Novel triadius-like N4 specie of iron nitride compounds under high pressure

Various nitrogen species in nitrides are fascinating since they often appear with these nitride as superconductors, hard materials, and high-energy density. As a typical complex, though iron nitride has been intensively studied, nitrogen species in the iron–nitrogen (Fe-N) compounds only have been confined to single atom (N) or molecule nitrogen (N2). Using a structure search method based on the CALYPSO methodology, unexpectedly, we here revealed two new stable high pressure (HP) states at 1:2 and 1:4 compositions with striking nitrogen species. The results show that the proposed FeN2 stabilizes by a break up of molecule N2 into a novel planar N4 unit (P63/mcm, >228 GPa) while FeN4 stabilizes by a infinite 1D linear nitrogen chains N∞ (P-1, >50 GPa; Cmmm, >250 GPa). In the intriguing N4 specie of P63/mcm-FeN2, we find that it possesses three equal N = N covalent bonds and forms a perfect triadius-like configuration being never reported before. This uniqueness gives rise to a set of remarkable properties for the crystal phase: it is identified to have a good mechanical property and a potential for phonon-mediated superconductivity with a Tc of 4–8 K. This discovery puts the Fe-N system into a new class of desirable materials combining advanced mechanical properties and superconductivity.

Nitrogen (N) is the most abundant element in the earth's atmosphere and is one of the least studied elements regarding the composition of the Earth 1 . At standard temperature and pressure (T = 298 K; P = 1 atm), elemental nitrogen is a gas, consisting of diatomic N 2 molecules that are bound by stiff covalent triple bonds. So the molecule is chemically inert and hardly dissociate and not many higher molecular or extended structures are known for nitrogen other than N 2 under normal conditions. Syntheses of useful nitrides with various nitrogen species rely on chemical methods via, e.g., photochemical reaction, electrochemical synthesis [2][3][4][5][6][7][8] . A few higher molecular units are known, such as photolytic cyclic N 3 3,4 , the tetrahedral N 4 molecule 5 , the N 5 − anion 6 , and the N 5 + in a crystalline phase of N 5 + SbF 6 −7,8 . Note that, among these units, though the tetrahedral N 4 has been a form of the N 4 unit for synthesis, it is observed as a metastable species with a lifetime exceeding one microsecond.
As a typical transition-metal nitride, the Fe-N system is extensively investigated to explore its compounds in the interior layers of earth following its first discovery since 19 century 2 . A rich Fe-N chemistry exists, most synthesized compounds have a Fe/N ratio higher than unity, such as α″-Fe 16 N 2 , α′-Fe 8 N, γ′-Fe 4 N, Fe 7 N 3 , Fe 3 N x (x = 0.75-1.4), Fe 3 N, Fe 2 N and FeN 4-10 . Among, the FeN compound is most nitrogen-rich iron nitride reported benign synthesized in a high pressure apparatus thus far 10 . This synthesis of FeN spurred the endeavors in search for Fe-N compounds with a more nitrogen content exceeding the FeN compound with other nitrogen species. However, in contrast to the Fe-rich compounds, there is little work on the N-rich iron nitrides, both from the experimental and theoretical sides. Only few theoretical investigations are available to report that an N-rich iron pernitride (FeN 2 ) crystallizes in the space group R-3m at 17 GPa (1000 K) 11 and transforms an orthorhombic Pnnm structure up to 22 GPa 12 obtained by assuming the parent metal under pressure. All these known Fe-N compounds adapt single N atom or molecule N 2 configuration and keep iron 6-coordination.
