Phonon-based determination of elastic coefficients in the Weyl semimetal TaAs

Technical Summary: Phonon-based Determination of Elastic Coefficients in the Weyl Semimetal TaAs

Problem Statement
Reliable determination of elastic properties in topological semimetals, such as the Weyl semimetal TaAs, is critical for understanding strain-related effects and potential device applications (straintronics). However, direct experimental measurements of elastic coefficients for TaAs are currently lacking. Theoretically, determining these properties via conventional Density Functional Theory (DFT) strain-stress methodologies faces significant hurdles. Specifically, calculating shear moduli ($C_{44}$ and $C_{66}$) requires applying finite shear strains, which break crystal symmetries and necessitate large supercells to remain within the linear-elastic regime. This approach introduces numerical difficulties, including symmetry breaking, Pulay stress corrections, and potential strain-induced inconsistencies in the electronic basis.

Methodology
The authors propose an alternative computational strategy that combines first-principles phonon calculations with an elastic continuum model to determine the full set of elastic moduli without applying finite shear strains.

• Computational Framework: Calculations were performed using the QUANTUM ESPRESSO package with Projector Augmented-Wave (PAW) pseudopotentials and the Perdew–Burke–Ernzerhof (PBE) and PBEsol functionals. Spin-orbit coupling (SOC) was included in both electronic and phonon calculations to accurately capture band topology and lattice dynamics.
• Phonon Approach: Instead of deforming the lattice, the authors calculated the phonon dispersion relations using Density Functional Perturbation Theory (DFPT). They extracted sound velocities ($v$) from the acoustic phonon branches in the long-wavelength limit near the $\Gamma$ point along high-symmetry directions.
• Continuum Model: These sound velocities were fed into an analytical elastic continuum model derived for a body-centered tetragonal lattice with $C_{4v}$ symmetry. This model relates the velocities of longitudinal and transverse acoustic modes to the six independent elastic coefficients ($C_{11}, C_{33}, C_{12}, C_{13}, C_{44}, C_{66}$).
• Validation: The results were compared against:
  1. Conventional DFT strain-stress calculations performed within the same study.
  2. Existing literature values from other DFT studies.
  3. Available experimental data for sound velocities (where applicable).

Key Results

• Structural and Electronic Properties: The calculated lattice parameters using PBE and PBEsol functionals showed excellent agreement with experimental values (relative deviations $\sim$0.5%). The electronic band structure confirmed that SOC gaps nodal lines, stabilizing isolated Weyl nodes, consistent with previous findings.
• Phonon Spectrum: The phonon dispersion revealed a clear energy gap ($\approx 3.69$ meV) between low-energy modes dominated by Ta vibrations and high-energy modes dominated by As vibrations, attributed to the large mass difference between the atomic species. The structure was confirmed to be dynamically stable (no imaginary frequencies).
• Sound Velocities: Extracted sound velocities for various propagation directions and polarizations showed good agreement with literature values and specific experimental measurements (e.g., $v_{xy,t2}$ along the [110] direction matched experimental data within error margins).
• Elastic Moduli:
  • The elastic coefficients derived from the phonon-based continuum model agreed well with conventional DFT strain-stress results and other literature values.
  • The largest discrepancies (approx. 20%) were observed in the off-diagonal coefficients $C_{12}$ and $C_{13}$, while other moduli (particularly $C_{33}$ and $C_{66}$) showed deviations below 10%.
  • All derived coefficients satisfied the mechanical stability criteria for tetragonal crystals.
• Macroscopic Parameters: Using the Voigt-Reuss-Hill averaging scheme, the authors derived bulk ($B$), shear ($G$), and Young's ($E$) moduli, as well as the Poisson ratio ($\nu$).
  • The calculated $B/G$ ratio (1.59–1.69) suggests a brittle behavior for TaAs, contrasting with some literature values that suggest malleability.
  • The Poisson ratio was found to be $\approx 0.497$, indicating strong central inter-atomic forces.

Significance and Claims
The paper claims that the proposed phonon-based approach offers a computationally efficient alternative to conventional strain-based methods. Its primary advantages are:

1. Avoidance of Shear Strain: It eliminates the need for large supercells and the associated numerical instabilities (Pulay stress, symmetry breaking) required when applying shear deformations.
2. Elastic Limit Compliance: By relying on acoustic phonon spectra, the method inherently operates within the elastic limit, avoiding the need to monitor static deformations for non-linear effects.
3. Experimental Connection: The method provides a direct route to compare theoretical predictions with experimental lattice dynamics characterizations (e.g., Raman scattering, Brillouin-Mandelstam scattering), which are feasible for TaAs, whereas direct mechanical testing is not yet reported.

The authors maintain a modest stance, noting that while their method avoids specific numerical pitfalls of strain-stress calculations, it relies on the accurate resolution of acoustic phonon branches and can be affected by numerical dispersion. They position this approach as complementary to, rather than a replacement for, direct DFT-based elastic calculations, suggesting that future comparisons across a broader pool of materials are necessary to fully assess the relative accuracy of both methods.