Tunable Dual-Type Weyl Points in Dirac-Weyl Semimetal CaAgBi

Based on first-principles calculations, this study identifies CaAgBi as a tunable Dirac-Weyl semimetal hosting distinct type-I and type-II Weyl points that can be manipulated through alloy engineering and strain to control their positions and annihilation for topological spintronic applications.

The Gist

Imagine a city built on a grid where the roads represent the paths electrons can take. In most materials, these roads are like a flat, boring highway system. But in a special class of materials called topological semimetals, the roads twist and turn in strange, magical ways. Some roads cross at a single point (like a 4-way intersection), while others cross in a way that creates a "one-way" traffic flow that can't be stopped.

This paper introduces a new material, CaAgBi (a mix of Calcium, Silver, and Bismuth), which acts like a unique traffic hub where two different types of these magical intersections exist at the same time.

Here is a simple breakdown of what the researchers found:

1. The Two Types of Intersections

In this material, electrons behave like particles called "fermions." The researchers found two distinct types of these particles coexisting:

• Type-I (The Standard Intersection): Imagine a perfect, symmetrical cone. Electrons can roll up or down this cone equally in all directions. This is the "standard" behavior.
• Type-II (The Tilted Intersection): Now, imagine that same cone, but someone has pushed it so hard it's leaning over. Electrons can only move in one direction easily, like water rushing down a steep, tilted slide.

The Discovery: Usually, a material has one type or the other. CaAgBi is special because it hosts both types simultaneously. The "standard" intersections are found on one layer of the material, while the "tilted" ones are found on a slightly different layer. It's like a building where the first floor has round tables, but the second floor has only long, slanted benches.

2. The "Ghost" Roads (Fermi Arcs)

In these materials, the electrons on the surface don't follow the usual rules. They create "ghost roads" called Fermi arcs.

• Analogy: Imagine a bridge that connects two islands. In normal materials, the bridge is a full loop. In CaAgBi, the bridge is a half-loop that starts at one intersection and ends at another, floating in the air without a return path.
• The researchers calculated that these bridges are wide and distinct, meaning scientists should be able to see them easily using a special camera (called ARPES) that takes pictures of electron paths.

3. Tuning the Material (The "Dial" and the "Stretch")

The most exciting part of this paper is that the researchers found they could change where these intersections happen, almost like tuning a radio or stretching a rubber band. They tested two methods:

• The "Recipe Change" (Alloy Engineering):
  They mixed the Bismuth (Bi) in CaAgBi with a lighter element called Antimony (Sb).

  • The Result: As they changed the recipe, the "intersections" moved around. Interestingly, the "tilted" (Type-II) intersections disappeared at a different mixing ratio than the "standard" (Type-I) ones. This means scientists could potentially create a material that has only one type of intersection by carefully choosing the recipe.

• The "Stretch" (Strain):
  They physically pulled the material apart (stretched it).

  • The Result: When they stretched it by about 2%, the "tilted" intersections on one layer vanished. However, the "standard" intersections on the other layers stayed put and remained stable even when stretched up to 6%. This shows the material is very tough and can handle physical stress without losing its special properties.

4. Why This Matters (According to the Paper)

The paper doesn't promise a new phone or a medical cure yet. Instead, it claims that CaAgBi is a versatile playground.

• It is the first time such a mix of "standard" and "tilted" intersections has been found naturally in a material without needing outside tricks to force it.
• Because the researchers can move these intersections around using simple changes (mixing ingredients or stretching), it gives scientists a new tool to study how these different types of electrons interact with each other.

In short: The researchers found a material that acts like a dual-mode traffic system for electrons. They showed that by changing the ingredients or stretching the material, they can control where the traffic flows, offering a new, robust platform for studying the strange physics of the quantum world.

Technical Summary

Technical Summary: Tunable Dual-Type Weyl Points in Dirac-Weyl Semimetal CaAgBi

Problem and Motivation
Topological semimetals, characterized by linear or quasi-linear band crossings near the Fermi level, have emerged as a central focus in condensed matter physics. While Dirac semimetals (protected by time-reversal and inversion symmetries) and Weyl semimetals (requiring the breaking of one of these symmetries) are well-studied, the exploration of materials hosting both Dirac and Weyl fermions simultaneously is essential for realizing topological spintronic devices. Recent theoretical work identified Dirac-Weyl semimetals in polar hexagonal LiGaGe-type materials (e.g., SrHgPb), where Dirac and Weyl points coexist. However, the tunability of these topological features and the specific coexistence of different Weyl point types (Type-I and Type-II) within a single intrinsic material remain areas requiring systematic investigation. This work addresses the need to identify and characterize a material candidate with tunable topological properties, specifically focusing on CaAgBi, a material already synthesized experimentally.

