I. INTRODUCTION
Polymeric nitrogen materials are environmentally friendly highenergydensity materials (HEDMs). Their high energy density comes from the huge energy difference between the N≡N triple bond (954 kJ/mol) on the one hand and the N=N double and N–N single bonds (418 and 160 kJ/mol, respectively) on the other. To date, a wide variety of novel polynitrogen structures have been reported in theoretical studies, such as the threedimensional (3D) network structures (cgN, Pnnm, Cccm, and CW),1–3 twodimensional (2D) layer structures (A7, ZS, LB, LP, HLP, PP, and BP phases),4–9 onedimensional (1D) chain structures (chN and Cmcm),10 and a zerodimensional (0D) cage structure (N_{10}).11 High temperature and high pressure are two important factors in obtaining polynitrogen materials. However, the ultrahigh pressure (>100 GPa) and temperature (>2000 K) in experiment required for the synthesis of cgN, LPN, HLPN, and BPN limit their development.12–15 An additional challenge is posed in quenching the highpressure phases to ambient conditions, owing to their intrinsic metastable properties. Therefore, reducing the synthesis pressure and enhancing the structural stability of polynitrogen materials are crucial for their practical application. Nitrogenrich compounds have been demonstrated to be promising candidates to achieve these goals.
In experiments, LiN_{5}, K_{2}N_{6}, MgN_{4}, Mg_{2}N_{4}, CsN_{5}, αZnN_{4}, βZnN_{4}, and trBeN_{4} have been synthesized at pressures of 45, 45, 50, 50, 60, 63.5, 81.7, and 85 GPa, respectively.16–21 Moreover, LiN_{5}, Mg_{2}N_{4}, and trBeN_{4} can be quenched down to ambient conditions. Clearly, compared with polynitrogen structures, nitrogenrich compounds have milder synthesis conditions and higher stability. Theoretically, they exhibit a wide variety of nitrogen configurations, ranging from N_{2} dumbbells,22,23 azide roots,24–26 N rings (N_{4},27,28 N_{5},29–34 and N_{6}35–37), 1D N chains,38–43 2D N layers,44–49 to 3D network structures.50,51 Recently, the lanthanide polynitrides $P\stackrel{\u0304}{1}$GdN_{6} and $P\stackrel{\u0304}{1}$ErN_{6} have been proposed as HEDMs with excellent explosive performance.52,53 As a typical lanthanide element, cerium (Ce) is the most abundant rare earth element in the Earth’s crust and much cheaper than most rare earth metals, such as Gd and Er. Additionally, Ce atoms possess flexible electronic properties that allow them to reach high coordination numbers in their compounds. Thus, they exhibit excellent performance in the selective catalytic reduction of N_{2} to NH_{3}.54–56 In light of these characteristics, we suggest that Ce may be an ideal candidate for inducing polynitrogen structures. However, up until now, only CeN has been reported in highpressure studies,57–59 which has motivated us to perform a systematic highpressure study on nitrogenrich Ce–N compounds.
In this work, we systematically study Ce–N highpressure compounds at pressures up to 100 GPa through firstprinciples swarmintelligence structural searches. Seven stoichiometric ratios of CeN_{n} (n = 0.5, 1, 2, 3, 4, 5, and 6) compounds are explored. Six new highpressure phases are proposed, and their stability is verified using phonon dispersion curves, elastic constants, and molecular dynamic simulations. Interestingly, $P\stackrel{\u0304}{1}$CeN_{6} not only possesses the lowest synthesis pressure of 32 GPa among layered metal polynitrides, but also can be quenched to ambient conditions. The stability mechanism of $P\stackrel{\u0304}{1}$CeN_{6} is clarified by electronic structure and bonding analyses, and its mechanical properties are analyzed. Its infrared spectra (IR) and Raman spectra are calculated for experimental reference. Significantly, $P\stackrel{\u0304}{1}$CeN_{6} shows excellent volumetric energy density (up to 11.20 kJ/cm^{3}), detonation pressure (128.95 GPa), and detonation velocity (13.60 km/s), opening up exciting avenues for the exploration of highnitrogenconcentration lanthanide polynitrides via the metal ligand effect.
