Diverse ruthenium nitrides stabilized under pressure: a theoretical prediction.

First-principles calculations were performed to understand the structural stability, synthesis routes, mechanical and electronic properties of diverse ruthenium nitrides. RuN with a new I-4m2 symmetry stabilized by pressure is found to be energetically preferred over the experimental NaCl-type and ZnS-type ones. The Pnnm-RuN2 is found to be stable above 1.1 GPa, in agreement with the experimental results. Specifically, new stoichiometries like RuN3 and RuN4 are proposed firstly to be thermodynamically stable, and the dynamical and mechanical stabilities of the newly predicted structures have been verified by checking their phonon spectra and elastic constants. A phase transition from P4/mmm-RuN4 to C2/c-RuN4 is also uncovered at 23.0 GPa. Drawn from bonding and band structure analysis, P4/mmm-RuN4 exhibits semi-metal-like behavior and becomes a semiconductor for the high-pressure C2/c-RuN4 phase. Meanwhile the P21/c-RuN3 shows metallic feature. Highly directional covalent N-N and Ru-N bonds are formed and dominating in N-enriched Ru nitrides, making them promising hard materials.

The search of hard or superhard materials is of great interest due to the fundamental importance and technological applications1234. Transition metal (TM) nitrides, because of the strong covalent bonding between TM and N atoms, are considered to be promising candidates for hard materials5678. The inert nature of noble metals (e.g., Os, Ir, Pt, Ru, Rh and Pd) used to hamper the reaction with nitrogen; however, the discovery of platinum pernitrides overcomes the chemical barrier91011. Later on, iridium, osmium and palladium were also found to form pernitrides under pressure12131415. Recently, marcasite-type rhodium and ruthenium pernitrides have been successfully synthesized as well1617. Among these nitrides, IrN2 was found to have bulk modulus of 428 GPa1218, higher than most of previously synthesized materials. PtN2 and OsN2 also possess greater incompressibility with bulk modulus of 372 GPa910 and 358 GPa1218, respectively, comparable to the traditional superhard materials (e.g., diamond and c-BN). The synthesis of the noble metal nitrides is a milestone that has set the stage of paradigm of novel superhard materials.

Within the binary system of Ru and N, a cubic NaCl-type RuN with the lattice parameter of a = 4.445 Å was reported in the earlier study19. ZnS-type RuN thin films were also reported in the later studies2021. Recently, RuN2 with marcasite-type structure and bulk modulus of 330 GPa was identified by Niwa et al.17. Despite of these efforts, the knowledge of basic structural type, mechanical and electronic properties of Ru nitrides is still lacking. Understanding this binary system requires comprehensive knowledge of their structural stability, composition changes, nitrogen bonding features and even pressure response. In this study, we are motivated to perform structural searching for stable phases in RuN (x = 1–4) in the pressure range of 0–50 GPa. We also examine the stability range of reported Ru nitrides and their phase transition under pressure. For Ru mononitrides, two structures (I-4m2-RuN and R-3m-RuN) are found to be more energetically stable than the previously reported NaCl-type and ZnS-type phases. With regard to Ru pernitrides, the stability of Pnnm-RuN2 is confirmed by phonon analysis and its behaviors under pressure are described. Moreover, Ru trinitrides and tetranitrides that have not yet been synthesized in laboratory were predicted with P21/c-RuN3, P4/mmm-RuN4, C2/c-RuN4, and Cmmm-RuN4 structures, guiding further experimental attempts to produce N-rich Ru nitrides.

