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材料的非简谐性描述符

李环娅 周柯 尹万健

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材料的非简谐性描述符

李环娅, 周柯, 尹万健

Quantitative descriptor of lattice anharmonicity in crystal

Li Huan-Ya, Zhou Ke, Yin Wan-Jian
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  • 非简谐效应是诸如软模相变、负热膨胀、多铁性和超低热导率等材料性质的根源. 已有的关于量化材料非简谐性的方法没有给出清晰准确的非简谐性描述符, 并且计算流程复杂, 需要极其耗时的分子动力学模拟. 故亟需提出一个可以快速计算的非简谐性描述符, 用来理解、评估、设计和筛选具有强非简谐性的功能材料. 本研究将晶格非谐性分解为单声子非谐性$ {\sigma }_{(\boldsymbol{q}, j)}^{A} $, 并提出温度依赖的晶格非谐性的定量描述符$ {A}_{{\mathrm{p}}{\mathrm{h}}}\left(T\right) $. 该描述符既可以定量描述从Si, GaAs, CdTe, NaCl到CsPbI3的晶格非谐性的变化趋势, 又可以成功预测非简谐效应驱动的体积模量和晶格热导率性质的变化. 本工作提出的非简谐性描述符能够快速量化材料非简谐性, 并且可直观地展现材料非简谐效应的声子模态分布. 本方法计算简单、高效且有效, 可为基于非简谐性筛选与设计材料打下基础.
    Anharmonic effect is often one of the physical root causes of some special material properties, such as soft mode phase transition, negative thermal expansion, multiferroicity, and ultra-low thermal conductivity. However, the existing methods of quantifying the anharmonicity of material do not give a clear and accurate anharmonicity descriptor. The calculation of the anharmonic effect requires extremely time-consuming molecular dynamics simulation, the calculation process is complex and costly. Therefore, a quantitative descriptor is urgently needed, which can be used to implement quick calculation so as to understand, evaluate, design, and screen functional materials with strong anharmonicity.In this paper, we propose a method to quantify the anharmonicity of materials by only phonon spectrum and static self-consistent calculation through calculating and analyzing the material composed of germanium and its surrounding elements. In this method, the lattice anharmonicity is decomposed into the anharmonic contribution of independent phonon vibration modes, and the quantitative anharmonicity descriptor $ {\sigma }_{\boldsymbol{q},j}^{A} $ of phonons is proposed. Combining it with the Bose-Einstein distribution, the quantitative descriptor $ {A}_{{\mathrm{p}}{\mathrm{h}}}\left(T\right) $ of temperature-dependent material anharmonicity is proposed. We calculate the bulk moduli and lattice thermal conductivities at 300 K of nine widely representative materials. There is a clear linear trend between them and our proposed quantitative descriptor $ {A}_{{\mathrm{p}}{\mathrm{h}}}\left(T\right) $, which verifies the accuracy of our proposed descriptor. The results show that the descriptor has the following functions. i) It can systematically and quantitatively classify materials as the strength of anharmonicity; ii) it intuitively shows the distribution of the anharmonic effect of the material on the phonon spectrum, and realizes the separate analysis of the phonon anharmonicity that affects the specific properties of the material; iii) it is cost-effective in first-principles molecular dynamics calculations and lays a foundation for screening and designing materials based on anharmonicity.This study provides an example for the high-throughput study of functional materials driven by anharmonic effect in the future, and opens up new possibilities for material design and application. In addition, for strongly anharmonic materials such as CsPbI3, the equilibrium position of the atoms is not fixed at high temperatures, resulting in a decrease in the accuracy of quantifying anharmonicity using our proposed descriptor. In order to get rid of this limitation, our future research will focus on the distribution of atomic equilibrium positions in strongly anharmonic materials at high temperatures, so as to propose a more accurate theoretical method to quantify the anharmonicity in strongly anharmonic materials.
      通信作者: 尹万健, wjyin@suda.edu.cn
    • 基金项目: 国家重点研发计划(批准号: 2020YFB1506400)和国家自然科学基金 (批准号: 11974257)资助的课题.
      Corresponding author: Yin Wan-Jian, wjyin@suda.edu.cn
    • Funds: Project supported by the National Key Research and Development Program of China (Grant No. 2020YFB1506400) and the National Natural Science Foundation of China (Grant No. 11974257).
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    Evans J S O, Mary T A, Sleight A W 1997 Physica B: Condensed Matter 241–243 311Google Scholar

