🔬 materials science

First-principles calculation of electronic and topological properties of low-dimensional tellurium

This study employs first-principles calculations to comprehensively map the structural, electronic, and topological landscape of tellurium across its dimensional hierarchy, revealing a rich variety of topological phases ranging from bulk Weyl semimetals and one-dimensional helical systems to two-dimensional quantum spin Hall insulators and incipient topological metals.

Original authors: Gabriel Elyas Gama Araujo, Andreia Luisa da Rosa

Published 2026-02-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Gabriel Elyas Gama Araujo, Andreia Luisa da Rosa

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine Tellurium (Te) not just as a rare, silvery metalloid found in the Earth's crust, but as a shape-shifting superhero with a secret superpower: the ability to change its personality depending on how you squeeze, stretch, or slice it.

This paper is a deep dive into what happens when we take this superhero and shrink it down from a big 3D block, to a flat 2D sheet, and finally to a tiny 1D wire. The researchers used powerful computer simulations (like a digital microscope) to see how the atoms arrange themselves and how electrons move, specifically looking for "topological" properties—essentially, special electronic states that are robust and hard to break, much like a knot that won't untie no matter how much you pull on the string.

Here is the story of Tellurium's journey across dimensions, explained simply:

1. The 3D Block: The "Spinning Top" (Bulk Te)

Think of bulk Tellurium as a stack of helical springs (like a corkscrew or a DNA strand) stacked on top of each other.

The Twist: These springs are chiral, meaning they are either right-handed or left-handed. They can't be superimposed on their mirror image.
The Magic: Because of this twist and a strong internal force called "Spin-Orbit Coupling" (imagine the electrons spinning while running), the material hosts Weyl nodes.
The Analogy: Imagine a highway where cars (electrons) usually drive in lanes. In Tellurium, at certain points, the lanes merge into a perfect, frictionless vortex. These are Weyl nodes. They are like "traffic hubs" where the rules of normal traffic break down, creating a unique, high-speed flow that is protected by the material's symmetry.

2. The 2D Sheets: The "Flatlanders" (Tellurene)

Now, imagine peeling that stack of springs apart to make a single, flat sheet. This is called Tellurene. The researchers tested many different ways to arrange the atoms on this sheet, like folding a piece of paper into different origami shapes.

The "Boring" Sheets (Alpha and Beta): Some shapes (the Alpha and Beta phases) are stable and act like normal semiconductors (good for making computer chips). However, they are "topologically trivial."
- Analogy: Think of these as a flat, smooth pond. You can throw a stone in, but the ripples die out quickly. There are no hidden, protected currents underneath.
The "Exciting" Sheets (Buckled Kagome and Square): When the researchers crumpled the sheet into specific 3D-like shapes (buckled kagome and buckled square), something magical happened. These became topologically non-trivial.
- Analogy: Imagine a Möbius strip (a loop with a twist). Even if you cut it, the connection remains. These sheets have a hidden "knot" in their electronic structure. Electrons can flow along the edges without getting stuck, even if the sheet has defects. This is the Quantum Spin Hall effect—a superhighway for electrons that only works on the edge.
The "Chemical Makeover" (Hydrogen-Passivated Hexagonal): They also tried sticking hydrogen atoms to one side of a hexagonal sheet. This chemical "makeover" forced the sheet to become a perfect insulator in the middle but a super-conductor on the edges.
- Analogy: It's like painting the inside of a pipe with insulation but leaving the rim conductive. The electricity is forced to run only along the rim, protected from the outside world.

3. The 1D Wire: The "Single Helix" (Nanowires)

Finally, they took a single strand of the helical spring from the 3D block to make a nanowire.

