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Shuyin Yu*^{ab},
Qingfeng Zeng^{ab},
Artem R. Oganov^{bcde},
Gilles Frapper^{f},
Bowen Huang^{f},
Haiyang Niu^{c} and
Litong Zhang^{a}
^{a}Science and Technology on Thermostructural Composite Materials Laboratory, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China. E-mail: yushuyin2014@gmail.com
^{b}International Center for Materials Discovery, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China
^{c}Department of Geosciences, Center for Materials by Design, Institute for Advanced Computational Science, State University of New York, Stony Brook, NY 11794-2100, USA
^{d}Skolkovo Institute of Science and Technology, 3 Nobel Street, Skolkovo 143025, Russia
^{e}Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region 141700, Russia
^{f}IC2MP UMR 7285, Université de Poitiers, CNRS, 4, rue Michel Brunet, TSA 51106, 86073 Poitiers Cedex 9, France

Received
23rd November 2016
, Accepted 2nd January 2017

First published on the web 17th January 2017

Using a variable-composition ab initio evolutionary algorithm, we investigate stability of various Zr–N compounds. Besides the known ZrN and Zr_{3}N_{4}, new candidate structures with Zr:N ratios of 2:1, 4:3, 6:5, 8:7, 15:16, 7:8 and 4:5 are found to be ground-state configurations, while Zr_{3}N_{2} has a very slightly higher energy. Besides Zr_{2}N, the newly discovered Zr_{x}N_{y} compounds adopt rocksalt structures with ordered nitrogen or zirconium vacancies. The electronic and mechanical properties of the zirconium nitrides are further studied in order to understand their composition–structure–property relationships. Our results show that bulk and shear moduli monotonically increase with decreasing vacancy content. The mechanical enhancement can be attributed to the occurrence of more Zr–N covalent bonds and weakening of the ductile Zr–Zr metallic bonds. These simulations could provide additional insight into the vacancy-ordered rocksalt phases that are not readily apparent from experiments.

Unlike carbides, not only nonmetal vacancies exist, but structure can also tolerate metal atom vacancies since metal atom oxidation state can reach +4.^{6} Experimental investigations of nonstoichiometric TM nitrides (TM = Ti, Zr, Hf, V, Nb and Ta) have been conducted intensely for more than thirty years.^{7–10} About twenty ordered carbides and nitrides have been found.^{1} However, it has not yet been possible to construct a single phase diagram of TM–C or TM–N systems at low temperatures (most of the available phase diagrams have been constructed above 1300–1500 K). In this paper, we explore stable compounds in the Zr–N system at ambient pressure and finite temperatures. To date, there is no comprehensive and inclusive computational investigation of phase stability in the Zr–N system.

Zirconium nitrides represent a rich family of phases where the stability and microstructures are still not completely understood. According to the phase diagram provided by Gribaudo et al.,^{11} ZrN and Zr_{3}N_{4} can be stable at ambient conditions. In 2003, c-Zr_{3}N_{4} with a Th_{3}P_{4} structure was synthesized by Zerr et al. using diamond-anvil cell experiments at 16 GPa and 2500 K.^{12} This compound was expected to exhibit a very high Vickers hardness around 30 GPa, similar to that of γ-Si_{3}N_{4}. However, Kroll showed that hardness is just slightly harder than 14 GPa.^{13} Besides c-Zr_{3}N_{4}, an orthorhombic Pnma modification of Zr_{3}N_{4} has been proposed.^{14} First-principles calculations show that o-Zr_{3}N_{4} is energetically more stable than c-Zr_{3}N_{4}.^{15} However, both structures are metastable considering decomposition into ZrN and N_{2}.^{13} Besides, two nitrogen-rich phases ZrN_{x} (1.06 < x < 1.23) with NaCl-type structures have been claimed by Juza et al. in 1964.^{16} However, precise stoichiometries and crystal structures are not known for their synthesized samples.

Here, we apply recently developed evolutionary algorithm USPEX to extensively explore the crystal structures and stoichiometries in the Zr–N system at ambient conditions, and then their phase stability at finite temperatures are evaluated. Furthermore, the electronic and mechanical properties of stable Zr_{x}N_{y} compounds are studied using density functional theory. Our work should provide guidance for experimental groups aiming to synthesize these new technologically useful materials.

