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Article Cite This: Cryst. Growth Des. XXXX, XXX, XXX−XXX

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Structures of WxNy Crystals and Their Intrinsic Properties: FirstPrinciples Calculations Zhixiong Kang,† H. Y. He,† Rui Ding,‡ Junling Chen,‡ and B. C. Pan*,†,§ †

Key Laboratory of Strongly Coupled Quantum Matter Physics, Department of Physics and §Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China ‡ Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031, People’s Republic of China S Supporting Information *

ABSTRACT: Recent experiments had shown that nitrogen (N) may react with the tungsten (W) atoms at the surface of W-based plasma facing materials (PFM) in a fusion reactor to form WN compounds. The formed WN compounds exhibit excellent performance in retention of deuterium, which is essentially correlated with the atomic structures of the WN compounds. Unfortunately, the structural features of WxNy crystals with different stoichiometric ratios are not systematically made clear. In this paper, we report our systematic study on WxNy structures (x = 1−6, y = 1−6) based on density functional theory (DFT) combined with a particle-swarm optimization (PSO) technique. The lowest-energy WxNy structures (x = 1−6, y = 1−6) are proposed. It is found that the concerned WN compounds characterize metallic feature, except for W1N2 which is a semiconductor. Meanwhile, the evolution of the mechanical properties of the compounds on the N-to-W ratio for WxNy structures is investigated. It reveals that the good mechanical properties of a compound is associated with the presence of covalence bonds besides the common existence of ionic and/or metallic bonds in the compound.

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on, were suggested. Recently, Balasubramanian et al.15 have investigated the stability of WN compounds by the firstprinciples method and found that WN is unstable in cubic structures, but mechanically stable in the NbO structure. It is worth noting that, since the tungsten nitride has a variety of phases over a large composition range, probably some of stable structures of WN crystals could not be observed in previous experiments. In this paper, we intensively seek the stable and metastable crystalline of WxNy over a large composition range with x = 1− 6 and y = 1−6. Our obtained candidates not only contain all of the crystalline structures proposed in the literature, but also provide many new structures which are more stable than the related ones suggested in early experiments. Furthermore, some intrinsic properties, such as elastic, electronic, and vibrational properties, are predicted for the typical stable crystalline structures. From these calculations, the relation between the elastic property and the content of N in the crystalline is revealed.

INTRODUCTION Controlled thermonuclear fusion is considered to be the best solution to the requirement of energy for human living. Many fusion reactors, such as ITER,1 EAST,2 ASDEX-U,3 JET,4 and so on, are utilized to conduct scientific and engineering attempts for controlled thermonuclear fusion. As we know, when a fusion reactor works, nitrogen (N) exists in the edge plasma, and these nitrogen atoms may impact the W-based plasma-facing materials (PFMs),5−9 leading to the formation of WN compounds in the W-PFMs. Experiments showed that both hydrogen and deuterium (D) stay in the WN compounds,10,11 in which the retention of D decreased abruptly with increasing the content of N, whereas the retention of H was insensitive to the content of N. More interestingly, unlike the existence of H blisters in W-PFMs, no H blister was observed in the WN compounds even though both H and D were present in the compounds. Basically, the different retention behaviors of D in between W and WN systems are tightly correlated with the different structural features of the two systems. Thus, it is a fundamental issue to reveal the structures of WN compounds with different contents of N. In experiments, it is established that the N-to-W ratio in the WN compounds could be variable, which corresponds to different WxNy crystals. Because of this, the structures of the synthesized WN compounds are very complicated. According to the early measurement, the structures of several WN compounds such as hexagonal WN, 12 hexagonal and rhombohedral W2N3, cubic W3N4,13 NaCl type WN,14 and so © XXXX American Chemical Society

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METHODS

The particle-swarm optimization (PSO) algorithm implemented in the CALYPSO package16 is powerful for predicting stable and metastable structures at given external conditions.17−22 In this work, the structures Received: December 7, 2017 Revised: January 26, 2018 Published: March 1, 2018 A

DOI: 10.1021/acs.cgd.7b01707 Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crystal Growth & Design

