${\mathrm{Ca}}_3{\mathrm{P}}_2$ and other topological semimetals with line nodes and drumhead surface states

As opposed to ordinary metals, whose Fermi surfaces are two dimensional, topological (semi)metals can exhibit protected one-dimensional Fermi lines or zero-dimensional Fermi points, which arise due to an intricate interplay between symmetry and topology of the electronic wave functions. Here, we study how reflection symmetry, time-reversal symmetry, SU(2) spin-rotation symmetry, and inversion symmetry lead to the topological protection of line nodes in three-dimensional semimetals. We obtain the crystalline invariants that guarantee the stability of the line nodes in the bulk and show that a quantized Berry phase leads to the appearance of protected surfaces states, which take the shape of a drumhead. By deriving a relation between the crystalline invariants and the Berry phase, we establish a direct connection between the stability of the line nodes and the drumhead surface states. Furthermore, we show that the dispersion minimum of the drumhead state leads to a Van Hove singularity in the surface density of states, which can serve as an experimental fingerprint of the topological surface state. As a representative example of a topological semimetal, we consider ${\mathrm{Ca}}_3{\mathrm{P}}_2$, which has a line of Dirac nodes near the Fermi energy. The topological properties of ${\mathrm{Ca}}_3{\mathrm{P}}_2$ are discussed in terms of a low-energy effective theory and a tight-binding model, derived from ab initio DFT calculations. Our microscopic model for ${\mathrm{Ca}}_3{\mathrm{P}}_2$ shows that the drumhead surface states have a rather weak dispersion, which implies that correlation effects are enhanced at the surface of ${\mathrm{Ca}}_3{\mathrm{P}}_2$.

Figure 1

Crystal structure and electronic bands of ${\mathrm{Ca}}_3{\mathrm{P}}_2$. (a) Crystal structure of ${\mathrm{Ca}}_3{\mathrm{P}}_2$, which contains two planes with three Ca atoms (blue) and three P atoms (red) that are separated by interstitial Ca atoms (black). The gray dashed lines indicate the unit cell. (b) Top and side view of the crystal structure. The P-$p_x$ and Ca-$d_{z^2}$ orbitals included in the tight-binding model are shown schematically. (c) Calculated electronic band structure of ${\mathrm{Ca}}_3{\mathrm{P}}_2$. The weights of the P-$p_x$ and Ca-$d_{z^2}$ orbitals that are located within the layers are indicated by the width of the corresponding band. The weight of the Ca-$d_{z^2}$ orbital is multiplied by 2 to make it more visible on the scale of the plot. (d) Fermi ring of ${\mathrm{Ca}}_3{\mathrm{P}}_2$ as obtained from the tight-binding model, Eq. (2.2). The bulk and surface Brillouin zones are outlined by the green and black lines, respectively.

Figure 2

Band structure of the tight-binding model. Panels (a) and (b) show the energy bands of Hamiltonian (2.2) along high-symmetry lines within the mirror planes $k_z=0$ and $k_z=\pi /c$, respectively [cf. Fig. 1]. The reflection eigenvalues of the bands are indicated by color, with blue and red corresponding to $R=+1$ and $R=-1$, respectively.

Figure 3

Drumhead surface states and Berry phase. (a) Surface band structure of ${\mathrm{Ca}}_3{\mathrm{P}}_2$ as obtained from the tight-binding model (2.2) for the (001) surface in slab geometry with 60 unit cells. The surface state is highlighted in green. (b) Momentum-resolved surface density of states of Hamiltonian (2.2) for the (001) surface. White and dark red correspond to high and low density, respectively. (c) Energy-resolved surface density of states. The dispersion minimum of the drumhead state gives rise to a Van Hove singularity, i.e., a kink at $E=-0.06$ eV. The inset shows the variation of the Berry phase (2.9) of Hamiltonian (2.2) along high-symmetry lines of the (001) surface Brillouin zone [see Fig. 1]. (d) Surface spectrum of the low-energy effective model (3.1) for the (001) face as a function of surface momenta $k_x$ and $k_y$. The bulk states at $k_z=0$ with reflection eigenvalues $R=+1$ and $R=-1$ are colored in blue and red, respectively. The drumhead surface state is indicated by the green area.

Figure 4

Arc surface state and spin Chern number. (a) $k_y$ dependence of the spin Chern number (2.14) of Hamiltonian (2.2) in the presence of the mirror and time-reversal symmetry breaking perturbation (2.13). (b) Surface and bulk spectra of the low-energy model (3.1) perturbed by the mass term (3.2) with $d=0.9\mathrm{eV}\AA$ and ${\theta }_0=-\pi /4$, which breaks reflection and time-reversal symmetry. The bulk states and the arc state at the (001) surface are indicated in gray and green, respectively.

Figure 5

Bulk bands and drumhead surface states of a spinful time-reversal breaking line node semimetal. Panels (a) and (b) show the bulk bands and the surface density of states of Hamiltonian (2.15) in the presence of the staggered Zeeman term (2.16) with $h_z=0.1$ eV. The momentum in panel (a) is varied within the mirror plane $k_z=0$ along high-symmetry lines of the Brillouin zone. (c) Energy isosurfaces of Hamiltonian (2.15) with $h_z=0.1$ eV at $E_F\pm 5$ meV and $k_z=0$. (d) Surface and bulk spectra of the low-energy effective model (3.3) perturbed by the time-reversal breaking term (3.4) with ${\nu }_{\parallel }{h}_{\text{eff}}^z=0.07\mathrm{eV}$. The drumhead states at the (001) surface are colored in green. The reflection eigenvalues of the bulk bands at $k_z=0$ in panels (a), (c), and (d) are indicated by color, with blue and red corresponding to $R=+1$ and $R=-1$, respectively.

Figure 6

Bulk bands and drumhead surface states of a spinful inversion-breaking line node semimetal. Panels (a) and (b) display the bulk bands and the surface density of states of tight-binding model (2.15) in the presence of the inversion-breaking term (2.17). The momentum in panel (a) is varied within the mirror plane $k_z=0$ along high-symmetry lines. (c) Energy isosurfaces of Hamiltonian (2.15) perturbed by Eq. (2.17) at $E_F\pm 5$ meV and $k_z=0$. (d) Surface and bulk spectra of the low-energy effective model (3.3) perturbed by the inversion-breaking term (3.6) with $\delta =0.025\mathrm{eV}\AA$. The drumhead states at the (001) surface are indicated in green. The mirror eigenvalues of the bulk bands at $k_z=0$ in panels (a), (c), and (d) are represented by color, with blue and red corresponding to $R=+1$ and $R=-1$, respectively.

Figure 7

(a) Definitions of hopping integrals between two Ca orbitals. (b) Definitions of hopping integrals between two P orbitals and one Ca and one P orbital. Orbitals in the first Brillouin zone are labeled. Dark blue and red color represent orbitals in the lower plane while light blue and pink orbitals lie in the upper plane.

Figure 8

The numerical result for the 1d inversion-symmetric topological insulator with 200 sites. We consider the half-filling scenario and compute the absolute value of charge accumulated on the first 10 sites under Gaussian disorder as fixed disorder average $m$ and disorder random deviation $\mathrm{\Delta }$ 3000 times. (a) As $\mathrm{\Delta }=0.02$, the end charge is not quantized when the average disorder is not zero. In the special condition that the average disorder vanishes so inversion symmetry on average is preserved, the end charge on average is $\pm e/2$. (b) The standard deviation of the disorder grows as the deviation of the end disorder grows.