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First-principles study of phase transition of BaF 2 under high pressue

Tian Cheng Lan Jian-Xiong Wang Cang-Long Zhai Peng-Fei Liu Jie

Jiang Cheng-Xin, Chen Ling-Xiu, Wang Hui-Shan, Wang Xiu-Jun, Chen Chen, Wang Hao-Min, Xie Xiao-Ming. Synthesis and pressure study of bubbles in hexagonal boron nitride interlayer. Acta Phys. Sin., 2021, 70(6): 069801. doi: 10.7498/aps.70.20201482
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First-principles study of phase transition of BaF 2 under high pressue

Tian Cheng, Lan Jian-Xiong, Wang Cang-Long, Zhai Peng-Fei, Liu Jie
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  • There have been some theoretical studies of high pressure phase transition behavior of BaF 2, while in most cases the attention is paid mainly to the optical and electrical properties of BaF 2 under increasing pressure. To date, there has been still a lack of theoretical explanation for the hysteresis phenomenon of high-pressure phase of BaF 2 when the pressure is released. In addition, the pressure-dependent behavior of the BaF 2 band gap is still under controversy, and there are few studies of its high-pressure Raman spectra. Therefore, first principle is used to make a supplementary calculation of the high pressure behavior of BaF 2. For a given pressure P and temperature T, the thermodynamic stable phase has the lowest Gibbs free energy. The calculations are performed at zero temperature and hence, the Gibbs free energy becomes equal to the enthalpy. Thus, the variation of enthalpy is calculated as a function of pressure to study the high-pressure phase stability of BaF 2 based on density functional theory as implemented in the Vienna ab initio simulation package (VASP). The results show that the BaF 2 undergoes two structural phase transitions from Fm3 m(cubic) to Pnma (orthorhombic) and then to P6 3/ mmc(hexagonal) with increasing pressure, and their corresponding transition pressures are 3.5 and 18.3 GPa, respectively. By calculating the evolution of lattice constant with pressure, it is found that at about 15 GPa (near the second phase transition pressure), the lattice constants of the Pnma structure show abnormal behavior (a slight increase in b o and a slight decrease in a o). We suggest that this behavior leads the band gap to decrease, indicated by analyzing the calculated results of Pnma structure of other materials. The Pnma structure completely transforms into P6 3/ mmc structure at about 20 GPa. By analyzing the phonon dispersion curves of BaF 2 as a function of pressure, the structural stability information of the material can also be obtained. Then the density functional perturbation theory (DFPT) is used to calculate the phonon dispersion curves of BaF 2 by VASP code and Phonopy code. The hysteresis phenomenon of the P6 3/ mmc structure, when the pressure is released, is explained by the kinetic stability. The results predict that the P6 3/ mmc structure can be stabilized at least to 80 GPa.
      Corresponding author: Zhai Peng-Fei, zhaipengfei@impcas.ac.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12075290, 12035019), the Youth Innovation Promotion Association of Chinese Academy of Sciences (Grant No. 2020412), and the Opening Fund of Key Laboratory of Silicon Device and Technology, Chinese Academy of Sciences (Grant No. KLSDTJJ2019-06)

    高压作为一种调控材料性质和发现物质新结构的重要手段, 被广泛应用于实验研究中. 例如, Snider等 [ 1] 在(267 ± 10) GPa压强和(287.7 ± 1.2) K温度下实现了C-S-H材料的室温超导转变; Xia等 [ 2] 利用高压手段对WSe 2-MoSe 2二维异质结层间距和层间耦合作用进行了调控; 徐波和田永君 [ 3] 在高温高压条件下合成了纳米孪晶结构的立方氮化硼和金刚石块材超硬材料. 随着金刚石对顶砧(DAC)技术的不断发展, 材料在高压下的行为正受到越来越多的关注.

    CaF 2, SrF 2, BaF 2是一类常见的碱土金属氟化物, 由于它们具有独特的光学和电学性质而受到了广泛的关注, 如BaF 2具有优异的闪烁性能, 可用于探测γ射线 [ 4] . 一方面, 该类材料结构简单, 常温常压下均为立方萤石结构, 它们在较窄的应力范围内有较大的应变, 对于发展高压状态方程很有帮助. 另一方面, 了解这些材料高压引起的相变也有助于理解地球深处的材料相变 [ 5] . 已有实验研究表明, 高压下它们会经历两次相变, 相变顺序均为 Fm $ \overline {3} $ m(立方萤石结构)- Pnma( α-PbCl 2型结构)- P6 3/ mmc (Ni 2In结构)相, 相应的阳离子配位数由8增加到9再增加到11 [ 6- 9] . 由于BaF 2相变压强较低, 在实验上更容易实现, 是研究这类物质高压行为的范例, 其结果可为其他碱土金属氟化物的高压行为提供参考. 拉曼光谱 [ 10- 12] 和XRD [ 6, 9] 研究表明, BaF 2Fm $ \overline {3} $ m- Pnma- P6 3/ mmc相变压强分别为约2.3—3.0 GPa和14—15 GPa. 在第二次相变压强附近, Pnma相晶轴压缩性出现异常, 并且在卸压过程中两种高压相均存在滞后现象, 其中 Pnma相在卸压后可与 Fm $ \overline {3} $ m相共存.