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Figure 1. Low-enthalpy structures of the compounds. Blue and yellow balls stand for W and N, respectively. (a) W1N1, (b) W2N2, (c) W3N3, (d) W5N5, (e) W1N2, (f) W1N3, (g) W1N4, (h) W1N5, (i) W1N6, (j) W2N1, (k) W4N2, (l) W6N3, (m) W2N3, (n) W4N6, (o) W2N5. with different stoichiometric tungsten nitrides are carefully searched by a PSO algorithm. In detail, we set the population size of each generation to be 30, and the maximum number of generations is 20. At each generation, 60% of the population is evolved from the lowestenergy structures available from the previous generation, and the others are generated randomly to enhance the global searching ability. For each of our searched candidates, we performed the structures relaxation by using the Vienna Ab-initio Simulation Package (VASP).23,24 In our calculations, the generalized gradient approximation of Perdew−Burke−Ernzerh functional25,26 was used. The cutoff energy for expansion of wave function was set to be 450 eV. The Brillion zone was sampled with the Monk horst-Pack scheme,27 where the error of total energy arising from the chosen k-point sampling was less than 1 meV/atom. Both the lattice constants and the atomic positions for a candidate are fully relaxed until the force on each atom is less than 0.02 eV/Å. In addition, the stability of all concerned crystals was further examined by evaluating the phonon spectra.28

that of W4N4, and the structure of W3N3 is the same as that of W6N6. Figure 1a−d displays their atomic structures. The details of all structures are listed in Supporting Information.29 The W1N1 structure is identical to the δ-WN structure, which is the same as that reported in the previous literature.13 The formation enthalpy of this structure is −0.11 eV/atom. Structurally, each W atom bonds with six N atoms, forming a triangular prism. The six W−N bonds are all 2.2 Å, with three sets of bond angles, namely, 81°, 83°, and 136°, in the W1N1 crystal. The formation enthalpy of the best candidate of W2N2 compound is −0.37 eV/atom, lower than the W1N1 by 0.26 eV/atom in energy. There are two distinctive bonding environments around W atoms in W2 N 2 : the 6-foldcoordinated W atoms and the 5-fold-coordinated W atoms. In the case of the 6-fold-coordinated W atoms, bond lengths of W−N are sorted to be 2.2, 2.1, and 2.16 Å, and in the case of the 5-fold-coordinated W atoms, the W−N bond lengths are to be 2.09, 2.14, and 2.15 Å. The W atoms as well as the bonded N atoms form two types of polyhedron, hexahedron, and pentahedron, respectively. The structure of W2N2 with NiAs type considered by Chen et al.30 was also found in our search. However, its energy is 20 meV/atom higher than ours. The W3N3 crystal is 0.3 eV/atom lower in energy than the W1N1, and hence it is energetically more stable than the others. The structure of W6N6 is the same as W3N3. The bond lengths of four W−N bonds are 2.06 Å, and the bond angles are all 90°. This structure also characterizes the feature of the NbO structure,15 which has only three cations and three anions per cubic unit cell, exhibiting a regular array of both cation vacancies and anion vacancies. For the W5N5 crystal, the total energy of its best candidate is lower by 0.18 eV/atom than that of W1N1, but higher by 0.12 eV/atom than that of W3N3. So the W3N3 crystal is the lowest enthalpy in the family of ratio W:N = 1:1. This is consistent with the previous literature in which Mehl et al.31 predicted that W3N3 has the lowest enthalpy in this ratio. Similar to the local structures of the above crystals, each W atom has either 4-fold or 5-fold coordination in W5N5 crystal. For the 4-fold-

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RESULTS AND DISCUSSION Low-Enthalpy Structures. To assess relative stabilities of the low-enthalpy WxNy compounds, we computed the formation enthalpies based on the definition E(WxNy) =

Etot − xE(W) − y NW + NN

E(N2) 2

(1)

where E(WxNy) is the binding energy of WxNy, Etot is the total energy of WxNy, E(W) is the energy of one W atom, E(N2) is the energy of a N2, and both NW and NN are respectively the number of W atoms and N atoms in the supercell we considered. The formation enthalpies of all the candidates predicted in present work were evaluated based on formula 1, from which the lowest-enthalpy candidate for a given stoichiometric ratio of W-to-N in a compound is identified. These lowest-enthalpy candidates are described below. W:N = 1:1. For the case of stoichiometric ratio W:N = 1:1, four low-enthalpy structures were obtained: the hexagonal, monoclinic, cubic, and triclinic crystals with space group P6m̅ 2, Bm, Pm3m, and P1, respectively. For convenience, these four solids are named as W1N1, W2N2 (W4N4), W3N3 (W6N6), and W5N5, respectively. Here, the structure of W2N2 is the same as B



