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氮化硅(β-Si3N4)是当下最具应用前景的热透波材料, 其基础物性高温介电函数的精准测量对加快氮化硅基热透波材料的设计, 解决高超声速飞行器“黑障”问题具有重要意义. 本文以第一性原理数据为基本输入, 基于深度神经网络训练得到深度学习势, 然后运用深度学习分子动力学方法计算了氮化硅高温微波介电函数. 与传统经验势相比, 深度学习势的计算结果与实验结果在数量级上保持一致; 同时发现, 深度学习分子动力学在计算速度方面表现优异. 此外, 建立了氮化硅弛豫时间温度依变性的物理模型, 揭示了弛豫时间温度依变性规律. 本研究通过实现大规模高精度的分子动力学模拟计算了氮化硅高温微波介电函数, 为推动氮化硅基材料在高温热透波领域的应用提供了基础数据支撑.Silicon nitride (β-Si3N4) is a most promising thermal wave-transparent material. The accurate measurement of its high-temperature dielectric function is essential to solving the “black barrier” problem of hypersonic vehicles and accelerating the design of silicon nitride-based thermal wave-transparent materials. Direct experimental measurement at high temperature is a difficult job and the accuracy of classical molecular dynamics (CMD) simulations suffers the choice of empirical potential. In this work, we build a β-Si3N4 model on a nanoscale, train the deep learning potential (DLP) by using first-principles data, and apply the deep potential molecular dynamics (DPMD) to simulate the polarization relaxation process. The predicted energy and force by DLP are excellently consistent with first-principles calculations, which proves the high accuracy of DLP. The RMSEs for β-Si3N4 are quite low (0.00550 meV/atom for energy and 7.800 meV/Å for force). According to the Cole-Cole formula, the microwave dielectric function in the temperature range of 300–1000 K is calculated by using the deep learning molecular dynamics method. Compared with the empirical potential, the computational results of the DLP are consistent with the experimental results in the sense of order of magnitude. It is also found that the DPMD performs well in terms of computational speed. In addition, a mathematical model of the temperature dependence of the relaxation time is established to reveal the pattern of relaxation time varying with temperature. The high-temperature microwave dielectric function of silicon nitride is calculated by implementing large-scale and high-precision molecular dynamics simulations. It provides fundamental data for promoting the application of silicon nitride in high-temperature thermal transmission.
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Keywords:
- thermal wave-transparent material /
- dielectric function /
- high temperature /
- deep learning potential
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图 1 深度学习势训练流程: 主要包含AIMD计算采样、深度神经网络训练以及深度学习分子动力学模拟三部分
Fig. 1. DLP training process: AIMD computational sampling, DNN training, and DPMD simulation.
图 2 β-Si3N4的晶体结构 (a) 用于第一性原理分子动力学计算的体系(原子个数为140个); (b) 用于深度学习分子动力学模拟的体系(原子个数为2200个), 其中绿色原子代表硅原子, 红色原子代表氮原子
Fig. 2. Crystal structure of β-Si3N4: (a) The system for first-principles molecular dynamics calculations (140 atoms); (b) the system for deep potential molecular dynamics simulations (2200 atoms). The green atoms represent silicon atoms and the red atoms for nitrogen atoms.
图 3 (a) 第一性原理计算体系能量和深度学习势计算体系能量对比关系图; (b)—(d) 第一性原理计算原子受力和深度学习势计算原子受力对比关系图. 其中图中直线代表y=x
Fig. 3. (a) Comparison between the energy calculated by first-principles and that by the deep learning potential; (b)–(d) comparison between the forces on the atoms calculated by first-principles and those by the deep learning potential. The straight line in the figure represents y = x.
图 4 (a)—(c) 300 K温度下DPMD与AIMD计算径向分布函数对比图; (d)—(f) 1000 K温度下DPMD与AIMD模拟径向分布函数对比图
Fig. 4. (a)−(c) Comparison of radial distribution function between DPMD and AIMD simulations at 300 K; (d)−(f) comparison of radial distribution function between DPMD and AIMD simulations at 1000 K.
图 5 运用深度学习分子动力学计算的700 K温度下氮化硅偶极矩自相关函数 (a) 偶极矩自相关函数随时间的变化; (b) 偶极矩自相关极值点对数值随时间的变化
Fig. 5. Silicon nitride dipole moment autocorrelation function at 700 K calculated by the deep learning potential: (a) Dipole moment autocorrelation function versus time; (b) dipole moment autocorrelation polar point logarithm versus time.
图 6 (a) β-Si3N4在不同温度时的频率相关介电函数实部; (b) β-Si3N4在不同温度时的频率相关介电函数虚部. 其中文献[34]仅给出了实部值, 未给出虚部值. 文献[35]给出了8—12 GHz频率范围内的介电函数实验测量值
Fig. 6. (a) Real part and (b) imaginary part of frequency-dependent dielectric function of β-Si3N4 at varying temperatures. Note that only the real part is given in Ref. [34]. The values of dielectric function in the frequency range of 8–12 GHz are given in Ref. [35]
图 7 弛豫时间随温度变化曲线, 其中蓝色直线代表弛豫时间温度依变性模型, 红色散点代表不同温度下弛豫时间变化曲线
Fig. 7. Relaxation time variation curves with temperature. The blue straight line corresponds to relaxation time temperature dependence model, and the red scattered points represent relaxation time variation curves at different temperatures.
图 8 DPMD (Nvidia RTX 3080 GPU计算)与AIMD (48个Intel Xeon Platinum 9242 CPU计算)计算速度结果
Fig. 8. Computational speed of DPMD (running with a Nvidia RTX 3080 GPU) and AIMD calculations (running with 48 Intel Xeon Platinum 9242 CPU cores).