    理论计算可以不受实验技术及条件的限制, 在较大的压强范围内对材料的性质和结构做出合理的预测 [ 13, 14] . 已有学者对BaF 2的高压相变行为进行了一定的理论计算研究. 计算结果表明, BaF 2两次相变压强分别为2.83—5.10 GPa, 11.2—15.0 GPa [ 4, 15- 17] . 其中, Ayala [ 4] 采用shell-model方法, 通过对BaF 2晶格常数、原子位置、介电常数以及高压相三棱柱倾角的计算, 得出 Pnma- P6 3/ mmc完全相变压强点为17 GPa; Kanchana等 [ 17] 利用紧束缚线性muffin-tin轨道法, 指出在33 GPa压强下, 电子从氟的p轨道转移到钡的s和d轨道, 从而导致BaF 2P6 3/ mmc相发生金属化; 然而, Jiang等 [ 16] 利用原子线性轨道方法, 引入半经验参数对带隙进行修正后, 指出在50 GPa压强下BaF 2仍不会发生金属化; Yang等 [ 15] 基于密度泛函理论, 利用CASTEP软件对高压下BaF 2的结构稳定性、电子结构、弹性性质进行了理论研究, 他们的计算表明, 在高达210 GPa的压强下, 不会发生能带交叠的金属化现象. 目前, BaF 2卸压过程中 P6 3/ mmc相的滞后现象还缺少相应的理论解释. 此外, BaF 2带隙随压强变化的行为还存有争议, 而且其高压拉曼光谱行为的研究也比较少. 因此, 本文利用第一性原理对BaF 2的拉曼峰位及其 P6 3/ mmc相的声子色散曲线随压强的变化进行计算, 以期对其高压拉曼光谱行为和卸压过程中 P6 3/ mmc相的滞后现象进行解释, 并结合晶格常数和带隙计算, 对其高压行为进行较为系统的理论研究.

    本文计算基于密度泛函理论(DFT), 采用投影缀加波(PAW)方法 [ 18] , 利用VASP [ 19] 软件进行计算. 交换关联势选用广义梯度近似(GGA)的PBE版本 [ 20] . 经过收敛性测试, 对于BaF 2的3种物相, 截断能均设为600 eV, 能量和力的收敛标准分别为10 –8 eV和0.001 eV/Å. 布里渊区 K点网格均采用Gamma取点方式, 对于BaF 2Fm $ \overline {3} $ m, PnmaP6 3/ mmc相, 网格密度分别为7 × 7 × 7, 7 × 11 × 6, 11 × 11 × 9.

    由于理论计算是基于 T(温度) = 0 K的, 故吉布斯自由能( G)等于焓( H), 即 G = H = E + PV, 其中 E, P, V分别为体系的内能、压强、体积. 所以在给定压强下, 焓值越低的结构越稳定. 于是,可以由计算结果做出BaF 2 3种物相的焓( H)-压强( P)关系图, 并由此推测相变压强 [ 15, 21- 25] .

    为了判断相变压强, 在一系列给定压强下对BaF 23种物相进行结构优化, 其中, Fm $ \overline {3} $ m相在0 GPa下优化后的晶格常数为 a c = 6.281 Å, 原子位置为Ba(0, 0, 0), F(0.25, 0.25, 0.25); Pnma相在12 GPa下优化后的晶格常数分别为 a o = 6.284 Å, b o = 3.926 Å, c o = 7.791 Å, 原子位置为Ba(0.2452, 0.25, 0.3814), F1(0.0391, 0.25, 0.6820), F2(0.1496, 0.25, 0.0668); P6 3/ mmc相在32 GPa下优化后的晶格常数为 a h = b h = 4.183 Å, c h = 5.260 Å, 原子位置为Ba(0.3333, 0.6667, 0.25), F1(0, 0, 0), F2(0.3333, 0.6667, 0.75). 以上晶格常数计算结果均与实验测量值 [ 10] 符合较好, 证明本工作的计算模型合理.

    图1为BaF 23种物相相对焓差随压力的变化, 其中插图为18—19 GPa的局部放大图. 可以看出, Fm $ \overline {3} $ m- Pnma- P6 3/ mmc的结构相变压强分别为约3.5和18.3 GPa. 对于 PnmaP6 3/ mmc相, 当压强达到20.0 GPa后, 两相焓值几乎相等, 表明20 GPa以后这两相可以共存 [ 26] , 或者存在着连续交替相变 [ 4] .

    Figure 1.  Variation of enthalpy difference with pressure for BaF 2

    图2为BaF 2相对体积随压强的变化. 可以看出, 当压强达到3.5 GPa时, 材料经历第一次相变( Fm $ \overline {3} $ m- Pnma), 此时相对体积发生突变, 急剧塌缩约8.68%; 而当压强达到第二次相变压强点时(约18.3 GPa), BaF 2的相对体积变化较小, 仅仅塌缩约1.35%. 两次相变过程均伴随着体积塌缩, 表明 Fm $ \overline {3} $ m- Pnma- P6 3/ mmc均为一级相变.