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Figure 2. Low-enthalpy structures of the compounds. Blue and yellow balls stand for W and N, respectively. (a) W3N1, (b) W6N2, (c) W3N2, (d) W3N4, (e) W3N5, (f) W4N1, (g) W4N3, (h) W4N5, (i) W5N1, (j) W5N2, (k) W5N3, (l) W5N4, (m) W5N6, (n) W6N1, (o) W6N5.

reported by Mehl et al.31 The space group of this compound is identical to W1N1 and W1N2. As seen in Figure 1f, each W atom bonds to five N atoms, with three W−N bonds of 1.97 Å and two W−N bonds of 2.01 Å. As for the N atoms, one type of N atoms has 3-fold coordination, and the other one is in the onefold coordination. W:N = 1:4. For the stoichiometric ratio W:N = 1:4, W1N4 is an orthorhombic system with the space group Pmmm. As shown in Figure 1g, each W atom in W1N4 bonds with six N atoms, featuring an octahedral geometry in local structure. Among these six W−N bonds, the two of them along a-axis are 1.98 Å, and the four of them in the bc plane are 2.11 Å in length. For the N atoms, half of N atoms are in 1-fold coordination, and the others are in 2-fold coordination. The best candidate with this ratio is that in the previous report,31 which is lower by 25 meV/atom than ours. W:N = 1:5. The best candidate of W1N5 structures is a monoclinic crystal with B2 space group (Figure 1h). In this structure, each W atom connects with seven N atoms, and there exists only one type of decahedrons formed by W and the surrounding N atoms. Of the seven bonds, the bond lengths range from 2.0 to 2.17 Å. 60% of N atoms in this crystal are in 1-fold coordination and the others are in 2-fold coordination. W:N = 1:6. In the case of the stoichiometric ratio W:N = 1:6, a stable triclinic crystal with space group P1 is searched. As seen in Figure 1i, each W atom in this crystal is coordinated with four N atoms, where a W atom lies in the center of the tetrahedron and the six N atoms locate at vertices. In addition, four W−N bonds are all 1.88 Å in length. W:N = 2:1. For the stoichiometric ratio W:N = 2:1, we obtained three low-enthalpy solids with different structures, namely, W2N1, W4N2, and W6N3. The lowest-enthalpy structure W4N2 is a hexagonal crystal with P63/mmc space

coordinated W atoms, the bond lengths are sorted to be 2.16 and 2.11 Å; for the 5-fold-coordinated W atoms, there are still two types of rectangular pyramids connect together, where the bond lengths range from 2.05 to 2.25 Å. The local bonding environment around the N atoms is very similar to that of W atoms. Particularly, the stoichiometric WN in the NaCl-type structure14 and the structures of W4N4 reported in experiment32 are also found in the present work. However, another W4N4 structure suggested from experiment33 has not been found in our extensive search. For convenience, the atomic structures of the crystals suggested in experiments14,32,33 are notated as W4N4‑exp, W4N4‑exp‑2, and W4N4‑exp‑3 respectively. Our calculations show that the formation enthalpies of W4N4‑exp, W4N4‑exp‑2, and W4N4‑exp‑3 are respectively 0.3 eV/ atom, −0.22 eV/atom, and 1.53 eV/atom, where the W4N4‑exp‑2 crystal is energetically more stable than the other two, and so this crystal will be discussed in the following work. Overall, the predicted W3N3 structure is the most stable in the family of W:N = 1:1. W:N = 1:2. For the stoichiometric ratio W:N = 1:2, three low enthalpy structures, W1N2, W2N4, and W3N6, are identified from our calculations. Careful examination finds that the three low enthalpy crystals are the same as each other. This solid is hexagonal crystal with space group P6m̅ 2, as seen in Figure 1e. In this crystal, each W atom connects with six N atoms, which is similar to the situation in W1N1, except for the 3-fold coordination of a N atom. The bond lengths of the six W−N bonds are all 2.1 Å. This structure is exactly same to W1N2 reported by Mehl et al.31 W:N = 1:3. In the structures with the stoichiometric ratio W:N = 1:3, the lowest-energy candidate of W1N3 is the same to that of W2N6, which is lower 0.17 eV/atom in energy than that C



