"Half-Bogoliubons" as the intermediate states for the phase coherence in underdoped cuprates

This study reports the observation of "half-Bogoliubons" in underdoped cuprates, identifying them as intermediate excited states arising from local hole pairs whose entanglement and charge exchange establish the phase coherence necessary for superconductivity.

The Gist

Imagine a superconductor as a grand ballroom where electrons are the dancers. In a normal superconductor (like those described by standard physics), the dancers pair up perfectly into "Cooper pairs" and then, in perfect unison, they all start dancing the same synchronized routine. This synchronization is called phase coherence, and it's what allows electricity to flow without any resistance.

In the high-temperature superconductors studied in this paper (a type of material called cuprates), the story is a bit more chaotic. The electrons still want to pair up, but they don't immediately synchronize their dance moves across the whole room. Instead, they form small, local groups that dance together, but these groups are out of sync with their neighbors.

Here is what the researchers discovered, explained through simple analogies:

1. The "Half-Step" Dancers

Usually, when you look at the energy of these electron pairs, you see a perfect mirror image: a "coherence peak" on the positive energy side and an identical one on the negative side. It's like seeing a dancer's reflection in a mirror—perfectly symmetrical.

However, in these under-doped cuprate crystals, the researchers found something strange. In some spots, they only saw the "positive" peak (the dancer moving forward). In other spots, they only saw the "negative" peak (the dancer moving backward). They never saw both at the same time in the same spot.

The authors call these "Half-Bogoliubons." Think of them as dancers who are only showing you half of their routine. One spot shows you the "forward" step, and a nearby spot shows you the "backward" step, but neither shows the full dance alone.

2. The Puzzle Pieces

The magic happens when the researchers took the "forward" step from one spot and the "backward" step from a nearby spot and put them together. Suddenly, they reconstructed the complete, perfect dance routine (the full Bogoliubov dispersion) that you would expect in a normal superconductor.

This suggests that the "half-steps" are actually two halves of the same whole, just separated in space.

3. The "Two-Hole" Neighborhood

To understand why this happens, the authors look at the structure of the material. Imagine the material is made of small, square neighborhoods (called $4a_0 \times 4a_0$ plaquettes).

• The Ground State: In these neighborhoods, there are usually exactly two "holes" (missing electrons, which act like positive charges). These two holes are tightly bound together, like a couple holding hands. This is the local pairing.
• The "Half-Bogoliubon" Event: Sometimes, one of these holes decides to hop out of its neighborhood to visit a neighbor.
  • If a hole leaves a neighborhood, that spot now has only one hole. It becomes easier to pull an electron out of this spot (creating a "negative" peak).
  • If a hole hops into a neighborhood that already had two, that spot now has three holes. It becomes easier to push an electron in (creating a "positive" peak).

These "visiting" holes create the asymmetric "Half-Bogoliubon" signals. They are the intermediate states—the moment of transition where the charge is moving from one local pair to another.

4. How the Dance Becomes Synchronized

The paper argues that this "hopping" is the secret sauce for superconductivity in these materials.

• In standard superconductors, pairing and synchronization happen at the same time.
• In these cuprates, the pairs form first (locally), but they are stuck in their own little neighborhoods.
• To get the whole room dancing in sync (global phase coherence), the holes must dynamically hop between these neighborhoods, exchanging charge.

The "Half-Bogoliubons" are the physical evidence of this hopping process. They are the "glue" that connects the local pairs. When these half-steps entangle and exchange charge freely, the local pairs finally lock into a single, synchronized rhythm, and the material becomes a true superconductor.

Summary

The researchers found that in these specific crystals, the electrons don't just pair up and stay put. Instead, they form local pairs, and then "half-particles" (the Half-Bogoliubons) act as messengers, hopping back and forth between these pairs. This dynamic exchange is what eventually allows the entire material to achieve the perfect synchronization needed for superconductivity. It's a unique process where the "middle step" of the dance is just as important as the final pose.

Technical Summary

Technical Summary: "Half–Bogoliubons" as the Intermediate States for Phase Coherence in Underdoped Cuprates

Problem Statement
In conventional Bardeen-Cooper-Schrieffer (BCS) superconductors, electron pairing and the establishment of macroscopic phase coherence occur simultaneously, resulting in a symmetric Bogoliubov quasiparticle dispersion with coherence peaks at both positive and negative energies ($\pm \Delta$). However, in underdoped cuprate superconductors, pairing and phase coherence are distinct phenomena that do not happen simultaneously. While preformed local pairs (Cooper pairs) are believed to exist in the pseudogap phase, the mechanism by which these local pairs establish global phase coherence remains a central mystery. Specifically, the nature of the intermediate electronic states that facilitate the transition from local pairing to a globally coherent superconducting condensate is not fully understood.

