A Lieb-like lattice in a covalent-organic framework and its Stoner ferromagnetism.

Lieb lattice has been extensively studied to realize ferromagnetism due to its exotic flat band. However, its material realization has remained elusive; so far only artificial Lieb lattices have been made experimentally. Here, based on first-principles and tight-binding calculations, we discover that a recently synthesized two-dimensional sp carbon-conjugated covalent-organic framework (spc-COF) represents a material realization of a Lieb-like lattice. The observed ferromagnetism upon doping arises from a Dirac (valence) band in a non-ideal Lieb lattice with strong electronic inhomogeneity (EI) rather than the topological flat band in an ideal Lieb lattice. The EI, as characterized with a large on-site energy difference and a strong dimerization interaction between the corner and edge-center ligands, quenches the kinetic energy of the usual dispersive Dirac band, subjecting to an instability against spin polarization. We predict an even higher spin density for monolayer spc-COF to accommodate a higher doping concentration with reduced interlayer interaction.