Two-dimensional valleytronic semiconductor with spontaneous spin and valley polarization in single-layer ${\mathrm{Cr}}_2{\mathrm{Se}}_3$

Two-dimensional (2D) valleytronic materials, exhibiting valley-contrasting physics, are both fundamentally intriguing and practically appealing to be used in nanoscale devices. In this work, through first-principles calculations and model analysis, we identify single-layer ${\mathrm{Cr}}_2{\mathrm{Se}}_3$ as a promising 2D valleytronic semiconductor. We reveal that single-layer ${\mathrm{Cr}}_2{\mathrm{Se}}_3$ is a ferromagnetic semiconductor and harbors valley features. In contrast to most previously reported similar systems suffering from a natural in-plane magnetization, it favors the robust out-of-plane magnetization. This, combined with the broken inversion symmetry and strong spin-orbit coupling strength, renders that a sizable valley polarization occurs spontaneously in single-layer ${\mathrm{Cr}}_2{\mathrm{Se}}_3$, thus facilitating the observation of intriguing anomalous valley Hall effect. More remarkably, by employing strain and ferroelectric substrate, the on-off switching of the valley polarization is realized in single-layer ${\mathrm{Cr}}_2{\mathrm{Se}}_3$. The underlying physics is discussed in detail. Our work provides a tantalizing candidate for realizing and manipulating the valley and spin physics.

Figure 1

(a) Crystal structure of SL ${\mathrm{Cr}}_2{\mathrm{Se}}_3$ from top and side views. The solid diamond indicates unit cell of SL ${\mathrm{Cr}}_2{\mathrm{Se}}_3$, and the inset in (a) shows the space diagram of two vertically adjacent Cr atoms with their ligands. (b) 2D Brillouin zone with marking the $\mathrm{\Gamma }$ and $K/K^{\prime }$ points. (c) Electron localization function of SL ${\mathrm{Cr}}_2{\mathrm{Se}}_3$. (d) The splitting of $d$ orbitals under the octahedral environment.

Figure 2

(a) Schematic diagram of the exchange interaction between Cr-$t_{2g}$ and Cr-$e_g$ in SL ${\mathrm{Cr}}_2{\mathrm{Se}}_3$. (b) Spin charge density distributions of SL ${\mathrm{Cr}}_2{\mathrm{Se}}_3$, wherein the purple and yellow isosurfaces correspond to the spin-up and spin-down states, respectively. (c) Angular dependence of MAE of SL ${\mathrm{Cr}}_2{\mathrm{Se}}_3$ with the spin axis in the $\mathit{ac}$ plane. The MAE is set to zero in the magnetic easy axis, which is the out-of-plane direction. Inset in (c) shows the spherical plot of the MAE arising from rotating the spin axis from the easy axis in SL ${\mathrm{Cr}}_2{\mathrm{Se}}_3$.

Figure 3

Spin-polarized band structures of SL ${\mathrm{Cr}}_2{\mathrm{Se}}_3$ (a) without SOC; (b) and (c) with SOC. $\perp M$ and $\parallel M$ represent out-of-plane and in-plane magnetizations, respectively. Inset in (b) shows the enlarged band edges of the conduction band. The Fermi level is set to 0 eV.

Figure 4

Berry curvature of SL ${\mathrm{Cr}}_2{\mathrm{Se}}_3$ (a) as a contour map over the whole 2D Brillouin zone and (b) along the high-symmetry points. (c) Schematic diagram of anomalous valley Hall effect in SL ${\mathrm{Cr}}_2{\mathrm{Se}}_3$. The voltmeter is used to detect the anomalous valley Hall effect.

Figure 5

(a) The variation of total energies of SL ${\mathrm{Cr}}_2{\mathrm{Se}}_3$ under FM and AFM1 configurations as a function of biaxial strain ranging from −5 to 5%. The light (dark) green background represents that SL ${\mathrm{Cr}}_2{\mathrm{Se}}_3$ favors out-of-plane (in-plane) magnetization. Band structures of ${\mathrm{Cr}}_2{\mathrm{Se}}_3/{\mathrm{In}}_2{\mathrm{Te}}_3$ under (b) $P\downarrow$ state and (c) $P\uparrow$ state. The bands contributed from ${\mathrm{Cr}}_2{\mathrm{Se}}_3$ and ${\mathrm{In}}_2{\mathrm{Te}}_3$ are marked by yellow and blue lines, respectively. Inset in (b) shows the enlarged band edges of the conduction band under $P\downarrow$ state.