Type-II Dirac surface states in topological crystalline insulators

Recently, it has been realized that topological Weyl semimetals come in two different varieties: (i) with standard Weyl cones with pointlike Fermi surfaces (type I) and (ii) with tilted Weyl cones that appear at the contact of electron and hole pockets (type II). These two types of Weyl semimetals have very different physical properties, in particular, in their thermodynamics and magnetotransport. Here, we show that Dirac cone surface states of topological crystalline insulators can be distinguished in a similar way. We demonstrate this in terms of a general surface theory and then apply this knowledge to a family of antiperovskites of the form $A_{3}E\mathrm{O}$, where $A$ denotes an alkaline earth metal, while $E$ stands for Pb or Sn. Using ab initio DFT calculations, we investigate the bulk and surface topology of these antiperovskites and show that they exhibit type-I as well as type-II Dirac surface states protected by reflection symmetry. We find that the type-II Dirac states, as opposed to the type-I Dirac states, exhibit characteristic van Hove singularities in their dispersion, which lead to different thermodynamic properties, and which can serve as an experimental fingerprint of type-II surface states. The different magnetotransport characteristics between type-I and type-II surface states are discussed. In addition, we show that both type-I and type-II surface states exhibit an unusual helical spin polarization, which could lead to topological surface superconductivity.

Figure 1

Crystal symmetries and surface terminations. (a) Crystal structure of ${\mathrm{Ca}}_3\mathrm{PbO}$. (b), (c), and (d) Surface Brillouin zones (BZs) for the (001), (011), and (111) surfaces, respectively. For the (001) and (011) surfaces, the surface BZs can be obtained from the bulk cubic BZs by projecting along the [001] and [011] directions, respectively. For the (111) surface BZ, this is not possible. Nevertheless, the symmetries of the surface BZ follow from the projection of the bulk reflection planes [see (d)]. The brown shaded areas in (b), (c), and (d) indicate the bulk mirror planes. The lower panels show the different surface terminations. The crystal symmetries of the (001), (011), and (111) surfaces are described by the 2D space groups $p4m$, $pmm$, and $p3m1$, respectively.

Figure 2

Density of states at the (001) surface. (a) and (b) Surface density of states along the high-symmetry lines $\overline{\mathrm{\Gamma }}\to \overline{X}$ and $\overline{\mathrm{\Gamma }}\to \overline{M}$ of the (001) surface BZ, respectively, for the Pb-Ca termination [lower left panel in Fig. 1]. The two high-symmetry directions correspond to the two inequivalent mirror lines of Fig. 1, i.e., $k_y=0$ and $k_x=k_y$. The spectrum along the mirror line $k_y=0$ [see panel (a)] exhibits two right-moving chiral modes with mirror eigenvalue $R_y=+1$ and two left-moving chiral modes with mirror eigenvalue $R_y=-1$. Similarly, the spectrum along the mirror line $k_y=k_x$ [see panel (b)] shows two right-moving chiral modes with $R_{x,-y}=+1$ and two left-moving chiral modes with $R_{x,-y}=-1$. We observe that the Dirac nodal points (i.e., the band crossings) for the upper chiral modes are hidden by the bulk bands. For the lower chiral modes, there is no band crossing at all, since the modes are too far apart.

Figure 3

Density of states at the (011) surface. (a) and (b) Surface density of states along the high-symmetry lines $\overline{\mathrm{\Gamma }}\to \overline{X}$ and $\overline{\mathrm{\Gamma }}\to \overline{Y}$ of the (011) surface BZ, respectively, for the Ca-Pb-O termination [lower left panel in Fig. 1]. The two high-symmetry lines represent the intersection of the (011) surface plane with the mirror planes $k_x=0$ and $k_y=k_z$ of Fig. 1. The corresponding mirror symmetries protect the surface band crossings that are visible in (a) and (b). (c) shows the energy resolved surface density of states, which exhibits several van Hove singularities as indicated by the green, orange, blue, and purple arrows. (d) and (e) display the energy- and momentum-resolved surface density of states for different energy ranges. The van Hove singularities are marked by the arrows, except for the light green one, which indicates the touching of electron and hole pockets. (f) shows the momentum-resolved surface density of states at the energy of the type-II Dirac point, which is marked by the light green arrow.

