Type-II Dirac line node in strained ${\mathrm{Na}}_3\mathrm{N}$

Dirac line node (DLN) semimetals are a class of topological semimetals that feature band-crossing lines in momentum space. We study the type-I and type-II classification of DLN semimetals by developing a criterion that determines the type using band velocities. Using first-principles calculations, we also predict that ${\mathrm{Na}}_3\mathrm{N}$ under an epitaxial tensile strain realizes a type-II DLN semimetal with vanishing spin-orbit coupling, characterized by the Berry phase, which is ${\mathbb{Z}}_2$ quantized in the presence of inversion and time-reversal symmetries. The surface energy spectrum is calculated to demonstrate the topological phase and the type-II nature is demonstrated by calculating the band velocities. We also develop a tight-binding model and a low-energy effective Hamiltonian that describe the low-energy electronic structure of strained ${\mathrm{Na}}_3\mathrm{N}$. The occurrence of a DLN in ${\mathrm{Na}}_3\mathrm{N}$ under strain is captured in the optical conductivity, which we propose as a means to experimentally confirm the type-II class of DLN semimetals.

Figure 1

Typical energy band dispersions and Fermi surfaces of type-I (first row) and type-II (second row) DLNs. (a), (b) Electronic energy bands of a type-I DLN on the $k_x\text{−}{k}_y$ and $k_{\rho }\text{−}{k}_z$ planes, respectively. Here $k_{\rho }=\sqrt{k_x^2+k_y^2}$ and the DLN lies on the $k_x\text{−}{k}_y$ plane. (c) Changes in the Fermi surface for a type-I DLN in 3D momentum space as the chemical potential is tuned from $\mu =E_{\mathrm{DLN}}$, ${\mu }_1$ to ${\mu }_2$. The locations of $E_{\mathrm{DLN}}$, ${\mu }_1$, and ${\mu }_2$ in energy are shown in (b). Blue circles in (c) indicate the position of the DLN. Green surfaces represent the Fermi surface at the corresponding chemical potentials. $\mathit{q}=\mathit{k}-{\mathit{k}}_{\mathrm{D}}$ is the relative momentum defined with respect to a point in the DLN at ${\mathit{k}}_{\mathrm{D}}$. The left panel in (c) defines the in-plane $q_{\rho }$ and out-of-plane $q_z$ components of the relative momentum $\mathit{q}$. (d), (e) Typical energy bands of a type-II DLN on the $k_x\text{−}{k}_y$ and $k_{\rho }\text{−}{k}_z$ planes. (f) Corresponding changes in the Fermi surface for a type-II DLN at $\mu =E_{\mathrm{DLN}}$, ${\mu }_1$, and ${\mu }_2$. Blue circles in (f) indicate the position of the DLN. Green (red) surfaces represent electron (hole) pockets. At $\mu =E_{\mathrm{DLN}}$, the DLN coexists with the electron and hole pockets that touch each other linearly at the DLN.

Figure 2

Atomic structures of (a) pristine and (b) strained ${\mathrm{Na}}_3\mathrm{N}$. Blue arrows illustrate the direction of strain, which lowers the cubic $Pm\overline{3}m$ space-group symmetry to the tetragonal $P4/mmm$ space-group symmetry. Brillouin zones of (c) pristine ${\mathrm{Na}}_3\mathrm{N}$ and (d) strained ${\mathrm{Na}}_3\mathrm{N}$. A nodal point in pristine ${\mathrm{Na}}_3\mathrm{N}$ and a DLN of strained ${\mathrm{Na}}_3\mathrm{N}$ are shown in red.

Figure 3

Electronic band structures of (a) pristine and (b) 5% strained ${\mathrm{Na}}_3\mathrm{N}$. Band structures are drawn along high-symmetry lines in the BZs of cubic and tetragonal unit cells, respectively. Conduction (valence) bands are represented by blue (red) lines. The triply degenerate point in ${\mathrm{Na}}_3\mathrm{N}$ and doubly degenerate points in strained ${\mathrm{Na}}_3\mathrm{N}$ are indicated by red circles. (c) Diagram showing the evolution of energy ordering from $\mathrm{\Gamma }$ to $X$ for pristine ${\mathrm{Na}}_3\mathrm{N}$ and strained ${\mathrm{Na}}_3\mathrm{N}$. Lifting of triple degeneracy leads to the formation of a DLN.

Figure 4

(a) A DNL (red) in energy-momentum space. A DLN is placed on the $k_z=0$ plane, encircling the $\mathrm{\Gamma }$ point. The energy bands disperse along the DLN (green line) in the energy range $\mathrm{\Delta }{E}_{\mathrm{DLN}}\approx 0.002$ eV. (b) Energy dispersions at a point (at ${\mathit{k}}_D$) in the type-II DLN drawn along the $q_{\rho }$ (left panel) and $q_z$ (right panel) directions. (c) Radial band velocities of the valence and conduction bands calculated as a function of the azimuthal angle $\theta$ on the $q_{\rho }\text{−}{q}_z$ plane [see Fig. 1]. The pink (blue) curve represents the velocity of the valence (conduction) band. The valence (conduction) band undergoes sign change outside (inside) of the DLN across the critical angle $\theta \approx \pm 13.0^{\circ }$ ($\theta \approx {180}^{\circ }\pm 16.0^{\circ }$).

Figure 5

Tight-binding and DFT band structures for (a) pristine and (b) 5% strained ${\mathrm{Na}}_3\mathrm{N}$.

Figure 6

Surface band structure of 5% strained ${\mathrm{Na}}_3\mathrm{N}$ calculated from the maximally localized Wannier Hamiltonian. Topological surface states appear inside the bulk gap near the $\mathrm{\Gamma }$ point as a bright branch inside a green circle. The color scheme shows the density of states at a given energy and $k$ points. Red (blue) indicates a high (low) density of the surface projected states.

Figure 7

Interband optical conductivities of ${\mathrm{Na}}_3\mathrm{N}$ as a function of the epitaxial tensile strain. Both (a) ${\sigma }_{xx}$ and (b) ${\sigma }_{zz}$ show a sudden jump, which occurs near the energy of the DLN. The optical conductivity vanishes in the pristine case because interband transitions are forbidden.

Figure 8

(a) GGA band structure (dashed line) of pristine ${\mathrm{Na}}_3\mathrm{N}$ and tight-binding band structure reproducing the GGA result (solid line). (b) HSE06 band structure (dashed line) of pristine ${\mathrm{Na}}_3\mathrm{N}$ and tight-binding band structure reproducing the HSE06 result (solid line). (c) Tight-binding band structure reproducing the SIC and $G_0{W}_0$ results in [87].

Figure 9

(a) Electronic band structure of strained ${\mathrm{Na}}_3\mathrm{N}$ with and without SOC. The SOC band is represented by the solid line; the non-SOC band, by the dashed line. Magnified views of the band structure on (b) $\mathrm{\Gamma }\text{−}M$ and (c) $\mathrm{\Gamma }\text{−}X$. The size of the SOC gap on $\mathrm{\Gamma }\text{−}X$ ($\mathrm{\Gamma }\text{−}M$) is calculated as $\sim$5.6 meV ($\sim$5.7 meV).