Introduction

The discovery of nitrogen polymerization under high pressures has significantly extended the nitrogen chemistry. While the polymeric single-bonded nitrogen allotropes are formed only at pressures above 110 GPa1,2,3, the introduction of electropositive elements facilitates breaking the N2 triple-bond and initiates nitrogen catenation at significantly lower pressures. Indeed, under high-pressure high-temperature conditions nitrogen easily reacts with metals and forms numerous compounds featuring charged nitrogen N2x− dimers4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19 at low-to-mild pressures (5-50 GPa), or various catenated nitrogen units (e.g. tetranitrogen N44− units20, pentazolate N5− rings21,22,23, N6 rings24,25,26, and N18 macrocycle27) and 1D-polynitrogen chains20,28,29,30,31,32,33 at mild-to-high pressures (>50 GPa). Some of the nitrogen species discovered under high-pressure (e.g. pentazolate-anion, whose first stabilization in bulk was achieved in CsN5 at 60 GPa22) were subsequently synthesized by conventional chemistry methods under ambient pressure34,35,36.

In addition to the discoveries of unique nitrogen entities that push the boundaries of fundamental nitrogen chemistry, nitrides and polynitrides synthesized under high pressure often possess key properties for functional applications such as high hardness7, unique electronic properties33, and high energy density37. Polynitrides with a high nitrogen content are especially promising as high-energy-density materials (HEDM) because their decomposition results in the formation of molecular nitrogen, which is accompanied by a large energy release. The latter is due to a large difference between the energy of the triple intramolecular bond in N2 and the energy of double and single bonds in polynitrogen units37. For HEDMs the molecular weight of the compound also matters: with other properties being similar, the lighter the elements in the solid, the higher the gravimetric energy density of the compound. Since scandium is the lightest transition metal, its polynitrides may be especially promising as HEDM.

Hitherto, only one binary Sc-N compound is known: cubic scandium nitride ScN with the rock salt structure, which exists at ambient conditions and is predicted to be stable up to ~250 GPa38. There are several theoretical studies39,40,41,42, where nitrogen-rich phases with ScN3, ScN5, ScN6, and ScN7 compositions have been predicted to be stable under 30–110 GPa and may have potential as HEDM (gravimetric energy density ranges from 2.40 kJ/g to 4.23 kJ/g).

In this study, we experimentally investigated the behavior of the Sc-N system at pressures between 50 to 125 GPa and high temperatures. Here we present the synthesis and characterization of four novel Sc-N phases, whose structures were solved and refined on the basis of single-crystal X-ray diffraction. The nitrogen-rich polynitrides Sc2N6, Sc2N8, and ScN5 feature a unique nitrogen catenation: previously unknown N66− and N86− nitrogen units and \({\!\,}_{\infty }{\!\,}^{2}({{{{{\rm{N}}}}}}_{5}^{3-})\) anionic 2D-polynitrogen layers consisting of fused N12 rings, respectively.

Results and discussion

In this study diamond anvil cells (DACs) loaded with scandium pieces embedded in molecular nitrogen were used (see Methods section for details). Samples were compressed to their target pressures and laser-heated at 2500(300) K. Laser-heating experiments were carried out at pressures of 50(1), 78(2), 96(2), and 125(2) GPa (Supplementary Table 1). After laser-heating, detailed X-ray diffraction maps were collected around the heated spot to pinpoint the location of crystallites most appropriate for single-crystal X-ray diffraction measurements (Fig. 1). Then single-crystal X-ray diffraction data (Supplementary Fig. 1) were collected at the selected positions to identify the phases’ crystal structure and chemical composition.

Fig. 1: Sample chamber of the diamond anvil cell at 96 GPa.
figure 1

a Micro-photo of the sample chamber. b 2D X-ray diffraction map (collected with 0.75 µm steps at the ID11 beamline of the ESRF) showing the distribution of the scandium nitrides phases (determined by single-crystal XRD) within the heated sample at 96 GPa. The color intensity is proportional to the intensity of the following reflections: the (1 1 1) and (3 3 1) of ScN for the green regions; the (1 1 1) of ScN5 for the red regions; the (0 2 1) of Sc2N8 for the purple regions.

