First-principles calculations of stabilities and physical properties of ternary niobium borocarbides and tantalum borocarbides

Hu Qian-Ku; Qin Shuang-Hong; Wu Qing-Hua; Li Dan-Dan; Zhang Bin; Yuan Wen-Feng; Wang Li-Bo; Zhou Ai-Guo

doi:10.7498/aps.69.20200234

Abstract

Transition-metal light-element compounds are potential candidates for hard materials. In the past, most of studies focused on the binary transition metal borides, carbides and nitrides, while the researches of ternary phases are relatively rare. In this paper, the structure units of the known Nb₃B₃C and Nb₄B₃C₂ phases are first analyzed to be Nb₆C octahedron and Nb₆B triangular prism, respectively. By stacking the Nb₆C octahedron and Nb₆B triangular prism, twenty ternary Nb-B-C and twenty ternary Ta-B-C configurations with different compositions are constructed. The chemical formula of these Nb-B-C and Ta-B-C configurations can be defined to be Nb_{(m + n + 2)}B_{(2m + 2)}C_n and Ta_{(m + n + 2)}B_{(2m + 2)}C_n, respectively. Using first-principles density functional calculations, thermodynamical, dynamical and mechanical stabilities of the constructed ternary Nb-B-C and Ta-B-C configurations are investigated through calculating their enthalpies of formation, phonon dispersions and elastic constants. Five Nb-B-C (Nb₃B₃C, Nb₄B₃C₂, Nb₆B₄C₃, Nb₇B₄C₄ and Nb₇B₆C₃) phases and six Ta-B-C (Ta₃B₃C, Ta₄B₃C₂, Ta₆B₄C₃, Ta₇B₄C₄, Ta₇B₆C₃ and Ta₃BC₂) phases are predicted to be stable by analyzing the constructed ternary Nb-B-C and Ta-B-C phase diagrams, in which the seven phases (Nb₆B₄C₃, Ta₃B₃C, Ta₄B₃C₂, Ta₆B₄C₃, Ta₇B₄C₄, Ta₇B₆C₃ and Ta₃BC₂) are first predicted to be stable. The Nb₆B₄C₃, Ta₆B₄C₃, Ta₄B₃C₂ and Ta₃B₃C phases are stable when temperature is higher than 1730, 210, 360 and 1100 K, respectively. And the Ta₃BC₂ phase is stable only when temperature is lower than 130 K. The calculated results about mechanical and electric properties show that these Nb-B-C and Ta-B-C phases are conductive materials with a high hardness in a range of 23.8–27.4 GPa.

Keywords:

Authors and contacts

Henan Key Laboratory of Materials on Deep-Earth Engineering, School of Materials Science and Engineering, Henan Polytechnic University, Jiaozuo 454000, China

Corresponding author: Zhou Ai-Guo, zhouag@hpu.edu.cn

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References

[1]	Tian Y J, Xu B, Zhao Z S 2012 Int. J. Refract. Met. Hard Mater. 33 93 Google Scholar
[2]	包括, 马帅领, 徐春红, 崔田 2017 66 036104 Google Scholar Bao K, Ma S L, Xu C H, Cui T 2017 Acta Phys. Sin. 66 036104 Google Scholar
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Cited By

图 1 (a), (b) Ta₃B₃C; (c) Ta₄B₃C₂; (d) Ta₃BC₂; (e) Ta₆B₄C₃; (f) Ta₇B₄C₄; (g) Ta₇B₆C₃的晶体结构. 棕球: Ta原子; 蓝球: B原子; 粉球: C原子. Ta₆B三棱柱和Ta₆C八面体分别用绿色和褐色表示

Figure 1. The crystal structures of (a), (b) Ta₃B₃C; (c) Ta₄B₃C₂; (d) Ta₃BC₂; (e) Ta₆B₄C₃; (f) Ta₇B₄C₄; (g) Ta₇B₆C₃. The light brown, blue and pink spheres represent Ta, B, and C atoms, respectively. The Ta₆B triangular prisms and Ta₆C octahedrons are painted green and dark brown.

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图 2 (a) Nb-B-C和(b) Ta-B-C三元相图. 红色, 稳定相; 蓝色, 亚稳相; 绿色, 不稳定相

Figure 2. Ternary phase diagrams of (a) Nb-B-C and (b) Ta-B-C. Red, stable; blue, metastable; green, unstable.