II. CALCULATION DETAILS
The structure searches were performed using the particle swarm optimization structure prediction method in the CALYPSO code.60 Seven stoichiometric ratios of CeN_{n} (n = 0.5, 1, 2, 3, 4, 5, and 6) compounds were considered in the structural prediction at 0, 20, 50, and 100 GPa. The simulation cells contained 1, 2, and 4 formula units (f.u.), in which a total of 75 600 structures were produced and were ranked according to the calculated enthalpy. The previously reported $Fm\stackrel{\u0304}{3}m$CeN and P4/nmmCeN were successfully reproduced in our search,58 validating the effectiveness of the method. The structural relaxations and property calculation were performed using the Vienna Ab initio Simulation Package (VASP).61 The generalized gradient approximation (GGA) was used with the Perdew–Burke–Ernzerhof (PBE) exchange correlation functional.62 The GGA + U method (U = 6 eV) was used to correct the strong onsite Coulomb repulsion of Ce_4f states.63–66 The valence electrons of Ce and N atoms in the projector augmented wave (PAW) pseudopotentials were 4f^{1}5d^{1}6s^{2}and 2s^{2}2p^{3}, respectively.67 The Monkhorst–Pack k mesh spacing density and plane wave energy cutoff were set to 2π × 0.03 Å^{−1} and 520 eV, respectively. The accurate band structure was obtained using a hybrid functional (HSE06).68 The enthalpies of formation ΔH_{f} of the CeN_{n} compounds were calculated using the equation$$\mathrm{\Delta}{H}_{\text{f}}(\mathrm{C}\mathrm{e}{\mathrm{N}}_{n})=\frac{H(\mathrm{C}\mathrm{e}{\mathrm{N}}_{n})H(\mathrm{C}\mathrm{e})nH(\mathrm{N})}{1+n}.$$
The phonon dispersion curves were obtained using the PHONOPY code with densityfunctional perturbation theory.69 The energy density was calculated by considering the following dissociation path under ambient pressure: $\mathrm{C}\mathrm{e}{\mathrm{N}}_{n}\to \mathrm{C}\mathrm{e}\mathrm{N}+\frac{1}{2}(n1){\mathrm{N}}_{2}$. The detonation velocity and detonation pressure were calculated from the Kamlet–Jacobs semiempirical equations70${V}_{\mathrm{d}}=1.01{(N{M}^{0.5}{E}_{\text{d}}^{0.5})}^{0.5}(1+1.30\rho )$ and ${P}_{\text{d}}=15.58{\rho}^{2}N{M}^{0.5}{E}_{\text{d}}^{0.5}$. The ab initio molecular dynamics (AIMD) simulations were performed in the isobaric–isothermic (NPT) ensemble with a total simulation time of 10 ps.71 The crystal orbital Hamilton population (COHP) was calculated using the LOBSTER package.72 For Raman and IR spectra, normconserving pseudopotentials were adopted, and these were calculated using the CASTEP module of the Material Studio package. Convergence criteria consisted of an energy change of <2 × 10^{−5} eV/atom and a maximum force of <0.05 eV/Å.73
III. RESULTS AND DISCUSSION
A. Phase diagram and structural stability
The convex hull of CeN_{n} compounds is calculated for screening the thermodynamically stable phases of optimal structures in the prediction, and the results show that Ce can promote the formation of various polynitrides under high pressure. The stable phases of solid cerium ($Fm\stackrel{\u0304}{3}m$ and Cmcm phases) and solid nitrogen ($Pa\stackrel{\u0304}{3}$, P4_{2}/mnm, P4_{1}2_{1}2, and cgN phases) are used as the references to calculate the formation enthalpy ΔH_{f}, and this is plotted in Fig. 1(a), where the solid and blank squares correspond to the thermodynamically stable and unstable/metastable phases, respectively. The phase diagrams of thermodynamically stable highpressure structures are obtained from the enthalpy difference analysis and are shown in Fig. 1(b) and Fig. S1 in the supplementary material. It can be seen that $Fm\stackrel{\u0304}{3}m$CeN is stable from 0 to 69 GPa and then transforms to P4/nmmCeN. This phase transition pressure is consistent with a previous experimental result (65 GPa).58I4/mmmCeN_{2} remains stable at 8–100 GPa. C2/mCeN_{3} is stable above 6 GPa, and then changes to $P\stackrel{\u0304}{1}$CeN_{3} at 82 GPa. As the nitrogen concentration increases, $P\stackrel{\u0304}{1}$CeN_{4} remains stable in the pressure range of 32–100 GPa. At 20 and 50 GPa, we predicted C2/cCeN_{6} and $P\stackrel{\u0304}{1}$CeN_{6} phases, which are energetically favorable at 9–32 GPa and 32–100 GPa, respectively. Interestingly, as far as we know, the synthesis pressure (32 GPa) of $P\stackrel{\u0304}{1}$CeN_{6} is the lowest among the layered metal polynitrides.