In order to systematically explore the possibility of obtaining new stable N-rich iron nitrides, and especially to examine the possibility of attaining new nitrogen species at HP, we here present extensive structure searches of stoichiometric Fe-N compounds under various pressures ranging from 0 to 300 GPa, using an unbiased particle swarm optimization (PSO) algorithms for crystal structure predictions 18 . This swarm-intelligence high-throughput searching has proven effective in revealing new compositions favorable to form in large sets of multicomponent Ca-H, Li-B, Xe-N, Cs-N systems 16,17,19,20 . The effectiveness has been also demonstrated by recent successes in predicting high-pressure structures of various systems, and their several experimental confirmations [21][22][23][24][25][26][27][28][29][30] . In this work, we proposed new N-rich iron nitrides at 1:4 and 1:2 compositions under HP. Identifying their nitrogen species, it is strikingly found that the nitrogen species evolve from a N 2 unit to a novel N 4 units, and eventually N∞ with the increase of N contents. In N 4 unit, we find that it possesses three N = N covalent bonds and one lone pair, which leads it to forms an unknown triadius-like configuration. Its structural uniqueness gives rise to a set of remarkable properties for the crystal P6 3 /mcm phase with an unexpectedly Tc of 4~8 K and a good mechanical property.

Methods
The developed CALYPSO structure prediction method designed to search for the stable structures of given compounds has been employed for the investigation of phase stability of Fe-N systems in N-rich stoichiometry under HP. We performed structure predictions of stoichiometric Fe 1-i N i (0 < i < 1) with simulation cell sizes of 1-4 formula units (f.u.) in a pressure range from 0 to 300 GPa. The local structural relaxations and electronic band structure calculations were performed in the framework of density functional theory within the generalized gradient approximation (GGA) and frozen-core all-electron projector-augmented wave (PAW) method 31,32 , as implemented in the VASP code 33 . The PAW pseudopotentials with 3d 7 4s 1 and 2s 2 2p 3 valence electrons were adopted for Fe and N, respectively. The kinetic energy cutoff for the plane-wave basis set is taken as 800 eV and a dense k-point grid with the spacing of 2π × 0.03 Å −1 was used to sample the Brillouin zone, which was shown to yield excellent convergence for total energies (within 1 meV/atom). The phonon calculations were carried out by using a finite displacement approach through the PHONOPY code 34 . The electron-phonon coupling (EPC) of P6 3 /mcm-FeN 2 was calculated within the framework of linear response theory through the Quantum-ESPRESSO code 35 . A 2 × 2 × 2 q mesh was used in the interpolation of the force constants for the phonon dispersion curve calculations. A MP grid of 12 × 12 × 12 was used to ensure k-point sampling convergence, which approximates the zero-width limits in the calculations of EPC parameter. We Elastic constants were calculated by the strain-stress method and the bulk modulus and shear modulus were thus derived from the Voigt-Reuss-Hill averaging scheme 36 .

Results and Discussions
We focused our structure search on the phase stabilities of Fe-N systems in N-rich stoichiometry by calculating the formation enthalpy of various Fe 1-i N i (0 < i < 1) compounds in a pressure range of 0 to 300 GPa. The formation enthalpy was calculated with respect to the decomposition into FeN and N, as Δh(  FeN is exceedingly stable with respect to the binary Fe-N system having a max nitrogen ratio below 50% (Fe 4 N, Fe 3 N, Fe 2 N) under HP, as revealed by our study (Fig. S1) and relative experimental studies [4][5][6][7]10 . The stable structure of FeN below 50 GPa has space group F-43m structure (Fig. S2, ref. 10 ). At 50 GPa, the FeN transforms into a Pnma structure (Fig. S3), followed by a cubic P2 1 3 structure above 150 GPa (Fig. S4). All these FeN structures takes on isolated N atomic sublattice and keeps six-fold coordinated by Fe forming edge-sharing FeN 6 octahedron in their corresponding stable pressure ranges.
