Thermodynamic and electronic properties of ReN$_2$ polymorphs at high-pressure

High pressure synthesis of rhenium nitride pernitride ReN$_2$ with crystal structure 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 P2$_1$/c phase of ReN$_2$ and compare them with two other polytypes, 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 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 ReN$_2$ at $\approx$175 GPa. Considering the effects of finite temperature in the quasi-harmonic approximation at p=100 GPa we demonstrate that the P2$_1$/c phase has the lowest free energy of formation at least up to 1000 K. Our analysis of the pressure dependence of the electronic structure of the rhenium nitride pernitride shows a presence of two electronic topological transitions around 18 GPa, when the Fermi surface changes its topology due to an appearance of a new electron pocket at the high-symmetry Y$_2$ point of the Brillouin zone while the disruption of a neck takes place slightly off from the $\Gamma$-A line.


I. INTRODUCTION
High pressure diamond anvil cell (DAC) experiment is a successful approach to establish wide variety of physical conditions for synthesizing materials [1][2][3]. Though, exploring metastable phases of materials, which is a challenging experimental task because one has to achieve control of the small free energy barriers separating different polymorphs. On the other hand, computational high-throughput approaches [4,5] with sophisticated structure prediction algorithms [6,7] have entered into the field to advance the materials discoveries.
The launch of the materials genome initiative (MGI) [8] has further accelerated this trend [9,10] and triggered the need to develop infrastructure to store and utilize the data calculated for different materials. Large databases, such as NOMAD [11], Materials Project [12], AFLOW [13], BioExcel [14], Topological Material DataBase [15], etc. have been created, which contain different properties of both, existing and hypothetical materials. Analyzing the data from Materials Project database Sun et al. observed that nitrides have the largest thermodynamic scale of metastability defined as the energy differences between stable and metastable structures, ≈190 meV/atom [16], making this class of materials very interesting for further experimental and theoretical exploration.
At the same time, DAC has been actively used to synthesize novel nitrogen rich phases of transition metal nitrides [2,17]. One applies high pressure and temperature in the chamber filled with molecular nitrogen or azides (AN 3 , A=Li, Na, etc.) and metals, such as rhenium, tungsten, osmium etc. [18,19]. The synthesized metastable and nitrogen rich materialspolymeric forms of nitrogen chains have also been observed [17], represent a class of highenergy-density materials often with superior mechanical properties [19].
One challenge of using the high-pressure experiment to discover novel materials with 4 attractive properties is to quench them to ambient conditions. The work by Bykov et al. [19] have shown that ReN 2 compound in the monoclinic P2 1 /c phase discovered in a DAC experiment can be also synthesized in larger amount in a large-volume press at lower pressure. This compound has a crystal structure unusual for transition metal dinitrides MN 2 . It contains covalently bound dinitrogen dumbbells and discrete nitrogen atoms and represents an example of a mixed nitride-pernitride compound. Quenched to ambient pressure, the rhenium nitride pernitride showed high mechanical properties, hardness of 36.7(8) GPa and very high value of the bulk moduli of 428(10)GPa. The P2 1 /c phase was not reported in earlier experiments. Remarkably, despite numerous theoretical studies of the Re-N system at this composition, it was not predicted theoretically even with the use of advanced structure prediction algorithms.
Kawamura et al. [20] has reported the synthesis of ReN 2 at 7.7 GPa and 1473-1873 K, with hexagonal P6 3 /mmc (MoS 2 type) structure. Elastic and mechanical properties of the phase have been investigated using T=0 K density functional theory (DFT) calculations [21]. In the same year, Du et al. [22] based on DFT calculations has proposed the tetragonal P4/mmm phase of ReN 2 underlined by the existence of the same phase for ReB 2 [23]. However, static 0 K DFT calculations by Wang et al. [24] have ruled out the P4/mmm phase as ground state of ReN 2 . Instead, the calculations have indicated that the monoclinic C2/m structure of ReN 2 is more stable at 0 K than the experimentally found P6 3 /mmc phase. Furthermore, it has been shown that above 130 GPa the P4/mbm phase becomes the favoured phase. A computational structural search for stable and metastable rhenium-nitrides up to 100 GPa pressures has been conducted [25] using a sophisticated evolutionary algorithm implemented into USPEX [6]. The study has confirmed the C2/m phase of ReN 2 as ground state between 5 the investigated pressure range.
In the present paper we investigate the thermodynamic stability of the P2 1 /c phase with respect to the competing tetragonal P4/mbm and C2/m monoclinic phases in the pressure range between 0 GPa and 180 GPa. We use first principles electronic structure calculations and a quasi-harmonic approximation for the lattice dynamics and establish that the former is indeed thermodynamically more stable then two other polymorphs at pressure up to ≈170 GPa. At higher pressure the calculations predict the stabilization of the P4/mbm phase. As this phase was not reported in earlier experiments, we carry out the high-pressure synthesis of ReN 2 in diamond anvil cell at ≈175 GPa. The theoretical prediction is verified by a characterization ot the synthesized sample, which confirms the stabilization of the P4/mbm ReN 2 . Moreover, we calculate electronic properties of the P2 1 /c phase and show the presence of two electronic topological transitions at ≈ 18 GPa.

