Weyl semimetals from noncentrosymmetric topological insulators

We study the problem of phase transitions from three-dimensional topological to normal insulators without inversion symmetry. In contrast with the conclusions of some previous work, we show that a Weyl semimetal always exists as an intermediate phase regardless of any constriant from lattice symmetries, although the interval of the critical region is sensitive to the choice of path in the parameter space and can be very narrow. We demonstrate this behavior by carrying out first-principles calculations on the noncentrosymmetric topological insulators ${\mathrm{LaBiTe}}_3$ and ${\mathrm{LuBiTe}}_3$ and the trivial insulator BiTeI. We find that a robust Weyl-semimetal phase exists in the solid solutions ${\mathrm{LaBi}}_{1-x}{\mathrm{Sb}}_x{\mathrm{Te}}_3$ and ${\mathrm{LuBi}}_{1-x}{\mathrm{Sb}}_x{\mathrm{Te}}_3$ for $x\approx 38.5\%\text{–}41.9\%$ and $x\approx 40.5\%\text{–}45.1\%$, respectively. A low-energy effective model is also constructed to describe the critical behavior in these two materials. In BiTeI, a Weyl semimetal also appears with applied pressure, but only within a very small pressure range, which may explain why it has not been experimentally observed.

Figure 1

(a) The lattice structure of ${\mathrm{LaBiTe}}_3,{\mathrm{LuBiTe}}_3,{\mathrm{LaSbTe}}_3$, and ${\mathrm{LuSbTe}}_3$. (b) The BZ of La(Lu)Bi(Sb)${\mathrm{Te}}_3$. (c) The lattice structure of BiTeI. (d) The BZ of BiTeI.

Figure 2

Smallest direct band gap in the BZ vs composition $x$ for (a) ${\mathrm{LaBi}}_{1-x}{\mathrm{Sb}}_x{\mathrm{Te}}_3$ and (b) ${\mathrm{LuBi}}_{1-x}{\mathrm{Sb}}_x{\mathrm{Te}}_3$. Dashed gray line marks our chosen threshold of 0.2 meV to signal a gap closure.

Figure 3

Determinant of the Jacobian matrix evaluated at Weyl nodes with positive (red) and negative (black) chirality vs composition $x$, for (a) ${\mathrm{LaBi}}_{1-x}{\mathrm{Sb}}_x{\mathrm{Te}}_3$ and (b) ${\mathrm{LuBi}}_{1-x}{\mathrm{Sb}}_x{\mathrm{Te}}_3$.

Figure 4

Trajectories of Weyl nodes in the $\left(k_x,k_y\right)$ plane (in units of Å${}^{-1}$). Dashed red lines indicate Weyl nodes of positive chirality; solid black lines are negative. The “*” and “$\oplus$” denote, respectively, the points of creation or annihilation of Weyl nodes. (a) For ${\mathrm{LaBi}}_{1-x}{\mathrm{Sb}}_x{\mathrm{Te}}_3$. (b) For ${\mathrm{LuBi}}_{1-x}{\mathrm{Sb}}_x{\mathrm{Te}}_3$.

Figure 5

(a), (b) Trajectories of Weyl nodes in the $k_z$ direction (in units of Å${}^{-1}$) for (a) ${\mathrm{LaBi}}_{1-x}{\mathrm{Sb}}_x{\mathrm{Te}}_3$ and (b) ${\mathrm{LuBi}}_{1-x}{\mathrm{Sb}}_x{\mathrm{Te}}_3$. Dashed red (solid black) lines refer to the Weyl nodes with positive (negative) chirality. $\theta$ is the azimuthal angle in the $\left(k_x,k_y\right)$ plane, as indicated in Fig. 4. The “*” and “$\oplus$” denote the creation and annihilation point of the Weyl nodes, respectively. (c), (d) Trajectories of Weyl nodes in the direction of impurity composition $x$ for (c) ${\mathrm{LaBi}}_{1-x}{\mathrm{Sb}}_x{\mathrm{Te}}_3$ and (d) ${\mathrm{LuBi}}_{1-x}{\mathrm{Sb}}_x{\mathrm{Te}}_3$.

Figure 6

Surface spectral function averaged around the Fermi level ($k_x$ and $k_y$ in units of Å${}^{-1}$) for (a) ${\mathrm{LaBi}}_{1-x}{\mathrm{Sb}}_x{\mathrm{Te}}_3$ at $x=0.405$ and (b) ${\mathrm{LuBi}}_{1-x}{\mathrm{Sb}}_x{\mathrm{Te}}_3$ at $x=0.43$.

Figure 7

Bulk band structures of (a) ${\mathrm{LaBiTe}}_3$ and (b) ${\mathrm{LuBiTe}}_3$.

Figure 8

Top: Schematic diagram of the interlayer spin-independent hopping terms in the six-band model. Orbitals on sites Te1, Te2, and $\mathrm{Te}1{}^{\prime }$ make up a quintuple layer; $A,B$, and $C$ label in-plane hexagonal positions. Bottom: Phase diagram for the topological behavior of the six-band model.

Figure 9

(a) Bulk bandstructure of the six-band model at $\delta =0.09$ eV. (b) Smallest direct band gap in the BZ vs. $\delta$. (c) Trajectory of Weyl nodes projected onto the $\left(k_x,k_y\right)$ plane. Dashed red (solid black) line refers to the Weyl node with positive (negative) chirality. The “*” and “$\oplus$” denote the creation and annihilation point of the Weyl nodes respectively. $\theta$ is the azimuthal angle in the $\left(k_x,k_y\right)$ plane. (d) Trajectory of Weyl nodes along $k_z$. Units of $k_x,k_y$ and $k_z$ are Å${}^{-1}$.

Figure 10

(a) Smallest direct band gap in the BZ of BiTeI vs the pressure-scaling variable $\eta$. (b) Surface spectral function of BiTeI in the WSM phase (at 55% of the full pressure).

Figure 11

(a) Trajectories of Weyl nodes in the $\left(k_x,k_y\right)$ plane (in units of Å${}^{-1}$). Dashed red (solid black) lines indicate the trajectories of Weyl nodes with positive (negative) chirality. The “*” and “$\oplus$” denote the creation and annihilation point of the Weyl nodes, respectively. (b) Trajectory of Weyl nodes in the $k_z$ direction (units of Å${}^{-1}$).