Figure 1.(a) Formation enthalpies ΔH of various CeN_{n} (n = 0.5, 1, 2, 3, 4, 5, and 6) compounds under high pressure. The stable phases are connected by solid lines and unstable/metastable phases by dashed lines. (b) Pressure–composition phase diagram of the predicted Ce–N phases.
The crystal structures of CeN_{n} compounds reveal that Ce has strong coordination ability under high pressure [Figs. 2 and S2 (supplementary material)], with each Ce atom coordinating with 10, 8, 10, 10, 10, and 12 N atoms in I4/mmmCeN_{2}, C2/mCeN_{3}, $P\stackrel{\u0304}{1}$CeN_{3}, $P\stackrel{\u0304}{1}$CeN_{4}, C2/cCeN_{6}, and $P\stackrel{\u0304}{1}$CeN_{6}, respectively [Fig. S3 (supplementary material)]. The Nstructure units in I4/mmmCeN_{2} and C2/cCeN_{6} are N_{2} dumbbells, and those in C2/mCeN_{3} and $P\stackrel{\u0304}{1}$CeN_{3} are N_{2} dumbbells and N_{4} molecular chains. The polymeric structures of $P\stackrel{\u0304}{1}$CeN_{4} and $P\stackrel{\u0304}{1}$CeN_{6} are N_{8} molecular chains and layered structures with an N_{14} ring, respectively [Fig. S4 (supplementary material)]. Calculations of phonon dispersion curves show that the six new CeN_{n} compounds described above are dynamically stable at predicted pressures owing to the absence of an imaginary frequency in the Brillouin zone [Fig. S5 (supplementary material)], and thus they are highpressure stable phases. The structural parameters are listed in Table SI (supplementary material). Excluding C2/cCeN_{6}, the average N–N bond length in CeN_{n} compounds is about 1.30–1.39 Å [Table SII (supplementary material)], which is much larger than that in nitrogen molecules (1.11 Å), suggesting that these represent a new type of nitrogenrich materials in which the N–N bonds are single or intermediate between single and double in nature. We deduce that they may have high energy capacity properties.
Among these CeN_{n} compounds, $P\stackrel{\u0304}{1}$CeN_{6} is particularly fascinating and worthy of further discussion because of its high nitrogen concentration, long N–N bonds, high coordination number, and low synthesis pressure. As shown in Fig. 2, the crystal structure of $P\stackrel{\u0304}{1}$CeN_{6} is layered, with an AαAα stacking form along the aaxis direction, where A and α represent the N plane and Ce plane, respectively. Interestingly, $P\stackrel{\u0304}{1}$CeN_{6} possesses unique N_{14} rings, the first to be reported up until now. At 32 GPa, the bond lengths of N1–N1, N1–N3, N3–N3, N3–N2, and N2–N2 are 1.40, 1.42, 1.47, 1.35, and 1.45 Å, respectively, indicating that the N3–N2 bond is intermediate in nature between a single bond (1.45 Å) and a double bond (1.25 Å), whereas all the other bonds are close to single bond in nature. The mechanical and dynamical stability of $P\stackrel{\u0304}{1}$CeN_{6} were verified at 32 GPa by calculating elastic constants and phonon dispersion curves [Fig. 2(c) and Table SIII (supplementary material)]. The phonon density of states (PHDOS) in Fig. 2(c) shows that the highfrequency vibrational modes come from N–N vibrations, while the lowfrequency vibrational modes consist of Ce–N collective motions. Moreover, the thermal stability of $P\stackrel{\u0304}{1}$CeN_{6} was evaluated using AIMD simulations. As shown in Fig. 2(d), the total energy of $P\stackrel{\u0304}{1}$CeN_{6} fluctuates around the equilibrium position (−751.92 eV), and the structural skeleton remains intact at the end of the simulation, suggesting that $P\stackrel{\u0304}{1}$CeN_{6} is thermally stable under a pressure of 32 GPa and a temperature of 300 K.
