Theoretical prediction of solution in ScxY1–x Fe2 and order-disorder transitions in V2x Fe2(1–x)Zr

Jiang Yong-Lin; He Chang-Chun; Yang Xiao-Bao

doi:10.7498/aps.70.20210998

Abstract

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1. 引　言
2. 计算方法
3. 结果与讨论
3.1 合金模型
3.2 ScxY1–x Fe2固溶曲线
3.3 V2x Fe2(1–x)Zr体系有序-无序转变
4. 结　论

References

Abstract

Alloying is an important way to increase the diversity of material structure and properties. In this paper, we start from Ising model considering nearest neighbor interaction, in which a ferromagnetic system corresponds to a low temperature phase separation and high temperature solid solution of binary alloy, while antiferromagnetic system corresponds to a low temperature ordered solid solution and a high temperature disorder. The high-throughput first-principles calculation based on the structure recognition is realized by the program SAGAR (structures of alloy generation and recognition) developed by our research group. By considering the contribution of structural degeneracy to the partition function, theoretical prediction of alloy materials can be carried out at finite temperature. Taking hydrogen storage alloy (Sc_xY_1–x Fe₂ and V_2x Fe_2(1–x)Zr) for example, the formation energy of ground state (at zero temperature) can be obtained by the first-principles calculations. It is found that the formation energy of Sc_xY_1–x Fe₂ is greater than zero, thereby inducing the phase separation at low temperature. The free energy will decrease with the temperature and concentration increasing, where the critical temperature of solid solution of alloy is determined according to the zero point of free energy. The formation energies of V_2x Fe_2(1–x)Zr are all lower than zero, and the ordered phase occurs at low temperature. The order-disorder transition temperature of V_0.5Fe_1.5Zr and V_1.5Fe_0.5Zr are both about 100 K, while the transition temperature of VFeZr is nearly 50 K. The calculation process will effectively improve the high throughput screening efficiency of alloy, and also provide relevant theoretical reference for experimental research.

Keywords:

PACS:

36.40.-c	(Atomic and molecular clusters)
31.15.A-	(Ab initio calculations)
71.15.Pd	(Molecular dynamics calculations (Car-Parrinello) and other numerical simulations)

Authors and contacts

Department of Physics, South China University of Technology, Guangzhou 510640, China

Corresponding author: Yang Xiao-Bao, scxbyang@scut.edu.cn

Funds: Project supported by the Key-Area Research and Development Program of Guangdong Province, China (Grant No. 2020B010183001) and the Guangdong Provincial Key Laboratory for Computational Science and Materials Design Program, China (Grant No. 2019B030301001)

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1. 引　言

材料的结构特性和宏观性能取决于材料的元素种类和原子分布^[1,2]. 借助基于密度泛函理论的第一原理计算方法, 可以在微观原子层面理解材料宏观性能, 对未知材料的合成开发做出合理的判断和预测. 对于周期性有序晶体结构或者呈过渡态形式的非晶态团簇结构, 结合全局优化算法和第一性原理计算的材料设计发展迅速, 成功预测到了各种具有新颖性质的结构, 并被后续实验确认^[3]. 采用群体智能算法的CALYPSO^[4]和USPEX^[5]被广泛应用于稳定结构的全局优化, 尤其在高压条件下的结构演化. 另外, 随机搜索从头算方法( $ab \; initio$ random structure searching algorithm)^[6]在材料预测中发现了一系列的新颖结构.

合金材料, 从一般的二、三元合金到现阶段的多元高熵合金, 如根据结构特性和功能属性划分的Heusler合金、Laves相合金、磁性合金和高温合金等, 由于在性能上具备多种优良特性, 引起了实验和理论工作者的广泛关注^[7-10]. 为了描述合金材料结构和稳定性的关联, 通常可以采用集团展开方法^[11]构造体系的模型哈密顿量, 通过参数拟合来提高计算速度并保持较高的精度, 而相互作用参数的选择对预测未知的基态结构有重要影响^[12]. 在完成总能拟合后, 通过蒙特卡罗模拟可以对不同浓度、温度的结构演化进行系统研究^[13]. 在高温极限下, 无序固溶结构可以通过特征准随机结构(special quasi-random structure, SQS)方法^[14]构造, 考虑混合熵的贡献, 可以计算体系自由能随温度的变化^[15].

