Strain-induced topological insulator phase transition in HgSe

Using ab initio electronic structure calculations we investigate the change of the band structure and the ${\nu }_0$ topological invariant in HgSe (noncentrosymmetric system) under two different types of uniaxial strain along the [001] and [110] directions, respectively. Both compressive [001] and [110] strain lead to the opening of a (crystal field) band gap (with a maximum value of about 37 meV) in the vicinity of $\Gamma$, and the concomitant formation of a camel-back- (inverse camel-back-) shaped valence (conduction) band along the direction perpendicular to the strain with a minimum (maximum) at $\Gamma$. We find that the ${\mathbb{Z}}_2$ invariant ${\nu }_0=1$ which demonstrates conclusively that HgSe is a strong topological insulator (TI). With further increase of the strain the band gap decreases, vanishing at a critical strain value (which depends on the strain type) where HgSe undergoes a transition from a strong TI to a trivial (normal) insulator. HgSe exhibits a similar behavior under a tensile [110] uniaxial strain. On the other hand, HgSe remains a normal insulator by applying a [001] tensile uniaxial strain. Complementary electronic structure calculations of the nonpolar (110) surface under compressive [110] tensile strain show two Dirac cones at the $\Gamma$ point whose spin chiral states are associated with the top and bottom slab surfaces.

Figure 1

Band structure relative to the Fermi energy of unstrained HgSe along the $\Gamma$-X ([100]) and $\Gamma$-L ([111]) symmetry directions of the BZ [${\vec{k}}_L=\frac{\pi }{a}(1,1,1)$ and ${\vec{k}}_X=\frac{\pi }{a}(1,0,0)$, where $a$ is the equilibrium lattice constant] calculated by (a) the PBE and (b) the MBJLDA approach,[29] $c_{\mathrm{MBJ}}=1.2$.

Figure 2

Band structure of HgSe (relative to the Fermi energy $E_F$) under uniaxial strain along the [001] direction of D${}_{2d}$ symmetry near the $\Gamma$ point along the symmetry directions parallel and perpendicular to the strain direction. (a) The compressive strain of ${\epsilon }_{\left[001\right]}=-5$$\%$ yields a TI (${\nu }_0=1$) with $E_{\mathrm{gap}}=37$ meV. (b) The tensile strain of ${\epsilon }_{\left[001\right]}=+3$$\%$ yields a NI (${\nu }_0=0$) with $E_{\mathrm{gap}}=13$ meV. Red and blue denote the two spin-polarized bands, while green refers to non-spin-polarized bands.

Figure 3

Band structure of HgSe under uniaxial strain along the [110] direction of C${}_{2v}$ symmetry near the $\Gamma$ point along the high symmetry directions parallel and perpendicular to the strain direction. (a) The compressive strain of ${\epsilon }_{\left[110\right]}=-5$$\%$ yields a TI (${\nu }_0=1$) with $E_{\mathrm{gap}}=31$ meV. (b) The tensile strain of ${\epsilon }_{\left[110\right]}=+3$$\%$ yields a TI (${\nu }_0=1$) with $E_{\mathrm{gap}}=9$ meV. (c) Critical tensile strain ${\epsilon }_{\left[110\right]}=+6$$\%$ where the conduction and valence bands of irreducible representation 4 cross at $E_F$ and HgSe undergoes a TI $\leftrightarrow$ NI transition. (d) At larger tensile strain of ${\epsilon }_{\left[110\right]}=+8$$\%$ HgSe becomes a NI (${\nu }_0=0$) with a band gap $E_{\mathrm{gap}}=25$ meV. Red and blue denote the spin-polarized bands.

Figure 4

Variation of the band gap $E_{\mathrm{gap}}$ and the energies of the conduction band and valence band states, $E\left({\Gamma }_n\right)$, at the $\Gamma$ point as a function of uniaxial strain along (a) the [001] direction of D${}_{2d}$ symmetry (where $n=6$ and 7) and (b) the [110] direction of C${}_{2v}$ symmetry (where $n=5c$ and $5v$). The gray shaded areas denote the nontrivial topological phase, where ${\nu }_0=1$.

Figure 5

Schematic of the strain-induced evolution of the band structure along an arbitrary direction ($\Gamma \leftrightarrow \pm K$) in the $k_z=0$ plane where the spin-split bands are denoted by red and blue. (a) The strain splits the originally fourfold degenerate frontier bands at $\Gamma$ in electron and hole bands which does not involve any band crossing and, hence, the bulk system is in a topologically trivial phase with $z_0=0$. (b) Band inversion of one of the spin-split bands (blue) at the critical strain. (c) Odd number (one) of band inversions for one of the bands (blue) at a larger strain, giving rise to a band gap opening away from $\Gamma$ and the concomitant formation of a camel-back- (inverse-camel-back-) shaped valence (conduction) band along the $\Gamma \leftrightarrow \pm K$ direction, rendering the system a strong TI ($z_0=1$). Note that the inverted bands exhibit both electron and hole characters.

Figure 6

Evolution of Wannier centers for HgSe along $k_y$ in the $k_z=0$ plane under uniaxial strain along the [001] direction of (a) ${\epsilon }_{\left[001\right]}=-5$$\%$ and (b) ${\epsilon }_{\left[001\right]}=+15$$\%$ and along the [110] direction of (c) ${\epsilon }_{\left[110\right]}=-5$$\%$ and (b) ${\epsilon }_{\left[110\right]}=+8$$\%$, respectively. The evolution lines cross any arbitrary reference line parallel to $k_y$ (for example, the red dotted line) an odd (even) number of times, yielding $z_0=1$ (0). In all cases the corresponding $z_{\pi }=0$; $N_x=48$.

Figure 7

Band structure of the (110) HgSe surface under a [110] uniaxial compressive strain (${\epsilon }_{\left[110\right]}=-$10$\%$) along the symmetry directions in the two-dimensional BZ showing a single Dirac cone at $\Gamma$ at 52 meV below the Fermi energy. The gray and blue bands correspond to bulk- and surface-derived states, where the latter are associated with the two top and bottom atomic layers.[35]

Figure 8

Constant energy contours of surface spin polarization vector of the uniaxially strained (110) HgSe surface in Fig. 7 within an energy of $\pm$25 meV about $E_F$ (all state above the Dirac point), projected onto the (a) top and (b) bottom (011) HgSe surfaces, respectively. The red arrow contours denote the spin polarization vector of states at $E_F$ separating those with energies higher and lower than $E_F$. The units of $k$ are $2\pi /a$.