Figure 2.(a) Crystal structure of $P\stackrel{\u0304}{1}$CeN_{6} at 32 GPa. (b) Nitrogen structural skeleton of $P\stackrel{\u0304}{1}$CeN_{6} at 32 GPa. The independent N atoms are marked as N1, N2, and N3 in the symmetric N_{14} rings. (c) Phonon dispersion curve and PHDOS of $P\stackrel{\u0304}{1}$CeN_{6} at 32 GPa. (d) Initial and terminal structures in AIMD simulations with total simulation time 10 ps and fluctuations of the total energy of $P\stackrel{\u0304}{1}$CeN_{6} at 32 GPa and 300 K.
B. Stability mechanism under high pressure
To pinpoint the stability mechanism of $P\stackrel{\u0304}{1}$CeN_{6}, we carefully analyzed the electron localization function (ELF). As shown in Fig. 3(a), the N1 atoms in $P\stackrel{\u0304}{1}$CeN_{6} hybridize in sp^{2} states with two σ bonds and a lone electron pair. Both the N2 and N3 atoms are sp^{3} hybridized. The hybrid orbital of N2 atoms contains two σ bonds and two lone electron pairs, and that of the N3 atom contains three σ bonds and a lone electron pair. Additionally, the strong localization of electrons between N atoms indicates a strong N–N covalent bond interaction, and the lonepair electrons of N atoms form strong coordination bonds with Ce atoms. Considering that the coordination number of Ce is up to 12 in $P\stackrel{\u0304}{1}$CeN_{6}, we deduce that electron transfer between the Ce atoms and Nstructures play a crucial role in stabilizing $P\stackrel{\u0304}{1}$CeN_{6} at the relatively low synthesis pressure of 32 GPa.
Figure 3.(a) ELF of $P\stackrel{\u0304}{1}$CeN_{6} (isovalue = 0.8). (b) The charge transfer and formation energy of CeN_{n} (n = 1, 2, 3, 4 and 6) compounds at 32 GPa.
To confirm our idea, Bader charges were calculated and clearly show the charge transfer for each Ce atoms and Nstructures in stable CeN_{n} (n = 1, 2, 3, 4, and 6) compounds at 32 GPa. The amount of charge transferred increases with nitrogen content from Fm3mCeN to C2/mCeN_{3} [Fig. 3(b)], and the formation energy decreases from −4.37 to −5.83 eV/f.u., because the Ce atoms need to contribute a greater amount of charge in the Nrich phases than in the Npoor ones to stabilize the structure. As the nitrogen concentration continues to increase, the amount of charge transferred no longer increases, because the Ce atom reaches its highest valence. In spite of this, the excess nitrogen atoms can share electrons with others and form coordination bonds with Ce atoms, and hence the formation energy just increases from −5.83 eV/f.u. (C2/mCeN_{3}) to −5.67 eV/f.u. ($P\stackrel{\u0304}{1}$CeN_{6}). Consequently, the donor electrons and the strong ligand effect of Ce atoms are two important factors stabilizing highnitrogen compounds.
To determine the unique ligand effect of Ce in $P\stackrel{\u0304}{1}$CeN_{6}, we performed a comparative analysis by simulating the highpressure polymerization behavior of N_{2} molecules under two reaction environments. The projection of the COHP (pCOHP), the integral of the COHP (ICOHP), and the projected density of states (PDOS) were calculated to analyze the bonding features and bonding strength. According to the lattice parameters and stoichiometric ratio of $P\stackrel{\u0304}{1}$CeN_{6}, we constructed a $P\stackrel{\u0304}{1}$N_{2} molecular crystal with six N atoms per cell, and made it more reasonable and stable by geometrical optimization at 32 GPa. The distance between the N_{2} molecules decreases dramatically to about 2.37–2.45 Å [Fig. 4(a)], but it is still larger than length of an N–N single bond (1.45 Å), suggesting that there is no bonding reaction between N_{2} molecules. The large −ICOHP (26.4) of the N–N bond and the strong hybridization interaction between N_2s and N_2p orbitals indicate strong covalent interaction in the N–N bond [Figs. 4(b) and 4(c)]. All the above points support the fact that the N≡N triple bonds in N_{2} molecules are hard to break down at 32 GPa.
Figure 4.(a) Crystal structure, (b) −pCOHP and −ICOHP, and (c) PDOS of N_{2} molecular crystal. (d) Crystal structure, (e) −pCOHP and −ICOHP, and (f) PDOS of transCeN_{6}. (g) Crystal structure, (h) −pCOHP and −ICOHP, and (i) PDOS of $P\stackrel{\u0304}{1}$CeN_{6} at 32 GPa.