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According to the synchrotron single-crystal X-ray diffraction data, only the well-known ScN phase (rock-salt type structure, a = 4.2492(7) Å, V = 76.72(4) Å3 at 50 GPa) was formed at 50 GPa. The obtained volume is in good agreement with the published ScN equation of state38. At 78 GPa, two novel scandium nitrides with chemical formulas Sc2N6 and Sc2N8 were obtained along with ScN. At 96 GPa, a mixture of ScN, Sc2N8, as well as the previously unobserved ScN5, was obtained. And, finally, at 125 GPa the collected synchrotron single-crystal X-ray diffraction data and the subsequent crystal structure solution and refinement revealed the formation of the ScN5 and Sc4N3 phases. Overall four novel Sc-N phases were synthesized by chemical reactions of Sc and N2 at 2500 K in the pressure range of 78 to 125 GPa (Supplementary Fig. 2).

Remarkably, at 50 GPa, scandium behaves like at ambient pressure producing only ScN, while at higher pressures a rich variety of phases was observed. In addition to a significant increase in the chemical potential of nitrogen under high pressure43, another possible reason explaining such difference in chemistry between 50 and 78 GPa is a significant drop in scandium’s electronegativity at 60 GPa (Supplementary Fig. 3a) and as a result, scandium is predicted to be the least electronegative atom in 60–110 GPa pressure range44. It leads to the significant increase of difference in electronegativity between N and Sc above 60 GPa (Supplementary Fig. 3b), which increases the chemical reactivity of scandium, decreases the potential kinetic barriers of reactions, and leads to the appearance of more local minima in the energy landscape.

The refinement against single-crystal X-ray diffraction data for all synthesized compounds resulted in very good agreement factors (Supplementary Tables 2–6). For cross-validation of the crystal structures, we performed density functional theory (DFT) calculations using the Vienna ab initio simulation package45 (see Methods section for details). We carried out variable cell structural relaxations for Sc2N6, Sc2N8, and ScN5 and found that the relaxed structural parameters closely reproduce the corresponding experimental values (Supplementary Tables 7–9).

Sc2N6 synthesized at 78 GPa (Fig. 2a) crystalizes in the triclinic crystal system (space group P−1 (#2)). The structure of Sc2N6 has one Sc and three N distinct atomic positions (see Supplementary Table 3 and the CIF for the full crystallographic data). Nitrogen atoms form isolated “zig-zag” N6 units (Fig. 2a, b). The existence of this phase was predicted at pressures of 30–100 GPa39.

Fig. 2: Crystal structure of Sc2N6 and Sc2N8 at 78 GPa.
figure 2

a A view of Sc2N6 along the a-axis; b an N6 unit; c structural formula of an N6 unit; d a view of Sc2N8 along the a-axis; e an N8 unit; f structural formula of an N8 unit. Sc atoms are purple, N atoms are blue; thin grey lines outline the unit cell. Values of bond lengths and angles obtained from the experiment are shown in black, while those obtained from the DFT calculations are shown in red.

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The structure of Sc2N8 (Fig. 2d) has the monoclinic space group P21/c (#14) with one Sc and four N distinct atomic positions (see Supplementary Table 4 and the CIF for the full crystallographic data). Nitrogen atoms form isolated “zig-zag” N8 units (Fig. 2d, e) that have never been observed or predicted.

The bond length analysis of the N6 and N8 units suggests that N1-N2, N2-N3 (in N6 unit) and N1-N2, N2-N3, N4-N4 (in N8 unit) are single-bonded, while N3-N4 (in N6 unit) and N3-N4 (in N8 unit) are double-bonded (Fig. 2b,c,e,f). Then, the charges can be described in a classic ionic approach: the total charge of [N6]6− and [N8]6− units is 6-, which corresponds to the +3 oxidation state of Sc atoms. The angle values and a small difference in bond length indicate the strong electron delocalization (indeed several different resonance Lewis formulas can be drawn for N6 and N8) and nitrogen atoms cannot be considered as purely sp2 or sp3 hybridized.

