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

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. Through performing both tight-binding and density functional theory calculations, we demonstrate that the experimentally realized graphitic carbon nitride (Adv. Mater., 22, 1004, 2010; Nat. Commun., 9, 3366, 2018), the HL-D-1/8 based C$_4$N$_3$, is a perfect system to host such flat bands. For the flat high-energy P-6m2 C$_4$N$_3$ structure, it displays the ferromagnetic half-metallicity which is not related to the isolated flat bands. However, the P-6m2 C$_4$N$_3$ structure is dynamically unstable. Using a structure searching method based on group and graph theory, we find that a new corrugated Pca21 C4N3 structure has the lowest energy among all known C$_4$N$_3$ structures. This Pca21 C$_4$N$_3$ structure is an intrinsic ferromagnetic half-semiconductor (Tc$\approx$241 K) with one semiconducting spin-channel (1.75 eV) and one insulating spin-channel (3.64 eV), which is quite rare in the two-dimensional (2D) systems. Its ferromagnetic semiconducting property originates from the isolated p$_z$-state flat-band as the corrugation shift the flat band upward to the Fermi level. Interestingly, this Pca21 C$_4$N$_3$ structure is found to be piezoelectric and ferroelectric, which makes C$_4$N$_3$ an unusual transition-metal-free 2D multiferroic.