Topological phases in the $\alpha -{\mathrm{Li}}_3\mathrm{N}$-type crystal structure of light-element compounds

Materials with tunable topological features, simple crystal structure, and flexible synthesis are in extraordinary demand towards the technological exploitation of unique properties of topological nodal points. The controlled design of the lattice geometry of light elements is determined by utilizing density functional theory and the effective Hamiltonian model together with the symmetry analysis. This provides an intriguing venue for reasonably achieving various distinct types of novel fermions. We, therefore, show that a nodal line (types I and II), Dirac fermion, and triple point fermionic excitation can potentially appear as a direct result of a band inversion in group-I nitrides with $\alpha -{\mathrm{Li}}_3\mathrm{N}$-type crystal structure. The imposed strain is exclusively significant for these compounds, and it invariably leads to the considerable modification of the nodal line type. Most importantly, a type-II nodal loop can be realized in the system under strain. These unique characteristics make $\alpha -{\mathrm{Li}}_3\mathrm{N}$-type crystal structure an ideal playground to achieve various types of novel fermions well-suited for technological applications.

Figure 1

Left panel: The crystal structure of $\alpha -{\mathrm{Li}}_3\mathrm{N}$ with $P6/mmm$ symmetry. The mirror reflection plane is shown by $M_{\mathrm{z}}$. Right panel: The Brillouin zone of the bulk and the projected surface Brillouin zones of the (001) and (100) surfaces.

Figure 2

The calculated bulk electronic band structure of ${\mathrm{Na}}_3\mathrm{N}$ using PBE. The irreducible representation of the selected bands at the $\mathrm{\Gamma }$ point and along the $\mathrm{\Gamma }-\mathrm{K}$ and $\mathrm{\Gamma }-\mathrm{M}$ lines are indicated. The orbital-projected band structures of ${\mathrm{Na}}_3\mathrm{N}$ shows that the band near the Fermi level is mostly composed of Na-$s$ and N-$p$ orbitals. The enlarged 3D plots show the existence of a type-I nodal line around the $\mathrm{\Gamma }$ point and along K-H in addition to TDNP along $\mathrm{\Gamma }$-A.

Figure 3

(a, b) The calculated bulk electronic band structure of ${\mathrm{Na}}_3\mathrm{N}$ with the SOC. Nl1 (Nl2) shows the nodal line around the $\mathrm{\Gamma }$ (A) point at $k_{\mathrm{z}}=0$ ($k_{\mathrm{z}}=\pi$). ${\mathrm{NL}}_{\mathrm{KH}}$ shows the nodal line along K-H, while ${\mathrm{NL}}_{\mathrm{H}}$ (${\mathrm{NL}}_{\mathrm{L}}$) shows the nodal line around the H (L) point at the (001) [(100)] plane. (c) The evolution of the Wannier charge center $z(k_{\mathrm{x}},k_{\mathrm{y}}$) for the occupied bands of ${\mathrm{Na}}_3\mathrm{N}$. (d) The calculated (100) Fermi surface of ${\mathrm{Na}}_3\mathrm{N}$ with the chemical potential at $-60$ meV. The red lines (green dots) indicates the Fermi arc (TDNP).

Figure 4

The calculated bulk electronic band structure of ${\mathrm{Li}}_2\mathrm{NaN}$. The $\mathrm{Nl}i$ ($i=1$, 2) (Nl3) shows the nodal line around the $\mathrm{\Gamma }$ ($A$) point at $k_{\mathrm{z}}=0$ ($k_{\mathrm{z}}=\pi$). The calculated (b) (001) and (c) (100) surface band structure of ${\mathrm{Li}}_2\mathrm{NaN}$ show the existence of the drumhead surface states and nodal lines. The calculated (d, e) (001) and (f)–(h) (100) Fermi surfaces of ${\mathrm{Li}}_2\mathrm{NaN}$ with the chemical potential at TDNP and NL1, Nl2. The isoenergy contour shows the nodal-line states and SS's.