The two novel catenated nitrogen units N66− and N86− discovered in this study—being intermediate non-cyclic species between dinitride and 1D-polynitrogen anions—significantly expand the list of anionic nitrogen oligomers (Fig. 3). Notably, all these units are built of an even number of nitrogen atoms suggesting their formation via the polymerization of dinitrogen molecules. The degree of polymerization increases with pressure: dinitrides are synthesized at low pressures (<50 GPa); N4, N6, N8 units are obtained at mild pressures (50–80 GPa), while 1D-polynitrogen chains are usually formed above 100 GPa. Since the dinitrogen ([N2]x− x = 0.66, 0.75, 1, 2, 3, 4), and 1D-polynitrogen ([N4]∞x−, x = 2–6) anions are able to accumulate different charges, one can expect that the N6 and N8 units can also exist in different charge states, and therefore can be found in other metal-nitrogen systems.

Fig. 3: Experimentally observed catenated nitrogen units and chains.
figure 3

The units in blue were first discovered in the present study.

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The structure of ScN5 has the monoclinic space group P21/m (#11) with one Sc and three N distinct atomic positions (see Supplementary Table 5 and the CIF for the full crystallographic data). Nitrogen atoms form corrugated 2D polymeric \({\!\,}_{\infty }{\!}^{2}({{{{{\rm{N}}}}}}_{5}^{3-})\) layers alternating along the a-axis built of fused N12 rings (Fig. 4a). Sc atoms are located in between the layers, in the way that the projection of Sc atoms along the a-axis is in the center of the N12 rings (Fig. 4b). Sc atoms are eight-fold coordinated (coordination number CN = 8, coordination polyhedron is a distorted square antiprism) by four N atoms of the lower layer and four N atoms of the upper layer (Fig. 4c).

Fig. 4: Crystal structure of ScN5 at 96 GPa.
figure 4

a A view of the crystal structure along the c-axis. b A view of the crystal structure along the a-axis. c The coordination environment of the Sc atom. d A specific view of N12 cycle along the a-axis. e A general view of N12 cycle. Sc atoms are purple, N atoms are blue; thin grey lines outline the unit cell. Values of bond lengths and angles obtained from the experiment are shown in black, while those obtained from the DFT calculations are shown in red.

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The analysis of N-N lengths in ScN5 suggests that all N-N bonds are single bonds (Fig. 4d). All N atoms can be considered as sp3-hybridized, which also explains that the values of N-N-N angles in the N12 cycles are close to the ideal tetrahedra angle (98.7°−114.5°, Fig. 4e). N1 atoms make three covalent N-N bonds, while N2 and N3 atoms make only two, therefore one can suggest a −1 charge on the N2 and N3 atoms. It corresponds to the +3 oxidation state of Sc atoms.

Despite the theoretical prediction of four different structures with the ScN5 composition39,40,41, the here observed structure was not predicted. Usually in polynitrides nitrogen prefers to form 1D polymeric chains20,28,29,30,31,32,33, and among all the experimentally synthesized polynitrides up-to-date there is only one discovered polynitride with 2D polynitrogen layers—monoclinic BeN433 with layers consisting of the fused N10 rings. The polynitrogen layers in ScN5 can be considered as distorted bp-N layers2, where 1/6 atoms are missing (Supplementary Fig. 4).

ScN5 is isostructural to a family of polyphosphides LnP5 (Ln=La-Lu, Y (except Eu and Pm)) known at ambient conditions46,47. It fully obeys the ninth high-pressure chemistry rule of thumb formulated in 1998: “Elements behave at high pressures like the elements below them in the periodic table at lower pressures” 48. The adoption of this structure type is also advantageous from a geometric point of view, since the ratio of ionic radii r(N3−)/r(Sc3+)=1.97 in ScN5 perfectly fits r(P3−)/r(Y3+) = 1.95 in the above-mentioned family member YP5.