在储氢合金中, 通过对组分的调节可以调控合金和氢的相互作用, 设计理想的储氢材料. 其中, ZrFe₂和YFe₂合金是典型的Laves相结构, 在过渡族储氢合金中是研究较多的AB₂型体系, 可以通过变换A, B位置的元素来调节合金的氢结合和脱吸附能力, 也可以通过引入异类原子的方式实现对其储氢性能的改善. 例如文献[16]构造了不同组分比的ZrFe_2–xV_x ( $x = 0.2,\; 0.4,\; 0.6,\; 0.8$ )三元合金以探究其氢化能力, 讨论了Y_1–x Sc_x Fe₂准二元合金^[17]的磁性和结构转变, 以及评估了呈立方晶格的Y-M-Fe-Al (M = Zr, Ti, V)^[18]四元合金的储氢性能. 在合金形成过程中, 由于不同元素间相互作用较弱时, 体系可能是相分离结构, 随着温度升高, 构型熵的贡献导致自由能降低, 使得体系出现固溶结构; 而不同元素间相互作用较强时, 体系可能自发形成固溶的有序结构, 随着温度的升高, 体系出现有序-无序转变. 然而, 如何快速判断体系能否形成合金以及合金中的原子分布是否有序, 在大规模材料筛选中仍有一定的难度. 比如, 基于集团展开方法, 需要先对给定的材料进行第一原理计算和参数拟合, 然后采用蒙特卡罗方法模拟相变; 而特征准随机结构对有限温度的自由能计算会带来较大的误差.

目前, 高通量计算已成为加速材料发现的有用工具, 在光伏太阳能吸收材料、光电化学电池、半导体发光二极管、透明导电材料等材料设计中取得了重要的进展^[19]. 值得注意的是, 当组分浓度固定时, 具体的原子分布对材料性能也有重要的影响. 在我们之前的工作中, 通过结构识别^[20]可以得到给定组分下的可能不等价结构, 预测了具有优异性能的低维纳米结构^[21,22]. 本文通过基于结构识别的高通量第一原理计算, 考虑合金中构型熵的贡献, 研究体系自由能、比热随温度的变化, 实现快速判断合金是否存在固溶相: 对于低温可固溶的体系, 给出有序-无序转变温度随原子配比的变化; 对于低温相分离结构, 研究给定原子配比时, 获得固溶相的最低温度.

2. 计算方法

本研究所用Laves相原胞结构为三角晶型, 如图1(a)所示. 从原胞出发, 可以生成不同体积的超胞: 给定体积的超胞, 存在大量不同形状的情况; 给定体积、形状的超胞, 存在大量可能的结构. 我们课题组开发了相关程序SAGAR (structures of alloy generation and recognition)[网页版地址: http://sagar.compphys.cn/sagar], 可以得到给定体积下所有不等价超胞^[23], 根据结构编码方法^[24], 可以得到所有不等价结构以及对应简并度. 例如, 图1(b)和图1(c)是V_2x Fe_2(1–x)Zr的2倍和4倍超胞, 图1(e)—(k)是通过图1(d)的Sc_xY_1–x Fe₂原胞变成4倍超胞的所有可能形状. 对于给定的超胞, 可能的准二元合金结构数目如表1所列.

