Anisotropic sub-band splitting mechanisms in strained HgTe: a first principles study

Mercury telluride is a canonical material for realizing topological phases, yet a full understanding of its electronic structure remains challenging due to subtle competing effects. Using first-principles calculations and $\mathbf{k}\cdot\mathbf{p}$ modelling, we study its topological phase diagram under strain. We show that linearly $k$-dependent higher-order $C_4$ strain terms are essential for capturing the correct low-energy behaviour. These terms lead to a nontrivial $k$-dependence of the sub-band splitting arising from the interplay of strain and bulk inversion asymmetry. This explains the camel-back feature in the tensile regime and supports the emergence of a Weyl semimetal phase under compressive strain.

💡 Research Summary

This paper presents a comprehensive first‑principles and k·p theoretical investigation of strained mercury telluride (HgTe), a prototypical material for realizing topological phases. The authors combine density‑functional theory (DFT) calculations performed with the hybrid HSE06 functional and a plane‑wave cutoff of 350 eV (using VASP) with an 8 × 8 × 8 Monkhorst‑Pack k‑mesh, including spin‑orbit coupling, to obtain accurate electronic band structures for both tensile (0.31 % biaxial) and compressive (−0.5 % biaxial) strain states. The tensile case reproduces the small‑gap topological insulator phase, while the compressive case is known to host a Weyl semimetal (WSM) phase.

To interpret the DFT results, the authors construct an 8 × 8 Kane Hamiltonian appropriate for zinc‑blende crystals and augment it with three additional contributions: (i) the conventional Pikus‑Bir strain Hamiltonian, (ii) a bulk inversion asymmetry (BIA) term reflecting the non‑centrosymmetric nature of HgTe, and crucially (iii) a previously neglected k‑linear “C₄” strain term that is linear both in momentum and in the strain tensor components. The C₄ term couples the strain difference (ε_yy − ε_zz) to the angular‑momentum matrices J_x, J_y, J_z (and similarly to U‑matrices for Γ₈‑Γ₇ coupling), thereby generating a momentum‑dependent splitting of the heavy‑hole (Γ₈^HH) bands.

By fitting the full Hamiltonian H_total = H_Kane + H_Pikus‑Bir + H_BIA + H_C₄ to the DFT band structure along several high‑symmetry paths (K‑Γ‑X, θ‑Γ‑X, etc.) using least‑squares regression, the authors extract robust values for the Luttinger parameters (γ₁, γ₂, γ₃), the BIA coefficient C, and the C₄ strain coefficient. γ₁, γ₂, and C₄ show negligible variation across different paths, while γ₃ and C fluctuate within ±10 % and ±20 %, respectively, indicating that the essential physics is captured by a relatively small set of parameters.

The analysis of band splitting proceeds by comparing three model variants: (a) the full Hamiltonian (both BIA and C₄), (b) a model without BIA (H_no BIA), and (c) a model without C₄ (H_no C₄). Along the Γ‑X direction (k_y = 0), the C₄ term yields a sizable splitting of ≈14 meV at k = 0.1 Å⁻¹, whereas the BIA contribution is symmetry‑forbidden (axial‑vector terms vanish). Consequently, the observed sub‑band splitting in this direction is dominated by C₄. In contrast, along the diagonal Γ‑K and Γ‑L directions, the C₄ contribution is strongly suppressed because ε_xx = ε_yy, leading to a negligible splitting of ≈1 meV, and the BIA term also remains small due to symmetry constraints. This directional anisotropy explains the “camel‑back” feature observed experimentally in the tensile‑strained regime.

For the compressively strained case, the interplay between C₄ and BIA leads to band crossings that evolve into Weyl nodes. Using WannierTools and Wannier90 to construct a tight‑binding model from the DFT data, the authors locate pairs of Weyl points and determine their chirality. Notably, the Weyl cones are slightly tilted, producing a type‑I Weyl semimetal with an enhanced Berry curvature dipole. This tilt can amplify nonlinear Hall responses and provides a plausible microscopic origin for the recently reported superconducting diode effect in strained HgTe heterostructures.

Overall, the paper demonstrates that inclusion of the k‑linear C₄ strain term is essential for reproducing the correct low‑energy dispersion of HgTe under both tensile and compressive strain. The resulting model accurately captures the experimentally observed sub‑band splittings, the camel‑back dispersion, and the emergence of a robust Weyl semimetal phase with tilted cones. The work resolves a long‑standing discrepancy between earlier k·p models and first‑principles calculations, and it offers a clear pathway for engineering strain‑tuned topological phases in HgTe‑based materials and heterostructures.