Ab initio study of thermodynamic, electronic, magnetic, structural, and elastic properties of Ni${}_4$N allotropes

We have employed parameter-free density functional theory calculations to study the thermodynamic stability and structural parameters as well as elastic and electronic properties of Ni${}_4$N in eight selected crystallographic phases. In agreement with the experimental findings, the cubic structure with Pearson symbol cP5, space group $Pm\overline{3}m$ (221) is found to be the most stable and it is also the only thermodynamically stable structure at $T=0$ K with respect to decomposition to the elemental Ni crystal and N${}_2$ gas phase. We determine structural parameters, bulk moduli, and their pressure derivatives for all eight allotropes. The thermodynamic stability and bulk modulus is shown to be anticorrelated. Comparing ferromagnetic and nonmagnetic states, we find common features between the magnetism of elemental Ni and studied ferromagnetic Ni${}_4$N structures. For the ground-state Ni${}_4$N structure and other two Ni${}_4$N cubic allotropes, we predict a complete set of single-crystalline elastic constants (in the equilibrium and under hydrostatic pressure), the Young and area moduli, as well as homogenized polycrystalline elastic moduli obtained by different homogenization methods. We demonstrate that the elastic anisotropy of the ground-state Ni${}_4$N is qualitatively opposite to that in the elemental Ni, i.e., these materials have hard and soft crystallographic directions interchanged. Moreover, one of the studied metastable cubic phases is found auxetic, i.e., exhibiting negative Poisson ratio.

Figure 1

Schematic figure of the experimentally observed ground state of Ni${}_4$N cubic allotrope (referred to as $\alpha$-Ni${}_4$N in the text below). The Ni atoms are shown as larger green and the N ones as smaller blue spheres.

Figure 2

Schematic visualization of the Lifshitz structural allotrope $\delta$-Ni${}_4$N before structural relaxation (left) and after full geometry optimization of both the cell shape and internal atomic positions (right) with the Ni atoms shown as larger green and the N ones as smaller blue spheres.

Figure 3

Lifshitz structural allotrope $\beta$-Ni${}_4$N (upper panel) and the $\gamma$-Ni${}_4$N phase (lower panel). The Ni atoms are shown as larger green and the N ones as smaller blue spheres.

Figure 4

Structural allotropes $\eta$-Ni${}_4$N (upper left), $\zeta$-Ni${}_4$N (upper right), and $\varepsilon$-Ni${}_4$N (lower panel) with the Ni atoms shown as larger green and the N ones as smaller blue spheres.

Figure 5

Visualization of structural allotrope $\theta$-Ni${}_4$N before structural relaxation (upper panel) and after full geometry optimization of both the cell shape and internal atomic positions (lower panel) with the Ni atoms shown as larger green and the N ones as smaller blue spheres.

Figure 6

Total energy with respect to the minimum of the ground-state energy $\Delta E_{\mathrm{TOT}}$ as a function of volume $V$ (in Å${}^3$ per atom) for different ferromagnetic (FM) and nonmagnetic (NM) Ni${}_4$N allotropes. Curves ${\eta }^{*}$ and ${\zeta }^{*}$ correspond to states with the internal parameters kept constant and equal to 0.25 (see the explanation in Sec. 3).

Figure 7

Ab initio calculated volumetric dependences of internal atomic-position parameter ${}^{\mathrm{N}}u$ and ${}^{\mathrm{Ni}}u$ (see also Table ) in $\beta$-Ni${}_4$N, $\varepsilon$-Ni${}_4$N, $\zeta$-Ni${}_4$N, $\gamma$-Ni${}_4$N, $\eta$-Ni${}_4$N, and $\theta$-Ni${}_4$N allotropes.

Figure 8

Ab initio calculated volumetric dependences of internal atomic-position parameter ${}^{\mathrm{Ni}}u$ (see also Table ) in $\gamma$-Ni${}_4$N and $\theta$-Ni${}_4$N allotropes.

Figure 9

Atomic volume $V$ (in Å${}^3$ per atom) as function of pressure of the various Ni${}_4$N phases.

Figure 10

Ab initio computed pressure dependence of the enthalpy $H$ for the various Ni${}_4$N phases visualized for each pressure relatively with respect to the enthalpy of the FM $\alpha$-Ni${}_4$N.

Figure 11

Relation between the total energy difference $\Delta E_{\mathrm{TOT}}$ with respect to the most stable structure $\alpha$-Ni${}_4$N and the bulk modulus $B$ (in GPa).

Figure 12

Ab initio calculated dependences of magnetic moments $\mu$ (per Ni atom) in ferromagnetic (FM) Ni${}_4$N allotropes and fcc and bcc FM Ni as functions of the first-nearest neighbor interatomic distance Ni-N (in case of nickel nitrides) or Ni-Ni (in case of cubic nickel).

Figure 13

Ab initio predicted directional dependence of single-crystalline Young modulus for FM $\alpha$-, NM $\zeta$-, and NM $\eta$-Ni${}_4$N, as well as for ferromagnetic fcc Ni.

Figure 14

Computed directional dependences of single-crystalline area modulus $A$ of FM $\alpha$-Ni${}_4$N, fcc FM Ni, NM $\zeta$-Ni${}_4$N, and NM $\eta$-Ni${}_4$N.

Figure 15

Ab initio calculated pressure dependences of single-crystalline elastic constants of the FM $\alpha$-Ni${}_4$N phase.

Figure 16

Ab initio calculated pressure dependences of single-crystal elastic constants of the NM $\zeta$-Ni${}_4$N phase.

Figure 17

Ab initio calculated pressure dependences of single-crystal elastic constants of the NM $\eta$-Ni${}_4$N phase.

Figure 18

Calculated pressure dependences of homogenized elastic moduli (Young $Y$ and shear $G$) together with the Poisson ratio $\nu$ of the FM $\alpha$-Ni${}_4$N phase as obtained employing Voigt, Reuss, and Reuss-Voigt-Hill-Gilvarry (RVHG) homogenization schemes. The data points using the Hershey's method are not shown as they are very similar to the RVHG ones.