Sc4N3 synthesized at 125 GPa has a well-known anti-Th3P4 structure type (space group I−43d (#220)) and contains only distinct, not-catenated N atoms (see Supplementary Table 6, Supplementary Fig. 5, and the CIF for the full crystallographic data), which we do not discuss in detail here. This Sc4N3 structure was predicted to be thermodynamically stable above 80 GPa39.

In order to get a deeper insight into the chemistry and the physical properties of the novel compounds, further DFT calculations were performed (see Methods section for details). As mentioned above, variable-cell structural relaxations for the Sc2N6, Sc2N8, and ScN5 compounds at the synthesis pressure closely reproduced structural parameters and bond lengths obtained from the experimental data. The phonon dispersion relations calculated in the harmonic approximation show that Sc2N6, Sc2N8, and ScN5 phases are dynamically stable at 96 GPa and remain dynamically stable at ambient pressure (Supplementary Figs. 6–8). Considering dynamical stability at 1 bar, we have attempted to quench Sc2N6, Sc2N8, ScN5 phases, however, due to technical limitations of the decompression experiment (see footnote Supplementary Table 1), no conclusion regarding their recoverability could be made. To trace the structures’ behavior during the pressure release and to get the equations of state of all synthesized nitrogen-rich high-pressure scandium polynitrides, the full variable-cell structure relaxation for the Sc2N6, Sc2N8, and ScN5 compounds were performed with 10 GPa pressure steps in the range of 0–150 GPa (Supplementary Fig. 9). The volume-pressure dependences of DFT-relaxed structures of Sc2N6, Sc2N8, and ScN5 were fitted with a 3rd order Birch-Murnaghan equation of state (Supplementary Fig. 10). The obtained bulk moduli (K0(Sc2N6) = 160 GPa, K0(Sc2N8) = 173 GPa, K0(ScN5) = 205 GPa) are lower than or comparable to the bulk modulus of known ScN (K0(ScN) = 207 GPa)38.

Under the same pressure, the volume per atom for all investigated nitrides monotonously linearly decreases with increasing nitrogen content (Supplementary Fig. 11a). Interestingly, the volume per nitrogen atom in the ScN-Sc2N6-Sc2N8-ScN5 series does not decrease with the degree of nitrogen polymerization (Supplementary Fig. 11b), so nitrogen polymerization probably is a way of crystal structure adaptation to closer N-N contacts.

While the structure of Sc2N6 has been predicted39, the crystal structures of Sc2N8 and ScN5 we observed have not been predicted. Remarkably, four different crystal structures with the ScN5 composition were proposed39,40,41, but the one we synthesized in the present study (P21/m ScN5) was not among them. Our calculations of relative formation enthalpies of ScN5 for various predicted structures (Cm ScN539, P−1 ScN539, C2/m ScN5, 40 and P21/c ScN5 41) with respect to P21/m ScN5 (Supplementary Fig. 12a) in the range of 0 to150 GPa have shown that above 46 GPa the P21/m ScN5 phase is thermodynamically more stable than all other predicted phases. Below 46 GPa P−1 ScN539 is more favorable. The C2/m ScN540 and P21/c ScN541 phases are not energetically competitive with P21/m ScN5 in the whole pressure range studied (Supplementary Fig. 12a).

To estimate the thermodynamic stability of the Sc2N6, Sc2N8, and ScN5 phases, the nitrogen-rich part of the static enthalpy convex hull was calculated at different pressures. Sc2N6 and ScN5 phases were found to be stable at the synthesis pressures (78 and 96 GPa, Supplementary Fig. 13a and Supplementary Fig. 12b), but Sc2N8 appears to be out of the convex hull (40 meV and 50 meV per atom above the convex hull at 78 and 96 GPa, respectively). Such insignificant departures from the convex hull, smaller than kBT at the synthesis temperature (2500 K, 215 meV), suggest that Sc2N8 may be thermodynamically stable at high temperatures and preserved as a metastable state under rapid T-quench to room temperature. ScN5 remains thermodynamically stable at least down to 40 GPa (Supplementary Fig. 13b), and Sc2N6—down to 30 GPa (Supplementary Fig. 13c), while at 20 GPa all nitrogen-rich scandium phases are out of the convex-hull (Supplementary Fig. 13d).

