First-principles study of vacancy ordered structures, mechanical properties and electronic properties of ternary Hf-C-N system

Peng Jun-Hui; Tikhonov Evgenii

doi:10.7498/aps.70.20210244

Abstract

The thermal-mechanical properties of transition metal carbonitrides can be affected by the concentration and ordering of vacancies besides the C/N atomic ratio. However, there are few reports on the vacancy ordered structure of ternary transition metal carbonitrides. In the present paper, the first-principles method is used to study the vacancy ordered structures, mechanical properties, electronic properties and the effect of vacancies on the ternary Hf-C-N system. Firstly, the crystal structures of Hf-C-N system is examined by the first-principles and evolutionary algorithms implemented in USPEX under ambient pressure, and eight thermodynamical stable vacancy ordered structures are found, each of which has a rock-salt structure, and is also dynamical and mechanical stable, which are verified by the calculations of their phonon dispersion curves and elastic constants. The vacancies are occupied at the [Hf₆] octahedral interstices, which replace the positions of non-metal atoms. Their crystallographic data such as space group, lattice constants are also predicted. To the best of our knowledge, there is no report on the Hf-C-N vacancy ordered structures and these structures investigated here in this work are all found for the first time. Then their mechanical properties are calculated. The Hf-C-N vacancy ordered structures all have very high bulk, shear and elastic modulus and hardness. It is found that except for C∶N = 1∶4, for the Hf-C-N system with the same C/N ratio the moduli, Vickers hardness values, and Pugh’s ratios decrease with the increase of the concentration of vacancy. However, the Vickers hardness of Hf₆CN₄ (the concentration of vacancy is equal to 1/6) is higher than that of Hf₅CN₄ (no vacancy), that is so-called vacancy hardening. Finally, the electronic density of states and the crystal orbital Hamilton populations are calculated. The chemical bonding of Hf-C-N vacancy ordered structure is analyzed, which is a mixture of covalence and metallic and is similar to that of binary transition metal carbides and nitrides. With the increase of the concentration of vacancy, the total bond strength decreases, and then the modulus decreases for Hf-C-N compound.

Keywords:

Authors and contacts

1.
International Center for Materials Discovery, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an 710072, China
2.
Department of Materials Engineering, Taiyuan Institute of Technology, Taiyuan 030008, China

Corresponding author: Tikhonov Evgenii, tikhonov.e@nwpu.edu.cn

Funds: Project supported by the Foreign Talents Introduction and Academic Exchange Program of China (Grant No. B08040)

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References

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图 1 (a) 常压下, 三元Hf-HfC-HfN体系的能量凸包图, 黑色球表示热力学稳定结构, 其他为亚稳结构; (b) Hf-C-N空位有序结构的X射线衍射模拟图谱, 衍射源为Cu Kα射线

Figure 1. (a) Enthalpy convex-hull of ternary Hf-HfC-HfN system at ambient pressure. The black sphere indicates stable structure, and others are metastable structure. (b) The simulated X-ray diffractions of Hf-C-N vacancy ordered structures with a copper Kα X-ray source.

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图 2 Hf-C-N空位有序结构在某一晶面上的空位分布　(a) Hf₆C₄N-$C2 $/m (0 0 1); (b) Hf₆C₃N-$C2 $ (1 0 0); (c) Hf₆C₃N₂-$C2 $/m (1 0 0); (d) Hf₃CN-$C2 $ (1 0 0); (e) Hf₆C₂N₃-$C2 $ (1 0 0); (f) Hf₄CN₂-Cmmm (0 0 1); (g) Hf₆CN₃-$C2 $/m (1 0 0); (h) Hf₆CN₄-$C2 $/m (0 0 1)

Figure 2. Vacancies on the crystallographic plane: (a) Hf₆C₄N-$C2 $/m (0 0 1); (b) Hf₆C₃N-$C2 $ (1 0 0); (c) Hf₆C₃N₂-$C2 $/m (1 0 0); (d) Hf₃CN-$C2 $ (1 0 0); (e) Hf₆C₂N₃-$C2 $ (1 0 0); (f) Hf₄CN₂-Cmmm (0 0 1); (g) Hf₆CN₃-$C2 $/m (1 0 0); (h) Hf₆CN₄-$C2 $/m (0 0 1).