Results and Discussion

The NaCl-type RuN (space group: Fm-3m) was suggested by Moreno-Armenta et al. using reactive pulsed laser ablation19, in which ruthenium target was laser ablated in N2 atmosphere in the pressure range of 1 × 10−9 to 0.16 Torr. Therefore, the NaCl-type RuN was initially chosen in our calculations. Interestingly, NaCl-type RuN is energetically unfavorable up to at least 50 GPa, as we can see from the convex hull of Ru-N system (Fig. 1) and relative formation enthalpies of RuN as a function of pressure (Fig. 2a). Furthermore, the calculated elastic constants (C) also rule out the NaCl-type RuN due to the mechanical instability with a negative C44 value (−94 GPa), according to the Born-Huang criterion22. Moreover, the phonon dispersion of NaCl-type RuN shows imaginary frequency in the Brillouin Zone (see Supplementary Fig. S1), indicating its dynamic instability. ZnS-type RuN was also reported to have been deposited by pulsed-DC magnetron sputtering at the pressure of 1 Pa and the temperature of about 50 °C2021. Similar to NaCl-type RuN, we found that ZnS-type RuN is also mechanically unstable with a negative C44 value (−170 GPa). All the results strongly motivate us to search a possible ground state structure for RuN. Two candidates, the I-4m2-RuN and the R-3m-RuN, were survived in our structure searches. Interestingly, ZnS-type structure is relaxed to be an I-4m2-RuN symmetry, consistent with our structure searching results. Nevertheless, the formation enthalpy of I-4m2-RuN is positive below 0.17 GPa, as shown in Fig. 1. As the pressure increases, I-4m2-RuN becomes energetically favorable in the pressure range of 0.17 to 10.5 GPa, and then R-3m-RuN stands out up to 50 GPa (see Fig. 2a). Nevertheless, as shown in the convex hull, the predicted I-4m2-RuN has a very narrow stable range, and then both I-4m2-RuN and R-3m-RuN become metastable with the increasing pressure. In I-4m2-RuN, as demonstrated in Fig. 3a, N atoms occupy the center and vertex positions of the tetragonal lattices. Ru and neighboring N atoms constitute a tetrahedron with Ru atom situated in the center, and the tetrahedrons are connected by sharing vertex. The bond length of Ru-N is found to be 1.968 Å, but the separation of N-N is relatively large, 3.068 Å, limiting its capability of forming polynitrogen. For R-3m-RuN (see Fig. 3b), Ru and N atoms constitute a puckered 2D graphene-like honeycomb structure paralleling to the xy plane, in which the bond distance of Ru-N is 2.007 Å. The honeycomb structure was also observed in the IIB selenides and tellurides23. The puckered honeycomb sheets are connected by bridging N-N bonds forming a Ru-N-N-Ru layer, stacking along c axis. The N-N bond length in R-3m-RuN is 1.361 Å, shorter than the typical N-N single bond (1.45 Å in N2H4), but much longer than the double bond (1.21 Å for N2F2) and triple bond (1.09 Å for N2)24.

Figure 1

Formation enthalpies (∆H) of the structures of N-rich Ru-N binary compounds at pressures of (a) 0 GPa, (b) 10 GPa, (c) 20 GPa and (d) 30 GPa. The convex hulls connecting stable phases (solid circles) are shown by solid lines. Unstable/metastable phases are shown by open circles.

Figure 2

(a,b) Formation enthalpies of RuN and RuN2 with respect to Ru and nitrogen as a function of pressure, respectively. (c,d) Relative formation enthalpies of RuN3 and RuN4 with respect to Pnnm-RuN2 and nitrogen as a function of pressure, respectively.

Figure 3

Crystal structures of Ru nitrides.

(a) I-4m2-RuN with wyckoff position Ru (2d) (0, 0.5, 0.75) and N (2a) (0, 0, 0). (b) R-3m-RuN with wyckoff position Ru (6c) (0, 0, 0.2322) and N (6c) (0, 0, 0.0374). (c) Pnnm-RuN2 with wyckoff position Ru (2a) (0, 0, 0) and N (4g) (0.1236, 0.4058, 0). (d) P21/c-RuN3 with wyckoff position Ru1 (4e) (0.5011, 0.1336, 0.1479), Ru2 (4e) (0.0313, 0.6289, 0.8824), N1 (4e) (0.6973, 0.3548, 0.1936), N2 (4e) (0.2439, 0.2475, 0.1333), N3 (4e) (0.3172, 0.9829, 0.1262), N4 (4e) (0.8148, 0.8595, 0.8150), N5 (4e) (0.2560, 0.7440, 0.8680) and N6 (4e) (0.1891, 0.4645, 0.8779). (e) P4/mmm-RuN4 with wyckoff position Ru (1c) (0.5, 0.5, 0) and N (4i) (0, 0.5, 0.3279). (f) C2/c-RuN4 with wyckoff position Ru (4e) (0, 0.0915, 0.25), N1 (8f) (0.2863, 0.0884, 0.6852) and N2 (8f) (0.3372, 0.2646, 0.2598). (g) Cmmm-RuN4 with wyckoff position Ru (2b) (0, 0.5, 0) and N (8q) (0.1606, 0.1930, 0.5). The Ru and N atoms are represented as big orange and small purple spheres, respectively.