    Figure 2.  Relative volume ( V/ V 0) variation of BaF 2 as a function of pressure.

    由于 P6 3/ mmc( a h, c h)可以视为 Pnma( a o, b o, c o)的一个supergroup [ 27] , 当两相晶格常数满足 a o = c h, b o = a h, c o = $ \sqrt{3} $ a h, 并且 Pnma相中的原子移动到相应位置时, 可以认为 Pnma相完全变成了 P6 3/ mmc[ 4, 28] , 由此可以推测完全相变压强点. 所以本文对BaF 2Fm $ \overline {3} $ m ( a c), Pnma ( a o, b o, c o), P6 3/ mmc ( a h, c h)相的晶格常数随压强变化进行了分析. 为了更好地比较晶格常数之间的关系, 利用上述等式关系对晶格常数做了一定的处理, 然后画出其随压强的变化图.

    图3所示, 一般情况下, Fm $ \overline {3} $ m, Pnma, P6 3/ mmc相的晶格常数均表现出随压强增加而平稳减小的特征. 但是在对 Pnma相的晶格优化中发现, 当压强约为15 GPa (接近相变压强18.3 GPa时), Pnma相开始向 P6 3/ mmc相弛豫, 晶轴压缩性出现异常变化, 表现为随压强增加晶格常数 b o轻微增加, a o略微减小. Smith等 [ 10] 在对BaF 2的高压实验中也发现了该现象, 他们将其称之为发生相变的前兆现象. Dorfman等 [ 9] 在对CaF 2, SrF 2, BaF 23种材料的高压实验中也观察到它们的 Pnma相随着压强增加晶格会朝向 P6 3/ mmc相扭曲.

    Figure 3.  Evolution of the lattice constants of BaF 2 with three structures under pressure.

    当压强达到约20 GPa后, a o = c h, b o = a h, c o/ $ \sqrt{3} $ = a h, 此时可以认为 Pnma完全弛豫为 P6 3/ mmc相, 即发生了完全相变. 在对CaF 2 [ 26] , BaF 2 [ 4] , CaH 2 [ 29] , CeO 2以及ThO 2 [ 30] Pnma相和Li 2O [ 31] , Mg 2Si [ 32] , Li 2S, Na 2S, K 2S [ 28] 等反 Pnma相的计算中, 均发现晶轴压缩性随压强变化会出现这种类似现象. 也许对于 Pnma- P6 3/ mmc结构相变的材料, 这种行为并不是特例.

    结合焓差和晶格常数随压强变化的关系图可知, Fm $ \overline {3} $ m- Pnma- P6 3/ mmc结构相变压强分别为3.5和18.3 GPa. 在约15 GPa时, Pnma相的晶轴压缩性出现异常(前兆现象), 并且在约20 GPa时完全转变为 P6 3/ mmc相. 相比于实验值 [ 6, 9, 10] (第一次相变压强2.3—3 GPa, 第二次相变压强14—15 GPa), 两次相变压强的计算结果均略高于实验. 通常广义梯度近似(GGA)会高估材料的晶格常数, 而局域梯度近似(LDA)则会低估, 导致实验结果位于两种近似的计算结果之间 [ 33] . 采用两种不同泛函对CaF 2 [ 26, 27] 计算也发现LDA泛函计算结果低于GGA泛函. 所以, 利用GGA泛函得出偏高的相变压强是可以理解的, 而对于 Pnma- P6 3/ mmc的相变, 计算的相变完全压强点(20 GPa)与开始相变压强点(18.3 GPa)的差值约为1.7 GPa, 与实验给出的结果(15 GPa开始相变, 17 GPa相变完成)较为一致 [ 6] .

    为了分析BaF 2的高压拉曼光谱行为以及对其 P6 3/ mmc相在卸压过程中的滞后现象进行理论解释, 本文基于密度泛函微扰理论 [ 34] (DFPT), 利用VASP [ 19] 和phonopy [ 35] 软件, 对BaF 2 3种物相的拉曼峰位以及 P6 3/ mmc相的声子色散曲线随压强的变化进行了计算.

    利用计算得到的布里渊区中心( Γ)声子特征向量来推导声子模的对称标记, 由群理论 [ 36] 分析, BaF 2 3种物相的拉曼振动可以分别表示为

    $$ \begin{split} &\varGamma_{Fm} \overline {3} _{m}={\rm{T}}_{2{\rm{g}}}, \\ & \varGamma_{Pnma}=6{\rm{A}}_{{\rm{g}}} + 3{\rm{B}}_{1{\rm{g}}} + 6{\rm{B}}_{2{\rm{g}}} + 3{\rm{B}}_{3{\rm{g}}},\\ &\varGamma_{P63/mmc}=2{\rm{E}}_{2{\rm{g}}} . \end{split} $$

    表1给出了计算所得BaF 2Pnma相在10 GPa压强下的拉曼峰位, 其中190.6—363.8 cm –1的波数范围与实验所测的6个拉曼峰(约210—380 cm –1)较为一致 [ 10] . 图4(a)图4(b)分别为 Fm $ \overline {3} $ m相T 2g模和 P6 3/ mmc相2E 2g模拉曼峰随压强变化的关系图. 在0 GPa时, Fm $ \overline {3} $ m相T 2g峰位约为235.8 cm –1, 与实验值(240—242 cm –1)较为符合 [ 10, 11] . 相对于其他计算 (约216 cm –1) [ 37] , 本文的计算结果更接近实验值. 随着压强增大, 峰位向高波数方向移动, 其斜率约为6.36 cm –1/GPa, 略低于实验值7.6—8 cm –1/GPa [ 10, 11] .

