Spontaneous valley polarization in two-dimensional organometallic lattices

Two-dimensional (2D) ferrovalley materials that exhibit spontaneous valley polarization are both fundamentally intriguing and practically appealing to be used in valleytronic devices. Usually, the research on 2D ferrovalley materials is mainly focused on inorganic systems, severely suffering from in-plane magnetization. Here, we alternatively show by k·p model analysis and high-throughput first-principles calculations that ideal spontaneous valley polarization is present in 2D organometallic lattice. We explore the design principle for organic 2D ferrovalley materials composed of (quasi-)planer molecules and transition-metal atoms in hexagonal lattice, and identify 12 promising candidates. These systems have a ferromagnetic or antiferromagnetic semiconducting state, and importantly they exhibit robust out-of-plane magnetization. The interplay between spin and valley, together with strong spin-orbit coupling of transition-metal atoms, guarantee the spontaneous valley polarization in these systems, facilitating the anomalous valley Hall effect. Our findings significantly broaden the scientific and technological impact of ferrovalley physics.

Figure 1

(a) Schematic structure of the organometallic framework with a formula of $\mathrm{T}{\mathrm{M}}_2{\left(\mathrm{OM}\right)}_3$, and 2D Brillouin zone with marking the Γ and $\pm K$ points. (b) Schematic band structures near the $\pm K$ valleys without (left) and with (right) SOC. Δc and Δv in (b) represent the valley polarizations at conduction and valence bands, respectively. (c) Organic molecules chosen for constructing organometallic frameworks. (d) The algorithm for screening 2D organic ferrovalley materials. The up and down arrows in (b), (d) represent up- and down-spin, respectively.

Figure 2

Band structures of (a) NbTa-benzene, (b) NbTa-pyrazine, (c) CrMn-pyrazine, (d) TiV-pyrazine, (e) VCr-pyrazine, (f) NbTa-DCB, (g) CrMn-DCA, and (h) MnFe-TTF. The two pairs of valleys evaluated from the deformed Dirac cones around the Fermi level for these systems are highlighted in orange shadows. Fermi level is set to 0 eV.

Figure 3

Band structures of (a) Ni-benzene, (b) V-pyrazine, (c) Cr-PDI, and (d) Ni-DCA. The two pairs of valleys evaluated from the deformed Dirac cones around the Fermi level for these systems are highlighted in orange shadows. Fermi level is set to 0 eV.

Figure 4

(a) Berry curvatures of NbTa-pyrazine, and the spin-up and spin-down valence bands of V-pyrazine along the high-symmetry points. (b) Berry curvatures of NbTa-pyrazine, and the spin-up and spin-down valence bands of V-pyrazine over the entire 2D Brillouin zone. (c) Diagrams of the anomalous valley Hall effect in NbTa-pyrazine under electron doping and linearly light irradiation. (d) Diagrams of the spin Hall effect under electron doping and the anomalous valley Hall effect under left circularly light irradiation in V-pyrazine. The $\text{``}+\text{"} \left(\text{``}–\text{"}\right)$ symbols represent electrons (holes), and the red and blue arrows represent spin-up and spin-down states, respectively.