II. COMPUTATIONAL DETAILS
Simulations of the phase stability at T=0 K have been performed using the Quantum Espresso (QE) program package [26] with PAW pseudo-potentials [27] using 60 Rydberg for kinetic energy cutoff and 350 Rydberg for the density and potential cutoff. The exchange correlation energy was approximated by the Perdew-Burke-Ernzerhof generalized gradient functional (PBE-GGA) [28]. In case of the P4/mbm structure we have used a (16 × 16 × the k-mesh density. Fermi surfaces have been calculated and visualized using Xcrysden [30] and VESTA [31].
In all VASP calculations the energy cutoff was set to 700 eV. Table I lists Table I. The deviation are less than 4 GPa. The largest error, 4%, has been observed for σ yy in case of comparing the P2 1 c and C2/m phases. In the enthalpy calculations the hcp phase of rhenium [37] and the cubic gauche phase of nitrogen (which is stable up to ≈150 GPa) have been used as the end-member states [38].

III. EXPERIMENTS
The piece of Re metal was placed in a sample chamber of a BX90 diamond anvil cell [39] equipped with Boehler-Almax type diamonds (40 µm culet diameters). Nitrogen served as a pressure-transmitting medium and as a reagent [40]. The DAC was compressed up to the target pressure of ≈ 175 GPa and laser-heated using double sided laser-heating systems installed at the at the Bayerisches Geoinstitute (BGI, Bayreuth, Germany) [41]. The sample was studied by means of powder and single-crystal X-ray diffraction at the synchrotron beamline ID11 of the ESRF using a monochromatic X-ray beam focused to ≈ 0.3 × 0.3 µm 2 .
For the single-crystal XRD measurements the sample was rotated around the vertical ω-axis in a range ±38 • . The diffraction images were collected with an angular step ∆ω = 0.5 • and 7 an exposure time of 10 s/frame. For analysis of the single-crystal diffraction data (indexing, data integration, frame scaling and absorption correction) we used the CrysAlisPro software package. To calibrate an instrumental model in the CrysAlisPro software, i.e., the sample-todetector distance, detector's origin, offsets of goniometer angles, and rotation of both X-ray beam and the detector around the instrument axis, we used a single crystal of orthoenstatite In Figure 2a we present the calculated formation enthalpy differences (∆H) for the P2 1 /c and P4/mbm phases in the pressure interval from 0 GPa to 180 GPa relative to the values obtained for the C2/m phase. The relevance of spin-orbit coupling (SOC) effects on the formation energy differences between the P2 1 /c and C2/m phases at 0 GPa can be ruled 8 out by the electronic band structures calculated without SOC, see Materials Project [12] IDs: mp-1077354 and mp-1019055. The figure shows that the recently sythesized P2 1 /c phase is the the most stable of the three up to ≈170 GPa at T= 0 K. Above this pressure the tetragonal phase is favored over the two monoclinic structures. Another interesting observation is that the P2 1 /c phase is favored over the C2/m monoclinic phase in the whole pressure range. The dynamical stability at 0 GPa has been proven for all the three phases earlier [19,25,46]. The effect of temperature on the relative stability of the three phases is analyzed in Fig.2b). The figure shows the Gibbs free energy of formation of the P2 1 /c, C2/m and P4/mbm phases at 100 GPa up to 1000 K. The dashed vertical line in Fig.2a) and b) is used to establish the correspondence between the zero temperature static calculations (triangles) and calculations including the effects of the lattice dynamics, e.g. the zeropoint motion (circles). One sees that the relative order of the three phases is not changed by temperature, though the formation energy differences become slightly smaller with increasing temperature. In Fig. 2b) Table III and supplementary CIF file [47] for details. Figure 4 shows the X-ray reflections corresponding to the (hk-2) reciprocal lattice plane of the P4/mbm phase. Re atoms are coordinated by eight nitrogen atoms forming ReN 8 rectangular prisms. These prisms are stacked along the c-axis (short edge of the rectangular prism) sharing faces and forming infinite columns. The columns also share common edges and additionally interconnected via N-N bonds as shown in Fig.1d). The refined N-N distance of 1.34Åis close to the expected value of a single N-N bond at this pressure [48] suggesting that nitrogen forms a pernitride anion [N-N] 4− , while Re has an oxidation state +4.
To understand the electronic properties of ReN 2 polymorphs at 0 GPa we have calculated the electronic structure of all three phases. The calculated total and partial density of states (DOS) are shown in Fig.3 a), b) and c). For each phase the first panel shows the total DOS curves including the total rhenium 5d partial DOS as shaded curves. One should note here that the tetragonal polymorph has half as many atoms than the other two, therefore the twice of DOS is plotted. The second panel compares the nitrogen 2p partial DOS for the differnt types (N1 and N2) nitrogens. In comparison, the total DOS of the three structures show different characteristics at the Fermi level. One observes nearly semimetallic behavior for P2 1 /c and significantly more typical metalic behaviour for the two competing phases.