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图 3 Hf-C-N空位有序结构的声子谱曲线　(a) Hf₆C₄N-$C2 $/m; (b) Hf₆C₃N-$C2 $; (c) Hf₆C₃N₂-$C2 $/m; (d) Hf₃CN-$C2 $; (e) Hf₆C₂N₃-$C2 $; (f) Hf₄CN₂-Cmmm; (g) Hf₆CN₃-$C2 $/m; (h) Hf₆CN₄-$C2 $/m

Figure 3. Phonon dispersion curves of (a) Hf₆C₄N-$C2 $/m, (b) Hf₆C₃N-$C2 $, (c) Hf₆C₃N₂-$C2 $/m, (d) Hf₃CN-$C2 $, (e) Hf₆C₂N₃-$C2 $, (f) Hf₄CN₂-Cmmm, (g) Hf₆CN₃-$C2 $/m, (h) Hf₆CN₄-$C2 $/m. They are all dynamical stable because no imaginary frequencies were found in Brillouin zone.

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图 4 三元Hf-HfC-HfN体系的力学性质-组分相图　(a) 体模量(B); (b) 剪切模量(G ); (c) 弹性模量(E ); (d) 维氏硬度(H_V); (e) Pugh比(G/B); (f) 泊松比(μ)

Figure 4. Mechanical properties-composition diagrams of ternary Hf-HfC-HfN system: (a) Bulk modulus (B); (b) shear modulus (G ); (c) elastic modulus (E ); (d) Vickers hardness (H_V); (e) Pugh’s ratio (G/B); (f) Poisson’s ratio (μ).

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图 5 (a) Hf₆C₄N-$C2 $/m, (b) Hf₆C₃N-$C2 $, (c) Hf₆C₃N₂-$C2 $/m, (d) Hf₃CN-$C2 $, (e) Hf₆C₂N₃-$C2 $, (f) Hf₄CN₂-Cmmm, (g) Hf₆CN₃-$C2 $/m和(h) Hf₆CN₄-$C2 $/m的态密度和分态密度; (i) Hf₃CN和Hf₂CN的总态密度对比; 其中Fermi能级位于0 eV

Figure 5. Density of state (DOS) and partial density of state (PDOS) normalized by per HfC_xN_y of (a) Hf₆C₄N-$C2 $/m, (b) Hf₆C₃N-$C2 $, (c) Hf₆C₃N₂-$C2 $/m, (d) Hf₃CN-$C2 $, (e) Hf₆C₂N₃-$C2 $, (f) Hf₄CN₂-Cmmm, (g) Hf₆CN₃-$C2 $/m and (h) Hf₆CN₄-$C2 $/m; (i) the total DOS of Hf₃CN and Hf₂CN normalized by per HfC_xN_y. The Fermi level is at 0 eV.

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图 6 Hf-C-N化合物的晶体轨道哈密顿分布(–COHP), Fermi能级位于0 eV

Figure 6. Crystal orbital Hamilton populations (–COHP) of Hf-C-N compounds. The Fermi level is at 0 eV.

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表 1 Hf-C-N空位有序结构的空间群、晶格常数、反应焓ΔH (eV/atom)、Hf原子的配位数(CN) 和空位浓度(CV)

Table 1. Space group, lattice constants, the enthalpy of reaction ΔH (eV/atom), coordination number (CN) of Hf and the concentration of vacancy (CV) of Hf-C-N vacancy ordered structures.

Compound	Space group	Lattice constants/Å	ΔH/(eV·atom^–1)	CN	CV
Hf₆C₄N	$C2 $/m	a = 5.679, b = 9.799, c = 5.671, β = 70.6^o	–0.0899	5	1/6
Hf₆C₃N	$C2 $	a = 5.658, b = 9.763, c = 9.262, β = 144.8^o	–0.0980	4	1/3
Hf₆C₃N₂	$C2 $m	a = 5.660, b = 9.783, c = 5.619, β = 109.6^o	–0.1038	5	1/6
Hf₃CN	$C2 $	a = 5.632, b = 9.705, c = 5.625, β = 109.8^o	–0.1107	4	1/3
Hf₆C₂N₃	$C2 $	a = 5.624, b = 9.725, c = 5.602, β = 109.6^o	–0.1047	5	1/6
Hf₄CN₂	Cmmm	a = 6.427, b = 9.147, c = 3.235	–0.1082	4/5	1/4
Hf₆CN₃	$C2 $/m	a = 5.592, b = 9.658, c = 6.455, β = 125.3^o	–0.0894	4	1/3
Hf₆CN₄	$C2 $/m	a = 5.580, b = 9.681, c = 5.587, β = 70.3^o	–0.0815	5	1/6

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表 2 Hf-C-N空位有序结构的弹性常数C_ij (单位: GPa)

Table 2. Calculated elastic constants C_ij (in GPa) of Hf-C-N vacancy ordered structures.