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    Gao W, Huang R 2014 J. Mech. Phys. Solids 66 42Google Scholar

    [4]

    Tyagi A K, Achary S N, Mathews M D 2002 J. Alloys Compd. 339 207Google Scholar

    [5]

    Pugachev A M, Zaytseva I V, Surovtsev N V, Krylov A S 2020 Ceram. Int. 46 22619Google Scholar

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    Poojitha B, Rubi K, Sarkar S, Mahendiran R, Venkatesan T, Saha S 2019 Phys. Rev. Mater. 3 024412Google Scholar

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    Klarbring J, Hellman O, Abrikosov I A, Simak S I 2020 Phys. Rev. Lett. 125 045701Google Scholar

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    Knoop F, Purcell T A R, Scheffler M, Carbogno C 2023 Phys. Rev. Lett. 130 236301Google Scholar

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    Zhao L D, Lo S H, Zhang Y, et al. 2014 Nature 508 373Google Scholar

    [10]

    Mukhopadhyay S, Bansal D, Delaire O, et al. 2017 Phys. Rev. B 96 100301Google Scholar

    [11]

    Xia Y, Pal K, He J, Ozoliņš V, Wolverton C 2020 Phys. Rev. Lett. 124 065901Google Scholar

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    Yang J, Li S 2022 Mater. Horiz. 9 1896Google Scholar

    [13]

    Knoop F, Purcell T A R, Scheffler M, Carbogno C 2020 Phys. Rev. Mater. 4 083809Google Scholar

    [14]

    Munson K T, Swartzfager J R, Asbury J B 2019 ACS Energy Lett. 4 1888Google Scholar

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    Li C, Wu Y, Pennycook T J, Lupini A R, Leonard D N, Yin W, Paudel N, Al-Jassim M, Yan Y, Pennycook S J 2013 Phys. Rev. Lett. 111 096403Google Scholar

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    Northrup J E, Zhang S B 1993 Phys. Rev. B 47 6791Google Scholar

    [17]

    Zhang D, Lou W, Miao M, Zhang S cheng, Chang K 2013 Phys. Rev. Lett. 111 156402Google Scholar

    [18]

    Miyata K, Meggiolaro D, Trinh M T, Joshi P P, Mosconi E, Jones S C, De Angelis F, Zhu X Y 2017 Sci. Adv. 3 e1701217Google Scholar

    [19]

    Batignani G, Fumero G, Srimath Kandada A R, Cerullo G, Gandini M, Ferrante C, Petrozza A, Scopigno T 2018 Nat. Commun. 9 1971Google Scholar

    [20]

    Nishida J, Breen J P, Lindquist K P, Umeyama D, Karunadasa H I, Fayer M D 2018 J. Am. Chem. Soc. 140 9882Google Scholar

    [21]

    Zhang M, Zhang X, Huang L Y, Lin H Q, Lu G 2017 Phys. Rev. B 96 195203Google Scholar

    [22]

    Chen Y, Yi H T, Wu X, Haroldson R, Gartstein Y N, Rodionov Y I, Tikhonov K S, Zakhidov A, Zhu X Y, Podzorov V 2016 Nat. Commun. 7 12253Google Scholar

    [23]

    Jong U G, Yu C J, Kye Y H, Kim Y S, Kim C H, Ri S G 2018 J. Mater. Chem. A 6 17994Google Scholar

    [24]

