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. 2018 Sep 20;3(9):11477–11485. doi: 10.1021/acsomega.8b01602

Prediction of Thermodynamically Stable Compounds of the Sc–N System under High Pressure

Muhammad Aamir Aslam 1, Z J Ding 1,*
PMCID: PMC6645223  PMID: 31459249

Abstract

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Materials under high pressure often exhibit unusual physical and chemical behaviors. We investigated the Sc–N system under high pressure in the range of 0–110 GPa using variable-composition methodology implemented in Universal Structure Predictor: Evolutionary Xtallograpgy (USPEX) in conjunction with Vienna Ab Initio Simulation Package (VASP). The calculation led to prediction of new thermodynamically stable compounds, Sc4N3, Sc8N7, ScN3, and ScN5, and also phase transition of ScN5 from triclinic to monoclinic at 104 GPa. These results are important to understand the structure of Sc–N compounds under high pressure and their elastic and electronic properties.

Introduction

Pressure is a fundamental thermodynamic variable that has a capacity to alter the interatomic interactions, electron density distribution, and bonding patterns of materials. These alterations in a material under pressure lead to phase transitions with unusual physical and chemical properties of the material.1 These phases cannot be recovered at ambient condition if the changes of properties are irreversible. Therefore, high-pressure study can be an effective way to discover new functional materials with exciting properties.2−5

The early transition-metal nitrides have significant importance due to their chemical stability and outstanding mechanical properties.6−8 However, scandium nitride (ScN) has attracted significant attention among the early transition-metal nitrides, which is due to its high melting point of 2600 °C, high hardness, extreme corrosive behavior, and electric conductivity.9−12 These remarkable properties categorize scandium nitride as a refractory material and also make it suitable for thermoelectric devices that can be operated at high temperature.13,14

A lot of work has been done on stoichiometric and nonstoichiometric scandium nitrides.15,16 However, most of the work highlights the electronic, optical, and chemical bonding properties in their rock-salt structure.17−19 Many transition metals have a tendency to form nitride compounds under extreme conditions of pressure,20−24 but the question that how many scandium nitride compounds exist is still a mystery. Therefore, we aim in this work to reveal this mystery by exploring crystal structures and all possible stoichiometries of the Sc–N system in the pressure range of 0–110 GPa. Using a state-of-the-art computation technique that has been recently developed, i.e., Universal Structure Predictor: Evolutionary Xtallograpgy (USPEX),25−28 we found several thermodynamically stable compounds with different stoichiometries, Sc4N3, Sc8N7, ScN3, and Sc1N5, under high pressure.

Results and Discussion

To reveal the energetically most favorable structure for the Sc–N compounds under high pressure, we systematically explored the stoichiometric compounds of ScxNy in the pressure range of 0–110 GPa. The formation enthalpy for the ScxNy compounds was computed relative to dissociation into constituent elements, scandium and nitrogen, by the equation

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where ΔH is the formation enthalpy per atom of a compound and H is the calculated enthalpy of each constituent element. We constructed a convex hull plot to incorporate all possible decomposition routes of the Sc–N system at pressures of 0, 30, 50, 75, 100, and 110 GPa, as shown in Figure 1. Only the structures located on the convex hull are considered stable structures, and those lying above the convex hull are unstable or metastable structures. At ambient pressure of 0 GPa, the ScN (Fm3̅m, space group 225) compound appears on the convex hull and its calculated lattice parameter a = 4.518 Å is in good agreement with the experimental values, 4.50129 and 4.512 Å.30 Hence, this calculation validates the methodology applied to the Sc–N system under zero-pressure condition. Although the Sc4N3, Sc8N7, ScN3, and ScN5 compounds are lying above the convex hull at 0 GPa, they become stable at higher pressures, as depicted in Figure 1. To reveal the pressure stability of these stoichiometric compounds, we constructed the pressure–composition phase diagram as shown in Figure 2a. Our search reproduced the scandium phases, i.e. (P63/mmc, space group 194) and (I4/mmm, space group 139),31 and also four phases of nitrogen, i.e., (Pa3̅ space group 205), (P21/c, space group 14), (P41212, space group 92), and (I213, space group 199);32 herein, scandium and nitrogen are the basic constituents for the stoichiometric compounds and the appearance of their phases in our search also validates our methodology. The ScN (Fm3̅m, space group 225) compound remains thermodynamically stable throughout the considered pressure range. The predicted pressure phase diagram also indicates that the Sc8N7 (C2/m, space group 12) compound is stable from 5 to 45 GPa, Sc4N3 (I4̅3d, space group 220) compound above 80 GPa, and triclinic ScN3 (P1̅, space group 2) above 29 GPa. One of the most interesting predicted structures ScN5 belongs to the family of pentazolate compounds, which have been investigated more closely in recent years than at any time since their first synthesis. CsN533 and NaN534 are newly explored pentazolate compounds; these nitrogen-rich compounds can decompose to dinitrogen N2 with a large energy release. We also explored phase transition of the stoichiometric compound ScN5 at 104 GPa from triclinic to monoclinic, as shown in Figure 2b.