$图 1 取代含量为半数时的一种构型下, V2x Fe2(1–x)Zr的(a)原胞结构和体积扩大为原胞(b) 2倍、(c) 4倍时对应的晶格; ScxY1–x Fe2合金体系的(d)原胞结构和(e)—(f)体积扩大为原胞4倍时对应的7种晶格(红色、绿色、黄色、青色和紫色小球分别代表 V, Zr, Fe, Y和Sc原子)\r\nFig. 1. Lattices of V2x Fe2(1–x)Zr (a) primitive cell and corresponding to volumes expanded respectively from primitive cell by (b) 2 times, (c) 4 times under a half of replacement content. (d) Primitive cell structure of ScxY1–x Fe2 alloys system and (e)−(f) the 7 kinds of lattices with 4 times volume of that of primitive one (red, green, yellow cyan and purple sphere represent respectively V, Zr, Fe, Y and Sc atom).$

图 1 取代含量为半数时的一种构型下, V_2x Fe_2(1–x)Zr的(a)原胞结构和体积扩大为原胞(b) 2倍、(c) 4倍时对应的晶格; Sc_xY_1–x Fe₂合金体系的(d)原胞结构和(e)—(f)体积扩大为原胞4倍时对应的7种晶格(红色、绿色、黄色、青色和紫色小球分别代表 V, Zr, Fe, Y和Sc原子)

Fig. 1. Lattices of V_2x Fe_2(1–x)Zr (a) primitive cell and corresponding to volumes expanded respectively from primitive cell by (b) 2 times, (c) 4 times under a half of replacement content. (d) Primitive cell structure of Sc_xY_1–x Fe₂ alloys system and (e)−(f) the 7 kinds of lattices with 4 times volume of that of primitive one (red, green, yellow cyan and purple sphere represent respectively V, Zr, Fe, Y and Sc atom).

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表 1 利用SAGAR对图1所示两种合金在不同晶格下生成的所有不等价结构数目

Table 1. Numbers of all nonequivalent structures from different lattices created by using SAGAR for the two kinds of alloy system shown as Fig. 1

晶型	(a)	(b)	(c)	(d)	(e)	(f)	(g)	(h)	(i)	(j)	(k)
结构数	5	28	531	3	43	34	43	21	34	30	16

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| 显示表格

在获得可能的原子分布后, 体系总能的计算采用VASP (vienna $ab\;initio$ simulation package)^[25,26] 程序包完成. 交换关联函数采用广义梯度近似下Perdew-Burke- Ernzerhof泛函^[27,28]. 经测试, 平面波截断能和能量收敛阈值分别设置为450 eV和0.001 eV/Å, K空间网格划分采用Monkhorst-Pack方法自动生成, 并保证K点的采样间隔小于0.1 Å^–1.

对于型如A_x B_1–x M的准二元合金, 形成能(E )计算公式如下^[29]:

$E = E_{\rm t} - x_A\times E_{AM} - (1-x_{AM})\times E_{BM},$

(1)

式中 $E_{\rm t}$ 是合金体系总能, $x_A$ 是体系中A组元的取代含量, $E_{AM}$ 和 $E_{BM}$ 分别对应两端纯二元相总能.

根据形成能和结构简并度, 体系配分的函数可表示为

$Z = \sum \varOmega {\rm e}^{-\beta E},$

(2)

式中Ω是形成能为E的体系所对应的简并度, 参数 $\beta = 1/(kT),\;k = 1.38 \times 10^{-23} \; {\rm{J}} \cdot {\rm{K}}^{-1}$ 为玻尔兹曼常数, 自由能与配分函数的关系为^[30]

$F = -kT\log Z.$

(3)

体系总能是对应所有可能微观状态能量的平均:

$\bar{E} = \dfrac{\displaystyle\sum E {\rm e}^{-\beta E_{\rm s}}}{\displaystyle\sum {\rm e}^{-\beta E_{\rm s}}}.$

(4)

体系热容为

$C_v = \frac{\overline{(E-\overline{E})}}{kT^2}.$

(5)

3. 结果与讨论

先采用考虑最近邻相互作用的Ising模型讨论两类典型情况: 1)低温相分离, 高温固溶; 2)低温有序固溶, 高温无序. 通过高通量第一原理计算, 发现Sc_xY_1–x Fe₂属于第一类情况, 可以确定固溶温度随组分变化; V_2x Fe_2(1–x)Zr属于第二类情况, 有序-无序转变温度同样依赖体系组分.