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图 3 不同温度下　(a) Nb-B-C和(b) Ta-B-C三元相分别和其相应最稳定竞争组合相的自由能之差

Figure 3. Energy differences of (a) Nb-B-C and (b) Ta-B-C ternary phases with respect to their most competing phases as a function of temperature.

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图 4 Nb-B-C和Ta-B-C三元相的声子色散曲线

Figure 4. Phonon dispersion curves of Nb-B-C and Ta-B-C ternary phases.

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图 5 Nb-B-C和Ta-B-C三元相的态密度图

Figure 5. Density of states of Nb-B-C and Ta-B-C ternary phases.

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表 1 不同成分Nb_{(m + n + 2)}B_{(2m + 2)}C_n和Ta_{(m + n + 2)}B_{(2m + 2)}C_n晶体的结构参数

Table 1. Structural parameters of Nb_{(m + n + 2)}B_{(2m + 2)}C_n and Ta_{(m + n + 2)}B_{(2m + 2)}C_n crystals.

m	n	空间群	模型	晶格参数/Å			模型	晶格参数/Å
m	n	空间群	模型	a	b	c	模型	a	b	c
0	1	Cmmm	Nb₃B₂C	3.254	13.808	3.141	Ta₃B₂C	3.240	13.697	3.127
0	2	Cmcm	Nb₂BC	3.235	18.330	3.153	Ta₂BC	3.220	18.165	3.140
0	3	Cmmm	Nb₅B₂C₃	3.225	22.903	3.153	Ta₅B₂C₃	3.199	22.659	3.138
0	4	Cmcm	Nb₃BC₂	3.214	27.376	3.156	Ta₃BC₂	3.198	27.132	3.150
1	1	Pmmm	Nb₄B₄C	3.290	18.994	3.145	Ta₄B₄C	3.277	18.878	3.127
1	2	Immm	Nb₅B₄C₂	3.267	23.600	3.150	Ta₅B₄C₂	3.248	23.377	3.138
1	3	Pmmm	Nb₆B₄C₃	3.243	28.028	3.154	Ta₆B₄C₃	3.225	27.872	3.141
1	4	Immm	Nb₇B₄C₄	3.242	32.545	3.158	Ta₇B₄C₄	3.224	32.315	3.147
2	1	Cmmm	Nb₅B₆C	3.302	24.414	3.134	Ta₅B₆C	3.289	24.208	3.122
2	2	Cmcm	Nb₃B₃C	3.284	28.877	3.144	Ta₃B₃C	3.267	28.688	3.133
2	3	Cmmm	Nb₇B₆C₃	3.264	33.364	3.148	Ta₇B₆C₃	3.246	33.164	3.136
2	4	Cmcm	Nb₄B₃C₂	3.257	37.874	3.153	Ta₄B₃C₂	3.243	37.609	3.141
3	1	Pmmm	Nb₆B₈C	3.309	14.889	3.137	Ta₆B₈C	3.298	14.788	3.122
3	2	Immm	Nb₇B₈C₂	3.290	34.247	3.144	Ta₇B₈C₂	3.276	34.007	3.131
3	3	Pmmm	Nb₈B₈C₃	3.276	19.350	3.148	Ta₈B₈C₃	3.258	19.235	3.135
3	4	Immm	Nb₉B₈C₄	3.268	43.255	3.151	Ta₉B₈C₄	3.252	42.977	3.138
4	1	Cmmm	Nb₇B₁₀C	3.312	35.192	3.131	Ta₇B₁₀C	3.299	34.990	3.116
4	2	Cmcm	Nb₄B₅C	3.296	39.694	3.139	Ta₄B₅C	3.280	39.441	3.125
4	3	Cmmm	Nb₉B₁₀C₃	3.281	44.206	3.142	Ta₉B₁₀C₃	3.263	43.924	3.130
4	4	Cmcm	Nb₅B₅C₂	3.273	48.729	3.145	Ta₅B₅C₂	3.257	48.400	3.134

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表 2 不同成分Nb-B-C相和Ta-B-C相的形成焓 (单位: eV/atom), $ \Delta{H}_{\rm{elements}} $表示单质为反应物, $ \Delta{H}_{\rm{comp}} $表示最稳定竞争组合为反应物

Table 2. Calculated formation enthalpies of different Nb-B-C and Ta-B-C phases (in eV/atom).$ \Delta{H}_{\rm{elements}} $ represents the elements as the reactants, and $\Delta{H}_{\rm{comp}}$ indicates the most stable composite as the reactants.

