Fermi surface and topology of multiband superconductor BeAu

This study presents a first-principles analysis of the multiband superconductor BeAu, revealing its complex Fermi surface topology with high-Chern-number Weyl points and surface Fermi arcs, which supports a $\nu=4$ topological superconducting phase and potentially explains its multigap behavior.

The Gist

Imagine a crystal called BeAu (made of Beryllium and Gold) not just as a solid object, but as a bustling, multi-layered city for electrons. This paper is a detailed map of that city, revealing why it's a special place for electricity and why it might hold secrets to a new kind of "super-highway" for energy.

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

1. The City's Architecture: A Twisted Labyrinth

Most crystals are like a standard grid of city blocks—symmetrical and predictable. BeAu is different. It belongs to a family of crystals called B20, which are chiral.

• The Analogy: Imagine a spiral staircase or a corkscrew. If you look at it in a mirror, it doesn't look the same as the original. BeAu has this "handedness" (it's either left-handed or right-handed).
• Why it matters: Because the city is twisted, the rules of the road for electrons are weird. Instead of smooth, straight highways, the electrons encounter strange intersections where roads merge, split, and cross in ways that shouldn't be possible in normal materials.

2. The Traffic Jams: "Multifold Fermions"

In this electron city, there are specific high-traffic intersections (called high-symmetry points) where multiple roads crash into each other at the exact same energy level.

• The Analogy: Think of a standard intersection where two roads cross (a 2-way). In BeAu, you have 4-way, 6-way, or even 8-way intersections happening all at once.
• The "Charge": These aren't just messy intersections; they are charged with "topological energy." The paper found that some of these intersections act like powerful magnets for the flow of electrons, creating a "charge" as high as +6.
• The Record: Finding a single intersection with a charge of +6 is like finding a single bridge that can handle six times the traffic of any other bridge ever built. It's a world record for this type of material.

3. The Ghost Roads: "Weyl Points" and "Fermi Arcs"

Usually, if you have a road that starts at point A and ends at point B, you need a return path. In physics, this is called a "pair." But in BeAu, the twisted architecture allows for unpaired roads.

• The Analogy: Imagine a subway line that starts at a station but never loops back to the start. It just disappears into a "ghost road" that only exists on the surface of the city.
• Fermi Arcs: These are the surface roads. The paper mapped out these "ghost roads" (called Fermi arcs) and found they are incredibly long and complex, spiraling around the surface of the crystal. They are like a rollercoaster track that only exists on the roof of the building, connecting different parts of the city in a way that doesn't happen inside.

4. The Superpower: Becoming a "Superconductor"

BeAu is a superconductor, meaning it can conduct electricity with zero resistance (no friction, no heat loss). But it's a special kind:

• The Multi-Gap Mystery: Usually, a superconductor has one "speed limit" for electrons. BeAu has two different speed limits (gaps). Some electrons zip along one path, while others take a slower, different path.
• The "s±" Pairing: The paper suggests that the electrons on these different paths are dancing to different tunes. One group is dancing to a positive beat, and the other to a negative beat. When these opposite beats meet, they create a special, stable state.

5. The Big Discovery: A Topological Superconductor

The authors asked: What happens if we combine these weird, twisted roads with the superconducting dance?

• The Result: They calculated that BeAu could become a Topological Superconductor.
• The Analogy: Imagine a highway where the cars (electrons) are so well-organized by the road's shape that they can't crash, even if there are potholes. The "topology" (the shape of the road) protects the flow.
• The "ν = 4" and "ν = 6" Numbers: These numbers represent how many "lanes" of this protected super-highway exist. The paper confirms BeAu has a 4-lane protected highway (which was predicted by theory) and hints that it might have a 6-lane highway (which would be a massive discovery, as no one has seen a 6-lane super-highway before).

6. Why is this happening? The "Be" Factor

The paper also looked at why the electrons behave this way.

• The Analogy: The city is made of two types of bricks: heavy Gold bricks and light Beryllium bricks. The light Beryllium bricks are vibrating a lot (like a drum).
• The Discovery: The electrons on the "6-lane highway" are mostly made of the light Beryllium material, while others are made of Gold. Because the Beryllium is vibrating so much, it helps the electrons pair up and become superconducting. This explains why the material has two different "speed limits" (gaps)—the Beryllium-heavy roads are super-fast, while the Gold-heavy roads are different.

Summary: Why Should We Care?

This paper is like finding a new type of terrain in a video game that allows for physics-defying movement.

1. It breaks records: It found the highest "topological charge" (traffic capacity) ever seen on a single surface.
2. It solves a puzzle: It explains why BeAu has two different superconducting gaps (the Beryllium vs. Gold mix).
3. Future Tech: If we can harness this "topological superconductivity," we might build computers that don't lose energy and are immune to errors (quantum computers). BeAu is a blueprint for how to build such a machine.

In short, BeAu is a twisted, chiral crystal where electrons take shortcuts on the surface, dance to opposite beats, and create a super-highway that might be the key to the next generation of technology.

Technical Summary

Here is a detailed technical summary of the paper "Fermi surface and topology of multiband superconductor BeAu."

1. Problem Statement

The paper addresses the complex interplay between superconductivity and topology in the chiral material BeAu (Beryllium-Gold). While BeAu is known as a multiband type-I superconductor with a critical temperature ($T_c$) of 3.2 K, the microscopic origin of its multigap superconductivity remains unclear. Specifically, two key questions are unresolved:

1. Which specific Fermi surface (FS) sheets are involved in the superconducting state?
2. What mechanism drives the different gap magnitudes observed experimentally?