However, when we insert a Ce atom into the bodycentered position (0.5, 0.5, 0.5) of $P\stackrel{\u0304}{1}$N_{2}, the transitional CeN_{6} (transCeN_{6}) with cisform N chains is formed after optimization at the same pressure [Fig. 4(d)]. The hybridization interaction between N_2s and N_2p orbitals is weakened. The pCOHP of N–N bonds changes remarkably, and the −ICOHP value (12.9) of the N–N bonds is reduced by half compared with its value (26.4) in the absence of Ce atoms [Fig. 4(e)]. At this time, the N–N bond strength is close to that in P2_{1}–LiN_{5} (−ICOHP = 15.3) at 32 GPa,74 indicating that N≡N triple bonds have become N=N double or N–N single bonds. The reaction mechanism is similar to that in which Ce atoms are able to catalyze the production of NH_{3} from N_{2} through the breaking of N≡N triple bonds.2 Although the Ce–N bonds exhibit an ionic bond interaction (−ICOHP = 2.1), the interaction between Ce_4f and N_2p orbitals plays a crucial role in the reaction [Figs. 4(e) and 4(f)]. Besides, each Ce atom provides 2.10e to N_{2} molecules to assist their transformation to a polymerized N chain. More interestingly, the reaction Ce + 3N_{2} → transCeN_{6} is exothermic, with an energy release of 3.19 eV/f.u. at 32 GPa, indicating that the above process tends to occur spontaneously under high pressure.
Laser heating is an effective method to produce metal polynitrides under high pressure. As shown in Fig. 4(g), $P\stackrel{\u0304}{1}$CeN_{6} should be obtainable from transCeN_{6} by laser heating. Because the reaction transCeN_{6} → $P\stackrel{\u0304}{1}$CeN_{6} is exothermic, with an energy release of 2.48 eV/f.u., the hightemperature environment could increase the anharmonic vibration of atoms and accelerate the conversion of thermodynamically unstable transCeN_{6} into the stable phase $P\stackrel{\u0304}{1}$CeN_{6} at 32 GPa [Fig. S1(e) supplementary material]. Although the average −ICOHP of N–N bonds in $P\stackrel{\u0304}{1}$CeN_{6} (11.2) is a bit smaller than that in transCeN_{6} (12.9) [Fig. 4(h)], the total number of N–N bonds increases from six to seven in a unit cell, and the total −ICOHP increases from 77.2 to 78.50. The larger −ICOHP (2.7) of Ce–N bonds and the greater amount of charge transferred (2.20e) for the Ce atom in $P\stackrel{\u0304}{1}$CeN_{6} mean that the Ce–N interaction becomes stronger than that in transCeN_{6}. As shown in Fig. 4(i), the PDOS near the Fermi level is remarkably different from that of transCeN_{6}, indicating that a new atomic coordinated environment is formed in $P\stackrel{\u0304}{1}$CeN_{6}, which exhibits a more stable configuration than that of transCeN_{6}. According to Fig. S6 (supplementary material), $P\stackrel{\u0304}{1}$CeN_{6} is an insulator with a bandgap of 3.25 eV. It is obvious that the bandgap increases with increasing pressure, which is similar to the behavior of AlN_{4}, AlN_{5}, and N_{10} cages.11,51,75 The reason for this is that the strong coupling interaction of sp^{2} or sp^{3} orbitals in adjacent N atoms reduces the energy of the bonding state (valence band) and increases the energy of the antibonding state (conduction band), resulting in an increase in the gap.
C. Stability under ambient conditions
Quenching down the highpressure phase to ambient conditions is of great significance for the practical application of HEDMs. Here, we further confirm the stability of $P\stackrel{\u0304}{1}$CeN_{6} under ambient conditions. The calculated phonon dispersion curve and elastic constants indicate that $P\stackrel{\u0304}{1}$CeN_{6} possess dynamical and mechanical stability at ambient pressure [Fig. 5(a) and Table SⅣ (supplementary material)]. An AIMD simulation was also performed. At 300 and 1000 K, the total energy of $P\stackrel{\u0304}{1}$CeN_{6} fluctuates about the equilibrium position and the nitrogen skeleton remains intact, indicating that this material possesses thermal stability under ambient conditions and 1000 K [Fig. S7 (supplementary material)]. In the radial distribution functions (RDFs), the first sharp peaks of each line demonstrate the nearest N–N, Ce–N, and Ce–Ce distances. The nearest N–N and Ce–N distances of $P\stackrel{\u0304}{1}$CeN_{6} are 1.44 and 2.59 Å at 0 GPa and 300 K, which are larger than those at 32 GPa and 300 K (1.41 and 2.47 Å, respectively) [Figs. 5(b) and S8 (supplementary material)]. The nearest N–N and Ce–N distances at 1000 K (1.43 and 2.58 Å) are comparable to those at 300 K. To sum up, $P\stackrel{\u0304}{1}$CeN_{6} can be quenched to ambient conditions if synthesized. In addition, C2/mCeN_{3}, $P\stackrel{\u0304}{1}$CeN_{3}, and $P\stackrel{\u0304}{1}$CeN_{4} are also dynamically, mechanically, and thermally stable under ambient conditions [Figs. S7 and S9 and Table SⅣ (supplementary material)], and they can maintain thermal stability up to 1000, 700, and 400 K, respectively.