Recently, Pnnm-RuN2 was synthesized by direct chemical reaction between ruthenium and molecular nitrogen above the pressure of 32 GPa17. The ground-state structure of RuN2 was confirmed by the formation enthalpy curves as a function of pressure (Fig. 2b), and its formation enthalpy turns negative above 1.1 GPa. The convex hull also shows that Pnnm-RuN2 is the most stable phase in Ru-N phase diagram above the pressure of 10 GPa. In Pnnm-RuN2 (shown in Fig. 3c), Ru atoms are located in the center and vertex sites of the orthorhombic lattice. Each Ru atom is coordinated by six N atoms forming a RuN6 octahedron. The four equatorial and two axial Ru-N distances in RuN6 octahedron are 2.102 and 2.057 Å, respectively. The RuN6 octahedrons that situated in the center of the unit cell and the ones in the vertexes are connected by sharing corners and N2 dimers. The N2 dimer has a bond length of 1.375 Å, shorter than that in the OsN2 (~1.4 Å)12, IrN2 (1.42 Å)18 and PtN2 (1.41 Å)10, but larger than that in RhN2 (1.30 Å)18. The strong covalent N–N bonding in N2 dimer provides a strengthening effect on the elastic modulus.

It is known that IrP325, IrAs326, IrSb326, CoP325, and RhP325 with cubic skutterudite CoAs3-type structure were synthesized in experiments. The chemical related compounds, including the corresponding nitrides IrN327, CoN328 and RhN328 with the same type structure were also suggested by first-principles calculations. Besides, Imm2-TcN3, P4/mmm-TcN4, Imm2-ReN3 and Cmmm-ReN4 were also proposed by Zhao et al.2930, together with ReN4, OsN4 and WN4 with ReP4-type structure by Aydin et al.31. To explore the possibility for Ru nitrides with higher nitrogen concentration, Ru trinitrides and tetranitrides were also searched in our calculations. Simultaneously, the calculated formation enthalpy-pressure curves with respect to Pnnm-RuN2 are given in Fig. 2c,d, respectively. For Ru trinitrides, a new P21/c type structure for RuN3 is found to be favored over RuN2+1/2N2 above 12.8 GPa and thermodynamically stable up to at least 50 GPa (see Fig. 2c). P21/c-RuN3 (Fig. 3d) contains two types of distorted RuN6 octahedrons and puckered S-shaped N6 units. In comparison with Pnnm-RuN2, P21/c-RuN3 shows a variety of Ru-N distances from 1.976 to 2.275 Å for distorted RuN6 octahedrons. Besides, different with the regular RuN6 octahedron in Pnnm-RuN2, the distorted RuN6 octahedrons in P21/c-RuN3 have the axis N-Ru-N angles of 166° and 170°, respectively. Moreover, in Pnnm-RuN2, there is only one unique N-N distance of 1.375 Å in N2 dimer, whereas the N-N distances in puckered N6 units in P21/c-RuN3 vary from 1.352 to 1.460 Å. Interestingly, the stacking of RuN6 octahedrons in P21/c-RuN3 becomes more packed than that in Pnnm-RuN2. Furthermore, N2 dimers (N6 units) in Pnnm-RuN2 (P21/c-RuN3) is paralleling to xy (yz) planes, which will be reflected on the great incompressibility along a and b axis (b and c axis) as expected.

With further increasing N concentration, a P4/mmm-RuN4 for Ru tetranitrides becomes energetically preferable relative to RuN2+N2 at 13.6 GPa, and transforms to C2/c-RuN4 at 23 GPa (see Fig. 2d). For comparison, the relative enthalpy of Cmmm-RuN4 with respect to C2/c-RuN4 as a function of pressure is shown in Supplementary Fig. S2. The C2/c-RuN4 is thermodynamically favorable up to 101 GPa and transforms to Cmmm-RuN4, which is preferable at least up to 150 GPa. For P4/mmm-RuN4 (Fig. 3e), Ru atoms occupy the center of the top and bottom of the tetragonal unit cell and N2 dimers locate in the center of the four sides of the lattice. In this structure, each Ru atom is surrounded by eight N atoms, forming RuN8 cuboids. The intra-layer cuboids are connected by sharing edges, and the interlayer ones are connected by vertical N2 dimers. The Ru-N bond length in RuN8 cuboid is 2.161 Å and the N-N bond length in N2 dimer is 1.244 Å, close to the typical double bond (1.21 Å).