The calculated electron localization functions for Sc2N6, Sc2N8, and ScN5 demonstrate a strong covalent bonding between nitrogen atoms within the N6, N8 units, and 2D-polynitrogen layers (Fig. 5a–c), while there is no covalent bonding between nitrogen and scandium atoms. The computed electron density of states (DOS) shows that Sc2N6 and Sc2N8 are metals (Fig. 5d, e) with an anion-driven metallicity10, since the main electronic contribution at the Fermi level comes from the nitrogen p-states. At the same time, ScN5 is an indirect semiconductor with a band gap of 1.8 eV at 96 GPa (Fig. 5f). One can explain such different electronic properties considering the chemical bonding in these compounds. In ScN5 there are only single N-N bonds, which means all π* antibonding nitrogen molecular orbitals are fully filled, whereas, in Sc2N6 and Sc2N8, containing delocalized π-bonds within N66− and N86− units, π* antibonding nitrogen states are partially filled and can conduct electrons through the π*-band. A similar trend of electronic properties with respect to the presence of N-N π-bonds is observed for many known polynitrides27,28,29,30,31,32,33. Among all known polynitrides there are only two compounds with solely σ N-N bonds: TaN5, which contains single-bonded branched polynitrogen chains31, and m-BeN4, which contains single-bonded 2D-polynitrogen layers33. Both compounds are semiconductors, as reported for TaN531, and calculated for m-BeN4 in the present study (Supplementary Fig. 14). Other polynitrides contain N-N π-bonds and the majority of them (tr-BeN4, FeN4, α-ZnN4, β-ZnN4, TaN4, ReN8·xN2, WN8·N2, Os5N28·3N2, Hf4N20·N2, Hf2N11, Y2N11, YN6)27,28,29,30,31,32,33 exhibit an anion-driven metallicity.

Fig. 5: Calculated electronic properties of Sc2N6 at 78 GPa, and Sc2N8, ScN5 at 96 GPa.
figure 5

Electron localization function calculated for (a) Sc2N6 in the (3 0 2) plane, (b) Sc2N8 in the (−2 4 1) plane, and (c) ScN5 in the (1 0 0) plane. The electron density of states of (d) Sc2N6, (e) Sc2N8, and (f) ScN5.

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Considering the dynamical stability of Sc2N6, Sc2N8, and ScN5 at ambient pressure, these phases might be preserved at ambient conditions as metastable and potentially can serve as high-energy-density materials. The key metrics of energetic materials’ performance49, such as volumetric and gravimetric energy densities, detonation velocity, and detonation pressure, were estimated for Sc2N6, Sc2N8, and ScN5 (Table 1) considering their decomposition to ScN and molecular nitrogen at 1 bar (see Methods section for details).

Table 1 Characteristics of Sc2N6, Sc2N8, ScN5 and TNT as energetic materials
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The energy densities and explosive performance increase from Sc2N6 to ScN5 along with the increase in nitrogen content. Due to the higher density of scandium nitrides compared to organic explosives, they possess extremely high volumetric energy densities that are higher than the typical energy density of TNT. The estimated gravimetric energy densities are lower than that of TNT, but higher than those of many other polynitrides31 since scandium is a light metal. The estimated detonation velocity and detonation pressure of scandium polynitrides are also higher than those of TNT. Thus, the Sc2N6, Sc2N8, and ScN5 are promising high-energy-density materials.

To summarize, in this study, four novel Sc-N phases—Sc2N6, Sc2N8, ScN5, and Sc4N3—were synthesized from Sc and N2 by laser-heating at 2500 K at pressures between 78 and 125 GPa. Nitrogen-rich scandium polynitrides Sc2N6, Sc2N8, and ScN5 demonstrate a unique nitrogen catenation: they feature N6 units, N8 units, and 2D polynitrogen \({\!\,}_{\infty }{\!\,}^{2}({{{{{\rm{N}}}}}}_{5}^{3-})\) layers consisting of N12 fused rings, respectively. DFT calculations showed that all three scandium polynitrides are dynamically stable at the synthesis pressure as well as at 1 bar. Sc2N6 and Sc2N8 are metals with the main electronic contribution at the Fermi level that comes from the nitrogen p-states, while ScN5 is an indirect semiconductor. Synthesized Sc2N6, Sc2N8, and ScN5 compounds are promising high-energy-density materials with volumetric energy densities, detonation velocities, and detonation pressures higher than those of TNT.

