Metal nitrides have attracted increasing attention in a wide range application due to their fascinating properties, such as hardness, superconductivity, various types of magnetism, etc.^[1–2] Recently, extensive efforts have focused on the synthesis and design of metal polynitrides with single and double nitrogen bonds under high pressure, as metal polynitrides are proposed to be promissing candidates for high-energy density materials (HEDMs)^[3–5]. The decomposition of these polynitrides is expected to release huge energy due to the conversion from a single/double-bonded polymeric nitrogen (160/418 kJ/mol) into a triple-bonded nitrogen (954 kJ/mol) molecule without producing pollutants. The atomic polymeric nitrogen with cubic gauche structure (cg-N) at a high pressure and high temperature (HPHT) condition (pressure above 110 GPa and temperature greater than 2 000 K) was successfully synthetized by Eremets et al.^[6–8]. Motivated by that synthesis, alkali metal azides AN₃, constructed by spherical cations and linear molecular ${\mathrm{N}}_3^-$ anions with N＝N double bonds, was proposed to be suitable precursor in the formation of polymeric nitrogen under pressure^[9]. Experimentally, the polymeric evolutionary behaviors of ${\mathrm{N}}_3^-$ anions in LiN₃^[10], NaN₃^[11–13], KN₃^[14–16], RbN₃^[17–18], and CsN₃^[19–20] have been fully investigated under high pressure. One of the most interesting findings is the reported polymeric nitrogen nets in NaN₃^[11] at 160 GPa. More recently, a fully sp²-hybridized layered polymeric nitrogen structure, featuring fused 18-membered rings, in potassium supernitride (K₂N₁₆) was successfully synthesized under HPHT condition using a laser heating diamond anvil cell (DAC) technique^[21]. In our previous works^[22–25], the ${\mathrm{N}}_3^-$ anions in LiN₃, NaN₃, and KN₃ all transform firstly to “N₆” molecular clusters under compression, and then to a polymerized nitrogen phase at high pressures.

Besides the well-known alkali metal azides AN₃, alkali metal diazenides A₂N₂ containing the N＝N double bonds, were also proposed as precursors of the polymeric atomic nitrogen under pressure. In 2010, two alkali metal diazenides, Li₂N₂ and Na₂N₂, were proposed to have orthorhombic Pmmm structure^[26]. However, an orthorhombic Immm structure of Li₂N₂^[27] was then synthesized under HPHT condition. Immediately after that, a serial pressure-induced structural phase transitions of Li₂N₂ were reported in two independent works^[28–29]. Moreover, a high-pressure tetragonal I4₁/acd phase containing the spiral nitrogen chain was uncovered at 242 GPa^[29], indicating that the polymerization of nitrogen is realized from Li₂N₂.

Compared to Li₂N₂ and Na₂N₂, K₂N₂ have not been reported so far, neither for ambient pressure nor for high pressure. Here, we will extensively explore the high-pressure structures of K₂N₂ up to 150 GPa by using the first principles swarm structure searching method. The structure evolutions and chemical bonding behaviors of K₂N₂ within different structures were then fully studied to provide an insight into the formation of polymeric nitrogen in alkali metal diazenides.

1. Computational Method

The variable-cell crystal structure predictions for K₂N₂ (1–4 formula unit (f. u.) in the simulation cell) up to 150 GPa were conducted by CALYPSO code^[30–31]. The effectiveness of this method has been fully confirmed in predicting high-pressure structures of various systems^[32–35]. During the structure search process, the 60% structures of each generation with lower enthalpies were selected to generate the structures for the next generation by particle swarm optimization (PSO) technique, and the other structures in new generation were randomly generated to increase the structural diversity. Usually, the structure searching simulation stopped at the 20th generation and 40 structures per generation were generated at each pressure point. After finishing the structure search, the most stable structures were used for further structural relaxation and property calculation, a process that is implemented in the Vienna ab Initio Simulation Package^[36]. The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional under the generalized gradient approximation^[37] and the projector-augmented wave (PAW)^[38] for ion-electron potentials were adopted herein. The van-der-Waals (vdW) correction was applied in the structural relaxation using Grimme’s method (DFT-D3)^[39]. The cutoff energy of 550 eV and Monkhorst-Pack k meshes^[40] of 2π×0.032 Å⁻¹ were used to ensure the good convergences of total energy and maximum force on each atom. The lattice dynamic stability of each structure was checked by phonon spectrum obtained by PHONOPY code^[41]. The charge transfer behavior of K₂N₂ was performed using Bader atoms-in-molecules (AIM) method^[42].

