Elastic and electronic properties of partially ordered and disordered Zr(C1−xNx) solid solution compounds: A first principles calculation study

Journal of Alloys and Compounds 619 (2015) 788–793

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Elastic and electronic properties of partially ordered and disordered Zr(C1xNx) solid solution compounds: A first principles calculation study Jiwoong Kim a,b, Hanjung Kwon a, Jae-Hee Kim c, Ki-Min Roh a, Doyun Shin a,b, Hee Dong Jang a,b,⇑ a

Rare Metals Research Center, Korea Institute of Geoscience and Mineral Resources, Gwahang-ro 92, Yuseong-gu, Daejeon 305-350, Republic of Korea Korea University of Science and Technology, Gajeong-ro, Yuseong-gu, Daejeon, Republic of Korea c Department of Structure and Materials, Korea Aerospace Research Institute, 45 Eoeun-Dong, Yuseong-Gu, Daejeon 305-333, Republic of Korea b

a r t i c l e

i n f o

Article history: Received 30 July 2014 Received in revised form 27 August 2014 Accepted 29 August 2014 Available online 18 September 2014 Keywords: First principles calculations Special quasi-random structure Atomic configuration Elastic properties Electronic properties

a b s t r a c t The elastic properties and electronic structures of partially ordered and disordered Zr(C1xNx) solid solution compounds were investigated using first principles calculations to understand the effects of nitrogen content and atomic distribution. To obtain a proper exchange–correlation energy, we used local density and generalized gradient approximations with Perdew–Burke–Ernzerhof (LDA and GGA-PBE) parametrization. Partially ordered and disordered structures of Zr(C1xNx) compounds were expressed using unit cell and special quasi-random structure (SQS) models, respectively. We demonstrated that although the disordered models have P1 symmetry with different model sizes and cell shapes compared with ordered models, they reproduce the equilibrium structure and elastic properties of the Zr(C1xNx) compounds with B1 (Fm-3m) symmetry. However, clear differences exist in the electronic structures. Therefore, the atomic configuration is essential for calculating the electronic structures of the Zr(C1xNx) compounds. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Computational materials science has been applied not only to materials but also in other research and industrial fields, such as in life sciences. This has occurred because of developments in density functional theory (DFT) and the various advantages of computational materials science, such as convenience, coverage, and capability [1–6]. The usefulness of computational materials science is increasing steadily for developing theories and computational systems. However, several problems need to be solved in computational materials science. One such emerging problem is that of expressing compositional disordered structures because many materials are multi-component and systems are becoming more complex. Unfortunately, most DFT methods rely on periodic boundary conditions to obtain bulk properties. Therefore, they normally require perfectly ordered models in their calculations. This problem is more serious for solid solution compounds [7–9]. The simplest way to express compositional disordered models is by using the supercell method. A supercell that is as large as possible is constructed with randomly distributed components. The model ⇑ Corresponding author at: Rare Metals Research Center, Korea Institute of Geoscience and Mineral Resources, Gwahang-ro 92, Yuseong-gu, Daejeon 305-350, Republic of Korea. Tel.: +82 42 868 3612; fax: +82 42 868 3415. E-mail address: [email protected] (H.D. Jang). http://dx.doi.org/10.1016/j.jallcom.2014.08.250 0925-8388/Ó 2014 Elsevier B.V. All rights reserved.

size in DFT calculations is limited to several hundreds of atoms despite remarkable developments in computational systems in recent decades. Therefore, a method must be developed in DFT calculations to express compositional disordered models. Recently, several alternative methods, such as the coherent potential approximation (CPA) and cluster expansion (CE) methods, have been developed to solve the problem [10–12]. Although these methods are effective, the CPA cannot account for local environmental relaxations and the CE method has computing power problems [10,13]. To overcome these problems, Zunger et al. [14] suggested special quasi-random structures (SQS) to mimic perfect disordered models with several crystal structures. The basic concepts of this method are pair and multi-site correlation functions to satisfy a random local environment within a few neighbors [15]. The optimal configurations for a random model are satisfied by:

hY

i k;m

where

SQS

hQ

ffi

k;m

hY

i k;m

i Random

Random

ð1Þ

is the correlation function of the random struc-

ture, which is expressed by (2x  1)k in the A1xBx-type solid solutions, and k and m indicate pairs of atoms (e.g., k = 1, 2, and 3 are a point, pairs, and triplets, respectively) and the mth-neighbor distance, respectively [14]. The SQS method has provided a good approximation of random models with several crystal structures,