3.1 合金模型

二元合金固溶的定性行为可以采用考虑最近邻原子相互作用能的Ising模型^[31,32]:

$E = -J\sum s_is_j,\; J = J_{AA}+J_{BB}-2J_{AB},$

(6)

式中 $s_i = \pm 1$ 分别表示A, B两类原子; $J_{AA}$ , $J_{BB}$ 和 $J_{AB}$ 分别为同种元素和不同元素直接的相互作用, $J$ 的正负值分别对应低温相分离和低温有序固溶情况.

选择包含16个格点的二维正方晶格, 通过结构遍历得到系统所有状态的能量和结构简并度, 可以计算体系的配分函数. 当 $J = 1$ 时, 所有A_x B_1–x $(0 < x < 1)$ 结构的能量都是大于零的, 在温度较低时, 体系自由能大于相分离的情况(相分离时 $F = 0$ ), 随着温度的升高, 给定组分体系的自由能将由正转负, 如图2(a)所示, 表明体系将过渡到较高温度下的固溶状态, 所处温度即为各组分互溶的临界温度; 随着浓度的增加, 临界温度先升高后降低, 如图2(b)所示. 当 $J = -1$ 时, 所有A_x B_1–x $(0 < x < 1)$ 结构的能量都小于零, 体系的固溶是放热反应; 图2(c)分别给出了模型计算的不同浓度体系热容随温度的变化, 其中比热峰的位置对应有序-无序转变的临界温度, 低温下体系处于有序稳定状态, 温度超过临界温度对应无序分布状态; 而取代浓度处于半数的时候对应最高的热容峰, 与实验上测试所得二元相图类似, 中部偏下区域对应有序稳定相区而外围则对应分散无序区域, 相变温度随浓度变化曲线如图2(d)所示.

$图 2 根据Ising模型拟合结果得出的体系　(a)自由能符号; (c)热容; (b), (d)相转变温度\r\nFig. 2. Systems obtained according to the fitting results of Ising model: (a) Free energy signal; (c) heat capacity; (b), (d) phase transition temperature.$

图 2 根据Ising模型拟合结果得出的体系　(a)自由能符号; (c)热容; (b), (d)相转变温度

Fig. 2. Systems obtained according to the fitting results of Ising model: (a) Free energy signal; (c) heat capacity; (b), (d) phase transition temperature.

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3.2 Sc_xY_1–x Fe₂固溶曲线

在实验研究中, 可以通过在YFe₂体系中掺入Sc得到准二元合金Sc_xY_1–x Fe₂. 通过SAGAR程序, 得到4倍体积下7个不等价超胞(如图1所示), 每个超胞分别有43, 34, 43, 30, 21, 34, 16 种不等价结构, 对应不同的原子分布. 对于Sc_xY_1–x Fe₂体系, 结构的形成能都是大于零的, 属于低温相分离而高温固溶体系, 随着温度升高, 构型熵对自由能的贡献增大, 实现晶格间原子的互溶. 在结构(k)的超胞中, 不同浓度体系的自由能符号随温度变化如图3(a)所示, 对应浓度为1/8的体系最先出现固溶, 而浓度为1/2的体系固溶临界温度最高. 如图3(b)所示, 在1/8浓度下不同形状的超胞自由能随温度升高均出现不同程度的下降, 其中超胞k最先达到0, 对应的临界温度在400 K附近; 在5/8浓度下, 超胞e最先达到0, 对应的临界温度在450 K附近 (如图3(c)所示). 考虑不同超胞的情况, 在同一浓度下选择最低的临界温度, 得到固溶曲线随浓度的变化, 如图3(d)所示. 体系自由能的降低来自构型熵的贡献, 而不同形状的超胞对应的结构多重度不同, 也将影响体系的固溶温度. 在周期性边界条件下, 特定形状的超胞无法包含能量较低的结构, 导致体系自由能升高而高估体系的固溶温度. 因此, 在同一浓度下选择可能的超胞中临界温度最低值是合理的.