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group, being consistent with the previous one;31 for W2N1, its space group is R3m, and the structure of W6N3 is a monoclinic crystal belonging to the B2 space group. These crystals are displayed in Figure 1j. In both W2N1 and W4N2, W and N atoms constitute a hexahedron-shaped structure. The lengths of W−N bonds are all around 2.19 Å. It is worth emphasizing that the W4N2 structure we obtained is exactly the same as the structure found in experiments.34 This once again shows that our theoretical search is very effective and reliable. There are two types of W atoms in W6N3: one type of the W atoms forms tetrahedron, where the bond lengths of W−N are about 2.20 Å, and the other type of the W atoms is formed by these atoms on the one side of the tetrahedron connecting another triangle unit. In this structure, all N atoms are in 6-fold coordination. The experimental W6N335 structure found by Khitrova is 0.64 eV/ atom higher than ours in energy. W:N = 2:3. Two low-enthalpy crystalline structures are found in the stoichiometric ratio W:N = 2:3. The lowest-enthalpy structure W4N6 is a hexagonal crystal belonging to the R32 space group. The second structure W2N3 is a triclinic crystal belonging to the P1 space group. As shown in Figure 1m, each W atom bonds with six N atoms, manifesting triangular prism in W2N3. The W−N bond lengths distribute to be 2.04, 2.16, 1.99, and 2.08 Å. In W2N3, 33.3% of N atoms are in 5-fold coordination, 33.3% in 4-fold coordination, and 33.3% in 3-fold coordination. The bonding shape for W and N atoms in W4N6 is similar as in W2N3, but bond lengths are all 2.05 Å and all N atoms are in 4-fold coordination. In Experiment, ‘r-W2N3’ and ‘h-W2N3’ proposed by Wang et al.13 were not found in our extensive search; but ours is consistent with the structures of W2N3 reported in ref 31. within energy difference of only 20 meV/atom. W:N = 2:5. W2N5 structure is a monoclinic crystal belonging to the B2/m space group, as shown in Figure 1o. W atoms are in 6-fold coordination and form two identical heptahedrons, which are aligned together. The bond lengths of W−N are 2.14, 2.0, 2.11, and 1.84 Å. The percentages of 2-fold-coordinated N atoms and 3-fold coordinated N atoms are 60% and 40%, respectively. W:N = 3:1. In the case W:N = 3:1, the lowest-enthalpy structure W6N2 is a monoclinic structure with the B2/b space group, and the second structure W3N1 is an orthorhombic structure, which are displayed in Figure 2a. This crystal found by us is nearly degenerate with the reported one31 within the error of 3 meV/atom in energy. In W3N1, 66.7% of W atoms are in 3-fold coordination. Around these W atoms, there are three types of W−N bonds with bond lengths of 2.15 and 2.22 Å respectively. The other W atoms do not connect with the N atoms. In W6N2 structure, all the N atoms are in 6-fold coordination, and W atoms are all in 2-fold coordination, causing W−N bond lengths of about 2.14 Å. As shown in Figure 2b, all N atoms in W6N2 form octahedron and bond with W atoms. W:N = 3:2. Both W3N2 and W6N4 are dramatically identical structures, with R3m̅ space-group symmetry (Figure 2c). In this crystal, 33.3% of the W atoms are in 6-fold coordination, forming an octahedron. The rest of W atoms are in 3-fold coordination, constituting tetrahedrons. The octahedron is in the middle of two tetrahedrons. For N atoms, all of them are in 6-fold coordination. The bond lengths of W−N bonds range from 2.16 to 2.21 Å. This structure differs from that proposed