Similar to P21/c-RuN3, C2/c-RuN4 (shown in Fig. 3f) is also composed of distorted RuN6 octahedrons and puckered S-shaped N∞ chains. The intra-layer RuN6 octahedrons are connected by sharing edges and the interlayer ones by the puckered N∞ chains that extend infinitely in the crystal. The Ru-N bond lengths in the distorted RuN6 octahedrons are between 1.993 to 2.203 Å, and the N-N bond lengths in the N∞ chains are 1.305, 1.449 and 1.473 Å, close to the 1.32, 1.39 and 1.43 Å in the spiral N∞ chains in C2/c-CsN232. The transition from P4/mmm-RuN4 to C2/c-RuN4 accompanies a decrease of the coordination number of Ru atoms from 8 to 6. Resembling P4/mmm-RuN4, RuN8 cuboids are also formed in Cmmm-RuN4 (Fig. 3g), but different with the intra-layer edge-sharing cuboids and N2 dimers in P4/mmm-RuN4, the intra-layer RuN8 cuboids are face-sharing and the interlayer cuboids are connected by planar N∞ chains in Cmmm-RuN4. The Ru-N bond length in the cuboids is 2.197 Å and the N-N bond distances in the planar N∞ chains are 1.416 and 1.426 Å, close to the typical N-N single bond length (1.45 Å). Also, the phase transition sequence of these Ru nitrides can also be reflected by the total energy-volume curves, which are given in Supplementary Fig. S3.

The calculated equilibrium lattice parameters of Ru nitrides with different stoichiometries and their formation enthalpies at 0 GPa are listed in Table 1 in comparison with available data171833. Our results agree well with the experimental lattice parameters within a maximum error of 1.0%, and also with the theoretical values, indicating the reliability of our calculations.

Table 1

Calculated equilibrium lattice parameters, a, b and c (Å), β (deg.); formation enthalpies ΔH (eV/atom) of Ru nitrides at 0 GPa.

	S. G. (No.)	a	b	c	β	ΔH	Ref.
RuN	I-4m2(119)	3.068		4.929		0.008
	R-3m(166)	2.835		18.220		0.078
RuN2	Pnnm(58)	4.115	4.910	2.689		0.062
		4.073	4.888	2.707			Exp17
		4.058	4.847	2.665			Cal18
		4.098	4.919	2.696			Cal33
RuN3	P21/c(14)	11.925	4.070	13.032	154.12	0.211
RuN4	P4/mmm(123)	3.6135		3.6137		0.263
	C2/c(15)	3.834	8.888	5.720	117.83	0.355
	Cmmm(65)	7.629	3.669	2.867		0.646

The mechanical stability of Ru nitrides is examined by calculating the individual elastic constants (see Table 2), all proposed phases are mechanically stable at 0 GPa with the satisfactions of Born-Huang stability criteria27. To evaluate the mechanical performance of Ru nitrides, the calculated bulk modulus (B), shear modulus (G), B/G ratio, Young’s modulus (E), Poisson’s ratios (ν), and Vicker’s hardness (Hv) are also listed in Table 2. The Cmmm-RuN4 has the highest C11 (738 GPa), comparable with that of IrN2 (739 GPa)27. C22 of Pnnm-RuN2 is 769 GPa, close to that of PtN2 (798 GPa)34. P4/mmm-RuN4 has a C33 value of 698 GPa, comparable to that of OsN2 (683 GPa)35. The elastic constant, C44, is another important parameter reflecting the hardness of material. Among these Ru-N compounds, Cmmm-RuN4 has the lowest C44 value (61 GPa), lower than that of RhN2 (80 GPa)36, but higher than that of PdN2 (43 GPa)37. P21/c-RuN3 has the highest value of C44 (188 GPa), thereby a relatively strong shear strength.

Table 2

Calculated elastic constants C (GPa), bulk modulus B (GPa), shear modulus G (GPa), B/G ratio, Poisson’s ratio ν, and Vickers hardness Hv (GPa) of Ru nitrides.

