## III. RESULTS AND DISCUSSIONS 

## A. Structural properties 

In experiment with the powder X-ray diffraction patterns, Cao et al 15 have reported that InN$_{y}$Co$_{3}$ and InN$_{y}$Ni$_{3}$ ( y ∼ 1.0 and 0.8, respectively) have the cubic anti-perovskite structure with the space group 221( Pm ¯ 3 m ) and the corresponding lattice parameters were 3.854 ˚ A and 3.844 ˚ A, respectively. Starting from the experimental data, we have calculated the total energies of unit cell at a series of volumes for each compound in the paramagnetic (PM) and ferromagnetic (FM) states. The results are presented in Fig. 1. It is found that the energy difference between the FM and PM states is -0.0397 eV (-0.226 eV) in the LDA (GGA) calculations for InNCo$_{3}$ and 0.0 eV for InNNi$_{3}$. The total magnetic moment of InNCo$_{3}$ is about 2.14 µ$_{B}$ (2.91 µ$_{B}$ ) and the local magnetic moment of each In ion is about 0.69 µ$_{B}$ (0.94 µ$_{B}$ ) in LDA (GGA) calculations. The total magnetic moment of InNNi$_{3}$ and the local magnetic moment of each Ni atom are zero. These indicate that the ferromagnetic state is energetically favorable to InNCo$_{3}$ and the ground state of InNNi$_{3}$ is paramagnetic state (non-magnetic). The obtained equilibrium lattice constant ( a$_{0}$ ), bulk modulus ( B ), and first pressure derivative of bulk modulus ( B ′ ) of InNCo$_{3}$ and InNNi$_{3}$ are listed in Table I. In our calculations, the predicted lattice constant of InNCo$_{3}$ is slightly larger than that of InNNi$_{3}$, which is oppsite to the trend reported in experiment $^{15}$. This may be due to the deviation of the Ni/N atomic ratio from the ideal stoichiometry in experiment for InN$_{y}$Ni$_{3}$. In addition, it can be seen that the deviations of the LDA (GGA) lattice constants of both the InNCo$_{3}$ and InNNi$_{3}$ with respect to the experimental values are less than 2.6% (1.0%). That is to say, the calculated equilibrium lattice constants of InNCo$_{3}$ and InNNi$_{3}$ are in excellent agreement with the experimental data $^{15}$. 

## B. Elastic properties 

The calculated independent elastic constants for single crystal of InNCo$_{3}$ and InNNi$_{3}$ are listed in Table II. Based on the Voigt-Reuss-Hill approximation $^{33–35}$, the elastic moduli of InNCo$_{3}$ and InNNi$_{3}$ are estimated and the results are listed in Table II. For the bulk moduli of InNCo$_{3}$ and InNNi$_{3}$, the estimations based on the independent elastic constants agree well those obtained by the fit of the Birch-Murnaghan 3rd-order EoS. To the best of 