in experiment.36 The W6N4 structure suggested by Mehl et al.31 is lower 20 meV/atom than ours in energy. W:N = 3:4. The W3N4 structure is an orthorhombic crystal belonging to Amm2 space group (seen in Figure 2d). Each W atom in this structure bonds with six N atoms, with bond lengths of 2.14 and 2.02 Å. In this structure, 50% of N atoms are in 5-fold coordination and others in 4-fold coordination. Compared to the reported one,31 our candidate of W3N4 is lower about 40 meV/atom in energy. The cubic W3N413 observed in experiment, notated as W3N4‑exp, is also found in our calculations. This compound has only three cations and four anions per cubic unit cell. However, this structure is not the lowest-energy one for the case of W:N = 3:4, and thus the structural feature of this crystal is not addressed in detail here. W:N = 3:5. The W3N5 compound is a triclinic crystal as displayed in Figure 2e, which is belonging to P1 space group. The W atoms in this crystal respectively bond with six atoms and seven atoms. For the 6-fold-coordinated W atoms, the bond lengths of W−N bonds are 2.07, 2.12, and 2.05 Å; for the 7-fold-coordinated W atoms, there are two types of octahedrons. The octahedrons link together. W:N = 4:1. The W4N1 structure is an orthorhombic crystal belonging to Fmm2 space group (Figure 2f). In W4N1, half of W atoms are in 3-fold coordination, where the bond lengths of W−N are 2.23 and 2.15 Å respectively, and the other W atoms do not connect with N atoms. All N atoms are in 6-fold coordination. The enthalpy of this candidate is lower by 70 meV/atom than that proposed by Mehl et al.31 So, our candidate probably is the ground state structure. W:N = 4:3. For W:N = 4:3, the W4N3 structure only has a stable crystal structure belonging to P1 space group, which is shown in Figure 2g. In this structure, half of W atoms are in 4fold coordination with the W−N bond length of 2.06 Å; the others are 2-fold coordination with bond lengths of around 2.04 Å, being slightly shorter than the former. In addition, all N atoms are in 4-fold coordination. Energetically, this structure is more stable than that reported in ref 31, lowering by 0.11 eV/ atom. W:N = 4:5. The best candidate of W4N5 structures is a monoclinic crystal belonging to B2 space group (Figure 2h). In this crystal, each W atom bonds with six N atoms. In addition, 20% N atoms are 6-fold coordination, 40% N atoms are 5-fold coordination, and the other N atoms are 4-fold coordination. For this kind of WN crystal, no previous research is available for comparison. W:N = 5:1. In the case of W:N = 5:1, the lowest-energy candidate is a monoclinic crystal belonging to B2/m space group (Figure 2i). In this structure, 60% of the W atoms possess 2-fold coordination, and the others do not bond with N atoms at all. Meanwhile, all N atoms show 6-fold coordination in the structure. W:N = 5:2. In the ratio W:N = 5:2, the best candidate of W5N2 compound is a monoclinic crystal belonging to B2/m space group (Figure 2j). The percentage of 3-fold-coordinated W atoms is less than that of 2-fold-coordinated ones, and all W−N bond lengths are about 2.15 Å. N atoms are in favor of the 6-fold coordination in this structure. Our calculations show that the energy of this structure is lower by 0.26 eV/atom than that predicted in previous theoretical research.31 W:N = 5:3. The optimal W5N3 structure is a triclinic crystal belonging to P1 space group (Figure 2k). More than half of W atoms possess 3-fold coordination. Regarding to the 3-foldcoordinated W atoms, the longest bond length of W−N in this D



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positive, this structure is dynamically stable, otherwise unstable. Our calculations indicate that the systems such as W1N1, W3N3, W4N4‑exp‑2, W5N5, W1N2, W2N1, W4N2, W2N5, W3N1, W6N2, W3N2, W3N4, W4N1, W4N3, W5N1, W5N2, W5N4, W5N6, and W6N1 are dynamically stable, but the others are dynamically unstable because the latter possess negative frequencies. The Supporting Information29 includes the phonon spectra of these structures. It should be emphasized that the NaCl-typed W4N4‑exp structure possesses negative phonon modes, and thus it is unstable dynamically. This is the same as the report from Karthik et al.15 In addition, the cubic W3N4‑exp,13 W4N4‑exp‑3,33 W6N3,35 W6N4,36 h-W2N3,13 and r-W2N313 proposed from the experiments are also dynamical unstable, but the structure of W4N4‑exp‑232 is dynamically stable (Figure 4h). The W2N2‑NiAs structure30 is regarded as the first intrinsic super hard metal by Lu et al.37 Unfortunately, our calculations show that this structure is dynamically unstable also (Figure 4i). Even so, this structure is probably stable at higher temperatures owing to the assistance of the excited phonon modes. Furthermore, we assess their relative stabilities at finite temperatures according to

