Weyl Semiconductor Tellurium Overview

• Weyl semiconductor tellurium is a narrow-gap, chiral semiconductor hosting symmetry-protected Weyl nodes that enable both conventional and topological electronic behavior.
• It exhibits unique hedgehog spin and orbital textures due to strong spin–orbit coupling in a noncentrosymmetric structure, which yield robust magnetotransport and nonlinear optical responses.
• External perturbations like pressure and optical pulses tune its band topology, offering a versatile platform for advanced quantum and optoelectronic device engineering.

Weyl semiconductor tellurium (Te) is a narrow-gap, chiral semiconductor that hosts symmetry-protected band crossings—Weyl nodes—endowing the material with both conventional semiconducting and exotic topological properties. In contrast to the archetypal Weyl semimetals, tellurium combines noncentrosymmetric crystal symmetry, strong spin–orbit interaction, tunable carrier concentration, and sensitivity to external perturbations (pressure, optical pulses, phononic excitation) to produce topologically robust electronic and spin transport phenomena. The distinctive hedgehog spin and orbital textures observed in Te underpin a suite of anomalous magnetotransport and nonlinear optical effects, establishing this elemental material as a prototypical Weyl semiconductor and a promising platform for advanced device engineering.

1. Crystal Structure, Symmetry, and Band Topology

Trigonal tellurium crystallizes in the chiral space groups P3₁21 (right-handed) and P3₂21 (left-handed), forming infinite helical chains oriented along the crystallographic c-axis. The absence of inversion and mirror symmetry, combined with threefold screw rotational symmetry, stabilizes degeneracies and protects topological band crossings in momentum-space. Ab initio calculations consistently show the emergence of Weyl nodes near high-symmetry points—especially at the H point—where conduction and valence bands cross with linear dispersion (Hirayama et al., 2014, Nakayama et al., 2017). The generic effective Weyl Hamiltonian in the vicinity of such crossings is

$H = \sum_{i,j=1}^3 a_{ij} \delta k_i \sigma_j$

where $\delta \mathbf{k}$ is measured from the node, $\sigma_j$ denote Pauli matrices, and $a_{ij}$ encode the velocity anisotropy. The resulting monopole (Berry) charge $N_{\pm} = \mp\, \mathrm{sgn}(\det a)$ assigns a topological chirality ($\pm 1$), ensuring robustness against weak perturbations.

In few-layer and nanostructured Te, covalent-like quasi-bonding (CLQB) between chains modulates the interlayer interactions, leading to pronounced band-edge sensitivity and bandgap tuning (Qiao et al., 2017). The effective quantum well model captures the reciprocal-square dependence of the bandgap on layer number: $E_\text{gap} \sim 1/n_L^2$ imparting significant flexibility for engineering Te’s optoelectronic response.

2. Spin–Orbit Coupling, Hedgehog Texture, and Topological Indices

Te’s strong spin–orbit interaction, amplified by the chiral crystal environment, induces unique spin and orbital textures in its bands. Near the Weyl nodes, instead of conventional Rashba-type tangential spin orientation, Te manifests hedgehog radial spin (and orbital) textures: $$H_\text{SO} = \Lambda\, \boldsymbol{\sigma} \cdot \mathbf{k}$$ locally locking the spin direction to momentum ($\mathbf{S}_\pm = \pm \mathbf{b}/|\mathbf{b}|$). The resulting Pontryagin index

$$P_{\pm} = \iint \left[\frac{dS_k}{4\pi|S_\pm|^3}\right]\, \epsilon_{ijk} S_\pm \cdot (\partial_{k_i} S_\pm \times \partial_{k_j} S_\pm) = \pm 1$$

measures the global winding number of the spin texture, exactly matching the Weyl node’s monopole charge (Hirayama et al., 2014).

In p-type Te, k-dependent spin–orbit coupling mixes trivial and Weyl-hosting valence bands, generating a full 3D “hedgehog” orbital magnetic texture even in the regime where Weyl nodes are split in energy rather than momentum (Maruggi et al., 2022). The orbital moment effectively locks to the carrier wavevector: $$\mathbf{m}^\pm_\text{orb}(\mathbf{k}) \propto \pm \mathbf{k}$$ which, together with the induced Berry curvature, governs nontrivial transport phenomena such as nonreciprocal anomalous and planar Hall effects.

3. Pressure and External Field-Induced Topological Transitions

At ambient conditions, bulk Te is a semiconductor with a narrow gap ($\sim$0.31–0.38 eV). Increasing hydrostatic pressure lowers the conduction band edge and closes the gap at the H point, inducing a topological phase transition into a Weyl semimetal (Hirayama et al., 2014, Ideue et al., 2019, Rodriguez et al., 2020).

Key experimental signatures of the transition include:

• Anomalous evolution of Shubnikov–de Haas (SdH) oscillation periods and phase, consistent with gap closure and formation of linear, massless dispersion.
• Reduction of cyclotron mass toward the critical pressure ($\sim$2 GPa), indicating emergent Weyl quasiparticles.
• Optical conductivity $\sigma_1(\omega)\sim \omega$, highlighting linear band dispersion and metallization under pressure (Rodriguez et al., 2020).

At even higher pressures ($\gtrsim$4.3 GPa), Te enters metallic Te-II/Te-III phases, which are robust metals but lack the delicate Weyl signatures seen at lower pressures.

