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

TL;DR

This study addresses how strain and inversion-asymmetry shape the electronic structure of HgTe, a canonical topological material. By coupling first-principles DFT results (calibrated with HSE06) to an extended $8\times8$ Kane $\mathbf{k}\cdot\mathbf{p}$ model that includes $H_{\mathrm{Pikus\text{--}Bir}}$, $H_{\mathrm{BIA}}$, and the linearly $k$-dependent $H_{C_4}$ terms, the authors reveal a competition between $C_4$ strain and BIA that drives anisotropic sub-band splitting along different crystallographic directions. The work explains the camel-back feature under tensile strain and confirms a Weyl semimetal phase under compressive strain, with Weyl nodes in the $k_y=0$ plane and a tilt-dependent transition between type-1 and type-2 regimes. Overall, the extended $\mathbf{k}\cdot\mathbf{p}$ framework provides a robust, first-principles-consistent description of HgTe’s topological phase diagram and offers a predictive tool for strain-engineered topological materials and related Berry-curvature phenomena.

Abstract

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.

Figures (7)

• Figure 1: (a) The first Brillouin zone of the HgTe lattice depicting the $k$-paths used in this study (b) The crystal structure of HgTe (c) The electronic band structure of unstrained HgTe calculated by first principles.
• Figure 2: The $\mathbf{k}\cdot\mathbf{p}$ electronic band structure fit to the DFT results including (a) both BIA and $C_4$ strain terms ($H_{\mathrm{Total}}$) (b) only $C_{4}$ strain terms ($H_{\mathrm{no\,BIA}}$) and (c) only BIA terms ($H_{\mathrm{no\,C_{4}}}$). The $\Gamma_{8v}^{HH}$ bands correspond to the $\ket{\Gamma_{8}, \pm\frac{3}{2}}$ basis whereas the $\Gamma_{8v}^{LH}$ bands correspond to the $\ket{\Gamma_{8}, \pm\frac{1}{2}}$ basis.
• Figure 3: (a) The isoenergetic surface at $E = -0.05$ eV of the hybridized $\Gamma_{8v}^{HH}$ bands: $(\Gamma_{8v}^{HH})_{1}$ and $(\Gamma_{8v}^{HH})_{2}$ obtained using $H_{\mathrm{no\,BIA}}$ in radial coordinates i.e $(r,\,\theta)$ on the $k_{z}$ = 0 plane, where $\vec{k}\,=\,(r\cos\theta,\,r\sin\theta)$. Here $\theta$ represents the angle subtended by a vector $\vec{k}$ with the $k_{x}$ axis and $r$ represents the magnitude of $\vec{k}$. Comparison between the electronic band structure calculated along the $\theta$-$\Gamma$-$X$ path, for (b) $\theta$ = 45$\degree$ and 135$\degree$ (both equivalent to $\Gamma$-$K$ direction), where the dotted line represents the energy at which the isoenergetic surface in (a) has been constructed, and (c) $\theta$ = 15$\degree$ and $\theta$ = 75$\degree$.
• Figure 4: Fit of the 8x8 model Hamiltonian to the electronic band structure calculated using DFT with BIA ($H_{\mathrm{Total}}$) and without BIA ($H_{\mathrm{no\,BIA}}$) along the (a) $\theta$-$\Gamma$-$X$ path at $\theta$ = 15$\degree$ (b) $Z$-$\Gamma$-$X$ (c) $L$-$\Gamma$-$X$ and (d) an arbitrary path inclined at 57.65$\degree$ to the $k_z$ direction whose in-plane projection forms an angle of 31.64$\degree$ with the $k_x$ axes
• Figure 5: (a) A fit of $H_{\mathrm{Total}}$ to the DFT electronic structure in the Weyl semimetal state along the $K$-$\Gamma$-$X$ path.(b) The DFT band dispersion obtained along the $K$-$\Gamma$-$W1$ path, where $\Gamma$-$W1$ represents the line through the origin and one of the Weyl points. (c) A comparison of the $\mathbf{k}\cdot\mathbf{p}$ band structure obtained using $H_{\mathrm{Total}}$ and $H_{\mathrm{no}\,C_{4}}$ along the $W1$-$\Gamma$-$X$ path (d) The 3D band structure in the $k_{y} = 0$ plane depicting a tilted type-1 Weyl cone and at the Fermi surface.