    Yang R X, Skelton J M, Da Silva E L, Frost J M, Walsh A 2020 J. Chem. Phys. 152 024703Google Scholar

    [25]

    Jung Y K, Lee J H, Walsh A, Soon A 2017 Chem. Mater. 29 3181Google Scholar

    [26]

    Guo Y, Yaffe O, Paley D W, et al. 2017 Phys. Rev. Mater. 1 042401Google Scholar

    [27]

    Yaffe O, Guo Y, Tan L Z, et al. 2017 Phys. Rev. Lett. 118 136001Google Scholar

    [28]

    Yang R X, Skelton J M, Da Silva E L, Frost J M, Walsh A 2017 J. Phys. Chem. Lett. 8 4720Google Scholar

    [29]

    Emin D 2018 J. Appl. Phys. 123 055105Google Scholar

    [30]

    Zhu X Y, Podzorov V 2015 J. Phys. Chem. Lett. 6 4758Google Scholar

    [31]

    Schlipf M, Poncé S, Giustino F 2018 Phys. Rev. Lett. 121 086402Google Scholar

    [32]

    Brennan M C, Draguta S, Kamat P V, Kuno M 2018 ACS Energy Lett. 3 204Google Scholar

    [33]

    Haruyama J, Sodeyama K, Han L, Tateyama Y 2015 J. Am. Chem. Soc. 137 10048Google Scholar

    [34]

    Karakus M, Jensen S A, D’Angelo F, Turchinovich D, Bonn M, Cánovas E 2015 J. Phys. Chem. Lett. 6 4991Google Scholar

    [35]

    Rehman W, McMeekin D P, Patel J B, Milot R L, Johnston M B, Snaith H J, Herz L M 2017 Energy Environ. Sci. 10 361Google Scholar

    [36]

    Gallop N P, Selig O, Giubertoni G, et al. 2018 J. Phys. Chem. Lett. 9 5987Google Scholar

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    Souvatzis P, Eriksson O, Katsnelson M I, Rudin S P 2008 Phys. Rev. Lett. 100 095901Google Scholar

    [38]

    Errea I, Rousseau B, Bergara A 2011 Phys. Rev. Lett. 106 165501Google Scholar

    [39]

    Errea I, Calandra M, Mauri F 2014 Phys. Rev. B 89 064302Google Scholar

    [40]

    Monacelli L, Bianco R, Cherubini M, Calandra M, Errea I, Mauri F 2021 J. Phys. Condens. Matter 33 363001Google Scholar

    [41]

    Monserrat B, Drummond N D, Needs R J 2013 Phys. Rev. B 87 144302Google Scholar

    [42]

    Bowman J M 1978 J. Chem. Phys. 68 608Google Scholar

    [43]

    Hellman O, Abrikosov I A 2013 Phys. Rev. B 88 144301Google Scholar

    [44]

    Hellman O, Steneteg P, Abrikosov I A, Simak S I 2013 Phys. Rev. B 87 104111Google Scholar

    [45]

    Togo A, Tanaka I 2015 Scrip. Mater. 108 1Google Scholar

    [46]

    Togo A 2023 J. Phys. Soc. Jpn. 92 012001Google Scholar

    [47]

    Togo A, Chaput L, Tanaka I 2015 Phys. Rev. B 91 094306Google Scholar

    [48]

    Haynes W M 2014 CRC Handbook of Chemistry and Physics (CRC Press

    [49]

    Inyushkin A V, Taldenkov A N, Yakubovsky A Y, Markov A V, Moreno-Garsia L, Sharonov B N 2003 Semicond. Sci. Technol. 18 685Google Scholar

    [50]

    Slack G A, Galginaitis S 1964 Phys. Rev. 133 A253Google Scholar

    [51]

    Guo Z, Wang J, Yin W J 2022 Energy Environ. Sci. 15 660Google Scholar

    [52]

    Varotsos P, Alexopoulos K 1981 Phys. Rev. B 24 904Google Scholar

    [53]

