Topological nodal line and superconductivity of highly thermally stable two-dimensional ${\mathrm{TiB}}_4$

By means of first-principles calculations, we study the electronic structures, lattice dynamics, electron-phonon coupling (EPC), and superconductivity of ${\mathrm{TiB}}_4$ monolayer (TBML) and ${\mathrm{TiB}}_4$ bilayer (TBBL). We find that both TBML and TBBL are nodal line semimetals, and the occurrences of their nodal lines are mainly due to the band inversions between B-$p_x+p_y$ and B-$p_z$ for TBML and between Ti-$d_{xz}+d_{yz}$ and Ti-$d_{z^2}$ for TBBL. The distortion of Ti atoms in the TBBL induces a horizontal glide mirror plane, which protects its nodal line against the spin-orbit coupling. The computed EPC constant $\lambda$ of TBML is 0.65, higher than that of the TBBL with $\lambda =0.35$. Both TBML and TBBL are identified to be phonon-mediated two-dimensional (2D) superconductors with the calculated $T_c=1.66\mathrm{K}$ and 0.82 K, respectively. The $T_c$ of the TBBL can be further enhanced to 6.43 K by applying a tensile strain of 11%. Moreover, they exhibit excellent thermal stability. The coexistence of the topological nodal-line states around the Fermi level and superconductivity in the square-lattice ${\mathrm{TiB}}_4$ monolayer may show more potential for realization of exotic physics.

Figure 1

The crystal structure of 2D ${\mathrm{TiB}}_4$. (a) The top and (b) the side view of the optimized TBML. The cyan atoms denote Ti atoms and the green atoms denote B atoms. (c) The top and (d) the side view of the optimized TBBL. The cyan and orange atoms denote Ti atoms of upper and bottom layer, respectively. The green and blue atoms denote B atoms of the upper and bottom layers, respectively.

Figure 2

The electronic structures of the TBML. (a) The band structure and (b) the orbital-resolved band structure of the TBML, where the red and the blue circles represent the projections of the B $p_z$ and $p_x+p_y$ orbits, respectively. (c) and (d) are the 3D band dispersion around the PNP1 and PNP2, respectively. (e) The gap between $N_{\mathrm{occ}}$ band and $N_{\mathrm{occ}}+$1 bands in the whole BZ. The black circle around $M$ is the nodal line shown in (a). The red $+$ and – are the mirror eigenvalues of time-reversal invariant points. (f) The projected edge band spectrum of the (010) edge of the TBML.

Figure 3

The electronic structures of the TBBL. (a) The band structure and the orbital-resolved band structure of the TBBL. (b) The 3D band of $N_{\mathrm{occ}}$ (red) and $N_{\mathrm{occ}}+$1 (blue) band in the whole BZ. The black 3D-snake-like ring is the nodal ring of the TBBL. The black ring at the bottom is the projection of the nodal ring. The red $+$ and – are the mirror eigenvalues of time-reversal invariant points. (c) The projected edge band spectrum of the (010) edge of the TBBL.

Figure 4

(a) The phonon dispersion of the TBML, the area of the red circles represents the strength of phonon linewidth ${\gamma }_{\mathit{q},\nu }$, (b) phonon DOS, and (c) Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ with accumulated EPC constant $\lambda \left(\omega \right)$.

Figure 5

(a) The phonon dispersion of the TBBL, the area of the red circles represents the strength of phonon linewidth ${\gamma }_{\mathit{q},\nu }$, (b) phonon DOS, and (c) Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ with accumulated EPC constant $\lambda \left(\omega \right)$.

Figure 6

(a) The phonon dispersion of the TBBL with a tensile strain 11%, the area of the red circles represents the strength of phonon linewidth ${\gamma }_{\mathit{q},\nu }$, (b) phonon DOS, and (c) Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ with accumulated EPC constant $\lambda \left(\omega \right)$.

Figure 7

The phonon dispersion relationships of TBBL with strain from –1% to 12%. The TBBL with strain –1% and 12% are dynamically unstable.

Figure 8

The electronic band structures of the TBBL with strain ranging from 0%–11%.

Figure 9

Snapshots of the final frame of each $abinitio$ molecular dynamics simulation from 500 to 3000 K (top and side views) after 10 ps of simulated annealing. Only those bonds within the simulated supercell are shown.