Magnetism plays a central role in iron and its compounds. Therefore it is necessary to confirm the role of magnetism on the stability of these Fe-N structures. From our spin-polarized calculations, we find that every Fe atom of Fe-N compounds possesses a magnetic moment of 0.21-1.68 μ B under pressure (<50 GPa), which is substantially lower than that of the pure Fe solid (2.2 μ B ). Meanwhile, the magnetic moment will decrease rapidly with increasing pressure and be completely quenched as pressure exceeds 50 GPa. As an example, we performed the energy calculations of FeN after considering magnetism and found that the magnetic effect did not change the phase transition sequence but slightly shifted the phase transition pressure. According to a model derived from a Slater-Pauling type behavior 37 , the magnetization with increasing amount of N becomes decrease in the Fe-N system. It thus is plausible to perform the structure search and enthalpy calculations without considering the magnetic effect under HP in the N-rich Fe-N compounds.
Analysis the convex hull for researching the thermodynamically stable in the Fig. 1a, we can get a main result as follows: at P = 0 GPa, the Δh of all N-rich stoichiometry are positive, meaning that the nitrogen ratio above 50% Fe-N system are not stable. This is consistent with the experimental observation that no Fe-N compound whose which the nitrogen content exceeds the iron content can form at ambient pressure; at 100 GPa, stable stoichiometries of FeN 2 and FeN 4 emerge on the convex hull as the most stable stoichiometry, this situation preserves up to 300 GPa. Detailed pressure-composition phase diagram for these two N-rich species is presented in Fig. 1b. Moreover, we performed phonon spectra calculations using the finite-displacement method to assess the dynamical structural stability of their structural phases at desired pressure. No imaginary frequency was found for their structures, which indicated that they are dynamically stable (Figs S5-10).
In the FeN 2 compound, the low pressure crystalline phase is a trigonal R-3m structure at approximately 22 GPa, above which an orthorhombic Pnnm structure becomes more favorable, consistent with the previous reports 11,12 . These two structures both contain a dinitrogen unit and N-sharing six-fold FeN 6 octahedrons (Figs S11,12). Analysis of the dinitrogen unit indicating a strong N-N covalent bond and the dinitrogen unit can be formulated as (N-N) 2− and (N = N) 2− in the R-3m and Pnnm structures, respectively 11,12 . Recently, these phases have been synthesized and verified by the experiment under high pressure and high temperature 38 . Upon to 228 GPa, an unknown energetically more favored hexagonal P6 3 /mcm structure was firstly discovered (Fig. 2a). Tracing the volume change of the phase transition from Pnnm to P6 3 /mcm, it is found that this transition is a first-order accompanied by a volume drop of 3.5%. Viewing the P6 3 /mcm structure, it contains two types of N atoms occupying two different 2c Wyckoff sites as middle N m and peripheral N p (Fig. 2a): the N p atom is shared by four Fe atoms forming a FeN 6 octahedron with Fe-N p distances of 1.76 Å, while the N m atom bonds with three N p atoms forming perfect N 4 unit with a bond length of ~1.25 Å (at 300 GPa). These Fe-N p distances are shorter than the sum (1.92 Å) of covalent radii of Fe and N atoms. In such exotic structure, each Fe forms 6 Fe-N p bonds with those 6 neighboring N p atoms and each N p atom has four neighbors Fe-N p bonds and a N p -N m bond. The N p -N m distance (1.25 Å) is slightly longer than the double N = N bonds (1.20 Å) can be verified as the N = N bonding nature.