Compounds	C₁₁	C₂₂	C₃₃	C₄₄	C₅₅	C₆₆	C₁₂	C₁₃	C₂₃
Hf₆C₄N-$C2 $/m	414.3	406.6	415.6	158.0	170.6	148.7	94.1	116.1	104.5
Hf₆C₃N-$C2 $	358.5	362.8	352.2	100.0	114.3	132.3	87.5	98.3	91.6
Hf₆C₃N₂-$C2 $/m	414.6	417.4	407.6	152.2	157.6	147.8	111.9	115.0	116.3
Hf₃CN-$C2 $	354.5	363.5	348.7	90.6	103.6	128.7	102.1	109.5	101.3
Hf₆C₂N₃-$C2 $	409.7	418.7	418.1	149.6	160.2	148.9	123.4	122.9	126.5
Hf₄CN₂-Cmmm	373.4	368.8	406.8	142.2	133.1	135.8	146.4	112.0	124.4
Hf₆CN₃-$C2 $/m	361.1	358.4	351.7	84.9	99.8	124.9	108.1	121.7	114.2
Hf₆CN₄-$C2 $/m	401.2	414.1	403.5	146.5	157.2	139.8	134.0	139.9	147.8

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表 3 Hf-C-N空位有序结构和HfC_1–xN_x^[19]的力学性质—体模量(B)、剪切模量(G )、弹性模量(E )、泊松比(μ)、Pugh比(G/B)、维氏硬度(H_V)等

Table 3. Mechanical properties—bulk modulus (B), shear modulus (G ), elastic modulus (E ), Poisson’s ratio (μ), Pugh’s ratio (G/B), Vickers hardness (H_V) of Hf-C-N vacancy ordered structures and HfC_1–xN_x^[19].

Compound	B /GPa	G /GPa	E /GPa	μ	G/B	H_V /GPa
Hf₆C₄N	229.0	140.8	350.6	0.2449	0.6148	17.5
Hf₅C₄N^[19]	260.6	201.3	480.3	0.1928	0.7727	29.9
Hf₆C₃N	180.9	121.5	297.9	0.2256	0.6717	17.8
Hf₄C₃N^[19]	262.2	202.1	482.4	0.1934	0.7707	29.9
Hf₆C₃N₂	214.0	151.1	366.9	0.2143	0.7059	22.1
Hf₃CN	188.0	113.4	283.3	0.2489	0.6031	14.6
Hf₂CN^[19]	268.1	198.5	477.6	0.2031	0.7403	28.1
Hf₆C₂N₃	221.3	149.7	366.6	0.2239	0.6766	20.7
Hf₄CN₂	212.7	132.8	329.7	0.2417	0.6242	17.1
Hf₃CN₂^[19]	272.8	185.1	452.8	0.2233	0.6786	23.9
Hf₆CN₃	195.4	108.9	275.6	0.2650	0.5574	12.7
Hf₄CN₃^[19]	276.2	179.6	442.8	0.2328	0.6504	22.2
Hf₆CN₄	207.2	156.1	374.4	0.1989	0.7535	24.6
Hf₅CN₄^[19]	279.0	171.5	427.0	0.2449	0.6147	20.0

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表 4 Hf-C-N化合物的晶体轨道哈密顿分布的积分值(–ICOHP)

Table 4. Integrated crystal orbital Hamilton populations (–ICOHP) of Hf-C-N compounds.

Compound	–ICOHP			Compound	–ICOHP
Compound	Hf—C	Hf—N	Hf—Hf	Compound	Hf—C	Hf—N	Hf—Hf
Hf₆C₄N	3.373	3.567	0.529	Hf₆C₃N₂	3.181	2.990	0.571
Hf₅C₄N	3.373	3.033	0.459	Hf₄CN₂	3.474	3.111	0.650
Hf₆C₃N	3.350	3.067	0.718	Hf₃CN₂	3.551	3.091	0.541
Hf₄C₃N	3.319	3.029	0.454	Hf₆CN₃	3.408	3.211	0.570
Hf₆C₃N₂	3.607	3.103	0.530	Hf₄CN₃	3.321	3.159	0.520
Hf₃CN	3.277	3.211	0.737	Hf₆CN₄	3.675	3.179	0.591
Hf₂CN	3.483	2.802	0.490	Hf₅CN₄	3.319	3.017	0.500

