Electric-field-tuned anomalous valley Hall effect in $A$-type hexagonal antiferromagnetic monolayers

The combination of antiferromagnetic (AFM) spintronics and anomalous valley Hall effect (AVHE) is of great significance for potential applications in valleytronics. Here, we propose a way for achieving AVHE in $A$-type hexagonal AFM monolayer. The proposed way involves the introduction of layer-dependent electrostatic potential caused by an out-of-plane external electric field, which can break the combined symmetry ($PT$ symmetry) of spatial inversion ($P$) and time reversal ($T$), producing spin splitting. The spin order of spin splitting can be reversed by regulating the direction of electric field. Based on first-principles calculations, the way can be verified in AFM ${\mathrm{Cr}}_2{\mathrm{CH}}_2$. The layer-locked hidden Berry curvature can give rise to layer-Hall effect, including a valley layer–spin Hall effect and layer-locked AVHE. Moreover, we propose Janus monolayer ${\mathrm{Cr}}_2\mathrm{CHF}$ with internal electric polarization, which can also realize the AVHE. Our works provide an experimentally feasible way to realize AVHE in AFM monolayer.

Figure 1

(a) A hexagonal $A$-type AFM monolayer with spin degeneracy but nonequivalent –K and K valleys; (b) by applying an out-of-plane electric field, the spin degeneracy is removed, and the –K and K valleys still are unequal; and (c) when the direction of electric field is reversed, the order of spin splitting is also reversed. The spin-up and spin-down channels are depicted in blue and red. The green arrow represents the electric field.

Figure 2

For monolayer ${\mathrm{Cr}}_2{\mathrm{CH}}_2$ with $a/a_0$ = 0.96, (a), (b) the top and side views of crystal structures, (c) the first BZ with high-symmetry points, and (d) the phonon dispersion spectrum.

Figure 3

Four magnetic configurations with side view of crystal structure: FM (a), AFM1 (b), AFM2 (c), and AFM3 (d) configurations. The blue (red) balls represent the spin-up (spin-down) Cr atoms.

Figure 4

For ${\mathrm{Cr}}_2{\mathrm{CH}}_2$ with $a/a_0=0.96$, the energy-band structures without SOC (a), and with SOC (b), (c), (d) for magnetization direction along the positive $z$, negative $z$, and positive $x$ direction, respectively. In (a), the blue (red) lines represent the band structure in the spin-up (spin-down) direction.

Figure 5

For ${\mathrm{Cr}}_2{\mathrm{CH}}_2$ with $a/a_0=0.96$, the distribution of Berry curvatures of total (a), spin-up (b), and spin-down (c). In the presence of a longitudinal in-plane electric field, an appropriate hole doping for three cases in Fig. 1 produces valley layer–spin Hall effect (d) and layer-locked anomalous valley Hall effect (e and f). The upper and lower planes represent the top and bottom Cr layers.

Figure 6

For ${\mathrm{Cr}}_2{\mathrm{CH}}_2$ with $a/a_0=0.96$, the spin-resolved energy-band structures near the Fermi level for the valence band with SOC at representative $E$ (0.00, 0.02, 0.04, and 0.06 $\mathrm{V}/\AA$). The blue (red) lines represent the band structure in the spin-up (spin-down) direction.

Figure 7

For ${\mathrm{Cr}}_2{\mathrm{CH}}_2$ with $a/a_0=0.96$, the valley splitting ($V_v$) and spin splitting ($S_{-K}$ and $S_K$ at –K and K valleys) for the valence band as a function of $E$.

Figure 8

For Janus ${\mathrm{Cr}}_2\mathrm{CHF}$ with $a/a_0=0.96$: the crystal structures (a), the phonon dispersion spectrum (b), the spin-resolved energy-band structures with SOC for magnetization direction along the positive $z$ direction (c), and the distribution of Berry curvatures of spin-up (d). In (c), the spin-up and spin-down are depicted in blue and red.