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A new set of lattice-models based on the hexagonal $\sqrt{N}\times\sqrt{N}$super-cells of the well-known honeycomb lattice with single-hole defect(HL-D-1/2N) are proposed to realize the nontrivial isolated flat-bands. Throughperforming both tight-binding and density functional theory calculations, wedemonstrate that the experimentally realized graphitic carbon nitride (Adv.Mater., 22, 1004, 2010; Nat. Commun., 9, 3366, 2018), the HL-D-1/8 basedC$_4$N$_3$, is a perfect system to host such flat bands. For the flathigh-energy P-6m2 C$_4$N$_3$ structure, it displays the ferromagnetichalf-metallicity which is not related to the isolated flat bands. However, theP-6m2 C$_4$N$_3$ structure is dynamically unstable. Using a structure searchingmethod based on group and graph theory, we find that a new corrugated Pca21C4N3 structure has the lowest energy among all known C$_4$N$_3$ structures.This Pca21 C$_4$N$_3$ structure is an intrinsic ferromagnetichalf-semiconductor (Tc$\approx$241 K) with one semiconducting spin-channel(1.75 eV) and one insulating spin-channel (3.64 eV), which is quite rare in thetwo-dimensional (2D) systems. Its ferromagnetic semiconducting propertyoriginates from the isolated p$_z$-state flat-band as the corrugation shift theflat band upward to the Fermi level. Interestingly, this Pca21 C$_4$N$_3$structure is found to be piezoelectric and ferroelectric, which makesC$_4$N$_3$ an unusual transition-metal-free 2D multiferroic.

Flat-band based ferromagnetic semiconducting state in the graphitic C$_4$N$_3$ monolayer