Large spontaneous valley polarization and high magnetic transition temperature in stable two-dimensional ferrovalley $\mathrm{Y}{X}_2(X=\mathrm{I}, \mathrm{Br}, \text{and} \mathrm{Cl}$)

Ferrovalley materials with spontaneous valley polarization are crucial to valleytronic application. Based on first-principles calculations, we demonstrate that two-dimensional (2D) $\mathrm{Y}{X}_2(X=\mathrm{I}, \mathrm{Br}, \text{and} \mathrm{Cl})$ in a 2H structure constitutes a series of promising ferrovalley semiconductors with large spontaneous valley polarization and high magnetic transition temperature. Our calculations reveal that $\mathrm{Y}{X}_2$ are dynamically, energetically, thermally and mechanically stable 2D ferromagnetic semiconductors with a magnetic transition temperature about 200 K. Due to the natural noncentrosymmetric structure, intrinsic ferromagnetic order and strong spin orbital coupling, the large spontaneous valley polarizations of 108.98, 57.70, and 22.35 meV are also predicted in single-layer $\mathrm{Y}{X}_2(X=\mathrm{I}, \mathrm{Br}, \text{and} \mathrm{Cl})$, respectively. The anomalous valley Hall effect is also proposed based on the valley contrasting Berry curvature. Moreover, the ferromagnetism and valley polarization are found to be effectively tuning by applying a biaxial strain. Interestingly, the suppressed valley physics of ${\mathrm{YBr}}_2$ and ${\mathrm{YCl}}_2$ can be switched on via applying a moderate compression strain. The present findings promise $\mathrm{Y}{X}_2$ as competitive candidates for the further experimental studies and practical applications in valleytronics.

Figure 1

Geometry and stability of SL $\mathrm{Y}{X}_2$. The (a) side and (b) top view of structure of SL $\mathrm{Y}{X}_2$. (c) The antiferromagnetic magnetic configuration in a $2\times 2$ supercell in which the atoms with different spin directions are labeled by different colors. Phonon spectrum of ${\mathrm{YI}}_2, {\mathrm{YBr}}_2$, and ${\mathrm{YCl}}_2$ are given in (d)–(f), respectively. Variations of the total potential energy of SL $\mathrm{Y}{X}_2$ in the ($4\times 4$) supercell with respect to simulation time during ab initio molecular dynamics simulations are shown in (g)–(i).

Figure 2

Electronic configuration and magnetic exchange interaction of single-layer ${\mathrm{YI}}_2$. (a) The splitting of $d$ orbitals under the trigonal environment. (b) The change in total energy in different magnetic phases of ${\mathrm{YI}}_2$ as a function of biaxial strain. (c) The paths for the mediation of nearest-neighbor magnetic exchange interactions in SL ${\mathrm{YI}}_2$.

Figure 3

The anisotropy-energy landscapes of single-layer $\mathrm{Y}{X}_2$. (a)–(c) represent the ${\mathrm{YI}}_2, {\mathrm{YBr}}_2$, and ${\mathrm{YCl}}_2$, respectively.

Figure 4

The change in magnetic moment and magnetic susceptibility as a functional of temperature. The (a)–(c) correspond to variance of magnetic moment of ${\mathrm{YI}}_2, {\mathrm{YBr}}_2$, and ${\mathrm{YCl}}_2$, respectively, whereas (d)–(f) correspond to the change in magnetic susceptibility.

Figure 5

The magnetic exchange coupling constants and Curie temperatures of $\mathrm{Y}{X}_2$ ($X$=I, Br, and Cl) under equilibrium and strain conditions. (a)–(c) represent the magnetic exchange coupling constants of ${\mathrm{YI}}_2, {\mathrm{YBr}}_2$, and ${\mathrm{YCl}}_2$ as a function of the number of neighbors, respectively. (d)–(f) represent the estimated Curie temperatures of ${\mathrm{YI}}_2, {\mathrm{YBr}}_2$, and ${\mathrm{YCl}}_2$ as a function of the number of neighbors considered. (g)–(i) represent the Curie temperature under different strain conditions of ${\mathrm{YI}}_2, {\mathrm{YBr}}_2$, and ${\mathrm{YCl}}_2$ with the 14th NN exchange interaction, respectively.

Figure 6

The band structure of SL ${\mathrm{YI}}_2$. (a) Without spin polarization and SOC. (b) With spin polarization but without SOC. Red and blue lines represent the spin-up and spin-down bands. (c) with spin polarization and SOC. (d) is the same as (c) but with the opposite magnetization orientation. The Fermi level is set to 0 eV. The color bar represents the spin contribution of the energy band, in which $-1$ (corresponding to the lower limit) and 1 (upper limit) represent two opposite spin directions, respectively.

Figure 7

The band structure of SL ${\mathrm{YBr}}_2$. (a) Without spin polarization and SOC. (b) With spin polarization but without SOC. Red and blue lines represent the spin-up and spin-down bands. (c) Both with spin polarization and SOC. (d) is the same as (c) but with the opposite magnetization orientation. The Fermi level is set to 0 eV.

Figure 8

The band structure of SL ${\mathrm{YCl}}_2$. (a) Without spin polarization and SOC. (b) With spin polarization but without SOC. Red and blue lines represent the spin-up and spin-down bands. (c) Both with spin-polarization and SOC. (d) is the same as (c) but with the opposite magnetization orientation. The Fermi level is set to 0 eV.

Figure 9

The $4d$ orbital-resolved band structure of SL ${\mathrm{YI}}_2$. (a) and (b) represent $d_{xy}(d_{x^2-y^2}$), and $d_{z^2}$, respectively. The color bar represents the weight of the orbital contributions in which the value of 1 means that the band is contributed by the band fully whereas the value of 0 means the band does not contain this orbital component at all.

Figure 10

The $4d$ orbital-resolved band structure of SL ${\mathrm{YBr}}_2$. (a) and (b) represent $d_{xy}(d_{x^2-y^2}$) and $d_{z^2}$, respectively.

Figure 11

The $4d$ orbital-resolved band structure of SL ${\mathrm{YCl}}_2$. (a) and (b) represent $d_{xy}(d_{x^2-y^2}$) and $d_{z^2}$, respectively.

Figure 12

The variation of valley splitting of $\mathrm{Y}{X}_2$ as a function of magnetization direction.

Figure 13

The comparison of energy band of ${\mathrm{YI}}_2$ calculated by PBE $+ U$ and Wannier fitting methods (a). The contour map for the Berry curvatures in the Brillouin zone of the SL ${\mathrm{YI}}_2$, and the corresponding Berry curvature along high-symmetry lines is shown in (b) and (c). Anomalous Hall conductivity as a function of energy is present in (d).

Figure 14

(a) The valley polarization, (b) magnetic exchange constants, and (c) Curie temperature of ${\mathrm{YI}}_2$ as a function of strains.

Figure 15

The variance of band structure of $\mathrm{Y}{X}_2$ as a functional of biaxial strains. (a)–(c) corresponds the results of ${\mathrm{YI}}_2, {\mathrm{YBr}}_2$, and ${\mathrm{YCl}}_2$.