Figure 5.(a) Phonon dispersion curve and PHDOS of $P\stackrel{\u0304}{1}$CeN_{6} at 0 GPa. (b) RDFs g(r) and structures of final states from the last 2 ps of AIMD simulations at ambient pressure and temperatures of 300 K (solid lines) and 1000 K (dotted lines). (c) PDOS of $P\stackrel{\u0304}{1}$CeN_{6} at 0 GPa. (d) Curves of charge transfer and enthalpy as functions of pressure during pressure release in $P\stackrel{\u0304}{1}$CeN_{6}, with the enthalpy of $P\stackrel{\u0304}{1}$CeN_{6} at 32 GPa being taken as the reference zero point.
Usually, high nitrogen content and stability under ambient conditions are mutually exclusive. According to the analysis above, the stability of $P\stackrel{\u0304}{1}$CeN_{6} in a highpressure environment results from the high coordination number of Ce atoms and from charge transfer. Interestingly, as the pressure decreases to ambient, the PDOS shows that the DOS of N_2p orbitals in the valence band increases slightly and can hold more electrons [Fig. 5(c)]. The Bader charge analysis supports the same conclusions, with the amount of charge transferred increasing as the pressure decreases in $P\stackrel{\u0304}{1}$CeN_{6}, indicating that the Ce atoms tend to provide more electrons at low pressure to enhance the stability of the N structure [Fig. 5(d)]. Meanwhile, the enthalpy decreases with decreasing pressure. Hence, charge transfer plays an important role in the stability of N structures at ambient pressure.
D. Energy density and explosive performance
The energy density, detonation velocity, and detonation pressure are important parameters to evaluate the explosive performance of an HEDM. Usually, these parameters are significantly related to the nitrogen content, the bonding types, and the mass of the coordination element. The mass energy density and volumetric energy density of $P\stackrel{\u0304}{1}$CeN_{6} are calculated using the dissociation path under ambient pressure: $P\stackrel{\u0304}{1}$CeN_{6} → CeN + $\frac{5}{2}$N_{2}. As shown in Table I, the mass energy density of $P\stackrel{\u0304}{1}$CeN_{6} is 2.00 kJ/g, which is comparable to that of the reported CNO (2.2 kJ/g)76 and LiN_{5} (2.72 kJ/g),74 and larger than that of C2/mFeN_{6} (1.83 kJ/g),38$P\stackrel{\u0304}{1}$GdN_{6} (1.62 kJ/g),52 and IbamBaN_{10} (1.33 kJ/g).30 Additionally, the volumetric energy density of $P\stackrel{\u0304}{1}$CeN_{6} is as high as 11.20 kJ/cm^{3}, which is larger than those of the high explosives TNT (7.05 kJ/cm^{3}) and HMX (10.83 kJ/cm^{3}) and those of some metal polynitrides (βBeN_{4}, γBeN_{4}, GdN_{6}, ReN_{8}, BeN_{10}, MgN_{10}, BaN_{10}, CaN_{10}, YN_{10}, GaN_{15}, ScN_{15}, and YN_{15}) (3.46–10.95 kJ/cm^{3}), and close to the reported maximum value in Be–N compounds (12.7 kJ/cm^{3}).29–32,39,40
Table 1. Mass density ρ, mass energy density E_{d}, volumetric energy density E_{v}, detonation velocity V_{d}, and detonation pressure P_{d} of $P\stackrel{\u0304}{1}$CeN_{6} compared with those of TNT and HMX.
Table 1. Mass density ρ, mass energy density E_{d}, volumetric energy density E_{v}, detonation velocity V_{d}, and detonation pressure P_{d} of $P\stackrel{\u0304}{1}$CeN_{6} compared with those of TNT and HMX.