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[1]	Squire T, Marschall J 2010 J. Eur. Ceram. Soc. 30 2239 Google Scholar
[2]	Opeka M M, Talmy I G, Zaykosk J A 2004 J. Mater. Sci. 39 5887 Google Scholar
[3]	Levine S R, Opila E J, Halbig M C, Kiser J D, Singh M, Salem J A 2002 J. Eur. Ceram. Soc. 22 2757 Google Scholar
[4]	Ushakov SV, Navrotsky A 2012 J. Am. Ceram. Soc. 95 1463 Google Scholar
[5]	Grill A, Aron P R 1983 Thin Solid Films 108 173 Google Scholar
[6]	Helmersson U, Todorova S, Barnett S A, Sundgren J E, Markert L C, Greene J E 1987 J. Appl. Phys. 62 481 Google Scholar
[7]	Mirkarimi P B, Hultman L, Barnett S A 1990 Appl. Phys. Lett. 57 2654 Google Scholar
[8]	Veprek S, Veprek-Heijman M G J, Karvankova P, Prochazka J 2005 Thin Solid Films 476 1 Google Scholar
[9]	Hultman L, Bareno J, Flink A, Soderberg H, Larsson K, Petrova V, Oden M, Greene J E, Petrov I 2007 Phys. Rev. B 75 155437 Google Scholar
[10]	Shin C S, Gall D, Hellgren N, Patscheider J, Petrov I, Greene J E 2003 J. Appl. Phys. 93 6025 Google Scholar
[11]	Jhi S H, Louie S G, Cohen M L, Ihm J 2001 Phys. Rev. Lett. 86 3348 Google Scholar
[12]	Shin C S, Rudenja S, Gall D, Hellgren N, Lee T Y, Petrov I, Greene J E 2004 J. Appl. Phys. 95 356 Google Scholar
[13]	Lee T, Ohmori K, Shin C S, Cahill D G, Petrov I, Greene J E 2005 Phys. Rev. B 71 144106 Google Scholar
[14]	Holleck H 1986 J. Vac. Sci. Technol., A 4 2661 Google Scholar
[15]	Yang Q, Lengauer W, Koch T, Scheerer M, Smid I 2000 J. Alloys Compd. 309 L5 Google Scholar
[16]	Jhi S H, Ihm J, Louie S G, Cohen M L 1999 Nature 399 132 Google Scholar
[17]	Feng W, Cui S, Hu H, Zhang G, Lü Z 2011 Physica B 406 3631 Google Scholar
[18]	Balasubramanian K, Khare S V, Gall D 2018 Acta Mater. 152 175 Google Scholar
[19]	Peng J, Tikhonov E 2021 Comput. Mater. Sci. 195 110464 Google Scholar
[20]	Gusev A I, Rempel A A, Magerl A J 2001 Disorder and Order in Strongly Nonstoichiometric Compounds (Berlin Heidelberg: Springer) pp179−243
[21]	Gusev A I 1991 Physical Chemistry of Non stoichiometric Refractory Compounds (Moscow: Nauka) (in Russian)
[22]	Rudy E 1965 Ternary Phase Equilibria in Transition Metal-boron-carbon-silicon Systems. Part II. Ternary Systems. Vol. I. Ta-Hf-C system (Air Force Materials Laboratory, Wright-Patterson Air Force Base) pp38−60
[23]	Lipatnikov V N, Lengauer W, Ettmayer P, Keil E, Groboth G, Kny E 1997 J. Alloys Compd. 261 192 Google Scholar
[24]	Yu X X, Thompson G B, Weinberger C R 2015 J. Eur. Ceram. Soc. 35 95 Google Scholar
[25]	Yu X X, Weinberger C R, Thompson G B 2016 Comput. Mater. Sci. 112 318 Google Scholar
[26]	Yu X X, Weinberger C R, Thompson G B 2014 Acta Mater. 80 341 Google Scholar
[27]	Xie C, Liu N, Cheng X, Li D, Zeng Q 2016 J. Eur. Ceram. Soc. 36 3593 Google Scholar
[28]	Xie C, Oganov A R, Li D, Debela T T, Liu N, Dong D, Zeng Q 2016 Phys. Chem. Chem. Phys. 18 12299 Google Scholar
[29]	Zhang Y, Liu B, Wang J 2016 Sci. Rep. 5 18098 Google Scholar
[30]	Gunda N S H, Van der Ven A 2018 Phys. Rev. Mater. 2 083602 Google Scholar
[31]	Connolly J W D, Williams A R 1983 Phy. Rev. B 27 5169 Google Scholar
[32]	Weinberger C R, Thompson G B 2018 J. Am. Ceram. Soc. 101 4401 Google Scholar
[33]	Gusev A I, Rempel A A 1994 J. Phys. Chem. Solids 55 299 Google Scholar
[34]	Yu S, Zeng Q, Oganov A R, Frapper G, Zhang L 2015 Phys. Chem. Chem. Phys. 17 11763 Google Scholar
[35]	Yu S, Zeng Q, Oganov A R, Frapper G, Huang B, Niu H, Zhang L 2017 RSC Adv. 7 4697 Google Scholar
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