    Table 1.  Calculated Raman shift of Pnma structure BaF 2 under 10 GPa.
    Mode ω/cm –1 Mode ω/cm –1 Mode ω/cm –1
    A g 81.2 A g 190.6 B 2g 268.7
    B 3g 81.4 B 1g 203.3 A g 283.3
    B 1g 90.2 A g 211.2 B 1g 304.4
    A g 112.9 B 3g 218.7 B 3g 309.6
    B 2g 151.5 B 2g 224.0 A g 321.6
    B 2g 174.7 B 2g 251.2 B 2g 363.8
     | Show Table
    DownLoad: CSV
    Figure 4.  Raman shift as a function of pressure for (a) T 2g of Fm $ \overline {3} $ m and (b) 2E 2g of P6 3/ mmc.

    图4(b)P6 3/ mmc相的两个E 2g模随压强的变化. 在14 GPa时, 两个拉曼峰分别位于约95.5和366.7 cm –1; 80 GPa时, 分别位于约123.0和537.8 cm –1, 与实验(14.1 GPa时95和325 cm –1, 77.1 GPa只有一个约525 cm –1的高频模) [ 10] 在相应压强下所得结果基本一致. 由 图4(b)可知, 高波数的振动模对压强的变化更为敏感, 两个振动模随压强变化均表现出非线性: 压强越低, 非线性程度越小, 压强越高, 非线性程度越高. 所以实验上, 在较小的压强范围内对拉曼峰随压强的变化关系进行线性拟合是可行的, 但是当压强范围较大时, 进行线性拟合的做法似乎不太合适.

    图5为不同压强下 P6 3/ mmc相的声子色散曲线. 在14 GPa时, 整个布里渊区没有虚频, 表明结构此时是动力学稳定的, 但是 M点(0.5, 0.0, 0.0)声学支声子有软化的迹象; 12 GPa时, 在 M点出现虚频, 表明此时结构是动力学不稳定的. 图6给出了 M点声学声子频率随压强的变化. 可以看出, 声子频率软化至0 THz的压强约为13.5 GPa. 这说明卸压过程中, P6 3/ mmc相可以存在至13.5 GPa, 由此可以解释卸压过程中, 该相存在滞后现象(加压过程, 18.3 GPa时才出现 P6 3/ mmc相), 滞后约4.8 GPa.

    Figure 5.  Phonon dispersion curves for P6 3/ mmc structure of BaF 2 at different pressures: (a) 12 GPa; (b) 14 GPa; (c) 40 GPa; (d) 80 GPa.
    Figure 6.  Phonon frequencies at M point as a function of pressure for P6 3/ mmc structure.

    当压强为80 GPa时, 整个布里渊区不存在虚频, 表明BaF 2P6 3/ mmc相至少可以稳定到80 GPa, 与实验所得77 GPa的结论一致 [ 10] . 对比40 GPa时的声子色散曲线可以发现, 此时 M点声子似乎又出现了软化迹象, 故预计超过80 GPa后, P6 3/ mmc相开始不稳定, 可能会再经历一次相变, 例如转变为BaH 2的AlB 2结构 [ 38] , 或Luo等 [ 39] 所预测的 P-3 m1结构.

    本文还对BaF 23种物相的带隙随压强的变化进行了计算, 结果如 图7所示. 0 GPa时, Fm $ \overline {3} $ m相带隙约为6.61 eV, 低于实验值 [ 40] 10 eV, 这可以被认为是密度泛函对带隙的计算误差. 当达到第一次相变压强时, 结构转变为 Pnma相, 带隙突然增加约0.22 eV, 且带隙刚开始与压力呈正相关关系, 但当压强为约16 GPa时, 带隙开始减小. 在18.3 GPa时, P6 3/ mmc相的带隙约为5.61 eV, 相比于同压强下 Pnma相的带隙, 降低约1.11 eV. BaF 2的这种带隙随压强的变化行为与其他碱土金属氟化物的行为一致 [ 26, 41- 43] , 本文的结果也与Kanchana等 [ 17] 的结论一致, 而在文献[ 15]和文献[ 44]中作者指出, BaF 2Pnma相的带隙随压强增加是一直增加的, 没有出现降低的情况, 但他们并没有给出相变点附近的带隙数据, 也没有对材料的晶格常数随压强变化做出分析. 可能是由于带隙突变的压强范围较小, 所以导致其他学者忽略了这一点.

    Figure 7.  Band gap as a function of pressure for three structures with GGA of BaF 2.

