Type-II Dirac fermions in the ${\mathbf{PtSe}}_{\mathbf{2}}$ class of transition metal dichalcogenides

Recently, a new “type-II” Weyl fermion, which exhibits exotic phenomena, such as an angle-dependent chiral anomaly, was discovered in a new phase of matter where electron and hole pockets contact at isolated Weyl points [Nature (London) 527, 495 (2015)]. This raises an interesting question about whether its counterpart, i.e., a type-II Dirac fermion, exists in real materials. Here, we predict the existence of symmetry-protected type-II Dirac fermions in a class of transition metal dichalcogenide materials. Our first-principles calculations on ${\mathrm{PtSe}}_2$ reveal its bulk type-II Dirac fermions which are characterized by strongly tilted Dirac cones, novel surface states, and exotic doping-driven Lifshitz transition. Our results show that the existence of type-II Dirac fermions in ${\mathrm{PtSe}}_2$-type materials is closely related to its structural $P\overline{3}m1$ symmetry, which provides useful guidance for the experimental realization of type-II Dirac fermions and intriguing physical properties distinct from those of the standard Dirac fermions known before.

Figure 1

Crystal structure and electronic structure of ${\mathrm{PtSe}}_2$. (a) Crystal structure of ${\mathrm{PtSe}}_2$ with $P\overline{3}m1$ symmetry. (b) BZ of bulk and the projected surface BZs of a (001) surface. The blue dots indicate the high-symmetry points of the BZ, and the red dots highlight the 3D Dirac point position. (c) and (d) The bulk band structures of ${\mathrm{PtSe}}_2$ with/without SOC. The valence bands are labeled as $v_i(i=1–4)$ in terms of their distance to the Fermi level. $E_D$ is the energy of the Dirac points. The irreducible representations of selected bands along the high-symmetric $\mathbf{k}$ path are indicated.

Figure 2

Type-II Dirac points in ${\mathrm{PtSe}}_2$. (a) and (b) Three-dimensional band structures on the (a) $k_x\text{−}{k}_z$ plane and (b) $k_z=k_z^c$ plane around the Dirac point. The bottom projection shows the isoenergy contour of electron and hole pockets with respect to $E_D$. (c)–(e) Three-dimensional isoenergy surface at (c) $E_F=E_D-30\mathrm{meV}$, (d) $E_D$, and (e) $E_D+30\mathrm{meV}$. Contact between electron and hole pockets occurs at the Dirac points (marked with blue arrows) when $E_F=E_D$.

Figure 3

The projected surface states and isoenergy surface for the (001) surface of ${\mathrm{PtSe}}_2$. (a) The projected surface density of states for the (001) surface. The projected Dirac points are denoted by green dots. The green arrows mark surface states around the projected Dirac point, whereas the white arrows represent the Dirac-cone surface state due to nontrivial ${\mathbb{Z}}_2$ topology. (b) The isoenergy surface at $E_D$ of the (001) surface.