In the FeN 4 compound, the energetically favored structure of FeN 4 (stable above 50 GPa) has a triclinic structure with a low P-1 symmetry (Fig. 2b). The nitrogen sublattice in this structure takes on a polyacetylene-like infinite linear chain structure with a closest NN separation (N∞) in the range of 1.32−1.34 Å. Polyhedral view of the P-1 structure, it forms octahedrons linked together by the NN bonds of N∞, with Fe atoms sitting at the center of octahedrons and being 6-fold bonded to the N atoms of N∞. Up to 250 GPa, a surprising transition from such the 6-fold P-1 structure to a 8-fold structure takes place. This 8-fold structure adopts a high symmetric orthorhombic Cmmm structure, which have similar N∞ structural character of the P-1 phase (Fig. 2c). Analysis of their ELF (Figs S15,16) suggests that the N atoms of N∞ are in sp 2 hybridization, each N forms two σ bonds with two neighboring N atoms and one Fe-N bond. Due to the N∞ units, the electronic structures of these two phases both exhibit the metal properties (Figs S17,18). Such N∞ units and electronic properties can be also found in works for LiN 3 , NaN 3 , and CsN 3 under high pressure [39][40][41] . For the two phases, their different is that the Cmmm phase are composed of a Fe 8-coordination decahedrons, in stark contrast with 6-coordination in the octahedrons of P-1 structure (Fig. 2b,c). Analyses of the coordination number of Fe, we note that conventional coordination chemistry of Fe consists of four-, five-, and six-coordinate metal ions, while coordination numbers higher than six are seldom observed in only discrete molecules and polynuclear metal clusters [42][43][44][45][46][47][48][49] . Despite much effort, the Fe atoms are found to be very resistive to become 8-fold coordinate in solids, and the search for solids containing 8-coordinate has so far been scarcely successful. To our knowledge, this is first time to identify the 8-fold coordination of Fe atoms in the Fe-N compounds.
Return to identify the nitrogen species of Fe-N compounds (FeN, FeN 2 and FeN 4 ), it is strikingly found that the nitrogen sublattice evolve from isolated N atom to in turn the N 2 unit, the N 4 unit and eventually N∞ with the increase of N contents. It is noted that these features, except for the N 4 units, can be often found in alkali metal azides (Li-N, Na-N, Cs-N) [39][40][41] . Tracing the history of the related N 4 units, the first investigation into N 4 units can be traced back to a reported about successfully isolated neutral N 4 molecule in the gas phase a decade ago 5 , but the lifetime of the N 4 molecule is only around 1 microsecond. Recently, a charged N 4 species as predicted in the CsN crystal is substantially stabilized by strong cation-anion interactions 17 . This predicted N 4 4− anion has an open-chain structure containing two terminal single-bonds and one internal double-bond. Being different that, we found here the N 4 units of P6 3 /mcm-FeN 2 has three equal N = N bond and forms plane N 4 units like as the triadius star (Fig. 3a). We also try to look for crystal structures that incorporate such special N 4 units in the other systems. But no crystal structure is found so far. Here the exotic N 4 unit having strong N-N covalent bonding can be clearly shown by its ELF (Fig. 3b,c). Each N m atom possesses one lone pair of electrons at its p z orbital and forms three N = N covalent bonds with peripheral N p (N1, N2, N3) atoms (Fig. 3c), owing to its sp 2 hybridization.
Notice that the p z orbital of N is much lower in energy than that of Fe and form a strong overlapping between p z orbitals of N. This fact justifies us to study the electronic structure of hypothetic neutral N 4 unit first before researching the electronic properties of its crystal P6 3 /mcm structure. The atomic model for neutral N 4 unit is sketched in Fig. 3d with several typical symmetry elements of point group D 3h marked out explicitly (Table S7). The linear combination of three p z orbitals of peripheral N p can form orbitals with A 2 ″ and E″ symmetry, while the p z orbital of N m belongs to A 2 ″ (Table S8). Therefore, for a bare N 4 unit, the four p z orbitals of N would constitute two nonbonding (E″), one bonding and one antibonding molecular orbitals. According to the diagram in Fig. 3e, the HOMO derives from the nonbonding p z orbitals of N p atoms, while the LUMO comes mainly from the antibonding of p z orbitals between N m and N p . Meanwhile, the bader charge analysis reveals the fact of electron abundant N 4 units and electron deficient Fe atoms (the less electronegative Fe loses 1.67 electrons per atom, and N p directly bonded to Fe obtains 1.13 electrons per atom, while the N m atom remains almost neutral). Such unoccupied antibonding orbitals between N m and N p can minimize the influence of excess electrons, which can explain why the P6 3 /mcm structure with the N 4 unit is fairly stable.