structure is 2.41 Å, and the shortest one is 2.08 Å. The rest W atoms are 4-fold-coordinated. The percentages of 5-foldcoordinated N atoms and 6-fold-coordinated N atoms are 66.7% and 33.3%, respectively. W:N = 5:4. For this ratio, the lowest-energy structure is a triclinic crystal belonging to P1 space group (Figure 2l). A quarter of the W atoms are in 2-fold coordination, with W−N bond length of 2.19 Å. Among the rest W atoms, half of them is 5-fold coordination, and the others are 3-fold coordination. Our proposed W5N4 is energetically lower approximately 60 meV/ atom than the structure reported in ref 31. W:N = 5:6. For the case of W:N = 5:6, the best structure is a hexagonal crystal belonging to P62m̅ space group (Figure 2m). From Figure 2m, we find that each W atom bonds with six N atoms, forming a triangular prism; and all N atoms are in 5-fold coordination. As compared with the structure from ref 31, our predicted structure of W5N6 as mentioned above is energetically lower by 20 meV/atom. So, the predicted structure in present work may be the best candidate for the W5N3 compound. W:N = 6:1. With regard to W:N = 6:1, the best W6N1 structure is a monoclinic crystal belonging to P2/m space group, which is shown in Figure 2n. 50% of W atoms in this crystal are in 2-fold coordination, and the others do not bond with N atoms. Meanwhile, all N atoms are in 6-fold coordination. W:N = 6:5. The last ratio we addressed is W:N = 6:5. For this case, the best candidate is a monoclinic crystal structure with B2/m space group symmetry (Figure 2o). The W atoms in this crystal are sorted as the 3-fold, 4-fold, and 5-fold atoms according to their coordination. The bond length of the W−N bonds ranges from 2.03 to 2.16 Å. Figure 3 summarizes the formation enthalpies of 23 kinds of compounds as mentioned above. From this figure, the relative

F=E+

1 ∑ ℏωj + kBT ∑ ln(1 − eℏωj /kBT ) 2 j j

(2)

Here, F represents the Helmholtz free energy averaged over each atom, E is the binding energy defined by Formula 1, T and ωj mean the temperature and the vibration frequency of the jth vibrational mode, respectively. At a given temperature, the lower the free energy of the system is, the more stable this structure is. From the computed enthalpy of the concerned structures, the W3N3 is the most stable structure in the ratio W:N = 1:1 at zero temperature. When we consider the effect of lattice vibration, the W3N3 crystal still has the smallest free energy at T = 0 K as displayed in Figure 5a. As the temperature increases, the relative stabilities of both W4N4‑exp‑2 and W1N1 crystals are always less than those of W3N3 and W5N5. For both the W4N2 crystal and the W2N1 crystal, the former one is more stable than the latter one even at higher temperatures, as displayed in Figure 5b. The similar situation occurs for both W3N1 and W6N2 crystals (Figure 5c). As we know, the structural feature of a crystal can be read in its X-ray diffraction (XRD) spectra. Because of this, the XRD spectra of all the stable structures are predicted in present work, which are summarized in the Supporting Information.29 These XRD spectra are available for further examination by experiments. Properties of Stable Structures. Electronic Properties. Here, we focus on the electronic and mechanical properties of the energetically and dynamically stable candidates, such as W1N1, W3N3, W5N5, W1N2, W2N1, W4N2, W2N5, W3N1, W6N2, W3N2, W3N4, W4N1, W4N3, W5N1, W5N2, W5N4, W5N6, W6N1, and W4N4‑exp‑2. The band structures and the density of states (DOS) for these systems are computed. We find most WN compounds exhibit a metallic feature, except for W1N2. Figure 6 only shows the band structures and the DOS of two representatives W1N1 and W1N2, the others are displayed in the Supporting Information.29 Figure 6 clearly shows that the best candidate of W1N2 is a semiconductor with an indirect band gap of 2.42 eV. It is known that the band gap of a semiconductor is commonly underestimated at the level of PBE functional. Hence, the real band gap of this crystal should be

Figure 3. Convex hull diagram for the WN compounds. The formation enthalpy, ΔH, is defined as ΔH = [H(WxNy) − xH(W) − yH(N)]/(x + y).

stabilities of the compounds are clearly exhibited. It is noted that all the best candidates above are proposed in terms of their energetic stability. In addition to the energetic stability, the dynamic stability of these candidates together with the related candidates proposed in experiments and from the previous theoretical predictions is also examined by evaluating their phonon spectra: if all the phonon frequencies of a structure are E



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Figure 5. Free energy at different temperatures for the candidates of (a) W:N = 1:1 and (b) W:N = 2:1, (c) W:N = 3:1.