First-principles electronic structure calculations were carried out within the generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof form.^{22} The interactions between ions and electrons were described by the projector-augmented wave method^{23} with a cutoff energy of 600 eV. Uniform Γ-centered k-points meshes with a resolution of 2π × 0.03 Å^{−1} and Methfessel–Paxton electronic smearing^{24} were adopted for the integration in the Brillouin zone. These settings ensure convergence of the total energies to within 1 meV per atom. Structure relaxation proceeded until all forces on atoms were less than 1 meV Å^{−1} and the total stress tensor was within 0.01 GPa of the target value.

Theoretical phonon spectra were calculated with the supercell method using the PHONOPY package.^{25} Hellmann–Feynman forces exerted on all atoms in supercells (2 × 2 × 2 of the unit cell) were calculated by finite atomic displacements of each symmetrically nonequivalent atom. Phonon dispersion relations were then obtained by the diagonalization of the dynamical matrix. We used the quasiharmonic approximation to calculate the free energy of zirconium nitrides at finite temperatures. Free energy of a crystal was obtained as a sum of the static total energy, vibrational energy and configurational energy. Computational details are described in ESI.†

ΔG(Zr_{x}N_{y}) = [G(Zr_{x}N_{y}) − xG(Zr) − yG(N)]/(x + y)
| (1) |

Any phase located on the convex hull is considered to be thermodynamically stable (at T = 0 K, G = H) and at least in principle synthesizable.^{26} In the case of zirconium nitrides, a series of stable compounds at various Zr:N ratios, i.e. 2:1, 4:3, 6:5, 8:7, 1:1, 15:16, 7:8, 4:5 have been discovered by our evolutionary searches at 0 K, shown in Fig. 1. The rocksalt ZrN with space group (SG) Fmm was found to have the lowest enthalpy of formation. Besides ZrN, substoichiometric Zr_{2}N (SG: P4_{2}/mnm), Zr_{4}N_{3} (SG: C2/m), Zr_{6}N_{5} (SG: C2/m) and Zr_{8}N_{7} (SG: C2/m) have also been found to be thermodynamically stable. For the missing composition Zr_{3}N_{2}, the lowest-energy structure is Immm with the enthalpy of formation lying very close to the convex hull at only 0.005 eV per atom, i.e. Zr_{3}N_{2} is a metastable phase at 0 K.

Additionally, Juza et al. in 1964 have discovered two nitrogen-rich phases ZrN_{x} (1.06 < x < 1.23) with rocksalt structures.^{16} The synthesized sample had a dark blue color and turned into metallic ZrN upon heating. Unfortunately, detailed stoichiometries and crystallographic information were not determined. Subsequent studies even questioned the existence of these two compounds.^{11} From our evolutionary searches, we found these two compounds could be Zr_{15}N_{16} (x = 1.07, SG: P) and Zr_{4}N_{5} (x = 1.25, SG: C2/m). Their structures are composed of edge-sharing ZrN_{6} and □N_{6} (□ means Zr vacancy) octahedra, similar to the rocksalt ZrN structure. To the best of our knowledge, such nitrogen-rich nitrides have never been reported in other TM–N systems. For Zr_{3}N_{4}, the most stable structure has the orthorhombic Pnma symmetry,^{27} which is energetically more favorable than the Th_{3}P_{4}-type structure by ∼0.019 eV per atom at 0 GPa and 0 K. We found it is thermodynamic metastable considering decomposition into ZrN and N_{2} at ambient conditions. First-principles calculation shows that o-Zr_{3}N_{4} will transform into c-Zr_{3}N_{4} at ∼2 GPa (Fig. S2†).

We have carefully calculated the temperature contribution to the phase stability of the new discovered zirconium nitrides from 0 K to 2000 K within the quasiharmonic approximation, as shown in Fig. 1. Note that for each stoichiometry, the space group/structure found at 0 K is kept for higher temperatures. The free energies of formation increase with increasing temperatures for all phases but at different rates, yielding a convex hull which changes with temperature. Our results show that Zr_{2}N, Zr_{8}N_{7}, ZrN and Zr_{15}N_{16} will not lose their stability in the whole studied temperature range. The unstable Zr_{3}N_{2} at 0 K will become stable at temperatures higher than ∼900 K, while for Zr_{4}N_{3}, Zr_{6}N_{5}, Zr_{7}N_{8} and Zr_{4}N_{5}, the temperature contributions have negative effect on their structural stability. For example, the formation enthalpy of reaction 2Zr_{4}N_{5} (s) → 8ZrN (s) + N_{2} (g) will become negative above 300 K, which means Zr_{4}N_{5} should decompose into ZrN and N_{2} gas at roughly room temperature if associated kinetic barrier allows this process, perfectly consistent with the results of Juza et al.^{16}