Phases	$ \Delta{H}_{\rm{elements}} $	$ \Delta{H}_{\rm{comp}} $	最稳定竞争组合	Phases	$ \Delta{H}_{\rm{elements}} $	$ \Delta{H}_{\rm{comp}} $	最稳定竞争组合
Nb₃B₂C	–0.620	0.070	Nb₃B₄ + 6NbB + Nb₆C₅ = 5Nb₃B₂C	Ta₃B₂C	–0.651	0.086	Ta₃BC₂ + 3TaB = 2Ta₃B₂C
Nb₂BC	–0.619	0.029	Nb₃B₄ + NbB + Nb₆C₅ = 5Nb₂BC	Ta₂BC	–0.664	0.035	Ta₃BC₂ + TaB = 2Ta₂BC
Nb₅B₂C₃	–0.586	0.036	3Nb₃B₄ + Nb₇B₄C₄ + 4Nb₆C₅ = 8Nb₅B₂C₃	Ta₅B₂C₃	–0.655	0.021	3Ta₃BC₂ + TaB = 2Ta₅B₂C₃
Nb₃BC₂	–0.586	0.019	Nb₃B₄ + 3Nb₇B₄C₄ + 4Nb₆C₅ = 16Nb₃BC₂	Ta₃BC₂	–0.660	–0.002	TaB + 2TaC = Ta₃BC₂
Nb₄B₄C	–0.679	0.030	3Nb₃B₄ + Nb₇B₄C₄ = 4Nb₄B₄C	Ta₄B₄C	–0.691	0.044	Ta₇B₄C₄ + 3Ta₃B₄ = 4Ta₄B₄C
Nb₅B₄C₂	–0.668	0.006	Nb₃B₄ + Nb₇B₄C₄ = 2Nb₅B₄C₂	Ta₅B₄C₂	–0.694	0.019	Ta₇B₄C₄ + Ta₃B₄ = 2Ta₅B₄C₂
Nb₆B₄C₃	–0.645	0.005	Nb₃B₄ + 3Nb₇B₄C₄ = 4Nb₆B₄C₃	Ta₆B₄C₃	–0.693	0.004	3Ta₇B₄C₄ + Ta₃B₄ = 4Ta₆B₄C₃
Nb₇B₄C₄	–0.632	–0.006	3Nb₃B₄ + 2C + 2Nb₆C₅ = 3Nb₇B₄C₄	Ta₇B₄C₄	–0.685	–0.017	3Ta₃B₄ + 4TaC = Ta₇B₄C₄
Nb₅B₆C	–0.697	0.015	3Nb₃B₄ + C + 2Nb₃B₃C = 3Nb₅B₆C	Ta₅B₆C	–0.697	0.024	C + Ta₅B₆ = Ta₅B₆C
Nb₃B₃C	–0.685	–0.001	3Nb₃B₄ + C + 3Nb₄B₃C₂ = 7Nb₃B₃C	Ta₃B₃C	–0.699	0.010	3Ta₇B₄C₄ + 9Ta₃B₄ + 4C = 16Ta₃B₃C
Nb₇B₆C₃	–0.664	0.0005	Nb₃B₃C + Nb₄B₃C₂ = Nb₇B₆C₃	Ta₇B₆C₃	–0.695	0.0008	5Ta₇B₄C₄ + 7Ta₃B₄ + 4C = 8Ta₇B₆C₃
Nb₄B₃C₂	–0.648	–0.001	5Nb₃B₄ + 4C + 7Nb₇B₄C₄ = 16Nb₄B₃C₂	Ta₄B₃C₂	–0.684	0.002	7Ta₇B₄C₄ + 5Ta₃B₄ + 4C = 16Ta₄B₃C₂
Nb₆B₈C	–0.695	0.019	2Nb₃B₄ + C = Nb₆B₈C	Ta₆B₈C	–0.685	0.034	2Ta₃B₄ + C = Ta₆B₈C
Nb₇B₈C₂	–0.683	0.008	3Nb₃B₄ + 2C + 4Nb₃B₃C = 3Nb₇B₈C₂	Ta₇B₈C₂	–0.686	0.020	Ta₇B₄C₄ + 7Ta₃B₄ + 4C = 4Ta₇B₈C₂
Nb₈B₈C₃	–0.665	0.008	C + 8Nb₃B₃C = 3Nb₈B₈C₃	Ta₈B₈C₃	–0.684	0.012	Ta₇B₄C₄ + 3Ta₃B₄ + 2C = 2Ta₈B₈C₃
Nb₉B₈C₄	–0.651	0.008	C + 5Nb₃B₃C + 3Nb₄B₃C₂ = 3Nb₉B₈C₄	Ta₉B₈C₄	–0.675	0.013	3Ta₇B₄C₄ + 5Ta₃B₄ + 4C = 4Ta₉B₈C₄
Nb₇B₁₀C	–0.693	0.021	C + 2Nb₂B₃ + Nb₃B₄ = Nb₇B₁₀C	Ta₇B₁₀C	–0.677	0.030	TaB₂ + 2Ta₃B₄ + C = Ta₇B₁₀C
Nb₄B₅C	–0.684	0.011	2C + Nb₃B₃C + 3Nb₃B₄ = 3Nb₄B₅C	Ta₄B₅C	–0.679	0.026	Ta₇B₄C₄ + 19Ta₃B₄ + 12C = 16Ta₄B₅C
Nb₉B₁₀C₃	–0.668	0.012	C + 2Nb₃B₃C + Nb₃B₄ = Nb₉B₁₀C₃	Ta₉B₁₀C₃	–0.677	0.019	3Ta₇B₄C₄ + 17Ta₃B₄ + 12C = 8Ta₉B₁₀C₃
Nb₅B₅C₂	–0.655	0.011	C + 5Nb₃B₃C = 3Nb₅B₅C₂	Ta₅B₅C₂	–0.670	0.018	5Ta₇B₄C₄ + 15Ta₃B₄ + 12C = 16Ta₅B₅C₂

