Introduction

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 Earth1. At standard temperature and pressure (T = 298 K; P = 1 atm), elemental nitrogen is a gas, consisting of diatomic N2 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 N2 under normal conditions. Syntheses of useful nitrides with various nitrogen species rely on chemical methods via, e.g., photochemical reaction, electrochemical synthesis2,3,4,5,6,7,8. A few higher molecular units are known, such as photolytic cyclic N33,4, the tetrahedral N4 molecule5, the N5− anion6, and the N5+ in a crystalline phase of N5+SbF6−7,8. Note that, among these units, though the tetrahedral N4 has been a form of the N4 unit for synthesis, it is observed as a metastable species with a lifetime exceeding one microsecond.

Besides the chemical methods, in fact, adopting HP technology nitrogen also does form innumerable stable and metastable chemical compounds with various nitrogen species9,10,11,12,13,14,15,16,17. These nitrogen species have various structural forms ranging from single atom (N) to molecular (N2, N3, N4, N5, N6) units and polynitrides13,14,15,16,17. To note, these stable nitrides have a variety of intriguing properties, such as superconductivity (MoN)13, high-energy density (LiN3, NaN3)14, and high hardness (WN2)15, as well as extraordinary chemical and thermal stability (Xe-N)16. These nitrides aroused our significant interest in the field of exploring new HP nitrogen species and their potential remarkable properties.

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 century2. A rich Fe-N chemistry exists, most synthesized compounds have a Fe/N ratio higher than unity, such as α″-Fe16N2, α′-Fe8N, γ′-Fe4N, Fe7N3, Fe3Nx(x = 0.75–1.4), Fe3N, Fe2N and FeN4,5,6,7,8,9,10. Among, the FeN compound is most nitrogen-rich iron nitride reported benign synthesized in a high pressure apparatus thus far10. 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 (FeN2) crystallizes in the space group R-3m at 17 GPa (1000 K)11 and transforms an orthorhombic Pnnm structure up to 22 GPa12 obtained by assuming the parent metal under pressure. All these known Fe–N compounds adapt single N atom or molecule N2 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 predictions18. 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 systems16,17,19,20. The effectiveness has been also demonstrated by recent successes in predicting high-pressure structures of various systems, and their several experimental confirmations21,22,23,24,25,26,27,28,29,30. 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 N2 unit to a novel N4 units, and eventually N∞ with the increase of N contents. In N4 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 P63/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 Fe1-iNi (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) method31,32, as implemented in the VASP code33. The PAW pseudopotentials with 3d74s1 and 2s22p3 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 code34. The electron-phonon coupling (EPC) of P63/mcm-FeN2 was calculated within the framework of linear response theory through the Quantum-ESPRESSO code35. 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 scheme36.

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 Fe1-i Ni (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(Fe1-i Ni) = h(Fe1−i Ni) − (1−i)h(FeN) − (2i−1)h(N), where the enthalpies h for Fe1-i Ni and FeN are obtained for the most stable structures as searched by the CALYPSO method at the desired pressures. The convex hulls are depicted in Fig. 1a for pressures at 0, 100, 200 and 300 GPa. The validity of using FeN instead of Fe in defining Δh is ensured by the fact that FeN is exceedingly stable with respect to the binary Fe-N system having a max nitrogen ratio below 50% (Fe4N, Fe3N, Fe2N) under HP, as revealed by our study (Fig. S1) and relative experimental studies4,5,6,7,10. 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 P213 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 FeN6 octahedron in their corresponding stable pressure ranges.

Figure 1
figure 1

(a) Formation enthalpies (ΔH) of various Fe-N compounds with respect to decomposition into constituent elemental solids at 0–300 GPa. Data points located on the convex hull (solid lines) represent stable species against any type of decomposition. (b) Pressure ranges in which the corresponding structures of FeN, FeN2 and FeN4 are stabilized.

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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 μB under pressure (<50 GPa), which is substantially lower than that of the pure Fe solid (2.2 μB). 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 behavior37, 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 FeN2 and FeN4 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 FeN2 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 reports11,12. These two structures both contain a dinitrogen unit and N-sharing six-fold FeN6 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, respectively11,12. Recently, these phases have been synthesized and verified by the experiment under high pressure and high temperature38. Upon to 228 GPa, an unknown energetically more favored hexagonal P63/mcm structure was firstly discovered (Fig. 2a). Tracing the volume change of the phase transition from Pnnm to P63/mcm, it is found that this transition is a first-order accompanied by a volume drop of 3.5%. Viewing the P63/mcm structure, it contains two types of N atoms occupying two different 2c Wyckoff sites as middle Nm and peripheral Np (Fig. 2a): the Np atom is shared by four Fe atoms forming a FeN6 octahedron with Fe-Np distances of 1.76 Å, while the Nm atom bonds with three Np atoms forming perfect N4 unit with a bond length of ~1.25 Å (at 300 GPa). These Fe-Np 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-Np bonds with those 6 neighboring Np atoms and each Np atom has four neighbors Fe-Np bonds and a Np-Nm bond. The Np-Nm 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.

Figure 2
figure 2

Structures of predicted stable FeN2 and FeN4 crystals: (a) The P63/mcm structure of FeN2, viewing of the FeN6 and N4 unit. (b) Low pressure P-1 structure of FeN4. (b) the HP Cmmm-FeN4.

