Sc、Ti、V、Zr掺杂Cr<sub>2</sub>B<sub>3</sub>高压力学性质的第一性原理计算

张畅; 孙小伟; 宋婷; 田俊红; 刘子江

doi:10.11858/gywlxb.20210916

Sc、Ti、V、Zr掺杂Cr₂B₃高压力学性质的第一性原理计算

doi: 10.11858/gywlxb.20210916

兰州交通大学数理学院, 甘肃兰州　730070

基金项目: 甘肃省高等学校创新基金（2020A-039）；甘肃省自然科学基金重点项目（20JR5RA427）；甘肃省杰出青年科学基金（145RJDA323）

详细信息

作者简介:
张　畅（1996-），女，硕士研究生，主要从事功能新材料高压物性研究.E-mail：zhangch_lzjtu@126.com

通讯作者:
孙小伟（1979-），男，博士，教授，主要从事材料高压物性、声子晶体和声学超材料研究.E-mail：sunxw_lzjtu@yeah.net

中图分类号: O521.2; O482.1
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出版历程
- 收稿日期: 2021-12-10
- 修回日期: 2022-02-06
- 录用日期: 2022-02-06
- 网络出版日期: 2022-06-30
- 刊出日期: 2022-07-28

First-Principles Study on Mechanical Properties of Sc, Ti, V, Zr-Doped Cr₂B₃ at High Pressure

School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China

摘要

摘要: 采用基于密度泛函理论的第一性原理方法，计算了掺杂Sc、Ti、V和Zr的Cr₂B₃在零压下的晶体结构和电子结构及其在0～150 GPa压力范围内的弹性常数和维氏硬度。结果表明：Cr₂B₃及其掺杂化合物均具备力学稳定性；在零压下，添加Sc、Ti、V和Zr元素均可提高Cr₂B₃的维氏硬度，其中Ti掺杂Cr₂B₃的硬度由26.3 GPa提高至40.2 GPa，提高52.9%，达到超硬材料标准，且 Ti和V掺杂Cr₂B₃的剪切模量分别提高14.3%和16.2%，杨氏模量分别提高8.2%和12.0%；由电子结构分析可知，Sc、Ti、V和 Zr元素可以加强B与B之间的电子局域化程度，从而增强共价键结合强度，使Cr₂B₃的硬度升高；Cr₂B₃的弹性常数、体积弹性模量、剪切模量、杨氏模量以及硬度随着压力的增加而增加，但其硬度仍较低，150 GPa下仅为28.3 GPa，而掺杂V的Cr₂B₃的硬度在整个压力范围内约为37 GPa。研究结果可为Cr₂B₃在高压等特殊条件下的应用提供理论参考。
- Cr₂B₃ /
- 掺杂 /
- 弹性常数 /
- 硬度 /
- 高压
Abstract: By using first-principles method based on density functional theory, the crystal structures and the electronic structures of Sc, Ti, V and Zr element doped Cr₂B₃ under zero pressure, and their elastic constants and hardness in the pressure range of 0−150 GPa are calculated. According to the calculation, Cr₂B₃ and its doped compounds all meet the mechanical stability. At zero pressure, the Vicker’s hardness of Cr₂B₃ can be improved by adding Sc, Ti, V and Zr elements, in which the hardness of Ti-doped Cr₂B₃ is increased from 26.3 GPa to 40.2 GPa, increasing by 52.9%, reaching the standard of superhard materials, and the shear moduli of Ti and V doped Cr₂B₃ increase by 14.3% and 16.2%, respectively, and the Young’s moduli of the Ti and V doped Cr₂B₃ increase by 8.2% and 12.0%, respectively. The analysis of electronic structures shows that Sc, Ti, V and Zr can enhance the degree of electronic localization between B and B, thus enhance the covalent bonding strength and increase the hardness of Cr₂B₃. In addition, the elastic constants, the bulk elastic modulus, the shear modulus, the Young’s modulus and the hardness of Cr₂B₃ increase with pressure augmentation, but even so, the hardness is still low, merely 28.3 GPa at 150 GPa, while the hardness of the V element doped Cr₂B₃ is nearly constant (about 37 GPa) in the pressure range of 0−150 GPa. This work provides a theoretical reference for an extending application of Cr₂B₃ in special conditions, such as high pressure.
- Cr₂B₃ /
- doping /
- elastic constants /
- hardness /
- high pressure

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图 1 (a) Cr₂B₃ 和 (b) CrMB₃（M = Sc, Ti, V, Zr）的晶体结构（蓝色、绿色和粉色小球分别代表 Cr、B 和过渡金属原子）

Figure 1. Crystal structures of (a) Cr₂B₃ and (b) CrMB₃ (M=Sc, Ti, V, Zr), where the blue, green and pink spheres represent Cr, B and transition metal atoms, respectively

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图 2 Cr₂B₃的弹性常数随压力的变化

Figure 2. Pressure dependence of the elastic constants for Cr₂B₃

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图 3 CrMB₃（M=Sc, Ti, V, Zr）的弹性常数随压力的变化

Figure 3. Pressure dependence of the elastic constants for CrMB₃ (M=Sc, Ti, V, Zr)

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图 4 Cr₂B₃在零压(a)和 150 GPa (b)下的声子色散曲线

Figure 4. Phonon-dispersion curves of Cr₂B₃ at (a) 0 GPa and (b) 150 GPa

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图 5 (a) CrScB₃、(b) CrTiB₃、(c) CrVB₃、(d) CrZrB₃在零压下的声子色散曲线

Figure 5. Phonon-dispersion curves of (a) CrScB₃, (b) CrTiB₃, (c) CrVB₃ and (d) CrZrB₃ at 0 GPa