Figure 1.

Figure 1

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Convex hull diagram for the Sc–N system under different pressures: (a) 0 GPa; (b) 30 GPa; (c) 50 GPa; (d) 75 GPa; (e) 100 GPa; and (f) 110 GPa. The red solid circles represent the thermodynamically stable compounds, while the open circles represent the unstable structures.

Figure 2.

Figure 2

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(a) Pressure–composition phase diagram of the Sc–N system in the pressure range of 0–110 GPa; (b) formation enthalpy for ScN5 from the triclinic to monoclinic structures; a phase transition occurs at 104 GPa.

The four new stoichiometric compounds found in our search are shown in Figure 3, and their structural details are summarized in Table 1. The Sc4N3 (I4̅3d, space group 220, four formula units) structure has the lattice parameter, a = 6.225 Å, at 100 GPa. Inequivalent Sc atoms occupy the Wyckoff position, 16c (0.182, 0.182, and 0.182), and the other inequivalent N atoms take the Wyckoff position, 12b (0.875, 0.0, and 0.25). The Sc–N bond length ranges from 2.049 to 2.268 Å. The scandium atom is coordinated with six nitrogen atoms.

Figure 3.

Figure 3

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Predicted structures: (a) ScN3 (P1̅) at 30 GPa; (b) ScN5 (P1̅) at 50 GPa; (c) ScN5 (Cm) at 110 GPa; (d) Sc8N7 (C2/m) at 30 GPa; and (e) Sc4N3 (I4̅3d) at 100 GPa, where the purple sphere represents Sc atom and the gray sphere represents N atom.

Table 1. Structural Details of the Predicted Structures of Sc–N Compounds under Pressure.

      lattice parameters a, b, c (Å)    
phases pressure (GPa) space group α, β, γ (deg) atom fractional atomic coordinates
        Wyckoff positions x y z
ScN 0 225 (Fm3̅m) a = b = c = 4.518 Sc (4a) 0.0 0.0 0.0
α = β = γ = 90.0 N (4b) 0.5 0.5 0.5
Sc4N3 100 220 (I4̅3d) a = b = c = 6.225 Sc (16c) 0.182 0.182 0.182
α = β = γ = 90.0 N (12b) 0.875 0.0 0.25
Sc8N7 30 12 (C2/m) a = 13.348 Sc1 (4i) –0.375 0.0 0.313
b = 3.068 Sc2 (4i) 0.126 0.0 0.064
c = 10.60 Sc3 (4i) 0.369 0.0 0.195
α = γ = 90.0 Sc4 (4i) 0.123 0.0 –0.435
β = 48.609 N1 (4i) –0.247 0.0 0.370
  N2 (4i) 0.499 0.0 0.247
  N3 (4i) 0.251 0.0 0.123
  N4 (2c) 0.0 0.0 0.500
ScN3 30 2 (P1̅) a = 4.486 Sc (2i) –0.208 0.194 0.412
b = 5.057 N1 (2i) –0.170 –0.324 –0.106
c = 3.034 N2 (2i) 0.360 –0.472 0.002
α = 102.192 N3 (2i) –0.291 –0.124 –0.249
β = 98.35, γ = 99.315        
ScN5 50 2 (P1̅) a = 5.084 Sc (2i) 0.187 0.757 0.677
b = 4.427 N1 (2i) 0.746 0.933 0.044
c = 4.345 N2 (2i) 0.598 0.430 0.585
α = 70.349 N3 (2i) 0.364 0.170 0.209
β = 101.130 N4 (2i) 0.737 0.563 0.814
γ = 103.458 N5 (2i) 0.906 0.780 0.265
ScN5 110 8 (Cm) a = 4.363 Sc (2a) 0.069 0.0 –0.300
b = 4.138 N1 (2a) –0.385 0.0 –0.149
c = 4.364 N2 (2a) 0.203 0.0 0.173
α = γ = 90.0 N3 (4b) 0.220 0.0 0.349
β = 106.244 N4 (2a) –0.376 0.0 0.158
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The Sc8N7 (C2/m, space group 12, two formula units) stoichiometry has the lattice parameters, a = 13.348 Å, b = 3.068 Å, c = 10.60 Å, α = γ = 90.0°, and β = 48.609° at 30 GPa. Four inequivalent Sc atoms occupy the Wyckoff positions, Sc1 4i (−0.375, 0.0, 0.313), Sc2 4i (0.126, 0.0, 0.064), Sc3 4i (0.369, 0.0, 0.195), and Sc4 4i (0.123, 0.0, −0.435), and four nitrogen atoms occupy the Wyckoff positions, N1 4i (−0.247, 0.0, 0.370), N2 4i (0.499, 0.0, 0.247), N3 4i (0.251, 0.0, 0.123), and N4 2c (0.0, 0.0, 0.500). The Sc atom is coordinated with six nearby nitrogen atoms, and the bond length of Sc–N is 2.176 Å.