Figure 4. Phonon spectra of the structures of (a) W4N4‑exp, (b) W3N4‑exp, (c) W4N4‑exp‑3, (d) W6N3, (e) W6N4, (f) h-W2N3, (g) rW2N3, (h) W4N4‑exp‑2, (i) W2N2‑NiAs.

Figure 6. Band structure and the density of states of the dynamical stability structure for (a) W1N1 and (b) W1N2.

larger than 2.42 eV. Unlike the electronic structure of the W1N2, no band gap exhibits in the band structures of W1N1, as well as all other crystals mentioned above, characterizing the metallic feature. Mechanical Properties. The mechanical properties, including the elastic constants, bulk modulus (B), shear modulus (G), Young’s modulus (Y), and Poisson’s ratio (ν), of the concerned crystals are carefully computed, which are summarized in Table 1. For comparison, the values of the elastic constants available from experiments38 are also listed in Table 1. In our calculations, the bulk modulus and shear modulus of polycrystalline are treated by using the Voigt−Reuss−Hill method,39−41 which are employed to estimate the values of the Young’s modulus and the Poisson’s ratio ν by using the following formulas42

Y=

9BG 3B + G

υ=

3B − 2G 6B + 2G

(3)

Physically, the bulk modulus B can well reflect the average strength of the chemical bonds in a material. The higher the bulk modulus of a material is, the stronger the ability to resist uniform compression is. From Table 1, we find that the bulk moduli of some tungsten nitrides are significantly higher than that of the polycrystalline W. In particular, the W1N1 bulk modulus is the highest. The next one is the W4N4‑exp‑2. Usually, the hardness of a material is more sensitive to the shear modulus than to the bulk modulus.43 Among the crystals we concerned above, the four ones, W3N3, W4N4‑exp‑2, W1N2, and W5N6, have highlights in their shear moduli, indicating the F



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Table 1. Calculated Elastic Constants (GPa), Bulk Modulus (B, GPa), Shear Modulus (G, GPa), Young’s Modulus (E, GPa), Poisson’s Ratio (μ, GPa) of Dynamical Stability Structure, Wexp is the Experimental Elastic Constants c11 Wexp W W1N1 W3N3 W4N4‑exp‑2 W5N5 W1N2 W2N1 W4N2 W2N5 W3N1 W6N2 W3N2 W3N4 W4N1 W4N3 W5N1 W5N2 W5N4 W5N6 W6N1

533 540 671 815 678 545 631 504 530 256 553 550 596 643 547 657 491 560 461 648 506

c22

c33

671

705

678 393 631 504 536 377 504 558 596 413 583 526 538 701 594 648 543

783 556 992 624 695 135 617 511 718 584 514 451 497 521 450 720 490

c12 205 208 195 138 244 142 183 308 332 58 252 233 278 179 235 83 225 180 205 189 204

c13

c23

294

294

236 243 213 249 229 90 233 214 243 201 223 174 269 287 198 178 266

236 138 213 249 210 55 212 250 243 198 221 211 219 209 178 178 227

c44 163 154 120 167 225 162 229 100 132 79 106 154 182 128 132 57 136 128 125 210 150

higher hardness of these crystal than the others. The ratio of B/ G is another important factor to represent the materials’ ductility. If the value of the B/G ratio of a material is larger than the threshold 1.75,44 this material is ductile; otherwise, the material is brittle. As listed in Table 1, three crystals, W3N3, W1N2, and W5N6, have B/G ratios of 1.64, 1.55, and 1.42 respectively. These B/G ratios are less than the threshold of 1.75, and thus these crystals are brittle. In contrast, the other crystals have B/G ratios larger than the value of 1.75, exhibiting ductile behaviors. Intuitively, the mechanical properties of the WxNy compounds do correlate with the component of N which could be presented by the ratio of x/(x + y). As displayed in Figure 7, the three kinds of the mechanical parameters, such as the Young’s modulus, the shear modulus, and the bulk modulus, varies with the ratio of x/(x + y) apparently. Strikingly, when the value of x/(x + y) is 0.30, the three kinds of mechanical parameters are quite small. For each ratio in

c55

c66

120

238

225 200 229 100 153 107 175 150 182 149 147 129 138 131 144 210 149

217 165 224 98 73 50 151 154 159 147 166 122 149 181 212 230 155

B

G

B/G

E

μ

314 319 400 364 396 273 377 360 367 122 340 334 382 305 333 285 328 348 294 345 326

