  First-principles based on density functional theory is used to study the
phase stability, elastic, magnetic, and electronic properties of cubic
(c)-Fe$_4$C. Our results show that c-Fe$_{4}$C has a ferromagnetic (FM) ground
state structure compared with antiferromagnetic (AFM) and nonmagnetic
(NM)states. To study the phase stability of c-Fe$_4$C, BCC Fe$_4$C, FCC
Fe$_4$C, and BCC Fe$_{16}$C, where C is considered at tetrahedral and
octahedral interstitial sites, are also considered. Although, the formation
energy of c-Fe$_4$C is smaller than BCC Fe$_4$C, but the shear moduli of
c-Fe$_4$C is negative in the FM and AFM states indicating that c-Fe$_4$C is
dynamically not stable in the magnetic (FM/AFM) states. However, NM state has
positive shear moduli which illustrates that instability in c-Fe$_4$C is due to
magnetism and can lead to soft phonon modes. The calculated formation energy
also shows that c-Fe$_4$C has higher formation energy compared with the FCC
Fe$_4$C indicating no possibility of c-Fe$_4$C in low carbon steels at low
temperature. The magnetic moment of Fe in c-Fe$_4$C is also sensitive to
lattice deformation. The electronic structure reveals the itinerant nature of
electrons responsible for metallic behavior of c-Fe$_4$C.