J. Kim et al. / Journal of Alloys and Compounds 619 (2015) 788–793

such as body-centered cubic (bcc), face-centered cubic, and hexagonal close packing [10,16–19]. The SQS models have shown excellent results, particularly in thermodynamic approaches. Jiang [16] reported on the lattice information and thermodynamic properties of ternary bcc-structured alloys using SQS models. To express the ternary random alloys, he used 32-, 36-, and 64-atom SQS models and calculated the thermodynamic interaction parameters between the ternary components to produce mixing properties. Shang et al. [17] investigated the structural and thermodynamic properties of Ni1xPtx binary disordered alloys with several space groups using 8-, 16-, and 32-atom SQS models. In their study, phonon calculations yielded finite temperature properties of Ni1xPtx alloys, such as the Helmholtz free energy and the phase transition temperature. Recently, the thermodynamic mixing properties and phase stabilities of titanium oxides and titanium oxi-carbides were investigated using 16- and 32-atom SQS models [20]. In this study, the effects of carbon monoxide partial pressure, carbon activity, and temperature on the phase stability were revealed and the reliability of the calculations was determined by comparing the calculated results with experimental data. Although the SQS models have focused on depicting disordered models for computation, only a few studies have been performed to investigate the elastic and electronic properties of disordered models expressed as SQS [21,22]. To the best of our knowledge, limited studies have been conducted on the elastic properties of transition metal carbonitrides with random structures using the SQS models. Therefore, we have investigated the elastic and electronic properties of ordered and disordered Zr(C1xNx) solid solution compounds using the unit cell and SQS models.

789

where E(V) is the energy of the model at a specific volume, and V and a to e are the fitting coefficients, respectively. The partially ordered and disordered models were expressed by the unit cell and SQS models, respectively. The SQS models were constructed using the gensqs code as implemented in the Alloy Theoretic Automation Toolkit (ATAT) package. Detailed methods for constructing the SQS models and practical computational examples have been reported in the ATAT user guide and in previous studies [10,16,20,29,30]. To express the disordered Zr(C1xNx) solid solution compounds with x = 0.25 and 0.50, we constructed 32- and 16-atom SQS models, respectively. The model for x = 0.75 was obtained by switching the atomic positions of C and N in the x = 0.25 32-atom SQS model. The calculation model structures for expressing the disordered Zr(C1xNx) solid solution compounds are shown in Fig. 1. The elastic properties of the partially ordered and disordered compounds were calculated using a stress–strain relationship [7,31]. To obtain the elastic constants, we conducted volume-conserving calculations using distorted models with tetragonal and trigonal shear distortions from 0.03 to +0.03 (step: 0.01) per model, respectively. For simplicity, we assumed that all of the ordered models are cubic, although the Zr(C0.5N0.5) model has a tetragonal structure. The elastic properties of the ordered compounds were calculated using the following equations [32–34]:

BV ¼

C 11 þ 2C 12 BV þ BR ; BR ¼ ð3S11 þ 6S12 Þ1 ; BVRH ¼ ; 3 2

ð3Þ

GV ¼

C 11  C 12 þ C 44 GV þ GR ; GR ¼ 5ð4S11  4S12 þ 3S44 Þ1 ; GVRH ¼ ; 5 2

ð4Þ

Ex ¼

9Bx Gx ; x ¼ V; R; and VRH; Gx þ 3Bx

ð5Þ

where B, G, and E are the bulk, shear, and Young’s modulus, respectively, and V, R, and VRH stand for Voigt, Reuss, and Voigt–Reuss–Hill, respectively. Sij is the compliance, which is the inverse matrix of the elastic constant matrix Cij as follows:

S11 ¼ ðC 11 þ C 12 Þ=fðC 11  C 12 ÞðC 11 þ 2C 12 Þg;

ð6Þ

S12 ¼ C 12 =fðC 11  C 12 ÞðC 11 þ 2C 12 Þg;