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In the FeN4 compound, the energetically favored structure of FeN4 (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 sp2 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 LiN3, NaN3, and CsN3 under high pressure39,40,41. 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 clusters42,43,44,45,46,47,48,49. 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, FeN2 and FeN4), it is strikingly found that the nitrogen sublattice evolve from isolated N atom to in turn the N2 unit, the N4 unit and eventually N∞ with the increase of N contents. It is noted that these features, except for the N4 units, can be often found in alkali metal azides (Li-N, Na-N, Cs-N)39,40,41. Tracing the history of the related N4 units, the first investigation into N4 units can be traced back to a reported about successfully isolated neutral N4 molecule in the gas phase a decade ago5, but the lifetime of the N4 molecule is only around 1 microsecond. Recently, a charged N4 species as predicted in the CsN crystal is substantially stabilized by strong cation-anion interactions17. This predicted N44− anion has an open-chain structure containing two terminal single-bonds and one internal double-bond. Being different that, we found here the N4 units of P63/mcm-FeN2 has three equal N = N bond and forms plane N4 units like as the triadius star (Fig. 3a). We also try to look for crystal structures that incorporate such special N4 units in the other systems. But no crystal structure is found so far. Here the exotic N4 unit having strong N-N covalent bonding can be clearly shown by its ELF (Fig. 3b,c). Each Nm atom possesses one lone pair of electrons at its pz orbital and forms three N = N covalent bonds with peripheral Np (N1, N2, N3) atoms (Fig. 3c), owing to its sp2 hybridization.

Figure 3
figure 3

(a) The structural feature of N4 unit with Fe in the P63/mcm-FeN2. (b) The ELF plots (001) of P63/mcm structure at 250 GPa with an isosurface value of 0.75. (c) The ELF plots of N4 unit in the P63/mcm structure at 250 GPa. (d) The atomic model for hypothetic N4 cluster with typical symmetry operations marked out. (e) The orbital interaction diagram of a N4 unit with Fe atom.

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Notice that the pz orbital of N is much lower in energy than that of Fe and form a strong overlapping between pz orbitals of N. This fact justifies us to study the electronic structure of hypothetic neutral N4 unit first before researching the electronic properties of its crystal P63/mcm structure. The atomic model for neutral N4 unit is sketched in Fig. 3d with several typical symmetry elements of point group D3h marked out explicitly (Table S7). The linear combination of three pz orbitals of peripheral Np can form orbitals with A2″ and E″ symmetry, while the pz orbital of Nm belongs to A2″ (Table S8). Therefore, for a bare N4 unit, the four pz 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 pz orbitals of Np atoms, while the LUMO comes mainly from the antibonding of pz orbitals between Nm and Np. Meanwhile, the bader charge analysis reveals the fact of electron abundant N4 units and electron deficient Fe atoms (the less electronegative Fe loses 1.67 electrons per atom, and Np directly bonded to Fe obtains 1.13 electrons per atom, while the Nm atom remains almost neutral). Such unoccupied antibonding orbitals between Nm and Np can minimize the influence of excess electrons, which can explain why the P63/mcm structure with the N4 unit is fairly stable.

To probe the electronic structures of the P63/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 FeN2, Fe0N2 (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 Fe0N2 system (red dash lines) with the realistic one of FeN2 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 Ef and the bands appears “flat band-steep band” characteristic50 around the Ef. These are typical features favorable for strong EPC and superconductors. Using the linear response theory, we calculated the phonon DOS (PHDOS), Eliashberg function α2F(ω) and the strength of the e-ph coupling λ(ω) of the P63/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-FeB4 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 N4 units (15% of λ) (Fig. 4d). Using the Allen-Dynes equation51, with the calculated ωlog of 643 K and typical µ* = 0.1~0.15, it reveals that the P63/mcm structure is a weak-coupling BCS-type superconductor with a superconducting Tc of 4~8 K. Moreover, the most commonly known transition-metal pernitrides crystallize can act as hard materials, such as MnN2 (HV = 19.9 GPa), CoN2 (16.5 GPa), and NiN2 (15.7 GPa)52. Fe in the same period as Mn, Co and Ni and the P63/mcm-FeN2 phase is as a typical transition-metal pernitride, so the P63/mcm-FeN2 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 Ym (597 GPa), and Vicker’s hardness (HV = 29 GPa) at ordinary pressure. As expect, the result shows that the P63/mcm-FeN2 phase exhibits highly incompressible. Bases on a correlation the covalent bond with hardness, we attribute the excellent mechanical properties of this structure to the N4 units with strong covalent bonds dominantly by providing coulomb repulsion between the nitrogen atoms as a result of charge transfer from Fe.

Figure 4
figure 4

(a) Electronic band structure and (b) the projected density of states for P63/mcm-FeN2 structure at 250 GPa, the red dash lines represent the band structure of the N sublattice with a uniform compensated background. (c) The calculated Eliashberg EPC spectral function α2F(ω) and its integral λ(ω) and (d) the projected phonon density of states for P63/mcm-FeN2 structure at 250 GPa.

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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 FeN4 and FeN2 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 P63/mcm structure was firstly discovered at above 228 GPa. Structurally, it is intriguing with the appearance of exotic triadius-like N4 unit. In the N4 unit, the Nm atom possesses one lone pair of electrons at its pz orbital and forms three N = N covalent bonds with peripheral Np atoms, owing to its sp2 hybridization. To probe the electronic structures of the P63/mcm structure, it reveals that its intriguing feature gives rise to a set of remarkable properties with an unexpectedly Tc of 4~8 K and a good mechanical property.