The Result: This wire kept the "twist" (chirality) of the original block. Because it's so thin, the electrons are squeezed into a tiny space (quantum confinement).
The Edge States: Just like the 2D sheets, this wire developed special states on its very ends.
- Analogy: Imagine a long, twisted rope. If you look at the middle, it's just a rope. But if you look at the very tips, the twist creates a unique "fringe" where things behave differently. The electrons like to hang out at these tips.
The Catch: Unlike the 2D sheets, this 1D wire doesn't have a "topological knot" that protects the whole system in the same way. The special states at the ends are more like a result of the rope being cut off, rather than a fundamental property of the rope itself. However, they are still very interesting for making tiny electronic devices.

The Big Picture: Why Does This Matter?

The researchers found that Tellurium is a universal playground for engineers.

Tunability: You don't need to invent a new material to get different properties. You just need to change the shape of Tellurium.
- Want a 3D traffic vortex? Use the bulk block.
- Want a 2D edge superhighway? Use the buckled square sheet.
- Want a 1D wire with special tips? Use the nanowire.
The Future: This makes Tellurium a prime candidate for next-generation electronics (spintronics). Instead of just moving charge (electricity), we can move "spin" (the electron's magnetic orientation) with almost no energy loss. This could lead to faster, cooler, and more efficient computers and sensors.

In summary: Tellurium is like a piece of clay that, depending on how you mold it (3D, 2D, or 1D), reveals different hidden superpowers. The scientists mapped out these powers, showing us that by simply changing the shape, we can turn a simple metalloid into a high-tech topological material.

1. Problem Statement

Tellurium (Te) is a Group-16 chalcogen known for its strong spin–orbit coupling (SOC), non-centrosymmetric bulk structure, and potential for topological phenomena. While the topological properties of bulk trigonal tellurium (Te-I) are relatively well-understood (hosting Weyl nodes), the evolution of these topological characteristics across its dimensional hierarchy—specifically in two-dimensional (2D) tellurene polymorphs and one-dimensional (1D) helical nanowires—remains fragmented. There is a lack of a unified theoretical framework connecting the 3D Weyl physics of bulk Te to the symmetry-driven topological phases in its low-dimensional derivatives. The study aims to systematically investigate how structural chirality, symmetry breaking, and SOC govern the electronic and topological properties of Te from 3D bulk down to 1D nanowires.

2. Methodology

The authors employed a comprehensive first-principles approach based on Density Functional Theory (DFT):

Software & Functionals: Calculations were performed using the VASP package. The exchange–correlation potential was treated using both the PBE-GGA functional and the HSE06 hybrid functional for improved band gap accuracy.
Spin–Orbit Coupling (SOC): Full inclusion of SOC was critical for capturing topological features.
Van der Waals Interactions: Long-range vdW forces were accounted for using the DFT-D3 correction.
Structural Models: The study covered:
- 3D: Bulk trigonal Te-I.
- 2D: $\alpha$ -Te ( $P\bar{3}m1$ ), $\beta$ -Te ( $P2/m$ ), and various proposed allotropes including buckled pentagonal, buckled kagome, buckled square, planar hexagonal, Lieb-like, planar kagome, and planar square lattices.
- 1D: Helical tellurium nanowires (Te-h) derived from decoupled bulk chains.
Stability Analysis: Dynamic stability was assessed via phonon dispersion calculations (finite-displacement method). Thermodynamic stability was evaluated using cohesive energies and specific heat ( $C_v$ ).
Topological Characterization:
- 2D Systems: Topological invariants ( $Z_2$ and Chern numbers) were calculated using Maximally Localized Wannier Functions (MLWFs) generated via WANNIER90 and analyzed with WANNIERTOOLS. Berry curvature distributions were computed to identify topological hotspots.
- 1D Systems: Topology was analyzed via the Zak phase (Berry phase) within the modern theory of polarization.
- Spin Textures: Spin polarization and spin-momentum locking were visualized to confirm topological nature.

