Layer-locked anomalous valley Hall effect in a two-dimensional $A$-type tetragonal antiferromagnetic insulator

Antiferromagnetic (AFM) spintronics provides a route towards energy-efficient and ultrafast device applications. Achieving anomalous valley Hall effect (AVHE) in AFM monolayers is thus of considerable interest for both fundamental condensed-matter physics and device engineering. Here we propose a route to achieve an AVHE in A-type AFM insulator composed of vertically stacked monolayer quantum anomalous Hall insulators with strain and electric field modulations. Uniaxial strain and electric field generate valley polarization and spin splitting, respectively. Using first-principles calculations, ${\mathrm{Fe}}_2\mathrm{BrMgP}$ monolayer is predicted to be a prototype hosting valley-polarized quantum spin Hall insulators in which AVHE and quantum spin Hall effect are synergized in a single system. Our findings reveal a route to achieve multiple Hall effects in two-dimensional tetragonal AFM monolayers.

Figure 1

Concept of layer-locked anomalous valley Hall effect in 2D tetragonal lattice. (a) 2D tetragonal lattice possesses equivalent valleys along $\mathrm{\Gamma }$-X and $\mathrm{\Gamma }$-Y lines with Berry curvature mainly occurring around $Y_1$ and $X_1$ valleys. (b) Applying uniaxial strain along the $x$ direction makes the $Y_1$ and $X_1$ valleys become unequal but spin degeneracy is still preserved. Simultaneous application of uniaxial strain and the out-of-plane electric field further breaks the spin degeneracy. Reversing the electric field leads to opposite spin splitting at both $Y_1$ and $X_1$ valleys. The spin-up and spin-down channels are depicted in blue and red, respectively. (c) Superposition of two tetragonal QAHIs with equal but opposite magnetic moments (A-type AFM order) leads to spin-degenerate equivalent $Y_1$ and $X_1$ valleys with net-zero Berry curvature in momentum space. However, the Berry curvatures for the spin-up and spin-down channels are positive and negative, respectively, yielding nonzero layer-locked hidden Berry curvature in real space. (d) The layer–spin Hall effect, the valley layer–spin Hall effect, and the layer-locked anomalous valley Hall effect.

Figure 2

Lattice and electronic structures of ${\mathrm{Fe}}_2\mathrm{BrMgP}$ monolayer. (a) and (b) Top and side views of monolayer ${\mathrm{Fe}}_2\mathrm{BrMgP}$. Energy band structures of ${\mathrm{Fe}}_2\mathrm{BrMgP}$ monolayer (c) without and (d) with SOC. In panel (c), the spin-up and spin-down channels are depicted in blue and red.

Figure 3

Layer-dependent Berry curvature of ${\mathrm{Fe}}_2\mathrm{BrMgP}$. The Berry curvatures are shown for the spin-up (a, c) and spin-down (b, d) channels for $a/a_0=1.00$ and $E=0.00 \mathrm{V}/\AA$ (a, b) and for $a/a_0=1.04$ and $E=0.02 \mathrm{V}/\AA$ (c, d).

Figure 4

Strain and electric field modulation. For ${\mathrm{Fe}}_2\mathrm{BrMgP}$, the related band gaps including the global gap ($G_{\mathrm{tot}}$) and gaps of $Y_1$ and $X_1$ valleys ($G_{Y_1}$ and $G_{X_1}$) (a, c), and valley splitting for both valence ($V$) and condition ($C$) bands (b, d) as a function of $a/a_0$ (a, b) and $E$ (c, d) with $a/a_0=1.04$.

Figure 5

Sign-reversible layer-locked anomalous valley Hall effect. For ${\mathrm{Fe}}_2\mathrm{BrMgP}$ with $a/a_0=1.04$, the spin-resolved energy band structures for $E=+0.02 \mathrm{V}/\AA$ (a) and $E=-0.02 \mathrm{V}/\AA$ (b) are shown. The spin-up and spin-down channels are depicted in blue and red.

Figure 6

Topological properties of ${\mathrm{Fe}}_2\mathrm{BrMgP}$. For ${\mathrm{Fe}}_2\mathrm{BrMgP}$ with $a/a_0=1.04$ at $E=+0.02 \mathrm{V}/\AA$: (a) the edge states along the [100] direction; (b) the evolution of WCCs along $k_y$.