Compound  ρ (g/cm^{3})  E_{d} (kJ/g)  E_{v} (kJ/cm^{3})  V_{d} (km/s)  P_{d} (GPa) 

$P\stackrel{\u0304}{1}$CeN_{6}  5.60  2.00  11.20  13.60  128.95  TNT  1.64^{a}  4.30^{b}  7.05^{c}  6.90^{d}  19.00^{a}  HMX  1.90^{a}  5.70^{b}  10.83^{c}  9.10^{d}  39.30^{a} 

Interestingly, $P\stackrel{\u0304}{1}$CeN_{6} possesses outstanding detonation velocity V_{d} and detonation pressure P_{d}. Its detonation pressure is 128.95 GPa, which is about seven times that of TNT (19.00 GPa) and more than three times that of HMX (39.30 GPa), and its detonation velocity is 13.60 km/s, which is twice that of TNT (13.60 km/s). Moreover, the detonation pressure and velocity of $P\stackrel{\u0304}{1}$CeN_{6} are also greater than those of typical metal polynitrides, such as BeN_{4}, ScN_{6}, ScN_{7}, GdN_{6}, SnN_{20}, and MN_{10} (M = Be, Mg, Ba, Ca, and Y) and MN_{15} (M = Al, Ga, Sc, and Y) compounds (15.81–100.96 GPa and 5.22–13.04 km/s).29–33,39,41 Notably, the excellent volumetric energy density, detonation pressure, and detonation velocity of $P\stackrel{\u0304}{1}$CeN_{6} are caused by its high mass density (up to 5.6 g/cm^{3}). Thus, $P\stackrel{\u0304}{1}$CeN_{6} exhibits excellent explosive properties as a novel HEDM under ambient pressure.
E. Mechanical properties
Mechanical properties are important for the practical application of Ce polynitrides. Hence, the bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio υ, and Vickers hardness H_{v} of $P\stackrel{\u0304}{1}$CeN_{6} were calculated at ambient pressure (Table II). The C_{33} (469 GPa) of $P\stackrel{\u0304}{1}$CeN_{6} is greater than its C_{11} (273 GPa) and C_{22} (371 GPa), indicating greater incompressibility along the [001] directions than the [100] and [010] directions. The hardness of $P\stackrel{\u0304}{1}$CeN_{6} is 20.7 GPa, which is greater than those of AlN_{5} (15.2 GPa), MnN_{4} (17.5 GPa), HfN_{10} (13.7 GPa), NbN_{4} (17.9 GPa), ScN_{5} (17.4 GPa), ScN_{3} (17.5 GPa), MReN_{8}, (13.9 GPa) and TʹReN_{8} (14.1 GPa), and comparable to those of FeN_{6} (24.9 GPa), RuN_{3} (23.4 GPa), and IrN_{4} (22.4 GPa),77–79 indicating that $P\stackrel{\u0304}{1}$CeN_{6} is a typical hard material (>20 GPa). The calculated Poisson’s ratio of $P\stackrel{\u0304}{1}$CeN_{6} is 0.2. This low Poisson’s ratio is the result of directional bonds, which increase the shear modulus and limit the movement of dislocations, increasing the hardness of the material.77 The small B/G ratio (<1.75) reveals that $P\stackrel{\u0304}{1}$CeN_{6} is a brittle material.
Table 2. Bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio ν, and Vickers hardness H_{v} of $P\stackrel{\u0304}{1}$CeN_{6}.
Table 2. Bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio ν, and Vickers hardness H_{v} of $P\stackrel{\u0304}{1}$CeN_{6}.
B (GPa)  G (GPa)  E (GPa)  ν  H_{v} (GPa)  B/G 

155.00  118.12  282.59  0.20  20.73  1.31 

Additionally, the material stiffness of $P\stackrel{\u0304}{1}$CeN_{6} is anisotropic. The 3D surface of the Young’s modulus of $P\stackrel{\u0304}{1}$CeN_{6} is deformed [Fig. 6(a)], and the Young’s modulus of $P\stackrel{\u0304}{1}$CeN_{6} along the c axis is higher than that along the other direction owing to the strong N–N covalent bonds, indicating that the layered structure with novel N_{14} rings along the [001] direction is more incompressible [Fig. 6(b)]. The degree of crystal anisotropy can be described quantitatively by the ratio E_{max}/E_{min}. The E_{max}/E_{min} value of $P\stackrel{\u0304}{1}$CeN_{6} is 2.40, which is less than those of trFeN_{4} (3.74) and ReN_{8} (4.95–13.49),40 indicating a weaker crystal anisotropy.