    对于晶体材料而言, 随着压强增加, 原子间距离减小, 电子云重叠程度增加, 造成晶体材料中存在离子键向共价键转变的趋势, 从而导致带隙增加; 当压强足够大时, 电子不再属于单个原子或者某个键, 形成离域电子, 导致带隙减小 [ 26, 45] . 为了进一步分析 Pnma相带隙降低的原因, 本文对其电子态密度随压强的变化进行了计算, 结果如 图8所示. 价带顶主要由F的p轨道电子贡献, 导带底主要由Ba的d轨道电子贡献, 压强增大时, 态密度所在区域能带宽度有所增加, 当压强由5 GPa增加到16 GPa时, 导带部分向高能量方向移动, 带隙变宽; 16—18 GPa时, 导带部分又向低能量方向移动, 导致价带和导带靠近, 使带隙变窄. 图9为不同压强下 Pnma相中F1(位置见 3.1)原子在p x 与p y + p z 轨道上的投影电子态密度, 16 GPa以后, F1原子的p x 和p y + p z 轨道电子在费米面附近出现了明显的分离, 在相变压强下, P6 3/ mmc相F原子的p z 与p x + p y 轨道电子也出现了分离, 且更为明显, 与CaF 2 [ 42] 的结果类似, 这也可以解释其带隙的减小.

    Figure 8.  DOS of BaF 2 for Pnma structure at different pressure
    Figure 9.  Projected DOS onto p y + p z and px orbitals of F1 atoms for Pnma and P6 3/ mmc structure.

    根据已有的CaF 2 [ 26, 42] , SrF 2 [ 43] , Na 2S [ 28] , ThO 2和CeO 2 [ 30] 等材料的计算结果, 可以发现这些材料 Pnma相的带隙随着压强增加, 均为先增加再减小. 仔细观察文中给出的晶格常数以及带隙随压强变化的关系图后, 可知 Pnma相带隙减小的位置处于相变前兆现象的压强范围内, 此时 Pnma相开始向 P6 3/ mmc相弛豫, 而 P6 3/ mmc相的带隙比较小, 所以 Pnma相的带隙减小也是可以理解的.

    对于 P6 3/ mmc相, 其带隙随压强增加而降低, 对数据进行线性拟合, 得到斜率为–0.0033 eV/GPa, 如果采用文献[ 23]中所提出的外推法推测其金属化压强, 则压强应达到约1700 GPa以上, 而结合声子谱分析, 可知BaF 2P6 3/ mmc相在80 GPa以后可能是不稳定的, 所以采用外推法来寻找材料金属化压强点的做法还有待商榷.

    第一性原理计算表明, BaF 2在高压下会经历两次一级相变, 相变压强分别为3.5和18.3 GPa. 对于 Pnma相, 在压强约为15 GPa时, 晶轴压缩性出现异常, 随压强增加其晶轴 b o轻微增加, a o略微减小. 16 GPa以后, 由于F1原子p x 与p y + p z 轨道电子离域而导致其带隙减小. 当压强约为20 GPa时, Pnma相完全转变为 P6 3/ mmc相. 另外还计算了BaF 2的拉曼峰位随压强的变化关系, 给出了 Pnma相在10 GPa下的拉曼峰, 为实验上利用拉曼光谱指认该相提供了理论依据. 对于 P6 3/ mmc相的两个E 2g模式, 其高波数拉曼峰对压强变化更敏感. 通过分析不同压强下的声子色散曲线, 指出卸压时 P6 3/ mmc相存在滞后现象, 滞后约4.8 GPa, 并且该物相至少可以稳定到80 GPa.

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    Snider E, Dasenbrock-Gammon N, McBride R, Debessai M, Vindana H, Vencatasamy K, Lawler K V, Salamat A, Dias R P 2020 Nature 586 373Google Scholar

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    Xia J, Yan J, Wang Z, He Y, Gong Y, Chen W, Sum T C, Liu Z, Ajayan P M, Shen Z 2021 Nat. Phys. 17 92Google Scholar

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    徐波, 田永君 2017 物理学报 66 036201Google Scholar

    Xu B, Tian Y J 2017 Acta Phys. Sin. 66 036201Google Scholar

    [4]

    Ayala A P 2001 J. Phys. Condens. Matter 13 11741Google Scholar

    [5]

    Kavner A 2008 Phys. Rev. B 77 224102Google Scholar

    [6]

    Leger J M, Haines J, Atouf A, Schulte O, Hull S 1995 Phys. Rev. B 52 13247Google Scholar

    [7]

    Wang J S, Ma C L, Zhou D, Xu Y S, Zhang M Z, Gao W, Zhu H Y, Cui Q L 2012 J. Solid State Chem. 186 231Google Scholar

    [8]

    Speziale S, Duffy T S 2002 Phys. Chem. Miner. 29 465Google Scholar

    [9]

    Dorfman S M, Jiang F M, Mao Z, Kubo A, Meng Y, Prakapenka V B, Duffy T S 2010 Phys. Rev. B 81 174121Google Scholar

    [10]