	S. G.	C11	C22	C33	C44	C55	C66	C12	C13	C23	B	G	B/G	v	Hv
RuN	I-4m2	371		305	99		37	177	234		260	66	3.94	0.38	—
	R-3m	406		530	87			205	173		271	105	2.58	0.33	—
RuN2	Pnnm	615	769	464	106	269	132	152	218	71	298	180	1.66	0.25	20.1
RuN3	P21/c	367	473	639	188	186	201	183	127	127	253	174	1.45	0.22	23.4
RuN4	P4/mmm	283		698	104		129	112	38		174	128	1.36	0.20	20.9
	C2/c	540	664	344	151	208	286	199	172	97	257	171	1.50	0.23	22.1
	Cmmm	738	496	374	61	73	257	132	184	135	269	133	2.02	0.29	12.3

Materials with high bulk modulus are expected to be strong in resisting uniform compression. As shown in Table 2, the calculated bulk modulus within GGA level is 298 GPa for Pnnm-RuN2, consistent with the previous theory 305.9 GPa33. This value is lower than the measured value 330 GPa17 and the calculated value 343 GPa18, caused by the difference between LDA and GGA methods938. The bulk modulus of Pnnm-RuN2 is higher than that of RhN2 (235 GPa)16, although lower than that of PtN2 (372 GPa)910, IrN2 (428 GPa)12, and OsN2 (358 GPa)1218. Besides the highest bulk modulus, Pnnm-RuN2 also has the highest shear modulus (180 GPa), close to that of PtN2 (187 GPa)34, while I-4m2-RuN has the lowest G value (66 GPa). Except RuN and Cmmm-RuN4, the calculated hardness of the ground-state and high-pressure phases of Ru nitrides are between 20.1–23.4 GPa, belonging to the class of hard materials. Poisson’s ratio is an important parameter of directionality of the covalent bonding, and low Poisson’s ratio points to a high degree of covalency. For the Pnnm-RuN2, P21/c-RuN3, P4/mmm-RuN4, C2/c-RuN4 and Cmmm-RuN4, v values are between 0.20 and 0.29, indicating their covalent bonding. The B/G ratio represents the ductility of the materials. The high (low) B/G ratio means that the material is ductile (brittle), and the critical value is about 1.7539. From Table 2 we can see that I-4m2-RuN, R-3m-RuN, and Cmmm-RuN4 are ductile and the others are brittle.

The dynamical stability of the newly predicted phases, P21/c-RuN3, P4/mmm-RuN4, C2/c-RuN4, and Cmmm-RuN4, and metastable I-4m2-RuN and R-3m-RuN, together with the synthesized Pnnm-RuN2, is checked by calculating the phonon spectra (see Supplementary Fig. S4). All these phases are dynamically stable at 0 GPa with no imaginary frequency found throughout the Brillouin zone.

The total and partial density of states (DOS and PDOS) of Ru nitrides are plotted in Fig. 4 to understand the electronic properties and bonding features. It can be seen that except Ru tetranitrides, all of the compounds exhibit the metal features because of the finite DOS at the Fermi level (E), which originates mostly from 4d electrons of Ru. Note that the significant hybridization between Ru 4d and N 2p orbitals is observed from about −8 to −5 eV in I-4m2-RuN, −10 to −5 eV in R-3m-RuN. For Pnnm-RuN2, the states located between about −8.5 eV and −4 eV mainly originate from N-2p orbitals with some contributions from Ru-4d states, and in the region from −4 eV to 2 eV, the Ru-4d states interacts mainly with the N-2p states. Furthermore, the arrangement of RuN6 octahedrons may derive Ru 4d orbitals splitting into triply degenerate t orbitals at lower energy and doubly degenerate e orbitals at higher energy. Moreover, the pseudogap near the Fermi level is observed, enhancing the stability. For P21/c-RuN3, in the range from −11.5 eV to −3 eV, the states are essentially dominated by N-2p orbitals due to the increase of N concentration, while from −3 eV to 2 eV, the states are mainly contributed by Ru-4d orbitals with admixture of N-2p orbitals.

Figure 4

Calculated total and partial density of states for Ru nitrides.

(a) I-4m2-RuN; (b) R-3m-RuN (c) Pnnm-RuN2; (d) P21/c-RuN3; (e) P4/mmm-RuN4; (f) C2/c-RuN4 and (g) Cmmm-RuN4. The vertical dash line at zero is the Fermi energy level.