$图 3 (a) ScxY1–x Fe2 体系立方晶格中不同取代浓度下随温度变化的自由能符号; (b) 1/8和(c) 5/8取代浓度下不同晶格结构随温度变化的自由能; (d) ScxY1–x Fe2体系温度-浓度相图\r\nFig. 3. (a) Free energy signal induced by temperature under different replacement concentration in ScxY1–x Fe2 cubic structure. Free energy versus temperature for different crystal structures at (b) 1/8 and (c) 5/8 replacement concentration, respectively. (d) Temperature-concentration phase diagram of ScxY1–x Fe2 alloy system$

图 3 (a) Sc_xY_1–x Fe₂ 体系立方晶格中不同取代浓度下随温度变化的自由能符号; (b) 1/8和(c) 5/8取代浓度下不同晶格结构随温度变化的自由能; (d) Sc_xY_1–x Fe₂体系温度-浓度相图

Fig. 3. (a) Free energy signal induced by temperature under different replacement concentration in Sc_xY_1–x Fe₂ cubic structure. Free energy versus temperature for different crystal structures at (b) 1/8 and (c) 5/8 replacement concentration, respectively. (d) Temperature-concentration phase diagram of Sc_xY_1–x Fe₂ alloy system

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3.3 V_2x Fe_2(1–x)Zr体系有序-无序转变

对于V_2x Fe_2(1–x)Zr合金体系, 结构的形成能都是小于零的, 在此基础上可进一步探讨该体系中有序-无序转变所依赖的温度和浓度关系, 以及在对应极值处热容与温度的关系. 以体积分别为原胞2倍和4倍的四方晶胞和立方惯胞为参考对象进行探讨, 分别对应28和531种不等价结构. 如图4(a)所示, 这两类超胞给出了相似的结果, 其中对应3个有序相分别是V_0.5Fe_1.5Zr, VFeZr, V_1.5Fe_0.5Zr. 本文考察了这些有序相附近的浓度, 可以看到: 浓度是1/4时, 比热峰比相邻浓度的要高出一倍; 浓度是1/2时, 比热峰比相邻浓度的要高出70%; 浓度是3/4时, 比热峰达到相邻浓度的4倍. 类比图2中模型计算的结果, 可以确认这3个有序相, 发现V_0.5Fe_1.5Zr和V_1.5Fe_0.5Zr的有序-无序转变温度在105.9 K和89.8 K附近, 而VFeZr的转变温度在57.5 K附近. 如果实验中材料制备的温度超过转变温度, 体系对应的原子分布将是无序的. 体系的有序-无序转变来自于形成能和构型熵的竞争. 在低温时, 自由能贡献主要来自于形成能, 因而有序相更稳定; 随着温度的升高, 构型熵对自由能的贡献逐渐变大, 结构多重度低的有序相将不再稳定, 体系进入无序分布. 当体系浓度为1/4, 1/2, 3/4时, 体系出现“完整”的有序相, 发生有序-无序转变时, 体系序参量变化最大, 对应的比热峰相对比较高; 当体系浓度不是1/4, 1/2, 3/4时, 体系无法对应“完整”的有序相, 体系初始的有序度不高, 相变对应的参量变化较小, 因而对应的比热峰将变低.

$图 4 (a) V2x Fe2(1–x)Zr体系的温度-浓度相图(图中浓度指代为$ x $); (b) V0.5Fe1.5Zr, (c) VFeZr和(d) V1.5Fe0.5Zr组分及邻近组分合金结构的热容\r\nFig. 4. (a) Phase diagram of temperature versus concentration in V2x Fe2(1–x)Zr system (concentration in Fig. 4 is devoted to x). Heat capacities of (b) V0.5Fe1.5Zr, (c) VFeZr and (d) V1.5Fe0.5Zr components and their adjacent components$

图 4 (a) V_2x Fe_2(1–x)Zr体系的温度-浓度相图(图中浓度指代为

$x$

); (b) V_0.5Fe_1.5Zr, (c) VFeZr和(d) V_1.5Fe_0.5Zr组分及邻近组分合金结构的热容

Fig. 4. (a) Phase diagram of temperature versus concentration in V_2x Fe_2(1–x)Zr system (concentration in Fig. 4 is devoted to x). Heat capacities of (b) V_0.5Fe_1.5Zr, (c) VFeZr and (d) V_1.5Fe_0.5Zr components and their adjacent components

