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

This first-principles study reveals that linearly $k$-dependent higher-order $C_4$ strain terms are crucial for accurately modeling the electronic structure of strained HgTe, explaining the camel-back feature in the tensile regime and supporting the emergence of a Weyl semimetal phase under compressive strain.

The Gist

The Big Picture: A Quantum Material with a Secret

Imagine Mercury Telluride (HgTe) as a very special, high-tech fabric. Scientists know this fabric can do "magic tricks" with electricity, acting as a Topological Insulator (a material that conducts electricity on its surface but acts like an insulator inside) or a Weyl Semimetal (a state where electrons behave like massless particles moving at the speed of light).

However, for years, scientists had a hard time predicting exactly how this fabric behaves when you stretch or squeeze it. They had a "map" (a mathematical model) to describe the fabric, but the map kept missing small, crucial details. It was like trying to navigate a city with a map that showed the main roads but missed the tiny, winding alleys where the real action happens.

The Problem: The "Missing Piece" in the Puzzle

The researchers in this paper realized that the old maps were missing a specific type of instruction.

• The Old Map: It knew about the material's natural lack of symmetry (like a glove that only fits a left hand, called Bulk Inversion Asymmetry or BIA). It also knew about the general stress of stretching the material.
• The Missing Piece: They discovered a subtle, higher-order effect called $C_4$ strain terms. Think of this as a "twist" that happens specifically when you stretch the material in certain directions. The old models ignored this twist, assuming it was too small to matter.

The Discovery: It's a Tug-of-War

The team used powerful supercomputers to simulate the material and then built a new, more detailed map (a k·p model) that included this missing "twist."

They found that the behavior of the electrons depends on a tug-of-war between two forces:

1. The Natural Twist (BIA): The material's inherent "left-handedness."
2. The Stretch-Induced Twist ($C_4$): The new effect they found, which depends heavily on the direction you are looking at.

The Analogy of the "Camel's Back":
Imagine the energy levels of the electrons as a landscape. In some directions, the old map predicted a smooth hill. The new map, however, revealed a "camel's back"—a landscape with two humps and a dip in the middle.

• Why it matters: This shape only appears because of the $C_4$ twist. Without it, the landscape looks flat and boring. The researchers found that if you look along the straight axes (like the X or Y direction), the stretch-induced twist ($C_4$) wins the tug-of-war and creates this split. But if you look at diagonal angles, the natural twist (BIA) takes over.

The Weyl Semimetal: A Tilted Ice Cream Cone

When the material is squeezed (compressed), it turns into a Weyl Semimetal. In this state, the energy bands cross each other, forming points called Weyl nodes.

• The Old View: Previous studies thought these nodes were like perfect, upright ice cream cones standing straight up.
• The New View: The researchers found that because of their new, more accurate model, these cones are actually tilted. They lean over like a tipped-over ice cream cone.

Why the tilt matters (according to the paper):
This tilt isn't just a cosmetic change. The paper notes that this "tilted" state is different from the "ideal" upright state. This specific tilt is known to enhance a property called the Berry curvature dipole (a complex quantum property related to how electrons curve in space) and can explain a phenomenon called the superconducting diode effect (where electricity flows easily in one direction but not the other, even without a magnetic field).

The Conclusion: What Changed?

1. For the "Stretchy" Phase (Topological Insulator): The new model is essential. If you want to understand the "camel's back" shape or the splitting of energy bands in stretched HgTe, you must include the $C_4$ twist. Without it, your map is wrong.
2. For the "Squeezed" Phase (Weyl Semimetal): The new model shows that the material is a tilted Weyl semimetal, not an ideal one. However, the existence of the Weyl state itself doesn't depend on this new twist; the twist just changes the angle of the cones.

In short: The researchers fixed the map of Mercury Telluride by adding a missing "twist" term. This revealed that the material's behavior is a directional tug-of-war between its natural shape and how it's being stretched, and it corrected our understanding of the "Weyl cones" from being perfectly upright to being slightly tilted.

Technical Summary

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

Problem Statement
Mercury telluride (HgTe) is a canonical material for realizing topological phases, yet a complete understanding of its electronic structure remains challenging due to subtle competing effects. While advanced $k \cdot p$ models are typically employed to describe the electronic band structure, they exhibit quantitative discrepancies compared to first-principles calculations, particularly regarding the magnitude of sub-band splitting along specific $k$-paths. These splittings are critical because the band gap in tensile-strained HgTe is very small, and the specific nature of these splittings underpins phenomena such as the "camel-back" feature in the tensile regime and the topological phase transition to a Weyl semimetal under compressive strain. Previous models often neglected higher-order strain terms, leading to an inability to capture the correct low-energy behavior and the competition between different symmetry-breaking mechanisms.

Methodology
The authors employ a combined approach of state-of-the-art Density-Functional Theory (DFT) calculations and a perturbed 8-band $k \cdot p$ model.