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图 6 Cr₂B₃和 CrMB₃（M=Sc, Ti, V, Zr）的体积弹性模量(a)、剪切模量(b)和杨氏模量(c)随压力的变化

Figure 6. Pressure dependence of (a) the bulk moduli, (b) the shear moduli, (c) the Young’s moduli for Cr₂B₃ and CrMB₃ (M=Sc, Ti, V, Zr)

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图 7 Cr₂B₃和 CrMB₃（M=Sc, Ti, V, Zr）的维氏硬度随压力的变化

Figure 7. Pressure dependence of the Vicker’s hardness for Cr₂B₃ and CrMB₃ (M=Sc, Ti, V, Zr)

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图 8 Cr₂B₃及 CrMB₃（M=Sc, Ti, V, Zr）的B/G 随压力的变化

Figure 8. Pressure dependence of the B/G for Cr₂B₃ and CrMB₃ (M=Sc, Ti, V, Zr)

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图 9 零压下 Cr₂B₃的体积弹性模量的三维表示(a)及其在 xy、xz 和 yz 面上的投影(b)

Figure 9. (a) 3D representation and (b) 2D projections on xy, xz and yz planes of the bulk modulus for Cr₂B₃ at 0 GPa

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图 10 零压下 Cr₂B₃的杨氏模量的三维表示(a)及其在 xy、xz 和 yz 面上的投影(b)

Figure 10. (a) 3D representation and (b) 2D projections on xy, xz and yz planes of the Young’s modulus for Cr₂B₃ at 0 GPa

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图 11 零压下 CrMB₃（M=Sc, Ti, V, Zr）的体积弹性模量在 xy、xz 和 yz 面的投影

Figure 11. 2D projections of the bulk modulus for CrMB₃ (M=Sc, Ti, V, Zr) on xy, xz and yz planes at 0 GPa

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图 12 零压下 CrMB₃（M=Sc, Ti, V, Zr）的杨氏模量在 xy，xz 和 yz 面的投影

Figure 12. 2D projections of the Young’s modulus for CrMB₃ (M=Sc, Ti, V, Zr) on xy, xz and yz planes at 0 GPa

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图 13 零压下 Cr₂B₃的总态密度以及分波态密度

Figure 13. Total density of states and partial density of states for Cr₂B₃ at 0 GPa

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图 14 零压下 CrMB₃（M=Sc, Ti, V, Zr）的总态密度以及分波态密度

Figure 14. Total density of states and partial density of states for CrMB₃ (M=Sc, Ti, V, Zr) at 0 GPa

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图 15 零压下(a) Cr₂B₃、(b) CrScB₃、(c) CrTiB₃、(d) CrVB₃、(e) CrZrB₃在(100)平面的 ELF 以及零压下(f) Cr₂B₃、(g) CrScB₃、(h) CrTiB₃、(i) CrVB₃和(j) CrZrB₃在( $00 \overline 1$ )平面的 ELF

Figure 15. Electronic local functions contours for (a) Cr₂B₃, (b) CrScB₃, (c) CrTiB₃, (d) CrVB₃ and (e) CrZrB₃ in plane (100) at 0 GPa, and electronic local functions contours for (f) Cr₂B₃, (g) CrScB₃, (h) CrTiB₃, (i) CrVB₃ and (j) CrZrB₃ in plane ( $00 \overline 1$ ) at 0 GPa

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表 1 零压下Cr₂B₃及掺杂结构CrMB₃（M=Sc, Ti, V, Zr）的晶格常数、形成焓及掺杂结构的形成能

Table 1. Lattice constants, formation enthalpy of Cr₂B₃ and CrMB₃ (M = Sc, Ti, V, Zr), and impurity formation energy ofCrMB₃ (M=Sc, Ti, V, Zr) at zero pressure

Compounds	Doping-site position	Space group	Lattice constants			E_f/eV	ΔH/(eV∙atom⁻¹)
Compounds	Doping-site position	Space group	a/Å	b/Å	c/Å	E_f/eV	ΔH/(eV∙atom⁻¹)
Cr₂B₃		Cmcm	2.8983	18.0464	2.9286		−0.4731
CrScB₃	Cr1	Cmcm	3.2207	18.5222	3.0293	−0.9713	−0.6653
CrScB₃	Cr2
CrTiB₃	Cr1	Cmcm	3.0648	18.1669	2.9746	−1.9192	−0.8561
CrTiB₃	Cr2	Cmcm	3.0541	18.5900	2.9883	−0.6876	−0.6118
CrVB₃	Cr1	Cmcm	2.9546	18.0874	2.9436	−1.1856	−0.7092
CrVB₃	Cr2	Cmcm	2.9547	18.2254	2.9489	−0.6797	−0.6101
CrZrB₃	Cr1	Cmcm	3.2972	18.5872	3.0702	−1.2236	−0.7181
CrZrB₃	Cr2	Cmcm	3.1763	20.0136	3.0524	0.1705	−0.4424

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表 2 Cr₂B₃中B―B键的键长以及布居数随压力的变化

Table 2. Pressure dependence of B―B bond length and population for Cr₂B₃

Pressure/GPa	Bond length/Å			Population
Pressure/GPa	B1―B1	B1―B2	B2―B3	B1―B1	B1―B2	B2―B3
0	1.69586	1.71971	1.75978	1.59	0.63	1.36
25	1.66181	1.68355	1.72369	1.62	0.64	1.40
50	1.63474	1.65559	1.69462	1.65	0.65	1.44
75	1.61237	1.63214	1.67012	1.68	0.66	1.47
100	1.59298	1.61225	1.64931	1.70	0.67	1.50
125	1.57589	1.59506	1.63098	1.73	0.68	1.53
150	1.56058	1.57950	1.61442	1.75	0.69	1.56

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