Crystal structures of the representative zirconium-rich Zr_{6}N_{5} and nitrogen-rich Zr_{4}N_{5} are schematically shown in Fig. 2, while other structures and their corresponding phonon dispersion curves are shown in Fig. S3 and S4.† No imaginary phonon frequencies are found, indicating their dynamical stability. The detailed crystallographic data, enthalpies and zero-point energies are listed in Table S1.† From Table S1,† we can find the computed lattice parameters for ZrN and Zr_{3}N_{4} are in good agreement with those obtained from other theoretical and experimental investigations, which confirms the accuracy of our calculations. From the structural point of view, ZrN has the ideal cubic rocksalt structure, while Zr_{n+1}N_{n} (n = 2, 3, 5, 7) and Zr_{m}N_{m+1} (m = 4, 7, 15) are versions of the rocksalt structure with ordered nitrogen or zirconium vacancies (Zr_{2}N has rutile-type structure).

In the structures of Zr-rich phases, the metal atoms form hexagonal close-packed (hcp) sublattices with N atoms filled in the octahedral voids, thus each N atom is coordinated by six Zr atoms, forming NZr_{6} octahedra. However, the concentration of filled octahedral voids in various Zr_{n+1}N_{n} structures is different. Two thirds of them are filled in Zr_{3}N_{2}; while seven eighths in Zr_{8}N_{7}. Similar nitrogen vacancy-ordered structures were also reported earlier to be stable for transition metal carbides M_{n+1}C_{n} (M = Hf and n = 2, 5;^{28} M = Zr and n = 1, 2, 3, 6;^{29} M = Ti and n = 1, 2, 5 (ref. 30)) and nitrides M_{n+1}N_{n} (M = Ti and n = 1, 2, 3, 5 (ref. 31)). For N-rich phases, one eighth of the metal atoms are replaced with vacancies in Zr_{7}N_{8}, while one fifth in Zr_{4}N_{5}.

The formation of such N-rich phases could be attributed to the enhanced stability of the +4 oxidation state of Zr and Hf compared to Ti due to the relativistic effects,^{32} leading to the coexistence of +3 in MN and +4 in M_{3}N_{4}, while in the Ti–N system, TiN has the highest nitrogen content under normal conditions. In the structures of Zr_{3}N_{4}, the hcp metal framework of the rocksalt structure is significantly distorted. For o-Zr_{3}N_{4}, there are three nonequivalent types of Zr atoms, one of them is octahedrally coordinated to six N atoms, one resides at the center of a trigonal prism, and the last one is located inside of a heavily distorted octahedron (Fig. S3†). Thus, the second and third nonequivalent Zr atoms are sevenfold coordinated, while in c-Zr_{3}N_{4}, each Zr atom is coordinated to eight N atoms.

Fig. 3 The calculated electronic density of states of (a) o-Zr_{3}N_{4}, (b) ZrN and (c) Zr_{2}N. (d) Crystal orbital Hamilton population (–COHP) curves of Zr_{2}N. |

For the well-known stoichiometric rocksalt Zr_{x}N_{y}, we will first briefly discuss their electronic properties, then analyze the electronic perturbation due to the creation of nitrogen or zirconium vacancies – empty octahedral sites in the fcc network – leading to symmetry-broken Zr_{n+1}N_{n} and Zr_{m}N_{m+1} structures. Similarly to o-Zr_{3}N_{4}, DOS of Zr_{x}N_{y} phases can be decomposed into three well-separated energy regions as shown in Fig. 3 and S5,† but here no gap separates the valence and conducting bands: (1) a deep lowest valence band, s_{N}; (2) hybridized Zr(4d)/N(2p) band, d_{M}p_{N}; (3) a partially filled higher-energy Zr(4d) band, d_{M}. The s_{N} band is dominated by the 2s orbitals of the nitrogen atoms and is nonbonding. The next group of valence bands, d_{M}p_{N}, results from strong hybridization of the 4d states of zirconium atoms with 2p states of nitrogen atoms. Also, one may see that for Zr_{x}N_{y} the bottom of the d_{M} band, dominated by 4d orbitals of zirconium atoms, responsible for metallicity.