The ScN3 (P1̅, space group 2, two formula units) has the lattice parameters, a = 4.486 Å, b = 5.057 Å, c = 3.034 Å, α = 102.192°, β = 98.35°, and γ = 99.315°, at 30 GPa. The scandium atom lies on the Wyckoff position, 2i (−0.208, 0.194, 0.412), and three nitrogen atoms occupy N1 2i (−0.170, −0.324, −0.106), N2 2i (0.360, −0.472, 0.002), and N3 2i (−0.291, −0.124, −0.249). In this structure, N atoms form a chain of six atoms and the shortest distance between them is 1.345 Å. These nitrogen atoms are coordinated with two nearest Sc atoms with a bond length of 2.181 Å.

The ScN5 (P1̅, space group 2, two formula units) has the lattice parameters, a = 5.084 Å, b = 4.427 Å, c = 4.345 Å, α = 70.349°, β = 101.130°, and γ = 103.458°, at 50 GPa; the Sc atom occupies the Wyckoff position, Sc 2i (0.187, 0.757, 0.677), while the N atoms occupy the Wyckoff positions, N1 2i (0.746, 0.933, 0.044), N2 2i (0.5987, 0.4306, 0.58545), N3 2i (0.364, 0.170, 0.209), N4 2i (0.737, 0.563, 0.814), and N5 2i (0.906, 0.780, 0.265). However, at 104 GPa, it is transformed to a monoclinic structure with the lattice parameters, a = 4.363 Å, b = 4.138 Å, c = 4.364 Å, α = γ = 90.0°, and β = 106.244° and the Wyckoff positions, Sc 2a (0.069, 0.0, −0.300), N1 2a (−0.385, 0.0, −0.149), N2 2a (0.203, 0.0, 0.173), N3 4b (0.220, 0.252, 0.349), and N4 2a (−0.376, 0.0, 0.158).

To confirm the dynamic stability of these predicted structures, we recorded the phonon dispersion spectra. It is mandatory for the dynamic stability to have the positive phonon frequencies in the whole Brillouin zone. Any structure having a negative or an imaginary frequency in the Brillouin zone is considered as dynamically unstable. Our recorded phonon spectra in Figure 4 show that there is no imaginary frequency in the whole Brillouin zone, which confirms the dynamic stability of our predicted structures.

Figure 4.

Figure 4

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Phonon dispersion band structure of the predicted structures: (a) ScN3 (P1̅) at 30 GPa; (b) Sc8N7 (C2/m) at 30 GPa; (c) ScN5 (P1̅) at 50 GPa; (d) Sc4N3 (I4̅3d) at 100 GPa; and (e) ScN5 (Cm) at 110 GPa.

The atomic interaction between Sc and N atoms in the Sc–N compounds was revealed by calculating the Bader charges for all predicted stoichiometries at respective pressures. The Bader charges transferred from Sc to N atoms for Sc4N3, Sc8N7, ScN3, and ScN5 are 1.3, 1.6, 1.7, and 1.8, respectively. This also indicates that as the N concentration is increased, the charge transfer from Sc to N also increases. To further explore the bonding nature of compounds, we calculated the electron localization function (ELF), which determines the likelihood of electrons in the nearby space of a crystal. The ELF values generally lie between 0 and 1; the values higher than 0.5 correspond to the covalent bonding, core electrons, and lone pairs. The value 0.5 corresponds to the electron gas, while the lower value corresponds to metallic or ionic bonds. As shown in Figure 5, the ELF values are higher between the nearest nitrogen atoms, which shows that the bonding between them is covalent. However, Sc and N formed metallic or ionic bonds because the ELF value is lower between them.

