Theoretical prediction of two-dimensional functionalized MXene nitrides as topological insulators

Recently, two-dimensional (2D) transition-metal carbides and nitrides, namely, MXenes have attracted a lot of attention for electronic and energy storage applications. Due to a large spin-orbit coupling and the existence of a Dirac-like band at the Fermi energy, it has been theoretically proposed that some of the MXenes will be topological insulators (TIs). Up to now, all of the predicted TI MXenes belong to transition-metal carbides, whose transition-metal atom is W, Mo, or Cr. Here, on the basis of first-principles and ${\mathbb{Z}}_2$ index calculations, we demonstrate that some of the MXene nitrides can also be TIs. We find that ${\mathrm{Ti}}_3{\mathrm{N}}_2{\mathrm{F}}_2$ is a 2D TI, whereas ${\mathrm{Zr}}_3{\mathrm{N}}_2{\mathrm{F}}_2$ is a semimetal with nontrivial band topology and can be turned into a 2D TI when the lattice is stretched. We also find that the tensile strain can convert ${\mathrm{Hf}}_3{\mathrm{N}}_2{\mathrm{F}}_2$ semiconductor into a 2D TI. Since Ti is one of the most used transition-metal elements in the synthesized MXenes, we expect that our prediction can advance the future application of MXenes as TI devices.

Figure 1

(a) Top and (b) side views of optimized crystal structure of ${\mathrm{Ti}}_3{\mathrm{N}}_2{\mathrm{F}}_2$. Here, Ti, N, and F atoms are represented by light blue, dark blue, and red spheres, respectively. Three possible positions, i.e., B (on top of N atom), T (on top of Ti atom in the outer layer), and A (on top of Ti atom in the central layer), decorating either side of the surfaces are shown in (b). The unit cell (black solid lines) with primitive lattice vectors $\vec{a}$ and $\vec{b}$ are indicated in (a). (c) The phonon dispersion spectrum for optimized ${\mathrm{Ti}}_3{\mathrm{N}}_2{\mathrm{F}}_2$.

Figure 2

The energy bands (solid lines) of ${\mathrm{Ti}}_3{\mathrm{N}}_2{\mathrm{F}}_2$ calculated using the PBE functional (a) without and (b) with the SOC. The energy bands calculated using the hybrid HSE06 functional with the SOC are also shown by dashed lines in (b). $E_{\mathrm{F}}$ is the Fermi energy. The irreducible representations of the energy bands at the $\mathrm{\Gamma }$ point close to the Fermi energy are indicated in (a). (c) A schematic figure of the band inversion mechanism in ${\mathrm{Ti}}_3{\mathrm{N}}_2{\mathrm{F}}_2$. The $d$ type atomic orbitals (AOs) are split due to the crystal field (CF) of lattice with $P\overline{3}m1$ symmetry. Band inversion (BI) occurs between states with different parities.

Figure 3

The energy bands onto $d$ orbitals of (a) two outer and (b) inner Ti atoms for ${\mathrm{Ti}}_3{\mathrm{N}}_2{\mathrm{F}}_2$ calculated using the PBE functional without the SOC. $E_{\mathrm{F}}$ is the Fermi energy.

Figure 4

The energy bands of ${\mathrm{Ti}}_3{\mathrm{N}}_2{\mathrm{F}}_2$ under the strain calculated using the PBE functional without the SOC. (a) The lattice is compressed by 12%. (b) The lattice is stretched by 12%. The red circle indicates the Dirac point in (b). The irreducible representations of the energy bands at the $\mathrm{\Gamma }$ points and along the $\mathrm{\Sigma }$ axis (the axis connecting the $\mathrm{\Gamma }$ and $M$ points) near the Fermi energy $E_{\mathrm{F}}$ are indicated. (c) The energy evolution of three bands at the $\mathrm{\Gamma }$ point with $A_{1g}$, $A_{2u}$, and $E_g$ symmetries, indicated in (a) and (b), when the lattice is compressed (negative strain) or stretched (positive strain). The red arrow indicates the compressive strain above which the band with $A_{2u}$ symmetry becomes lower in energy than that with $E_g$ symmetry, and thus the system becomes topologically trivial. The blue arrow implies the stretching strain above which the accidental degeneracy along the $\mathrm{\Sigma }$ axis occurs. (d) The enlarged energy bands indicated by the red circles in (b). The accidental degeneracy at the Dirac point is lifted with the introduction of the SOC (red lines).

Figure 5

The edge states of ${\mathrm{Ti}}_3{\mathrm{N}}_2{\mathrm{F}}_2$ cut along the $a$ axis [see Fig. 1]. The Fermi energy is located at zero energy. The inset shows the enlarged energy bands around the $\overline{X}$ point. It is clearly observed that the edge states cross the Fermi energy five times and are doubly degenerate at the $\overline{X}$ point.

Figure 6

The energy bands of ${\mathrm{Zr}}_3{\mathrm{N}}_2{\mathrm{F}}_2$ with no strain (a) in the absence of SOC and (b) in the presence of SOC, and with (c) 5% compression and (d) 7% stretching in the absence of SOC. The irreducible representations of the energy bands at the $\mathrm{\Gamma }$ point near the Fermi energy $E_{\mathrm{F}}$ are indicated. (e) The energy evolution of the bands at the $\mathrm{\Gamma }$ point with $A_{2u}$, $A_{1g}$, and $E_g$ symmetries. The red and blue arrows imply the occurrence of bands crossing with varying the strain. (f) The enlarged energy bands (black lines) indicated by the red circle in (d). The accidental degeneracy at the Dirac point is lifted with the SOC (red lines). All calculations are performed using the PBE functional.

Figure 7

The energy bands of ${\mathrm{Hf}}_3{\mathrm{N}}_2{\mathrm{F}}_2$ calculated using the HSE06 hybrid functional (a) without any strain and the SOC, (b) with 5% stretching strain but no SOC, and (c) with 5% stretching strain and the SOC. The irreducible representations of the energy bands at the $\mathrm{\Gamma }$ point near the Fermi energy $E_{\mathrm{F}}$ are indicated.

Figure 8

The phonon spectra of ${\mathrm{Ti}}_3{\mathrm{N}}_2{\mathrm{F}}_2$, ${\mathrm{Zr}}_3{\mathrm{N}}_2{\mathrm{F}}_2$, and ${\mathrm{Hf}}_3{\mathrm{N}}_2{\mathrm{F}}_2$ with different tensile strains.