Figure 4

Density of states at the (111) surface. (a) and (b) Surface density of states along the high-symmetry lines $\overline{\mathrm{\Gamma }}\to \overline{K}$ and $\overline{\mathrm{\Gamma }}\to \overline{M}$ of the (111) surface BZ, respectively, for the Ca-Pb termination. The $\overline{\mathrm{\Gamma }}\to \overline{M}$ direction corresponds to the mirror lines of the (111) surface. Thus the spectrum along $\overline{\mathrm{\Gamma }}\to \overline{M}$ is gapless and there appear two type-II Dirac states protected by reflection symmetry. The spectrum along the $\overline{\mathrm{\Gamma }}\text{−}\overline{K}$ line, on the other hand, is gapped, since it is not a mirror line. (c) Surface density of states which exhibits several van Hove singularities as indicated by the orange, blue, and green arrows. The van Hove singularity at $E=5.76$ eV (orange peak) stems from the back bending of the type-II Dirac state. (d) Schematic illustration of the six type-II Dirac states on the (111) surface. (e) and (f) Energy- and momentum-resolved surface density of states for different energy ranges. The van Hove singularities are indicated by the blue and orange arrows. The red arrow indicates the type-II Dirac point at $E=5.70$ eV, where the electron and hole pockets meet.

Figure 5

Spin polarization of the Dirac states at the (001) surface. (a) and (b) Momentum-resolved surface density of states and spin polarization for the (001) face (Ca-Pb termination) at the energies $E=5.60$ and 5.67 eV, respectively. The spin polarization, which is purely in-plane, is indicated by the arrows. The direction of the spin polarization is correlated with the momentum, forming a ${90}^{\circ }$ angle.

Figure 6

Landau level spectrum for type-I and type-II Dirac surface states. The left and right columns show the energy spectrum and Landau level structure, respectively, for the tight-binding model given by Eq. (3.4). The magnetic field strength is chosen to be $B=0.01{\varphi }_0/a^2$, where ${\varphi }_0\equiv h/e$ is the flux quantum and $a^2$ is the area of a plaquette of the square lattice, see Fig. 8. The tilting of the Dirac cones is controlled by the ratio $t_1/t$. The four rows correspond to (a) $t_1/t=0$ (type-I Dirac cone), (b) $t_1/t=0.2$ (type-I Dirac cone), (c) $t_1/t=0.5$ (transition between type-I and -II Dirac cones), and (d) $t_1/t=0.6$ (type-II Dirac cone), respectively. All dispersive curves in the Landau level structure arise from states localized at the edge of the sample, and thus are not of interest here.

Figure 7

Comparison between a type-I (dashed lines) and a type-II (solid lines) Dirac fermion. Shown are the energy dispersions of Eq. (D1) with $\eta =0.5$ (type-I) and $\eta =2$ (type-II). The valence bands are marked by yellow and light blue for type-I and type-II Dirac fermions, respectively.

Figure 8

Illustration of the square lattice, on which the tight-binding model (D4) is defined. The lattice is bipartite with each unit cell containing two inequivalent atoms labeled by $A$ (red) and $B$ (black). The second neighbor hopping integral along all dashed lines is $t_1$. The nearest-neighbor hopping is specified as $t_a=t_b=t_c=-t$, $t_d=t$. The gray rectangle marks the ribbon geometry in our calculation, which is periodic along the $k_2$ direction and finite along the $k_1$ direction.

Figure 9

(a) Bulk energy spectrum of model (D7) as a function ok $k_2$ with $k_1=0$. (b) Energy spectrum of Hamiltonian (D7) in a ribbon geometry with the edges along the $k_2$ direction. (c) Landau level spectrum of Hamiltonian (D7) with $\lambda \equiv B{a}^2/{\varphi }_0=0.05$. We note that all dispersive curves in the Landau level structure arise from states localized at the edge of the ribbon.

Figure 10

Surface density of states along the high-symmetry lines (a) $\overline{\mathrm{\Gamma }}\to \overline{X}$ and (b) $\overline{\mathrm{\Gamma }}\to \overline{M}$ of the (001) surface BZ for the Ca-O termination [lower right panel of Fig. 1]. In both the plots, a small surface band crossing, which is barely visible, is located at the bulk gap of $\overline{\mathrm{\Gamma }}$ point.

Figure 11

Surface density of states along the high-symmetry lines (a) $\overline{\mathrm{\Gamma }}\to \overline{X}$ and (b) $\overline{\mathrm{\Gamma }}\to \overline{Y}$ of the (011) surface BZ for the Ca termination.