    Yang J, Jain A, Fan L, Ang Y S, Li H, Ong W L 2023 Chem. Mater. 35 5185Google Scholar

    [54]

    Allen P B 2015 Phys. Rev. B 92 064106Google Scholar

    [55]

    Fanggao C, Cankurtaran M, Saunders G A, Al-Kheffaji A, Almond D P, Ford P J 1991 Supercond. Sci. Technol. 4 13Google Scholar

    [56]

    高占鹏 1981 物理学报 30 679Google Scholar

    Gao Z P 1981 Acta Phys. Sin. 30 679Google Scholar

    [57]

    Gu H Y, Yin W J, Gong X G 2021 Appl. Phys. Lett. 119 191101Google Scholar

    [58]

    Yang K K, Yang H, Sun Y J, et al. 2021 arXiv: 2102.12619

    [59]

    刘娜娜, 宋仁伯, 孙翰英, 杜大伟 2008 物理学报 57 7145Google Scholar

    Liu N N, Song R B, Sun H Y, Du D W 2008 Acta Phys. Sin. 57 7145Google Scholar

  • 图 1  不同振动模式的声子的非简谐性示意图, 图中红色圆点表示完全简谐情况下声子振动对体系自由能的贡献$ V\left({\boldsymbol{r}}_{\boldsymbol{q}, j}\right) $随着振动幅度$ \left|{\boldsymbol{r}}_{\boldsymbol{q}, j}\right| $的变化趋势, 蓝色圆点则表示真实情况下其变化趋势

    Fig. 1.  The schematic diagram of phonon anharmonicity of different vibration modes. The red dots represent the change trend of the contribution of phonon vibration to the free energy of the system $ V\left({\boldsymbol{r}}_{\boldsymbol{q}, j}\right) $ with the vibration amplitude $ \left|{\boldsymbol{r}}_{\boldsymbol{q}, j}\right| $ in harmonic approximation, and the blue dots represent the change trend in the real case.

    图 2  (a) 所选材料的组成元素分布; (b) $ Fd\bar{3}m $ (C, Si, Ge, Sn); (c) $ F\bar{4}3 m $ (GaAs, CdTe, NaCl ); (d) $ I4/mcm $((β-CsPbI3); (e) Pnma (γ-CsPbI3); (f)计算流程图

    Fig. 2.  (a) The distribution of constituent elements of typical materials we selected; (b) $ F{{d}}\bar{3}m $(C, Si, Ge, Sn); (c) $ F\bar{4}3 m $(GaAs, CdTe, NaCl); (d) $ I4/mcm $(β-CsPbI3); (e) Pnma (γ-CsPbI3); (f) calculation flow diagram.

    图 3  所选材料的声子谱, 图中点的颜色值为对应声子模态的非简谐性描述符$ {\sigma }_{\boldsymbol{q}, j}^{A} $ (a) C; (b) Si; (c) Ge; (d) Sn; (e) GaAs; (f) CdTe; (g) NaCl; (h) γ-CsPbI3; (i) β-CsPbI3

    Fig. 3.  The phonon spectrum of the selected material, the color value of the point in the graph is the anharmonic descriptor $ {\sigma }_{\boldsymbol{q}, j}^{A} $ of the corresponding phonon mode: (a) C; (b) Si; (c) Ge; (d) Sn; (e) GaAs; (f) CdTe; (g) NaCl; (h) γ-CsPbI3; (i) β-CsPbI3.