For Ru tetranitrides, Ru-4d orbitals and N-2p orbitals are hybridized in the region from about −3 eV to the Fermi level. The difference is that P4/mmm-RuN4 and Cmmm-RuN4 are semimetal-like while C2/c-RuN4 is semiconductor, which can be seen from their electronic band structure (Fig. 5). P4/mmm-RuN4 exhibits Dirac cones near the Fermi level in the A−M, M−Γ directions and at the R point of the Brillouin zone (BZ). Cmmm-RuN4 has a bulk Dirac cone below the Fermi level in the Γ−Z direction, giving rise to the small overlap of the valence band and the conduction band near the Γ point in the BZ. For C2/c-RuN4, the bottom of the conduction band is located at the Г point of BZ, and the top of the valence band at a point in the Z-Г direction, indicating semiconductor character with an indirect band gap of 0.84 eV. The band structures of the other Ru-N compounds are also computed and depicted in Supplementary Fig. S5, indicating their metallic character.

Figure 5

Band structure of (a) P4/mmm-RuN4; (b) C2/c-RuN4 and (c) Cmmm-RuN4. The P4/mmm-RuN4 and Cmmm-RuN4 have semimetallic feature, and C2/c-RuN4 is a semiconductor with an indirect band gap of 0.84 eV.

To gain deeper insights into the bonding nature in Ru-N compounds, we computed the distributions of valence electron density, as presented in Fig. 6. The Mulliken overlap populations (MOP) are also calculated to evaluate the relative bond strength. As we can see that for all these compounds, the nearly spherical charge distribution around Ru atoms indicates that the bonding between Ru and N atoms has partially ionic characteristic. As to I-4m2-RuN, the valence electrons are mainly located in the centre of Ru and N atoms, forming a zig-zag chain along b axis. The MOP for Ru-N bonding is 0.38, reflecting the moderate covalent bonding characteristic. The N-N distance is 3.068 Å, too far to form a covalent bonding. Different from I-4m2-RuN, σ covalent bonding of N-N are present in R-3m-RuN. The electron density at the N-N bond is higher with MOP of 0.83, indicating the strong N-N interactions along c axis. Compared with the N-N bonding, the Ru-N bonding is much weaker with a lower MOP value of 0.38, which is close to I-4m2-RuN.

Figure 6

Calculated valence electron density distributions of Ru nitrides.

(a) I-4m2-RuN in (100) plane; (b) R-3m-RuN in (110) plane; (c) Pnnm-RuN2 in (001) plane; (d) P21/c-RuN3 in (101) plane; (e) P4/mmm-RuN4 in (010) plane; (f) C2/c-RuN4 in (001) plane and (g) Cmmm-RuN4 in (001) plane. The big orange and small purple spheres represent Ru and N atoms, respectively.

For Pnnm-RuN2, the electron density is higher at the center of the N2 dimer with the MOP value of 0.76, resulting in the highest bulk modulus. The interactions between the Ru and N atoms are much weaker, as can be reflected by the MOP values 0.33 and 0.50 for Ru-N bonding. With regard to P21/c-RuN3, the strong N-N covalent bonding can be found in the puckered N6 unit, with MOP values between 0.60 and 0.87. Due to the distortion and tilt of the RuN6 octahedrons, the length and strength of Ru-N bonding are irregular compared with Pnnm-RuN2 with MOP ranging from 0.28 to 0.47.

Similar to the R-3m-RuN and Pnnm-RuN2, N-N dimer in the P4/mmm-RuN4 is also characteristic of σ covalent bond, which contributes to a largest MOP value of 1.23, indicating the strong interactions between the N atoms. The largest C33 value comes from the contribution of strong directional covalent N-N bonding along c axis. While MOP for Ru-N bonding is 0.30, demonstrating weak interactions between Ru and N atoms. For C2/c-RuN4 and Cmmm-RuN4, the valence electrons are mainly located in the S-shaped N∞ chains. The difference is that Ru atoms are sandwiched in between two N∞ chains in C2/c-RuN4, while the Ru atoms and N∞ chains are not in the same plane in Cmmm-RuN4. The MOP is between 0.55 and 1.01 for N-N bonding, between 0.28 and 0.44 for Ru-N bonding in C2/c-RuN4, while the electronic density is in a narrow range for Cmmm-RuN4, with MOP values of 0.64 and 0.67 for N-N bonding, 0.27 for Ru-N bonding. Compared to C2/c-RuN4, the strength of the N-N bonding and the distribution of Ru-N bonding are more homogeneous in Cmmm-RuN4.