    Smith J S, Desgreniers S, Tse J S, Sun J, Klug D D, Ohishi Y 2009 Phys. Rev. B 79 134101Google Scholar

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    Kourouklis G A, Anastassakis E 1989 Phys. Status Solidi B 152 89Google Scholar

    [12]

    Kessler J R, Monberg E, Nicol M 1974 J. Chem. Phys. 60 5057Google Scholar

    [13]

    Gao G Y, Oganov A R, Li P F, Li Z W, Wang H, Cui T, Ma Y M, Bergara A, Lyakhov A O, Iitaka T, Zou G T 2010 Proc. Natl. Acad. Sci. U. S. A. 107 1317Google Scholar

    [14]

    Jin X L, Meng X, He Z, Ma Y M, Liu B, Cui T A, Zou G T, Mao H K 2010 Proc. Natl. Acad. Sci. U. S. A. 107 9969Google Scholar

    [15]

    Yang X C, Hao A M, Wang X M, Liu X, Zhu Y 2010 Comput. Mater. Sci. 49 530Google Scholar

    [16]

    Jiang H T, Pandey R, Darrigan C, Rerat M 2003 J. Phys. Condens. Matter 15 709Google Scholar

    [17]

    Kanchana V, Vaitheeswaran G, Rajagopalan M 2003 J. Alloys Compd. 359 66Google Scholar

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    Blochl P E 1994 Phys. Rev. B 50 17953Google Scholar

    [19]

    Kresse G, Furthmuller J 1996 Comput. Mater. Sci. 6 15Google Scholar

    [20]

    Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865Google Scholar

    [21]

    Dai J J, Feng Q G 2020 Phys. Status Solidi B 257 1900726Google Scholar

    [22]

    Xiao H Y, Jiang X D, Duan G, Gao F, Zu X T, Weber W J 2010 Comput. Mater. Sci. 48 768Google Scholar

    [23]

    Cui S X, Feng W X, Hua H Q, Feng Z B, Wang Y X 2009 Comput. Mater. Sci. 47 41Google Scholar

    [24]

    Kessair S, Arbouche O, Amara K, Benallou Y, Azzaz Y, Zemouli M, Bekki M, Ameri M, Bouazza B S 2016 Indian J. Phys. 90 1403Google Scholar

    [25]

    Boudjemline A, Louail L, Islam M M, Diawara B 2011 Comput. Mater. Sci. 50 2280Google Scholar

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    Guo Y, Fang Y M, Li J 2021 Chin. Phys. B 30 030502Google Scholar

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    Wu X, Qin S, Wu Z Y 2006 Phys. Rev. B 73 134103Google Scholar

    [28]

    Verma A K, Modak P, Sharma S M 2017 J. Alloys Compd. 710 460Google Scholar

    [29]

    Tse J S, Klug D D, Desgreniers S, Smith J S, Flacau R, Liu Z, Hu J, Chen N, Jiang D T 2007 Phys. Rev. B 75 134108Google Scholar

    [30]

    Song H X, Liu L, Geng H Y, Wu Q 2013 Phys. Rev. B 87 184103Google Scholar

    [31]

    Kunc K, Loa I, Syassen K 2008 Phys. Rev. B 77 094110Google Scholar

    [32]

    Ji D P, Chong X Y, Ge Z H, Feng J 2019 J. Alloys Compd. 773 988Google Scholar

    [33]

    Liu G, Wang H, Ma Y M, Ma Y M 2011 Solid State Commun. 151 1899Google Scholar

    [34]

    Gonze X, Lee C 1997 Phys. Rev. B 55 10355Google Scholar

    [35]

    Togo A, Oba F, Tanaka I 2008 Phys. Rev. B 78 134106Google Scholar

    [36]

    Kroumova E, Aroyo M I, Perez-Mato J M, Kirov A, Capillas C, Ivantchev S, Wondratschek H 2003 Phase Transitions 76 155Google Scholar

    [37]

    Soni H R, Gupta S K, Talati M, Jha P K 2011 J. Phys. Chem. Solids 72 934Google Scholar

    [38]

    Kinoshita K, Nishimura M, Akahama Y, Kawamura H 2007 Solid State Commun. 141 69Google Scholar

    [39]

    Luo D B, Wang Y C, Yang G C, Ma Y M 2018 J. Phys. Chem. C 122 12448Google Scholar

    [40]

    Rubloff G W 1972 Phys. Rev. B 5 662Google Scholar

    [41]

    Kanchana V, Vaitheeswaran G, Rajagopalan M 2003 Physica B 328 283Google Scholar

    [42]

    Shi H, Luo W, Johansson B, Ahujia R 2009 J. Phys. Condens. Matter 21 415501Google Scholar

    [43]

    Hao A M, Yang X C, Li J, Xin W, Zhang S H, Zhang X Y, Liu R P 2009 Chin. Phys. Lett. 26 077103Google Scholar

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    朱春野, 刘欢欢, 刘艳辉 2011 延边大学学报(自然科学版) 37 19Google Scholar

    Zhu C Y, Liu H H, Liu Y H 2011 J. Yanbian Univ. (Natural Science Edition) 37 19Google Scholar

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    吴成国, 武文远, 龚艳春, 戴斌飞, 何苏红, 黄雁华 2015 物理学报 64 114213Google Scholar

    Wu C G, Wu W Y, Gong Y C, Dai B F, He S H, Huang Y H 2015 Acta Phys. Sin. 64 114213Google Scholar

  • 图 1  BaF 2相对焓差随压强的变化

    Figure 1.  Variation of enthalpy difference with pressure for BaF 2

    图 2  BaF 2的相对体积随压强的变化

    Figure 2.  Relative volume ( V/ V 0) variation of BaF 2 as a function of pressure.