    图 4  (a) 所选材料横模声学声子支(TA)的声子非谐性描述符差值, $ \Delta {\sigma }_{\boldsymbol{q}, j}^{A}={\sigma }_{\boldsymbol{q}, j}^{A}\left({\mathrm{G}}{\mathrm{a}}{\mathrm{A}}{\mathrm{s}}\right)-{\sigma }_{\boldsymbol{q}, j}^{A}\left({\mathrm{C}}{\mathrm{d}}{\mathrm{T}}{\mathrm{e}}\right) $; (b) 所选材料的总声子非简谐贡献(红色)和Γ点附近的声子非简谐贡献(蓝色); (c) 材料非简谐性描述符$ {A}_{{\mathrm{p}}{\mathrm{h}}}\left(300{\mathrm{K}}\right) $与材料体模量$ {B}_{{\mathrm{L}}} $的关联图; (d) 所选材料$ {A}_{{\mathrm{p}}{\mathrm{h}}}\left(300~{\mathrm{K}}\right) $与晶格热导率$ {\kappa }_{{\mathrm{L}}}\left(300~{\mathrm{K}}\right) $的关联图. 图(c), (d)中红色标记晶体为金刚石结构, 蓝色标记晶体为闪锌矿结构, 绿色标记为钙钛矿结构; 其中空心标签是工作中的DFT模拟结果, 实心标签则是引用实验测得的数据, 包括C (expt. )[48], Si (expt.)[48], Ge (expt.)[48], Sn (expt.)[48], GaAs (expt.)[49], CdTe (expt.)[50](数据详见补充材料(https://wulixb.iphy.ac.cn/article/doi/10.7498/aps.73.20231428)表S1)

    Fig. 4.  (a) The phonon anharmonicity descriptor value difference of the transverse acoustic phonon branch (TA) of the selected material, $ \Delta {\sigma }_{\boldsymbol{q}, j}^{A}={\sigma }_{\boldsymbol{q}, j}^{A}\left({\mathrm{G}}{\mathrm{a}}{\mathrm{A}}{\mathrm{s}}\right)-{\sigma }_{\boldsymbol{q}, j}^{A}\left({\mathrm{C}}{\mathrm{d}}{\mathrm{T}}{\mathrm{e}}\right) $; (b) the total phonon anharmonic contribution in all K-points (red) and the phonon anharmonic contribution near the Γ point (blue) of the selected material; (c) the correlation graph of $ {A}_{{\mathrm{p}}{\mathrm{h}}}\left(300~{\mathrm{K}}\right) $ and Bulk Modulus $ {B}_{{\mathrm{L}}} $ of the material; (d) the correlation graph of $ {A}_{{\mathrm{p}}{\mathrm{h}}}\left(300~{\mathrm{K}}\right) $ and lattice thermal conductivity $ {\kappa }_{{\mathrm{L}}}\left(300~{\mathrm{K}}\right) $. In Fig. (c)(d), the red label crystal is a diamond structure, the blue label crystal is a zinc blende structure, and the green label is a perovskite structure. The hollow tag is the DFT simulation results in our work, and the solid tag is the data measured by the citation experiment, including C (expt.)[48], Si (expt.)[48], Ge (expt.)[48], Sn (expt.)[48], GaAs (expt.)[49], CdTe (expt.)[50] (The data are shown in Table S1 of the support material (https://wulixb.iphy.ac.cn/article/doi/10.7498/aps.73.20231428)).

  • [1]

    Demiroglu I, Sevik C 2021 Phys. Rev. B 103 085430Google Scholar

    [2]

    Evans J S O, Mary T A, Sleight A W 1997 Physica B: Condensed Matter 241–243 311Google Scholar

    [3]

    Gao W, Huang R 2014 J. Mech. Phys. Solids 66 42Google Scholar

    [4]

    Tyagi A K, Achary S N, Mathews M D 2002 J. Alloys Compd. 339 207Google Scholar

    [5]

    Pugachev A M, Zaytseva I V, Surovtsev N V, Krylov A S 2020 Ceram. Int. 46 22619Google Scholar

    [6]

    Poojitha B, Rubi K, Sarkar S, Mahendiran R, Venkatesan T, Saha S 2019 Phys. Rev. Mater. 3 024412Google Scholar

    [7]

    Klarbring J, Hellman O, Abrikosov I A, Simak S I 2020 Phys. Rev. Lett. 125 045701Google Scholar

    [8]