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4. 结　论

本文采用基于结构识别的高通量第一原理计算, 考虑结构简并度对配分函数的贡献, 借助自主开发的SAGAR程序实现了对合金材料进行有限温度下的快速预测. 以两类典型储氢合金为例, 通过第一原理计算得到基态(零温下)形成能, 发现: Sc_xY_1–x Fe₂形成能大于零, 在低温下相分离, 根据自由能符号确定合金固溶的临界温度, 并得到临界温度随浓度的变化; V_2x Fe_2(1–x)Zr形成能小于零, 在低温下倾向于形成有序相, 其中V_0.5Fe_1.5Zr和V_1.5Fe_0.5Zr的有序-无序转变温度在100 K附近, 而VFeZr的转变温度在50 K附近. 本文的计算流程将有效提高合金的高通量筛选效率, 也为实验研究提供相关的理论参考.

References

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Figure 1. Lattices of V_2x Fe_2(1–x)Zr (a) primitive cell and corresponding to volumes expanded respectively from primitive cell by (b) 2 times, (c) 4 times under a half of replacement content. (d) Primitive cell structure of Sc_xY_1–x Fe₂ alloys system and (e)−(f) the 7 kinds of lattices with 4 times volume of that of primitive one (red, green, yellow cyan and purple sphere represent respectively V, Zr, Fe, Y and Sc atom).

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图 2 根据Ising模型拟合结果得出的体系　(a)自由能符号; (c)热容; (b), (d)相转变温度

Figure 2. Systems obtained according to the fitting results of Ising model: (a) Free energy signal; (c) heat capacity; (b), (d) phase transition temperature.

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Figure 3. (a) Free energy signal induced by temperature under different replacement concentration in Sc_xY_1–x Fe₂ cubic structure. Free energy versus temperature for different crystal structures at (b) 1/8 and (c) 5/8 replacement concentration, respectively. (d) Temperature-concentration phase diagram of Sc_xY_1–x Fe₂ alloy system

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图 4 (a) V_2x Fe_2(1–x)Zr体系的温度-浓度相图(图中浓度指代为 $x$ ); (b) V_0.5Fe_1.5Zr, (c) VFeZr和(d) V_1.5Fe_0.5Zr组分及邻近组分合金结构的热容

Figure 4. (a) Phase diagram of temperature versus concentration in V_2x Fe_2(1–x)Zr system (concentration in Fig. 4 is devoted to x). Heat capacities of (b) V_0.5Fe_1.5Zr, (c) VFeZr and (d) V_1.5Fe_0.5Zr components and their adjacent components

DownLoad: Full-Size Img PowerPoint

表 1 利用SAGAR对图1所示两种合金在不同晶格下生成的所有不等价结构数目

Table 1. Numbers of all nonequivalent structures from different lattices created by using SAGAR for the two kinds of alloy system shown as Fig. 1

晶型	(a)	(b)	(c)	(d)	(e)	(f)	(g)	(h)	(i)	(j)	(k)
结构数	5	28	531	3	43	34	43	21	34	30	16

