Weyl semimetal engineering by symmetry control in NiTe$_2$

Through first-principles calculations and tight-binding methods, this study demonstrates that controlled inversion symmetry breaking in 1T-NiTe$_2$ induces the emergence of three distinct sets of Weyl points, offering a viable strategy for engineering topological states for Weyltronics applications.

The Gist

Imagine the world of electrons inside a solid material as a bustling city. Usually, electrons move around in predictable lanes, but in special materials called Weyl Semimetals, the roads twist and turn in a way that creates "highways" where electrons can zip along without any friction, behaving like massless particles.

This paper is about a specific material called NiTe₂ (Nickel Telluride) and how scientists can turn it from a standard city into a high-tech, friction-free super-highway system just by tweaking the rules of the road.

Here is the breakdown of their discovery, using simple analogies:

1. The Starting Point: The "Four-Lane Highway"

Think of the electrons in normal NiTe₂ as cars driving on a four-lane highway that is perfectly symmetrical. Because the city is perfectly balanced (it has "inversion symmetry"), the lanes are stacked on top of each other.

• The Problem: In this state, the electrons are stuck in a "Dirac Semimetal" phase. It's like a four-lane road where the traffic is so dense and symmetrical that you can't really tell the lanes apart. It's stable, but not very exciting for new technology.

2. The Trick: Breaking the Mirror

The researchers realized that if they could break the symmetry of this city, they could split that four-lane highway into two separate, two-lane roads.

• The Analogy: Imagine holding a mirror up to the city. If the city looks exactly the same in the mirror, it's symmetrical. The scientists decided to "tilt" the city slightly (by applying an electric field or shifting the atoms).
• The Result: This "tilt" breaks the mirror image. Suddenly, the four-lane highway splits into two distinct two-lane roads. These new roads are the Weyl Points.

3. The Surprise: Finding Hidden Highways

Here is the most exciting part. The scientists expected that breaking the symmetry would just split the original highway into two lanes.

• The Analogy: It's like they went to remodel a house, expecting to just open up a closet door. Instead, they found that by pushing on one wall, three different secret rooms appeared!
• The Discovery:
  1. Set A: The original split highway (4 points). This was expected.
  2. Set B: A brand new set of highways that appeared from a "gap" in the city (24 points).
  3. Set C: Another brand new set of highways, even deeper in the city (24 points).

The number of these new "highways" depends on how hard they push (the "weight" of the symmetry breaking). A small push creates a few; a bigger push creates a whole fleet of them.

4. The "Chirality" (The Handedness)

Every Weyl point has a property called chirality, which is like a "handedness" (left-handed or right-handed).

• The Analogy: Imagine the electrons are like screws. Some are right-handed screws, and some are left-handed. The laws of physics (Nielsen-Ninomiya theorem) say that in the whole city, the number of left-handed screws must equal the number of right-handed screws. They always come in pairs.
• The scientists mapped these out and confirmed that for every "left-handed" highway, there is a matching "right-handed" one nearby.

5. The "Fermi Arcs": The Surface Bridges

One of the coolest features of Weyl semimetals is something called Fermi Arcs.

• The Analogy: Imagine the city is a 3D block of buildings. Inside the block, the roads are complex. But on the surface of the block, the electrons can't follow the normal rules. Instead, they form a bridge that connects two specific points on the surface.
• These bridges are like a "shortcut" that only exists on the surface. The paper shows that by controlling the symmetry, they can turn these bridges on and off.

Why Does This Matter? (The "Weyltronics" Dream)

The authors call the potential application "Weyltronics."

• The Vision: Just as we use transistors to turn electricity on and off in computers today, this research suggests we could use these "symmetry controls" to turn these special electron highways on and off.
• The Benefit: This could lead to super-fast, low-energy electronics, better sensors, or new ways to process information, all by simply tweaking the material with an electric field or a substrate.

Summary

In short, the scientists took a material that was already interesting (NiTe₂), gave it a little "nudge" to break its perfect symmetry, and discovered that instead of just getting two new paths, they unlocked three distinct sets of super-highways for electrons. They proved they can control exactly how many highways appear and where they are, opening the door to a new era of electronic devices that manipulate these exotic electron flows.

Technical Summary

1. Problem Statement

Weyl semimetals (WSMs) are topological materials characterized by massless Weyl fermions and unique surface states known as Fermi arcs. While Weyl points (WPs) typically emerge from the symmetry breaking of Dirac semimetals (DSMs), most known WSMs are intrinsically non-centrosymmetric (e.g., TaAs, MoTe2). The transition metal dichalcogenide 1T-NiTe2 is a centrosymmetric Dirac semimetal. The challenge addressed in this work is how to engineer and manipulate distinct sets of Weyl points in this centrosymmetric system without altering its chemical composition, specifically by controlling symmetry breaking to transition it from a DSM to a WSM. Furthermore, the authors aim to determine if symmetry breaking can induce Weyl points not only from the original Dirac crossing but also from gapped regions of the band structure.