2. Results and Discussion

2.1 Ground-State and High-Pressure Structures of K₂N₂

The ground-state structure for K₂N₂ at ambient pressure in our fully structural search (in Fig. 1(a)) is a monoclinic structure with the space group C2/m, which is different from those reported for Li₂N₂ and Na₂N₂^[26]. In Fig. 1(a), the N₂ quasimolecule dimers in C2/m structure are parallel to each other, and are tilted by a certain angle within a-b plane. The N-N bond length is 1.192 Å, which is a typical value for double bond. Also, the calculated Mayer bond order^[43] for the N₂ anion in the C2/m is 2.18, being consistent with double bond. Due to the weak interaction between the N₂ quasimolecules in the C2/m phase, the application of gigapascal pressure leads to the orientation change of N₂ quasimolecules and the structure transformation. Indeed, two stable orthorhombic structures, i.e. Pmmm and Cmmm, for K₂N₂ were uncovered at 2 and 5 GPa, respectively, as shown in Fig. 1(b) and Fig. 1(c). Remarkably, the unveiled Pmmm phase and the previously reported ground-state structure of Na₂N₂ are exactly isostructrual (the Pmmm phase denoted as the Na₂N₂-type phase hereafter), and the Cmmm phase of K₂N₂ and the high-pressure metastable structures of Na₂N₂ are isostructrual as well^[26]. Furthermore, both Na₂N₂-type and Cmmm orthorhombic phases have the similar atomic arrangement that the parallel N₂ dimers are perpendicular to the K atomic layer. In the Na₂N₂-type and Cmmm phases, the calculated N-N bond lengths are 1.220 and 1.254 Å, while the characteristics of the N＝N double bond are as that of the C2/m structure in the ground-state. As the pressures rises to 150 GPa, a stable monoclinic C2/c structure, containing a folded one-dimensional nitrogen chains (see Fig. 1(d)), is identified for K₂N₂ for the first time. The lengths of two nonequivalent N-N bond in C2/c structure are 1.474 and 1.373 Å at 135 GPa, indicating genuine N―N single bond, and thus the nature of the structure is similar to that of the high-pressure phase of NaN₃^[24]. Table 1 lists the structural details of the lattice parameters and the atomic positions of various K₂N₂ structures at selected pressure points.

Figure  1.  Crystal structure of C2/m, Na₂N₂-type, Cmmm, and C2/c phases (The large and small spheres represent K and N atoms, respectively.)

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Table  1.  Optimized structural parameters of the C2/m, Na₂N₂-type, Cmmm, and C2/c phases of K₂N₂

Phase	Pressure/GPa	Lattice parameters	dN-N/Å	Atomic fractional coordinates
C2/m	0	a = 7.562 Å, b = 3.912 Å, c = 10.923 Å, α = γ = 90°, β= 134.955°	1.192	K 4i (0.613, 0, 0.260) N 4i (0.519, 0, 0.456)
Na2N2-type	2.5	a = 3.326 Å, b = 4.375 Å, c = 5.694 Å, α = β = γ = 90°	1.220	K1 1e (0, 0.500, 0) K2 1c (0, 0, 0.500) N 2t (0.500, 0.500, 0.393)
Cmmm	20.0	a = 6.896 Å, b = 5.104 Å, c = 2.855 Å, α = β = γ = 90°	1.254	K 4g (0.694, 0, 0) N 4j (0, 0.877, 0.500)
C2/c	135.0	a = 6.971 Å, b = 4.099 Å, c = 4.510 Å, α = γ = 90°, β= 81.342°	1.474, 1.373	K 8f (0.669, 0.909, 0.198) N 8f (0.041, 0.887, 0.102)

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The phonon dispersion curves of the four predicted structures are illustrated in Fig. 2 in order to check their dynamic stabilities. Fig. 2 shows that all the K₂N₂ phases are dynamically stable as the eigen frequencies of their lattice vibrations are positive for all wavevectors in the Brillouin zone. It should be noted that the predicted C2/m, Na₂N₂-type, and Cmmm phases exhibit higher eigen frequencies of around 50 THz, which correspond to the vibrations of N₂ quasimolecules with stronger N＝N double bonds. By contrast, the C2/c phase shows relative lower vibration frequencies of around 30−35 THz, corresponding to the vibration frequencies of weaker N―N single bonds.

Figure  2.  Calculated phonon curves of the C2/m (a), Na₂N₂-type (b), Cmmm (c), and C2/c (d) phases at selected pressure points

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Herein, we evaluated the energy involved in the decomposition of the C2/c phase with polymerized N―N single bond chains into metal K and gaseous N₂. The energy involved in the C2/c-K₂N₂ decompostion was determined to be 0.344 kJ/g, which is about three times of that of conventional energetic materials (0.104−0.105 kJ/g)^[44], but much smaller than those reported for high-energy density materials (3.61 kJ/g for BeN₄, 5.30 kJ/g for GaN₅, 4.81 kJ/g for GaN₁₀)^[45–46]. This may be due to the low nitrogen content in K₂N₂, so the nitrogen-rich KN_x (x> 1) is expected to be potential HEDM under high pressure.

