Thermodynamic and electronic properties of ${\mathrm{ReN}}_2$ polymorphs at high pressure

The high-pressure synthesis of rhenium nitride pernitride with a crystal structure that is unusual for transition metal dinitrides and high values of hardness and bulk modulus attracted significant attention to this system. We investigate the thermodynamic and electronic properties of the $P{2}_1/c$ phase of ${\mathrm{ReN}}_2$ and compare them with two other polytypes, the $C2/m$ and $P4/mbm$ phases, suggested in the literature. Our calculations of the formation enthalpy at zero temperature show that the former phase is the most stable of the three up to a pressure $p=170$ GPa, followed by the stabilization of the $P4/mbm$ phase at higher pressure. The theoretical prediction is confirmed by diamond anvil cell synthesis of the $P4/mbm$ ${\mathrm{ReN}}_2$ at $\approx 175$ GPa. Considering the effects of finite temperature in the quasiharmonic approximation at $p=100$ GPa we demonstrate that the $P{2}_1/c$ phase has the lowest free energy of formation at least up to 1000 K. Our analysis of the pressure dependence of the electronic structure of rhenium nitride pernitride shows the presence of two electronic topological transitions around 18 GPa, when the Fermi surface changes its topology due to the appearance of an electron pocket at the high-symmetry $Y_2$ point of the Brillouin zone while the disruption of the neck takes place slightly off from the $\mathrm{\Gamma }\text{−}A$ line.

Figure 1

The $P{2}_1/c$, $C2/m$, and $P4/mbm$ phases of ${\mathrm{ReN}}_2$. Big circles show the Re atoms, while the small ones the nitrogen singles (${\mathrm{N}}_1$) and dumbbells (${\mathrm{N}}_2$). (d) shows the structure of the $P4/mbm$ structure and the formed ${\mathrm{ReN}}_8$ rectangular prisms at $\approx 175$  GPa as seen along the $c$ axis.

Figure 2

(a) Formation enthalpy differences of the $P{2}_1/c$, $C2/m$, and $P4/mbm$ phases of ${\mathrm{ReN}}_2$ with respect to the $C2/m$ phase. (b) Gibbs free energy of formation of the $P{2}_1/c$, $C2/m$, and $P4/mbm$ phases at 100 GPa within the quasiharmonic approximation up to 1000 K. The dashed vertical line is used to make a correspondence between (a) and (b)—see text.

Figure 3

Reconstructed precession images showing the $\left(hk\overline{2}\right)$ reciprocal lattice plane of $P4/mbm$ ${\mathrm{ReN}}_2$ at $\approx 175$ GPa. The indexed reflections are encircled. Nonindexed reflections correspond to other grains of ${\mathrm{ReN}}_2$, which are present in the sample.

Figure 4

Calculated total and partial density of states in (a) $P{2}_1/c$, (b) $C2/m$, and (c) $P4/mbm$ phases of ${\mathrm{ReN}}_2$ at 0 GPa and $T=0$ K.

Figure 5

Electronic band structure of $P{2}_1/c$ ${\mathrm{ReN}}_2$ at (a) $p=0$ GPa and (b) 23 GPa. The lines shows the electronic bands parallel to the $\mathrm{\Gamma }\text{−}A$ and $\mathrm{\Gamma }\text{−}{Y}_2$ paths in the Brillouin zone.

Figure 6

Fermi surfaces of the $P{2}_1/c$ phase of ${\mathrm{ReN}}_2$ at 0 and 23 GPa.

Figure 7

Relative lattice parameters of the $P{2}_1/c$ phase of ${\mathrm{ReN}}_2$ calculated with a $k$ mesh of $(26\times 14\times 20)$.