Nitrogen species in these compounds have various structural forms, such as single N atom, N2 dimers, N6 units and N∞ chains. Despite of the diverse features, the polynitrogens are in close correlation with the N-sp hybridization, characterized by σ covalent N-N bond. Mulliken charges analysis reveals a transferred charge of 0.55 e and 0.34 e from Ru to N atom for I-4m2-RuN and R-3m-RuN, respectively, 0.7 e from Ru to two N atoms for Pnnm-RuN2, 0.75 e from Ru to three N atoms for P21/c-RuN3, 0.9 e, 0.81 e, and 0.83 e from Ru to four N atoms for P4/mmm-RuN4, C2/c-RuN4, and Cmmm-RuN4, respectively. The charge transfer also suggests the ionic feature of these compounds.

Conclusions

In summary, we have systematically investigated the structures and properties of Ru nitrides at pressures of 0–50 GPa based on the density functional theory. We found two structures (I-4m2-RuN and R-3m-RuN) energetically prior to the previously reported NaCl-type and ZnS-type RuN. Besides the experimentally synthesized Pnnm-RuN2, new stoichiometries of P21/c-RuN3, P4/mmm-RuN4, and C2/c-RuN4 are suggested for possible synthesis. The P21/c-RuN3, P4/mmm-RuN4, and C2/c-RuN4 become stable at pressures above 12.8, 13.6, and 23 GPa, respectively. A new structure, Cmmm-RuN4 is also predicted to be stable above 101 GPa. The Cmmm-RuN4, Pnnm-RuN2, and P4/mmm-RuN4 possess the greatest incompressibility. The Pnnm-RuN2, P21/c-RuN3, P4/mmm-RuN4, and C2/c-RuN4 are potential hard materials with the Vickers hardness between 20.1 and 23.4 GPa. The P4/mmm-RuN4 and Cmmm-RuN4 exhibit semi-metal-like properties and C2/c-RuN4 shows semiconductor features, while the Pnnm-RuN2 and P21/c-RuN3 exhibit electronic characteristics of metals. Except I-4m2-RuN, high directional covalent N-N bonds are presented in all the other nitrides. A charge transfer from Ru to N atoms is predicted to occur, crucial to the stability of the Ru-N bonding in these compounds. These results are expected to stimulate the exploration and discovery of the newly predicted Ru nitrides, which may have practical technology applications due to their interesting mechanical and electronic properties.

Methods

The stable structures of RuN (x = 1, 2, 3 and 4) with up to six (x = 1, 2) and four (x = 3, 4) formula units (f. u.) were searched at the pressures of 0, 20, 50 and 100 GPa using the particle swarm optimization method as implemented in the CALYPSO code4041. Stable stoichiometries were determined by the construction of the convex hull: a compound is thermodynamically stable when its formation enthalpy falls on the line. The calculations of formation enthalpy and geometry optimizations presented in this study were carried out in the framework of density functional theory (DFT) with the Perdew-Burke-Ernzerhof (PBE) parameterization of the generalized gradient approximation (GGA)42 using CASTEP package43. An energy cutoff of 500 eV and dense k-point grids within the Monkhorst-Pack44 scheme were adopted for the sampling Brillouin zone of different structures, yielding excellent convergence for total energies (within 1 meV/atom). When the individual elastic constants were derived, the bulk (B), Young’s (E) and shear (G) moduli and Poisson’s ratio (ν) were obtained by using Voigt-Reuss-Hill approximation (VRH)45. The theoretical Vickers hardness was estimated by using the empirical model46, H = 2.0(k2G)0.585 − 3.0, where k = G/B. The global stability of Ru nitrides can be quantified by constructing the thermodynamic convex hull within considered pressures, which is defined as the average atomic formation enthalpy of the most stable phases at each composition:

where H is the enthalpy of either a compound or a constituent element at a specific pressure. Here α-N2 and ε-N2 are adopted as the reference structure at below 10 GPa and 10–50 GPa for nitrogen, respectively. Phonon spectra of new proposed phases were calculated by finite displacement methods to examine their dynamical stabilities. The structures were visualized by VESTA47.

Additional Information

How to cite this article: Zhang, Y. et al. Diverse ruthenium nitrides stabilized under pressure: a theoretical prediction. Sci. Rep. 6, 33506; doi: 10.1038/srep33506 (2016).