Figure 5

The calculated (a) (001) and (b) (100) Fermi surfaces of ${\mathrm{Na}}_3\mathrm{N}$ with the chemical potential at $-200$ meV. The isoenergy contour is showing the nodal-line states (NL) and surface states. (c) The nodal lines and points in the bulk BZ and their projections on the (100) and (001) surface planes in ${\mathrm{Na}}_3\mathrm{N}$. The type-I nodal lines NL1 (Nl2) are located inside the bulk BZ at $k_{\mathrm{z}}=0$ ($k_{\mathrm{z}}=\pi$) and marked in brown color. The type-I nodal lines (${\mathrm{NL}}_{\mathrm{KH}}$) are located along the hinges and shown in the solid, green color. The type-I nodal lines around the H (${\mathrm{NL}}_{\mathrm{H}}$) and around the L (${\mathrm{NL}}_{\mathrm{L}}$) are marked in blue and purple colors, respectively. The calculated (d) (001) and (e) (100) surface band structure of ${\mathrm{Na}}_3\mathrm{N}$ show the existence of the drumhead surface states and nodal lines.

Figure 6

(a) The calculated bulk band structure of $-4\%$-strained ${\mathrm{Na}}_3\mathrm{N}$. The enlarged plot shows the type-II nodal line around the $\mathrm{\Gamma }$ point at $k_{\mathrm{z}}=0$. The calculated (b) (001) surface band structure of $-4\%$-strained ${\mathrm{Na}}_3\mathrm{N}$ show the existence of the drumhead surface states (DSS) is nestled inside the projected nodal loop.

Figure 7

(a) Top view and (b) side view of the crystal structure of the ${\mathrm{Li}}_2\mathrm{N}$ monolayer. (c) The first BZ and (1, 0) edge of ${\mathrm{Li}}_2\mathrm{N}$ monolayer. (d) The calculated orbital-projected band structure of ${\mathrm{Li}}_2\mathrm{N}$ monolayer. The irreducible representation of the selected bands at the $\mathrm{\Gamma }$ and along the $\mathrm{\Gamma }-\mathrm{K}$ and $\mathrm{\Gamma }-\mathrm{M}$ lines are indicated. (e) The phonon dispersions of optimized ${\mathrm{Li}}_2\mathrm{N}$ monolayer.

Figure 8

(a) The calculated (1, 0) edge band structure of ${\mathrm{Li}}_2\mathrm{N}$ shows the existence of the DSS which is nestled outside the projected nodal loop. (b) The enlarged 3D plots show the existence of the type-II nodal line around the $\mathrm{\Gamma }$. (c) The calculated nodal line in the $E-k_{\mathrm{x}}-k_{\mathrm{y}}$ space using the effective Hamiltonian of Eqs. (2) and (4).

Figure 9

(a) Calculated bulk band structure of ${\mathrm{Li}}_3\mathrm{N}$ using PBE. (b, c) The electronic band structure of ${\mathrm{Li}}_2\mathrm{NaN}$ and ${\mathrm{Li}}_2\mathrm{N}$ monolayer when the reflection symmetry is broken by moving the Li atom slightly along the $z$-direction.

Figure 10

The calculated bulk electronic band structure and the orbital-projected band structures of tensile-strained ${\mathrm{Li}}_3\mathrm{N}$. The irreducible representation of the selected bands at the $\mathrm{\Gamma }$ point and along the $\mathrm{\Gamma }-A, \mathrm{\Gamma }-\mathrm{K}, \mathrm{\Gamma }-M$ lines are indicated. (b) Calculated CBM and VBM of ${\mathrm{Li}}_3\mathrm{N}$ as a function of the lattice constant $\mathbf{c}$ using ab initio band structure (dashed line with a triangle) and effective model (dashed line without triangle). (c) The dashed lines show that the effective model reproduces quite well the ab initio band structure (solid blue line) along the $\mathrm{\Gamma }-A$.