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表 3 Nb-B-C和Ta-B-C三元相的弹性常数C_ij、体模量B、剪切模量 G和维氏硬度H_v (单位: GPa)

Table 3. Elastic constants C_ij, bulk modulus B, shear modulus G, Vickers hardness H_v of Nb-B-C and Ta-B-C ternary phases (in GPa).

结构	弹性常数									力学性能^a			硬度
结构	C₁₁	C₂₂	C₃₃	C₄₄	C₅₅	C₆₆	C₁₂	C₁₃	C₂₃	B	G	B/G	H_Chen	H_Tian
Nb₃B₃C	544.3	479.8	522.8	181.5	171.9	245.3	170.9	132.9	162.2	275.3	189.7	1.45	24.8	24.7
Nb₄B₃C₂	551.5	499.2	548.5	184.0	175.1	257.1	183.2	132.7	157.8	282.9	195.8	1.44	25.5	25.4
Nb₆B₄C₃	533.3	493.8	548.1	174.9	161.3	255.2	175.4	138.9	151.7	278.5	189.5	1.47	24.4	24.3
Nb₇B₄C₄	535.9	505.9	526.4	172.2	161.3	259.1	184.0	142.8	152.6	280.6	188.3	1.49	23.9	23.8
Nb₇B₆C₃	553.1	494.5	563.2	188.7	179.6	255.6	176.4	132.1	157.7	282.5	198.9	1.42	26.3	26.2
Ta₃B₃C	569.6	514.4	563.5	194.1	180.0	261.8	187.1	147.3	173.9	295.9	200.8	1.47	25.3	25.3
Ta₄B₃C₂	581.1	535.3	602.1	197.3	185.1	275.8	200.3	146.0	170.2	305.7	209.0	1.46	26.2	26.2
Ta₃BC₂	550.0	547.7	550.0	159.8	159.5	292.1	216.7	160.0	149.2	299.6	191.8	1.56	22.7	22.9
Ta₆B₄C₃	584.7	539.6	614.2	203.0	189.9	279.9	195.5	168.0	144.1	305.9	213.9	1.43	27.4	27.3
Ta₇B₄C₄	563.1	547.5	571.5	183.6	170.4	281.4	200.2	162.0	164.3	303.9	200.8	1.51	24.4	24.5
Ta₇B₆C₃	584.7	540.0	614.2	203.0	190.0	280.0	195.5	168.0	144.1	305.9	213.9	1.43	27.4	27.3
TaB₂										302	200	1.51	24.4	24.5
NbB₂										287	195	1.47	24.8	24.8
TaC										324	215	1.51	25.6	25.9
NbC										239	161	1.48	21.6	21.4
SiC										213	187	1.14	33.6	32.2
Al₂O₃										232	147	1.58	18.7	18.7
TiN										259	180	1.44	24.3	24.0
注: ^a二元相力学性能数据来自Materials Project网站.