2.2 Pressure-Induced Structural Phase Transitions of K₂N₂

To determine the phase transition pressure of K₂N₂, we plotted in Figs. 3(a) and 3(b) the enthalpy difference of the predicted C2/m, Cmmm, and C2/c structures relative to the Na₂N₂-type phase in the pressure range of 0−150 GPa. The inset of Fig. 3(a) shows that above 1.7 GPa the Na₂N₂-type phase becomes energetically preferable with respect to the ground-state C2/m structure and is enthalpically stable up to about 3.6 GPa, above which it transforms into the Cmmm structure. From Fig. 3(b), it can be seen that the Cmmm structure persists up to 122 GPa and then transforms to the C2/c structure according to our structural predictions and enthalpy difference calculations. The wide pressure range of stable Cmmm phase may be attributed to the relative smaller ionic radius and atomic mass of K atom，in contrast with the heavier Rb and Cs atoms. It is thus expected that the chemical precompression exerted by heavier atoms, such as Rb and Cs, in A₂N₂ (as for Cs₂N₂^[43]) can yield lower pressures for the polymerization of one-dimension N chains. From the calculated pressure dependence of the volume per f. u. in Figs. 3(c) and 3(d), the volume collapses of the first-order transformations C2/m→Na₂N₂-type, Na₂N₂-type→Cmmm, and Cmmm→C2/c are 14.4%, 22.5%, and 4.0% respectively. Thus, the Cmmm phase is much stiffer than low-pressure C2/m and Na₂N₂-type phases under compression. This is consistent with the increasing coordination number of the K atom in the different K₂N₂ phases under compression in Figs. 3(c) and 3(d). The densification of K₂N₂ under pressure is realized by increasing the cation coordination number from 5 in ground-state C2/m phase to 8 in Na₂N₂-type (K2 atoms are coordinated by eight N atoms and K1 atoms are fourfold coordinated by four N atoms) and Cmmm phases, and finally to 10 in C2/c phase.

Figure  3.  (a)–(b) Enthalpy differences of the different predicted structures relative to the Na₂N₂-type structure under pressure;(c)–(d) pressure dependence of the volume per f. u. of each structure for K₂N₂

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2.3 Electronic Structures of K₂N₂ Phases

To get deeper insight into the electronic structure evolutions of these K₂N₂ phases under pressure, we calculated the total density of states (DOS) and the site projected DOS, as shown in Fig. 4, where the Fermi level is the vertical dotted line. In contrast with the semiconducting nature of C2/c phase (Fig. 4(d)), the C2/m, Na₂N₂-type, and Cmmm phases are all metallic with the N-p electron crossing the Fermi level. Thus, the K₂N₂ possesses a metal-semiconductor transition, that is similar to that of Li-N system in recent work^[28]. From Fig. 4(a)–Fig. 4(c), the C2/m, Na₂N₂-type, and Cmmm phases exhibit similar projected DOS profiles in the valence band region, the K-s and K-p atomic orbitals are located at the deep energy levels (around −32 and −16 eV, respectively), and the N-s atomic orbitals mainly lie around −22 − −10 eV. Moreover, the interaction between K and N atoms is extremely weak due to the absence of overlapping regions in their atomic projected DOSs. From the inspection of the DOS profiles, a clear broadening of different atomic decomposed states is seen in the high-pressure C2/c phase (Fig. 4(d)), leading to clear hybridizations between K-p and N-p atomic orbitals. In contrast to the first three low-pressure K₂N₂ phases, this distinction is partially due to the increasing coordination number of K atoms in C2/c phase and to the resulting chemical bonding change that will be illustrated in the following Bader charge transfer analysis.

Figure  4.  Total and site projected DOSs of C2/m (a), Na₂N₂-type (b), Cmmm (c), and C2/c (d) phases

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Fig. 5 presents the PBE-resulted energy band structures of the four K₂N₂ phases, the projected weights of K-s, K-p, and N-s orbital electrons are not taken into account herein because of their exceptionally tiny contributions around the Fermi level in Fig. 4. Compared to the strong metallic nature of the Cmmm phase, which is mainly dominated by N-p_x/N-p_z at the considered energy range of −2 − 2 eV (Fig. 5(c)), the C2/m ( Na₂N₂-type) phase exhibits a relative weaker metallic behavior that two bands of N-p_y/N-p_z (N-p_x/N-p_y) crossing the Fermi level in Fig. 5(a) (Fig. 5(b)). The indirect semiconducting nature of C2/c phase (band gap of 2.0 eV) is demonstrated in Fig. 5(d), with the valence band maximum (VBM) and conduction band minimum (CBM) is located at A and E points, respectively. It is further found that the VBM and CBM of the C2/c phase are principally composed of N-p_x and N-p_z orbitals, respectively.