When nitrogen vacancies are created in substoichiometric Zr_{n+1}N_{n} (n = 1, 2, 3, 5 and 7), notice that obviously the formal oxidation state of Zr decreases as the number of nitrogen vacancies increases, going from Zr^{3+} d^{1} in ZrN to Zr^{1.5+} d^{2.5} in Zr_{2}N. Therefore, one may expect the occupation of the Zr 4d levels in substoichiometric Zr_{n+1}N_{n} compounds. This is what happens: Zr–Zr bonding and nonbonding (slightly antibonding) Zr–N levels appear just below the Fermi level, mainly metal 4d in character (see Fig. 3d and S6†). In Zr_{n+1}N_{n}, Zr atoms are no longer all in the MN_{6} octahedral environment; some of them are in MN_{5} square pyramidal configurations. Therefore, one may expect the stabilization of antibonding Zr–N levels when going from formally octahedral ZrN_{6} to square pyramidal ZrN_{5} environment due to the lack of a Zr(4d)–N(2p) antibonding component. The occupation of these Zr–N nonbonding levels may explain the mechanical properties of these substoichiometric Zr_{n+1}N_{n} compounds.

Fig. 3c displays the total and projected DOS of Zr_{2}N, but also the projected d states of a hypothetical ZrN structure within the Zr_{2}N structure (all N vacancies are filled in the so-called perfect structure). One can see that nitrogen vacancies give rise to additional states just below the Fermi level compared to its corresponding perfect structure, which originates from the Zr–Zr bonds passing through a nitrogen vacancy site. Such “vacancy states” usually lead to a drastic increase in the density of states at the Fermi level (0.076 in ZrN; 0.090 in Zr_{6}N_{5}; 0.112 in Zr_{2}N, states per eV per electron). The increasing density of the d state at the Fermi level can be interpreted as an increase in the Zr(4d_{σ})–Zr(4d_{σ}) bonding or metallic bonds between the zirconium atoms.

Phase | C_{1} |
C_{12} |
C_{13} |
C_{15} |
C_{22} |
C_{23} |
C_{25} |
C_{33} |
C_{35} |
C_{44} |
C_{46} |
C_{55} |
C_{66} |
B | G | E | ν | B/G | A^{U} |
H_{S} |
H_{C} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

a B = 249,^{34} G = 157,^{35} H_{v} = 16.^{36}b B = 203,^{37} 238,^{38} H_{v} = 12.4.^{37}c B = 224,^{37} H_{v} = 11.7,^{39} 11.4.^{37} |
|||||||||||||||||||||

Zr_{2}N |
305 | 142 | 108 | 362 | 119 | 114 | 187 | 109 | 274 | 0.256 | 1.721 | 0.116 | 14.8 | 13.5 | |||||||

Zr_{3}N_{2} |
293 | 132 | 130 | 325 | 93 | 384 | 112 | 96 | 84 | 190 | 100 | 255 | 0.276 | 1.896 | 0.170 | 15.5 | 11.0 | ||||

Zr_{4}N_{3} |
336 | 147 | 140 | 8 | 349 | 127 | 2 | 371 | 13 | 127 | −10 | 95 | 109 | 209 | 108 | 277 | 0.280 | 1.931 | 0.091 | 15.2 | 11.3 |

Zr_{6}N_{5} |
376 | 135 | 159 | −25 | 384 | 153 | 29 | 364 | 2 | 118 | 25 | 134 | 145 | 224 | 122 | 309 | 0.270 | 1.845 | 0.233 | 15.3 | 13.2 |

Zr_{8}N_{7} |
394 | 137 | 171 | −28 | 396 | 162 | 31 | 385 | 2 | 131 | 35 | 154 | 147 | 235 | 129 | 328 | 0.267 | 1.815 | 0.301 | 13.3 | 14.1 |

ZrN^{a} |
556 | 123 | 113 | 267 | 147 | 372 | 0.268 | 1.818 | 0.534 | 15.6 | 15.4 | ||||||||||

Zr_{15}N_{16} |
437 | 153 | 155 | −16 | 467 | 117 | 1 | 468 | 20 | 154 | −13 | 130 | 161 | 247 | 148 | 369 | 0.251 | 1.673 | 0.323 | 15.5 | 17.3 |