One can expect that the N6 and N8 units will be stabilized at ambient conditions in the future, considering a positive example of CsN5 high-pressure synthesis and subsequent stabilization of the N5− anion at atmospheric pressure. It may not only open access to novel high-energy-density materials but also to analogues of Li- and Mg- metalorganic compounds that are currently widely used in organic synthesis. N6 and N8 units, if used as building blocks in organic chemistry, may provide new routes for the targeted synthesis of novel N-heteroatomic organic, metalorganic, and coordination compounds.

Methods

Sample preparation

The BX90-type large X-ray aperture DACs52 equipped with Boehler-Almax type diamonds53 (culet diameters are 250, 120, and 80 μm) were used in the experiments. The sample chambers were formed by pre-indenting of rhenium gaskets to 20, 18, and 15 μm thickness and laser-drilling a hole of 115, 60 and 40 μm, respectively, in diameter in the center of the indentation. A DAC equipped with 250-μm culet anvils was used for the experiment at 50(1) GPa; a DAC equipped with 120-μm culet anvils was used for experiments at 78(2) and 96(2); and a DAC equipped with 80-μm culet anvils was used for the experiment at 125(2) GPa. A piece of scandium (99.9%, Sigma Aldrich) was placed in a sample chamber, then molecular nitrogen (purity grade N5.0) was loaded using a BGI high-pressure gas loading system (1300 bars)54. The sizes of the scandium pieces were 40 × 40 × 8 μm3 for 250 μm culet anvils and not bigger than 15 × 15 × 5 μm3 for DACs with anvils of all other sizes. The samples were compressed to target pressure (50(1), 78(2), 96(2), and 125(2) GPa) and then laser-heated up to 2500(200) K using a home-made double-sided laser-heating system equipped with two YAG lasers (λ = 1064 nm) and the IsoPlane SCT 320 spectrometer with a 1024 × 2560 PI-MAX 4 camera for the collection of thermal emission spectra from the heated spot55. The temperature during the laser heating was determined by fitting of sample’s thermal emission spectra to the grey body approximation of Planck’s radiation function in a given wavelength range (570–830 nm). The pressure in the DACs was determined using the Raman signal from the diamond anvils56 and monitored additionally by X-ray diffraction of the Re gasket edge using the rhenium equation of state57.

X-ray diffraction

The X-ray diffraction studies were done at the ID11 beamline (λ = 0.2843 Å and λ = 0.2846 Å) and ID15b beamline (λ = 0.4100 Å) of the Extreme Brilliant Source European Synchrotron Radiation Facility (EBS-ESRF) as well as at the GSECARS 13IDD beamline of the APS (λ = 0.2952 Å). At ID11 beamline of ESRF the X-ray beam was focused down to 0.75 × 0.75 μm2 and data was collected with Eiger2X CdTe 4 M hybrid photon counting pixel detector. At ID15b beamline of ESRF the X-ray beam was focused down to 1.5 × 1.5 μm2 and data was collected with Eiger2X CdTe 9 M hybrid photon counting pixel detector. At 13IDD beamline of APS the X-ray beam was focused down to 2 × 2 μm2 and data was collected with Pilatus 1 M detector. In order to determine the position of the polycrystalline sample on which the single-crystal X-ray diffraction acquisition is obtained, a full X-ray diffraction mapping of the pressure chamber was achieved. The sample position displaying the most and the strongest single-crystal reflections belonging to the phase of interest was chosen for the collection of single-crystal data, collected in step-scans of 0.5° from −36° to +36°. The CrysAlisPro software package58 was used for the analysis of the single-crystal XRD data (peak hunting, indexing, data integration, frame scaling, and absorption correction). To calibrate an instrumental model in the CrysAlisPro software, i.e., the sample-to-detector distance, detector’s origin, offsets of the goniometer angles, and rotation of both the X-ray beam and detector around the instrument axis, we used a single crystal of orthoenstatite [(Mg1.93Fe0.06)(Si1.93,Al0.06)O6, Pbca space group, a = 8.8117(2) Å, b = 5.18320(10) Å, and c = 18.2391(3) Å]. The DAFi program was used for the search of reflection’s groups belonging to the individual single crystal domains59. Using the OLEX2 software package60, the structures were solved with the ShelXT structure solution program61 using intrinsic phasing and refined with the ShelXL62 refinement package using least-squares minimization. Crystal structure visualization was made with the VESTA software63.