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[1]	Tian Y J, Xu B, Zhao Z S 2012 Int. J. Refract. Met. Hard Mater. 33 93 Google Scholar
[2]	包括, 马帅领, 徐春红, 崔田 2017 66 036104 Google Scholar Bao K, Ma S L, Xu C H, Cui T 2017 Acta Phys. Sin. 66 036104 Google Scholar
[3]	Zhou X F, Sun J, Fan Y X, Chen J, Wang H T, Guo X J, He J L, Tian Y J 2007 Phys. Rev. B 76 100101 Google Scholar
[4]	Wu Q H, Hu Q K, Hou Y M, Wang H Y, Zhou A G, Wang L B 2018 J. Phys. Condens. Matter 30 385402 Google Scholar
[5]	Tian Y J, Xu B, Yu D L, Ma Y M, Wang Y B, Jiang Y B, Hu W T, Tang C C, Gao Y F, Luo K, Zhao Z S, Wang L M, Wen B, He J L, Liu Z Y 2013 Nature 493 385 Google Scholar
[6]	Huang Q, Yu D L, Xu B, Hu W T, Ma Y M, Wang Y B, Zhao Z S, Wen B, He J L, Liu Z Y, Tian Y J 2014 Nature 510 250 Google Scholar
[7]	徐波, 田永君 2017 66 036201 Google Scholar Xu B, Tian Y J 2017 Acta Phys. Sin. 66 036201 Google Scholar
[8]	Wu Q H, Hu Q K, Hou Y M, Wang H Y, Zhou A G, Wang L B, Cao G H 2018 Mater. Des. 140 45 Google Scholar
[9]	Cumberland R W, Weinberger M B, Gilman J J, Clark S M, Tolbert S H, Kaner R B 2005 J. Am. Chem. Soc. 127 7264 Google Scholar
[10]	Chung H Y, Weinberger M B, Levine J B, Kavner A, Yang J M, Tolbert S H, Kaner R B 2007 Science 316 436 Google Scholar
[11]	Gregoryanz E, Sanloup C, Somayazulu M, Badro J, Fiquet G, Mao H K, Hemley R J 2004 Nat. Mater. 3 294 Google Scholar
[12]	Young A F, Sanloup C, Gregoryanz E, Scandolo S, Hemley R J, Mao H K 2006 Phys. Rev. Lett. 96 155501 Google Scholar
[13]	Ivanovskii A L 2012 Prog. Mater. Sci. 57 184 Google Scholar
[14]	陶强, 马帅领, 崔田, 朱品文 2017 66 036103 Google Scholar Tao Q, Ma S L, Cui T, Zhu P W 2017 Acta Phys. Sin. 66 036103 Google Scholar
[15]	Hillebrecht H, Gebhardt K 2001 Angew. Chem. Int. Ed. 40 1445 Google Scholar
[16]	胡前库, 侯一鸣, 吴庆华, 秦双红, 王李波, 周爱国 2019 68 096201 Google Scholar Hu Q K, Hou Y M, Wu Q H, Qin S H, Wang L B, Zhou A G 2019 Acta Phys. Sin. 68 096201 Google Scholar
[17]	Wang P F, Weng M Y, Xiao Y, Hu Z X, Li Q H, Li M, Wang Y D, Chen X, Yang X N, Wen Y R, Yin Y X, Yu X Q, Xiao Y G, Zheng J X, Wan L J, Pan F, Guo Y G 2019 Adv. Mater. 31 1903483 Google Scholar
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Vol. 69, Issue 11 - 2020-06-05

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