Furthermore, BeAu belongs to the B20 crystal family (space group $P2_13$), which hosts exotic topological features like multifold fermions, unpaired Weyl points, and nodal surfaces. Previous theoretical studies on topological superconductivity in this family relied on simplified effective models that neglected material-specific details. The authors aim to provide a rigorous, first-principles characterization of BeAu's electronic structure to determine if it supports a topological superconducting phase (specifically with a non-trivial topological invariant $\nu$) under $s_{\pm}$ pairing conditions.

2. Methodology

The authors employed a comprehensive first-principles approach combining Density Functional Theory (DFT) with topological analysis:

• DFT Calculations: Full-relativistic calculations were performed using the FPLO code with the Generalized Gradient Approximation (GGA) for exchange-correlation. Spin-orbit coupling (SOC) was included.
• Wannierization: To facilitate surface and topological calculations, Kohn-Sham eigenstates were projected onto a basis of 80 spinful orbitals (Be 2s, 2p and Au 6s, 5d) to generate a Wannier representation.
• Fermi Surface Analysis: A dense $k$-mesh ($24 \times 24 \times 24$) was used to compute the Fermi surface. The authors identified 12 distinct FS sheets (labeled$S_1$to$S_{12}$).
• Topological Charge Calculation: Unlike conventional Weyl semimetals where Chern numbers can be inferred by counting enclosed Weyl points, the presence of unpaired Weyl points and nodal surfaces in BeAu required a direct numerical integration of the Berry curvature over each Fermi surface sheet ($C_i = \frac{1}{2\pi} \oint \Omega_i \cdot d^2k$).
• Surface Spectral Function: Slab calculations were performed to visualize surface states (Fermi arcs) and their connectivity.
• Pressure Dependence: The stability of Weyl points under hydrostatic pressure (up to 3 GPa) was analyzed using experimental lattice parameter curves.

3. Key Contributions and Results

A. Electronic Structure and Topological Crossings

• Multifold Fermions: The study confirms the presence of 4-fold degenerate fermions at the $\Gamma$ point (Chern number $C = -4$) and 6-fold fermions at the $R$ point ($C = +4$), consistent with space group $P2_13$ symmetries.
• Unpaired Weyl Points: The authors identified 81 unpaired Weyl points in the irreducible Brillouin zone. These are compensated by nodal surfaces (nodal walls) pinned at the Brillouin zone boundaries, allowing for isolated Weyl points without surface Fermi arcs connecting them to opposite chirality partners.
• Charge-2 Dirac Point: A double Weyl point (charge-2 Dirac point) was identified at the $M$ point with $C = -2$.
• Highest Reported Chern Number: A significant discovery is a Fermi surface sheet ($S_6$) centered around the $M$ point carrying a Chern number of $C = +6$. This is the highest Chern number reported to date for a single Fermi surface sheet.
• Surface Fermi Arcs: The $C=\pm4$ crossings at $\Gamma$ and $R$ generate extremely long and complex surface Fermi arcs on the (001) termination. The authors observed a dynamic connectivity exchange of these arcs as a function of energy, including the formation of U-shaped structures indicative of type-II van Hove singularities.

B. Topological Superconductivity Implications

Using the calculated Chern numbers ($C_i$) and assuming an $s_{\pm}$ pairing state (where the superconducting gap changes sign between different Fermi surfaces), the authors calculated the topological invariant $\nu$ using the formula:
$$\nu = \frac{1}{2} \sum_{i \in FS} C_i \cdot \text{sgn}(\Delta_i)$$

• $\nu = 4$ Phase: If the gap changes sign between the Fermi surfaces at $\Gamma$ and $R$ (which have $C = \pm 4$), the system realizes a topological superconducting phase with $\nu = 4$. This confirms theoretical predictions made by minimal models but validates them with material-specific first-principles data.
• $\nu = 6$ Possibility: Due to the discovery of the $C=6$ sheet, a topological phase with $\nu = 6$ is theoretically possible if the gap sign changes between the $\Gamma$ and $M$ pockets. While the authors note this is less likely due to the small energy separation between these pockets, it represents a unique possibility for richer topological phases.

C. Origin of Multigap Superconductivity

• Orbital Inhomogeneity: The analysis of atomic-resolved band structures reveals a strong inhomogeneity in orbital character at the Fermi level. Specifically, the Fermi pockets around the $M$ point ($S_5, S_6$) are dominated by Be character, whereas pockets near $\Gamma$ and $R$ show significant Au hybridization.
• Mechanism: Since superconductivity in BeAu is driven by electron-phonon coupling involving light Be atoms, this spatial separation of Be-character suggests that the electron-phonon interaction strength varies significantly across different Fermi sheets. This anisotropy provides a plausible explanation for the experimentally observed multigap superconductivity (strongly coupled vs. weakly coupled gaps).

4. Significance

• Material-Specific Validation: This work bridges the gap between abstract effective models and real materials, proving that the complex topology of B20 compounds (specifically BeAu) supports high-order topological superconductivity.
• Record-Breaking Topology: The identification of a Fermi surface with $C=6$ sets a new benchmark for topological charges in condensed matter systems, suggesting that B20 compounds are a fertile ground for discovering exotic quantum phases.
• Multigap Mechanism: The link between orbital composition (Be-rich vs. Au-rich) and gap magnitude offers a concrete theoretical framework for understanding multiband superconductivity in non-centrosymmetric materials.
• Experimental Guidance: The detailed mapping of Weyl points, nodal surfaces, and Fermi arc connectivity provides specific targets for future Angle-Resolved Photoemission Spectroscopy (ARPES) and scanning tunneling microscopy (STM) experiments to verify the predicted topological superconducting state.

In conclusion, the paper establishes BeAu as a prime candidate for a $\nu=4$ topological superconductor and highlights the unique role of orbital-selective electron-phonon coupling in driving its multigap behavior, all underpinned by a highly complex and rich topological band structure.