163 159 172 223 228 162 243 67 128 77 148 153 121 153 153 124 138 156 154 243 146

1.97 2.01 2.32 1.64 1.74 1.69 1.55 5.39 2.86 1.58 2.29 2.19 3.15 2 2.18 2.29 2.38 2.22 1.91 1.42 2.23

419 408 451 555 573 405 599 189 345 191 388 398 328 393 397 325 363 408 393 589 381

0.28 0.29 0.31 0.25 0.26 0.25 0.24 0.41 0.34 0.24 0.31 0.3 0.36 0.29 0.3 0.31 0.32 0.3 0.28 0.22 0.31

Figure 7, we choose larger shear modulus as the ordinate, which is less messy for the graphic. When the ratio is in between 0.33 and 0.5, these parameters become larger significantly, and once the ratio is more than 0.5, the variation of these parameters becomes slight. The features of the curves displayed in Figure 7 imply that the WxNy compounds with the ratio x/(x + y) between 0.33 and 0.5, where the crystals of W1N2, W5N6, and W4N4‑exp‑2 are included, possess superior elastic properties. Considering the values of both ductility and strength for our concerned crystal, we conclude that the W4N4‑exp‑2 crystal shows the best performance in the mechanical properties. Basically, the mechanical behaviors shown in the different crystals above are originated from the chemical bonding character, which are reflected in the feature of charge density around each atom in the crystals. Figure 8 displays the distribution of the calculated charge density for six typical crystals of W1N2, W5N6, and W4N4‑exp‑2, W3N4, W4N3, and W6N2. It is found from Figure 8 that no covalent bond exists in W3N4, W4N3, or W6N2. However, for the crystals of W1N2, W5N6, W4N4‑exp‑2, besides the ionic bonds between N and W ions, such as mark C in W4N4‑exp‑2, there are covalent bonds marked with A and B. For example, for W5N6, there exists the covalent bonds A locating in the middle of three W atoms, and B in the middle of two W atoms, as shown in Figure 8e. Essentially, such covalent bonds are stemmed from the overlap of the localized d orbitals between the two neighboring W atoms. These covalent bonds enhance the Shear modulus and the Young’s modulus of the crystal significantly. So, W1N2, W5N6, and W4N4‑exp‑2 possess superior elastic properties.

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CONCLUSIONS By performing theoretical calculations, the best candidates for W1N1, W3N3, W5N5, W1N2, W2N1, W4N2, W2N5, W3N1, W6N2, W3N2, W3N4, W4N1, W4N3, W5N1, W5N2, W5N4, W5N6, and W6N1 are achieved. Meanwhile, our calculations support the structure of W4N4‑exp‑2 proposed in experiments. The detailed

Figure 7. Bulk modulus (B, GPa), Shear modulus (G, GPa), Young’s modulus (E, GPa) of WxNy as a function of x/(x + y). G



Article

Figure 8. Differential charge density of (a) W1N2, (b) W3N4, (c) W4N3, (d) W6N2, (e) W5N6, (f) W4N4‑exp‑2. Blue and yellow balls stand for W and N, respectively.

11275191). The computational center of USTC is acknowledged for computational support.

atomic structures and the electronic structures as well as the crystal vibrations of these best candidates are predicted. It is found that only the W1N2 crystal is a semiconductor and the others are metallic. Furthermore, the mechanical properties such as the elastic parameters, bulk modulus, Young’s modulus, shear modulus, and the ductility of WxNy (x = 1−6, y = 1−6) are systemically evaluated, from which the evolution of the mechanical property of the WxNy on the x-to-y ratio is obtained, and the crystals with good performances in mechanical properties are proposed. We, furthermore, find that the covalence bonds in a compound are reliable for the good mechanical property of the WN compounds.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.cgd.7b01707. Detailed structure information; phonon spectra; XRD spectra; band structures and the DOS (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

B. C. Pan: 0000-0002-5128-7860 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is financially supported by the National Magnetic Confinement Fusion Program (Grant No. 2013GB107004), the National Natural Science Foundation of China (No. H



Article

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