    图 3  BaF 23种物相晶格常数随压强的变化

    Figure 3.  Evolution of the lattice constants of BaF 2 with three structures under pressure.

    图 4  BaF 2(a) Fm $ \overline {3} $ m结构T 2g模和(b) P6 3/ mmc结构2E 2g模拉曼峰位随压强的变化

    Figure 4.  Raman shift as a function of pressure for (a) T 2g of Fm $ \overline {3} $ m and (b) 2E 2g of P6 3/ mmc.

    图 5  不同压强下BaF 2P6 3/ mmc相声子谱 (a) 12 GPa; (b) 14 GPa; (c) 40 GPa; (d) 80 GPa

    Figure 5.  Phonon dispersion curves for P6 3/ mmc structure of BaF 2 at different pressures: (a) 12 GPa; (b) 14 GPa; (c) 40 GPa; (d) 80 GPa.

    图 6  P6 3/ mmc结构 M点声学支声子振动频率随压强的变化

    Figure 6.  Phonon frequencies at M point as a function of pressure for P6 3/ mmc structure.

    图 7  GGA泛函计算BaF 23种物相带隙随压强的变化

    Figure 7.  Band gap as a function of pressure for three structures with GGA of BaF 2.

    图 8  Pnma相电子态密度随压强的变化

    Figure 8.  DOS of BaF 2 for Pnma structure at different pressure

    图 9  Pnma相F1原子投影电子态密度(PDOS)随压强的变化

    Figure 9.  Projected DOS onto p y + p z and px orbitals of F1 atoms for Pnma and P6 3/ mmc structure.

    表 1  10 GPa压强下 Pnma结构BaF 2拉曼峰位计算结果

    Table 1.  Calculated Raman shift of Pnma structure BaF 2 under 10 GPa.

    Mode ω/cm –1 Mode ω/cm –1 Mode ω/cm –1
    A g 81.2 A g 190.6 B 2g 268.7
    B 3g 81.4 B 1g 203.3 A g 283.3
    B 1g 90.2 A g 211.2 B 1g 304.4
    A g 112.9 B 3g 218.7 B 3g 309.6
    B 2g 151.5 B 2g 224.0 A g 321.6
    B 2g 174.7 B 2g 251.2 B 2g 363.8
    DownLoad: CSV
  • [1]

    Snider E, Dasenbrock-Gammon N, McBride R, Debessai M, Vindana H, Vencatasamy K, Lawler K V, Salamat A, Dias R P 2020 Nature 586 373Google Scholar

    [2]

    Xia J, Yan J, Wang Z, He Y, Gong Y, Chen W, Sum T C, Liu Z, Ajayan P M, Shen Z 2021 Nat. Phys. 17 92Google Scholar

    [3]

    徐波, 田永君 2017 物理学报 66 036201Google Scholar

    Xu B, Tian Y J 2017 Acta Phys. Sin. 66 036201Google Scholar

    [4]

    Ayala A P 2001 J. Phys. Condens. Matter 13 11741Google Scholar

    [5]

    Kavner A 2008 Phys. Rev. B 77 224102Google Scholar

    [6]

    Leger J M, Haines J, Atouf A, Schulte O, Hull S 1995 Phys. Rev. B 52 13247Google Scholar

    [7]

    Wang J S, Ma C L, Zhou D, Xu Y S, Zhang M Z, Gao W, Zhu H Y, Cui Q L 2012 J. Solid State Chem. 186 231Google Scholar

    [8]

    Speziale S, Duffy T S 2002 Phys. Chem. Miner. 29 465Google Scholar

    [9]

    Dorfman S M, Jiang F M, Mao Z, Kubo A, Meng Y, Prakapenka V B, Duffy T S 2010 Phys. Rev. B 81 174121Google Scholar

    [10]

    Smith J S, Desgreniers S, Tse J S, Sun J, Klug D D, Ohishi Y 2009 Phys. Rev. B 79 134101Google Scholar

    [11]

    Kourouklis G A, Anastassakis E 1989 Phys. Status Solidi B 152 89Google Scholar

    [12]

    Kessler J R, Monberg E, Nicol M 1974 J. Chem. Phys. 60 5057Google Scholar

    [13]

    Gao G Y, Oganov A R, Li P F, Li Z W, Wang H, Cui T, Ma Y M, Bergara A, Lyakhov A O, Iitaka T, Zou G T 2010 Proc. Natl. Acad. Sci. U. S. A. 107 1317Google Scholar

    [14]