Zr_{7}N_{8} |
422 | 132 | 146 | 30 | 421 | 157 | −30 | 396 | 2 | 150 | 28 | 166 | 160 | 234 | 146 | 362 | 0.242 | 1.605 | 0.199 | 15.5 | 18.2 |

Zr_{4}N_{5} |
338 | 120 | 159 | −11 | 425 | 116 | −9 | 378 | 21 | 139 | −12 | 106 | 118 | 215 | 120 | 303 | 0.264 | 1.789 | 0.211 | 15.0 | 13.7 |

o-Zr_{3}N_{4}^{b} |
209 | 159 | 164 | 469 | 167 | 422 | 95 | 63 | 130 | 214 | 91 | 239 | 0.314 | 2.359 | 1.043 | 12.4 | 7.3 | ||||

c-Zr_{3}N_{4}^{c} |
423 | 146 | 133 | 238 | 135 | 341 | 0.261 | 1.761 | 0.002 | 11.0 | 15.2 |

The effect of vacancy concentration on bulk and shear moduli of the rocksalt Zr_{x}N_{y} structures is shown in Fig. 4. It can be seen that bulk and shear moduli monotonically decrease with increasing vacancy concentration. When N vacancies increase in a rocksalt structures, the number of Zr–N bonds obviously decreases. Moreover, the computed Zr–N separations are increasing when N vacancies increase, i.e. from 2.24 to 2.27 Å in Zr_{8}N_{7} and Zr_{2}N, reflecting the weakening of the Zr–N bonding. Therefore, one may understand our findings, i.e. the loss of B and G is mainly attributed to the disappearance of some strong covalent Zr–N bonds.

For brittle materials, B/G ratio is smaller than 1.75 (ref. 45) (for example, for diamond B/G = 0.8). From Table 1, we can find that B/G values decreases in the following sequence: o-Zr_{3}N_{4} > Zr_{4}N_{3} > Zr_{3}N_{2} > Zr_{6}N_{5} > ZrN > Zr_{8}N_{7} > Zr_{4}N_{5} > c-Zr_{3}N_{4} > Zr_{2}N > Zr_{15}N_{16} > Zr_{7}N_{8}. B/G values of Zr_{2}N (1.721), Zr_{15}N_{16} (1.673) and Zr_{7}N_{8} (1.605) are smaller than 1.75, which indicate that these are brittle or borderline materials. For the other compounds, B/G values are larger than 1.75, which suggest that they are ductile materials. For ZrN, B/G value is slightly larger than 1.75, making a good compromise between hardness and ductility, which is mainly due to a peculiar interplay between metallicity and covalency. Besides, we can find that B/G values of Zr-rich phases are larger than N-rich ones except Zr_{2}N and o-Zr_{3}N_{4}. Obviously, the higher metal content, the more ductile the material. Surprisingly, semiconducting o-Zr_{3}N_{4} has the largest B/G value (2.359) due to the low C_{11} and C_{44}, and o-Zr_{3}N_{4} also possesses remarkable elastic anisotropy. Here, we used the Ranganathan and Ostoja-Starzewski method^{46} to estimate anisotropy:

(2) |

The calculated anisotropy parameters A^{U} of Zr_{x}N_{y} phases are listed in Table 1. Elastic anisotropy decreases in the following sequence: o-Zr_{3}N_{4} > ZrN > Zr_{15}N_{16} > Zr_{8}N_{7} > Zr_{6}N_{5} > Zr_{4}N_{5} > Zr_{7}N_{8} > Zr_{3}N_{2} > Zr_{2}N > Zr_{4}N_{3} > c-Zr_{3}N_{4}. Fig. 5 shows the directional dependence of Young's moduli for the selected Zr_{x}N_{y} compounds (see eqn (8) in ESI†). For an isotropic system, one would see a spherical shape. The degree of elastic anisotropy can be directly reflected from the degree of deviation in shape from a sphere. From Fig. 5, we can find that Young's modulus is more anisotropic in o-Zr_{3}N_{4}, while Zr_{4}N_{3} and Zr_{2}N show more isotropic features. The anisotropy of o-Zr_{3}N_{4} is due to low C_{11} and high C_{22}, C_{33} values, resulting in a flat shape of Young's modulus.

The Vickers hardness of zirconium nitrides was estimated by using Chen's model,^{47} as follows:

H_{C} = 2(κ^{2}G)^{0.585} − 3
| (3) |

(4) |

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra27233a |

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