ð7Þ

2. Calculation methods The Vienna ab initio simulation package was used to calculate the elastic properties and electronic structure of the Zr(C1xNx) solid solution compounds [23,24]. Exchange–correlation effects were treated in the framework of the local density approximation (LDA), generalized gradient approximation (GGA) with Perdew– Burke–Ernzerhof (PBE) parametrization, and GGA-PBEsol exchange correlation (for solids calculations) to obtain equilibrium structures [25–28]. The GGA-PBEsol exchange correlation was only used to obtain the elastic properties and electronic structures because it yielded the most accurate values in structural optimizations compared with experimental values. Integration in the Brillouin zone was carried out using Monkhorst Pack 13  13  13 k-points for the 8-atom ordered models and 9  7  5 and 7  7  13 k-points were used for the 32- and 16-atom disordered models, respectively. To improve the accuracy of the results, we used a high energy cutoff of 500 eV with an energy convergence of 0.01 eV/Å and the tetrahedron method with Blochl correction for the energy calculations. Structural optimization was conducted by energy minimization of the model structures with different volumes. The calculated energy–volume curves were fitted using the Birch–Murnaghan equation of state with five parameters:

EðVÞ ¼ a þ bV

2=3

þ cV 4=3 þ dV

2

þ eV 8=3

ð2Þ

Fig. 2. Lattice parameter variations of Zr(C1xNx) compounds with increasing nitrogen content (x).

Fig. 1. SQS models of Zr(C0.75N0.25) and Zr(C0.5N0.5) solid solution compounds. Zr(C0.25N0.75) is described by swapping the C and N positions of Zr(C0.75N0.25). (a) Zr(C0.75N0.25) and (b) Zr(C0.5N0.5).

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Table 1 Elastic properties of Zr(C1xNx) solid solution compounds from the stress–strain relationship and the energy–volume (E–V) fitting of the ordered models: parentheses indicate the E–V fitting value. x

C11

C12

C44

Poisson’s ratio

Bulk modulus

Shear modulus

Young’s modulus

GGA-PBE 0.00 0.25 0.50 0.75 1.00

452.4 476.3 499.4 503.5 512.8

104.7 104.6 104.3 109.6 113.5

149.8 160.0 156.5 127.6 97.6

0.210 0.202 0.207 0.240 0.275

220.6(216.3) 228.4(222.2) 235.3(229.5) 241.2(234.6) 246.6(241.5)

159.0(156.0) 169.9(165.2) 171.1(166.8) 152.0(147.8) 130.5(127.8)

384.6(377.2) 408.4(397.2) 413.1(402.8) 376.8(366.4) 332.7(325.8)

GGA-PBEsol 0.00 0.25 0.50 0.75 1.00

487.2 516.0 542.4 552.0 567.9

106.9 106.2 106.3 112.1 116.0

152.0 163.6 157.1 131.8 112.4

0.212 0.204 0.211 0.241 0.264

233.6(229.0) 242.8(241.3) 251.8(245.6) 258.7(252.6) 266.7(266.6)

166.3(163.1) 179.0(178.0) 180.4(175.9) 162.0(158.1) 149.3(149.2)

403.3(395.3) 431.1(428.5) 436.8(426.0) 402.0(392.5) 377.2(377.2)

LDA 0.00 0.25 0.50 0.75 1.00

510.6 541.3 567.0 583.2 601.0

111.0 109.5 112.3 116.2 120.3

155.7 167.0 159.3 135.3 115.4

0.215 0.206 0.215 0.243 0.266

244.2(233.0) 253.4(249.1) 263.5(258.0) 271.8(259.0) 280.5(274.6)

172.0(164.3) 184.8(181.7) 185.1(181.3) 168.6(160.6) 155.5(152.1)

418.0(399.1) 446.1(438.5) 449.8(440.6) 419.1(399.4) 393.5(385.2)

207a 240b

162a 154c

386a 376c

Experimental 0.00 (ZrC) 1.00 (ZrN) x: N content in Zr(C1xNx) compounds. a Ref. [42]. b Ref. [9]. c Ref. [43].

Fig. 3. Changes in elastic properties with respect to nitrogen content (x). (a) Bulk, (b) shear, (c) Young’s moduli, and (d) Poisson’s ratio.

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Fig. 4. Electronic band structure and density of states of ZrC and ZrN. (a) ZrC and (b) ZrN. (Symmetry: W (0.50 0.25 0.75), L (0.50 0.50 0.50), g (0.00 0.00 0.00), X (0.50 0.00 0.50), K (0.375 0.375 0.375)).

Fig. 5. ZrC electronic band structure changes by applying shear strain, exy = eyz = +0.1. (a) Strain-free and (b) strained model. (Symmetry: F (0.00 0.50 0.00), g (0.00 0.00 0.00), B (0.50 0.00 0.00), G (0.00 0.00 0.50)).