Methodology
The study employs a multi-faceted computational approach:

1. First-Principles Calculations: Density Functional Theory (DFT) calculations were performed using the Vienna Ab Initio Simulation Package (VASP) with the Projector-Augmented Wave (PAW) method and the Perdew–Burke–Ernzerhof (PBE) functional. Structural optimization yielded lattice constants consistent with experimental values.
2. Tight-Binding Modeling: To explore topological properties in detail, a tight-binding model was constructed using Maximally Localized Wannier Functions (MLWF) derived from DFT results, focusing on Ca-d, Ag-s, and Bi-p orbitals.
3. Topological Analysis: The chirality of Weyl points was determined by integrating the Berry curvature over spherical regions. Surface Fermi arcs were calculated using the recursive Green's function method implemented in WannierTools.
4. Effective Hamiltonian: A low-energy effective $k \cdot p$ model was developed around the $\Gamma$ point to capture the essential features of the band structure and Dirac/Weyl points.
5. Tunability Studies: The robustness and tunability of the topological states were investigated through alloy engineering (CaAgBixSb1−x using the Virtual Crystal Approximation) and the application of external biaxial tensile strain and hydrostatic pressure.

Key Results

• Crystal and Electronic Structure: CaAgBi crystallizes in the non-centrosymmetric hexagonal space group $P6_3mc$. The band structure reveals three pairs of Dirac points along the $\Gamma$–A rotational axis, protected by combined $C_{3z}$ (or $S_{2z}$) and $M_{yz}$ symmetries.
• Dual-Type Weyl Points: Beyond the previously reported Dirac points, the calculations identify 18 additional pairs of Weyl points distributed across three distinct planes in the Brillouin zone:
  • Type-II Weyl Points: Located in the $k_z = 0$ plane and the $k_z = \pm 0.0698 \frac{2\pi}{c}$ planes. These exhibit strongly tilted cones and line-like Fermi surfaces.
  • Type-I Weyl Points: Located in the $k_z = \pm 0.0698 \frac{2\pi}{c}$ planes. These exhibit point-like Fermi surfaces with Lorentz-symmetric dispersion.
  • This configuration represents the first observation of intrinsic Type-I and Type-II Weyl fermion coexistence in a Dirac-Weyl semimetal without external tuning.
• Surface States: The study confirms the existence of surface Fermi arcs connecting Weyl points of opposite chirality on the (001) surface. The separation between Weyl points in the $k_z = 0$ plane is approximately $0.09 \text{ \AA}^{-1}$, which is larger than in comparable materials like SrHgPb and MoTe$_2$, suggesting they may be more readily detectable via Angle-Resolved Photoemission Spectroscopy (ARPES).
• Alloy Engineering Tunability: In the CaAgBixSb1−x alloy system, the positions of Dirac and Weyl points are tunable via Bi concentration ($x$).
  • As $x$ decreases, Weyl points shift toward the $\Gamma$ point due to the weakening of Spin-Orbit Coupling (SOC) by lighter Sb atoms.
  • Differential Annihilation: Type-II Weyl points in the $k_z = 0$ plane annihilate at $x = 0.6$, while Type-I Weyl points in the $k_z \neq 0$ planes annihilate at $x = 0.67$. This creates a concentration range ($0.61 \le x \le 0.67$) where only Type-II Weyl points coexist with Dirac points, mimicking the SrHgPb phase.
• Strain Tunability: Under biaxial tensile strain:
  • Dirac points remain stable but shift along the $k_z$ axis.
  • Type-II Weyl points in the $k_z = 0$ plane transform into Type-I at strains exceeding 1.96% and annihilate along the $\Gamma$–M path at 2.1% strain.
  • Weyl points in the $k_z \neq 0$ planes remain robust up to 6% strain, shifting outward from the $\Gamma$ point.
  • The Dirac-Weyl phase remains stable under hydrostatic pressure up to 10 GPa.
• Candidate Material: The study also identifies CaAuBi in its high-pressure phase ($P6_3mc$) as a potential Dirac-Weyl semimetal, though its Weyl points are predominantly Type-I.

Significance and Claims
The paper establishes CaAgBi as a versatile platform for manipulating Dirac and Weyl interactions. The primary significance lies in the discovery of a material that intrinsically hosts both Type-I and Type-II Weyl points alongside Dirac points, a feature not previously reported in other topological materials. The work demonstrates that the positions and annihilation of these topological nodes can be systematically controlled through alloy engineering and strain, offering distinct "annihilation concentrations" for different Weyl types. This provides novel experimental strategies for manipulating Weyl points. Furthermore, the development of a low-energy effective Hamiltonian captures the unique topological phase, and the large separation of Weyl points suggests high detectability via ARPES. The authors conclude that SOC-driven alloy control and strain-selective engineering open avenues for topological electronics within this material system.