Figure  5.  Projected weights of N-p orbitals in the band structures of C2/m (a), Na₂N₂-type (b), Cmmm (c), and C2/c phases (d) (The Fermi level is indicated by horizontal lines and the black solid lines denote the energy structures of each phase of K₂N₂)

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2.4 Atomic Chemical Bonding of K₂N₂ Phases

Fig. 6 shows the projected two-dimensional (2D) electron localization functions (ELF) of the four K₂N₂ phases to further reveal the atomic chemical bonding. The ELF is used to characterize the localized distribution of electrons of different atoms. The ELF values of 1, 0.5, and close to 0 indicate strong electron localization, electron gas, and nonelectron localization, respectively. From Fig. 6(a)–Fig. 6(c), the high-ELF regions are distributed between two N atoms, indicating the strong covalent N-N bonding in the C2/m, Na₂N₂-type, and Cmmm phases. Lone-pair electrons of K remain from the C2/m to Cmmm phases under pressure, as illustrated by the ELF value of 0 around K atoms, which is in conformity with the projected atomic orbital DOSs described in Fig. 4(a)–Fig. 4(c). For the compressed C2/c phase, some lone-pair electrons of K still remain, yet some of them are transferred to the regions of the K-N bond. Hence, the pressure compels the K-p lone-pair electrons to participate in the chemical bonding with N-p electrons under compression, realizing the high pressure C2/c phase with increased coordination number of K atoms. The electron localization on N-N bond in the C2/c phase (Fig. 6(d)) is weaker than that in the other three low-pressure K₂N₂ phases, confirming that the N―N single bond of the polymerized one-dimensional nitrogen chains is weaker than the N＝N double bond of the N₂ quasimolecules.

Figure  6.  Contours of ELF for the C2/m on the (010) plane (a), Na₂N₂-type on the (010) plane (b), Cmmm on the (100) plane (c), and C2/c on the (010) plane (d)

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Finally, the Bader charges in the four phases are listed in Table 2 by using the Bader AIM method. For the ground-state C2/m phase, each N atom accepts 0.491e which corresponds to the electron loss per K atom. When phase transits into the Na₂N₂-type and Cmmm phases, a relatively stronger ionic bonding feature of K-N bond is disclosed because that the charge transfer is raised to approximately 0.645e − 0.690e as a result of the electron gain of N. Contrarily, it is noted that the atomic charge transfer from the K atom to N atom decrease to 0.623e in the compressed C2/c phase. This variation may be originated from that partial K atoms participate in covalent bonding with the N atom under compression, as indicated by the orbital hybridizations of K-p and N-p in Fig. 4(d). It then can be concluded that the chemical bonding in the C2/c phase containing one-dimensional polymerized nitrogen chains is a complex mixture of covalent and ionic characteristics.

Table  2.  Calculated Bader charges of K and N atoms in C2/m, Na₂N₂-type, Cmmm, and C2/c phases

Phase	Pressure/GPa	Atom	Charge value/e	Charge transfer/e
C2/m	0	K N	8.509 (×2) 5.491 (×2)	+0.491 −0.491
Na2N2-type	2.5	K1 K2 N	8.407 8.303 5.645 (×2)	+0.593 +0.697 −0.645
Cmmm	20.0	K N	8.310 (×2) 5.690 (×2)	+0.690 −0.690
C2/c	135.0	K N	8.377 (×2) 5.623 (×2)	+0.623 −0.623

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3. Conclusions

In summary, we have performed a systematic crystal structure search for alkali metal diazenide K₂N₂ under pressure using the advanced swarm intelligence structure simulations. The ground-state phase of K₂N₂ is determined to be a new monoclinic C2/m structure with N₂ quasimolecules, and three high-pressure structures of K₂N₂, including Na₂N₂-type (1.7−3.6 GPa), Cmmm (3.6−122.0 GPa), and C2/c (above 122.0 GPa), are identified. Remarkably, the pressure-induced polymerization of one-dimensional nitrogen chains in the high-pressure C2/c semiconducting phase is firstly reported, attributed to the participation of K-p lone-pair electrons in the chemical bonding under pressure. The occurrences of the high-pressure phases of Na₂N₂-type, Cmmm, and C2/c follow the increased coordination numbers (from 5 to 10) of K atoms with the incresed pressures, accompanied by significant N-N bonding modification from N＝N dimers to polymerized N―N single bond one-dimensional chains. Furthermore, a metal-semiconductor transition of K₂N₂ under pressure is demonstrated by the electronic structures calculations. The present findings provide fundamental insights into the high-pressure structures and the polymerization of N atoms of alkali metal diazenides, supplying useful guidance for the syntheses of K₂N₂ in further experimental efforts.