in the inner planes, the superconducting gap is enhanced rather than suppressed. The AF order appears to stabilize the pocket formation without competing destructively with superconductivity in this clean environment.

Significance and Claims
The paper claims to provide a long-sought microscopic foundation for the $d$-wave pairing mechanism in doped AF-Mott insulators. By utilizing the nearly disorder-free inner layers of a four-layer cuprate, the authors demonstrate that:

• High-$T_c$ superconductivity does not strictly rely on antinodal electronic states; a large gap and strong pairing can emerge from small Fermi pockets.
• The BCS-BEC crossover is accessible in cuprates, characterized by a dimensionless pairing strength ($\Delta/\varepsilon_F$) reaching theoretical limits.
• The crossover can be driven by increasing carrier density in the presence of AF order, a behavior distinct from other known systems.
• The clean inner planes of multilayer cuprates serve as a unique benchmark system, bridging the gap between theoretical models of strongly correlated systems and experimental reality, free from the disorder that plagues single- and double-layer analogs.

The authors conclude that these findings shed light on the fundamental nature of high-$T_c$ superconductivity, suggesting that the interplay between magnetic order and kinetic energy in clean CuO$_2$ planes creates a regime where minute carrier tuning can switch the system between BCS-like and BEC-like behaviors.