表 1 不同温度条件下的弛豫时间
Table 1. Relaxation time under different temperatures by DLP and Tersoff potential.
温度/K 弛豫时间/ps DLP Tersoff potential 300 29.27 52.89 400 26.48 33.17 500 16.14 36.08 600 14.39 27.87 700 7.14 16.76 800 6.04 20.05 900 4.15 0.88 1000 2.83 0.27 
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[1] Mehra N, Singh R K, Bera S C 2015 Prog. Electromagn. Res. B 63 161
Google Scholar
[2] Hartunian R A, Stewart G E, Fergason S D 2007 Aerospace Corp. 5309 1
[3] Jayaraman B, Shyy W 2008 Prog. Aerosp. Sci. 44 139
Google Scholar
[4] 钟汶帆, 吴孟强 2014 压电与声光 36 1004
Zhong W F, Wu M Q 2014 Piezoelectr. Acoustoopt. 36 1004
[5] Zhang T, Zhang S R, Wu M Q, Sang W J, Gao Z P, Li Z P 2007 J. Electron. Sci. Technol. 5 4
[6] Neumann M 1983 Mol. Phys. 50 841
Google Scholar
[7] Neumann M, Steinhauser O 1983 Chem. Phys. Lett. 102 508
Google Scholar
[8] Neumann M, Steinhauser O 1984 Chem. Phys. Lett. 106 563
Google Scholar
[9] Afify N D, Sweatman M B 2018 J. Chem. Phys. 148 024508
Google Scholar
[10] Cardona J, Fartaria R, Sweatman M B, Lue L 2016 Mol. Simul. 42 370
Google Scholar
[11] Blank T B, Brown S D, Calhoun A W, Doren D J 1998 J. Chem. Phys. 103 4129
[12] Behler J, Parrinello M 2007 Phys. Rev. Lett. 98 146401
Google Scholar
[13] Bartók A P, Payne M C, Kondor R, Csányi G 2010 Phys. Rev. Lett. 104 136403
Google Scholar
[14] Behler J 2011 J. Chem. Phys. 134 074106
Google Scholar
[15] Novikov I S, Gubaev K, Podryabinkin E V, Shapeev A V 2021 Mach. Learn. Sci. Technol. 2 025002
Google Scholar
[16] Zhang L, Han J, Wang H, Car R, Weinan E 2018 Phys. Rev. Lett. 120 143001
Google Scholar
[17] Chen W, Li L S 2021 J. Appl. Phys. 129 244104
Google Scholar
[18] Kühne T D, Iannuzzi M, Del Ben M, Rybkin V V, Seewald P, Stein F, Laino T, Khaliullin R Z, Schütt O, Schiffmann F, Golze D, Wilhelm J, Chulkov S, Bani-Hashemian M H, Weber V, Borštnik U, Taillefumier M, Jakobovits A S, Lazzaro A, Pabst H, Müller T, Schade R, Guidon M, Andermatt S, Holmberg N, Schenter G K, Hehn A, Bussy A, Belleflamme F, Tabacchi G, Glöß A, Lass M, Bethune I, Mundy C J, Plessl C, Watkins M, VandeVondele J, Krack M, Hutter J 2020 J. Chem. Phys. 152 194103
Google Scholar
[19] VandeVondele J, Krack M, Mohamed F, Parrinello M, Chassaing T, Hutterc J 2005 Comput. Phys. Commun. 167 103
Google Scholar
[20] Del Ben M, Hutter J, VandeVondele J 2012 J. Chem. Theory Comput. 8 4177
Google Scholar
[21] Wang H, Zhang L, Han J, E W 2018 Comput. Phys. Commun. 228 178
Google Scholar
[22] Abadi M, Barham P, Chen J, Chen Z, Davis A, Dean J, Devin M, Ghemawat S, Irving G, Isard M, Kudlur M, Levenberg J, Monga R, Moore S, Murray D G, Steiner B, Tucker P, Vasudevan V, Warden P, Wicke M, Yu Y, Zheng X 2015 arXiv:1605.08695 [cs.DC]
[23] Behler J 2011 Phys. Chem. Chem. Phys. 13 17930
Google Scholar
[24] Jia W, Wang H, Chen M, Lu D, Car R, E W, Zhang L 2018 Adv. Neural Inf. Process. Syst. 31 1
Google Scholar
[25] Kingma DP, Ba J 2017 arXiv. 1412 6980
[26] Plimpton S 1995 J. Comput. Phys. 117 1
Google Scholar
[27] Martyna G J, Klein M L, Tuckerman M 1992 J. Chem. Phys. 97 2635
Google Scholar
[28] Evans D J, Holian B L 1985 J. Chem. Phys. 83 4069
Google Scholar
[29] Tersoff J 1989 Phys. Rev. B 39 5566
Google Scholar
[30] Tersoff J 1988 Phys. Rev. B 37 6991
Google Scholar
[31] Cole K S, Cole R H 1941 J. Chem. Phys. 9 341
Google Scholar
[32] Petousis I, Mrdjenovich D, Ballouz E, Liu M, Winston D, Chen W, Graf T, Schladt T D, Persson K A, Prinz F B 2017 Sci. Data 4 160134
Google Scholar
[33] Wang M, Zhang Y, Liu X, Wang X 2013 Ceram. Int. 39 2069
Google Scholar
[34] Shao S, Luo H, Deng L, He J, Huang S 2018 AIP Adv. 8 075127
Google Scholar
[35] Saleem A, Zhang Y J, Gong H Y, Majeed M K, Lin X, Jing J, Sheng M M, Zhao C C 2020 J. Mater. Sci. Mater. Electron. 31 2918
Google Scholar
[36] Yang J Y, Xu M, Liu L H 2016 J. Quant. Spectrosc. Radiat. Transf. 184 111
Google Scholar
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