Type-II Symmetry-Protected Topological Dirac Semimetals

The recent proposal of the type-II Weyl semimetal state has attracted significant interest. In this Letter, we propose the concept of the three-dimensional type-II Dirac fermion and theoretically identify this new symmetry-protected topological state in the large family of transition-metal icosagenides, $M{A}_3$ ($M=\mathrm{V}$, Nb, Ta; $A=\mathrm{Al}$, Ga, In). We show that the ${\mathrm{VAl}}_3$ family features a pair of strongly Lorentz-violating type-II Dirac nodes and that each Dirac node can be split into four type-II Weyl nodes with chiral charge $\pm 1$ via symmetry breaking. Furthermore, we predict that the Landau level spectrum arising from the type-II Dirac fermions in ${\mathrm{VAl}}_3$ is distinct from that of known Dirac or Weyl semimetals. We also demonstrate a topological phase transition from a type-II Dirac semimetal to a quadratic Weyl semimetal or a topological crystalline insulator via crystalline distortions.

Figure 1

(a) The crystal structure of ${\mathrm{VAl}}_3$. The blue and red spheres represent the V and Al atoms, respectively. (b) The local structure of ${\mathrm{VAl}}_3$. (c) The bulk BZ of ${\mathrm{VAl}}_3$. (d) The calculated bulk band structure of ${\mathrm{VAl}}_3$ in the presence of spin-orbit coupling. The green shaded region shows the energy gap between the lowest conduction and valence bands. (e),(f) Enlarged calculation of the band dispersion along the (e) $k_z$ and (f) $k_x/k_y$ directions in the vicinity of the type-II Dirac node. (g) An enlarged view of the area highlighted by the gray box in (d). (h) Bulk Fermi surface of ${\mathrm{VAl}}_3$ with the red and blue pockets denoting the electron and hole bands, respectively. (i) Bulk constant energy contour in $(k_y,k_z)$ space at $k_x=0$ and at the energy of the type-II Dirac nodes.

Figure 2

(a) Surface and bulk band structure of ${\mathrm{VAl}}_3$ along the $\overline{\mathrm{\Gamma }}\text{−}\overline{N}$ direction on the (100) surface BZ. (b) The calculated $k_z\text{−}{k}_y$ surface and bulk electronic structure at the energy of the bulk Dirac node ($\sim 28\mathrm{meV}$ above Fermi level). (c) Enlarged view of the area highlighted by the black box that surrounds the projected type-II Dirac node in (b). (d),(e) The bulk BZ is projected onto the (100) surface form ($k_x$, $k_z$), (d) side view and (e) tilted view.

Figure 3

(a)–(c) A schematic of the phase transition and corresponding Fermi surface in ${\mathrm{Na}}_3\mathrm{Bi}$. (a) A topological insulating phase by breaking the $C_{3z}$ rotational symmetry. (b) A schematic Fermi surface of ${\mathrm{Na}}_3\mathrm{Bi}$. (c) A single Weyl phase by applying a Zeeman field along the rotational symmetry preserving axis. (d)–(g) Schematic illustration of the topological phase transitions and corresponding Fermi surface in ${\mathrm{VAl}}_3$. (d) A topological crystalline insulator phase by breaking the $C_{4z}$ rotational symmetry. (e) An illustration of surface Fermi arcs and type-II Dirac nodes. (f) Type-II Dirac node splits into a pair of double Weyl nodes with chiral charge $\pm 2$ by applying an Zeeman field along the $k_z$ axis. (g) By further applying a $C_{4z}$ symmetry breaking perturbation, each double Weyl node splits into two single Weyl nodes with $\pm 1$ chiral charge. The net Chern number for the regions defined by dashed lines is shown. (h)–(m) The calculated band dispersion in different topological phases of ${\mathrm{VAl}}_3$. (h),(i) The band dispersion along the $k_z$ directions (h) without and (i) with a Zeeman field. (j) The band dispersion along the $k_x$ and $k_y$ directions for the type-II double Weyl cone in panel (i). (k) By breaking the $C_{4z}$ symmetry, the double Weyl node in panel (j) splits into two single Weyl nodes with an equal chiral charge of $-1$. The enlarged band dispersions along the (l) $k_x$ and (m) $k_z$ directions for the single Weyl nodes.

Figure 4

(a) Landau-level spectrum of the type-II Dirac fermions in ${\mathrm{VAl}}_3$. The magnetic field is along the tilting direction of the type-II Dirac fermions, which is the $k_z$ direction for ${\mathrm{VAl}}_3$. (b) Landau-level spectrum of the type-I Dirac fermions and (c) type-II Weyl fermions. (d)–(f) The band dispersion with varying the magnetic field direction within the $(k_x,k_z)$ plane. $\theta$ defines the angle between the magnetic field and $k_z$ shown in panel (g). (d) The band dispersion along $k_x$. (e) The band dispersion along $k_{\theta 1}$. (f) The band dispersion along $k_{\theta 2}$. (g) Schematic illustration of different magnetic field directions and also the $k$ directions ($k_x$, $k_{\theta 1},k_{\theta 2}$) that are perpendicular to the magnetic fields. The three insets in the top-right corner show the constant energy contours in the $k$ plane that is perpendicular to the magnetic field.