DownLoad: CSV

[1]	Le T, Epa V C, Burden F R, Winkler D A 2012 Chem. Rev. 5 112 Google Scholar
[2]	Oganov A R, Pickard C J, Zhu Q, Needs R J 2019 Nat. Rev. Mater. 5 4 Google Scholar
[3]	Woodley S M, Catlow R 2008 Nat. Mater. 12 7 Google Scholar
[4]	Wang Y, Lv J, Zhu L, Ma Y 2010 Phys. Rev. B 82 094116 Google Scholar
[5]	Lyakhov A O, Oganov A R, Stokes H T, Zhu Q 2013 Comput. Phys. Commun. 4 184 Google Scholar
[6]	Pickard C J, Needs R J 2011 J. Phys. Condens. Matter 5 23 Google Scholar
[7]	黄文军, 乔君威, 陈顺华, 王雪姣, 吴玉程 2021 物理学报 70 106201 Google Scholar Huang W J, Qiao J W, Chen S H, Wang X J, Wu Y C 2021 Acta Phys. Sin. 70 106201 Google Scholar
[8]	Li Z M, Wang H, Ouyang L Z, Liu J W, Zhu M 2016 J. Alloys Compd. 689 154865 Google Scholar
[9]	王鹏程, 曹亦, 谢红光, 殷归, 王伟, 王泽蓥, 马欣辰, 王琳, 黄维 2020 物理学报 69 117501 Google Scholar Wang P C, Cao Y, Xie H G, Yin Y, Wang W, Wang Z Y, Ma X C, Wang L, Huang W 2020 Acta Phys. Sin. 69 117501 Google Scholar
[10]	王大能, Olsen A, 叶恒强 1985 物理学报 34 681 Google Scholar Wang D N, Olsen A, Ye H Q 1985 Acta Phys. Sin. 34 681 Google Scholar
[11]	van de Walle A 2008 Nat. Mater. 7 455 Google Scholar
[12]	Hart G L W, Blum V, Walorski M J, Zunger A 2005 Nat. Mater. 4 391 Google Scholar
[13]	Yuge K 2009 Phys. Rev. B 79 144109 Google Scholar
[14]	Zunger A, Wei S H, Ferreira L G, Bernard J E 1990 Phys. Rev. Lett. 65 353 Google Scholar
[15]	Xia Z G, Liu G K, Wen J G, Mei Z G, Balasubramanian M, Molokeev M S, Peng L C, Gu L, Miller D J, Liu Q L, Poeppelmeier K R 2016 J. Am. Chem. Soc. 138 1158 Google Scholar
[16]	Banerjee S, Kumar A, Pillai C G S 2014 Intermetallics 51 30 Google Scholar
[17]	Budzyński M, Sarzyński J, Wiertel M, Surowiec Z 2001 Acta Phys. Pol. A 100 717 Google Scholar
[18]	Li X X, Yang C, Lu H Z, Luo X, Li Y Y, Ivasishin O M 2019 J. Alloys Compd. 787 112 Google Scholar
[19]	Luo S L, Li T S, Wang X J, Faizan M, Zhang L J 2021 Comput. Mol. Sci. 11 7690 Google Scholar
[20]	何长春, 廖继海, 杨小宝 2017 物理学报 66 163601 Google Scholar He C C, Liao J H, Yang X B 2017 Acta Phys. Sin. 66 163601 Google Scholar
[21]	Xu S G, Li X T, Zhao Y J, Liao J H, Xu W P, Yang X B, Xu H 2017 J. Am. Chem. Soc. 134 48 Google Scholar
[22]	Cheng Y H, Liao J H, Zhao Y J, Ni J, Yang X B 2019 Carbon 154 140 Google Scholar
[23]	Hart G L W, Forcade R W 2008 Phys. Rev. B 77 224115 Google Scholar
[24]	Cheng Y H, Liao J H, Zhao Y J, Yang X B 2017 Sci. Rep. 7 16211 Google Scholar
[25]	Kresse G, Furthmüller J 1996 Phys. Rev. B 54 11169 Google Scholar
[26]	Kresse G, Joubert D 1999 Phys. Rev. B 59 1758 Google Scholar
[27]	Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865 Google Scholar
[28]	Perdew J P, Burke K, Ernzerhof M 1998 Phys. Rev. Lett. 80 891 Google Scholar
[29]	Yuan S R, Ouyyang L Z, Zhu M, Zhao Y J 2018 J. Magn. Magn. Mater. 460 61 Google Scholar
[30]	汪志诚 2013 热力学·统计物理 (北京: 高等教育出版社) 第256页 Wang Z C 2013 Thermodynamics·Statistical Physics (5th Ed.) (Beijing: Higher Education Press) p256 (in Chinese)
[31]	Sivardière J, Lajzerowicz J 1975 Phys. Rev. A 11 2090 Google Scholar
[32]	Jiang Y L, Chen Y Z, Wang H, Yang X B 2020 Phys. Lett. A 284 126658 Google Scholar

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