Ligand-Controlled Valley and Spin Properties in Ni-Based Two-Dimensional Metal-Organic Frameworks

First-principles calculations based on density functional theory are employed to investigate the interplay among charge, orbital, lattice, valley, and porosity degrees of freedom in ligand-substituted two-dimensional metal-organic frameworks, Ni₃C₁₂X₁₂ (X = O, S, and Se). Depending on the ligand configuration and the level of substitution (half or full), the structures fall into three groups: cis-like, trans-like, and homogeneous. Modulating the charge degrees of freedom shifts the Fermi level into spin-orbit coupling gaps, enabling the emergence of nontrivial topological features across most of the families. In cis-like structures, broken space inversion symmetry simultaneously tunes the lattice and orbital degrees of freedom, opening a valley Hall gap accompanied by spin splitting. The magnitude of this spin splitting scales with the inner potential differences induced by ligands of varying electronegativity. This potential difference also reshapes the Berry curvature so that larger potential contrasts suppress its peak value while broadening its distribution across the Brillouin zone. Trans-like configurations with broken mirror symmetry, preserve nontrivial topological characteristicsincluding quantized spin Hall conductivity (in most cases), a finite Z₂ invariant, and helical edge statesunder low electron doping. Nevertheless, trans-like structures with O-ligand can undergo a transition to a Z₂ metallic phase at higher electron doping concentrations, driven by band gap closure. Homogeneous structures maintain their two-dimensional topological-insulator character except in two cases with large tensile tension.