3. Key Contributions & Results

A. Bulk Trigonal Tellurium (Te-I)

Structure & Stability: Confirmed as a dynamically and thermodynamically stable narrow-gap semiconductor with a chiral helical chain structure.
Electronic Topology: Identified as a Weyl semimetal. The band structure reveals Weyl nodes near the Fermi level arising from broken inversion symmetry and strong SOC.
Spin Texture: The nodes exhibit a characteristic "hedgehog" spin texture in momentum space, confirming spin-momentum locking.
Vibrational: Degenerate phonon modes at the $\Gamma$ point suggest chiral phonon behavior, though explicit Weyl phonon identification requires further Berry curvature analysis.

B. Two-Dimensional Tellurene Polymorphs

The study reveals a rich hierarchy of topological phases dependent on lattice geometry and symmetry:

Trivial Phases:
- $\alpha$ -Te and $\beta$ -Te: Both are confirmed as topologically trivial ( $Z_2 = 0$ ). Despite strong SOC, no band inversion occurs in the occupied manifold. $\beta$ -Te undergoes an indirect-to-direct band gap transition under SOC, while $\alpha$ -Te remains indirect.
Non-Trivial Phases (Emergent Topology):
- Buckled Kagome & Buckled Square: These metallic lattices exhibit a non-trivial $Z_2 = 1$ topology. The buckling enhances SOC-induced hybridization, lifting degeneracies and opening gaps at near-crossings.
  - Kagome: Shows extreme Berry curvature hotspots due to multi-band crossings.
  - Square: Exhibits a more moderate, symmetry-localized Berry curvature.
- Hydrogen-Passivated Hexagonal Te: A planar hexagonal lattice is dynamically unstable in isolation but becomes a fully gapped Quantum Spin Hall (QSH) insulator ( $Z_2 = 1$ ) upon one-side hydrogen passivation or application of 5% strain. This phase features valley-localized Berry curvature and Rashba-type spin splitting.
Stability: While some non-trivial phases (like buckled kagome/square) show soft phonon modes suggesting a need for substrate stabilization, their topological character is intrinsic to the lattice geometry.

C. One-Dimensional Helical Nanowires (Te-h)

Structure: Preserves the bulk chirality and screw symmetry but removes interchain interactions.
Electronic Properties: Acts as a wide-gap semiconductor ( $E_g \approx 2.23$ eV with SOC) with quasi-flat bands.
Topology:
- The periodic nanowire is topologically trivial ( $Zak \approx 0$ ) because it lacks the symmetries required to quantize the Zak phase.
- However, finite nanowires exhibit edge-localized states within the band gap. These are interpreted as boundary manifestations inherited from the 3D Weyl parent phase rather than protected 1D topological states.
Transport: Shows pronounced anisotropy in carrier effective masses, suggesting high mobility potential.

D. Thermodynamic and Mechanical Properties

Stability: Bulk Te-I is the most stable phase. Among 2D phases, $\alpha$ -Te is the most stable, followed by $\beta$ -Te.
Specific Heat: Most phases exhibit a specific heat ( $C_v$ ) around 73 J/(K·mol) at 300 K, significantly higher than graphene, indicating strong in-plane atomic interactions.

4. Significance

Unified Framework: The paper bridges the gap between 3D Weyl physics, 2D QSH phases, and 1D helical systems, demonstrating that tellurium is a versatile platform for engineering topology across dimensions.
Tunability: It establishes that topological phases in Te are not fixed but can be engineered via strain, chemical functionalization (e.g., hydrogen passivation), and substrate interaction.
Material Discovery: The identification of non-trivial $Z_2$ phases in metallic buckled kagome and square tellurene offers a realistic experimental route to QSH physics without requiring complex chemical substitution.
Applications: The findings highlight Te nanostructures as promising candidates for next-generation spintronics, optoelectronics, and quantum devices, leveraging their tunable band gaps, high carrier mobilities, and robust topological edge states.

In conclusion, this work provides a definitive theoretical map of tellurium's topological landscape, proving that structural flexibility and symmetry breaking are the primary levers for switching between trivial and non-trivial topological states in this elemental semiconductor.