• DFT Calculations: Calculations were performed using the Vienna Ab-initio Simulation Package (VASP) with the hybrid HSE06 functional, which was previously validated for its superior performance in describing HgTe photoemission spectra. The study considers unstrained HgTe, as well as systems under 0.31% in-plane tensile strain (to simulate the topological insulator state) and 0.5% compressive biaxial strain (to simulate the Weyl semimetal state). Spin-orbit coupling was included, and Weyl points were identified using the WannierTools code.
• $k \cdot p$ Modeling: An 8-band Kane Hamiltonian was constructed and fitted to the DFT band structures. Crucially, the model extends beyond standard formulations by including:
  1. The standard Kane Hamiltonian ($H_{Kane}$).
  2. The Pikus-Bir strain Hamiltonian ($H_{Pikus-Bir}$).
  3. The Bulk Inversion Asymmetry (BIA) Hamiltonian ($H_{BIA}$), accounting for the non-centrosymmetric nature of the zincblende lattice.
  4. Linearly $k$-dependent higher-order $C_4$ strain terms ($H_{C4}$): These terms, derived from perturbation theory, describe interactions between $\Gamma_8$ and $\Gamma_7$ bands that are linear in momentum and depend on the strain tensor.

The total Hamiltonian ($H_{Total}$) was fitted to DFT data along various crystallographic paths (e.g., $\Gamma$-X, $\Gamma$-K, $\Gamma$-L) using least-squares regression to extract robust parameters ($\gamma_1, \gamma_2, \gamma_3, C, C_4$). The study isolates the contributions of $C_4$ and BIA terms by comparing fits with and without specific Hamiltonian components.

Key Contributions and Results

1. Identification of $C_4$ Strain Terms: The study establishes that linearly $k$-dependent $C_4$ strain terms are essential for capturing the correct low-energy behavior of strained HgTe. Previous models, which neglected these terms or treated strain as $k$-independent, failed to reproduce the observed sub-band splitting.
2. Mechanism of Anisotropic Splitting: The paper demonstrates that sub-band splitting arises from a competition between the $C_4$ strain terms and the BIA terms.
   • Along $\Gamma$-X (and axes): The splitting is dominated by the $C_4$ strain terms. Symmetry analysis reveals that BIA-induced splitting is strongly suppressed along these directions due to the vanishing of linear-in-$k$ axial vector terms. In contrast, $C_4$ terms mix valence bands, inducing significant splitting (e.g., $\sim 14.46$ meV at $k=0.1$ Å$^{-1}$).
   • Along $\Gamma$-K and $\Gamma$-L: The splitting is dominated by BIA terms. Here, the $C_4$ contribution is negligible (e.g., $\sim 1.37$ meV) due to symmetry cancellations, while BIA terms induce large splittings (e.g., $\sim 37.6$ meV).
3. Four-Fold Symmetry in Splitting: In the $k_z=0$ plane, the $C_4$-induced splitting exhibits four-fold rotational symmetry, being maximal along the $k_x$ and $k_y$ axes and minimal along the $\Gamma$-K direction ($\theta = 45^\circ$). This explains the "camel-back" feature observed in the tensile strain regime.
4. Weyl Semimetal State Characterization:
   • The inclusion of $C_4$ terms does not alter the existence or topological nature of the Weyl semimetal phase under compressive strain; the Weyl nodes remain robust.
   • However, the refined model reveals that the Weyl semimetal state at high compressive strain (e.g., -0.5%) is a tilted type-1 Weyl semimetal, rather than the ideal (untilted) type-1 Weyl semimetal predicted by prior work.
   • At very low strain magnitudes, the system transitions to a type-2 Weyl semimetal phase where the cones are tilted enough to form charge pockets at the Fermi surface.
   • The tilt of the Weyl cones in the high-strain regime is quantified ($\kappa_{max} \approx 0.846$), indicating a finite tilt that enhances the Berry curvature dipole.

Significance and Claims
The paper claims that incorporating the $C_4$ strain terms into the $k \cdot p$ Hamiltonian is necessary to quantitatively describe the photoemission spectra and sub-band splitting of HgTe, resolving long-standing discrepancies between theoretical models and first-principles results. The authors emphasize that while $C_4$ terms are crucial for accurately modeling the band structure near the $\Gamma$ point and the formation of the camel-back feature in the topological insulator phase, they act as higher-order corrections in the Weyl semimetal phase, modifying the dispersion and anisotropy (tilt) without changing the topological character.

The identification of a tilted type-1 Weyl semimetal state in strained HgTe is significant because such tilting is theoretically linked to an enhanced Berry curvature dipole and the superconducting diode effect. The study provides a more accurate framework for understanding how symmetry-breaking terms (BIA and $C_4$) influence the topological phase diagram of HgTe as a function of strain, offering a refined tool for materials design and the interpretation of experimental data in topological semimetals.