Defects-driven magnetism in bulk $alpha$-Li$ _{3}$N

textit{Ab-initio} calculations based on density functional theory with local spin density approximation are used to study defects-driven magnetism in bulk $alpha$-Li$ _{3}$N. Our calculations show that bulk Li$ _{3} $N is a non-magnetic semiconductor. Two types of Li vacancies (Li-I and Li-II) are considered, and Li-vacancies (either Li-I or Li-II type) can induce magnetism in Li$ _{3}$N with a total magnetic moment of 1.0 $mu_{rm B}$ which arises mainly due to partially occupied N-$p$-orbitals around the Li vacancies. The defect formation energies dictate that Li-II vacancy, which is in the Li$ _{2}$N plane, is thermodynamically more stable as compared with Li-I vacancy. The electronic structures of Li-vacancies show half-metallic behavior. On the other hand N-vacancy does not induce magnetism and has a larger formation energy than Li-vacancies. N vacancy derived bands at the Fermi energy are mainly contributed by the Li atoms. Carbon is also doped at Li-I and Li-II sites, and it is expected that doping C at Li-I site is thermodynamically more stable as compared with Li-II site. Carbon can induce metallicity with zero magnetic moment when doped at Li-I site, whereas magnetism is observed when Li-II site is occupied by the C impurity atom and C-driven magnetism is spread over the N atoms as well. Carbon can also induce half-metallic magnetism when doped at N site in Li$ _{3}$N, and has a smaller defect formation energy as compared with Li-II site doping. The ferromagnetic (FM) and antiferromagnetic (AFM) coupling between the C atoms is also investigated, and we conclude that FM state is more stable than the AFM state.