To probe the electronic structures of the P6 3 /mcm structure, we calculated the band structure and density of states (DOS), finding that it exhibit metallic features. A comparison of the band structures of FeN 2 , Fe 0 N 2 (all Fe atoms removed out of the lattice and a uniform compensated background charge (8e/Fe) is applied to preserve the total valence electrons of the system) is performed (Fig. 4a). The difference between the resultant band structure of a hypothetical Fe 0 N 2 system (red dash lines) with the realistic one of FeN 2 indicates that the Fe atoms not only act as electron donors but also bond with N. As shown in Fig. 4b, the DOS reveal that the Fe-d and N-p states are energetically degenerate in the valence bands region, which facilitates the Fe−N hybridization and the formation of covalent bond. These results offer further support for the ionic and covalent bonding nature of Fe-N bonds as described above. We noted that the large bands crosses the E f and the bands appears "flat band-steep band" characteristic 50 around the E f . These are typical features favorable for strong EPC and superconductors. Using the linear response theory, we calculated the phonon DOS (PHDOS), Eliashberg function α 2 F(ω) and the strength of the e-ph coupling λ(ω) of the P6 3 /mcm structure (Fig. 4c). The Eliashberg function integrates to a λ = 0.62 and gives the logarithmic average ~400 cm −1 , being much closer to the oP10-FeB 4 value of ~430 cm −1 . The main contributor to the EPC originates from the mixed Fe-N modes below 1000 cm −1 (85% of λ) and the high-frequency vibrations from N 4 units (15% of λ) (Fig. 4d). Using the Allen-Dynes equation 51 , with the calculated ω log of 643 K and typical µ* = 0.1~0. 15, it reveals that the P6 3 /mcm structure is a weak-coupling BCS-type superconductor with a superconducting T c of 4~8 K. Moreover, the most commonly known transition-metal pernitrides crystallize can act as hard materials, such as MnN 2 (H V = 19.9 GPa), CoN 2 (16.5 GPa), and NiN 2 (15.7 GPa) 52 . Fe in  the same period as Mn, Co and Ni and the P6 3 /mcm-FeN 2 phase is as a typical transition-metal pernitride, so the P6 3 /mcm-FeN 2 is also regarded as hard material here. We calculated and obtained its mechanical properties including bulk modulus B (341 GPa), shear modulus G (247 GPa), young's modulus Y m (597 GPa), and Vicker's hardness (H V = 29 GPa) at ordinary pressure. As expect, the result shows that the P6 3 /mcm-FeN 2 phase exhibits highly incompressible. Bases on a correlation the covalent bond with hardness, we attribute the excellent mechanical properties of this structure to the N 4 units with strong covalent bonds dominantly by providing coulomb repulsion between the nitrogen atoms as a result of charge transfer from Fe.

Conclusion
Using a structure search method based on CALYPSO methodology and density functional total energy calculations, we systematically studied the phase stabilities and the structures of Fe-N systems in the N-rich regime. We identify two stoichiometric FeN 4 and FeN 2 compounds with unexpected structures that might be experimentally synthesizable under pressure. At 1:4 composition, the energetically favored structure stabilizes in a low P-1 symmetry at low pressure and adopts a high symmetric orthorhombic Cmmm structure at high pressure, both having a infinite 1D linear nitrogen chains. Differently, the Cmmm phase has Fe 8-coordination decahedrons, in contrast with Fe 6-coordination in the octahedrons of P-1 structure. At 1:2 composition, an unknown energetically favored hexagonal P6 3 /mcm structure was firstly discovered at above 228 GPa. Structurally, it is intriguing with the appearance of exotic triadius-like N 4 unit. In the N 4 unit, the N m atom possesses one lone pair of electrons at its p z orbital and forms three N = N covalent bonds with peripheral N p atoms, owing to its sp 2 hybridization. To probe the electronic structures of the P6 3 /mcm structure, it reveals that its intriguing feature gives rise to a set of remarkable properties with an unexpectedly T c of 4~8 K and a good mechanical property.