    Jin X L, Meng X, He Z, Ma Y M, Liu B, Cui T A, Zou G T, Mao H K 2010 Proc. Natl. Acad. Sci. U. S. A. 107 9969Google Scholar

    [15]

    Yang X C, Hao A M, Wang X M, Liu X, Zhu Y 2010 Comput. Mater. Sci. 49 530Google Scholar

    [16]

    Jiang H T, Pandey R, Darrigan C, Rerat M 2003 J. Phys. Condens. Matter 15 709Google Scholar

    [17]

    Kanchana V, Vaitheeswaran G, Rajagopalan M 2003 J. Alloys Compd. 359 66Google Scholar

    [18]

    Blochl P E 1994 Phys. Rev. B 50 17953Google Scholar

    [19]

    Kresse G, Furthmuller J 1996 Comput. Mater. Sci. 6 15Google Scholar

    [20]

    Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865Google Scholar

    [21]

    Dai J J, Feng Q G 2020 Phys. Status Solidi B 257 1900726Google Scholar

    [22]

    Xiao H Y, Jiang X D, Duan G, Gao F, Zu X T, Weber W J 2010 Comput. Mater. Sci. 48 768Google Scholar

    [23]

    Cui S X, Feng W X, Hua H Q, Feng Z B, Wang Y X 2009 Comput. Mater. Sci. 47 41Google Scholar

    [24]

    Kessair S, Arbouche O, Amara K, Benallou Y, Azzaz Y, Zemouli M, Bekki M, Ameri M, Bouazza B S 2016 Indian J. Phys. 90 1403Google Scholar

    [25]

    Boudjemline A, Louail L, Islam M M, Diawara B 2011 Comput. Mater. Sci. 50 2280Google Scholar

    [26]

    Guo Y, Fang Y M, Li J 2021 Chin. Phys. B 30 030502Google Scholar

    [27]

    Wu X, Qin S, Wu Z Y 2006 Phys. Rev. B 73 134103Google Scholar

    [28]

    Verma A K, Modak P, Sharma S M 2017 J. Alloys Compd. 710 460Google Scholar

    [29]

    Tse J S, Klug D D, Desgreniers S, Smith J S, Flacau R, Liu Z, Hu J, Chen N, Jiang D T 2007 Phys. Rev. B 75 134108Google Scholar

    [30]

    Song H X, Liu L, Geng H Y, Wu Q 2013 Phys. Rev. B 87 184103Google Scholar

    [31]

    Kunc K, Loa I, Syassen K 2008 Phys. Rev. B 77 094110Google Scholar

    [32]

    Ji D P, Chong X Y, Ge Z H, Feng J 2019 J. Alloys Compd. 773 988Google Scholar

    [33]

    Liu G, Wang H, Ma Y M, Ma Y M 2011 Solid State Commun. 151 1899Google Scholar

    [34]

    Gonze X, Lee C 1997 Phys. Rev. B 55 10355Google Scholar

    [35]

    Togo A, Oba F, Tanaka I 2008 Phys. Rev. B 78 134106Google Scholar

    [36]

    Kroumova E, Aroyo M I, Perez-Mato J M, Kirov A, Capillas C, Ivantchev S, Wondratschek H 2003 Phase Transitions 76 155Google Scholar

    [37]

    Soni H R, Gupta S K, Talati M, Jha P K 2011 J. Phys. Chem. Solids 72 934Google Scholar

    [38]

    Kinoshita K, Nishimura M, Akahama Y, Kawamura H 2007 Solid State Commun. 141 69Google Scholar

    [39]

    Luo D B, Wang Y C, Yang G C, Ma Y M 2018 J. Phys. Chem. C 122 12448Google Scholar

    [40]

    Rubloff G W 1972 Phys. Rev. B 5 662Google Scholar

    [41]

    Kanchana V, Vaitheeswaran G, Rajagopalan M 2003 Physica B 328 283Google Scholar

    [42]

    Shi H, Luo W, Johansson B, Ahujia R 2009 J. Phys. Condens. Matter 21 415501Google Scholar

    [43]

    Hao A M, Yang X C, Li J, Xin W, Zhang S H, Zhang X Y, Liu R P 2009 Chin. Phys. Lett. 26 077103Google Scholar

    [44]

    朱春野, 刘欢欢, 刘艳辉 2011 延边大学学报(自然科学版) 37 19Google Scholar

    Zhu C Y, Liu H H, Liu Y H 2011 J. Yanbian Univ. (Natural Science Edition) 37 19Google Scholar

    [45]

    吴成国, 武文远, 龚艳春, 戴斌飞, 何苏红, 黄雁华 2015 物理学报 64 114213Google Scholar

    Wu C G, Wu W Y, Gong Y C, Dai B F, He S H, Huang Y H 2015 Acta Phys. Sin. 64 114213Google Scholar

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Metrics
  • Abstract views:  5274
  • PDF Downloads:  140
  • Cited By: 0
Publishing process
  • Received Date:  21 June 2021
  • Accepted Date:  29 September 2021
  • Available Online:  25 December 2021
  • Published Online:  05 January 2022

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