For P2 1 /c phase the calculations predict low DOS value at Fermi energy. In the case of C2/m, one sees instead a peak close to the Fermi energy, though the Fermi level is located in a valley between the two peaks. This could indicate smaller contribution from the oneelectron energy to the structural stability of the C2/m phase relative to P2 1 /c phase. In comparison, for P4/mbm one observes a finite value of the total DOS with a plateau in a vicinity of the Fermi energy. It also indicates smaller contribution from the one-electron term to the structural stability of the phase. For each of the polymorphs, one notes significant hybridization between Re 5d and N 2p orbitals. However, the strongly distorted trigonal prismatic local environments of the Re atoms do not allow any deeper analysis of the Re 5d orbitals using crystal field theory.
To analyze the nearly semimetallic DOS of the P2 1 /c phase we have calculated the electronic band structure in the Brillouin zone parallel to the Γ-A and Γ-Y 2 paths, see the inset figure of the Brillouin zone in Fig. 5. Fig. 5a) shows the band structures at ambient pressure while b) shows it for 23 GPa. One observes that the unoccupied band at Y 2 and the occupied band along Γ-A line could be responsible for the semimetallic behavior at p=0 GPa.
However, there are small electron pockets along the lines in the Γ-A-E-Z plane (not shown) which are responsible for the finite DOS at the Fermi energy. Interestingly, with increasing pressure (Fig. 5b) the band at Y 2 passes the Fermi energy. Besides this, one sees that the band along the Γ-A path crosses the Fermi energy, but not in the Γ-A-Y 2 plane, where the valence band just touches the Fermi energy. Accrodingly, one expects that the Fermi surface consists of a simple hole pocket around the Γ-A line at low pressures, while at higher pressure an additional electron pocket is expected around the Y 2 point in the Brillouin zone.
Thus, an electronic topological transition (ETT), that is the change of the Fermi surface topology [49] should be observed with increased pressure.
To investigate the topology of the Fermi surface and to calculate the pressure at which ETT occurs, we have increased the accuracy of the calculations by increasing the density of k points in the electronic structure calculations. We have selected the k point sampling (26 × 14 × 20) which provides sufficient resolution for the study of the ETT and its influence on the materials properties. Figure 6 shows the Fermi surface of the P2 1 /c phase of ReN 2 at 0 GPa and 23 GPa. The figure underlines the appearence of the additional electron pocket around Y 2 with increasing pressure. Importantly, one sees the second ETT associated with a disruption of the neck between the two shits of the Fermi surface slightly off from the Γ-A line. The ETT is connected to the band which touches the Fermi energy along this line at p=0 GPa (Fig. 6) and shifts below it with increasing pressure (Fig. 6b). Based on the chosen 1 GPa pressure grid the calculations have shown that the two ETTs occur at 18±0.5 GPa.
Experimental identification of the pressure-induced ETT is a non-trivial task, as exem-plified by cases of Zn [50][51][52][53] Os [54][55][56][57][58] and Fe [59] The point is that the thermodynamic potential and its first derivatives are not affected by an ETT, the second derivatives may show weak square-root shaped peculiarities, while the strong peculiarities are observed only for the third derivatives of the thermodynamic potential, leading to a classification of ETTs as the so-called "2 1/2" order phase transitions. Indeed, Fig. 7 shows that, as expected, the pressure dependence of the lattice parameters ratios c/a and b/a obtained in highly converged calculations at T=0 K does not show any peculiarities. However, as pointed out in [59], the ETT should lead to peculiarities of the thermal expansion, and it can show up in the lattice parameters ratios measured at finite temperatures due to anisotropy of the thermal expansion. The effect was indeed observed experimentally in hcp Fe [59] and Os [58]. Interestingly, comparing the calculated zero temperature lattice parameters ratios in Fig. 7 with room temperature experiment of Ref. [19] we observe good agreement between the two data sets. But the experimental information at pressure around 18 GPa is, unfortunately, missing. Therefore, careful examination of the lattice parameters of P2 1 /c phase of ReN 2 can be used to investigate the effect of the predicted ETT on the properties of this compound.

V. CONCLUSIONS
We have investigated the thermodynamic and electronic properties of the novel P2 1 /c phase of ReN 2 in comparison with previously suggested, competing phases. Our density functional theory calculations at T=0 K have shown that the P2 1 /c phase is the most stable polymorph of the three studied modifications of the compound up to ≈170 GPa. Above this pressure the tetragonal P4/mbm becomes more stable. This calculation is supported by the experiment. Using the quasi-harmonic approximation we have shown that the P2 1 /c phase is aslo stable phase up to 1000 K at p=100 GPa. Moreover, our electronic structure calculations have shown that two nearly co-existing electronic topological transitions occur in the P2 1 /c phase of ReN 2 with increasing pressure. We propose additional experiments that should verify the theoretically predicted ETT. We are grateful to Pavel Sedmak at the ESRF for providing assistance in using beamline.