S44 ¼ 1=C 44 :

ð8Þ

To obtain the elastic properties of the disordered compounds, we calculated 36 elastic constants (Cij:i and j = 1 to 6) and 21 of the elastic constants were used to calculate the elastic properties because Cij = Cji. Details of the methods used to calculate the elastic properties of the disordered compounds (triclinic) have been published previously [35,36]. The electronic structures were calculated using single-point energy calculations of the optimized models for the Zr(C1xNx) compounds. The other conditions for the calculations are identical to previous settings.

3. Results and discussion 3.1. Structural optimization and elastic properties of Zr(C1xNx) compounds The equilibrium lattice parameters of Zr(C1xNx) solid solution compounds are obtained from the energy–volume curves. Fig. 2 shows the lattice parameter variations of the solid solutions with

792

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with increasing nitrogen content. However, the maxima for the shear and Young’s moduli occur between 0.25 and 0.5. A bulk modulus implies a resistance against external pressure that is strongly affected by electron density [8]. However, the shear and Young’s modulus are related to the bonding characteristics between the metal-3d and nonmetal-2p orbitals [44]. Therefore, we can explain the variation in Poisson’s values, because a lower Poisson’s value implies a higher degree of directional bonding between constituent elements. Details of the relationship between the moduli and the bonding character will be discussed later; we only used the results from the GGA-PBEsol exchange correlation functional in the residual sections, because it yielded the most accurate results in the structural optimization.

3.2. Effect of atomic configuration on elastic properties of Zr(C1xNx) compounds Fig. 6. Total density of states changes of Zr(C1xNx) compounds with respect to nitrogen content and atomic configuration.

To examine the effect of the atomic configuration on the elastic properties of the compounds and to establish the reliability of the disordered models, we compared the variations in the elastic properties of the compounds with ordered and disordered structures in Fig. 3. The ordered models show slightly higher moduli regardless of the nitrogen content of the compounds. However, the differences are negligible and the models show identical trends in changes in the elastic properties, regardless of the atomic configuration. Compared with the experimental results [9,42,43], the calculated values agree well in terms of the variation in trends and the absolute values. This indicates that the atomic configuration of the compounds has a smaller influence on the elastic properties. Therefore, the disordered structures implemented here are suitable models for estimating the elastic properties of the compounds, although their shapes do not seem to be those of B1 structures.

3.3. Electronic properties of Zr(C1xNx) compounds Fig. 7. Variation in Fermi level energy of Zr(C1xNx) compounds with respect to nitrogen content and atomic configuration.

increasing nitrogen content. The calculated lattice parameters agree well with the experimental results for ZrC and ZrN and the deviation is within 1% [37–40]. The results indicate that the GGA-PBE and LDA yield over- and underestimated lattice parameters, respectively. GGA-PBEsol shows intermediate values between the results obtained from the GGA-PBE and LDA. This agrees with previously reported calculation results for solid phases [7,8,41]. A comparison with the experimental lattice parameters for ZrC and ZrN [38,40,42,43] shows that the GGA-PBEsol yields the most accurate values of the three exchange correlation functionals. The values of the equilibrium lattice parameters of the disordered compounds in Fig. 2 are slightly higher than those of the ordered compounds. However, the discrepancy is small, which indicates that the effect of the atomic configuration of the C and N atoms in the B1 lattice of the Zr(C1xNx) solid solution compounds is a minor factor in determining the lattice parameters. Table 1 shows the elastic constants and modulus of the Zr(C1xNx) solid solution compounds using the stress–strain relationship and the energy–volume (E–V) fitting, respectively [9,42,43]. The fitting method yields slightly lower values than the stress–strain relationship. The elastic moduli increase in the order of GGA < GGA-PBEsol < LDA. This trend is also apparent in the variation of lattice parameters (Fig. 2). The bulk moduli increase continuously