Recent advances demonstrate ultrafast optical control of the Peierls distortion amplitude using light pulses, enabling transitions between Weyl semiconductor, Weyl metal, and non-Weyl metal states (Ning et al., 2022). Time-dependent density functional theory (TDDFT) and time-resolved SHG experiments confirm the reversibility and dynamism of these light-induced topological phase transitions.

4. Magnetotransport, Quantum Hall Effect, and Nonlinear Responses

Tellurium supports a spectrum of quantum and topological transport phenomena:

• Negative longitudinal magnetoresistance (NLMR) and planar Hall effect (PHE), arising from the chiral anomaly under parallel electric and magnetic fields; the conductance follows $\sigma(B) = [1 + C_w B^2]\sigma_{WAL} + \sigma_N$ (Zhang et al., 2019).
• Well-resolved quantum Hall effect in n-type tellurene, with Landau level sequences controlled by valley and spin degeneracy at the conduction band edge, and a robust $\pi$ Berry phase extracted from Landau fan diagrams (Qiu et al., 2019).
• Bilayer quantum Hall states in wide Te quantum wells, where interlayer tunneling hybridizes Landau levels, and compound or charge-transferable quantum Hall plateaus reflect strongly correlated Weyl physics in double-layer systems (Niu et al., 2021).

Magnetotransport under extreme conditions reveals robust metallic surface states on the (10$\bar{1}$0) planes, manifesting 2D carrier densities and persistent quantum oscillations even in bulk semiconducting samples (Akiba et al., 2020). High-temperature SdH oscillations with a nontrivial Berry phase ($\gamma = 1/2 - \Phi_B/2\pi$) and ultralight carrier masses signal topologically protected “hot” 2D hole gas states, stable even against strong disorder and magnetic/nonmagnetic doping (Zhang et al., 2022).

5. Nonlinear Phononics, Optical Modulation, and Topological Transport

Strong THz laser excitation can dynamically control the electronic structure in tellurium via nonlinear phononics (Chen et al., 28 Nov 2024). Excitation of the infrared-active $A_2$ phonon mode couples to the anharmonic Raman-active $A_1$ mode, inducing lattice distortions that modulate band extrema and switch the bandgap from direct to indirect. Near the Weyl point (H), Berry curvature dipole $D_{zz}(\epsilon)$ is highly sensitive to this distortion: $$D_{zz}(\epsilon) = \frac{1}{(2\pi)^3} \sum_n \int_{E_{k,n} = \epsilon} dS\, v^z_{k,n} \Omega^z_{k,n}$$ Laser-driven lattice distortion leads to rearrangement—and reversal—of $D_{zz}$, dynamically switching the direction and magnitude of the nonlinear Hall effect. Electron doping aligns the Fermi level with these topologically active regions, enabling ultrafast non-equilibrium control of Hall-like transport without external magnetic fields. This framework provides a route for precise modulation of topological quasiparticle dynamics and high-speed device applications.

6. Thermoelectric and Optoelectronic Properties

2D Te nanofilms, leveraging Weyl band topology and robust spin–orbit coupling, display outstanding thermoelectric performance: Seebeck coefficients exceeding 400 $\mu$V/K, room-temperature power factor (PF) $\sim$31.7 $\mu$W/cm$\cdot$K, and $ZT\sim0.63$ (Qiu et al., 2018). Accumulation-type metal contacts (e.g., Pd) permit efficient carrier collection by minimizing interface barriers, a property rarely found in p-type semiconductors. Nanostructuring and reduced dimensionality further boost phonon scattering and quantum confinement, lowering thermal conductivity and sharpening the density of states for increased Seebeck response.

Few-layer Te displays high hole mobility (%%%%23$\sigma(B) = [1 + C_w B^2]\sigma_{WAL} + \sigma_N$24%%%% cm$^2$/Vs), strong per-layer absorption ($\sim$9%), and better environmental stability than black phosphorus, making it suitable for high-performance FETs, photodetectors, and thermoelectric microgenerators (Qiao et al., 2017). Polarization-resolved photoconductivity in Te nanowires demonstrates strong anisotropy and temperature-dependent carrier dynamics, with unique selection rules and optically induced spin-polarized currents, hinting at future spintronic and valleytronic applications (Li, 2020).

7. Implications and Prospects for Weyl Semiconductor Tellurium

Weyl semiconductor tellurium encompasses multiple domains of contemporary condensed matter research—topological band theory, nonlinear phononics, quantum transport, and optoelectronics. Its symmetry-protected Weyl nodes, hedgehog spin-orbital textures, and tunable phase transitions (via doping, gating, pressure, light, and phononic control) position it as a canonical material for exploring nontrivial quantum phenomena and engineering multifunctional devices.

Observed phenomena—Berry curvature-driven transport signatures, chiral anomaly, nonreciprocal Hall responses, light-induced phase transitions, ultrafast topological modulation—are consistently rooted in the material’s crystal symmetry, spin–orbit coupling, and topological invariants. The ability to control not only electronic conduction but also the underlying topological state via external fields or optical pulses distinguishes Te from other elemental and compound semiconductors.

A plausible implication is that further advances in the manipulation of lattice order, electron correlation, and phonon–electron coupling in Te may enable the next generation of ultrafast, low-dissipation, topology-driven quantum devices. Research into coupled quasiparticle dynamics, dynamic band topology control, and robust high-temperature quantum transport remains a promising frontier grounded in the foundational properties of Weyl semiconductor tellurium.