Methodology
The authors investigated underdoped single crystals of the high-temperature superconductor $\text{Bi}_2\text{Sr}_{2-x}\text{La}_x\text{CuO}_{6+\delta}$ (La-Bi2201) with a nominal La doping level of $x=0.75$, corresponding to a hole doping level of $p \approx 0.114$. This doping level places the sample in the underdoped regime, just above the insulator-to-superconductor transition threshold.

The study employed Scanning Tunneling Microscopy and Spectroscopy (STM/STS) under ultra-high vacuum conditions at temperatures around 1.5 K. The researchers performed:

1. Topographic Imaging: To visualize the atomic structure and identify the $4a_0 \times 4a_0$ plaquette structures known to host local pairing.
2. Differential Conductance ($dI/dV$) Mapping: Measured at positive (+15 mV) and negative (-15 mV) biases to map spatial variations in the electronic density of states (DOS).
3. Point Spectroscopy: Acquired tunneling spectra at specific locations to analyze the energy dependence of the DOS, focusing on the coherence peak regions.
4. Environmental Controls: The temperature dependence of the spectra was tested from 0.3 K to 20 K, and magnetic field dependence was tested up to 9 T (perpendicular to the $ab$-plane) to distinguish superconducting features from other electronic states.

Key Results

1. Observation of Asymmetric Spectra: The researchers identified regions within the sample exhibiting extremely particle-hole asymmetric tunneling spectra. Instead of the standard symmetric Bogoliubov coherence peaks, these spectra displayed a sharp "coherence peak" on only one side of the Fermi energy (either at $+11$ meV or $-11$ meV), while the opposite side showed only a weak kink or no peak.
2. Spatial Correlation with Plaquettes: These asymmetric spectra were spatially correlated with the $4a_0 \times 4a_0$ plaquette structures. Specifically, regions with "bright" stripes at positive bias showed peaks at $+11$ meV, while regions bright at negative bias showed peaks at $-11$ meV.
3. Reconstruction of Bogoliubov Dispersion: By merging the positive-bias side of a spectrum showing a $+11$ meV peak with the negative-bias side of a spectrum showing a $-11$ meV peak, the authors reconstructed a complete, symmetric Bogoliubov dispersion with two sharp coherence peaks. This suggests that the two asymmetric spectra represent the two separate branches of the Bogoliubov dispersion that are spatially separated in the underdoped regime.
4. Temperature and Field Dependence: The asymmetric peaks were strongly suppressed by increasing temperature (from 0.3 K to 20 K), indicating they are intrinsic to the superconducting state. However, unlike standard superconducting coherence peaks, these asymmetric peaks were not suppressed by a magnetic field up to 9 T, distinguishing them from the global phase-coherent state.
5. Local Pairing Context: The data supports a model where each $4a_0 \times 4a_0$ plaquette hosts approximately two holes in a local bound state. The asymmetric spectra correspond to excited states where a hole is either added to or removed from this local pair, breaking the local symmetry.

Key Contributions and Proposed Mechanism
The paper introduces the concept of "half-Bogoliubons" to describe these intermediate electronic states. The authors propose the following mechanism for establishing phase coherence in underdoped cuprates:

• Ground State: The system consists of local "two-hole" bound states within $4a_0 \times 4a_0$ plaquettes.
• Intermediate States: The "half-Bogoliubons" represent excited states where a hole hops dynamically between plaquettes.
  • A peak at $+11$ meV corresponds to a state where an extra hole is added to a plaquette (creating a three-hole state), resulting in a purely hole-like excitation.
  • A peak at $-11$ meV corresponds to a state where a hole is removed (leaving a one-hole state), resulting in a purely electron-like excitation.
• Phase Coherence: The entanglement and dynamic hopping of charge freedom between these local paired states (the "half-Bogoliubons") facilitate the establishment of global phase coherence. The authors argue that in cuprates, unlike BCS superconductors, phase coherence is built through the exchange of charge between preformed local pairs, rather than occurring simultaneously with pairing.

Significance
The authors claim that this work unravels a unique process for establishing phase coherence in cuprate superconductors. By identifying "half-Bogoliubons" as the intermediate states for phase coherence, the study provides a microscopic picture of how local pairing evolves into a macroscopic superconducting state. The findings suggest that the superconducting state in underdoped cuprates emerges from the dynamic hopping of holes between local "two-hole" bound states, a process distinct from the simultaneous pairing and coherence seen in conventional BCS theory. This offers a new avenue for understanding the origin of high-temperature superconductivity in the presence of strong correlations and preformed pairs.