Fig. 4 shows the electronic band structures and density of states of ZrC and ZrN. The four lowest bands in ZrC and ZrN are contributed to by the nonmetal (C and N) 1s. The bands located above 6 eV are formed by Zr-3d orbitals. The bands around 3 eV originate mainly from hybridization of nonmetal-2p and Zr-3d orbitals. These bands contribute mainly to strong directional bonding in the compounds. To clarify this observation, we applied a shear strain (exy) to ZrC. Fig. 5 shows the change in band structure around the Fermi level of the ZrC caused by shear strain of exy = eyz = +0.1. The result shows a remarkable change in bands I and II (solid lines in Fig. 6) along the g (0.0 0.0 0.0)–B (0.5 0.0 0.0)–G (0.0 0.0 0.5) symmetry. Bands I and II are composed mainly from the C-2p and Zr-3d hybridization with a small contribution from the Zr-3p orbital. That is, the hybridization of C-2p and Zr-3d orbitals is the dominant contribution to the resistance against shear deformation. The Fermi level of ZrC is located in the pseudo-band gap around 5.4 eV and the Fermi level of ZrN is 7.48 eV (Fig. 4). This difference occurs because the number of valence electrons in C and N is 8 and 9, respectively. The difference in Fermi levels indicates that an initial increase in nitrogen content enforces strong directional bonding by filling bands I and II, which consist of the Zr-3d and C-2p orbitals around the pseudo-band gap. However, an increase in nitrogen content in the Zr(C1xNx) compounds results in Zr-3d and Zr-3d metallic bonding. The creation of metallic bonding reduces the directional character of the Zr(C1xNx) compounds. This explains the variations in shear and Young’s moduli with respect to the nitrogen content in the compounds, as mentioned above.

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3.4. Effects of atomic configuration on the electronic structure of Zr(C1xNx) compounds The effects of atomic configuration on the electronic structure are shown in Fig. 6. The overall peak shapes are similar in the ordered and disordered models. The energies of the Fermi levels increase gradually with increasing nitrogen content. However, the Fermi level positions differ depending on the compound model structure. The ordered models show higher Fermi levels than those of the disordered models. To explain the effects of atomic configuration on the electronic properties, we plot the variation in Fermi level energies in Fig. 7. The ordered and disordered models show slightly higher and lower Fermi level energies compared with the predicted values from the lever rule. The energy states of the electrons in the compounds vary depending on the atomic configuration of the model structures, and the random distribution of C and N in the compounds produces lower energetic states for the electrons. The random atomic distribution of C and N in the Zr(C1xNx) compounds is favorable from an electronic energy point of view. 4. Conclusion The elastic properties and electronic structure of ordered and disordered Zr(C1xNx) solid solution compounds were investigated via a first principles calculation method. Three types of exchange correlation functionals, namely LDA, GGA-PBE, and GGA-PBEsol, were used to obtain accurate calculation values for the Zr(C1xNx) compounds. Compared with the experimental results, the GGAPBEsol functional yielded the most accurate values for ZrC and ZrN. The elastic properties of the Zr(C1xNx) compounds varied with nitrogen content and the variation was interpreted using the electronic band structures. The effects of atomic configuration on the elastic properties and electronic structure were also evaluated using ordered and disordered models. The SQS models can be used to determine the disordered B1 structure of the Zr(C1xNx) compounds. The atomic configuration is a minor factor affecting the elastic properties. However, it is important to obtain the electronic structure of the Zr(C1xNx) compounds. Therefore, the atomic configuration must be considered to examine the accurate electronic structure of the Zr(C1xNx) compounds. Acknowledgements This work was supported by the Basic Research Project of the Korea Institute of Geoscience and Mineral Resources (KIGAM) funded by the Ministry of Knowledge Economy. References [1] R.P. Stoffel, C. Wessel, M.W. Lumey, R. Dronskowski, Ab initio thermochemistry of solid-state materials, Angew. Chem. Int. Ed. 49 (2010) 5242–5266. [2] X.-G. Lu, M. Selleby, B. Sundman, Theoretical modeling of molar volume and thermal expansion, Acta Mater. 53 (2005) 2259–2272. [3] M. Blanco, E. Francisco, V. Luana, GIBBS: isothermal–isobaric thermodynamics of solids from energy curves using a quasi-harmonic Debye model, Comput. Phys. Commun. 158 (2004) 57–72. [4] D. Cheng, S. Wang, H. Ye, First-principles calculations of the elastic properties of ZrC and ZrN, J. Alloys Comp. 377 (2004) 221–224. [5] T. Van Mourik, First-principles quantum chemistry in the life sciences, Philos. Trans. R. Soc., A 362 (2004) 2653–2670. [6] M.J. Louwerse, P. Vassilev, E.J. Baerends, Oxidation of methanol by FeO2+ in water: DFT calculations in the gas phase and ab initio MD simulations in water solution, J. Phys. Chem. A 112 (2008) 1000–1012. [7] J. Kim, S. Kang, Elastic and thermo-physical properties of TiC, TiN, and their intermediate composition alloys using ab initio calculations, J. Alloys Comp. 528 (2012) 20–27. [8] J. Kim, S. Kang, First principles investigation of temperature and pressure dependent elastic properties of ZrC and ZrN using Debye–Gruneisen theory, J. Alloys Comp. 540 (2012) 94–99.

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