2. Methodology

The study employs a combination of first-principles calculations and topological invariant analysis:

• Density Functional Theory (DFT): Calculations were performed using VASP and Quantum Espresso with the Generalized Gradient Approximation (GGA-PBE). Spin-orbit coupling (SOC) was included via fully relativistic pseudopotentials within the noncollinear spin-DFT formalism.
• Symmetry Breaking Simulation: To break inversion symmetry while preserving $C_3$ rotation symmetry, the authors simulated an external electric field along the $z$-axis. This was modeled by displacing the Ni atomic plane relative to the Te layers ($\Delta z$), ranging from 1% to 5% of the lattice parameter $c$. This displacement creates a potential difference ($\Delta V_z$) equivalent to an electric field of 0.12–0.65 eV/Å.
• Topological Analysis:
  • Tight-Binding Models: Constructed using the Wannier90 package based on DFT results.
  • Topological Invariants: Calculated Weyl chirality ($\chi$), Berry curvature, and the evolution of Wannier Charge Centers (WCCs) using WannierTools.
  • Surface States: Fermi arcs were investigated by constructing a 35-layer slab and calculating momentum-resolved and energy-resolved surface densities of states (DOS) on the (001) surface.

3. Key Contributions

• Symmetry-Controlled Engineering: Demonstrated a method to engineer Weyl points in a centrosymmetric material (1T-NiTe2) by breaking mirror $\sigma_{xy}$ symmetry (equivalent to breaking inversion symmetry) via atomic displacement or external fields.
• Discovery of Multiple Weyl Sets: Identified three distinct sets of Weyl crossings emerging under symmetry breaking, rather than just the splitting of the original Dirac point.
• Origin of New Weyl Points: Revealed that Weyl points can emerge not only from the splitting of the Dirac semimetal node but also from gapped regions of the Brillouin zone, dependent on the magnitude of the symmetry breaking.
• Type-I and Type-II Classification: Characterized the different sets as Type-I and Type-II Weyl semimetals based on their band dispersion and tilting.

4. Key Results

The study identified three specific sets of Weyl points (Sets A, B, and C) as a function of the symmetry-breaking weight ($\Delta z$):

• Set A (From Dirac Semimetal):

  • Origin: The fourfold degenerate Dirac crossing at the $\Gamma$-$A$ path splits into a pair of Weyl points.
  • Characteristics: These are Type-II Weyl points located near the Fermi level. They appear even at low symmetry breaking.
  • Location: Near the Fermi level (e.g., at $\Delta z = 5\%$, coordinates approx. $k_z \approx 0.36$).
  • Chirality: Opposite chiralities ($\chi = \pm 1$) for the pair.

• Set B (From Gapped Region):

  • Origin: Emerges from a trivial gapped region when $\Delta z \geq 1.75\%$.
  • Characteristics: These are Type-I Weyl points (linear touching).
  • Energy: Located at approximately -0.89 eV below the Fermi level for $\Delta z = 5\%$.
  • Quantity: 6 pairs of Weyl nodes appear in the Brillouin zone.

• Set C (From Gapped Region):

  • Origin: Emerges at higher distortion, $\Delta z \geq 3\%$.
  • Characteristics: These are Type-II Weyl points, evidenced by tilted band crossings.
  • Energy: Located at approximately -1.41 eV below the Fermi level for $\Delta z = 5\%$.
  • Quantity: 6 pairs of Weyl nodes appear in the Brillouin zone.

• Topological Verification:

  • Chirality: The sum of chiral charges over the entire Brillouin zone vanishes, satisfying the Nielsen-Ninomiya theorem.
  • Berry Curvature: Visualized as sources and drains corresponding to positive and negative chirality.
  • Fermi Arcs: Surface state calculations confirmed the existence of non-closed Fermi arcs connecting bulk WPs of opposite chirality on the (001) surface for Sets B and C.

5. Significance

• Tunability: The work establishes a "knob" (symmetry breaking weight via electric field or substrate strain) to turn Weyl points on/off and control their energy positions and types (Type-I vs. Type-II).
• Weyltronics Applications: The ability to manipulate multiple sets of Weyl points and their associated surface states (Fermi arcs) provides a platform for "Weyltronics," potentially enabling novel devices based on chiral anomaly, negative magnetoresistance, and topological superconductivity.
• Fundamental Physics: It challenges the conventional view that Weyl points only arise from DSM splitting, showing they can be engineered from gapped regions in centrosymmetric systems, expanding the search space for topological materials.

In conclusion, the paper provides a comprehensive theoretical framework for transforming 1T-NiTe2 into a multi-set Weyl semimetal through controlled symmetry breaking, offering a pathway for the design of tunable topological devices.