Built-in electric field and strain tunable valley-related multiple topological phase transitions in $\mathrm{VSi}X{\mathrm{N}}_4(X=\mathrm{C},\mathrm{Si},\mathrm{Ge},\mathrm{Sn},\mathrm{Pb})$ monolayers

Valley-related multiple topological phase transitions have attracted significant attention because they provide significant opportunities for fundamental research and practical applications. Unfortunately, however, there is no real material as of yet that can realize valley-related multiple topological phase transitions. Here, through first-principles calculations and model analysis, we investigate the structural, magnetic, electronic, and topological properties of ${\text{VSi}X\text{N}}_4(X=\mathrm{C},\mathrm{Si},\mathrm{Ge},\mathrm{Sn},\mathrm{Pb})$ monolayers. ${\text{VSi}X\text{N}}_4$ monolayers are stable and intrinsically ferrovalley materials. Intriguingly, we found that built-in electric field and strain can induce valley-related multiple topological phase transitions in materials from valley semiconductor to valley half-semimetal, to valley quantum anomalous Hall insulator, to valley half-semimetal, and to valley semiconductor (or to valley metal). The nature of the topological phase transition is the built-in electric field and strain-induced band inversion between the $d_{xy}/d_{x^2-y^2}$ and $d_{z^2}$ orbitals at $K$ and $K^{\prime }$ valleys. Our findings not only reveal the mechanism of multiple topological phase transitions, but they also provide an ideal platform for the multifield manipulating the spin, valley, and topological physics. This will lead to alternative perspectives for spintronic, valleytronic, and topological nanoelectronic applications based on these materials.

Figure 1

(a) The top and side views of the crystal structure for ${\text{VSi}X\text{N}}_4$ ($X=\text{C,}\text{Si,}\text{Ge,}\text{Sn,}\text{Pb}$) monolayers. The red, gray, blue, and green balls represent V, N, Si, and $X$ elements, respectively. (b) The Brillouin zone (BZ) of the honeycomb lattice with the reciprocal-lattice vectors ${\vec{b}}_1$ and ${\vec{b}}_2$. $\mathrm{\Gamma }, K$, and $M$ are the high-symmetry points in the BZ, and $\overline{\mathrm{\Gamma }}$ and $\overline{X}$ are the high-symmetry points in the one-dimensional BZ. (c) The splitting of $d$ orbitals under the trigonal prismatic crystal field.

Figure 2

(a) Spin charge densities with spin directions are indicated (yellow and cyan correspond to spin-up and spin-down, respectively). The isovalue surface level is at 0.005 $e/{\AA }^3$. (b) Calculated total energies of ${\text{VSi}X\text{N}}_4$ different magnetic structures, which are defined relative to that of the FM state. (c) The magnetic anisotropy energy as a function of strain. These results were obtained with $U_{\mathrm{eff}}=3$ eV.

Figure 3

Band structures and edge state of ${\text{VSi}X\text{N}}_4$ monolayer obtained with the $\mathrm{GGA}+U$ ($U_{\mathrm{eff}}=3$ eV) method. (a) Spin-polarized band structures of ${\text{VSi}X\text{N}}_4$ monolayer. The red and blue lines represent spin-up and spin-down bands, respectively. (b) Band structure with SOC of ${\text{VSi}X\text{N}}_4$ monolayer. (c) Orbital-resolved band structure with SOC of ${\text{VSi}X\text{N}}_4$ monolayer. (d) The edge state of ${\text{VSi}X\text{N}}_4$ monolayer.

Figure 4

(a) The band gap of ${\text{VSi}X\text{N}}_4$ ($X=\text{C,}\text{Si,}\text{Ge,}\text{Sn,}\text{Pb}$) at $K$ and $K^{\prime }$ points. The dotted line is the fitting $K$ and $K^{\prime }$ point gap variation trend. (b) The built-in electric field ($E_{\text{in}}$) as functions of strain for ${\text{VSi}X\text{N}}_4$ monolayer. (c),(d) The schematic illustration of energy level for V-$d$ orbital in ${\mathrm{VSiGeN}}_4$ (c) and ${\mathrm{VSiSnN}}_4$ (d). $E_F$ represents the Fermi level. These results were obtained with $U_{\mathrm{eff}}=3$ eV.

Figure 5

Band structures and edge state of ${\text{VSi}X\text{N}}_4$ monolayer obtained with the $\mathrm{GGA}+U$ ($U_{\mathrm{eff}}=3$ eV) method. (a) Spin-polarized band structures of ${\mathrm{VSiGeN}}_4$ monolayer with the biaxial strain. The red and blue lines represent spin-up and spin-down bands, respectively. (b) Band structure with SOC of ${\mathrm{VSiGeN}}_4$ monolayer with the biaxial strain. (c) Orbital-resolved band structure with SOC of ${\mathrm{VSiGeN}}_4$ monolayer with the biaxial strain. (d) The edge state of ${\mathrm{VSiGeN}}_4$ monolayer with the biaxial strain.

Figure 6

Band structures and edge state of ${\text{VSi}X\text{N}}_4$ monolayer obtained with the $\mathrm{GGA}+U$ ($U_{\mathrm{eff}}=3$ eV) method. (a) The band gap of ${\text{VSi}X\text{N}}_4$ at the $K$ and $K^{\prime }$ points with $-5\%\sim 5\%$ strain, and the orange and light blue shade denotes the VQAHI and VSC states, respectively. (b) The yellow shading corresponds to the edge states. (c) Schematic diagram of the evolution of the band structures and Berry curvatures of ${\text{VSi}X\text{N}}_4$ with the various strain. The green solid line, red solid line, and dotted lines represent the valence band, conduction band, and Berry curvatures of ${\text{VSi}X\text{N}}_4$ with the various strain, respectively.

Figure 7

Schematic diagrams of valley-dependent topological phase transitions. (a) Schematic diagram of the evolution of the band structures with the atomic number of $X$ element increasing for ${\text{VSi}X\text{N}}_4$ monolayers. (b) Schematic diagram of the evolution of the band structures for the ${\mathrm{VSiGeN}}_4$ monolayer as a function of strain. These results were obtained with $U_{\mathrm{eff}}=3$ eV.

Figure 8

(a) The Berry curvatures of ${\mathrm{VSiGeN}}_4$ with the strain of $-1\%$ in the Brillouin zone and along the high-symmetry line (b). (c) Calculated AHC ${\sigma }_{xy}$ as a function of Fermi energy. The light blue shadows denote the valley splitting between the $K$ and $K^{\prime }$ valley. (d) Schematic diagram of tunable VQAHE in hole-doped ${\mathrm{VSiGeN}}_4$ monolayer at the $K$ and $K^{\prime }$ valley, respectively. The holes are denoted by the $+$ symbol. Upward and downward arrows refer to spin-up and spin-down carriers, respectively. These results were obtained with $U_{\mathrm{eff}}=3$ eV.