Figure 6.(a) 3D surface and (b) 2D projected profiles of Young’s modulus in $P\stackrel{\u0304}{1}$CeN_{6}.
F. IR and Raman spectra
The IR and Raman spectra of $P\stackrel{\u0304}{1}$CeN_{6} were calculated for experimental reference. From a grouptheoretical analysis, the irreducible representation of $P\stackrel{\u0304}{1}$CeN_{6} is Γ = 12A_{u}^{I} + 9A_{g}^{R}. Therefore, $P\stackrel{\u0304}{1}$CeN_{6} has 18 vibrational modes after the removal of three acoustic modes (3A_{u}^{I}). In Fig. S10 (supplementary material), the modes of IR and Raman activity are denoted by I and R, respectively. The corresponding vibrational modes are presented in Figs. S11–S14 (supplementary material). $P\stackrel{\u0304}{1}$CeN_{6} contains 9 IRactive modes (9A_{u}) and 9 Ramanactive modes (9A_{g}). At 32 GPa, both the A_{u} modes (274, 308, 355, 389, 509, 671, 683, 1009, and 1107 cm^{−1}) and A_{g} modes (465, 550, 674, 811, 947, 1011, 1036, 1078, and 1169 cm^{−1}) correspond to the outofplane N–N bending vibrations of folded N_{14} rings. At 0 GPa, both the A_{u} modes (169, 215, 220, 278, 439, 605, 625, 859, and 1003 cm^{−1}) and A_{g} modes (322, 481, 571, 699, 820, 875, 934, 967, and 1059 cm^{−1}) also correspond to outofplane N–N bending vibration in the layered Nstructure.
IV. CONCLUSION
A systematic highpressure study of CeN_{n} (n = 0.5, 1, 2, 3, 4, 5, and 6) compounds has been performed using firstprinciples swarmintelligence structural searches. The phase diagram of CeN_{n} compounds has been enriched by the proposal of six new stable highpressure phases (I4/mmmCeN_{2}, C2/mCeN_{3}, $P\stackrel{\u0304}{1}$CeN_{3}, $P\stackrel{\u0304}{1}$CeN_{4}, C2/cCeN_{6}, and $P\stackrel{\u0304}{1}$CeN_{6}). The stability of the new phases has been verified using the phonon dispersion curve, elastic constants, and AIMD simulations. The proposed layered structure of $P\stackrel{\u0304}{1}$CeN_{6} is composed of novel N_{14} ring. Not only does $P\stackrel{\u0304}{1}$CeN_{6} possess the lowest synthesis pressure of 32 GPa among the layered metal nitrides, but also it can be quenched down to ambient conditions. The reaction path Ce + 3N_{2} → transCeN_{6} → $P\stackrel{\u0304}{1}$CeN_{6} has been proposed to clarify the formation mechanism of $P\stackrel{\u0304}{1}$CeN_{6} under high pressure. The calculated results for COHP and electronic structure reveal that the charge transfer and orbital hybridization of Ce and N atoms play a crucial role in stabilizing $P\stackrel{\u0304}{1}$CeN_{6}. The volumetric energy density of $P\stackrel{\u0304}{1}$CeN_{6} reaches 11.20 kJ/cm^{3}, which is much higher than those of TNT and HMX. The detonation pressure (128.95 GPa) and detonation velocity (13.60 km/s) of $P\stackrel{\u0304}{1}$CeN_{6} are respectively almost seven times and twice those of TNT. Hence, $P\stackrel{\u0304}{1}$CeN_{6} has great application potential as an explosive material owing to its high stability and excellent explosive properties under ambient conditions.
SUPPLEMENTARY MATERIAL
ACKNOWLEDGMENTS
Acknowledgment. This work was supported financially by the National Key R&D Program of China (Grant Nos. 2018YFA0305900 and 2018YFA0703404), the National Natural Science Foundation of China under Grant Nos. 21905159, 11634004, 51320105007, 11604116, and 51602124, the Program for Changjiang Scholars and Innovative Research Team in the University of the Ministry of Education of China under Grant No. IRT1132, the Higher Educational Youth Innovation Science and Technology Program Shandong Province (Grant No. 2022KJ183), and GHfund B (Grant No. 202202026143).