Figure 5.

Figure 5

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Mapping of electron localization function (ELF) for the Sc–N system: (a) ScN3 (P1̅) at 30 GPa; (b) Sc1N5 (P1̅) at 50 GPa; (c) Sc4N3 (I4̅3d) at 50 GPa; (d) ScN5 (Cm) at 110 GPa; and (e) Sc8N7 (C2/m) at 30 GPa, where the purple sphere represents Sc atom and the gray sphere represents N atom.

For the electronic properties of the Sc–N system, we calculated the band structure of all of the predicted structures at respective pressures, as shown in Figure 6. The horizontal dashed line in the figure illustrates the Fermi energy level, and bands of all predicted structures are crossing the Fermi level, which shows that our predicted stable structures are metallic in nature. However, we also calculated the band structure of ScN5 with hybrid function Heyd–Scuseria–Ernzerhof used in density functional theory (DFT), and it opened a narrow band gap 0.24 eV, as shown in Figure 6c.

Figure 6.

Figure 6

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Electronic band structure of the predicted structures: (a) ScN3 at 30 GPa; (b) Sc8N7 at 30 GPa; (c) ScN5 at 50 GPa; and (d) Sc4N3 at 100 GPa.

We also investigated the mechanical stability of the predicted structures because it is the necessary condition for a structure to exist. Elastic constants contain all of the necessary information to understand the mechanical properties and are also helpful for estimating the hardness of a material. We calculated the elastic constants of the predicted structures, which are tabulated in Table 2. These calculated elastic constants satisfy the mechanical stability criteria,35 and the elastic matrix for each compound contains positive eigenvalues, which further proves the elastic stability of the considered structures. Bulk modulus and shear modulus depict the material resistance to volumetric changes and shape deformation; materials with high bulk modulus are considered strong in resisting uniform compression, while materials with high shear modulus can have high hardness because shear modulus is related to hardness. The calculated values of B, G, and Hv are tabulated in Table 2, which shows that the bulk modulus increases as the nitrogen content increases in the compound. For the ScN cubic structure, our calculated elastic constants C11, C12, and C44(36) and bulk modulus of 198.38 GPa are in good agreement with the literature value, 198 GPa,12 which supports our calculation for other structures as well because there is no published data on these compounds to the best of our knowledge. The estimated hardnesses of ScN, Sc8N7, and ScN5 compounds are 25, 24, and 23.6 GPa, respectively, while those of ScN3, ScN5, and Sc4N3 are 17.5, 17.4, and 17.0 GPa, respectively. The bulk modulus-to-shear modulus ratio, k = G/B, indicates the ductility and brittleness of a material; the critical value of k is about 1.75.37 The materials having k-values greater than 1.75 are ductile and those having lower values are brittle. The calculated Pugh ratios for ScN, Sc8N7, and ScN3 are 1.28, 1.44, and 1.69, respectively, indicating that these compounds are brittle. However, the Pugh ratios for ScN5 and Sc4N3 are 1.79 and 2.0, respectively, indicating that they are ductile. The ductility and brittleness of a material can also be differentiated by Poisson’s ratio, whose value greater than 0.26 indicates better ductile materials. Poisson’s ratio v also gives insights into the characteristics of bonding forces: for covalent materials, the value is 0.1, while for ionic materials, it is 0.25. For a central force solid, the value is between 0.25 and 0.5.38Table 2 shows our calculated values of v and indicates that for ScN5 and Sc4N3, the interatomic force is central while for ScN, Sc8N7, and ScN3, it has ionic bonding characteristics.

Table 2. Calculated Elastic Stiffness Constants Cij, Bulk Modulus B, Shear Modulus G, Young’s Modulus E, Poisson’s Ratio v, and Vickers Hardness Hv of the Sc–N System.

