Strain-induced half-valley metals and topological phase transitions in $M{\mathrm{Br}}_2$ monolayers $(M=\mathrm{Ru},\mathrm{Os})$

The target of valleytronics developments is to manipulate the valley degree of freedom and utilize it in microelectronics as charge and spin degrees of freedom. Based on first-principles calculations, we demonstrate that $M{\mathrm{Br}}_2$ $(M=\mathrm{Ru},\mathrm{Os})$ monolayers are intrinsically ferrovalley materials with large valley polarization up to 530 meV. Compressive strain can induce phase transitions in the materials from ferrovalley insulators to complete valley-polarized metals, called half-valley metals, in analogy to the concept of half metals in spintronics. With the increase of the strain, the materials become Chern insulators, whose edge states are chiral-spin-valley locking. The phase transition is caused by sequent band inversions of the $d_{xy}/d_{x^2-y^2}$ and $d_{z^2}$ orbitals at $K-$ and $K+$ valleys, analyzed based on a strained $\mathit{k}·\mathit{p}$ model. Our work provides a pathway for carrying out low-dissipation electronics devices with complete spin and valley polarizations.

Figure 1

Top and side views of the ${\mathrm{RuBr}}_2$ and ${\mathrm{OsBr}}_2$ monolayers. The black and dashed lines indicate the unit cells. The first Brillouin zone is shown in the upper right corner.

Figure 2

(a), (b) Phonon spectrum for the ${\mathrm{RuBr}}_2$/${\mathrm{OsBr}}_2$ monolayer. (c), (d) Energy variation with the increase of the time at 300 K/170 K for the ${\mathrm{RuBr}}_2$/${\mathrm{OsBr}}_2$ monolayer obtained from molecular-dynamics simulations. The insets in (c) and (d) display the snapshots of the final frames of the materials at the time of 6 ps.

Figure 3

(a) Band structure of the ${\mathrm{RuBr}}_2$ monolayer without SOC. (b) $4d$ orbital-resolved band structure of the ${\mathrm{RuBr}}_2$ monolayer without SOC. The blue, green, and red colors indicate $d_{xz}/d_{yz}$, $d_{xy}/d_{x^2-y^2}$, and $d_{z^2}$ orbitals, respectively. (c) Band structure of the ${\mathrm{RuBr}}_2$ monolayer with SOC. The red and blue colors in (a) and (c) indicate the spin-up and spin-down bands, respectively. (d)–(f) The same as (a)–(c), respectively, except for ${\mathrm{OsBr}}_2$.

Figure 4

(a)–(e) Band structures of the ${\mathrm{RuBr}}_2$ monolayer with SOC under compressive strain with different magnitudes. The red and blue colors indicate the spin-up and spin-down bands, respectively. The magenta curves denote the Berry curvatures of the corresponding system.

Figure 5

(a), (c) The phase diagram of the ${\mathrm{RuBr}}_2$/${\mathrm{OsBr}}_2$ monolayer as a function of compressive strain $\eta$. The HVM states are between the ferrovalley state and the topological nontrivial state. The blue and red curves are the band gaps for the $K-$ and $K+$ valleys, respectively. (b) Schematic diagram of the evolution of the band structures and Berry curvatures for the ${\mathrm{RuBr}}_2$ monolayer with the increase of the compressive strain.

Figure 6

(a) The calculated nontrivial chiral edge states for a semifinite ${\mathrm{RuBr}}_2$ monolayer with 2% compressive strain. The 100% spin-polarized chiral edge state is locked with the valley index and spin direction. (b) The same as in (a), except the magnetization direction is reversed. (c) Schematic diagrams of the electronic device rudiments made of the ${\mathrm{RuBr}}_2$ monolayer tuned by the compressive strain. The dotted red arrows in the middle panel of (c) give the edge states with the reversed magnetization direction.