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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. -
Keywords:
- anharmonic effect /
- anharmonicity of phonon /
- lattice thermal conductivity /
- first-principles calculation
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Google Scholar
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[55] Fanggao C, Cankurtaran M, Saunders G A, Al-Kheffaji A, Almond D P, Ford P J 1991 Supercond. Sci. Technol. 4 13
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Google Scholar
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Google Scholar
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Liu N N, Song R B, Sun H Y, Du D W 2008 Acta Phys. Sin. 57 7145
Google Scholar
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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
Figure 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
Figure 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
Figure 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)
Figure 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)).
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[1] Demiroglu I, Sevik C 2021 Phys. Rev. B 103 085430
Google Scholar
[2] Evans J S O, Mary T A, Sleight A W 1997 Physica B: Condensed Matter 241–243 311
Google Scholar
[3] Gao W, Huang R 2014 J. Mech. Phys. Solids 66 42
Google Scholar
[4] Tyagi A K, Achary S N, Mathews M D 2002 J. Alloys Compd. 339 207
Google Scholar
[5] Pugachev A M, Zaytseva I V, Surovtsev N V, Krylov A S 2020 Ceram. Int. 46 22619
Google Scholar
[6] Poojitha B, Rubi K, Sarkar S, Mahendiran R, Venkatesan T, Saha S 2019 Phys. Rev. Mater. 3 024412
Google Scholar
[7] Klarbring J, Hellman O, Abrikosov I A, Simak S I 2020 Phys. Rev. Lett. 125 045701
Google Scholar
[8] Knoop F, Purcell T A R, Scheffler M, Carbogno C 2023 Phys. Rev. Lett. 130 236301
Google Scholar
[9] Zhao L D, Lo S H, Zhang Y, et al. 2014 Nature 508 373
Google Scholar
[10] Mukhopadhyay S, Bansal D, Delaire O, et al. 2017 Phys. Rev. B 96 100301
Google Scholar
[11] Xia Y, Pal K, He J, Ozoliņš V, Wolverton C 2020 Phys. Rev. Lett. 124 065901
Google Scholar
[12] Yang J, Li S 2022 Mater. Horiz. 9 1896
Google Scholar
[13] Knoop F, Purcell T A R, Scheffler M, Carbogno C 2020 Phys. Rev. Mater. 4 083809
Google Scholar
[14] Munson K T, Swartzfager J R, Asbury J B 2019 ACS Energy Lett. 4 1888
Google 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 096403
Google Scholar
[16] Northrup J E, Zhang S B 1993 Phys. Rev. B 47 6791
Google Scholar
[17] Zhang D, Lou W, Miao M, Zhang S cheng, Chang K 2013 Phys. Rev. Lett. 111 156402
Google 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 e1701217
Google Scholar
[19] Batignani G, Fumero G, Srimath Kandada A R, Cerullo G, Gandini M, Ferrante C, Petrozza A, Scopigno T 2018 Nat. Commun. 9 1971
Google Scholar
[20] Nishida J, Breen J P, Lindquist K P, Umeyama D, Karunadasa H I, Fayer M D 2018 J. Am. Chem. Soc. 140 9882
Google Scholar
[21] Zhang M, Zhang X, Huang L Y, Lin H Q, Lu G 2017 Phys. Rev. B 96 195203
Google 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 12253
Google 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 17994
Google Scholar
[24] Yang R X, Skelton J M, Da Silva E L, Frost J M, Walsh A 2020 J. Chem. Phys. 152 024703
Google Scholar
[25] Jung Y K, Lee J H, Walsh A, Soon A 2017 Chem. Mater. 29 3181
Google Scholar
[26] Guo Y, Yaffe O, Paley D W, et al. 2017 Phys. Rev. Mater. 1 042401
Google Scholar
[27] Yaffe O, Guo Y, Tan L Z, et al. 2017 Phys. Rev. Lett. 118 136001
Google Scholar
[28] Yang R X, Skelton J M, Da Silva E L, Frost J M, Walsh A 2017 J. Phys. Chem. Lett. 8 4720
Google Scholar
[29] Emin D 2018 J. Appl. Phys. 123 055105
Google Scholar
[30] Zhu X Y, Podzorov V 2015 J. Phys. Chem. Lett. 6 4758
Google Scholar
[31] Schlipf M, Poncé S, Giustino F 2018 Phys. Rev. Lett. 121 086402
Google Scholar
[32] Brennan M C, Draguta S, Kamat P V, Kuno M 2018 ACS Energy Lett. 3 204
Google Scholar
[33] Haruyama J, Sodeyama K, Han L, Tateyama Y 2015 J. Am. Chem. Soc. 137 10048
Google Scholar
[34] Karakus M, Jensen S A, D’Angelo F, Turchinovich D, Bonn M, Cánovas E 2015 J. Phys. Chem. Lett. 6 4991
Google 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. 10 361
Google Scholar
[36] Gallop N P, Selig O, Giubertoni G, et al. 2018 J. Phys. Chem. Lett. 9 5987
Google Scholar
[37] Souvatzis P, Eriksson O, Katsnelson M I, Rudin S P 2008 Phys. Rev. Lett. 100 095901
Google Scholar
[38] Errea I, Rousseau B, Bergara A 2011 Phys. Rev. Lett. 106 165501
Google Scholar
[39] Errea I, Calandra M, Mauri F 2014 Phys. Rev. B 89 064302
Google Scholar
[40] Monacelli L, Bianco R, Cherubini M, Calandra M, Errea I, Mauri F 2021 J. Phys. Condens. Matter 33 363001
Google Scholar
[41] Monserrat B, Drummond N D, Needs R J 2013 Phys. Rev. B 87 144302
Google Scholar
[42] Bowman J M 1978 J. Chem. Phys. 68 608
Google Scholar
[43] Hellman O, Abrikosov I A 2013 Phys. Rev. B 88 144301
Google Scholar
[44] Hellman O, Steneteg P, Abrikosov I A, Simak S I 2013 Phys. Rev. B 87 104111
Google Scholar
[45] Togo A, Tanaka I 2015 Scrip. Mater. 108 1
Google Scholar
[46] Togo A 2023 J. Phys. Soc. Jpn. 92 012001
Google Scholar
[47] Togo A, Chaput L, Tanaka I 2015 Phys. Rev. B 91 094306
Google 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. 18 685
Google Scholar
[50] Slack G A, Galginaitis S 1964 Phys. Rev. 133 A253
Google Scholar
[51] Guo Z, Wang J, Yin W J 2022 Energy Environ. Sci. 15 660
Google Scholar
[52] Varotsos P, Alexopoulos K 1981 Phys. Rev. B 24 904
Google Scholar
[53] Yang J, Jain A, Fan L, Ang Y S, Li H, Ong W L 2023 Chem. Mater. 35 5185
Google Scholar
[54] Allen P B 2015 Phys. Rev. B 92 064106
Google Scholar
[55] Fanggao C, Cankurtaran M, Saunders G A, Al-Kheffaji A, Almond D P, Ford P J 1991 Supercond. Sci. Technol. 4 13
Google Scholar
[56] 高占鹏 1981 物理学报 30 679
Google Scholar
Gao Z P 1981 Acta Phys. Sin. 30 679
Google Scholar
[57] Gu H Y, Yin W J, Gong X G 2021 Appl. Phys. Lett. 119 191101
Google 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 7145
Google Scholar
Liu N N, Song R B, Sun H Y, Du D W 2008 Acta Phys. Sin. 57 7145
Google Scholar
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