    Knoop F, Purcell T A R, Scheffler M, Carbogno C 2023 Phys. Rev. Lett. 130 236301Google Scholar

    [9]

    Zhao L D, Lo S H, Zhang Y, et al. 2014 Nature 508 373Google Scholar

    [10]

    Mukhopadhyay S, Bansal D, Delaire O, et al. 2017 Phys. Rev. B 96 100301Google Scholar

    [11]

    Xia Y, Pal K, He J, Ozoliņš V, Wolverton C 2020 Phys. Rev. Lett. 124 065901Google Scholar

    [12]

    Yang J, Li S 2022 Mater. Horiz. 9 1896Google Scholar

    [13]

    Knoop F, Purcell T A R, Scheffler M, Carbogno C 2020 Phys. Rev. Mater. 4 083809Google Scholar

    [14]

    Munson K T, Swartzfager J R, Asbury J B 2019 ACS Energy Lett. 4 1888Google Scholar

    [15]

    Li C, Wu Y, Pennycook T J, Lupini A R, Leonard D N, Yin W, Paudel N, Al-Jassim M, Yan Y, Pennycook S J 2013 Phys. Rev. Lett. 111 096403Google Scholar

    [16]

    Northrup J E, Zhang S B 1993 Phys. Rev. B 47 6791Google Scholar

    [17]

    Zhang D, Lou W, Miao M, Zhang S cheng, Chang K 2013 Phys. Rev. Lett. 111 156402Google Scholar

    [18]

    Miyata K, Meggiolaro D, Trinh M T, Joshi P P, Mosconi E, Jones S C, De Angelis F, Zhu X Y 2017 Sci. Adv. 3 e1701217Google Scholar

    [19]

    Batignani G, Fumero G, Srimath Kandada A R, Cerullo G, Gandini M, Ferrante C, Petrozza A, Scopigno T 2018 Nat. Commun. 9 1971Google Scholar

    [20]

    Nishida J, Breen J P, Lindquist K P, Umeyama D, Karunadasa H I, Fayer M D 2018 J. Am. Chem. Soc. 140 9882Google Scholar

    [21]

    Zhang M, Zhang X, Huang L Y, Lin H Q, Lu G 2017 Phys. Rev. B 96 195203Google Scholar

    [22]

    Chen Y, Yi H T, Wu X, Haroldson R, Gartstein Y N, Rodionov Y I, Tikhonov K S, Zakhidov A, Zhu X Y, Podzorov V 2016 Nat. Commun. 7 12253Google Scholar

    [23]

    Jong U G, Yu C J, Kye Y H, Kim Y S, Kim C H, Ri S G 2018 J. Mater. Chem. A 6 17994Google Scholar

    [24]

    Yang R X, Skelton J M, Da Silva E L, Frost J M, Walsh A 2020 J. Chem. Phys. 152 024703Google Scholar

    [25]

    Jung Y K, Lee J H, Walsh A, Soon A 2017 Chem. Mater. 29 3181Google Scholar

    [26]

    Guo Y, Yaffe O, Paley D W, et al. 2017 Phys. Rev. Mater. 1 042401Google Scholar

    [27]

    Yaffe O, Guo Y, Tan L Z, et al. 2017 Phys. Rev. Lett. 118 136001Google Scholar

    [28]

    Yang R X, Skelton J M, Da Silva E L, Frost J M, Walsh A 2017 J. Phys. Chem. Lett. 8 4720Google Scholar

    [29]

    Emin D 2018 J. Appl. Phys. 123 055105Google Scholar

    [30]

    Zhu X Y, Podzorov V 2015 J. Phys. Chem. Lett. 6 4758Google Scholar

    [31]

    Schlipf M, Poncé S, Giustino F 2018 Phys. Rev. Lett. 121 086402Google Scholar

    [32]

    Brennan M C, Draguta S, Kamat P V, Kuno M 2018 ACS Energy Lett. 3 204Google Scholar

    [33]

    Haruyama J, Sodeyama K, Han L, Tateyama Y 2015 J. Am. Chem. Soc. 137 10048Google Scholar