  ScN cubic Sc8N7 monoclinic ScN3 triclinic ScN5 triclinic ScN5 monoclinic Sc4N3 cubic
elastic stiffness Cij constants C11 385.81 549.92 426.18 599.99 772.45 788.63
C22   498.73 443.61 457.08 821.69  
C33   464.91 563.28 414.57 861.50  
C44 163.74 162.13 138.72 215.40 426.77 165.11
C55   212.21 192.74 129.65 329.19  
C66   159.67 240.07 237.03 311.42  
C12 104.70 110.58 224.26 227.76 329.12 221.93
C13   124.52 137.47 266.51 373.19  
C14     30.12 –179.03    
C15   0.2770 32.47 42.87 109.77  
C16     32.94 –125.22    
C23   166.43 168.97 152.66 343.03  
C24     7.5572 –16.25    
C25   –0.071 –16.75 –53.96 –75.21  
C26     133.77 –51.62    
C34     80.56 –47.59    
C35   –0.019 –8.31 –27.54 0.74  
C36     29.81 31.63    
C45     55.13 10.41    
C46   0.002 8.91 34.17 145.83  
C56     0.95 –21.99    
B (GPa) 198.38 257.32 262.56 307.28 484.49 410.94
G (GPa) 154.46 178.44 154.53 171.4 269.94 205.33
E (GPa) 367.89 434.81 387.57 433.58 682.98 528.04
v 0.191 0.218 0.254 0.265 0.265 0.286
Hv (GPa) 25 24 17.5 17.4 23.6 17.0
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Conclusions

In summary, we systematically investigated the Sc–N system in the pressure range of 0–110 GPa using variable-composition methodology implemented in USPEX in conjunction with VASP. The calculation enables us to find new thermodynamically stable compounds, Sc4N3, Sc8N7, ScN3, and ScN5. We also explored the phase transition of ScN5 from triclinic to monoclinic at 104 GPa. Our results are also important for understanding the structures, electronic properties, and elastic properties of Sc–N compounds under high pressures.

Computational Details

To discover all potential ScxNy structures under high pressure for the Sc–N system, an extensive search for stable compounds was performed using the evolutionary algorithm and variable-composition strategy implemented in USPEX code in conjunction with Vienna Ab Initio Simulation Package (VASP).39

The variable-composition evolutionary search was carried out at pressures of 0, 30, 50, 75, 100, and 110 GPa by keeping 8–16 atoms in a primitive cell to find stable compositions. Then, we performed a fixed-composition search of each of these stable compositions with different formula units. The first-generation structures were produced randomly, and the number of structures in this generation was kept large. The consequent generations were obtained by applying different operators in the following percentages: heredity, 40%; lattice mutation, 20%; and atom transmutation, 20%, while 20% randomly produced structures were also added in each generation. The USPEX calculation efficiency is found to increase by applying these key parameters.

Total energy calculations were executed in the density functional theory (DFT) framework within the Perdew–Burke–Ernzerhof parametrization of generalized gradient approximation,40 as implemented in VASP code. The projector-augmented wave method described the interaction between the ions and the electrons with a cutoff energy of 520 eV. The 3d14s2 (cutoff radius, 2.5 au)41 and 2s22p3 (cutoff radius, 1.5 au)42 were treated as valence states in the Sc and N atoms, respectively. A uniform Γ-centered Monkhorst–Pack grid with a higher resolution of 2π × 0.03 Å–1 was used to sample the Brillouin zone, which also confirmed that the total energies were ≤10–3 eV/atom. Moreover, all forces on atoms were converged to ≤10–3 eV/Å and the total stress tensor deviated from the target pressure by ≤10–2 GPa. The phonon dispersion curve was calculated to determine the dynamical stability of the predicted structures using the finite displacement methodology, as implemented in the PHONOPY code.43 The charge transferred was calculated using Bader’s quantum theory.44 All structures were visualized and plotted using VESTA.45 The elastic stiffness constants were calculated using the strain–stress methodology;46 we used the ELATE47 code to obtain the bulk modulus B, shear modulus G, Young’s modulus E, and Poisson’s ratio. There are several models to calculate the theoretical Vickers hardness, but Chen’s model48 is considered more accurate.49 We used Chen’s model to estimate the theoretical Vickers hardness Hv by the relation, Hv = 2(k2G)0.585 – 3, where k = G/B is the Pugh ratio.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 11574289), “111” Project of the Education Ministry of China (No. B07033), and Special Program for Applied Research on Supercomputation of the NSFC-Guangdong Joint Fund (2nd phase) under Grant No. U1501501. The authors also thank Dr H. M. Li and acknowledge the supercomputing center of USTC for the support of parallel computing.

The authors declare no competing financial interest.

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