Theoretical calculations

First-principles calculations were performed using the framework of density functional theory (DFT) as implemented in the Vienna Ab initio Simulation Package (VASP)64. The Projector-Augmented-Wave (PAW) method65 was used to expand the electronic wave function in plane waves. The Generalized Gradient Approximation (GGA) functional is used for calculating the exchange-correlation energies, as proposed by Perdew–Burke–Ernzerhof (PBE)66. The recommended PAW potentials “Sc_sv” and “N” with the following valence configurations of 3s23p64s23d1 for Sc and 2s22p3 for N were used. We used the Monkhorst–Pack scheme with 10 × 10 × 10 for ScN, 12 × 8 × 8 for Sc2N6, 10 × 6 × 4 for Sc2N8, 12 × 6 × 12 for ScN5 k-points for Brillouin zone sampling, and the plane-wave kinetic energy cutoff was set to 800 eV, with which total energies are converged to better than 2 meV/atom. The electronic convergence criterion was set to ΔE = 10−8 eV, this minimized the interatomic forces to Fatom < 10−3 eV/Å. For electron band structure calculations the 1.5−2 fold denser k-points grids were used. The finite displacement method, as implemented in PHONOPY67, was used to calculate phonon frequencies and phonon band structures. The 4×3×3, 3×2×2, and 3 × 2 × 3 supercells with 4 × 4 × 4 k-points grids for Sc2N6, Sc2N8, and ScN5, respectively, were used for phonon calculations and displacement amplitudes were of 0.01 Å.

The gravimetric and volumetric energy densities of Sc2N6, Sc2N8, and ScN5 were calculated considering the enthalpy change ΔH for the following chemical decomposition reactions at ambient pressure at 0 K (the Fm−3m-ScN and α-N2 structures of products were used in the calculations since they are the most stable polymorphs at such conditions):

$${{{{{{\rm{Sc}}}}}}}_{2}{{{{{{\rm{N}}}}}}}_{6}\to 2\,{{{{{\rm{ScN}}}}}}+2\,{{{{{{\rm{N}}}}}}}_{2}$$
$${{{{{{\rm{Sc}}}}}}}_{2}{{{{{{\rm{N}}}}}}}_{8}\to 2\,{{{{{\rm{ScN}}}}}}+3\,{{{{{{\rm{N}}}}}}}_{2}$$
$${{{{{{\rm{ScN}}}}}}}_{5}\to {{{{{\rm{ScN}}}}}}+2\,{{{{{{\rm{N}}}}}}}_{2}$$

The detonation velocity (Vd, km/s) and detonation pressure (Pd, GPa) of the Sc2N6, Sc2N8, and ScN5 were estimated by the Kamlet-Jacobs empirical equations50:

$${V}_{d}={(N{{\cdot }}{M}^{0.5}{{\cdot }}{{GED}}^{0.5})}^{0.5}{{\cdot }}(1.011+1.312\rho )$$
(1)
$${P}_{d}=1.588{{\cdot }}N{{\cdot }}{M}^{0.5}{{\cdot }}{{GED}}^{0.5}{{\cdot }}{\rho }^{2}$$
(2)

where N is the number of moles of gaseous detonation product (nitrogen gas) per gram of explosive, M is the molar mass (28 g/mol) of nitrogen gas, GED is gravimetric energy density in cal/g, and ρ is density in g/cm3.