    [34]

    Karakus M, Jensen S A, D’Angelo F, Turchinovich D, Bonn M, Cánovas E 2015 J. Phys. Chem. Lett. 6 4991Google Scholar

    [35]

    Rehman W, McMeekin D P, Patel J B, Milot R L, Johnston M B, Snaith H J, Herz L M 2017 Energy Environ. Sci. 10 361Google Scholar

    [36]

    Gallop N P, Selig O, Giubertoni G, et al. 2018 J. Phys. Chem. Lett. 9 5987Google Scholar

    [37]

    Souvatzis P, Eriksson O, Katsnelson M I, Rudin S P 2008 Phys. Rev. Lett. 100 095901Google Scholar

    [38]

    Errea I, Rousseau B, Bergara A 2011 Phys. Rev. Lett. 106 165501Google Scholar

    [39]

    Errea I, Calandra M, Mauri F 2014 Phys. Rev. B 89 064302Google Scholar

    [40]

    Monacelli L, Bianco R, Cherubini M, Calandra M, Errea I, Mauri F 2021 J. Phys. Condens. Matter 33 363001Google Scholar

    [41]

    Monserrat B, Drummond N D, Needs R J 2013 Phys. Rev. B 87 144302Google Scholar

    [42]

    Bowman J M 1978 J. Chem. Phys. 68 608Google Scholar

    [43]

    Hellman O, Abrikosov I A 2013 Phys. Rev. B 88 144301Google Scholar

    [44]

    Hellman O, Steneteg P, Abrikosov I A, Simak S I 2013 Phys. Rev. B 87 104111Google Scholar

    [45]

    Togo A, Tanaka I 2015 Scrip. Mater. 108 1Google Scholar

    [46]

    Togo A 2023 J. Phys. Soc. Jpn. 92 012001Google Scholar

    [47]

    Togo A, Chaput L, Tanaka I 2015 Phys. Rev. B 91 094306Google Scholar

    [48]

    Haynes W M 2014 CRC Handbook of Chemistry and Physics (CRC Press

    [49]

    Inyushkin A V, Taldenkov A N, Yakubovsky A Y, Markov A V, Moreno-Garsia L, Sharonov B N 2003 Semicond. Sci. Technol. 18 685Google Scholar

    [50]

    Slack G A, Galginaitis S 1964 Phys. Rev. 133 A253Google Scholar

    [51]

    Guo Z, Wang J, Yin W J 2022 Energy Environ. Sci. 15 660Google Scholar

    [52]

    Varotsos P, Alexopoulos K 1981 Phys. Rev. B 24 904Google Scholar

    [53]

    Yang J, Jain A, Fan L, Ang Y S, Li H, Ong W L 2023 Chem. Mater. 35 5185Google Scholar

    [54]

    Allen P B 2015 Phys. Rev. B 92 064106Google Scholar

    [55]

    Fanggao C, Cankurtaran M, Saunders G A, Al-Kheffaji A, Almond D P, Ford P J 1991 Supercond. Sci. Technol. 4 13Google Scholar

    [56]

    高占鹏 1981 物理学报 30 679Google Scholar

    Gao Z P 1981 Acta Phys. Sin. 30 679Google Scholar

    [57]

    Gu H Y, Yin W J, Gong X G 2021 Appl. Phys. Lett. 119 191101Google Scholar

    [58]

    Yang K K, Yang H, Sun Y J, et al. 2021 arXiv: 2102.12619

    [59]

    刘娜娜, 宋仁伯, 孙翰英, 杜大伟 2008 物理学报 57 7145Google Scholar

    Liu N N, Song R B, Sun H Y, Du D W 2008 Acta Phys. Sin. 57 7145Google Scholar

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出版历程
  • 收稿日期:  2023-09-04
  • 修回日期:  2023-11-18
  • 上网日期:  2023-12-05
  • 刊出日期:  2024-03-05

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